Compactifications of PEL-Type Shimura Varieties and Kuga Families with Ordinary Loci 9789813207325

Table of contents :
Contents
Preface
0. Introduction
0.1 Background and Aim
0.2 Overview
0.3 Outline of the Constructions
0.4 What is Known, What is New, and What Can Be Studied Next
0.5 What to Note and to Skip in Special Cases
0.6 Notation and Conventions
0.7 Acknowledgements
1. Theory in Characteristic Zero
1.1 PEL-type Moduli Problems and Shimura Varieties
1.1.1 Linear Algebraic Data for PEL Structures
1.1.2 PEL-type Moduli Problems
1.1.3 PEL-type Shimura Varieties
1.2 Linear Algebraic Data for Cusps
1.2.1 Cusp Labels
1.2.2 Cone Decompositions
1.2.3 Rational Boundary Components
1.2.4 Parameters for Kuga Families
1.3 Algebraic Compactifications in Characteristic Zero
1.3.1 Toroidal and Minimal Compactifications of PEL-Type
Moduli Problems
1.3.2 Boundary of PEL-Type Moduli Problems
1.3.3 Toroidal Compactifications of PEL-Type Kuga Families and
Their Generalizations
1.3.4 Justification for the Parameters for Kuga Families
1.4 Automorphic Bundles and Canonical Extensions in
Characteristic Zero
1.4.1 Automorphic Bundles
1.4.2 Canonical Extensions
1.4.3 Hecke Actions
1.5 Comparison with the Analytic Construction
2. Flat Integral Models
2.1 Auxiliary Choices
2.1.1 Auxiliary Choices of Smooth Moduli Problems
2.1.2 Auxiliary Choices of Toroidal and Minimal Compactifications
2.2 Flat Integral Models as Normalizations and Blow-Ups
2.2.1 Flat Integral Models for Minimal Compactifications
2.2.2 Flat Integral Models for Projective Toroidal Compactifications
2.2.3 Hecke Actions
2.2.4 The Case When p is a Good Prime
3. Ordinary Loci
3.1 Ordinary Semi-Abelian Schemes and Serre's Construction
3.1.1 Ordinary Abelian Schemes and Semi-Abelian Schemes
3.1.2 Serre's Construction for Ordinary Abelian Schemes
3.1.3 Extensibility of Isogenies
3.2 Linear Algebraic Data for Ordinary Loci
3.2.1 Necessary Data for Ordinary Reductions
3.2.2 Maximal Totally Isotropic Submodules at p
3.2.3 Compatibility with Cusp Labels
3.3 Level Structures
3.3.1 Level Structures Away from p
3.3.2 Hecke Twists Defined by Level Structures Away from p
3.3.3 Ordinary Level Structures at p
3.3.4 Hecke Twists Defined by Ordinary Level Structures at p
3.3.5 Comparison with Level Structures in Characteristic Zero
3.3.6 Valuative Criteria
3.4 Ordinary Loci
3.4.1 Naive Moduli Problems with Ordinary Level Structures
3.4.2 Ordinary Loci as Normalizations
3.4.3 Properties of Kodaira{Spencer Morphisms
3.4.4 Hecke Actions
3.4.5 The Case When p is a Good Prime
3.4.6 Quasi-Projectivity of Coarse Moduli
4. Degeneration Data and Boundary Charts
4.1 Theory of Degeneration Data
4.1.1 Degenerating Families of Type (PE, )
4.1.2 Common Setting for the Theory of Degeneration
4.1.3 Degeneration Data for Polarized Abelian Schemes with
Endomorphism Structures
4.1.4 Degeneration Data for Principal Ordinary Level Structures
4.1.5 Degeneration Data for General Ordinary Level Structures
4.1.6 Comparison with Degeneration Data for Level Structures in
Characteristic Zero
4.2 Boundary Charts of Ordinary Loci
4.2.1 Constructions with Level Structures but without Positivity
Conditions
4.2.2 Toroidal Embeddings, Positivity Conditions, and Mumford
Families
4.2.3 Extended Kodaira{Spencer Morphisms and Induced
Isomorphisms
5. Partial Toroidal Compactifications
5.1 Approximation and Gluing Along the Ordinary Loci
5.1.1 Ordinary Good Formal Models
5.1.2 Ordinary Good Algebraic Models
5.1.3 Gluing in the Etale Topology
5.2 Partial Toroidal Compactifications of Ordinary Loci
5.2.1 Main Statements
5.2.2 Hecke Actions
5.2.3 The Case When p is a Good Prime
5.2.4 Boundary of Ordinary Loci
6. Partial Minimal Compactifications
6.1 Homogeneous Spectra and Their Properties
6.1.1 Construction of Quasi-Projective Models
6.1.2 Local Structures and Stratifications
6.2 Partial Minimal Compacti cations of Ordinary Loci
6.2.1 Main Statements
6.2.2 Hecke Actions
6.2.3 Quasi-Projectivity of Partial Toroidal Compactifications
6.3 Full Ordinary Loci in p-Adic Completions
6.3.1 Hasse Invariants
6.3.2 Nonordinary and Full Ordinary Loci
6.3.3 Nonemptiness of Ordinary Loci
7. Ordinary Kuga Families
7.1 Partial Toroidal Compactifications
7.1.1 Parameters for Ordinary Kuga Families
7.1.2 Boundary of Ordinary Loci, Continued
7.1.3 Ordinary Kuga Families and Their Generalizations
7.1.4 Main Statements
7.2 Main Constructions of Compactifications and Morphisms
7.2.1 Partial Toroidal Boundary Strata
7.2.2 Justification for the Parameters
7.2.3 Extensibility of f
7.2.4 Properness of ftor
7.2.5 Log Smoothness of ftor
7.2.6 Equidimensionality of ftor
7.2.7 Hecke Actions
7.3 Calculation of Formal Cohomology
7.3.1 Setting
7.3.2 Formal Fibers of ftor
7.3.3 Relative Cohomology and Local Freeness
7.3.4 Degeneracy of the (Relative) Hodge Spectral Sequence
7.3.5 Extended Gauss–Manin Connections
7.3.6 Identification of Rbf*tor ( Nord,tor)
8. Automorphic Bundles and Canonical Extensions
8.1 Constructions over the Ordinary Loci
8.1.1 Technical Assumptions
8.1.2 Automorphic Bundles
8.1.3 Canonical Extensions
8.1.4 Hecke Actions
8.2 Higher Direct Images to the Minimal Compactifications
8.2.1 Some Vanishing Theorems
8.2.2 Formal Fibers of ord
8.2.3 Relative Cohomology of Formal Fibers of ord
8.2.4 Formal Fibers of Canonical Extensions
8.2.5 End of the Proof
8.3 Constructions over the Total Models
8.3.1 Principal Bundles
8.3.2 Automorphic Bundles
8.3.3 Canonical Extensions
8.3.4 Compatibility with the Constructions over the Ordinary Loci
8.3.5 Pushforwards to the Total Minimal Compactifications
8.3.6 Hecke Actions
Bibliography
Index

Citation preview

Compactifications of PEL-Type Shimura Varieties and Kuga Families with Ordinary Loci

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Compactifications of PEL-Type Shimura Varieties and Kuga Families with Ordinary Loci Kai-Wen Lan University of Minnesota, Twin Cities, USA

World Scientific NEW JERSEY

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

COMPACTIFICATIONS  OF  PEL-TYPE  SHIMURA  VARIETIES  A ND KUGA  FAMILIES  WITH  ORDINARY  LOCI Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-3207-32-5

Printed in Singapore

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Preface

In the summer of 2011, Michael Harris, Richard Taylor, Jack Thorne, and I found a method for attaching p-adic Galois representations to all cohomological automorphic representations of general linear groups over CM or totally real fields, without any polarizability conditions. We were very excited about such a breakthrough. At that time, the arguments were based on my earlier works [57] (which was revised and published as [62]) and [61], which required the prime p to satisfy certain assumptions ensuring that good reduction p-integral models of Shimura varieties and their toroidal compactifications exist at some bottom levels at p. A natural question was whether such assumptions could be removed by improving my earlier works. Fortunately, soon afterwards, I was able to convince myself that the answer is in the affirmative, although a lot still had to be written down to fully justify the ideas. We then decided that we would split up our writing into two parts. One part would remain a four-author collaboration—which ended up being the long article [39]—explaining the most important new arguments under the assumption that certain geometric objects can be constructed in some special cases. The other part would be only by myself—which ended up being this even longer book—detailing the constructions of the geometric objects in the generality they deserve. The ordinary loci in the title of this book are natural generalizations of what are called Igusa towers in Hida’s works (see, for example, [41]), which are the loci where certain multiplicative-type subgroup schemes of the p-power torsion subgroup schemes of the universal abelian schemes can be rigidified (as in Katz’s article [46] on p-adic modular forms). (This is, admittedly, an abuse of language—we do not mean the full loci where the geometric fibers of the universal abelian schemes are ordinary.) Historically, such ordinary loci were almost always built over some good reduction p-integral models of Shimura varieties and their toroidal compactifications, which unavoidably imposed some conditions on the residue characteristic p. However, as we shall see in this book, even without such conditions, it is still possible to construct and describe in precise detail some ordinary loci and their partial compactifications, without relying on any good understanding of the whole p-integral models of Shimura varieties and their compactifications. (We shall see some more detailed explanations in the introduction of the main text.) This is good enough v

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for many applications to the construction of p-adic automorphic forms, and justifies the removal of all assumptions on p in the above-mentioned collaboration. It has been more than five years since the writing started. Although the manuscript already achieved its current form and complexity by the summer of 2013, it has undergone another three years of intermittent reviewing and revising. Despite all the flaws that still remain after all of these, I hope this book will nevertheless provide a clear and reasonably complete picture of a useful theory. January, 2017, in Minneapolis.

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Contents

Preface

v

0.

1

Introduction 0.1 0.2 0.3 0.4 0.5 0.6 0.7

1.

Background and Aim . . . . . . . . . . . . . . . . Overview . . . . . . . . . . . . . . . . . . . . . . Outline of the Constructions . . . . . . . . . . . . What is Known, What is New, and What Can Be What to Note and to Skip in Special Cases . . . Notation and Conventions . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . .

Theory in Characteristic Zero 1.1

1.2

1.3

1.4

. . . . . . . . . . . . . . . . . . . . . . . . Studied Next . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

1 2 8 13 17 18 18 21

PEL-type Moduli Problems and Shimura Varieties . . . . . . . . . 21 1.1.1 Linear Algebraic Data for PEL Structures . . . . . . . . . 21 1.1.2 PEL-type Moduli Problems . . . . . . . . . . . . . . . . . . 23 1.1.3 PEL-type Shimura Varieties . . . . . . . . . . . . . . . . . 26 Linear Algebraic Data for Cusps . . . . . . . . . . . . . . . . . . . 26 1.2.1 Cusp Labels . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.2.2 Cone Decompositions . . . . . . . . . . . . . . . . . . . . . 33 1.2.3 Rational Boundary Components . . . . . . . . . . . . . . . 38 1.2.4 Parameters for Kuga Families . . . . . . . . . . . . . . . . 43 Algebraic Compactifications in Characteristic Zero . . . . . . . . . 57 1.3.1 Toroidal and Minimal Compactifications of PEL-Type Moduli Problems . . . . . . . . . . . . . . . . . . . . . . . . 57 1.3.2 Boundary of PEL-Type Moduli Problems . . . . . . . . . . 66 1.3.3 Toroidal Compactifications of PEL-Type Kuga Families and Their Generalizations . . . . . . . . . . . . . . . . . . . 95 1.3.4 Justification for the Parameters for Kuga Families . . . . . 106 Automorphic Bundles and Canonical Extensions in Characteristic Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 vii

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1.4.1 Automorphic Bundles 1.4.2 Canonical Extensions 1.4.3 Hecke Actions . . . . Comparison with the Analytic

. . . . . . . . . . . . . . . . . . . . . . . . Construction

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

Flat Integral Models 2.1

2.2

3.

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1.5 2.

Compactifications with Ordinary Loci

Auxiliary Choices . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Auxiliary Choices of Smooth Moduli Problems . . . 2.1.2 Auxiliary Choices of Toroidal and Minimal Compactifications . . . . . . . . . . . . . . . . . . . Flat Integral Models as Normalizations and Blow-Ups . . . 2.2.1 Flat Integral Models for Minimal Compactifications 2.2.2 Flat Integral Models for Projective Toroidal Compactifications . . . . . . . . . . . . . . . . . . . 2.2.3 Hecke Actions . . . . . . . . . . . . . . . . . . . . . 2.2.4 The Case When p is a Good Prime . . . . . . . . .

125 . . . . 125 . . . . 125 . . . . 132 . . . . 141 . . . . 141 . . . . 147 . . . . 149 . . . . 152

Ordinary Loci 3.1

3.2

3.3

3.4

117 120 123 124

Ordinary Semi-Abelian Schemes and Serre’s Construction . . . . 3.1.1 Ordinary Abelian Schemes and Semi-Abelian Schemes . . 3.1.2 Serre’s Construction for Ordinary Abelian Schemes . . . 3.1.3 Extensibility of Isogenies . . . . . . . . . . . . . . . . . . Linear Algebraic Data for Ordinary Loci . . . . . . . . . . . . . . 3.2.1 Necessary Data for Ordinary Reductions . . . . . . . . . 3.2.2 Maximal Totally Isotropic Submodules at p . . . . . . . . 3.2.3 Compatibility with Cusp Labels . . . . . . . . . . . . . . Level Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Level Structures Away from p . . . . . . . . . . . . . . . 3.3.2 Hecke Twists Defined by Level Structures Away from p . 3.3.3 Ordinary Level Structures at p . . . . . . . . . . . . . . . 3.3.4 Hecke Twists Defined by Ordinary Level Structures at p 3.3.5 Comparison with Level Structures in Characteristic Zero 3.3.6 Valuative Criteria . . . . . . . . . . . . . . . . . . . . . . Ordinary Loci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Naive Moduli Problems with Ordinary Level Structures . 3.4.2 Ordinary Loci as Normalizations . . . . . . . . . . . . . . 3.4.3 Properties of Kodaira–Spencer Morphisms . . . . . . . . 3.4.4 Hecke Actions . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 The Case When p is a Good Prime . . . . . . . . . . . . 3.4.6 Quasi-Projectivity of Coarse Moduli . . . . . . . . . . . .

155 . . . . . . . . . . . . . . . . . . . . . .

155 155 157 160 164 164 168 171 172 172 175 177 183 192 194 197 197 203 206 208 212 213

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4.

215

4.1

215 215 216

Theory 4.1.1 4.1.2 4.1.3

of Degeneration Data . . . . . . . . . . . . . . . . . . . . . Degenerating Families of Type (PE, O) . . . . . . . . . . . Common Setting for the Theory of Degeneration . . . . . . Degeneration Data for Polarized Abelian Schemes with Endomorphism Structures . . . . . . . . . . . . . . . . . . 4.1.4 Degeneration Data for Principal Ordinary Level Structures 4.1.5 Degeneration Data for General Ordinary Level Structures . 4.1.6 Comparison with Degeneration Data for Level Structures in Characteristic Zero . . . . . . . . . . . . . . . . . . . . . Boundary Charts of Ordinary Loci . . . . . . . . . . . . . . . . . . 4.2.1 Constructions with Level Structures but without Positivity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Toroidal Embeddings, Positivity Conditions, and Mumford Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Extended Kodaira–Spencer Morphisms and Induced Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . .

Partial Toroidal Compactifications 5.1

5.2

6.

ix

Degeneration Data and Boundary Charts

4.2

5.

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Approximation and Gluing Along the Ordinary Loci 5.1.1 Ordinary Good Formal Models . . . . . . . . 5.1.2 Ordinary Good Algebraic Models . . . . . . ´ 5.1.3 Gluing in the Etale Topology . . . . . . . . . Partial Toroidal Compactifications of Ordinary Loci 5.2.1 Main Statements . . . . . . . . . . . . . . . . 5.2.2 Hecke Actions . . . . . . . . . . . . . . . . . 5.2.3 The Case When p is a Good Prime . . . . . 5.2.4 Boundary of Ordinary Loci . . . . . . . . . .

6.2

6.3

242 248 248 267 272 279

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. . . . . . . . .

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. . . . . . . . .

Partial Minimal Compactifications 6.1

216 217 235

Homogeneous Spectra and Their Properties . . . . . . . . . . . . 6.1.1 Construction of Quasi-Projective Models . . . . . . . . . 6.1.2 Local Structures and Stratifications . . . . . . . . . . . . Partial Minimal Compactifications of Ordinary Loci . . . . . . . 6.2.1 Main Statements . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Hecke Actions . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Quasi-Projectivity of Partial Toroidal Compactifications . Full Ordinary Loci in p-Adic Completions . . . . . . . . . . . . . 6.3.1 Hasse Invariants . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Nonordinary and Full Ordinary Loci . . . . . . . . . . . . 6.3.3 Nonemptiness of Ordinary Loci . . . . . . . . . . . . . . .

279 279 285 288 294 295 300 305 310 331

. . . . . . . . . . .

331 331 338 345 345 353 359 361 361 365 369

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Ordinary Kuga Families 7.1

7.2

7.3

8.

Compactifications with Ordinary Loci

Partial Toroidal Compactifications . . . . . . . . . . . . . . . . 7.1.1 Parameters for Ordinary Kuga Families . . . . . . . . . 7.1.2 Boundary of Ordinary Loci, Continued . . . . . . . . . 7.1.3 Ordinary Kuga Families and Their Generalizations . . . 7.1.4 Main Statements . . . . . . . . . . . . . . . . . . . . . . Main Constructions of Compactifications and Morphisms . . . 7.2.1 Partial Toroidal Boundary Strata . . . . . . . . . . . . 7.2.2 Justification for the Parameters . . . . . . . . . . . . . 7.2.3 Extensibility of fκ . . . . . . . . . . . . . . . . . . . . . 7.2.4 Properness of f tor . . . . . . . . . . . . . . . . . . . . . 7.2.5 Log Smoothness of f tor . . . . . . . . . . . . . . . . . . 7.2.6 Equidimensionality of f tor . . . . . . . . . . . . . . . . 7.2.7 Hecke Actions . . . . . . . . . . . . . . . . . . . . . . . Calculation of Formal Cohomology . . . . . . . . . . . . . . . . 7.3.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Formal Fibers of f tor . . . . . . . . . . . . . . . . . . . 7.3.3 Relative Cohomology and Local Freeness . . . . . . . . 7.3.4 Degeneracy of the (Relative) Hodge Spectral Sequence 7.3.5 Extended Gauss–Manin Connections . . . . . . . . . . . 7.3.6 Identification of Rb f∗tor (ON ~ ord,tor ) . . . . . . . . . . . . .

371 . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

Automorphic Bundles and Canonical Extensions 8.1

8.2

8.3

Constructions over the Ordinary Loci . . . . . . . . . . . . . . . . 8.1.1 Technical Assumptions . . . . . . . . . . . . . . . . . . . 8.1.2 Automorphic Bundles . . . . . . . . . . . . . . . . . . . . 8.1.3 Canonical Extensions . . . . . . . . . . . . . . . . . . . . 8.1.4 Hecke Actions . . . . . . . . . . . . . . . . . . . . . . . . Higher Direct Images to the Minimal Compactifications . . . . . 8.2.1 Some Vanishing Theorems . . . . . . . . . . . . . . . . . H ord 8.2.2 Formal Fibers of ~ H . . . . . . . . . . . . . . . . . . . . H ord 8.2.3 Relative Cohomology of Formal Fibers of ~ H . . . . . . 8.2.4 Formal Fibers of Canonical Extensions . . . . . . . . . . 8.2.5 End of the Proof . . . . . . . . . . . . . . . . . . . . . . . Constructions over the Total Models . . . . . . . . . . . . . . . . 8.3.1 Principal Bundles . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Automorphic Bundles . . . . . . . . . . . . . . . . . . . . 8.3.3 Canonical Extensions . . . . . . . . . . . . . . . . . . . . 8.3.4 Compatibility with the Constructions over the Ordinary Loci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.5 Pushforwards to the Total Minimal Compactifications . .

371 371 378 390 392 407 407 411 421 425 430 440 443 444 444 445 453 469 469 471 477

. . . . . . .

477 477 480 483 488 490 490

. 492 . . . . . . .

496 501 508 508 508 510 512

. 513 . 514

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Hecke Actions . . . . . . . . . . . . . . . . . . . . . . . . . 518

Bibliography

525

Index

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Chapter 0

Introduction

0.1

Background and Aim

In [62] (which is a published revision of [57]), based on the theories developed in [82] and [28], we studied the theory of degeneration of abelian varieties with PEL structures, and applied this theory to the construction of toroidal and minimal compactifications of moduli problems defining integral models of PEL-type Shimura varieties, under the assumption that each residue characteristic p is good in the sense that it is unramified in all linear algebraic data involved in the definition of the moduli problem of abelian varieties with PEL structures, and under the assumption that the level structures are defined by open compact subgroups of the adelic points of the associated reductive groups that are hyperspecial (maximal) at p. In [61], we also constructed toroidal compactifications of PEL-type Kuga families under the same assumptions on the residue characteristics, by realizing such toroidal compactifications in the toroidal boundary of larger PEL-type Shimura varieties. While these have been carried out for all PEL-type Shimura varieties, for practical reasons it is also natural to consider integral models when p is ramified in the linear algebraic data and when the level structures are defined by smaller open compact subgroups. Since the theory of degeneration developed in [62] works as long as the generic characteristic is good (and as long as the base of degenerations are noetherian normal), there is, a priori, no reason that we cannot consider compactifications with bad residue characteristics. However, without the assumption that p is good and that the level structures are defined by an open compact subgroup hyperspecial at p, it is not clear what integral models really mean in general (although reasonably natural definitions can still be made in many special cases). The answer may depend on the applications. For applications involving counting points over finite fields, it seems necessary to have integral models with a specific kind of moduli interpretations, but usually even the flatness of such models are difficult to prove. For studying intersections of cycles, it is desirable to have models that are regular and flat, and we might consider the closures of the generic fibers in moduli problems as a general source of flat models, but the regularity of such models can be beyond reach already at very low levels. In 1

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both cases, it is difficult to say much about the integral models of Shimura varieties themselves, let alone their compactifications. On the other hand, for studying modular forms using coherent sheaves, there is already a rich theory using mainly the ordinary loci , or more precisely the ordinary loci where (multiplicative-type) canonical subgroups can be defined , when p is good in the above sense, and when the levels are certain (analogues of) “Γ1 (pr ) levels”. The aim of this book is to show that, without insisting on the (perhaps still desirable) moduli interpretations, such a theory can be generalized without the assumption that p is good, after adding sufficiently many p-power roots of unity to the base rings. For such ordinary loci, we will construct partial toroidal and minimal compactifications which admit descriptions analogous to (and compatible with) their analogues in characteristic zero (and in mixed characteristics in the hyperspecial smooth case in [62]). We will also construct partial toroidal compactifications of Kuga families over such partial toroidal and minimal compactifications. Our construction works for all PEL-type Shimura varieties that can possibly admit ordinary loci. We allow p to be ramified (i.e., p is not good in the sense described above), and we allow (analogues of) arbitrarily high “Γ1 (pr ) levels”. We need the ordinary loci to be defined, but we do not assume that they are nonempty (although the theory is uninteresting otherwise). (In some special cases, we can easily show the nonemptiness of the ordinary loci using the partial toroidal compactifications we construct. See Section 6.3.3.) Unsurprisingly, we started the construction in this work because of some interesting cases in which the nonemptiness of ordinary loci is clear. (See, for example, [39].) As in characteristic zero and in the good reduction case, we will not need to answer difficult questions about p-divisible groups or p-adic Hodge theory in such a theory. As we shall see below, the difficulty in the constructions lies mainly in the sheer number of objects, morphisms, combinatorial data, and small subtle steps involved. It is not about proving some well-known conjecture that can be readily stated—rather, we want to know as much as possible about the constructions, and (at least for some applications we know) the theorems are useful only when they are detailed enough. It is fair to say that this is just another long exercise like [62]. Assuming that this is still interesting, the marathon begins.

0.2

Overview

Let us briefly describe the various objects to be constructed. (We say briefly but it still spans over many pages.)

Algebraic Constructions in Characteristic Zero Starting with an integral PEL datum (O, ?, L, h · , · i, h0 ), as in [62], we can construct the following canonical objects (in characteristic zero):

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(1) A group scheme G over Spec(Z), which is smooth and reductive over Spec(Z(p) ) when p is a good prime. (2) A number field F0 , which is defined as a subfield of C, called the reflex field . (3) A moduli problem MH over S0 = Spec(F0 ) for each open compact subgroup ˆ parameterizing abelian schemes with PEL structures defined by H ⊂ G(Z), the integral PEL datum, which is an algebraic stack separated, smooth, and of finite type over S0 := Spec(F0 ). When H is neat (see [89, 0.6] or [62, Def. 1.4.1.8]), MH is quasi-projective over Spec(F0 ). (In particular, MH is a scheme.) (4) A finite ´etale surjection [g] : MH0 → MH over S0 for all g ∈ G(A∞ ) and open compact subgroups H and H0 such that H0 ⊂ gHg −1 . This can be interpreted as the Hecke action of G(A∞ ) on the collection {MH }H . (5) A toroidal compactification Mtor H,Σ of MH over S0 for each compatible choice Σ of admissible smooth rational polyhedral cone decomposition data for MH , which is a collection of combinatorial data which can be defined using only the integral PEL datum. For technical reasons, we assume that Σ is smooth (even in characteristic zero) and satisfies some mild conditions, in which case we can show that Mtor H,Σ is a proper smooth algebraic stack, and that the boundary is a simple normal crossings divisor. If H is neat, then Mtor H,Σ is an algebraic space. The toroidal compactification admits a stratification (defined in terms of G and Σ), and the structure along its boundary can be described in detail. These are useful for defining and studying modular forms using coherent sheaf cohomology. (See [60] for a survey on this topic.) tor tor (6) A proper log ´etale surjection [g] : Mtor H0 ,Σ0 → MH,Σ over S0 extending ∞ [g], for all g ∈ G(A ) and open compact subgroups H and H0 such that H0 ⊂ gHg −1 , and for each Σ0 that is a g-refinement of Σ in a suitable sense. This can be interpreted as the Hecke action of G(A∞ ) on the collection 0 {Mtor etale H,Σ }H,Σ . When g = 1 and H = H, this means we have proper log ´ tor tor tor 0 surjections [1] : MH,Σ0 → MH,Σ when Σ is a refinement of Σ. (7) A minimal compactification Mmin H over S0 of the coarse moduli [MH ] of MH ˆ which is a normal scheme profor each open compact subgroup H ⊂ G(Z), jective over S0 , which admits a canonical (proper and surjective) morphism from the toroidal compactification Mtor H,Σ for each Σ. The stratification of any such toroidal compactification induces a stratification of the minimal compactification, which is independent of the choice of Σ. The strata in such a stratification are called cusps. min min (8) A finite surjection [g] : Mmin over S0 extending the finite surH0 → MH jection [[g]] : [MH0 ] → [MH ] between coarse moduli for all g ∈ G(A∞ ) and open compact subgroups H and H0 such that H0 ⊂ gHg −1 . This can be interpreted as the Hecke action of G(A∞ ) on the collection {Mmin H }H . (9) If H is neat and if Σ is (smooth and) projective, then we show that Mtor H,Σ is

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(smooth and) projective over S0 by showing that it is the normalization of the blowup of some coherent ideal sheaf JH,d0 pol on Mmin H , defined by some integer d0 ≥ 1 and some compatible collection pol of polarization functions for Σ. In particular, Mtor H,Σ is a scheme in this case. (10) A collection of Kuga families over MH , which is a collection of abelian schemes including the self-fiber products of the tautological (i.e., universal) abelian scheme as special members, together with toroidal compactifications projective over S0 and satisfying a long list of desirable compatibilities, including in particular the existence (up to refinements of cone decompositions) of compatible proper log smooth morphisms from toroidal compactifications of PEL-type Kuga families to Mtor H,Σ . We can enlarge the collection and include objects which are torsors under PEL-type Kuga families over MH , which we call generalized Kuga families over MH . They share the same nice properties enjoyed by PEL-type Kuga families. (11) A collection of automorphic bundles over MH , and their canonical and subcanonical extensions over Mtor H,Σ . The (algebraic) construction of such canonical and subcanonical extensions uses the toroidal compactifications of PEL-type Kuga families. The above constructions use only the theory of degeneration data and standard techniques in algebraic geometry. We call them the algebraic constructions over S0 . Analytic Constructions and Comparison with Them There is also the analytic constructions of analogous objects over S0 , which precedes the algebraic constructions in history. (These are algebraic objects constructed using transcendental arguments crucially in their constructions. Such analytic constructions use GAGA [95], but when we compare them to the algebraic constructions, we are not talking about a problem of GAGA anymore.) In [59] we showed that, for suitable H and Σ, these analytically constructed objects admit canonical open and closed immersions to the algebraically constructed objects above, respecting all stratifications and descriptions of local structures. (See also [63] for the relation between rational boundary components and cusp labels.) The Case When p is a Good Prime As explained in [62], when p is a good prime for the (O, ?, L, h · , · i, h0 ), and when H ˆ p ) × G(Zp ) = G(Z), ˆ (note that in this case G(Zp ) is of the form H = Hp G(Zp ) ⊂ G(Z is a hyperspecial maximal open compact subgroup of G(Qp ),) the above algebraically constructed objects admit analogues over Spec(OF0 ,(p) ), which we denote by min MHp , Mtor Hp ,Σp , MHp , etc. (Such notation makes sense because in the construction of these objects we only use Hp and an analogue Σp of Σ involving only adelic objects tor away from p.) Then there are canonical morphisms MH → MHp , Mtor H,Σ → MHp ,Σp , min min MH → MHp , etc, compatible with each other, and respecting all stratifications

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and descriptions of local structures. (But we will not assume that p is a good prime for the (O, ?, L, h · , · i, h0 ) in what follows.) Total Models in Mixed Characteristics Assuming no longer that p is good, we will construct the following objects (in mixed characteristics (0, p)): ~ H for each open compact subgroup H ⊂ G(Z), ˆ (1) A normal algebraic stack M flat over ~S0 = Spec(OF0 ,(p) ) for each open compact subgroup H as above, ~ H . The coarse moduli which admits a canonical morphism MH → M ~ H ] of M ~ H is a normal scheme quasi-projective and flat over space [M ~S0 = Spec(OF ,(p) ), which admits a canonical morphism [MH ] → [M ~ H ]. 0 ~ ~ ~ ~ (2) A finite surjection [g] : MH0 → MH over S0 for each g = (g0 , gp ) ∈ G(A∞,p ) × G(Zp ) ⊂ G(A∞ ) and two open compact subgroups H and H0 such that H0 ⊂ gHg −1 . This can be interpreted as the Hecke action of ~ H }H . G(A∞,p ) × G(Zp ) on the collection {M min ~ (3) A normal scheme MH projective and flat over ~S0 for each open compact ~ H ] of M ~ H as an open ˆ containing the coarse moduli [M subgroup H ⊂ G(Z), dense subscheme. ~ min : M ~ min ~ min over ~S0 extending the fi(4) A finite surjection [g] → M H0 H ~ : [M ~ H0 ] → [M ~ H ] between coarse moduli for each nite surjection [[g]] g ∈ G(A∞,p ) × G(Zp ) and two open compact subgroups H and H0 such that H0 ⊂ gHg −1 . This can be interpreted as the Hecke action of ~ min }H . G(A∞,p ) × G(Zp ) on the collection {M H tor ~ ~ (5) A normal scheme M H,d0 pol projective and flat over S0 , which is defined when H is neat, when Σ is (smooth and) projective with a compatible collection pol of polarization functions, and when d0 ≥ 1 is sufficiently large, which is the normalization of the blowup of some coherent ideal sheaf J~H,d0 pol on ~ min defined by J~H,d pol is the schematic ~ min , such that the subscheme of M M 0 H H closure of the subscheme of Mmin H defined by the JH,d0 pol above. ~ H , and their canonical and (6) A collection of automorphic bundles over M tor ~ ~ tor subcanonical extensions over MH,d0 pol when M H,d0 pol is defined. (7) For each integer i ≥ 0, we define S0,i := Spec(F0 [ζpi ]) and ~S0,i := ~ H,i (resp. M ~ min , resp. M ~ tor Spec(OF0 ,(p) [ζpi ]), and define M H,i H,d0 pol,i ) to be the min tor ~ ~ ~ ~ ~ normalization of MH × S0,i (resp. MH × S0,i , resp. MH,d0 pol × ~S0,i ). ~ S0

~ S0

~ S0

These constructions require noncanonical auxiliary choices. A priori, it is unclear whether the objects thus constructed are independent of the choices, although it can be proved that they are indeed so. The quasi-projectivity of certain objects that will ~ ord,min over ~S0,r = Spec(OF ,(p) [ζprH ]), be canonically constructed below, such as M H 0 H ~ min over ~S0 . is proved using the projectivity of such a noncanonically constructed M H (We do not know any other method for proving such quasi-projectivity.) Such

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quasi-projectivity over mixed characteristics bases is important for many practical reasons. In particular, it allows us to talk about congruences (between algebrogeometrically defined automorphic forms) using its affine subsets. The Ordinary Loci in Mixed Characteristics This is the main theme of this work. With a suitable choice of a maximal totally isotropic filtration D on L ⊗ Zp , we will construct the following canonical objects (in Z

mixed characteristics (0, p)): (1) A subgroup scheme Pord of G ⊗ Zp stabilizing the filtration D. (We do not D Z

say that Pord is parabolic because G ⊗ Zp is not smooth in general. But D Z

when G ⊗ Qp is connected, which is the case when O ⊗ Q involves no factor Z

Z

⊗ Qp is indeed a parabolic of type D in the sense of [62, Def. 1.2.1.15], Pord D Zp

subgroup scheme of the reductive group scheme G ⊗ Qp in the usual sense.) Z

ˆ of the form H = (2) A collection of open compact subgroups H ⊂ G(Z) p p ˆ ) × G(Zp ) = G(Z), ˆ such that Hp satisfies U bal (pr ) ⊂ Hp ⊂ H Hp ⊂ G(Z p,1 bal r Up,0 (pr ) for some integer r ≥ 0. Here Up,1 (p ) and Up,0 (pr ) are open compact subgroups of G(Zp ) defining the (analogues of) “balanced Γ1 (pr )” and “Γ0 (pr )” levels at p. ...ord (3) A naive moduli problem MH over Spec(Z(p) ), parameterizing abelian schemes with PEL structures away from p, and with certain ordinary level structures at p, but without the determinantal condition for Lie algebras in the definition of MH . (Since p is not assumed to be a good prime, such a con...ord dition is not useful.) This MH is an algebraic stack separated and of finite over Spec(Z(p) ), with completions of strict local rings the same as those of a group scheme of multiplicative type of finite type over Spec(Z(p) ). (Hence, it is not smooth in general, but the singularity is mild.) (4) An integer rH determined by the integral PEL datum (O, ?, L, h · , · i, h0 ), the data D. This integer rH stays as a constant rD and does not increase with r if Up,1 (pr ) ⊂ Hp , where Up,1 (pr ) is an open compact subgroup of G(Zp ) defining the (analogue of) “Γ1 (pr )” levels at p. But it increases with bal r r (and is equal to max(rD , r)) if, for example, Hp = Up,1 (p ). ord (5) An algebraic stack MH separated, smooth, and of finite type over S0,rH := Spec(F0 [ζprH ]) parameterizing certain ordinary level structures in characteristic zero, which is canonically isomorphic to MH × S0,rH , but with the S0

understanding that the usual level structures are turned into the ordinary level structures it parameterizes (with the help of roots of unity in S0,rH ). ...ord The universal property of MH induces a canonical quasi-finite morphism ...ord Mord H → MH .

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~ ord separated, smooth, and of finite type over ~S0,r = (6) An algebraic stack M H H ...ord Spec(OF0 ,(p) [ζprH ]), which admits canonical finite morphism to MH exten...ord ding the quasi-finite morphism Mord H → MH . ~ ord : M ~ ord0 → M ~ ord over ~S0,r for each (7) A quasi-finite flat surjection [g] H

H

H

∞ g = (g0 , gp ) ∈ G(A∞,p ) × Pord D (Qp ) ⊂ G(A ) and two open compact sub0 0 −1 groups H and H such that H ⊂ gHg , satisfying some reasonable additional conditions. This can be interpreted as the Hecke action of a semi~ ord subgroup of G(A∞,p ) × Pord D (Qp ) on the collection {MH }H . (And there ord ~ are conditions for the morphisms [g] to be finite or ´etale.) ~ ord,tor ~ ord over ~S0,r for each com(8) A partial toroidal compactification M of M H H H,Σord patible choice Σord of admissible smooth rational polyhedral cone decom~ ord , which is a collection of combinatorial data defined position data for M H in a way similar to the case of Σ above. In fact, each Σ as above induces ~ ord,tor a Σord . This M is an algebraic stack separated, smooth, and of finite H,Σord type over ~S0,rH , and the boundary is a simple normal crossings divisor. If ~ ord,tor Hp is neat, then M ord is an algebraic space. The partial toroidal comH,Σ

pactification admits a stratification (defined in terms of G, D, and Σord ), and the structure along its boundary can be described in detail; both are as in the case of Mtor H,Σ . ord,tor ~ ~ ord ~ ord,tor ~ ord,tor (9) A surjection [g] : M → M over ~S0,rH extending [g] H0 ,Σord,0 H,Σord ~ ord is defined, and for each Σord,0 that is for each g as above such that [g] a g-refinement of Σord in a suitable sense. This can be interpreted as the Hecke action of the same semi-subgroup of G(A∞,p ) × Pord D (Qp ) as above ord,tor ~ ord,min ~ on the collection {M ord }H,Σord . (And there are conditions for [g] H,Σ

to be proper, finite, flat, log ´etale, or ´etale.) ~ ord,min over ~S0,r of the coarse moduli (10) A partial minimal compactification M H H ord ord ~ ] of M ~ , which is a normal scheme quasi-projective and flat over [M H H ~S0,r , which admits a canonical proper (and surjective) morphism from the H ~ ord,tor partial toroidal compactification M for each Σord . The stratification H,Σord of any such partial toroidal compactification induces a stratification of the partial minimal compactification, which is independent of the choice of Σord . The strata in such a stratification are called ordinary cusps. ~ ord,min : M ~ ord,min ~ ord,min over ~S0,r exten(11) A quasi-finite surjection [g] →M H H0 H ord ~ ~ ord0 ] → [M ~ ord ] between coarse moding the quasi-finite surjection [[g] ] : [M H

H

~ ord is defined. This can be interpreted duli for each g as above such that [g] as the Hecke action of the same semi-subgroup of G(A∞,p ) × Pord D (Qp ) as ord,min ~ ord,min ~ above on the collection {M }H . (And there are conditions for [g] H

to be finite.) (12) If Hp is neat and if Σord is (smooth and) projective, then we show that

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~ ord,tor M is (smooth and) quasi-projective over ~S0,rH by showing that it is H,Σord the normalization of the blowup of some coherent ideal sheaf J~H,d0 polord on ~ ord,min , defined by some integer d0 ≥ 1 and some compatible collection M H ~ ord,tor polord of polarization functions for Σord . In particular, M ord is a quasiH,Σ

projective scheme in this case. ~ ord , which is a collection of (13) A collection of ordinary Kuga families over M H abelian schemes containing the self-fiber products of the tautological abelian scheme, together with partial toroidal compactifications quasi-projective over ~S0,rH and satisfying a long list of desirable compatibilities, including in particular the existence (up to refinements of cone decompositions) of compatible proper log smooth morphisms from partial toroidal compactifi~ ord,tor cations of ordinary PEL-type Kuga families to M . We can enlarge the H,Σord collection and include objects which are torsors under Kuga families over ~ ord . They ~ ord , which we call generalized ordinary Kuga families over M M H H share the same nice properties enjoyed by ordinary PEL-type Kuga families. ~ ord , and their canonical and (14) A collection of automorphic bundles over M H ord,tor ~ subcanonical extensions over MH,Σord . The (algebraic) construction of such canonical and subcanonical extensions uses the partial toroidal compacti~ ord ). (The class of fications of ordinary PEL-type Kuga families (over M H ~ ord is more restrictive than automorphic bundles we can construct over M H that over MH .) These objects are compatible with the algebraically constructed objects in characteristic zero, such as MH , and with the total models in mixed characteristics, such ~ H. as M 0.3

Outline of the Constructions

The objects above are not constructed in the same order as they are listed. The logical steps we need are as follows: Algebraic Constructions in Characteristic Zero min We start with all algebraically constructed objects MH , Mtor H,Σ , MH , etc over tor S0 = Spec(F0 ). The algebraic construction of MH,Σ by the theory of degeneration endows it with a semi-abelian scheme with PEL structures, which is universal among semi-abelian degenerations of abelian varieties with PEL structures of a particular degeneration pattern given by Σ. We call such semi-abelian schemes degenerating families. These are done in [62]. We review them in Chapter 1, because it might be hard to get used to the fact that even the characteristic zero theory can be done so noncanonically with various choices. (We still consider the theory canonical because

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it hardly favors any particular choices.) Auxiliary Choices of Good Reduction Models We make a (noncanonical) auxiliary choice of an integral PEL datum (Oaux , ?aux , Laux , h · , · iaux , h0,aux ) for which p is a good prime. This allows us to define a group scheme Gaux over Spec(Z) such that Gaux (Zp ) is a hyperspecial maximal open compact subgroup of Gaux (Qp ), and to construct for each open comp ˆ p ) the objects MHp , Mtorp , Mmin , etc over pact subgroup Haux ⊂ Gaux (Z aux Haux ,Σp Hp aux aux ~S0,aux = Spec(OF ,(p) ). (Here the superscript “p” means “away from p”.) The 0,aux

point is that MHpaux is a moduli problem, Mtor p carries a tautological degeneraHp aux ,Σ min ~ ting family, and M p is projective over S0,aux . Haux

The auxiliary choices are made in Section 2.1. The constructions of the geometric objects are done in [62]. We do not explicitly review them because they are only auxiliary in nature, and because their behaviors are almost identical to those of min MH , Mtor above. We will simply cite [62], with the “2” there filled with H,Σ , MH “p”, and with each object there attached with a subscript “aux” and a superscript “p”. The auxiliary objects are chosen so that there is a homomorphism G → Gaux of group schemes over Z, and so that we have morphisms MH → MHpaux , Mmin H → tor tor → M Mmin , M , etc compatible with each other when H is of the form p p p H,Σ Haux ,Σ Haux p p p ˆ ˆ ˆ H = H Hp ⊂ G(Z ) × G(Zp ) = G(Z) and is mapped into H G(Zp ) ⊂ Gaux (Z). aux

Total Models in Mixed Characteristics min ~ H (resp. M ~ min ) to be the normalization of M We define M ˆ p ) (resp. MG H ˆp ) Gaux (Z aux (Z ) ~ tor in MH (resp. Mmin H ). Then we define MH,d0 pol as described above as a normalization ~ min (depending on the choices of pol and d0 ). These are of a suitable blowup of M H algebraic stacks or schemes over ~S0 = Spec(OF ,(p) ), depending on the choice of 0

(Oaux , ?aux , Laux , h · , · iaux , h0,aux ). The Hecke actions are induced by the universal property of normalizations. These are done in Sections 2.2.1, 2.2.2, and 2.2.3. These total models should be considered as auxiliary in nature. Although they can be shown to be canonical by an indirect argument, based on certain techniques developed in [58], their constructions are noncanonical, and we cannot say much about their local structures. We will have to construct the ordinary loci separately, map them to these total models, and then show that suitable normalizations of these total models (after ramified base changes) have smooth open subschemes given by the images of the ordinary loci. ~ H and M ~ min can be Nevertheless, when p is a good prime, the schemes M H canonically constructed (without the auxiliary objects). This is explained in Section 2.2.4. Such special cases are important because p is a good prime for (Oaux , ?aux , Laux , h · , · iaux , h0,aux ). In what follows, we can often reduce the proof

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of important facts to the case of the auxiliary models, and prove them by more direct methods. ~ ord Construction of M H ~ ord over ~S0,r = Spec(OF ,(p) [ζprH ]) as follows: We construct M H H 0 (1) We investigate, roughly speaking, what happens when an abelian scheme with PEL structures over a scheme over MH extends to an ordinary abelian ~ H . We write down the necessary linear algescheme over a scheme over M braic data for this to happen, and turn them into formal definitions. This gives, in particular, a filtration D on L ⊗ Zp satisfying certain properties. Z

These are done in Sections 3.2.1 and 3.2.2. (2) We develop the notion of ordinary level structures at p defined by D, accompanied by usual level structures away from p. (We do not assume that the polarization degree is prime to p.) This is done in Section 3.3. ...ord (3) We define MH over Spec(Z(p) ) as a naive moduli problem for abelian schemes with polarizations, endomorphism structures, usual level structures away from p, and with ordinary level structures at p defined by D. The ...ord local structures of MH can be studied in two ways. At points of characteristic zero, it is the same as in the case of MH . At points of characteristic p, since all abelian schemes involved are ordinary, we use the Serre–Tate deformation theory explained in [47]. This is done in Section 3.4.1. r (4) We define Mord H over S0,rH = Spec(F0 [ζp H ]) by turning the level structures at p parameterized by the moduli problem MH into ordinary level ...ord ~ ord over ~S0,r = structures parameterized by MH . Then we define M H H ...ord . The point Spec(OF0 ,(p) [ζprH ]) to be the normalization of MH in Mord H of making the (ramified) base change to ~S0,rH is that the normalization ~ ord is smooth over ~S0,r and regular. These are done in Section 3.4.2. M H H When p is a good prime, in which case MHp is defined over ~S0 = Spec(OF0 ,(p) ), ~ ord can be defined by taking the schematic closure of Mord (the latter we show that M H H being just a base change of MH ) in a moduli problem schematic and quasi-finite over MHp . This is done in Section 3.4.5. ~ ord ] over ~S0,r as follows: Then one can show the quasi-projectivity of [M H H (1) Using an auxiliary choice of the filtration Daux at p for the ~ ord as above, toget(Oaux , ?aux , Laux , h · , · iaux , h0,aux ) above, we define M Haux ~ ord → M ~ ord . her with a quasi-finite morphism M H Haux (2) Since p is a good prime for (Oaux , ?aux , Laux , h · , · iaux , h0,aux ) by assump~ ord → M ~ ord . (See above.) tion, we obtain a quasi-finite morphism M Haux ˆp ) G(Z ~ ord → M ~ ord , (3) Combining the above, we obtain a quasi-finite morphism M H

ˆp ) G(Z

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~ ord ] → [M ~ ord ] between noetherwhich induces a quasi-finite morphism [M H ˆp ) G(Z ian normal schemes. Then Zariski’s Main Theorem implies that we have ~ ord ] ,→ [M ~ H,r ], which shows that [M ~ ord ] is quasian open immersion [M H H H ~ projective over S0,rH . These are done in Section 3.4.6. ~ ord,tor Construction of M H,Σord ~ ord,tor We construct M over ~S0,rH as follows: H,Σord (1) Following [28] and [62], we develop a theory of degeneration for abelian varieties with ordinary level structures. This will be used in the construction of toroidal boundary charts, and in showing that what we obtained satisfy certain universal property among all degenerations over normal schemes. (This will, in particular, provide us with a valuative criterion over complete discrete valuation rings.) This is done in Section 4.1. (2) Using the theory of degeneration, we can construct naive toroidal boundary charts over ~S0 parameterizing the degeneration data for the degeneration ...ord of objects parameterized by MH . These naive toroidal boundary charts are similar to their analogues constructed algebraically over S0 (as in [62]). We take the normalization of the naive objects in the base changes of the ~ ord above). characteristic zero objects to S0,rH (as in the construction of M H ~ We can show that these normalizations are smooth over S0,rH and regular. This is done in Section 4.2. (3) Then we show that suitable algebraizations of the formal completions of ~ ord in the ´etale topology. This gives these normalizations can be glued to M H ord,tor ~ us the desired MH,Σord . This is done in Sections 5.1 and 5.2. The outline here is simple, but of course these constructions are central to the whole work, without which other technical considerations make no sense. (Otherwise we could have also included the nonordinary loci in our study.) Fortunately, since the theory in [62] is developed in sufficient generality, there is no surprising difficulty in this part of the theory. ~ ord,min Construction of M H ~ ord,min over ~S0,r as follows: We construct M H H ~ ord,tor (1) We start with a partial toroidal compactification M carrying a semiH,Σord abelian scheme G, and we define the so-called Hodge invertible sheaf top ∼ top e∗ Ω1 ord,tor ωM Lie∨ ~ ord,tor := ∧ ~ ord,tor = ∧ G G/M ~ G/M H,Σord

H,Σord

H,Σord

as usual. By imitating the construction of Mmin H , we define   ord,min ord,tor ⊗k ~ ~ M := Proj ⊕ Γ(M ord , ω ord,tor ) . H

k≥0

H,Σ

~ M

H,Σord

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~ ord,tor ~ ord,min is However, since M is not proper, we cannot assert that M H H,Σord projective over ~S0,rH . The question is whether we can show that it is quasi-projective over ~S0,rH , and whether we can show that the canonical ~ ord,tor ~ ord,min is proper. (We will outline the steps below.) morphism M ord → M H,Σ

H

(2) Once we know this last properness, the familiar arguments for studying ~ ord,min by considering the Stein factorization of the local structures of M H ord,tor ord,min ~ ~ MH,Σord → MH (which coincide with itself) work as in [62]. These are done in Sections 6.1 and 6.2.1. ~ ord,tor ~ ord,min and the quasi-projectivity The proof for the properness of M →M H H,Σord ~ ord,min over ~S0,r is somewhat indirect. Therefore, we would like to summarize of M H H the steps here too: (1) We show that the statements can be proved by replacing H with a higher level that is equally deep at p, and by replacing Σord with a refinement. (2) Take Haux to be as deep as H. Using the assumption that p is good for (Oaux , ?aux , Laux , h · , · iaux , h0,aux ), we explain that, for Σord aux induced by ~ ord,tor ord by taking the schematic closure of some Σp , we can build M aux

Haux ,Σaux

tor Mord . p Haux in a moduli problem schematic and quasi-finite over MHp aux ,Σaux ord,0 p 0 (3) Take Haux to be Haux Gaux (Zp ), and take the Σaux also induced by Σpaux . ~ ord,tor ord → Mtorp Then, the quasi-finite morphism M factors through Haux ,Σp Haux ,Σaux aux ord,tor tor ~ an open immersion M , and there is an induced open p ord,0 ,→ M p H0aux ,Σaux

Haux ,Σaux

~ ord,min ~ ord,min immersion M ,→ Mmin . This shows that M is quasi-projective H0aux H0aux Hp aux over ~S0,rHaux . ~ ord,tor ord,0 is the (4) We can use the theory of degeneration to show that M 0 Haux ,Σaux

~ ord,min precise preimage of M under the proper morphism Mtor → p H0aux Hp aux ,Σaux ord,tor ord,min min ~ ~ M p . This shows what M is proper and surjective. ord,0 → M 0 Haux

H0aux ,Σaux

Haux

~ ord,tor ord → (5) By studying the fibers of the quasi-finite morphism M Haux ,Σaux ~ ord,tor ord,0 , we also obtain the properness of the morphism M ~ ord,tor ord → M 0 Haux ,Σ Haux ,Σaux

aux

~ ord,min . The usual Stein factorization argument then shows that M ~ ord,min M Haux Haux ~ min ~ ord,min is embedded as an open subscheme in M . This shows that M Haux ,rH Haux is quasi-projective over ~S0,rH . (6) Under the assumption that H and Haux are equally deep at p, by refining Σord if necessary (so that it is compatible with Σord aux ), we obtain a proper ord,tor ~ ord,tor ~ morphism M → M . Then we can finish the proof by the H,Σord Haux ,Σord aux usual Stein factorization argument. These are done in Sections 5.2.3 and 6.1.1. (When we show such quasi-projectivity, we need the noncanonically constructed total models above.)

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If Hp is neat and if Σord is (smooth and) projective, then we can construct ord,tor ~ ~ ord,min as the normalization of a blow-up, as explained above, which MH,Σord → M H ~ ord,tor ~ implies that M in this case. This is done in ord is quasi-projective over S0,r H,Σ

H

Section 6.2.3. Other Constructions As in [61], the partial toroidal compactifications of ordinary Kuga families (and their generalizations) are realized as closures of toroidal boundary strata in the partial toroidal boundary of ordinary loci for a larger MH . This is done in Chapter 7. (See also Sections 1.3.2, 5.2.4, and 7.1.2, where we collect geometric objects appearing along the toroidal boundaries of MH and generalized Kuga families, and interpret them as universal spaces for certain degeneration data.) The constructions of automorphic bundles and their canonical and subcanonical extensions in mixed characteristics are delicate because the group is ramified at p, but some ad hoc constructions are still possible. They are carried out in Chapter 8 (extending the more canonical theory in characteristic zero in Section 1.4). The constructions of Hecke actions are scattered in Sections 2.2.3, 3.4.4, 5.2.2, 7.2.7, 8.1.4, and 8.3.6, using various universal properties (of moduli problems, normalizations, universal spaces for degenerations, etc). They are all based on the same idea of modifying the tautological abelian or semi-abelian schemes by quasiisogenies (and by forgetting part of the data on the level structures) which we call Hecke twists. On the p-adic completions of the total models of integral models we constructed, we also compare the ordinary loci we use (which are the loci where canonical subgroups can be defined and rigidified by linear algebraic data) and the ordinary loci defined by the subscheme whose geometric points define ordinary abelian varieties on the auxiliary model. (In particular, we provide a simple criterion which guarantees that our theory is not empty in the applications we have in mind.) This is done in Section 6.3. 0.4

What is Known, What is New, and What Can Be Studied Next

Let us relate our techniques of construction to what is known in the literature. Ordinary Level Structures and “Balanced Γ1 (pr ) Levels” The consideration of the ordinary loci carrying canonical subgroups is influenced by [46] and works of Hida (see, for example, the book [41] and the citations there). Our use of “balanced Γ1 (pr ) levels”, and the strategy of studying structures near infinity (i.e., the cusps) after adding sufficiently many roots of unity, are both influenced by Katz and Mazur [49]. We do not know whether “balanced Γ1 (pr ) levels” have been seriously considered in general.

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In this work we only consider the ordinary loci where the (multiplicative-type) canonical subgroups can be defined. If we also consider maximal totally isotropic subgroups which admits a filtration with graded pieces given by groups of ´etale and multiplicative (but no other) types, then we can extend the definition of ordinary loci and have a richer theory. The “balanced Γ1 (pr ) levels” should be the ideal context for studying such “full ordinary loci”. It is also possible to consider ordinary level structures of increasing depth along a flag of subgroup schemes. However, both of these require much heavier notation. We have chosen not to carry this out, because it complicates an already lengthy story. Theory of Degeneration and Partial Toroidal Compactifications The theory of degeneration in this work is built on those developed in [82], [28], and [62]. In order to study the ordinary level structures without the assumption that p is good, which means, in particular, that the polarization degree might not be prime to p, we introduced the “balanced Γ1 levels” and studied the ordinary level structures on the abelian scheme and its dual in a parallel way. This is consistent with the fact that the theory of degeneration data is also “balanced” in the sense that most objects in the theory of degeneration appear in pairs (one for the degenerating abelian scheme, one for its dual). ~ ord,tor is heavily built on [62], which include Our boundary construction for M H,Σord considerations not readily available in [28]. Since we take normalizations of certain naive models in the models in characteristic zero algebraically constructed in [62], our work requires [62] but does not replace it. (The ordinary level structures in characteristic p is hardly more complicated than the principal levels in characteristic zero. The main reason we need to take normalizations is because of the ramification at p. We first introduced some of the ideas of working with arbitrary ramifications in [58], which we further developed in this work.) Our method of gluing (for the ~ ord,tor construction of M ) is the same as in [28] and [62]. H,Σord Our construction of the partial toroidal compactifications of Kuga families (and their generalizations) is the same as that of [61], which is different from that of [28]. (It is not clear that [61] and [28, Ch. VI] even construct the same objects.) It is close in spirit to the construction of toroidal compactifications of mixed Shimura varieties in [89], although the construction techniques can hardly be directly compared. (The construction in [89] has arithmetic quotients of symmetric spaces as local charts, which has been developed along the lines of [5] and [4]. On the other hand, the purely algebraic construction in [61] is based on the theory of degeneration in [62]. It was not until [59] that we know these two constructions are compatible.) The same techniques in this work allow the generalization of the theory of degeneration and the boundary construction to the “full ordinary loci” mentioned above, although one will need to add more roots of unity to the base rings. They also allow the generalization to the case of ordinary level structures of increasing depth along a flag of subgroup schemes. However, as we mentioned above, both will

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require much heavier notation. We have chosen not to carry them out even though the method is almost identical. There are still many other cases where one can consider the construction of (total or partial) toroidal compactifications. Our rather simple-minded techniques do not seem to be useful when one seriously considers the nonordinary loci. (See the following discussion on local models.)

Local Models We learned the Serre–Tate deformation theory of ordinary abelian varieties from [73] and [47]. Together with the deformation theory in the good reduction case in [62] (with no levels at p), these are all that we need for our main constructions. Although we allow ramification and level structures at p, our consideration is disjoint from the theory of local models (involving also nonordinary abelian varieties) in, for example, [91], [85], [86], [87], and [88]. In general, our integral models of Shimura varieties are not even the same. By giving up the moduli interpretation, we obtain normality and flatness for free, but we no longer have enough information about the nonordinary loci. We note that Stroh’s constructions of compactifications of the Siegel moduli with parahoric levels at p (generalizing the “Γ0 (p) levels”; see [97], [98], [99]), unlike ours, used the same integral models as in the works mentioned in the previous paragraph, and indeed used results from the theory of local model to deduce the normality he needs. The strength of his work is that he also considered the nonordinary loci. (If the ordinary loci is all one wants, one can just take the normalization of some relatively representable moduli problems of canonical subgroups over the toroidal compactifications with no level at p. In the Siegel case, there is a nice “bottom level” to start with, with no ramification at p at all.) However, since it is unclear to us what “Γ1 (pr ) levels” mean at the nonordinary loci (especially) when r > 1, we have not generalized his work to the higher levels we want. Nevertheless, there are special cases where our models at the “bottom level” at p indeed agree with the ones considered in, for example, [85] and [86], in which case we can also describe the local structures of the nonordinary loci of the boundary. For example, we can show that certain toroidal and minimal compactifications are normal and Cohen–Macaulay, and have geometrically normal reductions mod p. (Some modification of our constructions would also allow us to study collections of isogenies defining parahoric levels.) See [65] for more details. (See also [68] and [64] for some more recent improvements.) After all, over the nonordinary loci, there is still very little we know, and there is ample room for further investigations. We believe that some new ideas might be needed, not just for solving the known difficult problems in the theory of local models, but also for seeing whether one should fundamentally revise the way we construct integral models.

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Use of Auxiliary Models The technical idea of using auxiliary models (such as the Siegel moduli) to study models of Shimura varieties (which are, a priori, analytically defined double coset spaces) has a long history. In characteristic zero, this can be traced back to the work of Shimura and Deligne (see [19] and [21], and their references). In mixed characteristic, Carayol [13] defined integral models of Shimura curves using integral models of unitary Shimura varieties, including the bad reduction case. For Hodge-type Shimura varieties, one can find such an idea in, for example, [78], [100], [79], [51], and [74]. In the PEL-type cases, with arbitrarily high levels at p, our approach is closer in spirit to the classical work of Deligne and Rapoport [24], in which models with higher levels at p are simply constructed as normalizations. This is the same approach taken in [15] (in which the model with no level at p is constructed by [90] and [23]). It is fair to say that we are influenced by both. (It is hard not to know the latter because of our upbringing; it is hard not to have heard of the former because of the current fashion trend.) One should keep in mind that we need the auxiliary models mainly as a source of quasi-projectivity—which is otherwise difficult to obtain! Adelic Language and Mixed Shimura Varieties The collections of geometric objects we construct do carry Hecke actions, as we have painstakingly gone through their constructions in all relevant sections, but our descriptions of them are somewhat indirect. For many applications, it is also desirable to adopt a language closer to the adelic formulations of double cosets (as in, for example, the theory for mixed Shimura varieties in [89]). We have chosen not to fully carry this out, mainly because in our proofs in mixed characteristics (especially for showing the universal properties of the partial toroidal compactifications, and for showing the quasi-projectivity of the partial minimal compactifications) we need the theory of degeneration, and we want to be able to cite the available results in our previous work [62] without much reformulation or generalization. But we certainly agree that it is helpful to develop a more convenient language after the proofs are done. We leave this as a potential future development. (We believe that, since we have shown that the algebraic picture in mixed characteristics is analogous to the complex analytic picture in characteristic zero, such a task can be done by a person with no knowledge of the theory of degeneration. There is no logical reason that the proofs and the applications have to be in the same language.) Roots of Unity in the Base Rings To obtain nice models in mixed characteristics, we added roots of unity to the base rings and performed normalizations whenever needed. We have made the effort to keep track of the precise exponents of roots of unity we need, but in practice, in mixed characteristics (0, p), it might be much easier to add all p-power roots

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of unity at once. We traded this convenience with some notational complication, partly because in many cases we only need roots of unity of a bounded exponent (and sometimes none at all), and it is still desirable to have a precise formula for such bounds. 0.5

What to Note and to Skip in Special Cases

Readers might naturally wonder whether some of the considerations can be ignored or more easily addressed, or whether the constructions can be shortened or simplified, in some special cases. In what follows we list the sections or subsections that can be skipped in each typical special case, and remark about some convenient special facts. (Certainly, in each of these cases, the work can be further shortened, at least typographically, by simplifying the notation system.) When p is a Good Prime The readers can safely skip Sections 2.1.1, 2.1.2, and 6.3.3, and most of Sections 3.1, 3.4.6, 6.1.1, 8.1, and 8.3, because most of the statements or constructions there can be easily achieved using the “bottom level” at p, which is the hyperspecial good reduction case already been explained in [62] and [61]. Whenever the auxiliary models are mentioned, the readers can safely assume that they are in the maximal hyperspecial good reduction case. Moreover, the reader should focus on the sections titled “The case when p is a good prime,” as they provide substantial shortcuts to the various constructions. For example, by Lemma 5.2.3.2, for most levels of practical interest, and for cone decompositions that are admissible also for a level hyperspecial at p defining a good reduction model, the partial toroidal compactifications can be easily constructed over the good reduction model as a relatively representable moduli problem of ordinary level structures. When the Pairing is Self-Dual at p In Sections 2.1.1, 2.1.2, 3.4.6, and 6.1.1, the reader can safely assume that L and Laux have exactly the same size, because no Zarhin’s trick is really necessary. The Siegel Cases For Siegel cases defined for abelian schemes with principal polarization, it is as in the case above when p is a good prime. (There are many other simplifications possible, but it is less clear how we should give the instructions on them.) Moreover, the nonemptiness of the ordinary loci on the characteristic p fibers is trivial. For Siegel cases defined for abelian schemes without principal polarizations, Zarhin’s trick is used in our work, and hence the auxiliary models are still essential. However, one can ignore all treatments concerning the ramification of p in O.

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In both cases, it is possible to describe the cone decompositions using a simpler combinatorial language. For example, it is simpler to focus on the point boundary strata (of the partial minimal compactification) given by the rational parabolic subgroup of “maximal rank” (with abelian unipotent radical, and with Levi the product of a general linear group with Gm ). Whether trivially true or not, Corollary 6.3.3.2 shows that the ordinary loci on the characteristic p fiber is nonempty. The “Easier” Unitary Cases By “easier” unitary groups we mean unitary groups defined by a Hermitian pairing over an imaginary quadratic or CM field F , but not over a noncommutative semisimple algebra. In addition to the above (concerning whether p is good, whether the pairing is self-dual, etc), the main simplification possible is that it is also possible to describe the cone decompositions using a simpler combinatorial language. For example, it is also simpler to focus on the point boundary strata (of the partial minimal compactification) given by the rational parabolic subgroup of “maximal rank” (with possibly non-abelian unipotent radical). All Cases with “Siegel Parabolics” By a “Siegel parabolic” subgroup we mean a rational parabolic subgroup with an abelian unipotent radical. (These include all kinds of cases involving general semisimple algebras with positive involutions. We do not just consider the Siegel and “easier” unitary cases.) In such cases, the nonemptiness of the ordinary loci on the characteristic p fibers follows from Corollary 6.3.3.2. (Note that many of these are cases with nonquasisplit groups.) 0.6

Notation and Conventions

We shall follow [62, Notation and Conventions] unless otherwise specified. By symplectic isomorphisms between modules with symplectic pairings, we always mean isomorphisms between the modules matching the pairings up to an invertible scalar multiple. (These are often called symplectic similitudes, but our understanding is that the codomains of pairings are modules rather than rings, which ought to be matched as well.) Sheaves on schemes, algebraic spaces, or (Deligne–Mumford) algebraic stacks are ´etale sheaves by default, although for coherent sheaves on schemes it would suffice to work in the Zariski topology. 0.7

Acknowledgements

I would like to thank (in alphabetical order) George Boxer, Brian Conrad, Christopher Skinner, Richard Taylor, Jack Thorne, and Yichao Tian for helpful comments and corrections. I would also like to thank the following people and institutions for

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the hospitality during my visits: Jian-Shu Li and Shou-Wu Zhang, and the Institute for Advanced Study of the Hong Kong University of Science and Technology (HKUST IAS); Iku Nakamura and Hokkaido University; Richard Taylor and the Institute for Advanced Study (in Princeton); and Chia-Fu Yu, and the National Center for Theoretical Sciences (NCTS) and Academia Sinica (in Taipei). During the preparation of this work, my research was partially supported by the Learned Society Travel Fund at Princeton University; by a setup funding at the University of Minnesota, Twin Cities; by the National Science Foundation under agreements Nos. DMS-1069154, DMS-1258962, and DMS-1352216; and by an Alfred P. Sloan Research Fellowship. Any opinions, findings, and conclusions or recommendations expressed here are those of mine, and do not necessarily reflect the views of the funding organizations.

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Chapter 1

Theory in Characteristic Zero

In this chapter, we review the main definitions and results in [62] and [61] (based on earlier results of others) and specialize them to the case of characteristic zero bases (over the reflex fields). (Also, we take this opportunity to correct or improve certain assertions in [62] and [61].) Readers who are already familiar with these results should feel free to skip this chapter (and return to here only for references). However, despite the similarity, the theory developed in [19], [21], [5], [4], [38], [89], etc (based on arithmetic quotients of Hermitian symmetric domains, for which the compactifications were constructed by gluing using the analytic coordinates) are not directly related to the results reviewed here in this chapter (based on the moduli of polarized abelian varieties, for which the compactifications were constructed by gluing using the theory of degeneration). (It is not completely obvious that the two kinds of theories are compatible along the boundary; see [59].) Some readers might find the definitions and results in this chapter unfamiliar, and might want to at least glance over the notation system and running assumptions. 1.1 1.1.1

PEL-type Moduli Problems and Shimura Varieties Linear Algebraic Data for PEL Structures

Let us begin with the usual (rational) PEL data, which suffice for the definition of complex analytic PEL-type Shimura varieties and their attached moduli problems in characteristic zero or in every good characteristic as in [53] (see Definition 1.1.1.6 below). √ Let Z(1) := ker(exp : C → C× ) = (2π −1)Z, which is a free Z-module of rank √ √ one. Each square-root −1 of −1 in C determines an isomorphism (2π −1)−1 : ∼ Z(1) → Z, but there is no canonical isomorphism between Z(1) and Z. For each Z-module M , we denote by M (1) the module M ⊗ Z(1), called the Tate twist of M . Z

Note that M (1) and M are noncanonically isomorphic as Z-modules. For the construction of compactifications using the theory of degeneration in [28] and [62], it is useful to start with a (noncanonical) choice of an integral PEL datum: Definition 1.1.1.1. An integral PEL datum is a tuple (O, ?, L, h · , · i, h0 ) consisting 21

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of: (1) An order O in a finite-dimensional semisimple Q-algebra with a positive involution ? stabilizing O. We shall denote the center of O ⊗ Q by F . Z

(Then F is a product of number fields.) (2) An O-lattice L; namely, a finite free Z-module L with the structure of an O-module. (3) An alternating pairing h · , · i : L × L → Z(1) satisfying hbx, yi = hx, b? yi for all x, y ∈ L and b ∈ O, together with an R-algebra homomorphism h0 : C → EndO ⊗ R (L ⊗ R), satisfying: Z

Z

(a) For all z ∈ C and x, y ∈ L ⊗ R, we have hh0 (z)x, yi = hx, h0 (z c )yi, Z

where z 7→ z c is complex conjugation. √ √ (b) The R-bilinear pairing (2π −1)−1 h · , h0 ( −1) · i on L ⊗ R is (symZ

metric and) positive definite. (See [62, Def. 1.2.1.3], where h0 was denoted by h.) Such a tuple (O, ?, L, h · , · i, h0 ) is an integral version of the PEL datum (B, ?, V, h · , · i, h0 ) in [53] and related works. Definition 1.1.1.2. The dual lattice L# of L (with respect to the pairing h · , · i) is L# := {x ∈ L ⊗ Q : hx, yi ∈ Z(1), ∀y ∈ L}. Z

One advantage of making the choice of an integral datum is that it fixes the choice of an integral model of the algebraic reductive group in the usual definition of Shimura varieties: Definition 1.1.1.3. (See [62, Def. 1.2.1.6].) Let O and (L, h · , · i) be given as above. For each Z-algebra R, set   (g, r) ∈ GLO ⊗ R (L ⊗ R) × Gm (R) : Z Z G(R) := .  hgx, gyi = rhx, yi, ∀x, y ∈ L ⊗ R  Z

The assignment is functorial in R and defines a group functor G over Spec(Z). The projection to the second factor (g, r) 7→ r defines a homomorphism ν : G → Gm , which we call the similitude character. For simplicity, we shall often denote elements (g, r) in G by simply g, and denote by ν(g) the value of r when we need it. (This is an abuse of notation, because the value of r is not always determined by g.) Then we have, for each rational prime number p > 0, definitions for G(Q), G(R), ˆ G(Z ˆ p ), G(A∞ ), G(A∞,p ), G(A), G(Ap ), G(Z), G(Z/nZ), G(Z), ˆ → G(Z/n ˆ Z) ˆ = G(Z/nZ)) U(n) := ker(G(Z)

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for each integer n ≥ 1, ˆ p ) → G(Z ˆ p /n0 Z ˆ p ) = G(Z/n0 Z)) U p (n0 ) := ker(G(Z for each integer n0 ≥ 1 prime to p. The homomorphism h0 : C → EndO ⊗ R (L ⊗ R) defines a Hodge structure of Z

weight −1 on L, with Hodge decomposition

Z

L ⊗ C = V0 ⊕ V0c ,

(1.1.1.4)

Z

such that h0 (z) acts by 1 ⊗ z on V0 , and by 1 ⊗ z c on V0c . One can check easily that V0 is (maximal) totally isotropic under the nondegenerate pairing h · , · i, and hence (1.1.1.4) induces canonically an isomorphism V0c ∼ = V0∨ (1) := HomC (V0 , C)(1).

(1.1.1.5)

Let F0 be the reflex field of the O ⊗ C-module V0 . Recall (see [53, p. 389] Z

or [62, Def. 1.2.5.4]) that F0 is the subfield of C generated over Q by the traces TrC (b|V0 ) for b ∈ O. Definition 1.1.1.6. We say that a rational prime number p > 0 is good (for the integral PEL datum (O, ?, L, h · , · i, h0 )) if it satisfies the following conditions (cf. [53, Sec. 5] or [62, Def. 1.4.1.1]): (1) p is unramified in O (as in [62, Def. 1.1.1.18]). (2) p = 6 2 if O ⊗ Q involves simple factors of type D (as in [62, Def. 1.2.1.15]). Z

(3) p - [L# : L] (see Definition 1.1.1.2). When p is good, G ⊗ Zp is smooth and unramified (cf. [62, Prop. 1.2.3.11 and Z

Cor. 1.2.3.12]). 1.1.2

PEL-type Moduli Problems

ˆ Let H be an open compact subgroup of G(Z). By [62, Def. 1.4.1.4] (with 2 = ∅ there), the data of (L, h · , · i, h0 ) and H define a moduli problem MH over S0 = Spec(F0 ), parameterizing tuples (A, λ, i, αH ) over schemes S over S0 of the following form: (1) (2) (3) (4)

A → S is an abelian scheme. λ : A → A∨ is a polarization. i : O ,→ EndS (A) is an O-endomorphism structure as in [62, Def. 1.3.3.1]. LieA/S with its O ⊗ Q-module structure given naturally by i satisfies the Z

determinantal condition in [62, Def. 1.3.4.1] given by (L ⊗ R, h · , · i, h0 ). Z

ˆ h · , · i) as (5) αH is an (integral) level-H structure of (A, λ, i) of type (L ⊗ Z, Z

in [62, Def. 1.3.7.6].

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Remark 1.1.2.1. By [62, Prop. 1.4.3.4], the definition agrees with the one in [53, Sec. 5] over S0 = Spec(F0 ). The choice of L in L ⊗ Q corresponds to the choice of a Z

tautological (or universal ) abelian scheme A over MH within its Q× -isogeny class. If we have chosen another PEL-type O-lattice L0 in L ⊗ Q which is also stabilized Z

by H, then we have the corresponding A0 (with additional structures) over a moduli problem M0H canonically isomorphic to MH (see [62, Cor. 1.4.3.8]), together with a Q× -isogeny A → A0 (if we identify M0H with MH ). In brief, the Q× -isogeny class of A is independent of the choice of L in L ⊗ Q. This is useful because every open Z

compact subgroup of G(A∞ ) stabilizes some PEL-type O-lattice L0 , and for every two PEL-type O-lattices L and L0 there are common open compact subgroups of G(A∞ ) stabilizing both lattices. Hence, we can form a collection {MH }H , indexed ˆ (with a canonical by all open compact subgroups H of G(A∞ ), not just those of G(Z) ∞ action of G(A ); see [62, Rem. 1.4.3.11]). By [62, Thm. 1.4.1.11 and Cor. 7.2.3.10], MH is an algebraic stack separated, smooth, and of finite type over S0 , which is representable by a scheme quasiprojective (and smooth) over S0 when H is neat. (See [89, 0.6] or [62, Def. 1.4.1.8] for the definition of neatness.) Let (A, λ, i, αH ) → MH be the tautological tuple over MH . Consider the relative de Rham cohomology H 1dR (A/MH ), with the dual 1 H dR 1 (A/MH ) := HomOM (H dR (A/MH ), OMH ) H

defined to be the relative de Rham homology. Consider the canonical pairing dR h · , · iλ : H dR 1 (A/MH ) × H 1 (A/MH ) → OMH (1)

(1.1.2.2)

defined by the pullback under Id ×λ∗ of the canonical perfect pairing dR ∨ H dR 1 (A/MH ) × H 1 (A /MH ) → OMH (1)

defined by the first Chern class of the Poincar´e invertible sheaf PA over A × A∨ . MH

(See, for example, [23, 1.5].) Since MH is defined over the characteristic zero base S0 = Spec(F0 ), we know that λ is separable, that λ∗ is an isomorphism, and hence that the pairing h · , · iλ above is perfect. Let h · , · iλ also denote the induced pairing on H 1dR (A/MH ) × H 1dR (A/MH ) by duality. By [6, Lem. 2.5.3], we have canonical short exact sequences dR 0 → Lie∨ A∨ /MH (1) → H 1 (A/MH ) → LieA/MH → 0

and 1 0 → Lie∨ A/MH → H dR (A/MH ) → LieA∨ /MH (−1) → 0. ∨ The submodules Lie∨ A∨ /MH and LieA/MH are maximal totally isotropic under the pairing h · , · iλ .

Remark 1.1.2.3. The Tate twists in Lie∨ A∨ /MH (1) and LieA∨ /MH (−1) are often omitted (and have also been omitted in most of this author’s earlier writings).

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They signify a sign convention in the (otherwise canonical) identification between H 1 (A, OA ) and LieA∨ /MH , which is nevertheless the same sign convention involved in the definition of the pairing (1.1.2.2). Hence, we shall carry such Tate twists in the notation when the pairing (1.1.2.2) is also involved. However, for the sake of simplicity, we shall still omit them when discussing the Gauss–Manin connections or Kodaira–Spencer morphisms below. ˜ (m) be the m-th infinitesimal neighborhood of the diagonal image of MH Let M H ˜ (m) → MH be the two projections. Then we have in MH × MH , and let pr1 , pr2 : M H S0

m by definition the canonical morphism OMH → PM := pr1,∗ pr∗2 (OMH ). The H /S0 ˜ (m) → M ˜ (m) over MH swapping two components of the fiber isomorphism s : M H H m . When m = 1, the kernel of product then defines an automorphism s∗ of PM H /S0 ∗ 1 the structural morphism str : PMH /S0 → OMH , canonically isomorphic to Ω1MH /S0 by definition, is spanned by the image of s∗ − Id∗ (induced by pr∗1 − pr∗2 ). An important property of the relative de Rham cohomology of a smooth mor˜ (1) and A˜2 → M ˜ (1) phism like A → MH is that, for every two smooth lifts A˜1 → M H H ∼ ˜ (1) ) → H 1 (A˜1 /M ˜ (1) ) of A → MH , there is a canonical isomorphism H 1 (A˜2 /M dR

dR

H

H

lifting the identity morphism on H 1dR (A/MH ). (See, for example, [62, Prop. 2.1.6.4].) If we consider A˜1 := pr∗1 A and A˜2 := pr∗2 A, then we obtain a canoni∼ ˜ (1) ) → ˜ (1) ) ∼ H 1dR (pr∗1 A/M cal pr∗2 H 1dR (A/MH ) ∼ = H 1dR (pr∗2 A/M = pr∗1 H 1dR (A/MH ), H H which we denote by Id∗ by abuse of notation. On the other hand, pulling back by ∼ ˜ (1) ˜ (1) → the swapping automorphism s : M MH defines another canonical isomorphism H (1) ∼ 1 1 ∗ ∗ ∗ ∼ ˜ ˜ (1) ) ∼ s : pr2 H dR (A/MH ) = H dR (pr2 A/MH ) → H 1dR (pr∗1 A/M = pr∗1 H 1dR (A/MH ). H This allows us to define the Gauss–Manin connection as follows (cf. [62, Rem. 2.1.7.4]): Definition 1.1.2.4. The Gauss–Manin connection ∇ : H 1dR (A/MH ) → H 1dR (A/MH ) ⊗ Ω1MH /S0 OMH

(1.1.2.5)

on H 1dR (A/MH ) is the composition ∗

∗

∗

pr −Id ˜ (1) ) s → H 1dR (A/MH ) →2 H 1dR (pr∗2 A/M H 1dR (A/MH ) ⊗ Ω1MH /S0 . H OMH

Definition 1.1.2.6. The composition (ignoring Tate twists; see Remark 1.1.2.3) 1 Lie∨ A/MH ,→ H dR (A/MH ) ∇

→ H 1dR (A/MH ) ⊗ Ω1MH /S0  LieA∨ /MH ⊗ Ω1MH /S0 OMH

OMH

defines by duality a morphism ∨ 1 KSA/MH /S0 : Lie∨ A/MH ⊗ LieA∨ /MH → ΩMH /S0 , OMH

(1.1.2.7)

which we call the Kodaira–Spencer morphism. (This definition is compatible with the definition by deformation theory in [62, Def. 2.1.7.9].)

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Definition 1.1.2.8. (See [62, Def. 2.3.5.1].) The sheaf KS(A,λ,i)/MH := KS(A,λ,i,αH )/MH is the quotient ∨ (Lie∨ A/MH ⊗ LieA∨ /MH )/



OMH

 λ∗ (y) ⊗ z − λ∗ (z) ⊗ y ∨ i(b)∗ (x) ⊗ y − x ⊗(i(b) )∗ (y)

x∈Lie∨ A/M , H

y,z∈Lie∨ , A∨ /M H

b∈O.

According to [62, Prop. 2.3.5.2], we have: Proposition 1.1.2.9. The Kodaira–Spencer morphism (1.1.2.7) factors through the ∨ canonical quotient Lie∨ A/MH ⊗ LieA∨ /MH  KS(A,λ,i)/MH and induces an isomorOMH

phism ∼

KS(A,λ,i)/MH → Ω1MH /S0 ,

(1.1.2.10)

which we call the Kodaira–Spencer isomorphism, and denote again (by abuse of notation) by KSA/MH /S0 . 1.1.3

PEL-type Shimura Varieties

Consider the (real analytic) set X = G(R)h0 of G(R)-conjugates h : C → EndO ⊗ R (L ⊗ R) of h0 : C → EndO ⊗ R (L ⊗ R). It is well known (see [53, Sec. Z

Z

Z

Z

8] or [59, Sec. 2]) that there exists a quasi-projective variety ShH over F0 , together with a canonical open and closed immersion ShH ,→ [MH ]

(1.1.3.1)

over S0 = Spec(F0 ), where [ · ] denotes the coarse moduli space of an algebraic stack (see [62, Sec. A.7.5]), such that the analytification of ShH ⊗ C (as a comF0

plex analytic space) can be canonically identified with the double coset space G(Q)\X × G(A∞ )/H. (Note that ShH ,→ [MH ] is not an isomorphism in general, due to the so-called failure of Hasse’s principle. See, for example, [53, Sec. 8] and [62, Rem. 1.4.3.12].) We call both ShH and ShH ⊗ C the PEL-type Shimura variety of level H assoF0

ciated to the integral PEL datum (O, ?, L, h · , · i, h0 ). (More precisely, we should call ShH ⊗ C the complex PEL-type Shimura variety, and call ShH the canonical F0

model.) We will not emphasis their roles in our constructions from now on. 1.2 1.2.1

Linear Algebraic Data for Cusps Cusp Labels

For technical reasons, we shall impose the following technical condition on all PEL-type O-lattices we use:

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Condition 1.2.1.1. (See [62, Cond. 1.4.3.10].) The PEL-type O-lattice (L, h · , · i, h0 ) is chosen such that the action of O on L extends to an action of some maximal order O0 in O ⊗ Q containing O. Z

Although there is no rational boundary components in the algebraic theory of toroidal and minimal compactifications (constructed by the theory of degeneration, as in [28] and in [62]), we have developed in [62, Sec. 5.4] the notion of cusp labels that serves a similar purpose. (While G(Q) plays an important role in the analytic theory over C, it does not play any obvious role in the algebraic theory of degeneration.) Unlike in the analytic theory over C, where boundary components are naturally parameterized by group-theoretic objects, the only algebraic machinery we have is the theory of semi-abelian degenerations of abelian varieties with PEL structures. The cusp labels are (by their very design) part of the parameters (which we call the degeneration data) for such (semi-abelian) degenerations. Definition 1.2.1.2. (See [62, Sec. 1.2.6].) Let R be any noetherian Z-algebra. Suppose we have an increasing filtration F = {F−i } on L ⊗ R, indexed by nonpositive Z

integers −i, such that F0 = L ⊗ R. Z

(1) We say that F is integrable if, for every i, GrF−i := F−i /F−i−1 is integrable in the sense that GrF−i ∼ = Mi ⊗ R (as R-modules) for some O-lattice Mi . Z

(2) We say that F is split if there exists (noncanonically) some isomorphism ∼ GrF := ⊕ GrF−i → F0 of R-modules. i

(3) We say that F is admissible if it is both integrable and split. (4) Let m be an integer. We say that F is m-symplectic with respect to (L, h · , · i) if, for every i, F−m+i and F−i are annihilators of each other under the pairing h · , · i on F0 . We shall only work with m = 3, and we shall suppress m in what follows. ˆ (or Z ˆ p for some rational prime number p) almost always involves The fact that Z bad primes (cf. Definition 1.1.1.6) is the main reason that we may have to allow nonprojective filtrations. Definition 1.2.1.3. (See [62, Def. 5.2.7.1].) We say that a symplectic admissible ˆ is fully symplectic with respect to (L, h · , · i) if there is a filtration Z on L ⊗ Z Z

symplectic admissible filtration ZA = {Z−i,A } on L ⊗ A that extends Z in the sense ˆ = Z−i in L ⊗ A for all i. that Z−i,A ∩ (L ⊗ Z) Z

Z

Z

Definition 1.2.1.4. (See [62, Def. 5.2.7.3].) A symplectic-liftable admissible filtration Zn on L/nL is called fully symplectic-liftable with respect to (L, h · , · i) if ˆ that is fully it is the reduction modulo n of some admissible filtration Z on L ⊗ Z symplectic with respect to (L, h · , · i) as in Definition 1.2.1.3.

Z

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Degenerations into semi-abelian schemes induce filtrations on Tate modules and on Lie algebras of the generic fibers. While the symplectic-liftable admissible filˆ induced by filtrations on trations represent (certain orbits of) filtrations on L ⊗ Z Z

Tate modules via the level structures, the fully symplectic-liftable ones are equipped with (certain orbits of) filtrations on L ⊗ R induced by the filtrations on Lie Z

algebras via the Lie algebra condition (see Section 1.1.2). (One may interpret the Lie algebra condition as the “de Rham” (or rather “Hodge”) component of a certain “complete level structure”, the direct product of whose “`-adic” components being a level structure in the usual sense.) Such (orbits of) filtrations are the crudest invariants of degenerations we consider. Definition 1.2.1.5. (See [62, Def. 5.4.1.3].) Given a fully symplectic admissible ˆ with respect to (L, h · , · i) as in Definition 1.2.1.3, a torus filtration Z on L ⊗ Z Z

argument for Z is a tuple Φ = (X, Y, φ, ϕ−2 , ϕ0 ), where the entries are as follows: (1) X and Y are O-lattices of the same O-multi-rank (see [62, Def. 5.2.2.6]), and φ : Y ,→ X is an O-equivariant embedding. ∼ ∼ ˆ Z(1)) ˆ ˆ are isomorphisms and ϕ0 : GrZ0 → Y ⊗ Z (2) ϕ−2 : GrZ−2 → HomZˆ (X ⊗ Z, Z

Z

ˆ ˆ (of O ⊗ Z-modules) such that the pairing h · , · i20 : GrZ−2 × GrZ0 → Z(1) Z

defined by Z is the pullback of the pairing ˆ Z(1)) ˆ ˆ → Z(1) ˆ h · , · iφ : HomZˆ (X ⊗ Z, ×(Y ⊗ Z) Z

Z

defined by the composition ˆ Z(1)) ˆ ˆ ×(Y ⊗ Z) HomZˆ (X ⊗ Z, Z

Z

Id ×φ

ˆ Z(1)) ˆ ˆ → Z(1), ˆ → HomZˆ (X ⊗ Z, ×(X ⊗ Z) Z

Z

with the sign convention that h · , · iφ (x, y) = x(φ(y)) = (φ(y))(x) for all ˆ Z(1)) ˆ ˆ x ∈ HomZˆ (X ⊗ Z, and y ∈ Y ⊗ Z. Z

Z

Definition 1.2.1.6. (See [62, Def. 5.4.1.4 and 5.4.1.5].) Given a fully symplecticliftable admissible filtration Zn on L/nL with respect to (L, h · , · i) as in Definition 1.2.1.4, a torus argument at level n for Zn is a tuple Φn = (X, Y, φ, ϕ−2,n , ϕ0,n ), where: (1) X and Y are O-lattices of the same O-multi-rank, and φ : Y ,→ X is an O-equivariant embedding.

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∼

∼

(2) ϕ−2,n : GrZ−2,n → Hom(X/nX, (Z/nZ)(1)) (resp. ϕ0,n : GrZ0,n → Y /nY ) is an isomorphism that is the reduction modulo n of some isomorphism ∼ ∼ ˆ ˆ Z(1)) ˆ such that (resp. ϕ0 : GrZ0 → (Y ⊗ Z)), ϕ−2 : GrZ−2 → HomZˆ (X ⊗ Z, Z

Z

Φ = (X, Y, φ, ϕ−2 , ϕ0 ) is a torus argument as in Definition 1.2.1.5. We say in this case that Φn is the reduction modulo n of Φ. Two torus arguments Φn = (X, Y, φ, ϕ−2,n , ϕ0,n ) and Φ0n = (X 0 , Y 0 , φ0 , ϕ0−2,n , ϕ00,n ) ∼ at level n are equivalent if there exists a pair of isomorphisms (γX : X 0 → X, γY : ∼ Y → Y 0 ) (of O-lattices) such that φ = γX φ0 γY , ϕ0−2,n = t γX ϕ−2,n , and ϕ00,n = γY ϕ0,n . In this case, we say that Φn and Φ0n are equivalent under the pair of ∼ isomorphisms γ = (γX , γY ), which we denote by γ = (γX , γY ) : Φn → Φ0n . The torus arguments record the isomorphism classes of the torus parts of degenerations of abelian schemes with PEL structures. These are the second crudest invariants of degenerations we consider. Definition 1.2.1.7. (See [62, Def. 5.4.1.9].) A (principal) cusp label at level n for a PEL-type O-lattice (L, h · , · i, h0 ), or a cusp label of the moduli problem Mn , is an equivalence class [(Zn , Φn , δn )] of triples (Zn , Φn , δn ), where: (1) Zn is an admissible filtration on L/nL that is fully symplectic-liftable in the sense of Definition 1.2.1.4. (2) Φn is a torus argument at level n for Zn . ∼ (3) δn : GrZn → L/nL is a liftable splitting. Two triples (Zn , Φn , δn ) and (Z0n , Φ0n , δn0 ) are equivalent if Zn and Z0n are identical, and if Φn and Φ0n are equivalent as in Definition 1.2.1.6. The liftable splitting δn in each triple (Zn , Φn , δn ) is noncanonical and auxiliary in nature. Such splittings are needed for analyzing the “degeneration of pairings” in general PEL cases (unlike in the special case in Faltings–Chai [28, Ch. IV, Sec. 6]). To proceed from principal cusp labels at level n to general cusp labels at level ˆ we form ´etale orbits of the H, where H is an open compact subgroup of G(Z), objects we have thus defined. The precise definitions are complicated (see [62, Def. 5.4.2.1, 5.4.2.2, and 5.4.2.4]) but the idea is simple: For each H as above, consider those n ≥ 1 sufficiently divisible such that U(n) ⊂ H. Then we have a compatible system of finite groups Hn = H/U(n), and an object at level H is simply defined to be a compatible system of ´etale Hn -orbits of objects at running levels n as above. Then we arrive at the notions of torus arguments ΦH = (X, Y, φ, ϕ−2,H , ϕ0,H ) at level H, and of representatives (ZH , ΦH , δH ) of cusp labels [(ZH , ΦH , δH )] at level H. (The liftability condition is implicit in such a definition, as in the definition of level structures we omitted.) By abuse of language, we call these H-orbits of Φ = (X, Y, φ, ϕ−2 , ϕ0 ), (Z, Φ, δ), and [(Z, Φ, δ)], respectively. (Note that the splitting δ was denoted δˆ in [62, Sec. 5.2.2].) For simplicity, we shall often omit ZH from the notation.

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Lemma 1.2.1.8. (See [62, Lem. 5.2.7.5].) Let Zn be an admissible filtration on L/nL that is fully symplectic-liftable with respect to (L, h · , · i). Let (GrZ−1 , h · , · i11 ) be induced by some fully symplectic lifting Z of Zn , and let (GrZ−1,R , h · , · i11,R , (h0 )−1 ) be determined by [62, Prop. 5.1.2.2] by any extension ZA in Definition 1.2.1.3 (which has the same reflex field F0 as (L ⊗ R, h · , · i, h0 ) does). Then there is associaZ

ted (noncanonically) a PEL-type O-lattice (LZn , h · , · iZn , hZ0n ) satisfying Condition 1.2.1.1 such that we have the following: (1) There exist (noncanonical) O-equivariant isomorphisms ∼ ˆ h · , · iZn ) (GrZ , h · , · i11 ) → (LZn ⊗ Z, −1

Z

ˆ and (over Z) ∼

(GrZ−1,R , h · , · i11,R , (h0 )−1 ) → (LZn ⊗ R, h · , · iZn , hZ0n ) Z

(over R). (2) The moduli problem MZnn defined by the noncanonical (LZn , h · , · iZn , hZ0n ) as in Section 1.1.2 is canonical in the sense that it depends (up to isomorphism) only on Zn , but not on the choice of (LZn , h · , · iZn , hZ0n ). In fact, Lemma 1.2.1.8 (or rather [62, Lem. 5.2.7.5]) is based on [62, Rem. 5.2.7.2], which asserts the existence of a (noncanonical) boundary lattice ˆ so that we (LZ , h · , · iZ , hZ0 ) for each fully symplectic admissible filtration Z of L ⊗ Z, Z

ˆ have, in particular, a canonical isomorphism GrZ−1 ⊗ R ∼ = L ⊗ R for each Z-algebra ˆ Z

Z

R. With any fixed (noncanonical) choice of such (LZ , h · , · iZ , hZ0 ), we can make the following: Definition 1.2.1.9. We define the group functor GZ = G(LZ ,h · , · iZ ) by (LZ , h · , · iZ ) ˆ is well defined and depends only on as in Definition 1.1.1.3, so that Gh,Z := GZ ⊗ Z Z

Z (but not on the choice of (LZ , h · , · iZ , hZ0 )).

ˆ Definition 1.2.1.10. For each Z-algebra R, let PZ (R) denote the subgroup of G(R) consisting of elements g such that g(Z−2 ⊗ R) = Z−2 ⊗ R and g(Z−1 ⊗ R) = Z−1 ⊗ R. ˆ Z

ˆ Z

ˆ Z

∼

ˆ Z

Each element g in PZ (R) defines an isomorphism GrZ−i (g) : GrZ−i ⊗ R → GrZ−i ⊗ R ˆ Z

ˆ Z

for each 0 ≤ i ≤ 2. Then, under the isomorphism GrZ−1 ⊗ R ∼ = L ⊗ R above, the ˆ Z

Z

isomorphism GrZ−1 (g) corresponds to an element of Gh,Z (R), and define a group homomorphism GrZ−1 : PZ (R) → Gh,Z (R). ˆ Definition 1.2.1.11. For each Z-algebra R, we also define the following quotients of subgroups of PZ (R) (see Definition 1.2.1.10): (1) ZZ (R) is the kernel of the canonical homomorphism GrZ−1 : PZ (R) → Gh,Z (R). Then any splitting δ as above canonically induces an isomorphism PZ (R) ∼ = Gh,Z (R) n ZZ (R).

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(2) UZ (R) is the subgroup of PZ (R) consisting of elements g such that GrZ (g) = IdGrZ ⊗ R (i.e., GrZ−i (g) = IdGrZ−i ⊗ R for all i). ˆ Z

ˆ Z

(3) U2,Z (R) is the subgroup of PZ (R) consisting of elements g which induces IdZ−1 ⊗ R and Id(Z0 ⊗ R)/(Z−2 ⊗ R) on Z−1 ⊗ R and (Z0 ⊗ R)/(Z−2 ⊗ R), reˆ Z

ˆ Z

ˆ Z

ˆ Z

ˆ Z

ˆ Z

spectively. any splitting δ as above, this means δ −1 ◦ g ◦ δ is of the  1 g(Using  20 form for some g20 ∈ HomO (GrZ0 ⊗ R, GrZ−2 ⊗ R).) 1 1

ˆ Z

ˆ Z

(4) U1,Z (R) := UZ (R)/U2,Z (R). (5) Gl,Z (R) := ZZ (R)/UZ (R), which is canonically isomorphic to the subgroup G0l,Z (R) of GLO (GrZ−2 ⊗ R) × GLO (GrZ0 ⊗ R) consisting of elements compaˆ Z

ˆ Z

ˆ tible with the morphism → HomZˆ (GrZ−2 , Z(1)) induced by h · , · i (which ∼ are therefore the elements compatible with φ : Y ,→ X, ϕ−2 : GrZ−2 → ∼ ˆ Z(1)), ˆ ˆ for any torus argument Φ of Z; and ϕ0 : GrZ0 → Y ⊗ Z HomZˆ (X ⊗ Z, GrZ0

Z

Z

see Definition 1.2.1.5). (6) P0Z (R) is the kernel of the canonical homomorphism (ν −1 GrZ−2 , GrZ0 ) : PZ (R) → G0l,Z (R); i.e., the subgroup of PZ (R) consisting of elements g such that GrZ−2 (g) = ν(g) IdGrZ−2 ⊗ R and GrZ0 (g) = IdGrZ0 ⊗ R . Then any splitting ˆ Z

ˆ Z

∼ δ : GrZ → Z canonically induces an isomorphism PZ (R) ∼ = Gl,Z (R) n P0Z (R). 0 (7) G1,Z (R) := PZ (R)/U2,Z (R), which is (under any splitting δ above) isomorphic to (Gh,Z n U1,Z )(R) := Gh,Z (R) n U1,Z (R). (8) G0h,Z (R) := G1,Z (R)/U1,Z (R) ∼ = Gh,Z (R). = PZ (R)/ZZ (R) ∼ = P0Z (R)/UZ (R) ∼

ˆ we define: Definition 1.2.1.12. For each open compact subgroup H of G(Z), (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

ˆ HPZ := H ∩ PZ (Z). ˆ HZZ := H ∩ ZZ (Z). HGh,Z := HPZ /HZZ . ˆ HUZ := H ∩ UZ (Z). ˆ HU2,Z := H ∩ U2,Z (Z). HU1,Z := HUZ /HU2,Z . HGl,Z := HZZ /HUZ . ˆ HP0Z := H ∩ P0Z (Z). HG1,Z := HP0Z /HU2,Z . HG0h,Z := HP0Z /HUZ . HG0l,Z := HPZ /HP0Z .

Then we have an exact sequence 1 → HU1,Z → HG1,Z → HG0h,Z → 1

(1.2.1.13)

compatible with the canonical exact sequence ˆ → G1,Z (Z) ˆ → G0 (Z) ˆ → 1. 1 → U1,Z (Z) (1.2.1.14) h,Z p ˆ We shall also extend this definition to the cases of Z - and Zp -valued groups above.

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Definition 1.2.1.15. (See [62, Def. 5.4.2.6 and the errata].) The PEL-type O-lattice (LZH , h · , · iZH , hZ0H ) is a fixed (noncanonical) choice of any of the PEL-type O-lattice (LZn , h · , · iZn , hZ0n ) in Lemma 1.2.1.8 for any element Zn in any ZHn (in ZH = {ZHn }, a compatible collection of ´etale orbits ZHn at various levels n such that U(n) ⊂ H). The elements of Hn leaving Zn invariant induce a subgroup of ˆ  GZ (Z/nZ). Let Hh be the preimage of this subgroup under G(LZn ,h · , · iZn ) (Z) G(LZn ,h · , · iZn ) (Z/nZ) (see Definition 1.2.1.9). Then we define MHh to be the moduli problem defined by (LZn , h · , · iZn , hZn ) with level-Hh structures as in Lemma 1.2.1.8. (The isomorphism class of MHh is well defined and independent of the choice of H (LZH , h · , · iZH , hZH ) = (LZn , h · , · iZn , hZn ).) We define MΦ H to be the quotient of ` Zn Mn by Hn , where the disjoint union is over representatives (Zn , Φn , δn ) (with the same (X, Y, φ)) in (ZH , ΦH , δH ), which is finite ´etale over MHh by construction. H (The isomorphism class of MΦ H is independent of the choice of n and the representatives (Zn , Φn , δn ) we use.) We then (abusively) define MZHH to be the quotient H of MΦ H by the subgroup of Γφ stabilizing ΦH (whose action factors through a finite index subgroup), which depends only on the cusp label [(ZH , ΦH , δH )], but not on the choice of the representative (ZH , ΦH , δH ). By construction, we have finite ´etale morZH H 0 00 phisms MΦ H → MH → MHh (which can be identified with MHh → MHh → MHh for 0 00 some canonically determined open compact subgroups Hh ⊂ Hh ⊂ Hh ; see Lemmas 1.3.2.1 and 1.3.2.5 below). Such boundary moduli problems MZHH are the fundamental building blocks in the construction of toroidal boundary charts for MH . (They actually appear in the boundary of the minimal compactification of MH , which we call cusps. They are parameterized by the cusp labels of MH .) It is important to study the relations among cusp labels of different multi-ranks. Definition 1.2.1.16. (See [62, Def. 5.4.1.14].) A surjection (Zn , Φn , δn )  (Z0n , Φ0n , δn0 ) between representatives of cusp labels at level n, where Φn = (X, Y, φ, ϕ−2,n , ϕ0,n ) and where Φ0n = (X 0 , Y 0 , φ0 , ϕ0−2,n , ϕ00,n ), is a pair (of surjections) (sX : X  X 0 , sY : Y  Y 0 ) (of O-lattices) such that we have the following: (1) Both sX and sY are admissible surjections (i.e., with kernels defining filtrations that are admissible as in Definition 1.2.1.2), and they are compatible with φ and φ0 in the sense that sX φ = φ0 sY . (2) Z0−2,n is an admissible submodule of Z−2,n (i.e., defining an admissible filtra0

tion as in Definition 1.2.1.2), and the natural embedding GrZ−2,n ,→ GrZ−2,n 0 satisfies ϕ−2,n ◦ (GrZ−2,n ,→ GrZ−2,n ) = s∗X ◦ ϕ0−2,n . (3) Z−1,n is an admissible submodule of Z0−1,n , and the natural surjection 0

0

GrZ0,n  GrZ0,n satisfies sY ◦ ϕ0,n = ϕ00,n ◦ (GrZ0,n  GrZ0,n ). In this case, we write s = (sX , sY ) : (Zn , Φn , δn )  (Z0n , Φ0n , δn0 ) By taking orbits as above, there is a corresponding notion for general cusp labels:

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Definition 1.2.1.17. (See [62, Def. 5.4.2.12].) A surjection (ZH , ΦH , δH )  0 (Z0H , Φ0H , δH ) between representatives of cusp labels at level H, where ΦH = (X, Y, φ, ϕ−2,H , ϕ0,H ) and where Φ0H = (X 0 , Y 0 , φ0 , ϕ0−2,H , ϕ00,H ), is a pair (of surjections) s = (sX : X  X 0 , sY : Y  Y 0 ) (of O-lattices) such that we have the following: (1) Both sX and sY are admissible surjections, and they are compatible with φ and φ0 in the sense that sX φ = φ0 sY . (2) Z0H and (ϕ0−2,H , ϕ00,H ) are assigned to ZH and (ϕ−2,H , ϕ0,H ) respectively under s = (sX , sY ) as in [62, Lem. 5.4.2.11]. 0 In this case, we write s = (sX , sY ) : (ZH , ΦH , δH )  (Z0H , Φ0H , δH ).

Definition 1.2.1.18. (See [62, Def. 5.4.2.13].) We say that there is a surjection from a cusp label at level H represented by some (ZH , ΦH , δH ) to a cusp label at 0 ) if there is a surjection (sX , sY ) from level H represented by some (Z0H , Φ0H , δH 0 0 0 (ZH , ΦH , δH ) to (ZH , ΦH , δH ). This is well defined by [62, Lem. 5.4.1.15]. The surjection among cusp labels can be naturally seen when we have the socalled two-step degenerations (see [28, Ch. III, Sec. 10] and [62, Sec. 4.5.6]). This notion will be further developed in Definitions 1.2.2.12, 1.2.2.18, and 1.2.2.19 below. 1.2.2

Cone Decompositions

For each torus argument Φn = (X, Y, φ, ϕ−2,n , ϕ0,n ) at level n, consider the finitely generated commutative group (i.e., Z-module)   ... y ⊗ φ(y 0 ) − y 0 ⊗ φ(y) 1 (1.2.2.1) S Φn := (( n Y ) ⊗ X)/ (b n1 y) ⊗ χ − ( n1 y) ⊗(b? χ) y,y0 ∈Y, Z χ∈X,b∈O

... ... and set SΦn := S Φn ,free , the free quotient of S Φn . (See [62, (6.2.3.5) and Conv. 6.2.3.20].) Then, for a general torus argument ΦH = (X, Y, φ, ϕ−2,H , ϕ0,H ) at level H, there is a recipe [62, Lem. 6.2.4.4] that gives a corresponding free commutative group SΦH (which can be identified with a finite index subgroup of some SΦn ). The group SΦH provides indices for certain “Laurent series expansions” near the boundary strata. In the modular curve case, it is canonically isomorphic to Z, which means there is a canonical parameter q near the boundary—i.e., the cusps. The expansion of modular forms with respect to this parameter then gives the familiar q-expansion along the cusps. The compactification of the modular curves can be described locally near each of the cusps by Spec(R[q i ]i∈Z ) ,→ Spec(R[q i ]i∈Z≥0 ) for some suitable base ring R. For MH , we would like to have an analogous theory in which the torus with the character group SΦH can be partially compactified by adding normal crossings divisors in a smooth scheme. This is best achieved by the theory of toroidal embeddings developed in [50]. Many terminologies in such a

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theory will naturally show up in our description of the toroidal boundary charts, and we will review them in what follows. ∨ ∨ Let S∨ ΦH := HomZ (SΦH , Z) be the Z-dual of SΦH , and let (SΦH )R := SΦH ⊗ R = Z

HomZ (SΦH , R). By the construction of SΦH , the R-vector space (SΦH )∨ R is isomorphic to the space of Hermitian pairings (| · , · |) : (Y ⊗ R) × (Y ⊗ R) → O ⊗ R, by Z

Z

Z

sending a Hermitian pairing (| · , · |) to the function y ⊗ φ(y 0 ) 7→ TrO ⊗ R/R (|y, y 0 |) in Z HomR ((Y ⊗ R) × (Y ⊗ R), R) ∼ = (SΦ )∨ . (See [62, Lem. 1.1.4.5].) H

Z

Z

R

Definition 1.2.2.2. (See [62, Sec. 6.1.1 and Def. 6.1.2.5].) (1) A subset of (SΦH )∨ R is called a cone if it is invariant under the natural ∨ multiplication action of R× >0 on the R-vector space (SΦH )R . ∨ (2) A cone in (SΦH )R is nondegenerate if its closure does not contain any nonzero R-vector subspace of (SΦH )∨ R. ∨ (3) A rational polyhedral cone in (SΦH )∨ R is a cone in (SΦH )R of the form ∨ σ = R>0 v1 + · · · + R>0 vn with v1 , . . . , vn ∈ (SΦH )∨ Q = SΦH ⊗ Q. Z

(4) A supporting hyperplane of σ is a hyperplane P in (SΦH )∨ R such that σ does not overlap with both sides of P . (5) A face of σ is a rational polyhedral cone τ such that τ = σ ∩ P for some supporting hyperplane P of σ. (Here an overline on a cone means its closure in the ambient space (SΦH )∨ R .) (6) The canonical pairing h · , · i : SΦH × S∨ ΦH → Z defines by extension of scalars a canonical pairing h · , · i : SΦH ×(SΦH )∨ R → R. Then we define for each rational polyhedral cone σ in (SΦH )∨ the following semisubgroups of R SΦH : σ ∨ := {` ∈ SΦH : h`, yi ≥ 0 ∀y ∈ σ}, σ0∨ := {` ∈ SΦH : h`, yi > 0 ∀y ∈ σ}, σ ⊥ := {` ∈ SΦ : h`, yi = 0 ∀y ∈ σ} ∼ = σ ∨ /σ ∨ . 0

H

(SΦH )∨ R

Let PΦH be the subset of corresponding to positive semi-definite Hermitian pairings (| · , · |) : (Y ⊗ R) × (Y ⊗ R) → O ⊗ R, with radical (namely the Z

Z

Z

annihilator of the whole space) admissible in the sense that it is the R-span of some admissible submodule Y 0 of Y . (Recall that we say that a submodule Y 0 of Y is admissible if Y 0 ⊂ Y defines an admissible filtration on Y ; cf. Definition 1.2.1.2. In particular, the quotient Y /Y 0 is also an O-lattice.) Definition 1.2.2.3. (See [62, Def. 6.2.4.1 and 5.4.1.6].) The group ΓΦH is the subgroup of GLO (X) × GLO (Y ) consisting of elements γ = (γX , γY ) satisfying φ = γX φγY , ϕ−2,H = t γX ϕ−2,H , and ϕ0,H = γY ϕ0,H (if we view the latter two as collections of orbits). The group ΓΦH acts on SΦH , and its induced action preserves the subset PΦH of (SΦH )∨ R . (The group ΓΦH is the automorphism group of the torus argument ΦH .

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Such automorphism groups show up naturally because torus arguments are only determined up to isomorphism.) Definition 1.2.2.4. (See [62, Def. 6.1.1.10].) A ΓΦH -admissible rational polyhedral cone decomposition of PΦH is a collection ΣΦH = {σj }j∈J with some indexing set J such that we have the following: (1) Each σj is a nondegenerate rational polyhedral cone. (2) PΦH is the disjoint union of all the σj ’s in Σ. For each j ∈ J, the closure of σj in PΦH is a disjoint union of σk ’s with k ∈ J. In other words, ` ` PΦH = σj is a stratification of PΦH . (Here “ ” only means a setj∈J ` theoretic disjoint union. The geometric structure of σj is still the one j∈J

inherited from the ambient space (SΦH )∨ R of PΦH .) (3) Σ is invariant under the action of ΓΦH on (SΦH )∨ R , in the sense that ΓΦH permutes the cones in Σ. Under this action, the set ΣΦH /ΓΦH of ΓΦH -orbits is finite. Definition 1.2.2.5. (See [62, Def. 6.1.1.11].) A rational polyhedral cone σ in ∨ (SΦH )∨ R is smooth with respect to the integral structure given by SΦH if we have σ = R>0 v1 + · · · + R>0 vn with v1 , . . . , vn forming part of a Z-basis of S∨ ΦH . Definition 1.2.2.6. (See [62, Def. 6.1.1.12].) A ΓΦH -admissible smooth rational polyhedral cone decomposition of PΦH is a ΓΦH -admissible rational polyhedral cone decomposition ΣΦH = {σj }j∈J of PΦH in which every σj is smooth. Definition 1.2.2.7. (See [62, Def. 7.3.1.1].) Let ΣΦH = {σj }j∈J be any ΓΦH -admissible rational polyhedral cone decomposition of PΦH . An (invariant) polarization function on PΦH for the cone decomposition ΣΦH is a ΓΦH -invariant continuous piecewise linear function polΦH : PΦH → R≥0 such that we have the following: (1) polΦH is linear (i.e., coincides with a linear function) on each cone σj in ΣΦH . (In particular, polΦH (tx) = tpolΦH (x) for all x ∈ PΦH and t ∈ R≥0 .) (2) polΦH ((PΦH ∩ S∨ ΦH ) − {0}) ⊂ Z>0 . (In particular, polΦH (x) > 0 for all nonzero x in PΦH .) (3) polΦH is linear (in the above sense) on a rational polyhedral cone σ in PΦH if and only if σ is contained in some cone σj in ΣΦH . (4) For all x, y ∈ PΦH , we have polΦH (x + y) ≥ polΦH (x) + polΦH (y). This is called the convexity of polΦH . If such a polarization function exists, then we say that the ΓΦH -admissible rational polyhedral cone decomposition ΣΦH is projective. Definition 1.2.2.8. An admissible boundary component of PΦH is the image ∨ of PΦ0H under the embedding (SΦ0H )∨ R ,→ (SΦH )R defined by some surjection 0 0 (ΦH , δH )  (ΦH , δH ). (See Definition 1.2.1.17.)

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We shall always assume that the following technical condition is satisfied: Condition 1.2.2.9. (See [62, Cond. 6.2.5.25]; cf. [28, Ch. IV, Rem. 5.8(a)].) The cone decomposition ΣΦH = {σj }j∈J of PΦH is chosen such that, for each j ∈ J, if γσ j ∩ σ j 6= {0} for some γ ∈ ΓΦH , then γ acts as the identity on the smallest admissible boundary component of PΦH containing σj . This condition is used to ensure that there are no self-intersections of toroidal boundary strata when the level H is neat. To describe the toroidal boundary of MH , we will need not only cusp labels but also the cones: 0 Definition 1.2.2.10. (See [62, Def. 6.2.6.1].) Let (ΦH , δH ) and (Φ0H , δH ) be two ∨ 0 representatives of cusp labels at level H, let σ ⊂ (SΦH )R , and let σ ⊂ (SΦ0H )∨ R . We 0 say that the two triples (ΦH , δH , σ) and (Φ0H , δH , σ 0 ) are equivalent if there exists ∼ ∼ a pair of isomorphisms γ = (γX : X 0 → X, γY : Y → Y 0 ) (of O-lattices) such that we have the following: 0 (1) The two representatives (ΦH , δH ) and (Φ0H , δH ) are equivalent under γ (as in [62, Def. 5.4.2.4], the general level analogue of Definition 1.2.1.7). ∼ ∨ 0 (2) The isomorphism (SΦ0H )∨ R → (SΦH )R induced by γ sends σ to σ. 0 In this case, we say that the two triples (ΦH , δH , σ) and (Φ0H , δH , σ 0 ) are equivalent under the pair of isomorphisms γ = (γX , γY ). 0 Definition 1.2.2.11. (See [62, Def. 6.2.6.2].) Let (ΦH , δH ) and (Φ0H , δH ) be two representatives of cusp labels at level H, and let ΣΦH (resp. ΣΦ0H ) be a ΓΦH -admissible (resp. ΓΦ0H -admissible) smooth rational polyhedral cone decomposition of PΦH (resp. 0 , ΣΦ0H ) are equivaPΦ0H ). We say that the two triples (ΦH , δH , ΣΦH ) and (Φ0H , δH 0 0 lent if (ΦH , δH ) and (ΦH , δH ) are equivalent under some pair of isomorphisms ∼ ∼ γ = (γX : X 0 → X, γY : Y → Y 0 ), and if under one (and hence every) such γ the cone decomposition ΣΦH of PΦH is identified with the cone decomposition ΣΦ0H of 0 PΦ0H . In this case, we say that the two triples (ΦH , δH , ΣΦH ) and (Φ0H , δH , ΣΦ0H ) are equivalent under the pair of isomorphisms γ = (γX , γY ).

The compatibility among cone decompositions over different cusp labels are described as follows: 0 Definition 1.2.2.12. (See [62, Def. 6.2.6.4].) Let (ΦH , δH ) (resp. (Φ0H , δH )) be a representative of a cusp label at level H, and let ΣΦH (resp. ΣΦ0H ) be a ΓΦH -admissible (resp. ΓΦ0H -admissible) smooth rational polyhedral cone decomposi0 tion of PΦH (resp. PΦ0H ). A surjection (ΦH , δH , ΣΦH )  (Φ0H , δH , ΣΦ0H ) is given 0 0 0 0 by a surjection s = (sX : X  X , sY : Y  Y ) : (ΦH , δH )  (ΦH , δH ) (see Definition 1.2.1.17) that induces an embedding PΦ0H ,→ PΦH such that the restriction ΣΦH |PΦ0 of the cone decomposition ΣΦH of PΦH to PΦ0H is the cone decomposition H ΣΦ0H of PΦ0H .

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This allows us to define: Definition 1.2.2.13. (See [62, Cond. 6.3.3.2 and Def. 6.3.3.4].) A compatible choice of admissible smooth rational polyhedral cone decomposition data for MH is a complete set Σ = {ΣΦH }[(ΦH ,δH )] of compatible choices of ΣΦH 0 (satisfying Condition 1.2.2.9) such that, for every surjection (ΦH , δH )  (Φ0H , δH ) of representatives of cusp labels, the cone decompositions ΣΦH and ΣΦ0H define a 0 surjection (ΦH , δH , ΣΦH )  (Φ0H , δH , ΣΦ0H ) as in Definition 1.2.2.12. Definition 1.2.2.14. (See [62, Def. 7.3.1.3].) We say that a compatible choice Σ = {ΣΦH }[(ΦH ,δH )] of admissible smooth rational polyhedral cone decomposition data for MH (see Definition 1.2.2.13) is projective if it satisfies the following condition: There is a collection pol = {polΦH : PΦH → R≥0 }[(ΦH ,δH )] of polarization functions labeled by representatives (ΦH , δH ) of cusp labels, each polΦH being a polarization function of the cone decomposition ΣΦH in Σ (see Definition 1.2.2.7), which are 0 ) compatible in the following sense: For every surjection (ΦH , δH )  (Φ0H , δH of representatives of cusp labels (see Definition 1.2.1.17) inducing an embedding PΦ0H ,→ PΦH , we have polΦH |PΦ0 = polΦ0H . H

The most important relations among cone decompositions and among compatible choices of them are the so-called refinements: 0 ) be two reDefinition 1.2.2.15. (See [62, Def. 6.2.6.3].) Let (ΦH , δH ) and (Φ0H , δH presentatives of cusp labels at level H, and let ΣΦH (resp. ΣΦ0H ) be a ΓΦH -admissible (resp. ΓΦ0H -admissible) smooth rational polyhedral cone decomposition of PΦH (resp. PΦ0H ). We say that the triple (ΦH , δH , ΣΦH ) is a refinement of the triple 0 0 (Φ0H , δH , ΣΦ0H ) if (ΦH , δH ) and (Φ0H , δH ) are equivalent under some pair of isomorphisms γ = (γX , γY ), and if under one (and hence every) such γ the cone decomposition ΣΦH of PΦH is identified with a refinement of the cone decomposition ΣΦ0H of PΦ0H . In this case, we say that the triple (ΦH , δH , ΣΦH ) is a refinement 0 of the triple (Φ0H , δH , ΣΦ0H ) under the pair of isomorphisms γ = (γX , γY ).

Definition 1.2.2.16. (See [62, Def. 6.4.2.2].) Let Σ = {ΣΦH }[(ΦH ,δH )] and Σ0 = {Σ0ΦH }[(ΦH ,δH )] be two compatible choices of admissible smooth rational polyhedral cone decomposition data for MH . We say that Σ is a refinement of Σ0 if the triple (ΦH , δH , ΣΦH ) is a refinement of the triple (ΦH , δH , Σ0ΦH ), as in Definition 1.2.2.15, for (ΦH , δH ) running through all representatives of cusp labels. Proposition 1.2.2.17. (See [62, Prop. 6.3.3.5 and 7.3.1.4].) (1) A compatible choice Σ of admissible smooth rational polyhedral cone decomposition data for MH , as in Definition 1.2.2.13, exists. Moreover, we may assume that Σ is projective as in Definition 1.2.2.14. (2) Given any Σ and Σ0 , we can find a common refinement for them, which we may require to be smooth as in Definition 1.2.2.13, or both smooth and

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projective as in Definition 1.2.2.14. The same is true if we allow varying levels or twists by Hecke actions (see [62, Def. 6.4.2.8 and 6.4.3.2]). We may assume that this common refinement is invariant under any choice of an open compact subgroup H0 of G(A∞ ) normalizing H. Proof. The first part has been explained in the proofs of [62, Prop. 6.3.3.5 and 7.3.1.4], by induction on magnitudes of cusp labels (i.e., by starting with cusp labels of smaller multiranks and building cone decompositions and polarization functions along them, which appear as rational boundary components of homogeneous cones attached to cusp labels of larger multiranks). Based on such inductive constructions (which builds the smaller dimensional cones first), the second part can be reduced to questions over each PΦH (with prescribed cone decompositions and polarization functions over PΦH − P+ ΦH ), which is then well known. (See the arguments in [89, 5.21, 5.23, 5.24, 5.25], where the crucial existence of smooth and projective refinements is in turn based on [50, Ch. I, Sec. 2, Thm. 11 on pp. 33–35].) Finally, we would like to describe the relations among the equivalence classes [(ΦH , δH , σ)], which will describe the “incidence relations” among (closures of) the toroidal boundary strata. Definition 1.2.2.18. (See [62, Def. 6.3.2.13].) Let (ΦH , δH ) be a representative of a cusp label at level H, and let σ ⊂ P+ ΦH be a nondegenerate smooth rational 0 polyhedral cone. We say that a triple (Φ0H , δH , σ 0 ) is a face of (ΦH , δH , σ), if: 0 ) is the representative of some cusp label at level H, such that there (1) (Φ0H , δH 0 ) as in Definition exists a surjection s = (sX , sY ) : (ΦH , δH )  (Φ0H , δH 1.2.1.17. (2) σ 0 ⊂ P+ Φ0H is a nondegenerate smooth rational polyhedral cone, such that for one (and hence every) surjection s = (sX , sY ) as above, the image of σ 0 under the induced embedding PΦ0H ,→ PΦH is contained in the ΓΦH -orbit of a face of σ.

Note that this definition is insensitive to the choices of representatives in the classes 0 , σ 0 )]. This justifies the following: [(ΦH , δH , σ)] and [(Φ0H , δH Definition 1.2.2.19. (See [62, Def. 6.3.2.14].) We say that the equivalence class 0 0 [(Φ0H , δH , σ 0 )] of (Φ0H , δH , σ 0 ) is a face of the equivalence class [(ΦH , δH , σ)] of 0 (ΦH , δH , σ) if some triple equivalent to (Φ0H , δH , σ 0 ) is a face of some triple equivalent to (ΦH , δH , σ). 1.2.3

Rational Boundary Components

Now we explain how to associate cusp labels with rational boundary components. This is mainly for readers who are familiar with the notion of rational boundary components of Hermitian symmetric domains. (See, for example, the summaries in [5] or [9].)

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Let X0 be the connected component of X containing h0 , and let G(R)0 (resp. G(Q)0 ) denote its stabilizer in G(R) (resp. G(Q)). Then G(R)0 (resp. G(Q)0 ) has finite index in G(R) (resp. G(Q)). Lemma 1.2.3.1. Let us fix a choice of an element g ∈ G(A∞ ). Let L(g) denote the ˆ ˆ corresponds naturally to the O ⊗ Z-submodule O-lattice in L ⊗ Q such that L(g) ⊗ Z Z

Z

Z

ˆ of L ⊗ A∞ . Consider the five sets formed respectively by the following five g(L ⊗ Z) Z

Z

types of data on (L, h · , · i, h0 ): (1) A rational boundary component of X0 (as in [5, 1.5]). (For compatibility with formation of products, it is necessary to include X0 itself as a rational boundary component.) (2) An O ⊗ Q-submodule V−2 of L ⊗ Q that is totally isotropic under the pairing Z

Z

h · , · i. (3) An increasing filtration V = {V−i }i∈Z of L ⊗ Q satisfying the following conZ

ditions: (a) V−3 = 0 and V0 = L ⊗ Q. Z

(b) Each graded piece GrV−i := V−i /V−i−1 is an O ⊗ Q-module. (In this Z

case, the filtration V is admissible.) (c) V−1 and V−2 are annihilators of each other under the pairing h · , · i. (In this case, the filtration V is symplectic.) (g)

(g)

(4) An O-sublattice F−2 of L(g) , with L(g) /F−2 torsion-free, that is totally isotropic under the pairing h · , · i. (g) (5) An increasing filtration F(g) = {F−i }i∈Z of L(g) satisfying the following conditions: (g)

(g)

(a) F−3 = 0 and F0 = L(g) . (g)

(g)

(g)

F (b) Each graded piece Gr−i := F−i /F−i−1 is an O-lattice, admitting an (g)

splitting ε(g) : GrF

(g)

∼

F := ⊕ Gr−i → L(g) . (In this case, the filtration −i∈Z

F(g) is admissible.) (g) (g) (g) (c) F−1 and F−2 are annihilators of each other under the pairing h · , · i : L(g) × L(g) → Z(1). (In this case, the filtration F(g) is symplectic.) (We allow parabolic subgroups to be the whole group, and we allow totally isotropic submodules to be zero.) Then the five sets are in canonical bijections with each other. Proof. As explained in [5, 1.5], the rational boundary components of X0 correspond bijectively to the rational parabolic subgroups of G ⊗ Q each of whose images in Z

the Q-simple quotients of G ⊗ Q is either a maximal proper parabolic subgroup Z

or the whole group. For simplicity, let us call temporarily such rational parabolic

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subgroups maximal. Given any such rational parabolic subgroup of G ⊗ Q, the Z

action of the Lie algebra of its unipotent radical defines an isotropic filtration V of L ⊗ Q. By maximality of the parabolic subgroup, we see that V is determined by Z

its largest totally isotropic filtered piece. Now the equivalences among the maximal rational parabolic subgroups and the remaining objects in the lemma is elementary. ˆ For each g ∈ G(A∞ ), let L(g) denote the O-lattice in L ⊗ Q such that L(g) ⊗ Z Z

Z

ˆ of L ⊗ A∞ . Then the asˆ g(L ⊗ Z) corresponds naturally to the O ⊗ Z-submodule Z

Z

Z

signment

V−2 7→ V = {V−i }i∈Z (g)

7→ F(g) := {F−i := V−i ∩ L(g) }i∈Z (g)

(1.2.3.2)

(g)

ˆ i∈Z 7→ Z(g) := {Z−i := g −1 (F−i ⊗ Z)} Z

ˆ i∈Z = {(g −1 (V−i ⊗ A∞ )) ∩(L ⊗ Z)} Q

Z

defines an injection from the set of rational boundary components of X0 to the set ˆ (See [62, Def. 5.2.7.1].) of fully symplectic admissible filtrations on L ⊗ Z. Z

The action of G(Q) on X × G(A∞ ) induces an action of G(Q) on {V} × G(A∞ ). Definition 1.2.3.3. A rational boundary component of X × G(A∞ ) is a G(Q)-orbit of some pair (V, g). By the explicit definition above, pairs in the G(Q)-orbit of (V, g) define the ˆ This induces a map from the same fully symplectic admissible filtration on L ⊗ Z. Z

set of rational boundary components of X × G(A∞ ) to the set of fully symplectic ˆ However, this map is generally far from injective. admissible filtrations on L ⊗ Z. Z

ˆ is an element preserving V−2,A∞ := V−2 ⊗ A∞ , then For example, if u ∈ G(Z) Q

(V, g) and (V, gu) define the same filtration Z(g) = Z(gu) . For the purpose of studying toroidal compactifications, it is important to distinguish between (V, g) and (V, gu) by supplying a rigidification on the rational structure of V−2 . For each given (V, g), (g) (g) let us define a torus argument Φ(g) = (X (g) , Y (g) , φ(g) , ϕ−2 , ϕ0 ) for Z(g) as follows: (g)

(g)

(1) X (g) := HomZ (F−2 , Z(1)) = HomZ (GrF−2 , Z(1)). (g)

(g)

(g)

(2) Y (g) := GrF0 = F0 /F−1 . (3) φ(g) : Y (g) ,→ X (g) is equivalent to the nondegenerate pairing (g)

(g)

(g)

F h · , · i20 : Gr−2 × Gr0F (g)

→ Z(1)

induced by h · , · i : L(g) × L(g) → Z(1), with the sign convention (g) (g) hx, yi20 = φ (y)(x).

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(g)

(g)

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41

∼

ˆ Z(1)) ˆ is the composition (4) ϕ−2 : GrZ−2 → HomZˆ (X (g) ⊗ Z, Z

(g)

GrZ−2

Gr−2 (g) ∼

→

∼ F(g) ˆ Z(1)). ˆ ˆ→ HomZˆ (X (g) ⊗ Z, Gr−2 ⊗Z Z

Z

(5)

(g) ϕ0

:

(g) ∼ GrZ0 →

ˆ is the composition Y (g) ⊗ Z Z (g)

Gr0Z

Gr0 (g) ∼

(g)

→ Gr0F

∼ ˆ→ ˆ Y (g) ⊗ Z. ⊗Z Z

Z

Finally, by Condition 1.2.1.1 and the fact that maximal orders over Dedekind domains are hereditary ([93, Thm. 21.4 and Cor. 21.5]), for each (V, g), the associated (g) ∼ filtration F(g) of L(g) is split by some splitting ε(g) : GrF → L(g) . Each split∼ ˆ = ˆ → ˆ : GrF(g) ⊗ Z L(g) ⊗ Z ting ε(g) defines by base change a splitting ε(g) ⊗ Z Z

Z

Z

ˆ and hence by pre- and post-composition with Gr(g) and g −1 a splitting g(L ⊗ Z), Z

(g)

δ (g) : GrZ

∼ ˆ This defines an assignment → L ⊗ Z. Z

(V, g, ε(g) ) 7→ (Z(g) , Φ(g) , δ (g) ).

(1.2.3.4)

Let us define two triples (V, g, ε(g) ) and (V0 , g 0 , (ε(g) )0 ) to be equivalent if V = V0 and g = g 0 , and define two triples (Z, Φ, δ) and (Z0 , Φ0 , δ 0 ) to be equivalent if Z = Z0 and if the torus arguments Φ = (X, Y, φ, ϕ−2 , ϕ0 ) and Φ0 = (X 0 , Y 0 , φ0 , ϕ0−2 , ϕ00 ) are ∼ equivalent in the sense that there exists some pair of isomorphisms (γX : X 0 → ∼ X, γY : Y → Y 0 ) matching the remaining data. By definition, the equivalence clas(g) ses [(V, g, ε )] of triples (V, g, ε(g) ) correspond exactly to the pairs (V, g) they define by forgetting the splitting ε(g) . On the other hand, let us denote by [(Z(g) , Φ(g) , δ (g) )] the equivalence class defined by (Z(g) , Φ(g) , δ (g) ), and let us call them the cusp labels for (L, h · , · i, h0 ). Now we have the assignment (V, g) 7→ [(Z(g) , Φ(g) , δ (g) )] induced by the assignment (V, g, ε(g) ) 7→ (Z(g) , Φ(g) , δ (g) ). This assignment is still not injective in general, but will suffice for our purpose. For each Q-algebra R, let us write V−i,R := V−i ⊗ R and GrV−i,R := Q (g)

(g)

V−i,R /V−i−1,R . Similarly, for each Z-algebra R, let us write F−i,R := F−i ⊗ R and Z

(g)

(g)

(g)

GrF−i,R := F−i,R /F−i−1,R . To each boundary component represented by (V, g), the symplectic filtration V induces a symplectic lattice (GrV−1 , h · , · i11 ), and the associated symplectic filtra(g)

(g)

tion F(g) on L(g) induces a symplectic lattice (GrF−1 , h · , · i11 ). It is clear that (g) (g) ∼ (GrF−1 ⊗ Q, h · , · i11 ) = (GrV−1 , h · , · i11 ). Z √ Each h ∈ X defines a complex structure h( −1) on L ⊗ R, inducing an isomorZ

∼

(g)

phism L ⊗ R → Vh = (L ⊗ C)/Ph of C-vector spaces. Since F−2,R is totally isotropic, Z

Z

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√ √ and since − sgn(h) −1 h · , h( −1) · i is positive definite for some sgn(h) ∈ {±1}, √ (g) (g) we have F−2,R ∩ h( −1)(F−2,R ) = {0}. Then h defines a C-linear embedding (g)

(g)

F−2,C ,→ Vh , such that the composition F−2,R

√ h( −1)

→

(g)

F L ⊗ R  Gr0,R is an isoZ

morphism of O ⊗ R-modules. By abuse of notation, we shall denote the image of Z

(g)

(g)

the above embedding F−2,C ,→ Vh by F−2,h(C) . Let (g)

(g)

(F−2,h(C) )⊥ := {x ∈ L ⊗ R : hx, yi = 0, ∀y ∈ F−2,h(C) }. Z

Then we obtain an orthogonal direct sum (L ⊗ R, h · , · i) Z (g) ∼ = (F−2,h(C) , h · , · i|F(g)

⊥

−2,h(C)

(1.2.3.5)

(g)

) ⊕((F−2,h(C) )⊥ , h · , · i|(F(g)

)⊥ −2,h(C)

),

which induces an isomorphism ∼

(g)

((F−2,h(C) )⊥ , h · , · i|(F(g)

(g)

(g)

F ) → (Gr−1,R , h · , · i11,R ) (1.2.3.6) √ (g) of symplectic O ⊗ R-modules. Since h( −1) preserves F−2,h(C) , the relation Z √ √ hh( −1)x, h( −1)yi = hx, yi √ (g) for every x, y ∈ L ⊗ R shows that h( −1) also preserves (F−2,h(C) )⊥ . As a result, the Z √ (g) restriction of h( −1) defines a complex structure on (F−2,h(C) )⊥ , which corresponds via the isomorphism (1.2.3.6) (and [59, Lem. 2.1.2]) to a sgn(h)-polarization h−1 of (g) (g) (GrF−1,R , h · , · i−1,R ) (see [59, Def. 2.1.1]), such that −2,h(C)

)⊥

(g)

((F−2,h(C) )⊥ , h · , · i|(F(g)

−2,h(C)

∼

(g)

)⊥

, h|(F(g)

−2,h(C)

)⊥

)

(g)

(1.2.3.7)

(g)

→ (GrF−1,R , h · , · i11,R , h−1 ) is an isomorphism of polarized symplectic O ⊗ R-modules.

Hence, the triple

Z (g)

(GrF−1 , h · , · i h = h0 .)

, h−1 ) is a PEL-type O-lattice. (In particular, this is the case for

Lemma 1.2.3.8. With notation as in [62, Rem. 5.2.7.2] (with h there replaced with (g) (g) h0 here), the PEL-type O-lattice (GrF−1 , h · , · i , (h0 )−1 ) qualifies as a (noncanoni(g) (g) (g) (g) (g) (g) cal) choice of (LZ , h · , · iZ , hZ ), so that (GrZ , h · , · i11 ) ∼ = (GrF ˆ , h · , · i ) (g) (GrZ−1,R , h · ,

0 ∼ (g) · i11,R , (h0 )−1 ) →

F(g) (Gr−1,R ,h·,

−1 (g) · i11 , (h0 )−1 ).

and below for the justification of such notation.)

−1,Z

11

(See Remark 1.2.3.9 In particular, at any neat le-

(g)

Z

vel H, the scheme MHH can be identified with the moduli problem defined by (g)

(g)

(GrF−1 , h · , · i

, (h0 )−1 ) at a suitable level (see Lemma 1.3.2.1 below).

Remark 1.2.3.9. The notation (h0 )−1 appeared twice in the second isomorphism in Lemma 1.2.3.8. Nevertheless, their constructions are identical because we have to (g) (g) Z(g) use F−2,R = HomR (X (g) ⊗ R, R(1)) ,→ L ⊗ R to define (h0 )−1 for (Gr−1,R , h · , · i11,R ) Z

Z

in [62, Prop. 5.1.2.2]. This is why we allow such an identification.

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43

Parameters for Kuga Families

For the considerations in Section 1.3.3, we would like to have parameter sets for the toroidal compactifications there. Let (O, ?, L, h · , · i, h0 ) be an integral PEL datum as in Definition 1.1.1.1. By our running assumption that O satisfies 1.2.1.1, the action of O on L extends to an action of some maximal order O0 in O ⊗ Q containing O. Let us fix the choice of Z

such a maximal order O0 . Let Q be an O-lattice. Consider Diff −1 = Diff −1 O/Z , the inverse different of O over Z [62, Def. 1.1.1.8] with its canonical left O-module structure. Since the trace pairing Diff −1 × O → Z : (y, x) 7→ TrO/Z (yx) is perfect by definition, for each O-lattice Q, we may identify Q∨ := HomZ (Q, Z) with HomO (Q, Diff −1 ). By ∼ composition with the involution ? : O → Oop , the natural right action of O on Diff −1 induced a left action of O on Diff −1 , which commutes with the natural left action of O on Diff −1 . Accordingly, the Z-module Q∨ is torsion-free and has a canonical ∼ left O-structure induced by the right action of Oop on Diff −1 (and ? : O → Oop ). In other words, Q∨ is an O-lattice. Then the trace pairing induces a perfect pairing h · , · iQ : Q∨ × Q → Z : (f, x) 7→ TrO/Z (f (x)). For all b ∈ O, f ∈ Q∨ , and x ∈ Q, we have hbf, xiQ = TrO/Z (f (x)b? ) = TrO/Z (b? f (x)) = TrO/Z (f (b? x)) = hf, b? xi. Lemma 1.2.4.1. (See [61, Lem. 2.5].) There exists an embedding jQ : Q∨ ,→ Q of ∼ O-lattices inducing an isomorphism jQ : Q∨ ⊗ Q → Q ⊗ Q of O ⊗ Q-modules such Z

that the pairing

Z

Z

−1 hjQ ( · ), · iQ : (Q ⊗ R) ×(Q ⊗ R) → R Z

Z

is positive definite. Let jQ : Q∨ ,→ Q be an embedding of O-lattices given by Lemma 1.2.4.1, and e h · , · ie, e let (L, h0 ) be the symplectic O-lattice given by the following data: (1) An O-lattice e := Q−2 ⊕ L ⊕ Q0 , L where ∨ Q−2 := HomO (Q, Diff −1 O 0 /Z (1)) ⊂ Q ⊗ Q(1) Z

0 has O -module structure inherited from the two-sided ideal Diff −1 O 0 /Z of O , and where 0

Q0 := O0 · Q ⊂ Q ⊗ Q. Z

(Then the perfect pairing h · , · iQ : Q∨ × Q → Z : (f, x) → 7 TrO/Z (f (x)) e induces a perfect pairing h · , · iQ : Q−2 × Q0 → Z(1), and L satisfies Condition 1.2.1.1 by construction.)

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e×L e → Z(1) defined (symbolically) by (2) A symplectic O-pairing h · , · ie : L the matrix     h · , · iQ x−2 y−2  y−1  , h·, ·i hx, yie := t x−1   t − h · , · iQ x0 y0 namely by hx, yie := hx−2 , y0 iQ + hx−1 , y−1 i − hy−2 , x0 iQ ,     x−2 y−2 e = Q−2 ⊕ L ⊕ Q0 where x = x−1  and y = y−1  are elements of L x0 y0 expressed (vertically) in terms of components in the direct summands. e ⊗ R) defined by (3) An R-algebra homomorphism e h0 : C → EndO ⊗ R (L Z Z √ z = z1 + −1 z2   √ −1 ) z1 IdQ−2 ⊗ R −z2 ((2π −1) ◦ jQ Z   , 7→ e h0 (z) :=  h (z) 0   √ −1 z2 (jQ ◦ (2π −1) ) z1 IdQ0 ⊗ R Z √ √ ∼ ∼ where 2π −1 : Z → Z(1) and (2π −1)−1 : Z(1) → Z stand for the √ isomorphisms defined by the choice  of −1 in C, and where the matrix acts x−2 e ⊗ R by left multiplication. In (symbolically) on elements x = x−1  of L Z x0 other words, √     −1 )(x0 ) z1 x−2 − z2 ((2π −1) ◦ jQ x−2 e . h0 (z) x−1  =  h (z)x √ 0 −1−1 x0 z2 (jQ ◦ (2π −1) )(x−2 ) + z1 x0 e h · , · ie) making (L, e h · , · ie, e Then e h0 is a polarization of (L, h0 ) a PEL-type e e O-lattice. Note that the reflex field of (L ⊗ R, h · , · ie, h) is also F0 . Z

Remark 1.2.4.2. If p is a good prime for (O, ?, L, h · , · i, h0 ) as in Definition 1.1.1.6, e h · , · ie, e then it is also a good prime for (O, ?, L, h0 ). e h · , · ie), there is a fully symplectic admissible filtration By the construction of (L, ˆ e on L ⊗ Z induced by Z

e 0 ⊂ Q−2 ⊂ Q−2 ⊕ L ⊂ Q−2 ⊕ L ⊕ Q0 = L. More precisely, we have e Z−3 := 0,

ˆ e Z−2 := Q−2 ⊗ Z, Z

ˆ ⊕(L ⊗ Z), ˆ e Z−1 := (Q−2 ⊗ Z) Z

ˆ ⊕(L ⊗ Z) ˆ ⊕(Q0 ⊗ Z) ˆ =L ˆ e ⊗ Z, e Z0 := (Q−2 ⊗ Z) Z

Z

Z

Z

Z

and

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so that there are canonical isomorphisms e ˆ GrZ−1 ∼ = L ⊗ Z,

e ˆ GrZ−2 ∼ = Q−2 ⊗ Z,

e ˆ GrZ0 ∼ = Q0 ⊗ Z

and

Z

Z

Z

ˆ ˆ matching the pairings → Z(1) and → Z(1) induced by h · , · ie with h · , · iQ and h · , · i, respectively. e := HomO (Q−2 , Diff −1 (1)) and Ye := Q0 . The pairing h · , · iQ : Let X e and there are caQ−2 × Q0 → Z(1) induces a canonical embedding φe : Ye ,→ X ∼ e e ∼ Z ˆ (of ˆ Z(1)) ˆ e ⊗ Z, nonical isomorphisms ϕ e−2 : Gr → Hom ˆ (X and ϕ e0 : GrZ → Ye ⊗ Z e GrZ−2

e × GrZ0

−2

e GrZ−1

e × GrZ−1

0

Z

Z

Z

eϕ ˆ e := (X, e Ye , φ, These data define a torus argument Φ e−2 , ϕ e0 ) for e Z O ⊗ Z-modules). Z

as in Definition 1.2.1.5. e Let δe be the obvious splitting of e Z induced by the equality Q−2 ⊕ L ⊕ Q0 = L. e e Let G be the group functor defined by (L, h · , · ie) as in Definition 1.1.1.3, with ee defined by e ee(R), the subgroup functor P Z as in Definition 1.2.1.10, and quotients Z Z Z 0 0 0 e e(R), U e e(R), U e e(R), G e e(R), G e (R), P e (R), G e e(R), and G e (R) of subU Z 2,Z 1,Z l,Z 1,Z e l,e Z Z h,e Z ˆ ee(R) defined for each Z-algebra groups of P R as in Definition 1.2.1.11. Note that Z we have, by Definition 1.2.1.9, a canonical isomorphism G e(R) ∼ = G0 (R) ∼ = h,Z

h,e Z

ˆ (G ⊗ Z)(R). Then we also define: Z

Definition 1.2.4.3. b e e(R) = U e e(R)/U e e(R). (1) U(R) := U 1,Z Z 2,Z 0 b e e(R) = P e (R)/U e e(R), which is (under the splitting δe above) (2) G(R) := G 1,Z 2,Z e Z b b isomorphic to (G n U)(R) := G(R) n U(R). Definition 1.2.4.4. (Compare with Definition 1.2.1.12.) For each open comˆ we define H ˆ H ˆ e∩P ee(Z), e∩Z ee(Z), e of G( e Z), e e := H e e := H pact subgroup H Z Z PeZ ZeZ ˆ H ˆ H e∩U e e(Z), ee , e e := H e e := H e∩U e e(Z), e e := H e e /H ee , H e e := H e e /H H Gh,eZ

ee H G

l,e Z

PeZ

ZeZ

UeZ

Z

Z

Z

e Z

2,Z

U2,eZ

ˆ H e e /H ee , H e e 0 := H e∩P e 0 (Z), ee := H e Ze Ue P G Z

1,e Z

U1,eZ

e e 0 /H ee , H e e0 := H P U e G e Z

2,Z

h,e Z

U2,eZ

UeZ

e e 0 /H ee , := H P Ue e Z

Z

e e 0 := H e e /H e e 0 as in Definition 1.2.1.12, and define and H G Pe P Z

l,e Z

e Z

b := H e b := H ee . (1) H G G1,eZ b e ee . (2) H b := H b := H U

U

U1,eZ

bG := H/ b H bb ∼ e e0 . (3) H U = HG h,e Z

Then we have an exact sequence bb → H b→H bG → 1 1→H U

(1.2.4.5)

compatible with the canonical exact sequence ˆ → G( ˆ → G(Z) ˆ → 1. b Z) b Z) 1 → U(

(1.2.4.6)

ˆ p - or Zp -valued groups above. We shall also extend this definition to the cases of Z

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ˆ and let H e be any neat open Let H be any open compact subgroup of G(Z), ˆ satisfying the following condition: e Z) compact subgroup of G( bG = GreZ (H e e 0 ) = GreZ (H e e ) ⊂ H. (The first equality Condition 1.2.4.7. H −1 −1 PeZ PeZ is just the definition, while the second equality is the essential condition. Then e e e ) is a direct factor of H e e /H e e .) GrZ−1 (H Pe Pe Ue Z

Z

Z

e or rather on H: b For later purposes, we define two more conditions on such H, bG = H. (Then Condition 1.2.4.7 is redundant, because we Condition 1.2.4.8. H e Z b e e 0 ) ⊂ GreZ−1 (H e e ) ⊂ H.) always have HG = Gr−1 (H P Pe Z

e Z

Condition 1.2.4.9. The splitting δe defines a (group-theoretic) splitting of the seˆ n U( ˆ which also ˆ ∼ b Z) b Z), quence (1.2.4.6) and induces an isomorphism G( = G(Z) defines a (group-theoretic) splitting of the sequence (1.2.4.5) and induces an isob ∼ b b . (This condition is equivalent to the condition that the morphism H = HnH U ˆ → G( ˆ defined by δe maps H to H.) b Z) b splitting G(Z) e satisfying these conditions, because Remark 1.2.4.10. For each H, there exists H the pairing h · , · ie is the direct sum of the pairings on Q−2 ⊕ Q0 and on L. e be defined as above, which induces a representative e δ) Let (e Z, Φ, eϕ e e = (X, e Ye , φ, e e , δe e )] at level H. e Let (e ZHe , Φ e−2,He , ϕ e0,He ), δeHe ) of a cusp label [(e ZHe , Φ H H H e Σ be any compatible choice of admissible smooth rational polyhedral cone decome e that is projective (see Definitions 1.2.2.13 and 1.2.2.14). Let position data for M

σ e ⊂ P+ e Φ

H

be any top-dimensional nondegenerate rational polyhedral cone in the

f H

e e in Σ. e cone decomposition Σ Φf H

Definition 1.2.4.11. e ++ is the set of all triples κ e Σ, e σ e satisfies (1) K e = (H, e) as above (such that H Q,H Condition 1.2.4.7). e + is the subset of K e ++ consisting of elements κ e Σ, e σ (2) K e = (H, e) such that Q,H Q,H e satisfies Condition 1.2.4.8. H e Q,H is the subset of K e + consisting of elements κ e Σ, e σ (3) K e = (H, e) such that Q,H

e also satisfies Condition 1.2.4.9. H ˘ e , δ˘ e , τ˘)] having [(Φ e e , δe e , σ The equivalence classes [(Φ H H H H e )] as a face (as in Defie nition 1.2.2.19) are H-orbits of data of the following form: ˆ satisfying e⊗Z ˘ = {Z ˘−i } on L (1) A fully symplectic admissible filtration Z Z

e ˘−2 ⊂ Z ˘−1 ⊂ e Z−2 ⊂ Z Z−1 .

(1.2.4.12)

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˘ induces a fully symplectic admissible filtration Z = Each such filtration Z ˆ ˘−2 /e ˘−1 /e Z−2 and Z−1 := Z Z−2 , so that there is a {Z−i } on L ⊗ Z by Z−2 := Z Z

canonical isomorphism ˘−1 . Z0 /Z−1 ∼ Z−1 /Z =e

(1.2.4.13)

ˆ induces Conversely, each fully symplectic admissible filtration Z on L ⊗ Z Z

ˆ satisfying (1.2.4.12) and e⊗Z ˘ on L a fully symplectic admissible filtration Z Z

(1.2.4.13). ˘ ϕ˘−2 , ϕ˘0 ) for Z ˘ = (X, ˘ Y˘ , φ, ˘ (as in Definition 1.2.1.5), to(2) A torus argument Φ ˘ e and s ˘ : Y˘  Ye satisfying gether with admissible surjections sX˘ : X  X Y e ˘ and other natural compatibilities with ϕ˘−2 , ϕ˘0 , ϕ sX˘ φ˘ = φs e−2 , and ϕ e0 . Y (See Definitions 1.2.1.16, 1.2.1.17, and 1.2.1.18.) ˘ s ˘ , and s ˘ determine a torus argument Φ = (X, Y, φ, ϕ−2 , ϕ0 ) Any Φ, X Y ˘ Y , so that there is a for Z by X := ker(sX˘ ), Y := ker(sY˘ ), and φ := φ| commutative diagram 0

0

/Y

/ Y˘

φ

˘ φ

 /X

 ˘ /X

sY˘

/ Ye

/0

(1.2.4.14)

e φ sX ˘

 e /X

/0

whose horizontal rows are exact sequences. ˘, inducing some liftable splitting δ˘ e (3) The existence of some splitting δ˘ of Z H ˘ ˘ ˘ e , δ˘ e )] at level ˘ e, Φ ˘ e , Φ e , δ e ) of cusp label [(Z defining the representative (Z H H H H H H e H. e the existence of some splitting δ˘ is equivalent to the Given the splitting δ, existence of some splitting δ of Z. Then, for forming compatible orbits, we have the following Lemma 1.2.4.15. There is a canonical assignment from the set of cusp ˘ e , δ˘ e )] at level H e admitting a surjection to [(e e e , δe e )], to ˘ e, Φ labels [(Z ZHe , Φ H H H H H the set of cusp labels [(ZH , ΦH , δH )] at level H. This assignment is bijective e e e ) = H; if we assume Condition 1.2.4.8, so that, in particular, GrZ−1 (H PeZ e Z e and is still surjective if we only assume Gr (H e ) ⊂ H. −1

PeZ

By definition, we have the following: e representing the cusp e δ) Lemma 1.2.4.16. With the fixed choice of (e Z, Φ, e ˘ of the e e ˘ δ) ˘, Φ, e label [(ZHe , ΦHe , δHe )] at level H, the choices of representatives (Z ˘ e ˘ e ˘ e , Φ e , δ e )] that are compatible with (e cusp label [(Z Z, Φ, δ) as above form an H H H b H-orbit up to equivalence. Therefore, it makes sense to have the following:

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b Definition 1.2.4.17. We shall denote any H-orbit as in Lemma 1.2.4.16 by ˘ ϕ˘ b , ϕ˘ b ), δ˘ b ), and denote its (well-defined) equi˘ b = (X, ˘ Y˘ , φ, ˘ b, Φ (Z H H −2,H 0,H H ˘ b , δ˘ b )]. We say in this case that Φ ˘ b is a torus ˘ b, Φ valence class by [(Z H H H H b We can generalize all notions for cusp la˘ b at level H. argument for Z H ˘ b , δ˘ b )] at ˘ b, Φ bels for MH to the context here, and consider a cusp label [(Z H H H b for (L, e h · , · ie, e level H h0 , e Z) (see Definition 1.2.1.7 and [62, Def. 5.4.2.1, ˘ e ” with “Φ ˘ b ” in 5.4.2.2, and 5.4.2.4]). We shall replace the subscripts “Φ H H b ˘ the notation for objects depending only on the H-orbit of Φ. ˘ e ) be the torus argument for ZH (resp. e (4) Let ΦH (resp. Φ ZHe ) at level H H e ˘ (resp. H) induced by Φ (resp. Φ). Then (1.2.4.14) induces morphisms SΦH ,→ SΦ˘ f  SΦ e f, H

(1.2.4.18)

H

where the first morphism is canonical, and where the second morphism is defined by sX˘ and sY˘ , whose composition is zero. (In general, the morphisms in (1.2.4.18) do not form an exact sequence.) The dual of (1.2.4.18) defines morphisms ∨ ∨ ∨ (SΦ ˘ f )R  (SΦH )R , e f )R ,→ (SΦ

(1.2.4.19)

H

H

where the first morphism is defined by sX˘ and sY˘ , and where the second morphism is canonical, whose composition is zero, inducing morphisms P+ ˘ f  PΦH . e ,→ PΦ Φ Then τ˘ ⊂ P+ ˘ Φ

c H

(1.2.4.20)

H

f H

e ˘ having a face σ is a cone in the cone decomposition Σ ˘ Φf H

that is a ΓΦ˘ f -translation (see Definition 1.2.2.3) of the image of σ e ⊂ P+ e Φ H

f H

under the first morphism in (1.2.4.20). Without loss of generality, let us set σ ˘ to be the image of σ e ⊂ P+ e Φ

under the

f H

first morphism in (1.2.4.20), and consider the following: Definition 1.2.4.21. e˘ e˘ e+ (1) Σ ˘ Φ c,˘ σ = ΣΦ σ (resp. ΣΦ f,˘ H

σ c,˘ H

H

e+ =Σ ˘ Φ

σ f,˘ H

e ˘ consisting of ) is the subset of Σ Φf H

cones τ˘ ⊂ PΦ˘ f (resp. τ˘ ⊂ P+ ˘ (not just a ΓΦ˘ f -translation) as ˘ ) having σ Φ H

(2) (3) (4) (5)

f H

H

a face. ΓΦ˘ f,ΦH is the subgroup of ΓΦ˘ f stabilizing (both) X and Y . H H ΓΦ˘ f,ΦH ,˘σ is the subgroup of ΓΦ˘ f,ΦH stabilizing σ ˘. H H ΓΦ˘ c is the kernel of the canonical homomorphism ΓΦ˘ f,ΦH → ΓΦ e f (induced H H H by sX˘ and sY˘ ). ΓΦ˘ c,ΦH is the kernel of the canonical homomorphism ΓΦ˘ f,ΦH → H H ΓΦH × ΓΦ e f , which coincides with the kernel of the canonical homomorH phism ΓΦ˘ c → ΓΦH . H

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49

By definition, we have the following compatible exact sequences 1 → ΓΦ˘ c,ΦH → ΓΦ˘ f,ΦH → ΓΦH × ΓΦ ef

(1.2.4.22)

1 → ΓΦ˘ c,ΦH → ΓΦ˘ c → ΓΦH .

(1.2.4.23)

H

H

H

and H

H

Remark 1.2.4.24. The notation of ΓΦ˘ c and ΓΦ˘ c,ΦH in Definition 1.2.4.21 is jusH H b of Γ ˘ depend only on the group H tified because the subgroups Γ ˘ and Γ ˘ ΦH c,ΦH

ΦH c

ΦH f

e determined by H. Lemma 1.2.4.25. The subgroups ΓΦ˘ f,ΦH ,˘σ and ΓΦ˘ c of ΓΦ˘ f,ΦH are identical. H

H

H

Proof. This is because the image of ΓΦ˘ f,ΦH ,˘σ in ΓΦ e f is ΓΦ e f,e σ , which is trivial by H H H Conditions 1.2.2.9 and [62, Lem. 6.2.5.27]. ˘ e , δ˘ e , τ˘) of equivalence classes Corollary 1.2.4.26. For choosing representatives (Φ H H ˘ e ˘ e ˘, [(ΦHe , δHe , τ˘)] having [(ΦHe , δHe , σ e)] as a face (as above), for any given choices of Z e it suffices to take one τ˘ from each ˘ s ˘ , s ˘ , and δ˘ (compatible with e e and δ), Φ, Z, Φ, X Y + e ΓΦ˘ c -orbit in ΣΦ˘ ,˘σ . H

c H

˘ e , δ˘ e , τ˘)] haProof. This follows from the above review on equivalence classes [(Φ H H e e e + , and ving [(ΦHe , δHe , σ e)] as a face, from the very definitions of ΓΦ˘ f,ΦH ,˘σ and Σ ˘ ,˘ Φ σ H

from Lemma 1.2.4.25.

c H

˘ X e and s ˘ : Y˘  Ye identify Γ ˘ Lemma 1.2.4.27. The surjections sX˘ : X Y ΦH c,ΦH e Ye ) to φ(Y ), by e X), whose elements map φ( as a finite index subgroup of HomO (X, e X) induced to the element in HomO (X, sending each element (γ ˘ , γ ˘ ) ∈ Γ ˘ X

Y

ΦH c,ΦH

˘ → X (which contains X in its kernel). by γX˘ − IdX˘ : X e X) defined in the stateProof. The homomorphism from ΓΦ˘ c,ΦH to HomO (X, H ment of this lemma is injective because γX˘ − IdX˘ = 0 exactly when γX˘ = IdX˘ . e Ye ) to φ(Y ) because e X) induced by γ ˘ − Id ˘ maps φ( The element in HomO (X, X X e e e its restriction to φ(Y ) defines the element in HomO (Y , Y ) induced by γY˘ − IdY˘ . e X) induces an element γ ˘ ∈ GLO (X) ˘ with Conversely, any element fX˘ ∈ HomO (X, X e e image in X by setting γX˘ = IdX˘ +sX˘ ◦ fX˘ , any element also mapping φ(Y ) to φ(Y ) induces an element γY˘ ∈ GLO (Y˘ ) with image in Y˘ by setting γY˘ = IdY˘ +sY˘ ◦ fY˘ , e and φ. Since a sufficiently divisible where fY˘ ∈ HomO (Ye , Y ) is induced by fX˘ , φ, e Ye ) to φ(Y ), and since a sufficiently e multiple of any element of HomO (X, X) maps φ( high power of the element (γX˘ , γY˘ ) defined as above has trivial reduction modulo any prescribed integer (which means it can be made to be contained in ΓΦ˘ f,ΦH ), the H e X), recipe in the lemma identifies Γ ˘ as a finite index subgroup of HomO (X, as desired.

ΦH c,ΦH

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Remark 1.2.4.28. This group ΓΦ˘ c,ΦH here is the replacement of the group ΓΦ e f,ΦH H H in [61, Sec. 4A], which was incorrectly defined. (The rest of the arguments in [61] can be fixed with ΓΦ ˘ c,ΦH here.) e f,ΦH there replaced with the group ΓΦ H

H

Definition 1.2.4.29. We shall denote the kernel of the second morphism in b ˘ , so that the first morphism in (1.2.4.19) induces a canonical iso(1.2.4.18) by S ΦH c morphism ∨ ∨ b ˘ )∨ ∼ (S ˘ f )R /(SΦ e f )R . Φ c R = (SΦ H

H

(1.2.4.30)

H

˘ ⊗Q  X e ⊗ Q (over By choosing some (noncanonical) splitting of sX˘ ⊗ Q : X Z

Z

Z

Q), we can decompose the real vector space (SΦ˘ f )∨ R (noncanonically) as a direct H sum ∨ ∨ ∨ (SΦ ˘ c,ΦH )R ⊕(SΦH )R e f )R ⊕(ΓΦ

(1.2.4.31)

H

H

(defined over Q), on which the action of ΓΦ˘ c,ΦH is realized by its canonical transH lation action on the second factor. In particular, such a (noncanonical) splitting defines a projection pr(Sb ˘

∨ Φ c )R H

∨ ∨ ∼ b ∨ : (SΦ˘ f )∨ ˘ c,ΦH )R ⊕(SΦH )R = (SΦ ˘ c )R R → (ΓΦ H

H

H

(1.2.4.32)

(x, y, z) 7→ (y, z) (the intermediate morphisms are defined over Q, while the whole composition is defined over Z and independent of the choices of splittings, by Definition 1.2.4.29). Let b ˘ := pr b P Φc (S ˘

(PΦ˘ f )

(1.2.4.33)

b + := pr b P ˘ (S ˘ Φ

(P+ ˘ ). Φ

(1.2.4.34)

∨ Φ c )R H

H

H

and ∨ Φ c )R H

c H

f H

Lemma 1.2.4.35. The canonical morphisms PΦ˘ f  PΦH and P+  P+ ΦH (in˘ Φ H

f H

duced by the second morphism in (1.2.4.20)) factor through the canonical morphism pr(Sb ˘ )∨ in (1.2.4.32) and induce canonical morphisms Φc R H

b ˘  PΦ P H Φc

(1.2.4.36)

b +  P+ , P ΦH ˘ Φ

(1.2.4.37)

H

and c H

respectively. Proof. This is because the second morphism in (1.2.4.20) is unchanged under trans∨ lation by an element of (SΦ e f )R . H

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in (1.2.4.32), the image

Lemma 1.2.4.38. Under the projection pr(Sb ˘ pr(Sb ˘

∨ Φ c )R H

e˘ e+ (˘ τ ) of each τ˘ in Σ ˘ Φ f,˘ σ (resp. ΣΦ

σ f,˘ H

H

page 51

∨ Φ c )R H

) is a nondegenerate rational polyhedral

b + ). b ˘ (resp. P cone in P ˘ Φc Φ H

c H

Proof. Since σ e is a (nondegenerate) top-dimensional smooth rational polyhedral cone in P+ ˘ = , we can find a minimal subset {v1 , . . . , vr } of (SΦ˘ f )∨ R such that σ e Φ H

f H

∨ R>0 v1 + · · · + R>0 vr and Rv1 + · · · + Rvr = (SΦ e f )R (which we view as a subset H e˘ (which has σ ˘ as a face), there is a minimal of (S ˘ )∨ ). Then, for each τ˘ ∈ Σ ΦH σ f,˘

ΦH f R

subset {vr+1 , . . . , vr+s } of (SΦ˘ f )∨ R such that H

τ˘ = R>0 v1 + · · · + R>0 vr + R>0 vr+1 + · · · + R>0 vr+s . Moreover, we can write each x in the closure τ˘ of τ˘ as x = c1 v1 + · · · + cr+s vr+s , where the coordinates (c1 , . . . , cr+s ) ∈ Rr+s ≥0 are uniquely determined by x. Suppose there are x = c1 v1 + · · · + cr+s vr+s and y = d1 v1 + · · · + dr+s vr+s in τ˘ such that pr(Sb ˘ )∨ (x) + pr(Sb ˘ )∨ (y) = 0. Then x + y = e1 v1 + · · · + er vr for Φc R H

Φc R H

∨ some e1 , . . . , er ∈ R, because {v1 , . . . , vr } spans the kernel (SΦ b˘ e f )R of pr(S H

∨ Φ c )R H

. By

choosing e ∈ R≥0 such that e + ei ≥ 0 for all 1 ≤ i ≤ r, we obtain an identity of elements x + y + (ev1 + · · · + evr ) = (e + e1 )v1 + · · · + (e + er )vr in τ˘, and the nonnegative coordinates of both sides must coincide because of the choices of v1 , . . . vr+s . Hence, we must have cr+1 = · · · = cr+s = 0 and dr+1 = · · · = dr+s = 0 (because they are nonnegative). Thus, the image pr(Sb ˘ )∨ (˘ τ ) of τ˘ cannot contain Φc R H

any nonzero R-vector subspace; that is, it is a nondegenerate rational polyhedral b ˘ (see (1.2.4.33)). If τ˘ ∈ Σ b ˘ )∨ is contained in P b + (see e + , then (S cone in P ˘ ,˘ ˘ Φc Φc R Φ σ Φ H

H

f H

(1.2.4.34)).

c H

Lemma 1.2.4.39. There exists a continuous section ∨ ∨ ∨ ∨ x ˜0 : (ΓΦ˘ c,ΦH )∨ ˘ c,ΦH )R ⊕(SΦH )R e f )R ⊕(ΓΦ R ⊕(SΦH )R → (SΦ H

H

H

(y, z) 7→ (x0 (y, z), y, z) such that (˜ x0 ◦ pr(Sb ˘

∨ Φ c )R H

e˘ )(˘ τ ) ⊂ τ˘ for all τ˘ ∈ Σ Φ f,˘ σ. H

Proof. Let {v1 , . . . , vr } be as in the proof of Lemma 1.2.4.38. Then we can write the desired function x0 (y, z) as x0 (y, z) = x0,1 (y, z) v1 + x0,2 (y, z) v2 + · · · + x0,r (y, z) vr , ∨ where each x0,j ( · , · ) is a R-valued continuous function on (ΓΦ˘ c,ΦH )∨ R ⊕(SΦH )R . H e˘ For each τ˘ ∈ Σ ΦH σ , let {vr+1 , . . . , vr+s } be as in the proof of Lemma 1.2.4.38. f,˘ For each i = r + 1, . . . , r + s, let wi = pr(Sb ˘ )∨ (vi ) and write wi0 := vi − (0, wi ) Φc R H

as a linear combination wi0 = ci1 v1 + ci2 v2 + · · · + cir vr , where ci1 , . . . , cir ∈ R. ˘ ˘ ˘ (y, z) v1 + xτ0,2 (y, z) v2 + · · · + xτ0,r (y, z) vr to be linear on By taking xτ0˘ (y, z) = xτ0,1

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˘ R≥0 wr+1 + · · · + R≥0 wr+s and zero elsewhere, and by taking xτ0,j (y, z) to satisfy τ˘ τ˘ 0 τ˘ x0,j (wi ) > cij for all i and j, we have (x0 (wi ), wi ) = (x0 (wi ) − wi , 0, 0) + (wi0 , wi ) ∈ σ ˘ + vi for all i, and hence x ˜τ0˘ (y, z) := (xτ0˘ (y, z), y, z) satisfies (˜ xτ0˘ ◦ pr(Sb ˘ )∨ )(˘ τ ) ⊂ τ˘. Φc R H

˘ The same is true if we replace each xτ0,j (y, z) with a function with (pointwise) e˘ greater value. If τ˘ and τ˘0 are cones in Σ b ˘ )∨ above Φ f,˘ σ meeting some fiber of pr(S H

Φc R H

∨ (y, z) ∈ (ΓΦ˘ c,ΦH )∨ R ⊕(SΦH )R , then the above argument shows that there exists some H v ∈ τ˘ such that v + (0, y, z) ∈ τ˘ ∩ τ˘0 , forcing τ˘ = τ˘0 . Hence, there is at most one e˘ τ˘ in Σ b ˘ )∨ , and so we can take x0,j (y, z) to be any Φ f,˘ σ meeting each fiber of pr(S H

Φc R H

˘ e˘ continuous function (pointwise) greater than xτ0,j for all τ˘ ∈ Σ ΦH σ . Then we have f,˘ e˘ τ ) ⊂ τ˘ for all τ˘ ∈ Σ (˜ x0 ◦ pr(Sb ˘ )∨ )(˘ Φ f,˘ σ , as desired. H

Φc R H

Corollary 1.2.4.40. The set b ˘ = {pr b Σ Φc (S ˘

∨ Φ c )R H

H

(˘ τ )}τ˘∈Σ e˘

Φ f,˘ σ H

of rational polyhedral cones defines a ΓΦ˘ c -admissible rational polyhedral cone H decomposition (cf. Definition 1.2.2.4) of   b ˘ = pr b P ∪ pr(Sb ˘ )∨ (˘ τ) (1.2.4.41) ˘ f) = Φc (S ˘ )∨ (PΦ H

Φc R H

H

e˘ τ˘∈Σ Φ

Φc R H

σ f,˘ H

in the sense that we have the following: (1) Each pr(Sb ˘

∨ Φ c )R H

(˘ τ ) is a nondegenerate rational polyhedral cone.

b˘ . (2) The union (1.2.4.41) is disjoint and defines a stratification of P ΦH c b (3) ΣΦ˘ c is invariant under the action of ΓΦ˘ c in the sense that ΓΦ˘ c permutes H H H the cones in it. Under this action, the set of ΓΦ˘ c -orbits is finite. H

Proof. Statement (1) is Lemma 1.2.4.38. As for statement (2), suppose w ∈ e˘ pr(Sb ˘ )∨ (˘ τ ) ∩ pr(Sb ˘ )∨ (˘ τ 0 ) 6= ∅ for some τ˘, τ˘0 ∈ Σ Φ f,˘ σ . Then it was shown in Φc R H

H

Φc R H

the proof of Lemma 1.2.4.39 that τ˘ = τ˘0 . (Alternatively, any continuous section x ˜0 as in the statement of Lemma 1.2.4.39 defines an element x ˜0 (w) ∈ τ˘ ∩ τ˘0 , forcing that τ˘ = τ˘0 .) Hence, the union (1.2.4.41) is disjoint. Consequently, the incidence e˘ relations in {pr(Sb ˘ )∨ (˘ τ )}τ˘∈Σ inherits exactly those in Σ e˘ Φ f,˘ σ , and hence the Φc R H

H

Φ f,˘ σ H

union (1.2.4.41) defines a stratification (cf. (1) of Definition 1). Finally, statement e˘ (3) follows from the corresponding statement that ΓΦ˘ c acts on Σ ΦH σ with a finite f,˘ H number of orbits (cf. Corollary 1.2.4.26). b ˘ is relevant in the context, we shall denote elements pr b When only Σ Φc (S ˘

∨ Φ c )R H

H

(˘ τ) ∈

b ˘ by τb, without reference to the original Σ e˘ Σ Φc Φ f,˘ σ. H

H

b = {Σ b ˘ } ˘ ˘ , where [(Φ ˘ b , δ˘ b )] runs Lemma 1.2.4.42. The collection Σ ΦH H H c [(ΦH c,δH c)] e b e e through cusp labels at level H for (L, h · , · ie, h0 , Z) (i.e., equivalence classes of

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˘ compatible with (Φ, e as in Definition 1.2.4.17, b ˘ δ) e δ) H-orbits of representatives (Φ, ˘ and e with Z Z suppressed in the notation), defines a compatible choice of admissible smooth rational polyhedral cone decomposition data analogous to the notion for MH in Definition 1.2.2.13. There is an obvious notion of refinements for such collections, analogous to that in [62, Def. 6.4.2.8]. ˘ b , δ˘ b , τb)] and their facial Then we can also talk about equivalence classes [(Φ H H relations as in Definitions 1.2.2.10 and 1.2.2.19, and their refinements as in [62, Def. 6.4.3.1]. e = {Σ e˘ } ˘ ˘ (with Proof. This follows from the corresponding facts for Σ ΦH f [(ΦH f,δH f)] indices running through all cusp labels). Remark 1.2.4.43. Here we omit the precise definition of a compatible choice of admissible smooth rational polyhedral cone decomposition data because we can only b defined by some construct toroidal compactifications of Kuga families for those Σ e Σ and σ e. e ++ (see Definition 1.2.4.11) Definition 1.2.4.44. We say that two κ e1 and κ e2 in K Q,H b Σ). b In this case, we shall abusively are equivalent if they determine the same κ = (H, b Σ), b write κ = [e κ1 ] = [e κ2 ]. Then we take K++ to be the set of all such κ = (H, Q,H

b0 , Σ b 0 )  κ = (H, b Σ) b when H b0 ⊂ H b 0 and when Σ b 0 is with a partial order κ0 = (H + b (see Definition 1.2.4.42). We also take the subset K a refinement of Σ Q,H (resp. ++ e + (resp. K e Q,H ) of K e ++ under KQ,H ) of K to be the image of the subset K Q,H

Q,H

Q,H

e ++  K++ , with an induced partial order denoted by the canonical surjection K Q,H Q,H the same symbol . ˆ is induced by some b of G(Z) Lemma 1.2.4.45. Every neat open compact subgroup H ˆ as in Definition 1.2.4.4. Moreover, we may e of G( e Z) neat open compact subgroup H e satisfies Condition 1.2.4.7. assume that H e b ⊂ H. b Consider the preiProof. Consider any integer n ≥ 3 such that U(n) G + 0 ˆ ˆ U ˆ Then b b e e 0 (Z)/ e e(Z). mage H of H under the canonical homomorphism PeZ (Z)  P 2,Z e Z e b as in Definition 1.2.4.4, and satisfies Condition 1.2.4.7. e := H b + U(n) H induces H ˆ e b + and H b have the same Since elements of U e(Z) are unipotent, the elements in H 2,Z

e=H b + U(n) e eigenvalues (up to multiplicity). Since the elements of H are congruent + e b to elements of H modulo n by definition, H is neat by definition (see [89, 0.6] or [62, Def. 1.4.1.8]), and by Serre’s lemma that no nontrivial root of unity can be congruent to 1 modulo n if n ≥ 3. ˆ there exists b of G(Z), Lemma 1.2.4.46. For each neat open compact subgroup H ++ + b b b satisfies some element κ = (H, Σ) ∈ KQ,H , which lies in KQ,H (resp. KQ,H ) if H Condition 1.2.4.8 (resp. both Conditions 1.2.4.8 and 1.2.4.9).

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b is induced by some neat H e as in Definition 1.2.4.4, Proof. By Lemma 1.2.4.45, H which we assume to also satisfy Condition 1.2.4.7. By Proposition 1.2.2.17, there e e . Let us take Σ e for M b to be induced by Σ e as in exists some compatible choice Σ H ++ b b Lemma 1.2.4.42, and take κ = (H, Σ). Then, by definition, we have κ ∈ KQ,H . The remaining statements of the lemma also follow by definition. + Lemma 1.2.4.47. The partial order  among elements in K++ Q,H (resp. KQ,H , resp. b Σ) b and κ0 = (H b0 , Σ b 0 ), then KQ,H ) is directed; that is, if we are given two κ = (H, 00 00 b 00 00 00 0 b there exists some κ = (H , Σ ) such that κ  κ and κ  κ . Moreover, we can b 00 to be any open compact subgroup of H b∩H b 0 (which can be H b∩H b 0 itself). take H

Proof. Let us begin with the set K++ Q,H . b Σ) b = [(H, e Σ, e σ b0 , Σ b 0 ) = [(H e0 , Σ e 0, σ Suppose κ = (H, e)] and κ0 = (H e0 )] are in K++ Q,H , ++ 0 0 0 0 e e Σ, e σ e ,Σ e ,σ b = H e b and H b = H e0 . where (H, e) and (H e ) are in K , so that H Q,H

G

b G

b 00 be any open compact subgroup of H b∩H b 0 (which can be H b∩H b 0 itself). Let H Then, as in the proof of Lemma 1.2.4.45, by choosing some integer n ≥ 3 such e b ⊂H b 00 and U(n) e e∩H e 0 , and by taking H b 00,+ to be the preimage of that U(n) ⊂H G 00 0 ˆ 0 ˆ ˆ we obtain a neat b e e e e(Z), H under the canonical homomorphism PeZ (Z)  PeZ (Z)/U 2,Z e 00 = U(n) e H b 00,+ of H e∩H e 0 satisfying Condition 1.2.4.7 open compact subgroup H 00 00 e b (with H = H ). b G

b (Hb 00 ) (resp. Σ b 0,(Hb 00 ) ) denote the collection induced by Σ b (resp. Σ b 0 ) at leLet Σ 00 b , as in [62, Constr. 7.3.1.6]. By definition, it is also induced by the colvel H e (He 00 ) (resp. Σ e 0,(He 00 ) ) induced by Σ e (resp. Σ e 0 ) at level H e 00 . Let Σ b 00 ,pre be lection Σ 00 00 b b b (H ) and Σ b 0,(H ) (which might not be determiany common refinement of both Σ ++ 00 e 00 00 e e b 00 ,pre of Σ b (Hb 00 ) defines (by ned by some (H , Σ , σ e ) in K ). The refinement Σ Q,H

taking preimages) certain subdivisions of cones in (the cone decompositions in) e (He 00 ) , which can be further subdivided into a projective smooth refinement Σ e 00 Σ e 00 ) (H 00 00 00 e e of Σ . Let σ e be a top-dimensional cone in Σ such that σ e ⊂ σ e, and let b 00 , Σ b 00 ) := [(H e 00 , Σ e 00 , σ b 00 is a refinement of Σ b 00 ,pre (and hence of κ00 := (H e00 )]. Then Σ b (Hb 00 ) and Σ b 0,(Hb 00 ) ). Thus, we have defined an element κ00 in K++ satisfying both Σ Q,H both κ00  κ and κ00  κ0 , as desired. Then the cases for the sets K+ Q,H and KQ,H also follow, because Condition 1.2.4.8 b and so is the condition (equivalence to is clearly compatible with intersections in H; ˆ ˆ defined by δe maps H to H. b Z) b Condition 1.2.4.9) that the splitting G(Z) → G( Now we consider some compatibility conditions between a collection Σ for MH e ++ or K++ . and elements of K Q,H Q,H e ++ : e Σ, e σ First consider the following condition on an element κ e = (H, e) in K Q,H ˘ e , δ˘ e , τ˘) as Condition 1.2.4.48. (Compare with [61, Cond. 3.8].) For each (Φ H H + e e above, where τ˘ ⊂ PΦ˘ is a cone in the cone decomposition ΣΦ˘ f (in Σ) having σ ˘ as f H

H

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a face, the image of τ˘ in PΦH under the (canonical) second morphism in (1.2.4.20) is contained in some cone τ ⊂ P+ ΦH in the cone decomposition ΣΦH (in Σ). By Lemma 1.2.4.35, if κ = [e κ] ∈ K++ e, then Q,H is the element determined by κ Condition 1.2.4.48 for κ e is equivalent to the following condition for κ: b˘ Condition 1.2.4.49. (Compare with [28, Ch. VI, Def. 1.3].) For each τb ∈ Σ ΦH c ˘ ˘ b in (˘ τ ) for some ( Φ , δ , τ ˘ ) is in the cone decomposition Σ (where τb = pr b ∨ ˘ e e H

(SΦ )R ˘ c H

H

ΦH c

P+ ΦH

b the image of τb in Σ), under (1.2.4.37) is contained in some cone τ ⊂ P+ ΦH in the cone decomposition ΣΦH (in Σ). Definition 1.2.4.50. For ? = ++, +, or ∅, let us take K?Q,H,Σ to be the subset of K?Q,H consisting of elements κ satisfying Condition 1.2.4.49. b with a refinement Since Condition 1.2.4.49 can be achieved by replacing any given Σ ? (in the same set), we see that each KQ,H,Σ is nonempty and has an induced directed partial order. Remark 1.2.4.51. (Compare with [61, Rem. 3.10].) Condition 1.2.4.49 is analogous to the condition in [89, 6.25(b)], which is used in, for example, [40, Lem. 1.6.5] and other related works based on [4]. ˆ Proposition 1.2.4.52. Suppose H is any open compact subgroup of G(Z). For ? ?1 1 e each ?1 = ++, +, or ∅, and for each ?2 = Σ or ∅, the sets K and K Q,H Q,H,?2 are nonempty and compatible with each other under the various canonical maps. 1 Common refinements for finite subsets exist in any sets of the form K?Q,H,? . We 2 may allow varying levels or twists by Hecke actions when doing so, and we may b Σ), b we may vary ?1 and ?2 as well (in any order). For any such refinement κ = (H, ˆ in the context, we b to be any allowed open compact subgroup of G( b Z) prescribe H 0 b b b to may require Σ to be finer than any cone decomposition Σ , and we may require Σ ∞ b be invariant under any choice of an open compact subgroup of G(A ) normalizing b H. Proof. These follow from the corresponding existence and refinement statements f for M e e. e and pol in Proposition 1.2.2.17 for collections Σ H For later references, let us conclude with the following definitions: ˘ be Definition 1.2.4.53. (Compare with Definitions 1.2.1.11 and 1.2.4.3.) Let Z ˆ e ⊗ Z satisfying (1.2.4.12). For each any fully symplectic admissible filtration of L Z

ˆ e 0 (R) ⊂ P e 0 (R) and U e e(R) ⊂ U e ˘ (R), we define the following Z-algebra R, since P 2,Z ˘ 2,Z e Z Z b e e(R) = P e 0 (R)/U e e(R): quotient of subgroups of G(R) =G 1,Z

b ˘ (R) := (P e ˘ (R) ∩ P e 0 (R))/U e e(R). (1) P Z Z 2,Z e Z b˘ (R) := (Z e˘ (R) ∩ P e 0 (R))/U e e(R). (2) Z Z

Z

e Z

2,Z

e Z

2,Z

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b ˘ (R) := P b ˘ (R)/Z b˘ (R) ∼ e ˘ (R) ∼ e ˘ (R)/Z e˘ (R). G =G =P h,Z Z Z h,Z Z Z b ˘ (R) := U e ˘ (R)/U e e(R). U Z Z 2,Z b ˘ (R) := U e ˘ (R)/U e e(R). U 2,Z 2,Z 2,Z b e ˘ (R). b b e ˘ (R) = U e ˘ (R)/U U1,˘Z (R) := U˘Z (R)/U2,˘Z (R) ∼ =U Z 2,Z 1,Z b ˘ (R) := Z b˘ (R)/U b ˘ (R) ∼ e˘ (R) ∩ P e 0 (R))/U e ˘ (R). G = (Z l,Z Z Z Z Z e Z b 0 (R) := P e 0 (R)/U e e(R). (8) P ˘ ˘ 2, Z Z Z ∼G b ˘ (R) := P b 0 (R)/U b ˘ (R) = e ˘ (R) = P e 0 (R)/U e ˘ (R). (9) G 1,Z 2,Z 1,Z 2,Z ˘ ˘ Z Z 0 0 0 0 ∼ b b b e e e (10) Gh,˘Z (R) := P˘Z (R)/U˘Z (R) = Gh,˘Z (R) = P˘Z (R)/U˘Z (R). b 0 (R) := P b ˘ (R)/P b 0 (R) ∼ e ˘ (R) ∩ P e 0 (R))/P e 0 (R). (11) G = (P (3) (4) (5) (6) (7)

l,˘ Z

Z

˘ Z

Z

˘ Z

e Z

b Then the canonical homomorphism G(R) → G(R) induces the following canonical homomorphisms: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

b ˘ (R) → PZ (R). P Z b˘ (R) → ZZ (R). Z Z b ˘ (R) → Gh,Z (R). G h,Z b ˘ (R) → UZ (R). U Z b ˘ (R) → U2,Z (R). U 2,Z b ˘ (R) → U1,Z (R). U 1,Z b ˘ (R) → Gl,Z (R). G l,Z b 0 (R) → P0 (R). P Z ˘ Z b ˘ (R) → G1,Z (R). G 1,Z b 0 (R) → G0 (R). G h,˘ Z

h,Z

b 0 (R) → G0 (R). (11) G l,Z l,˘ Z b b := (H ee ∩ H e e 0 )/H ee Hence, it makes sense to define H P˘ P˘ P U Z

Z

e Z

2,e Z

b=H e b , so etc when H G

b b → HP etc when H bG ⊂ H. that we have H Z P˘ Z

Definition 1.2.4.54. With the setting as in Definition 1.2.4.53, consider e ˘ e(R) := P e ˘ (R) ∩ P ee(R), P Z Z,Z Z and define e ˘ e(R) := P e ˘ e(R)/P e ˘0 (R), G Z,Z l,Z,Z Z e ˘ (R) consisting of elements preserving the filtrations which is the subgroup of G l,Z ˘ X e and s ˘ : Y˘  Ye . Let induced by the admissible surjections sX˘ : X Y ˆ e e := (H e∩P e ˘ e(Z)) H Z,Z P˘ e Z,Z

and ee H G

l,˘ Z,e Z

e e /H e e0 . := H P˘ e P Z ,Z

˘ Z

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Then there are canonical homomorphisms e ˘ e(R)/U e ˘ (R) → PZ (R)/U2,Z (R) P 2,Z Z,Z and e e /H ee inducing H P˘ e U Z,Z

1.3

2,˘ Z

e ˘ e(R) → G0 (R) = PZ (R)/P0 (R), G l,Z Z l,Z,Z ee e 0 → HPZ /HU2,Z and H → H Gl,Z = HPZ /HP0Z when HG ⊂ H. G ˘e l,Z,Z

Algebraic Compactifications in Characteristic Zero

By algebraic compactifications, we mean compactifications as algebraic varieties, algebraic spaces, or algebraic stacks constructed using the (algebraic) theory of degeneration developed in [82], [28], and [62]. (We do not consider the constructions in [89] and [38] algebraic, because they are based on the analytic construction in [4] and on the theory of canonical models.) 1.3.1

Toroidal and Minimal Compactifications of PEL-Type Moduli Problems

Definition 1.3.1.1. (See [62, Def. 5.3.2.1].) Let S be a normal locally noetherian algebraic stack. A tuple (G, λ, i, αH ) over S is called a degenerating family of type MH , or simply a degenerating family when the context is clear, if there exists a dense subalgebraic stack S1 of S, such that S1 is defined over S0 = Spec(F0 ), and such that we have the following: (1) By viewing group schemes as relative schemes (cf. [37]), G is a semi-abelian scheme over S whose restriction GS1 to S1 is an abelian scheme. In this case, the dual semi-abelian scheme G∨ exists (up to unique isomorphism; cf. [80, IV, 7.1] or [62, Thm. 3.4.3.2]), whose restriction G∨ S1 to S1 is the dual abelian scheme of GS1 . (2) λ : G → G∨ is a group homomorphism that induces by restriction a polarization λS1 of GS1 . (3) i : O → EndS (G) is a homomorphism that defines by restriction an O-structure iS1 : O → EndS1 (GS1 ) of (GS1 , λS1 ). (4) (GS1 , λS1 , iS1 , αH ) → S1 defines a tuple parameterized by the moduli problem MH . Definition 1.3.1.2. (See [62, Def. 6.3.1].) Let (G, λ, i, αH ) be a degenerating family of type MH over S (as in Definition 1.3.1.1 ) over S0 = Spec(F0 ). Let ∨ ∗ 1 ∗ 1 Lie∨ G/S := eG ΩG/S be the dual of LieG/S , and let LieG∨ /S := eG ΩG∨ /S be the dual of LieG∨ /S . Note that λ : G → G∨ induces an O-equivariant morphism λ∗ : ∨ ∨ Lie∨ G∨ /S → LieG/S . (Here the O-action on LieG/S is a left action after twisted by ? the involution .) Then we define the OS -module KS = KS(G,λ,i)/S = KS(G,λ,i,αH )/S by setting   ∗ λ (y) ⊗ z − λ∗ (z) ⊗ y ∨ ∨ KS := (LieG/S ⊗ LieG∨ /S )/ . x∈Lie∨ G/S , (b? x) ⊗ y − x ⊗(by) OS ∨ y,z∈LieG∨ /S , b∈O

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Analogues of the OS -module KS appear naturally in the deformation theory of abelian varieties with PEL structures (without degenerations). The point of Definition 1.3.1.2 is that it extends the conventional definition (for abelian schemes with PEL structures) to the context of (semi-abelian) degenerating families (see Definition 1.3.1.1). The algebraically constructed toroidal compactifications in characteristic zero can be described as follows: Theorem 1.3.1.3. (See [62, Thm. 6.4.1.1].) To each compatible choice Σ = {ΣΦH }[(ΦH ,δH )] of admissible smooth rational polyhedral cone decomposition data as tor in Definition 1.2.2.13, there is associated an algebraic stack Mtor H = MH,Σ proper and smooth over S0 = Spec(F0 ), which is an algebraic space when H is neat (see [89, 0.6] or [62, Def. 1.4.1.8]), containing MH as an open dense subalgebraic stack, together with a degenerating family (G, λ, i, αH ) over Mtor H (as in Definition 1.3.1.1) such that we have the following: (1) The restriction (GMH , λMH , iMH , αH ) of (G, λ, i, αH ) to MH is the tautological tuple over MH . (2) Mtor H has a stratification by locally closed subalgebraic stacks a Mtor Z[(ΦH ,δH ,σ)] , H = [(ΦH ,δH ,σ)]

with [(ΦH , δH , σ)] running through a complete set of equivalence classes of (ΦH , δH , σ) (as in Definition 1.2.2.10) with σ ⊂ P+ ΦH and σ ∈ ΣΦH ∈ Σ. (Here ZH is suppressed in the notation by our convention. The notation ` “ ” only means a set-theoretic disjoint union. The algebro-geometric structure is still that of Mtor H .) 0 0 ,σ 0 )] lies in the cloIn this stratification, the [(Φ0H , δH , σ 0 )]-stratum Z[(Φ0H ,δH sure of the [(ΦH , δH , σ)]-stratum Z[(ΦH ,δH ,σ)] if and only if [(ΦH , δH , σ)] is a 0 , σ 0 )] as in Definition 1.2.2.19 (see also [62, Rem. 6.3.2.15]). face of [(Φ0H , δH The [(ΦH , δH , σ)]-stratum Z[(ΦH ,δH ,σ)] is smooth over S0 and isomorphic to the support of the formal algebraic stack XΦH ,δH ,σ /ΓΦH ,σ for every representative (ΦH , δH , σ) of [(ΦH , δH , σ)], where the formal algebraic stack XΦH ,δH ,σ (before quotient by ΓΦH ,σ , the subgroup of ΓΦH formed by elements mapping σ to itself; see [62, Def. 6.2.5.23]) admits a canonical structure as the completion of an affine toroidal embedding ΞΦH ,δH (σ) (along its σ-stratum ΞΦH ,δH ,σ ) of a torus torsor ΞΦH ,δH over an abelian scheme torsor ZH H CΦH ,δH over a finite ´etale cover MΦ H of the algebraic stack MH (separated, smooth, and of finite over S0 ) in Definition 1.2.1.15. (Note that ZH and the isomorphism class of MZHH depend only on the cusp label [(ZH , ΦH , δH )], but not on the choice of the representative (ZH , ΦH , δH ).) In particular, MH is an open dense stratum in this stratification. (3) The complement of MH in Mtor H (with its reduced structure) is a relative Cartier divisor D∞,H with normal crossings, such that each irreducible com-

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ponent of a stratum of Mtor H − MH is open dense in an intersection of irreducible components of D∞,H (including possible self-intersections). When H is neat, the irreducible components of D∞,H have no self-intersections (cf. Condition 1.2.2.9, [62, Rem. 6.2.5.26], and [28, Ch. IV, Rem. 5.8(a)]). (4) The extended Kodaira–Spencer morphism [62, Def. 4.6.3.44] for G → Mtor H induces an isomorphism ∼

→ Ω1Mtor /S0 [d log ∞] KSG/Mtor : KS(G,λ,i)/Mtor H /S0 H H

(see Definition 1.3.1.2). Here the sheaf Ω1Mtor /S0 [d log ∞] is the sheaf of H modules of log 1-differentials on Mtor H over S0 , with respect to the relative Cartier divisor D∞,H with normal crossings. (5) For every representative (ΦH , δH , σ) of [(ΦH , δH , σ)], the formal completion ∧ tor (Mtor H )Z[(ΦH ,δH ,σ)] of MH along the [(ΦH , δH , σ)]-stratum Z[(ΦH ,δH ,σ)] is canonically isomorphic to the formal algebraic stack XΦH ,δH ,σ /ΓΦH ,σ . (To form the formal completion along a given locally closed stratum, we first remove the other strata appearing in the closure of this stratum from the total space, and then form the formal completion of the remaining space along this stratum.) This isomorphism respects stratifications in the sense that, given any ´etale (i.e., formally ´etale and of finite type; see [35, I, 10.13.3]) morphism Spf(R, I) → XΦH ,δH ,σ /ΓΦH ,σ inducing a morphism Spec(R) → ΞΦH ,δH (σ)/ΓΦH ,σ , the stratification of Spec(R) (inherited from ΞΦH ,δH (σ)/ΓΦH ,σ ; see [62, Prop. 6.3.1.6 and Def. 6.3.2.16]) makes the induced morphism Spec(R) → Mtor H a strata-preserving morphism. The pullback of the degenerating family (G, λ, i, αH ) over Mtor H ∧ ♥ ♥ ♥ ♥ to (Mtor H )Z[(ΦH ,δH ,σ)] is the Mumford family ( G, λ, i, αH ) over XΦH ,δH ,σ /ΓΦH ,σ (see [62, Sec. 6.2.5]) after we identify the bases using the isomorphism. (Here both the pullback of (G, λ, i, αH ) and the Mumford family ( ♥ G, ♥ λ, ♥ i, ♥ αH ) are considered as relative schemes with additional structures; cf. [37].) (6) Let S be an irreducible noetherian normal scheme over S0 , and suppose † that we have a degenerating family (G† , λ† , i† , αH ) of type MH over S † † † † as in Definition 1.3.1.1. Then (G , λ , i , αH ) → S is the pullback of tor (G, λ, i, αH ) → Mtor H via a (necessarily unique) morphism S → MH (over S0 ) if and only if the following condition is satisfied at each geometric point s¯ of S: Consider any dominant morphism Spec(V ) → S centered at s¯, where V is a complete discrete valuation ring with fraction field K, algebraically ‡ closed residue field k, and discrete valuation υ. Let (G‡ , λ‡ , i‡ , αH ) → † † † † Spec(V ) be the pullback of (G , λ , i , αH ) → S. This pullback family defines an object of DEGPEL,MH (V ), which corresponds to a tu\,‡ ple (B ‡ , λB ‡ , iB ‡ , X ‡ , Y ‡ , φ‡ , c‡ , c∨,‡ , τ ‡ , [αH ]) in DDPEL,MH (V ) under [62,

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Thm. 5.3.1.19]. Then we have a fully symplectic-liftable admissible filtra\,‡ tion Z‡H determined by [αH ]. Moreover, the ´etale sheaves X ‡ and Y ‡ are necessarily constant, because the base ring V is strict local. Hence, it makes sense to say we also have a uniquely determined torus argument Φ‡H at level H for Z‡H . On the other hand, we have objects ΦH (G‡ ), SΦH (G‡ ) , and B(G‡ ) (see [62, Constr. 6.3.1.1]), which define objects Φ‡H , SΦ‡ , and in particular H

B ‡ : SΦ‡ → Inv(V ) over the special fiber. Then υ ◦ B ‡ : SΦ‡ → Z defines H H , where υ : Inv(V ) → Z is the homomorphism induced an element of S∨ Φ‡ H

by the discrete valuation of V . Then the condition is that, for each Spec(V ) → S as above (centered at ‡ s¯), and for some (and hence every) choice of δH , there is a cone σ ‡ in the cone decomposition ΣΦ‡ of PΦ‡ (given by the choice of Σ; cf. Definition H

H

1.2.2.13) such that σ ‡ contains all υ ◦ B ‡ obtained in this way. Statement (1) means the tautological tuple over MH extends to a degenerating tor family (G, λ, i, αH ) over Mtor H . (Since MH is noetherian normal, this extension is unique up to unique isomorphism, by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5].) Statements (2), (3), (4), and (5) are self-explanatory. Statement (6) can be interpreted as a universal property for the degenerating family (G, λ, i, αH ) → Mtor H among degenerating families over normal locally noetherian bases, as in Definition 1.3.1.1, satisfying moreover some conditions describing the degeneration patterns over pullbacks to complete discrete valuation rings with algebraically closed residue fields. (This universal property was crucially used in the proof of Theorem 1.3.3.15 below in [61].) Remark 1.3.1.4. (Compare with Remark 1.1.2.1.) If we have chosen another PEL-type O-lattice L0 in L ⊗ Q which is also stabilized by H, so that MH carries Z

the corresponding abelian scheme A0 (with additional structures) as in Remark 1.1.2.1, with a Q× -isogeny f : A → A0 , then a sufficiently divisible multiple N f of f is an isogeny with finite ´etale kernel, which we denote by K. Since A = GMH (see (1) of Theorem 1.3.1.3), we can take the schematic-closure K ext of K in G, which is 0 ext quasi-finite ´etale over Mtor by [62, H,Σ . Then we can form the quotient G := G/K × ext Lem. 3.4.3.1], which is a semi-abelian scheme with an Q -isogeny f : G → G0 . By [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5], G0 is (up to unique isomorphism) independent of the choice of N , and the additional structures λ, i, 0 αH of G naturally induce the additional structures λ0 , i0 , and αH of G0 , which 0 extend those of A (based on the moduli interpretation of the moduli problem M0H defined by L0 ). Hence, the Q× -isogeny class of G extends that of A, and carries 0 well-defined additional structures. (It can be verified that (G0 , λ0 , i0 , αH ) → Mtor H,Σ satisfies the corresponding universal property of the toroidal compactification of M0H defined by the corresponding collection of cone decompositions as in (6) of

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Theorem 1.3.1.3, so that the theory does not really depend on the choice of L. Then, as in Remark 1.1.2.1, we can define the collection {Mtor H,Σ }H,Σ , indexed by all open compact subgroups H of G(A∞ ) and collections Σ for the corresponding MH , with a canonical action of G(A∞ ); see Proposition 1.3.1.15 below.) The algebraically constructed minimal compactifications in characteristic zero can be described as follows: Theorem 1.3.1.5. (See [62, Thm. 7.2.4.1].) There exists a normal scheme Mmin H projective and flat over S0 = Spec(F0 ), such that we have the following: (1) Mmin H contains the coarse moduli space [MH ] of MH (see [62, Sec. A.7.5]) as an open dense subscheme. (2) Let (GMH , λMH , iMH , αH ) be the tautological tuple over MH . Let us define top ∗ eGM Ω1GM /MH over the invertible sheaf ωMH := ∧top Lie∨ GM /MH = ∧ H

H

H

⊗ N0 MH . Then there is a smallest integer N0 ≥ 1 such that ωM is the pullback H min of an ample invertible sheaf O(1) over MH . If H is neat, then MH → [MH ] is an isomorphism, and induces an embedding of MH as an open dense subscheme of Mmin H . Moreover, we have N0 = 1 with a canonical choice of O(1), and the restriction of O(1) to MH is isomorphic to ωMH . We shall denote O(1) by ωMmin , and interpret it as H min an extension of ωMH to MH . By abuse of notation, for each integer k divisible by N0 , we shall denote ⊗k O(1)⊗ k/N0 by ωM min , even when ωMmin itself is not defined. H H

(3) For each (smooth) arithmetic toroidal compactification Mtor H of MH as in Theorem 1.3.1.3, with a degenerating family (G, λ, i, αH ) over Mtor H extending the tautological tuple (GMH , λMH , iMH , αH ) over MH , let ωMtor := H ∨ top ∗ 1 tor top ∧ LieG/Mtor = ∧ eG ΩG/Mtor be the invertible sheaf over MH extending H

H

⊗k ωMH naturally. Then the graded algebra ⊕ Γ(Mtor H , ωMtor ), with its natural k≥0

H

algebra structure induced by tensor products, is finitely generated over F0 , and is independent of the choice (of the Σ used in the definition) of Mtor H . min The normal scheme  MH (projective  and flat over S0 ) is canonically iso-

⊗k morphic to Proj ⊕ Γ(Mtor ) , and there is a canonical morphism H , ωMtor H k≥0 H min : Mtor determined by ωMtor and the universal property of Proj, H → MH H H H∗ ⊗ N0 tor ∼ such that H O(1) = ωMtor over MH , and such that the canonical morH H tor is an isomorphism. Moreover, when we vary phism OMmin → O M H,∗ H H H the choices of Mtor H ’s, the morphisms H ’s are compatible with the canonical morphisms among the Mtor HH’s as in [62, Prop. 6.4.2.3]. H ∗ ∼ ∼ When H is neat, we have H ωMmin and H,∗ ωMtor . = ωMtor = ωMmin H H H H min (4) MH has a natural stratification by locally closed subschemes a Mmin Z[(ΦH ,δH )] , H = [(ΦH ,δH )]

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with [(ΦH , δH )] running through a complete set of cusp labels (see Definition 0 0 )] 1.2.1.7 and [62, Def. 5.4.2.4]), such that the [(Φ0H , δH )]-stratum Z[(Φ0H ,δH lies in the closure of the [(ΦH , δH )]-stratum Z[(ΦH ,δH )] if and only if there is 0 a surjection from the cusp label [(Φ0H , δH )] to the cusp label [(ΦH , δH )] as in ` Definition 1.2.1.18. (The notation “ ” only means a set-theoretic disjoint union. The algebro-geometric structure is still that of Mmin H .) Each [(ΦH , δH )]-stratum Z[(ΦH ,δH )] is canonically isomorphic to the coarse moduli space [MZHH ] (which is a scheme) of the corresponding algebraic stack MZHH (separated, smooth, and of finite type over S0 ) as in Definition 1.2.1.15. Let us define the O-multi-rank of a stratum Z[(ΦH ,δH )] to be the O-multirank of the cusp label represented by (ΦH , δH ) (see [62, Def. 5.4.2.7]). The only stratum with O-multi-rank zero is the open stratum Z[(0,0)] ∼ = [MH ], and those strata Z[(ΦH ,δH )] with nonzero O-multi-ranks are called cusps. (This explains the Hname of the cusp labels.) (5) The restriction of H to the stratum Z[(ΦH ,δH ,σ)] of Mtor H is a surjection to the stratum Z[(ΦH ,δH )] of Mmin . This surjection is smooth when H is neat, H ∨ and is proper if σ is top-dimensional in P+ ⊂ (S ) . ΦH R ΦH ∼ Under the above-mentioned identification [MZHH ] → Z[(ΦH ,δH )] on the target, this surjection can be viewed as the quotient by ΓΦH ,σ (see [62, Def. 6.2.5.23]) of a torsor under a torus EΦH ,σ over an abelian scheme torH sor CΦH ,δH (see Remark 1.3.1.6 below) over the finite ´etale cover MΦ H ZH ZH of the algebraic stack MH over the coarse moduli space [MH ] (which is a scheme). More precisely, this torus EΦH ,σ is the quotient of the torus EΦH := HomZ (SΦH , Gm ) corresponding to the subgroup SΦH ,σ := {` ∈ SΦH : h`, yi = 0, ∀y ∈ σ} of SΦH . (See [62, Lem. 6.2.4.4] for the definition of EΦH , and see [62, Def. 6.1.2.7] for the definition of σ-stratum.) Remark 1.3.1.6. In [62, Sec. 6.2.4; see also the errata], we should have considered b is a subquotient of Hn which is an extension of Hn,Gess by Hn,Uess (just as H 1,Zn h,Zn bG by H b b in (1.2.4.5)), which is not necessarily the semi-direct an extension of H U

ess product Hn,Gess = Hn,Gess n Hn,Uess used there. Therefore, it is incorrect 1,Zn h,Zn nU1,Zn h,Zn

H to conclude from [62, Lem. 6.2.4.5] that the further quotient CΦH ,δH → MΦ H is also an abelian scheme. (The identity section might not descend under the quotient by Hn,Uess .) Accordingly, in [62, Prop. 6.2.4.7 and later sections], we should only 1,Zn H assert that CΦH ,δH → MΦ H is an abelian scheme torsor. This does not affect the constructions of torus torsors and toroidal embeddings because the existence of identity sections is not logically necessary.

Now suppose H is neat. Let Σ = {ΣΦH }[(ΦH ,δH )] be any projective compatible choice of smooth rational polyhedral cone decomposition data, with a compatible collection pol = {polΦH }[(ΦH ,δH )] of polarization functions as in Definition 1.2.2.14.

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tor Definition 1.3.1.7. (See [62, Def. 7.3.3.1].) Let Σ, pol, and Mtor H = MH,Σ be as tor above. By (3) of Theorem 1.3.1.3, the complement D∞,H of MH in MH = Mtor H,Σ (with its reduced structure) is a relative Cartier divisor with normal crossings, each of whose irreducible components is an irreducible component of some Z[(ΦH ,δH ,σ)] that is the closure of some strata Z[(ΦH ,δH ,σ)] labeled by the equivalence class [(ΦH , δH , σ)] of some triple (ΦH , δH , σ) with σ a one-dimensional cone in the cone decomposition ΣΦH of PΦH . Let H,pol be the invertible sheaf of ideals over Mtor H supported on D∞,H such that the order of H,pol along each Z[(ΦH ,δH ,σ)] is the value of polΦH at the Z>0 -generator of σ ∩ S∨ ΦH for some (and hence every) representative (ΦH , δH , σ). This is well defined because of the compatibility condition for pol = {polΦH }[(ΦH ,δH )] as in Definition 1.2.2.14. For each integer d ≥ 1, let dpol denote the collection of polarization functions defined by multiplying all polarization functions in the collection pol by d. Then we d have a canonical isomorphism H,dpol ∼ = ⊗ H,pol .

Definition 1.3.1.8. For each integer d ≥ 1, let H H (d) d ∼ JH,dpol := JH,pol := H,∗ (⊗ H,pol ) = H,∗ (H,dpol ), H min where H : Mtor H → MH is the canonical morphism (as in (3) of Theorem 1.3.1.5). H (d) d (We introduced the two intermediate objects JH,pol and H,∗ (⊗ H,pol ) because this is what was done in [28, Ch. V] and [62, Sec. 7.3]. Later we will mainly use JH,dpolH and H,dpol in our exposition. Note that JH,dpol is a coherent OMmin -ideal because H is H H tor proper and because the canonical morphism OMmin → O is an isomorphism.) H,∗ MH H Let us introduce the following condition for Σ = {ΣΦH }[(ΦH ,δH )] and pol = {polΦH }[(ΦH ,δH )] (cf. [62, Lem. 7.3.1.7]): Condition 1.3.1.9. (See [62, Cond. 7.3.3.3]; cf. [4, Ch. IV, Sec. 2, p. 329] and [28, Ch. V, Sec. 5, p. 178].) For each representative (ΦH , δH ) of cusp label and each ∨ vertex `0 of Kpol corresponding to a top-dimensional cone σ0 , we have Φ H

h`0 , xi < hγ · `0 , xi for all x ∈ σ 0 ∩ P+ ΦH and all γ ∈ ΓΦH such that γ 6= 1. Theorem 1.3.1.10. (See [62, Thm. 7.3.3.4]; cf. [4, Ch. IV, Sec. 2.1, Thm.] and [28, Ch. V, Thm. 5.8].) Suppose H is neat, and suppose Σ is projective with a compatible collection pol of polarization functions as in Definition 1.2.2.14. For tor each integer d ≥ 1, suppose H,dpol is defined over Mtor H = MH,Σ as in Definition 1.3.1.7, and suppose JH,dpol is defined over Mmin as in Definition 1.3.1.8. Then H there exists an integer d0 ≥ 1 such that the following are true: H −1 (1) The canonical morphism H JH,d0 pol · OMtor → H,d0 pol of coherent H OMtor -ideals is an isomorphism, which induces a canonical morphism H H min NBlJH,d0 pol ( H ) : Mtor H → NBlJH,d0 pol (MH )

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by the universal property of the normalization of blow-up (see [62, Def. 7.3.2.2]). (Here NBl · ( · ) denotes the normalization of the blow-up, or the morphism induced by its universal property.) H (2) The canonical morphism NBlJH,d0 pol ( H ) above is an isomorphism. In particular, Mtor H is a scheme projective (and smooth) over S0 . If Condition 1.3.1.9 is satisfied, then the above two statements are true for all d0 ≥ 3. For technical reasons, we shall enlarge the collection of smooth toroidal compactifications we have in Theorem 1.3.1.3 to the following setup, including certain projective but nonsmooth toroidal compactifications. Proposition 1.3.1.11. With assumptions as in Theorem 1.3.1.10, suppose H0 is an open compact subgroup of H, with Σ0 (resp. pol0 ) at level H0 induced by Σ (resp. pol) as in [62, Constr. 7.3.1.6]. (Note that Σ0 is not necessarily smooth.) For each integer d ≥ 1, let min ∗ JH0 ,dpol0 := (Mmin H0 → MH ) JH,dpol .

Suppose d0 ≥ 1 is any integer such that the statements in Theorem 1.3.1.10 are true. Then we define min Mtor H0 ,d0 pol0 := NBlJH0 ,d0 pol0 (MH0 ).

With this definition, there is a canonical morphism tor ∼ tor Mtor H0 ,d0 pol0 → MH,Σ = MH,d0 pol

(1.3.1.12)

which is finite. Moreover, Mtor H0 ,d0 pol0 is canonically isomorphic to the normalization 0 of Mtor in M under the composition of canonical morphisms MH0 → MH ,→ H H,d0 pol tor ∼ Mtor , and is independent of the choices of pol and d0 . M H,Σ = H,d0 pol 0 If Σ is smooth, then we have a canonical isomorphism ∼

tor Mtor H0 ,Σ0 → MH0 ,d0 pol0 ,

(1.3.1.13)

where Mtor H0 ,Σ0 is given by Theorem 1.3.1.3. Proof. Since JH0 ,d0 pol0 is the pullback of JH,d0 pol under the finite morphism min Mmin H0 → MH , the canonical morphism (1.3.1.12) exists and is finite by Theorem 1.3.1.10 and by the universal property of the normalization of blow-up. Since tor Mtor H0 ,d0 pol0 is normal, it is canonically isomorphic to the normalization of MH,d0 pol in MH0 by Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11]). If Σ0 is smooth, then we have Mtor H0 ,Σ0 given by Theorem 1.3.1.3. Moreover, 0 H0 ,d0 pol is defined (as in Definition 1.3.1.7) and is the pullback of H,d0 pol under tor the canonical morphism Mtor H0 ,Σ0 → MH,Σ . Hence, we have a canonical morphism tor tor MH0 ,Σ0 → MH,d0 pol by Theorem 1.3.1.10, inducing a canonical morphism Mtor H0 ,Σ0 → Mtor , both of which follow from the universal property of normalization of H0 ,d0 pol0 blowup. By Zariski’s main theorem again, this last morphism is an isomorphism and gives (1.3.1.13), as desired.

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Then we can describe the so-called Hecke actions of G(A∞ ) as follows: Proposition 1.3.1.14. (See [62, Prop. 7.2.5.1].) Suppose we have an element ˆ g ∈ G(A∞ ), and suppose we have two open compact subgroups H and H0 of G(Z) such that H0 ⊂ gHg −1 . Then there is a canonical finite surjection [g]

min

min : Mmin H0 → MH

(over S0 = Spec(F0 )) extending the canonical finite surjection [[g]] : [MH0 ] → [MH ] induced by the canonical finite surjection [g] : MH0 → MH ⊗k ⊗k min defined by the Hecke action of g, such that ωM is pulled back to ωM min over MH min H0

H

over Mmin H0 (up to canonical isomorphism) whenever the former is defined. min 0 Moreover, the surjection [g] maps the [(Φ0H0 , δH 0 )]-stratum Z[(Φ0 ,δ 0 )] of H0 H0 min min MH0 to the [(ΦH , δH )]-stratum Z[(ΦH ,δH )] of MH if and only if there are repre0 0 0 sentatives (ΦH , δH ) and (Φ0H0 , δH 0 ) of [(ΦH , δH )] and [(ΦH0 , δH0 )], respectively, such 0 that (ΦH , δH ) is g-assigned to (Φ0H0 , δH 0 ) as in [62, Def. 5.4.3.9]. If g = g1 g2 , where g1 and g2 are elements of G(A∞ ), each having a setup min similar to that of g, then we have [g] = [g2 ] ◦ [g1 ], [[g]] = [[g2 ]] ◦ [[g1 ]], and [g] = min min [g2 ] ◦ [g1 ] . Proposition 1.3.1.15. (See [62, Prop. 6.4.3.4].) With the same setting as in Proposition 1.3.1.14, suppose Σ = {ΣΦH }[(ΦH ,δH )] and Σ0 = {Σ0Φ0 0 }[(Φ0 0 ,δ0 0 )] are two H H H compatible choices of admissible smooth rational polyhedral cone decomposition data for MH and MH0 , respectively, such that Σ0 is a g-refinement of Σ as in [62, Def. 6.4.3.3]. Then there is a canonical proper surjection tor

[g]

tor : Mtor H0 ,Σ0  MH,Σ

(over S0 = Spec(F0 )) compatible with the canonical finite surjection min

[g]

min : Mmin H0  MH

H min in Proposition 1.3.1.14 and the canonical proper surjections H0 : Mtor H0 ,Σ0 → MH0 H tor min tor and H : MH,Σ → MH , such that ωMtor over MH,Σ is pulled back to ωMtor0 0 over H,Σ H ,Σ

Mtor H0 ,Σ0 (up to canonical isomorphism). tor 0 0 Moreover, the surjection [g] maps the [(Φ0H0 , δH 0 , σ )]-stratum Z[(Φ0 ,δ 0 ,σ 0 )] of H0 H0 tor tor MH0 ,Σ0 to the [(ΦH , δH , σ)]-stratum Z[(ΦH ,δH ,σ)] of MH,Σ if and only if there are 0 0 0 0 0 representatives (ΦH , δH , σ) and (Φ0H0 , δH 0 , σ ) of [(ΦH , δH , σ)] and [(ΦH0 , δH0 , σ )], 0 0 0 respectively, such that (ΦH0 , δH0 , σ ) is a g-refinement of (ΦH , δH , σ) as in [62, Def. 6.4.3.1]. If g = g1 g2 , where g1 and g2 are elements of G(A∞ ), each having a setup similar tor tor tor to that of g, then we have [g] = [g2 ] ◦[g1 ] , extending [g] = [g2 ]◦[g1 ] and lifting min min min [g] = [g2 ] ◦ [g1 ] .

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Remark 1.3.1.16. While Proposition 1.3.1.14 is a logical consequence of Proposition 1.3.1.15, they were stated in the reversed order, because the former is easier to describe and understand than the latter. The last statements of Propositions 1.3.1.15 and 1.3.1.14 were not explicitly stated in [62, Prop. 6.4.3.4 and 7.2.5.1], but were implicit in the proofs there. 1.3.2

Boundary of PEL-Type Moduli Problems

Let us describe the building blocks of Mtor H,Σ in more detail. In particular, we H would like to describe and characterize the algebraic stacks MZHH , MΦ H , CΦH ,δH , ∼ ΞΦH ,δH , ΞΦH ,δH (σ), ΞΦH ,δH ,σ , Z[(ΦH ,δH ,σ)] = ΞΦH ,δH ,σ /ΓΦH ,σ and the formal algebraic stacks XΦH ,δH ,σ and XΦH ,δH ,σ /ΓΦH ,σ in (2) of Theorem 1.3.1.3 (and (5) of Theorem 1.3.1.5), and to describe canonical Hecke actions on collections of these geometric objects (compatible with those in Proposition 1.3.1.15). Throughout this subsection, let us fix the choice of a fully symplectic admisˆ as in Definitions 1.2.1.2 and 1.2.1.3. Let us also fix a sible filtration Z of L ⊗ Z Z

(noncanonical) choice of (LZ , h · , · iZ , hZ0 ), so that GZ can be defined as in Definition 1.2.1.9. ˆ we can define the boundary moduli For each open compact subgroup H of G(Z), ΦH ZH problems MHh , MH , and MH as in Definition 1.2.1.15. By definition, MHh parameterizes (B, λB , iB , ϕ−1,H ) appearing as the abelian part in degeneration data. ` Zn H By the construction of MΦ Mn by Hn = H/U(n) (for any H as the quotient of integer n ≥ 1 such that U(n) ⊂ H), where the disjoint union is over representatives (Zn , Φn , δn ) (with the same (X, Y, φ)) in (ZH , ΦH , δH ), the finite ´etale co∼ ∼ H ver MΦ H → MHh parameterizes the twisted objects (ϕ−2,H , ϕ0,H ) inducing both (ϕ−2,H , ϕ0,H ) and ϕ−1,H over MHh . Therefore, by the definition of MZHH as the H quotient of MΦ H by ΓΦH (see Definition 1.2.2.3), we have the following: Lemma 1.3.2.1. Let us fix the choice of a representative (Z, Φ, δ) in (ZH , ΦH , δH ). ˆ ∼ ˆ as in Let HGh,Z = HPZ /HZZ be the open compact subgroup of GZ (Z) = Gh,Z (Z) ˆ Definition 1.2.1.12, and let HGh,Z ,Φ denote the image in Gh,Z (Z) of the stabilizer HPZ ,Φ of Φ = (X, Y, φ, ϕ−2 , ϕ0 ) in HPZ , which is an open compact subgroup of HGh,Z isomorphic to HPZ ,Φ /HZZ . Then MHh ∼ = MHGh,Z and there is a canonical isomorphism MZHH ∼ = MHGh,Z ,Φ ,

(1.3.2.2)

where MHGh,Z ,Φ is defined by (LZ , h · , · iZ , hZ0 ) as in Section 1.1.2. If H0 is an open compact subgroup of H, then the corresponding morphism Z

MHH0 0 → MZHH

(1.3.2.3)

can be canonically identified with the finite ´etale morphism MH0G

h,Z ,Φ

→ MHGh,Z ,Φ .

(1.3.2.4)

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The collection {MHGh,Z ,Φ }HGh,Z ,Φ naturally carries a Hecke action by elements gh ∈ GZ (A∞ ) ∼ = Gh,Z (A∞ ), realized by finite ´etale surjections pulling tautological objects back to Hecke twists. If moreover H0 is a normal subgroup of H, then (1.3.2.4) is 0 an HGh,Z ,Φ /HG -torsor. h,Z ,Φ Lemma 1.3.2.5. With the same setting as in Lemma 1.3.2.1, let HG0h,Z be the open ˆ ∼ ˆ as in Definition 1.2.1.12, which is a normal compact subgroup of GZ (Z) = G0h,Z (Z) subgroup of HGh,Z ,Φ (by definition). Then there is a canonical isomorphism H ∼ MΦ H = MHG0 ,

(1.3.2.6)

h,Z

which is compatible with (1.3.2.2) and with Hecke actions as in Lemma 1.3.2.1. ZH H The canonical morphisms MΦ H → MH → MHh can be identified with the canonical finite ´etale morphisms MHG0 → MHGh,Z ,Φ → MHGh,Z , on which ΓΦH acts h,Z

equivariantly (and trivially on the latter two objects) via the canonical homomorphism ΓΦH → HG0l,Z /HGl,Z ∼ = HGh,Z /HG0h,Z with image HGh,Z ,Φ /HG0h,Z . In particular, ZH ΦH H MΦ etale and an HGh,Z ,Φ /HG0h,Z -torsor. H → MH = MH /ΓΦH is finite ´ H The abelian scheme torsor CΦH ,δH → MΦ H is, by the construction in [62, Sec. 6.2.3–6.2.4] (see also the correction in Remark 1.3.1.6), the quotient of a a CΦn ,δn → MZnn

by Hn = H/U(n) (for any integer n ≥ 1 such that U(n) ⊂ H), where each CΦn ,δn → MZnn has a canonical structure of an abelian scheme which is preserved under the ∼ action of Hn,Uess = HU1,Z /U(n)U1,Z . Therefore, we have the following: 1,Zn H Lemma 1.3.2.7. The quotient CΦH ,δH → MΦ depends only on HG1,Z , is an H abelian scheme when the splitting of (1.2.1.14) defined by any splitting δ also splits (1.2.1.13) (and induces an isomorphism HG1,Z ∼ = HG0h,Z n HU1,Z ), and is a torsor under the abelian scheme CΦgrp := C defined by any H0 with Φ ,δ H0 H0 H ,δH 0 × HG1,Z ∼ = HG0h,Z n HU1,Z , which is canonically Q -isogenous to HomO (X, B)◦ . (This H clarifies the abelian scheme torsor structure of CΦH ,δH → MΦ H .) We deduce from this that there is a canonical isomorphism ∼ 1 ∼ Ω1 = Ω1 grp ZH = Ω Φ Φ

CΦH ,δH /MH

CΦH ,δH /MHH

CΦ

H ,δH

/MHH

∼ Z ). = (CΦH ,δH → MZHH )∗ HomO (X, Lie∨ B/M H

(1.3.2.8)

H

If we fix the choice of (Zn and) Φn , then the canonical morphism CΦn ,δn → CΦH ,δH

(1.3.2.9)

ess -torsor (see [62, Sec. 6.2.4; see also the eris an HG1,Z /U(n)G1,Z ∼ = Hn,Gess h,Zn nU1,Zn ˆ as in Derata]), where HG1,Z and U(n)G1,Z are open compact subgroups of G1,Z (Z) finition 1.2.1.12, and induces an isomorphism

∼

CΦn ,δn /(HG1,Z /U(n)G1,Z ) → CΦH ,δH .

(1.3.2.10)

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n Lemma 1.3.2.11. The abelian scheme torsor S := CΦn ,δn → MΦ := MZnn is n ∨ universal for the additional structures (cn , cn ) satisfying certain symplectic and liftability conditions, which we review as follows:

(1) The homomorphism c : XS → B ∨ induced by cn : n1 XS → B ∨ (by restriction) is equivalent to the data of a semi-abelian scheme G\ that is an extension of B by the split torus T with character group X, and the lifting cn of c is equivalent to the data of a splitting of the canonical short exact sequence 0 → T [n] → G\ [n] → B[n] → 0. 1 (2) The homomorphism c∨ : YS → B induced by c∨ n : n YS → B (by restriction) is equivalent to the data of a semi-abelian scheme G∨,\ that is an extension ∨ of B ∨ by the split torus T ∨ with character group Y , and the lifting c∨ n of c is equivalent to the data of a splitting of the canonical short exact sequence

0 → T ∨ [n] → G∨,\ [n] → B ∨ [n] → 0. (3) The homomorphisms c, c∨ , φ : Y ,→ X, and λB : B → B ∨ satisfy the compatibility λB c∨ = cφ, and hence defines a homomorphism λ\ : G\ → G∨,\ inducing λT = φ∗S : T → T ∨ and λB : B → B ∨ . All of these are compatible with their O-structures. (4) The splittings defined by cn and c∨ n are not necessarily compatible under the canonical morphism G\ → G∨,\ induced by λB : B → B ∨ and φ : Y ,→ X. The failure of such a compatibility can be identified with the nontriviality of the pairing d10,n : B[n] ×( n1 Y /Y )S → µn,S (cf. [62, Lem. 5.2.3.12]), which sends (a, n1 y) to 1 eB[n] (a, (λB c∨ n − cn φn )( n y)),

where eB[n] : B[n] × B ∨ [n] → µn,S is the canonical perfect pairing between B[n] and B ∨ [n], for any functorial points a of B[n] and n1 y of ( n1 Y /Y )S . (5) The symplectic condition for (cn , c∨ n ) is that, under ϕ−1,n and ϕ0,n , the pairing d10,n above is matched with the pairing h · , · i10,n : GrZ−1,n × GrZ0,n → ((Z/nZ)(1))S induced by h · , · i and δn . (6) The liftability condition for (cn , c∨ n ) is that, for each integer m ≥ 1 such that n|m, and for any lifting δm of δn , there exists a finite ´etale covering of S over which there exist ϕ−1,m , (ϕ−2,m , ϕ0,m ), and (cm , c∨ m ) lifting ϕ−1,n , ∨ (ϕ−2,n , ϕ0,n ), and (cn , cn ), respectively, and satisfying the symplectic condition defined by h · , · i and δm as above.

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Zn n These can be re-interpreted as follows: S = CΦn ,δn → MΦ n = Mn parameterizes tuples

(G\ , λ\ : G\ → G∨,\ , i\ , βn\ ), where: (a) G\ (resp. G∨,\ ) is an extension of B (resp. B ∨ ) by T (resp. T ∨ ) as above, and λ\ : G\ → G∨,\ induces λT = φ∗ : T → T ∨ and λB : B → B ∨ . (b) i\ is a pair of homomorphisms O → EndS (G\ ) and O → EndS (G∨,\ ) compatible with each other under λ\ : G\ → G∨,\ , inducing compatible O-structures on B, B ∨ , T , and T ∨ . (c) βn\ = (βn\,0 , βn\,#,0 , νn\ ) is a principal level-n structure of (G\ , λ\ , i\ ) of type ∼ ∼ ∨,\ ˆ h · , · i, Z), where β \,0 : (Z−1,n )S → G\ [n] and βn\,#,0 : (Z# [n] (L ⊗ Z, n −1,n )S → G Z

are O-equivariant isomorphisms preserving filtrations on both sides and inducing on the graded pieces the given data ϕ−2,n , ϕ−1,n , and ϕ0,n (by duality), ∼ respectively; and where νn\ : ((Z/nZ)(1))S → µn,S is an isomorphism, which are compatible with λ\ and the canonical morphism Z−1,n → Z# −1,n induced by # # ˆ h · , · i. (Here Z is the filtration on L ⊗ Z canonically dual to the filtration on Z

ˆ equipped with a canonical morphism Z → Z# , respecting the filtration deL ⊗ Z, Z

grees, induced by h · , · i. Then the splitting δ corresponds under βn\ to splittings of 0 → T [n] → G\ [n] → B[n] → 0 and 0 → T ∨ [n] → G∨,\ [n] → B ∨ [n] → 0.) Moreover, βn\ satisfies the liftability condition that, for each integer m ≥ 1 such that n|m, there exists a finite ´etale covering of S over which there exists an \ analogous triple βm lifting the pullback of βn\ . Proof. The statements are self-explanatory. H Proposition 1.3.2.12. The abelian scheme torsor S := CΦH ,δH → MΦ H is universal for the additional structures (cH , c∨ H ) satisfying certain symplectic and liftability conditions, which can be interpreted as parameterizing tuples

\ (G\ , λ\ : G\ → G∨,\ , i\ , βH ),

(1.3.2.13) \ βH

where G\ , G∨,\ , λ\ , and i\ are as in Lemma 1.3.2.11, and where is a level-H \ \ \ \ ˆ structure of (G , λ , i ) of type (L ⊗ Z, h · , · i, Z), which is a collection {βH } , n n Z

\ where n ≥ 1 runs over integers such that U(n) ⊂ H, such that each βH (where n Hn := H/U(n)) is a subscheme of a ∨,\ [n]) IsomS ((Z−1,n )S , G\ [n]) × IsomS ((Z# −1,n )S , G S
 × IsomS ((Z/nZ)(1))S , µn,S S

over S, where the disjoint union is over representatives (Zn , Φn , δn ) (with the same (X, Y, φ)) in (ZH , ΦH , δH ), that becomes the disjoint union of all elements

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in the Hn -orbit of some principal level-n structure βn\ of (G\ , λ\ , i\ ) of type ˆ h · , · i, Z), as in Lemma 1.3.2.11, for any Z lifting Zn ; and where β \ is (L ⊗ Z, Hm Z

\ mapped to βH (under the canonical morphism, which we omit for simplicity) when n n|m. H Proof. This follows from the construction of CΦH ,δH → MΦ as a quotient of H ` ` Zn CΦn ,δn → Mn (over the same index set).

Proposition 1.3.2.14. (Compare with Proposition 1.3.2.12.) Fix any lifting (Z, Φ = (X, Y, φ, ϕ−2 , ϕ0 ), δ) of a representative (ZH , ΦH , δH ) of [(ZH , ΦH , δH )]. The × H abelian scheme torsor CΦH ,δH → MΦ H is universal for Q -isogeny classes of tuples (G\ , λ\ : G\ → G∨,\ , i\ , j \ , j ∨,\ , [βˆ\ ]HG1,Z )

(1.3.2.15)

over locally noetherian base schemes S, where: (1) G\ (resp. G∨,\ ) is a semi-abelian scheme which is the extension of an abelian scheme B (resp. B ∨ ) by a split torus T (resp. T ∨ ) over S, which is equivalent to a homomorphism c : X(T ) → B ∨ (resp. c∨ : X(T ∨ ) → B). (2) λ\ : G\ → G∨,\ is a Q× -isogeny (i.e., a (Q× >0 )S -multiple of a quasi-finite surjective homomorphism; or cf. [62, Def. 1.3.1.16]) of semi-abelian schemes over S, inducing a Q× -isogeny λT : T → T ∨ between the torus parts, which is dual to a Q-isomorphism λ∗T : X(T ∨ ) ⊗ Q → X(T ) ⊗ Q, Z

Z

and a Q× -polarization λB : B → B ∨ between the abelian parts (cf. [62, Def. 1.3.2.19 and the errata]), so that c(N λ∗T ) = (N λB )c∨ when N is any locally constant function over S valued in positive integers such that (N λ∗T )(X(T ∨ )) ⊂ X(T ) and such that N λ\ : G\ → G∨,\ is an isogeny. (3) i\ : O ⊗ Q → EndS (G\ ) ⊗ QS is a homomorphism inducing O ⊗ Q-actions Z

Z

Z

on G∨,\ , T , T ∨ , B, and B ∨ up to Q× -isogeny, compatible with each other under the homomorphisms between the objects introduced thus far. In particular, the induced homomorphism iB : O ⊗ Q → EndS (B) ⊗ QS satisfies Z

Z

the Rosati condition defined by λB (cf. [62, Def. 1.3.3.1]). ∼ ∼ (4) j \ : X ⊗ QS → X(T ) ⊗ QS and j ∨,\ : Y ⊗ QS → X(T ∨ ) ⊗ QS are isoZ

Z

Z

Z

morphisms of O ⊗ Q-modules, such that there exists a section r(j \ , j ∨,\ ) of Z

\ \ ∨,\ ∗ (Q× )λT ◦ j ∨,\ . >0 )S such that j ◦ φ = r(j , j \ ˆ (5) [β ]HG1,Z is a rational level-H structure of (G\ , λ\ , i\ , j \ , j ∨,\ ) of type (L ⊗ A∞ , h · , · i, Z ⊗ A∞ , Φ), which is an assignment to each geometric Z

ˆ Z

point s¯ of S a rational level-H structure of (G\ , λ\ , i\ , j \ , j ∨,\ ) of type (L ⊗ A∞ , h · , · i, Z ⊗ A∞ , Φ) based at s¯ (cf. [62, Def. 1.3.8.7]), which is Z

ˆ Z

a π1 (S, s¯)-invariant HG1,Z -orbit [βˆs¯\ ]HG1,Z of triples βˆs¯\ = (βˆs¯\,0 , βˆs¯\,#,0 , νˆs¯\ ), such that the assignments at any two geometric points s¯ and s¯0 of the same

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connected component of S determine each other (cf. [62, Lem. 1.3.8.6]), where: ∼ ∞ ∼ are → V G∨,\ (a) βˆs¯\,0 : Z−1 ⊗ A∞ → V G\s¯ and βˆs¯\,#,0 : Z# s¯ −1 ⊗ A ˆ Z

ˆ Z

O ⊗ A∞ -equivariant isomorphisms preserving filtrations on both sides, Z

which are compatible with λ\ and the canonical morphism Z−1 ⊗ A∞ → ˆ Z

∞ induced by h · , · i. Z# −1 ⊗ A ˆ Z

∼

(b) νˆs¯\ : A∞ (1) → V Gm,¯s is an isomorphism of A∞ -modules such that ˆ r(j \ , j ∨,\ )s¯ νˆs¯\ maps Z(1) to T Gm,¯s , where r(j \ , j ∨,\ )s¯ is the value \ at s¯ of the above section r(j \ , j ∨,\ ) of (Q× >0 )S such that j ◦ φ = \ ∨,\ ∗ ∨,\ r(j , j ) λT ◦ j . ∼ (c) The induced morphisms Gr−2 (βˆs¯\,0 ) : GrZ−2 ⊗ A∞ → V Ts¯ and ˆ Z

# ∼ Gr−2 (βˆs¯\,#,0 ) : GrZ−2 ⊗ A∞ → V Ts¯∨ coincide with the compositions

ˆ Z

ϕ−2 ⊗ A∞ ˆ Z

∼

GrZ−2 ⊗ A∞

HomA∞ (X ⊗ A∞ , A∞ (1))

→

ˆ Z

Z

((j \ )−1 ⊗ A∞ )∗ Q

∼

HomA∞ (X(T ) ⊗ A∞ , A∞ (1))

→

Z

ν ˆs¯\ ∼

∼

→ HomA∞ (X(T ) ⊗ A∞ , V Gm,¯s ) → V Ts¯. Z

and ∞ ϕ# −2 ⊗ A ˆ Z

∼

#

GrZ−2 ⊗ A∞

HomA∞ (Y ⊗ A∞ , A∞ (1))

→

ˆ Z

Z

((j

∨,\ −1

)

∼

⊗A

∞ ∗

)

Q

→

HomA∞ (X(T ∨ ) ⊗ A∞ , A∞ (1)) Z

ν ˆs¯\ ∼

∼

→ HomA∞ (X(T ∨ ) ⊗ A∞ , V Gm,¯s ) → V Ts¯∨ , Z

Z# ∼ ˆ ˆ respectively, where ϕ# ˆ (Y ⊗ Z, Z(1)) is induced by ϕ0 −2 : Gr−2 → HomZ Z

by duality. (d) Together with νˆ−1,¯s := νˆs¯\ , the induced morphisms ∼ ϕˆ−1,¯s := Gr−1 (βˆs¯\,0 ) : GrZ−1 ⊗ A∞ → V Bs¯ ˆ Z

and ∼

#

ˆ\,#,0 ) : GrZ ⊗ A∞ → V B ∨ ϕˆ# −1 s¯ −1,¯ s := Gr−1 (βs¯ ˆ Z

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determine each other by duality. By varying s¯ over geometric points of S, the (π1 (S, s¯)-invariant) HGh,Z -orbits of (ϕˆ−1,¯s , νˆ−1,¯s ) determine a tuple (B, λB , iB , ϕ−1,H ) whose Q× -isogeny class is parameterized by MHh (cf. [62, Prop. 1.4.3.4]), while the (π1 (S, s¯)-invariant) HG1,Z -orbits of (ϕˆ−1,¯s , νˆ−1,¯s , ϕ−2 , ϕ0 ) determine a tuple ∼ ((B, λB , iB , ϕ−1,H ), (ϕ∼ −2,H , ϕ0,H )) H whose Q× -isogeny class is parameterized by MΦ H .

The Q× -isogenies (G\ , λ\ : G\ → G∨,\ , i\ , j \ , j ∨,\ , [βˆ\ ]HG1,Z ) ∼Q× -isog. (G\,0 , λ\,0 : G\,0 → G∨,\,0 , i\,0 , j \,0 , j ∨,\,0 , [βˆ\,0 ]HG1,Z ) between tuples as in (1.3.2.15) are given by pairs of Q× -isogenies (f \ : G\ → G\,0 , f ∨,\ : G∨,\,0 → G∨,\ ) such that we have the following: \ \ ∨,\ ∨,\ (i) There exists a section r(f \ , f ∨,\ ) of (Q× )f ◦ >0 )S such that λ = r(f , f \,0 \ λ ◦f . (ii) f \ and f ∨,\ respect the compatible O ⊗ Q-actions on G\ , G\,0 , G∨,\ , and G∨,\,0 Z

(defined by i\ and i\,0 ). (iii) j \ = (f \ )∗ ◦ j \,0 and j ∨,\,0 = (f ∨,\ )∗ ◦ j ∨,\ . ∼ (iv) For each geometric point s¯, the morphisms V(f \ ) : V G\s¯ → V Gs\,0 ¯ and ∨,\,0 ∼ ∨,\ ∨,\ V(f ) : V Gs¯ → V Gs¯ satisfy the condition that, for any representatives βˆs¯\ = (βˆs¯\,0 , βˆs¯\,#,0 , νˆs¯\ ) and βˆs¯\,0 = (βˆs¯\,0,0 , βˆs¯\,#,0,0 , νˆs¯\,0 ) of [βˆ\ ]HG1,Z and [βˆ\,0 ]H , respectively, the HG -orbits of G1,Z

1,Z

(V(f \ ) ◦ βˆs¯\,0 , V(f ∨,\ )−1 ◦ βˆs¯\,#,0 , r(f \ , f ∨,\ )s−1 ˆs¯\ ) ¯ ν and (βˆs¯\,0,0 , βˆs¯\,#,0,0 , νˆs¯\,0 ) coincide, where r(f \ , f ∨,\ )s¯ is the value at s¯ of the above section r(f \ , f ∨,\ ) of \ \ ∨,\ ∨,\ (Q× )f ◦ λ\,0 ◦ f \ . >0 )S such that λ = r(f , f

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Proof. As in [62, Sec. 1.4.3], this can be proved by replacing any tuple as in (1.3.2.15) up to Q× -isogeny, as in the statement of this proposition, with a tu∼ ∼ ple such that j \ : X ⊗ Q → X(T ) ⊗ Q (resp. j ∨,\ : Y ⊗ Q → X(T ∨ ) ⊗ Q) maps X Z

Z

Z

Z

(resp. Y ) to X(T ) (resp. X(T ∨ )), and such that, at each geometric point s¯ of S, the assigned βˆs¯\ = (βˆs¯\,0 , βˆs¯\,#,0 , νˆs¯\ ) satisfies the condition that βˆs¯\,0 (resp. βˆs¯\,#,0 , resp. ∨,\ \ ˆ νˆs¯\ ) maps Z−1 (resp. Z# s ). Then the −1 , resp. Z(1)) to T Gs¯ (resp. T Gs¯ , resp. T Gm,¯ tuple determines and is determined by a tuple as in (1.3.2.13), as desired. (These can be simultaneously achieved because of the existence of the section r(j \ , j ∨,\ ) of (Q× >0 )S . The proof is similar to that of [62, Prop. 1.4.3.4], and hence omitted.) Construction 1.3.2.16. Suppose H is neat. Consider the degenerating family (G, λ, i, αH ) → Mtor H,Σ

(1.3.2.17)

of type MH as in Theorem 1.3.1.3. Let Z = Z[(ΦH ,δH ,σ)] be any stratum of Mtor H,Σ such that σ ⊂ P+ ΦH is a top-dimensional cone in ΣΦH (in Σ). Let (G\Z , λZ\ , i\Z ) → Z

(1.3.2.18)

denote the pullback of the (G, λ, i) in (1.3.2.17) to Z, the closure of Z in Mtor H,Σ . Since σ is top-dimensional, the canonical morphism Z → CΦH ,δH is an isomorphism. Since αH is defined only over MH , its pullback to Z is undefined. The goal of this construction is to define a partial pullback, which still retains some information of αH . Let n ≥ 1 be any integer such that U(n) ⊂ H, and let us fix any choice of (Zn , Φn , δn ). Consider any top-dimensional cone σ 0 contained in σ that is smooth for the integral structure defined by SΦn , we have a canonical morphism XΦn ,δn ,σ0 → XΦH ,δH ,σ (which might not be finite ´etale), inducing a morphism from the σ 0 -stratum Zn = Z[(Φn ,δn ,σ0 )] of the source to the σ-stratum Z = Z[(ΦH ,δH ,σ)] of the target (although the scheme-theoretic preimage of latter might not be the former), which can be identified with the canonical morphism (1.3.2.9). Let us denote the pullback of (1.3.2.18) to Zn by (G\Zn , λ\Zn , i\Zn ) → Zn .

(1.3.2.19)

Over each affine open formal subscheme Spf(R, I) of XΦn ,δn ,σ0 , such that S0 = Spec(R/I) is the σ 0 -stratum of S = Spec(R), where both R and R/I are regular domains, we have a degenerating family (GS , λS , iS , αn,η ) → S of type Mn = MU (n) . A priori, the level structure αn,η is defined only over the generic point η of S (and it only extends to the largest open subscheme of S over which the pullback of GS is an abelian scheme). Nevertheless, as explained in [62, Prop. 5.2.2.1], \ Gη [n] (resp. G∨ η [n]) admits a canonical filtration 0 ⊂ Tη [n] ⊂ Gη [n] ⊂ Gη [n] (resp. ∨ ∨,\ ∨ 0 ⊂ Tη [n] ⊂ Gη [n] ⊂ Gη [n]), with notation as in Lemma 1.3.2.11, where the subscripts “η” (and similar usages later) mean pullbacks. By the construction of ∼ XΦn ,δn ,σ0 , the symplectic isomorphism αn,η : L/nL → Gη [n] sends the filtration Zn to the above geometric filtration on Gη [n], and induces the pair (ϕ−2,n , ϕ0,n ) in Φn

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when restricted to the top and bottom filtered pieces. By duality, and by using the ∼ isomorphism νn,η : ((Z/nZ)(1))η → µn,η (which is part of the data of αn,η ), it also ∼ # defines a symplectic isomorphism αn,η : L# /nL# → G∨ η [n] which sends the dual # ∨ filtration Zn to the above geometric filtration on Gη [n], which induces (in particular) an object ϕ# −2,n dual to ϕ0,n in the obvious sense. These two isomorphisms αn,η and # αn,η are compatible under the canonical morphisms L ,→ L# and λη : Gη → G∨ η. \ ∨,\ Since GS [n] (resp. GS [n], resp. µn,S ) is finite ´etale over S, the restriction of αn,η # to Z−1 (resp. the restriction of αn,η to Z# −1 , resp. the isomorphism νn,η ) over η ex∼ ∼ \,0 \,#,0 ∨,\ tends to an isomorphism βn,S : (Z−1,n )S → G\S [n] (resp. βn,S : (Z# −1,n )S → GS [n], ∼ \ resp. νn,S : ((Z/nZ)(1))S → µn,S ) over the whole normal scheme S. These \,0 \,#,0 two isomorphisms βn,S and βn,S are compatible under the canonical morphisms # \ \ \ \,0 \,#,0 \ (Z−1,n )S → (Z−1,n )S and λS : GS → G∨,\ S . Let βn,S := (βn,S , βn,S , νn,S ), and con\ \,0 \,#,0 \ \ sider its pullback βn,S := (βn,S , βn,S , νn,S ) to S0 . By analyzing βn,S as in the 0 0 0 0 \ case of αn,η as in [62, Sec. 5.2.2–5.2.3], we see that βn,S retains almost all information of αn,η , including the pairing e10,n to be compared with d10,n , as in [62, Lem. 5.2.3.12 and Thm. 5.2.3.14], except that it loses information about the pairing e00,n to be compared with d00,n . Hence, if we denote the pullback of (1.3.2.18) to S0 \ by (G\S0 , λ\S0 , i\S0 ) → S0 , then (G\S0 , λ\S0 , i\S0 , βn,S ) → S0 determines and is deter0 mined by (the prescribed (Zn , Φn , δn ) and) the pullback to S0 of the tautological object ((B, λB , iB , ϕ−1,n ), (cn , c∨ n )) over CΦn ,δn (up to isomorphisms inducing automorphisms of Φn ; i.e., elements of ΓΦn ; see Lemma 1.3.2.11). By patching over varying S, we obtain (with (G\Zn , λ\Zn , i\Zn ) already defined as in (1.3.2.19)) a tuple (G\ , λ\ , i\ , β \ ) → Zn ∼ (1.3.2.20) = CΦ ,δ Zn

Zn

Zn

n,Zn

n

n

such that the previous sentence is true with S0 replaced with Zn . \ Since HG1,Z /U(n)G1,Z acts compatibly on βn,Z and (ϕ−1,n , cn , c∨ n ), the latter n action being compatible with the HG1,Z /U(n)G1,Z -torsor structure of (1.3.2.9), by \ \ forming the HG1,Z /U(n)G1,Z -orbit βH,Z of βn,Z , we can descend (1.3.2.20) to a n n tuple \ ∼ CΦ ,δ , (G\Z , λ\Z , i\Z , βH,Z )→Z= (1.3.2.21) H H where the first three entries form the pullback of (1.3.2.18) to Z, which determines and is determined by (the prescribed (ZH , ΦH , δH ) and) the tautological object
∼ ∨ (B, λB , iB , ϕ−1,H ), (ϕ∼ (1.3.2.22) −2,H , ϕ0,H ), (cH , cH ) → CΦH ,δH (up to isomorphisms inducing automorphisms of ΦH ; i.e., elements of ΓΦH ). Since the tautological object (1.3.2.22) is independent of the choice of n, so is the tuple (1.3.2.21). By abuse of language, we say that \ )→Z (1.3.2.23) (GZ\ , λ\Z , i\Z , βH,Z is the pullback of the degenerating family (1.3.2.17) to Z, with the convention that \ (as in the case of (G, λ, i, αH ) itself) βH,Z is defined only over Z, while (G\ , λ\ , i\ ) is defined over all of Z as in (1.3.2.18). (This finishes Construction 1.3.2.16.)

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Proposition 1.3.2.24. (Compare with Proposition 1.3.1.15.) By considering compatible Q× -isogenies (f : G\ → G\,0 , f ∨ : G∨,\,0 → G∨,\ ) inducing isomorphisms on the torus parts, we can define Hecke twists of the tautological object \ (G\ , λ\ , i\ , βH ) → CΦH ,δH by elements of G1,Z (A∞ ), and define the Hecke action of ∞ G1,Z (A ) on the collection {CΦH ,δH }HG1,Z , realized by finite ´etale surjections pulling tautological objects back to Hecke twists, which is compatible with the Hecke H action of G0h,Z (A∞ ) on the collection {MΦ under the canonical morphisms H }HG0 h,Z

∞ H CΦH ,δH → MΦ H (with varying H) and the canonical homomorphism G1,Z (A ) → G0h,Z (A∞ ). Over the subcollection indexed by HG1,Z with neat H, the Hecke action of G1,Z (A∞ ) on {CΦH ,δH }HG1,Z is compatible with the Hecke action of P0Z (A∞ ) on the collection of strata {Z[(ΦH ,δH ,σ)] } above {CΦH ,δH }HG1,Z (cf. Proposition 1.3.1.15) under the canonical homomorphism P0Z (A∞ ) → G1,Z (A∞ ) = P0Z (A∞ )/U2,Z (A∞ ). By also considering Q× -isogenies (f : G\ → G\,0 , f ∨ : G∨,\,0 → G∨,\ ) inducing Q× -isogenies on the torus parts, we can also define Hecke twists of the tautological \ object (G\ , λ\ , i\ , βH ) → CΦH ,δH by elements of PZ (A∞ )/U2,Z (A∞ ), and define the ` Hecke action of PZ (A∞ )/U2,Z (A∞ ) on the collection { CΦH ,δH }HPZ /HU2,Z , realized by finite ´etale surjections pulling tautological objects back to Hecke twists, where the disjoint unions are over classes [(ZH , ΦH , δH )] sharing the same ZH , which induces an action of G0l,Z (A∞ ) = PZ (A∞ )/P0Z (A∞ ) on the index sets {[(ZH , ΦH , δH )]}, which is compatible with the Hecke action of PZ (A∞ )/UZ (A∞ ) ∼ = G0l,Z (A∞ ) × G0h,Z (A∞ ) ` ΦH on the collection { MH }HG0 (with the same index sets and the same induced h,Z

H action of G0l,Z (A∞ )) under the canonical morphisms CΦH ,δH → MΦ H (with varying ∞ ∞ 0 ∞ H) and the canonical homomorphism PZ (A )/U2,Z (A ) → Gl,Z (A ) × G0h,Z (A∞ ). Over the subcollection indexed by HPZ /HU2,Z with neat H, the Hecke action of ` PZ (A∞ )/U2,Z (A∞ ) on { CΦH ,δH }HPZ /HU2,Z is compatible with the Hecke action of ` PZ (A∞ )(A∞ ) on the collection of strata {Z[(ΦH ,δH ,σ)] } above { CΦH ,δH }HPZ /HU2,Z (cf. Proposition 1.3.1.15) under the canonical homomorphism PZ (A∞ )(A∞ ) → PZ (A∞ )/U2,Z (A∞ ). In the Q× -isogeny class language as in Proposition 1.3.2.14, the morphism

[g] : CΦH0 ,δH0 → CΦH ,δH , 0 0 for g ∈ PZ (A∞ )/U2,Z (A∞ ) such that HP /HU ⊂ g(HPZ /HU2,Z )g −1 and such that Z 2,Z [(ΦH , δH )] is g-assigned to [(ΦH0 , δH0 )] with a pair isomorphisms ∼

∼

(fX : X ⊗ Q → X 0 ⊗ Q, fY : Y 0 ⊗ Q → Y ⊗ Q) Z

Z

Z

Z

as in [62, Prop. 5.4.3.8], is characterized by [g]∗ (G\ , λ\ : G\ → G∨,\ , i\ , j \ , j ∨,\ , [βˆ\ ]HG1,Z ) ∼Q× -isog. (G\,0 , λ\,0 : G\,0 → G∨,\,0 , i\,0 , fX ◦ j \,0 , fY−1 ◦ j ∨,\,0 , [βˆ\,0 ◦ g]HG1,Z ) over CΦH0 ,δH0 , where (G\ , λ\ : G\ → G∨,\ , i\ , j \ , j ∨,\ , [βˆ\ ]HG1,Z )

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and (G\,0 , λ\,0 : G\,0 → G∨,\,0 , i\,0 , j \,0 , j ∨,\,0 , [βˆ\,0 ]H0G ) 1,Z

are representatives of the universal Q× -isogeny classes over CΦH ,δH and CΦH0 ,δH0 , respectively, and where the rational level-H structure [βˆ\,0 ◦ g]HG1,Z of (G\,0 , λ\,0 , i\,0 , fX ◦ j \,0 , fY−1 ◦ j ∨,\,0 ) of type (L ⊗ A∞ , h · , · i, Z ⊗ A∞ , Φ) is determined ˆ Z

Z

at each geometric point s¯ of CΦH0 ,δH0 by the HG1,Z -orbit of βˆs¯\,0 ◦ g, where βˆs¯\,0 is any representative of the rational level-H0 structure [βˆs¯\,0 ]H0G of (G\,0 , λ\,0 , i\,0 , j \,0 , j ∨,\,0 ) 1,Z of type (L ⊗ A∞ , h · , · i, Z ⊗ A∞ , Φ0 ) based at s¯ (assigned to s¯ by [βˆ\,0 ]H0 ). Z

G1,Z

ˆ Z

Proof. The first assertions in both of the first two paragraphs, and the whole third paragraph, can be justified as in the case of MH , which we omit for simplicity. As for the second assertions in both of the first two paragraphs, it suffices to note that the pullback of the Hecke twist of (1.3.2.17) is the Hecke twist of (1.3.2.21), the latter of which can be identified with the tautological object over CΦH ,δH under the ∼ canonical isomorphism Z[(ΦH ,δH ,σ)] → CΦH ,δH (for any top-dimensional σ, when H is neat). The torus torsor ΞΦH ,δH → CΦH ,δH is, by the construction in [62, Sec. 6.2.3– 6.2.4; see also the errata], the quotient of a a ΞΦn ,δn → CΦn ,δn by Hn = H/U(n) (for any integer n ≥ 1 such that U(n) ⊂ H), where each ΞΦn ,δn → CΦn ,δn has a canonical structure of a torsor under the torus EΦn with character ∼ group SΦn (see Section 1.2.2), which is preserved under the action of Hn,Uess = 2,Zn HU2,Z /U(n)U2,Z . Therefore, we have the following: Lemma 1.3.2.25. The quotient ΞΦH ,δH depends only on HP0Z , and is a torsor under the torus EΦH with character group SΦH . If we fix the choice of (Zn and) Φn , then the canonical morphism ΞΦn ,δn → ΞΦH ,δH

(1.3.2.26)

ess -torsor (see [62, Sec. 6.2.4; see also the eris an HP0Z /U(n)P0Z ∼ = Hn,Gess h,Zn nUZn ˆ as in Definition rata]), where HP0Z and U(n)P0Z are open compact subgroups of P0Z (Z) 1.2.1.12, and induces an isomorphism

∼

ΞΦn ,δn /(HP0Z /U(n)P0Z ) → ΞΦH ,δH .

(1.3.2.27)

Lemma 1.3.2.28. (Compare with Lemma 1.3.2.7.) The torus torsor S := ΞΦn ,δn → CΦn ,δn is universal for the additional structure τn satisfying certain symplectic and liftability conditions, which we review as follows:

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∼

∗ ⊗ −1 (1) τn : 1 n1 Y × X,S → (c∨ is an O-compatible trivialization of biexn × c) PB tensions (as in [62, Lem. 5.2.3.2 and Def. 5.2.7.8]), where PB is the Poincar´e invertible sheaf of B, which defines an O-equivariant homomorphism 1 ιn : n1 YS → G\ lifting c∨ n : n YS → B (via the canonical homomorphism ∼ \ ∗ ⊗ −1 G → B). Its restriction τ : 1Y × X,S → (c∨ is a trivialin × c) PB ∗ zation of biextensions such that (IdY × φ) τ is symmetric, and such that (iY (b) × IdX )∗ τ = (IdY × iX (b? ))∗ τ for all b ∈ O; and defines homomorphisms ι : YS → G\ and ι∨ : XS → G∨,\ compatible with each under φ : Y ,→ X and λ\ : G\ → G∨,\ . (2) Let [n] : G\ → G\ denote the multiplication by n morphism on G\ . Then we define

G[n] := [n]−1 (ι(Y ))/ι(Y ), where ι(Y ) is the image of the O-equivariant homomorphism ι : Y → G\ induced by ιn by restriction. Note that we defined G[n] without actually having a quotient G = G\ /ι(Y ) over S (cf. [20, p. 57]). Then we have an exact sequence 0 → G\ [n] → G[n] → (Y /nY )S → 0

(1.3.2.29)

of finite flat group schemes over S. Similarly, we define G∨ [n] := [n]−1 (ι∨ (X))/ι∨ (X) without defining G∨ , together with an exact sequence 0 → G∨,\ [n] → G∨ [n] → (X/nX)S → 0

(1.3.2.30)

of finite flat group schemes over S, which is then equipped with a homomorphism λ : G[n] → G∨ [n] without defining λ : G → G∨ , respecting the filtrations defined by (1.3.2.29) and (1.3.2.30). We note that there is a canonical duality between G[n] and G∨ [n], just as in the case of usual abelian schemes, but we will not explicitly use this canonical duality for our purpose. (3) The lifting ιn of ι defines a splitting of (1.3.2.29). Together with the splittings defined by (cn , c∨ n ) in Lemma 1.3.2.11, we obtain a splitting ∼

ςn : T [n] ⊕ B[n] ⊕(Y /nY )S → G[n]. On the other hand, the biextension properties of PB allows τn to induce a dual trivialization ∼

τn∨ : 1 1

n X × Y,S

⊗ −1 → (cn × c∨ )∗ PB ,

by setting ∼

⊗ −1 τn∨ ( n1 χ, y) : 1S → (cn ( n1 χ, c∨ (y))∗ PB

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to be 1S

1 τn ( n y,χ) ∼

→

⊗ −1 ∗ ⊗ −n ∼ 1 1 (c∨ . = (cn ( n1 χ, c∨ (y))∗ PB n ( n y), cn ( n χ)) PB

Then τn∨ induces a lifting ι∨ n of ι, which defines a splitting of (1.3.2.29). Together with the splittings defined by (cn , c∨ n ) in Lemma 1.3.2.11, we obtain a splitting ∼

ςn∨ : T ∨ [n] ⊕ B ∨ [n] ⊕(X/nX)S → G∨ [n]. However, the two splittings have no reason to be compatible with each other under λ : G[n] → G∨ [n]. While the failure measured by the induced homomorphisms B[n] → T ∨ [n] and (Y /nY )S → B ∨ [n] can be identified (up to a sign convention) with the pairing d10,n : B[n] ×( n1 Y /Y )S → µn,S defined in Lemma 1.3.2.11, the failure measured by the induced homomorphism (Y /nY )S → T ∨ [n] can be identified (up to a sign convention) with a pairing d00,n : ( n1 Y /Y )S ×( n1 Y /Y )S → µn,S which sends ( n1 y1 , n1 y2 ) to τn ( n1 y1 , φ(y2 ))τn ( n1 y2 , φ(y1 ))−1 for any functorial points y1 and y2 of ( n1 Y /Y )S (cf. [62, Lem. 5.2.3.12]). (4) The symplectic condition for τn (or ιn ) is that, under ϕ0,n , the pairing d00,n above is matched with the pairing h · , · i00,n : GrZ0,n × GrZ0,n → ((Z/nZ)(1))S induced by h · , · i and δn . Then, together with the symplectic condition for (cn , c∨ n ) in Lemma 1.3.2.11, under (ϕ−2,n , ϕ0,n ) and ϕ−1,n (equipped with ∼ ν−1,n : ((Z/nZ)(1))S → µn,S ), we obtain an O-equivariant isomorphisms ∼

βn0 : (L/nL)S → G[n] and ∼

βn#,0 : (L# /nL# )S → G∨ [n] respecting filtrations on both sides, together with the isomorphism νn = ν−1,n , which are compatible with the canonical morphisms L ,→ L# and λ : G[n] → G∨ [n]. (5) The liftability condition for τn is that, for each integer m such that n|m, and for any lifting δm of δn , there exists a finite ´etale covering of S over which there exist ϕ−1,m , (ϕ−2,m , ϕ0,m ), (cm , c∨ m ), and τm lifting ϕ−1,n , ∨ (ϕ−2,n , ϕ0,n ), (cn , cn ), τn , respectively, and satisfying the symplectic condition defined by h · , · i and δm as above.

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... ... (6) The group of multiplicative type E Φn with character group S Φn as in (1.2.2.1) define a subgroup of
Hom ( 1 Y ) ⊗ X)S , Gm,S ∼ = Hom ( 1 YS , T ) n

Z

Z n

Z

over S, which acts on the collection of O-compatible τn , possibly not satisfying the symplectic and liftability conditions above (but preserving the symmetry and O-compatibility of the induced τ ), inducing a translation action on the collection of O-equivariant homomorphisms ιn : n1 YS → G\ . ... ... The subgroup EΦn of E Φn with character group SΦn = S Φn ,free , the free ... quotient of S Φn , preserves in addition the symplectic and liftability conditions satisfied by τn (see [62, (6.2.3.5) and Conv. 6.2.3.20], and the proofs leading to there), and makes S = ΞΦn ,δn → CΦn ,δn a torsor under the torus EΦn , which is equipped with a homomorphism SΦn → Pice (CΦn ,δn /MZnn ) : ` 7→ ΨΦn ,δn (`) (by the torus torsor structure; see [62, Prop. 6.2.3.21 and (6.2.3.22)]), assigning to each ` ∈ SΦn a rigidified invertible sheaf ΨΦn ,δn (`) over CΦn ,δn , such that   ∼ Spec ⊕ ΨΦ ,δ (`) . ΞΦ ,δ = n

OCΦ

n

n ,δn

`∈SΦn

n

n

When ` = [ n1 y ⊗ χ] for some y ∈ Y and χ ∈ X, we have a canonical isomorphism ΨΦ ,δ (`) ∼ = (c∨ ( 1 y), c(χ))∗ PB . n

n

n n

∨ ∼ ˆ By construction, we have S∨ Φ1 /SΦn = U2,Z (Z)/U(n)U2,Z .

These can be re-interpreted as follows: S = ΞΦn ,δn → CΦn ,δn parameterizes tuples (G\ , λ\ : G\ → G∨,\ , i\ , τ, βn ), where: (a) G\ , G∨,\ , λ\ , and i\ are as in Lemma 1.3.2.11. ∼ ⊗ −1 (b) τ : 1Y × X → (c∨ × c)∗ PB is a trivialization of biextensions such that ∗ (IdY × φ) τ is symmetric, and such that (iY (b) × IdX )∗ τ = (IdY × iX (b? ))∗ τ for all b ∈ O. Then τ induces homomorphisms ι : Y → G\ and ι∨ : X → G∨,\ compatible with the homomorphisms φ : Y ,→ X and λ\ : G\ → G∨,\ , and induces an O-equivariant homomorphism λ : G[n] → G∨ [n]. (c) βn = (βn0 , βn#,0 , νn ) is a principal level-n structure of (G\ , λ\ , i\ , τ ) of type ∼ ∼ ˆ h · , · i, Z), where β 0 : (L/nL)S → (L ⊗ Z, G[n] and βn#,0 : (L# /nL# )S → G∨ [n] n Z

are O-equivariant isomorphisms respecting the canonical filtrations on both si∼ des, and νn : ((Z/nZ)(1))S → µn,S is an isomorphism, which are compatible with the canonical morphisms L ,→ L# and λ : G[n] → G∨ [n], and induce on the graded pieces the given data ϕ−2,n , ϕ−1,n , and ϕ0,n . Moreover, βn satisfies the liftability condition that, for each integer m ≥ 1 such that n|m, there exists a finite ´etale covering of S over which there exists an analogous triple βm lifting the pullback of βn .

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Proof. The statements are self-explanatory. Proposition 1.3.2.31. (Compare with Proposition 1.3.2.12.) The torus torsor S := ΞΦH ,δH → CΦH ,δH is universal for the additional structure τH satisfying certain symplectic and liftability conditions, which can be interpreted as parameterizing tuples (G\ , λ\ : G\ → G∨,\ , i\ , τ, βH ), where G\ , G∨,\ , λ\ , and i\ are as in Lemma 1.3.2.11, where τ is as in Lemma 1.3.2.28, and where βH is a level-H structure of (G\ , λ\ , i\ , τ ) of type ˆ h · , · i, Z), which is a collection {βH }n , where n ≥ 1 runs over integers such (L ⊗ Z, n Z

that U(n) ⊂ H, such that each βHn (where Hn := H/U(n)) is a subscheme of a

IsomS (L/nL)S , G[n] × IsomS (L# /nL# )S , G∨ [n] S
 × IsomS ((Z/nZ)(1))S , µn,S S

over S, where the disjoint union is over representatives (Zn , Φn , δn ) (with the same (X, Y, φ)) in (ZH , ΦH , δH ), that becomes the disjoint union of all elements in the Hn -orbit of some principal level-n structure βn of (G\ , λ\ , i\ , τ ) of type ˆ h · , · i, Z), as in Lemma 1.3.2.28, for any Z lifting Zn ; and where βH is (L ⊗ Z, m Z

mapped to βHn (under the canonical morphism, which we omit for simplicity) when n|m. ∨ ∼ ˆ Let SΦH be the unique lattice in SΦ1 ⊗ Q such that S∨ Φ1 /SΦH = U2,Z (Z)/HU2,Z . Z

Then S = ΞΦH ,δH → CΦH ,δH is torsor under the split torus EΦH with character group SΦH , equipped a homomorphism SΦH → Pic(CΦH ,δH ) : ` 7→ ΨΦH ,δH (`) (by the torus torsor structure; see [62, Prop. 6.2.4.7 and (6.2.4.8); see also the errata]), assigning to each ` ∈ SΦH an invertible sheaf ΨΦH ,δH (`) over CΦH ,δH (up to isomorphism), together with isomorphisms ∆∗ΦH ,δH ,`,`0 : ΨΦH ,δH (`)

⊗

OCΦ

∼

ΨΦH ,δH (`0 ) → ΨΦH ,δH (` + `0 )

H ,δH

0

for all `, ` ∈ SΦH , satisfying the necessary compatibilities with each other making ⊕ ΨΦH ,δH (`) an OCΦH ,δH -algebra, such that `∈SΦH

ΞΦH ,δH

∼ = SpecOC

 ⊕ ΦH ,δH

`∈SΦH

 ΨΦH ,δH (`) .

When ` = [y ⊗ χ] for some y ∈ Y and χ ∈ X, we have a canonical isomorphism ΨΦH ,δH (`) ∼ = (c∨ (y), c(χ))∗ PB . Proof. This follows from the construction of ΞΦH ,δH → CΦH ,δH as a quotient of ` ` ΞΦn ,δn → CΦn ,δn (over the same index set).

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For each rational polyhedral cone σ ⊂ (SΦH )∨ R as in Definition 1.2.2.2, we have an affine toroidal embedding   ΞΦH ,δH ,→ ΞΦH ,δH (σ) := SpecO ⊕ ΨΦH ,δH (`) , (1.3.2.32) CΦ ,δ H H

`∈σ ∨

both sides being relative affine over CΦH ,δH , where ΞΦH ,δH (σ) → CΦH ,δH is smooth when the cone σ is smooth, with a closed subalgebraic stack defined by   ΞΦH ,δH ,σ := SpecO ⊕ ΨΦH ,δH (`) , (1.3.2.33) CΦ

`∈σ ⊥

H ,δH

which we call the σ-stratum (cf. [62, Def. 6.1.2.7]), which is by itself a torsor under the torus EΦH ,σ with character group σ ⊥ . For each ΓΦH -admissible rational polyhedral cone decomposition ΣΦH as in Definition 1.2.2.4, we have a toroidal embedding ΞΦH ,δH ,→ ΞΦH ,δH = ΞΦH ,δH ,ΣΦH ,

(1.3.2.34)

the right-hand side being only locally of finite type over CΦH ,δH , with an open covering ΞΦH ,δH =

∪

σ∈ΣΦH

ΞΦH ,δH (σ),

(1.3.2.35)

ΞΦH ,δH ,σ .

(1.3.2.36)

inducing a stratification a

ΞΦH ,δH =

σ∈ΣΦH

` (The notation “ ” only means a set-theoretic disjoint union. The algebro-geometric structure is still the one inherited from ΞΦH ,δH .) Concretely, if σ is a face of ρ, then ρ∨ ⊂ σ ∨ and ΞΦH ,δH (σ) ⊂ ΞΦH ,δH (ρ), but ΞΦH ,δH ,ρ is contained in the closure of ΞΦH ,δH ,σ . The closure of ΞΦH ,δH ,σ in ΞΦH ,δH (ρ) is   ⊕ ΨΦH ,δH (`) . (1.3.2.37) ΞΦH ,δH ,σ (ρ) := SpecO CΦ ,δ H H

`∈σ ⊥ ∩ ρ∨

In this case, the open embedding ΞΦH ,δH ,σ ,→ ΞΦH ,δH ,σ (ρ)

(1.3.2.38)

is an affine toroidal embedding (as in [62, Def. 6.1.2.3]) for the torus torsor ΞΦH ,δH ,σ → CΦH ,δH . Let XΦH ,δH ,σ := (ΞΦH ,δH (σ))∧ ΞΦ

H ,δH ,σ

,

(1.3.2.39)

the formal completion of ΞΦH ,δH (σ) along its σ-stratum ΞΦH ,δH ,σ . When σ ⊂ P+ ΦH appears in ΣΦH ∈ Σ, the quotient XΦH ,δH ,σ /ΓΦH ,σ is isomorphic to the formal ∼ completion of Mtor H,Σ along its [(ΦH , δH , σ)]-stratum Z[(ΦH ,δH ,σ)] = ΞΦH ,δH ,σ /ΓΦH ,σ , 0 0 0 as in Theorem 1.3.1.3. If there is a surjection (ZH , ΦH , δH )  (ZH , ΦH , δH ) such that + 0 σ is mapped to a face of a cone ρ ⊂ P+ Φ0 under the canonical mapping PΦH → PΦH , H

0 ,ρ)] is contained in the closure Z[(Φ ,δ ,σ)] and if ρ ∈ ΣΦ0H ∈ Σ, then Z[(Φ0H ,δH H H

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0 ,ρ)] is of Z[(ΦH ,δH ,σ)] in Mtor H,Σ , and the completion of Z[(ΦH ,δH ,σ)] along Z[(Φ0H ,δH canonically isomorphic to

XΦH ,δH ,σ,ρ := (ΞΦH ,δH ,σ (ρ))∧ ΞΦ

H ,δH ,ρ

,

(1.3.2.40)

the formal completion of ΞΦH ,δH ,σ (ρ) along its ρ-stratum ΞΦH ,δH ,ρ . Lemma 1.3.2.41. Let XΦH ,δH = XΦH ,δH ,ΣΦH be the formal completion of ΞΦH ,δH along the union of the σ-strata ΞΦH ,δH ,σ for σ ∈ ΣΦH and σ ⊂ P+ ΦH . Then we have a canonical morphism XΦH ,δH → Mtor H,Σ

(1.3.2.42)

inducing a canonical isomorphism ∼

∧ XΦH ,δH /ΓΦH → (Mtor H,Σ )∪ Z[(Φ

H ,δH ,σ)]

,

(1.3.2.43)

where ∪ Z[(ΦH ,δH ,σ)] is the union of all strata Z[(ΦH ,δH ,σ)] with σ ∈ ΣΦH (and ∨ σ ⊂ P+ (resp. Lie∨ G∨ /Mtor , resp. ΦH ), under which the pullback of LieG/Mtor λ∗ : Lie∨ G∨ /Mtor

H,Σ

H,Σ

H,Σ

→ Lie∨ G/Mtor ) can be canonically identified with the pullH,Σ

∨ \ ∗ : Lie∨ back of Lie∨ G\ /CΦH ,δH (resp. LieG∨,\ /CΦH ,δH , resp. (λ ) G∨,\ /CΦH ,δH → ∨ LieG\ /CΦ ,δ ). For each stratum Z[(ΦH ,δH ,σ)] , the isomorphism (1.3.2.43) is compaH

H

∼

∧ tible with the isomorphism XΦH ,δH ,σ /ΓΦH ,σ → (Mtor H,Σ )Z[(ΦH ,δH ,σ)] in (5) of Theorem 1.3.1.3 (under the canonical morphisms XΦH ,δH ,σ /ΓΦH ,σ → XΦH ,δH /ΓΦH ∧ tor ∧ and (Mtor H,Σ )Z[(ΦH ,δH ,σ)] → (MH,Σ )∪ Z[(ΦH ,δH ,σ)] ). (Such isomorphisms are induced by strata-preserving isomorphisms from ´etale neighborhoods of points of ΞΦH ,δH ,σ in ΞΦH ,δH (σ) to ´etale neighborhoods of points of Z[(ΦH ,δH ,σ)] in Mtor H,Σ .)

Proof. The formal algebraic stack XΦH ,δH admits an open covering by open formal algebraic substacks Uσ , where each Uσ is the formal completion of the smooth algebraic stack ΞΦH ,δH (σ) along its closed sub-algebraic stack formed by the union of ΞΦH ,δH ,τ such that τ is a face of σ (which can be σ itself), τ ∈ ΣΦH , and τ ⊂ P+ ΦH . Using the Mumford family carried by XΦH ,δH (see [62, Sec. 6.2.5]), by (6) of Theorem 1.3.1.3, three exist morphisms Uσ → Mtor H,Σ which patch together and form the desired canonical morphism (1.3.2.42), which is unchanged under the canonical action of ΓΦH and hence factors through a canonical morphism ∧ XΦH ,δH /ΓΦH → (Mtor H,Σ )∪ Z[(Φ

H ,δH ,σ)]

.

(1.3.2.44)

On the other hand, by the construction of Mtor H,Σ by gluing good algebraic models (see [62, Sec. 6.3]) in the ´etale topology, the pullback of the tautological object tor ∧ (G, λ, i, αH ) → Mtor H,Σ to (MH,Σ )∪ Z[(ΦH ,δH ,σ)] define degeneration data parameterized by XΦH ,δH /ΓΦH , and hence there is a canonical morphism giving the inverse of (1.3.2.44) and induces the canonical isomorphism (1.3.2.43). (This explains the last parenthetical remark in the statement of the lemma, because the good algebraic models carry approximations of the degeneration data, which include in particular

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trivializations of the invertible sheaves ΨΦH ,δH (`), which determine the stratifica\ ∨,\ tions.) Moreover, since the pullbacks of (G, G∨ , λ) → Mtor , λ\ ) → H,Σ and (G , G ∨ CΦH ,δH induce canonically isomorphic formal completions (Gfor , Gfor , λfor ) → Uσ \ and (G\for , G∨,\ for , λfor ) → Uσ (by the theory of degeneration) over each Uσ , the pul∨ ∨ ∨ ∗ lback of LieG/Mtor (resp. Lie∨ G∨ /Mtor , resp. λ : LieG∨ /Mtor → LieG/Mtor ) can be H,Σ

H,Σ

H,Σ

H,Σ

∨ canonically identified with the pullback of Lie∨ G\ /CΦH ,δH (resp. LieG∨,\ /CΦH ,δH , resp. ∨ (λ\ )∗ : Lie∨ G∨,\ /CΦH ,δH → LieG\ /CΦH ,δH ) under (1.3.2.43). Since (1.3.2.43) and the canonical isomorphism in (5) of Theorem 1.3.1.3 are both defined by the universal properties given in terms of degeneration data, they are naturally compatible with each other.

Proposition 1.3.2.45. (Compare with Propositions 1.3.1.15 and 1.3.2.24.) By considering compatible Q× -isogenies (f : G\ → G\,0 , f ∨ : G∨,\,0 → G∨,\ ) compatible with the homomorphisms (ι : Y → G\ , ι∨ : X → G∨,\ ) inducing isomorphisms on the torus parts T and T ∨ and on the domains of ι and ι∨ , we can define Hecke twists of the tautological object (G\ , λ\ , i\ , τ, βH ) → ΞΦH ,δH by elements of P0Z (A∞ ), and define the Hecke action of P0Z (A∞ ) on the collection {ΞΦH ,δH }HP0 , realized Z by finite ´etale surjections pulling tautological objects back to Hecke twists, which is compatible with the Hecke action of G1,Z (A∞ ) on the collection {CΦH ,δH }HG1,Z under the canonical morphisms ΞΦH ,δH → CΦH ,δH (with varying H) and the canonical homomorphism P0Z (A∞ ) → G1,Z (A∞ ) = P0Z (A∞ )/U2,Z (A∞ ). By also considering Q× -isogenies (f : G\ → G\,0 , f ∨ : G∨,\,0 → G∨,\ ) compatible with the homomorphisms (ι : Y → G\ , ι∨ : X → G∨,\ ) inducing Q× -isogenies on the torus parts T and T ∨ and on the domains of ι and ι∨ (possibly varying the isomorphism classes of the O-lattices X and Y ), we can also define Hecke twists of the tautological object (G\ , λ\ , i\ , τ, βH ) → CΦH ,δH by elements of PZ (A∞ ), and ` define the Hecke action of PZ (A∞ ) on the collection { ΞΦH ,δH }HPZ , realized by finite ´etale surjections pulling tautological objects back to Hecke twists, where the disjoint unions are over classes [(ZH , ΦH , δH )] sharing the same ZH , which induces an action of G0l,Z (A∞ ) = PZ (A∞ )/P0Z (A∞ ) on the index sets {[(ZH , ΦH , δH )]}, which is compatible with the Hecke action of PZ (A∞ )/U2,Z (A∞ ) on the collection ` { CΦH ,δH }HPZ /HU2,Z (with the same index sets and the same induced action of G0l,Z (A∞ )) under the canonical morphisms ΞΦH ,δH → CΦH ,δH (with varying H) and the canonical homomorphism PZ (A∞ ) → PZ (A∞ )/U2,Z (A∞ ). Any such Hecke action 0 [g] : ΞΦ0H0 ,δH → ΞΦH ,δH 0 0 covering [g] : CΦ0H0 ,δH → CΦH ,δH induces a (finite ´etale) morphism 0 0 ΞΦ0H0 ,δH → ΞΦH ,δH 0

× CΦH ,δH

0 CΦ0H0 ,δH 0

0 , which is equivariant with the morphism between torus torsors over CΦ0H0 ,δH 0 0 EΦH0 → EΦH dual to the homomorphism SΦH → SΦ0H0 induced by the pair of mor-

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∼

phisms (fX : X ⊗ Q → X 0 ⊗ Q, fY : Y 0 ⊗ Q → Y ⊗ Q) defining the g-assignment Z

Z

Z

Z

0 (Z0H0 , Φ0H0 , δH 0 ) →g (ZH , ΦH , δH ) of cusp labels (cf. [62, Def. 5.4.3.9]). 0 If g ∈ PZ (A∞ ) is as above and if (Φ0H0 , δH 0 , ρ) is a g-refinement of (ΦH , δH , σ) as in [62, Def. 6.4.3.1], then there is a canonical morphism 0 (ρ) → ΞΦ ,δ (σ) [g] : ΞΦ0H0 ,δH H H 0

(1.3.2.46)

0 0 covering [g] : CΦ0H0 ,δH → CΦH ,δH , extending [g] : ΞΦ0H0 ,δH → ΞΦH ,δH , mapping 0 0 0 0 ΞΦH0 ,δH0 ,ρ to ΞΦH ,δH ,σ , and inducing a canonical morphism 0 ,ρ → XΦ ,δ ,σ . [g] : XΦ0H0 ,δH H H 0

(1.3.2.47)

0 0 If g ∈ PZ (A∞ ) is as above and if (Φ0H0 , δH 0 , ΣΦ0 ) is a g-refinement of (ΦH , δH , ΣΦH ) H0 as in [62, Def. 6.4.3.2], then morphisms like (1.3.2.46) patch together and define a canonical morphism 0 ,Σ0 [g] : ΞΦ0H0 ,δH 0 0 Φ

H0

→ ΞΦH ,δH ,ΣΦH

(1.3.2.48)

0 0 covering [g] : CΦ0H0 ,δH → CΦH ,δH , extending [g] : ΞΦ0H0 ,δH → ΞΦH ,δH , and inducing 0 0 a canonical morphism

0 ,Σ0 [g] : XΦ0H0 ,δH 0 0 Φ

H0

→ XΦH ,δH ,ΣΦH

(1.3.2.49)

compatible with each (1.3.2.47) as above (under canonical morphisms). If g ∈ PZ (A∞ ) and if we have a collection Σ0 for MH0 that is a g-refinement of a collection Σ for MH as in [62, Def. 6.4.3.3], then the canonical morphism tor

[g]

tor : Mtor H0 ,Σ0 → MH,Σ

0 as in Proposition 1.3.1.15 is compatible with (1.3.2.47) when (Φ0H0 , δH 0 , ρ) is a g-refinement of (ΦH , δH , σ), under the canonical isomorphisms as in (5) of Theorem 0 0 1.3.1.3; and is compatible with (1.3.2.49) when (Φ0H0 , δH 0 , ΣΦ0 ) is a g-refinement H0 of (ΦH , δH , ΣΦH ), under the canonical isomorphisms as in Lemma 1.3.2.41.

Proof. The assertions in the first two paragraphs can be justified as in the case of MH . (We omit the details for simplicity.) The third paragraph follows by comparing the torus torsor actions of sufficiently divisible multiples of elements, for which we have explicit descriptions in Lemma 1.3.2.28 and Proposition 1.3.2.31. As for the last paragraph, since the canonical morphisms are defined by universal properties given in terms of degeneration data, their compatibility follows from the fact that (by the theory of degeneration [62, Thm. 5.3.1.19] based on [62, Thm. 5.2.3.14], in particular) the Hecke twist of the tautological tuple over Mtor H0 ,Σ0 by g defined using the level structure αH0 over MH0 is compatible with the Hecke twist of the 0 (ρ) by g defined using the level structure βH0 over tautological tuple over ΞΦ0H0 ,δH 0 0 0 ΞΦH0 ,δH0 .

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e etc be chosen as in Section 1.2.4. Let κ e h · , · ie, e e δ), Now let (L, h0 ), (e Z, Φ, e = e ++ as in Definition 1.2.4.11. The data of e Σ, e σ (H, e) be any element in the set K Q,H e e as in Section 1.1.2. ˆ define a moduli problem M e h · , · ie, e e ⊂ G( e Z) O, (L, h0 ), and H H e is neat and Σ e is projective (and smooth), by Theorems 1.3.1.3 and 1.3.1.10, Since H e tor = M e tor of M e e which is projective and we have a toroidal compactification M e H

H

eΣ e H,

smooth over S0 . We are mainly interested in comparing the boundary structures of e tor and Mtor under suitable conditions. M H,Σ eΣ e H, ˘ satisfying In the remainder of this subsection, let us fix the choice of a Z (1.2.4.12), so that we have the groups and homomorphisms defined in Definitions 1.2.4.53 and 1.2.4.54. Suppose that the cusp label [(ZH , ΦH , δH )] at level H is canonically assigned (as ˘ e , δ˘ e )] at level H e admitting a surjection ˘ e, Φ in Lemma 1.2.4.15) to a cusp label [(Z H H H e e to [(e ZHe , ΦHe , δHe )], so that we have (1.2.4.18), (1.2.4.19), and (1.2.4.20), and the definitions following them. Lemma 1.3.2.50. (Compare with [61, Lem. 4.9; see also the errata].) By comparing the universal properties, we obtain a canonical morphism e ˘ ˘ → CΦ ,δ , C H H Φ f,δ f H

(1.3.2.51)

H

etale-locally-defined pairs by sending (˘ cHe , c˘∨ e ), which is an orbit of ´ H (˘ cn :

1 ˘ nX

→ B ∨ , c˘∨ n :

1 ˘ nY

→ B)

e e to the orbit (cH , c∨ ) of ´etale-locallyfor some integer n ≥ 1 such that U(n) ⊂ H, H defined pairs (cn :

1 nX

→ B ∨ , c∨ n :

1 nY

→ B),

with (cn , c∨ cn , c˘∨ and n1 Y , where X and Y are n ) induced by (˘ n ) by restrictions to ˘ X e and s ˘ : Y˘  Ye , respectively. the kernels of the admissible surjections sX˘ : X Y (This definition canonically extends to a compatible definition in the Q× -isogeny class language in Proposition 1.3.2.14, which we omit for simplicity.) bG = H, The morphisms (1.3.2.51) and (1.3.2.52) are proper and smooth. If H ˘f Φ e H∼ then M = MΦH and there is a canonical homomorphism 1 nX

e H

H

grp e grp C ˘ ,δ˘ → CΦH ,δH Φ f H

(1.3.2.52)

f H

H of abelian schemes over MΦ H , which can be identified with the canonical homomorphism

˘ B)◦ → Hom (X, B)◦ HomO (X, O

(1.3.2.53)

H e grp - and C grp -torsor strucup to canonical Q× -isogenies over MΦ H , and the CΦ ΦH ,δH ˘ ,δ˘ f H

f H

e ˘ ˘ → MΦH and CΦ ,δ → MΦH , respectively, are compatible with tures of C H H H H ΦH f,δH f each other under (1.3.2.51) and (1.3.2.52). (See [62, Def. 5.2.3.8 and Prop. 5.2.3.9]

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˘ B)◦ and for the formation of the fiberwise geometric identity components HomO (X, ◦ HomO (X, B) . See also the beginning of Section 1.3.3 below.) Moreover, the kernel × H of (1.3.2.52) is an abelian scheme over MΦ H , which is canonically Q -isogenous to e B)◦ of (1.3.2.53), and (1.3.2.51) is a torsor under the pullback the kernel HomO (X, bG = H or to CΦH ,δH of this abelian scheme. We deduce from these that, whether H not, we have Ω1Ce

˘ /CΦH ,δH ˘ ,δ Φ f H f H

∼ e ˘ ˘ → MZH )∗ Hom (X, e Lie∨ ZH ), = (C O H Φ f,δ f B/M H

H

(1.3.2.54)

H

and the canonical short exact sequence e ˘ ˘ → CΦ ,δ )∗ Ω1 0 → (C H H Φ f,δ f C H

H

ZH ΦH ,δH /MH

→ Ω1Ce

ZH ˘ /MH ˘ ,δ Φ f H f H

→ Ω1Ce

˘ /CΦH ,δH ˘ ,δ Φ f H f H

→0

e ˘ ˘ → MZH of the canonical short exact can be identified with the pullback under C H ΦH f,δH f sequence ∨ ˘ 0 → HomO (X, Lie∨ Z ) → HomO (X, Lie Z ) B/M H B/M H H

H

e Lie∨ ZH ) → 0 → HomO (X, B/M H

under canonical morphisms (as in (1.3.2.8) and (1.3.2.54)). ˘f ˘f e ΦH e ΦH e˘ ˘ → M etale covering M The abelian scheme torsor C e and the finite ´ e → Φ f,δ f H H H

H

f e ˘ZH b e b and (Z ˘ b , δ˘ b ) (see ˘ b, Φ M e depend (up to canonical isomorphism) only on H = HG H H H H

˘

˘

c c c b ΦH b ΦH b ˘ZH b˘ ˘ → M Definition 1.2.4.17). We shall denote them as C b and MH b → MH b ΦH c,δH c H when we want to emphasize this (in)dependence.

Proof. The first paragraph is self-explanatory. As for the second paragraph, by e= Lemma 1.3.2.7, it suffices to verify the statements in the case H = U(n) and H e U(n) for some integer n ≥ 1. (The third paragraph also follows by Lemma 1.3.2.7.) e grp e ˘ ˘ and CΦ ,δ = C grp e˘ ˘ = C In this case, C = C H H ΦH ,δH = CΦn ,δn are ˘ ,δ˘ Φ f,δ f Φn ,δn Φ H

H

˘

f H

f H

f Zn e ΦH e ˘Zn ∼ ΦH abelian schemes over M e = Mn = MH = Mn . For simplicity, let us denote the H kernel of (1.3.2.51) by C, viewed as a scheme over MZnn . e ˘ ˘ → MZnn parameterizes liftings (to level While the abelian scheme torsor C Φn ,δn ˘ → B ∨ , c˘∨ : Y˘ → B) satisfying the compatibility n) of pairs of the form (˘ c : X c˘φ˘ = λB c˘∨ and the liftability and pairing conditions, and while the abelian scheme

torsor CΦn ,δn → MZnn parameterizes liftings (to level n) of pairs of the form (c : X → B ∨ , c∨ : Y → B) satisfying the compatibility cφ = λB c∨ and the liftability and pairing conditions, the scheme C → MZnn parameterizes liftings of pairs of e → B∨, e the form (e c : X c∨ : Ye → B) satisfying the compatibility e cφe = λB e c∨ and the liftability and pairing conditions induced by the ones of the pairs over e ˘ ˘ → MZnn . Therefore, the same (component annihilating) argument in [62, Sec. C Φn ,δn e B)◦ . 6.2.3–6.2.4] shows that C is an abelian scheme Q× -isogenous to Hom (X, O

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Consequently, all geometric fibers of the morphism (1.3.2.51) are smooth and e˘ ˘ have the same dimension (as the relative dimension of C → MZnn ). Since both C Φn ,δn and CΦn ,δn are smooth over S0 , the morphism (1.3.2.51) is smooth by [35, IV3, 15.4.2 e’)⇒b), and IV-4, 17.5.1 b)⇒a)]. By [10, Sec. 2.2, Prop. 14], smooth morphisms between schemes have sections ´etale locally. This shows that (1.3.2.51) is a torsor under the pullback of C to CΦn ,δn . (Regardless of this argument, the e ˘ ˘ → MZnn is.) morphism (1.3.2.51) is proper because the morphism C Φn ,δn Proposition 1.3.2.55. Under the canonical morphisms as in (1.3.2.51) (with e and H), and under the canonical homomorphisms G b ˘ (A∞ ) → varying H 1,Z e ˘ e(A∞ )/U e ˘ (A∞ ) → PZ (A∞ )/U2,Z (A∞ ), the Hecke action of G1,Z (A∞ ) and P 2,Z Z,Z b ˘ (A∞ ) on the collection {C b˘ ˘ } b is compatible with the Hecke action G 1,Z Φ c,δ c H b H

H

G

1,˘ Z

of G1,Z (A∞ ) on the collection {CΦH ,δH }HG1,Z (see Proposition 1.3.2.24); the e ˘ e(A∞ )/U e ˘ (A∞ ) (see Definition 1.2.4.54) on the collection Hecke action of P 2,Z Z,Z `e is compatible with the Hecke action of PZ (A∞ )/U2,Z (A∞ ) { C˘ ˘ } e e ΦH e f,δH f HP

˘ Z,e Z

/HU e

˘

`2,Z e ˘ e(A∞ ) on on the collection { CΦH ,δH }HPZ /HU2,Z ; and the induced action of G l,Z,Z e e , δe e )]} is compatible with the induced action of G0 (A∞ ) the index sets {[(e ZHe , Φ l,Z H H on the index sets {[(ZH , ΦH , δH )]} (again see Proposition 1.3.2.24) under the canoe ˘ e(A∞ ) → G0 (A∞ ). nical homomorphism G l,Z l,Z,Z b ˘ (A∞ )/U b ˘ (A∞ ) These Hecke actions induce a Hecke action of the subgroup P Z 2,Z ` e ˘ e(A∞ )/U e ˘ (A∞ ) on the collection { C b˘ ˘ } b b of P , which is compatible 2, Z Z,Z ΦH b /HU b c,δH c HP ˘ Z 2,˘ Z ` with the Hecke action of PZ (A∞ )/U2,Z (A∞ ) on the collection { CΦH ,δH }HPZ /HU2,Z b ˘ ˘ → CΦ ,δ (with varying H b and H) and the under the canonical morphisms C ΦH c,δH c

H

H

b ˘ (A∞ )/U b ˘ (A∞ ) → PZ (A∞ )/U2,Z (A∞ ); and the inducanonical homomorphism P Z 2,Z 0 ∞ b e ˘ e(A∞ ) on the index sets {[(Z ˘ b , δ˘ b )]} ˘ b, Φ ced action of the subgroup Gl,˘Z (A ) of G l,Z,Z H H H 0 ∞ is compatible with the induced action of Gl,Z (A ) on the index sets {[(ZH , ΦH , δH )]} b 0 (A∞ ) → G0 (A∞ ). under the canonical homomorphism G l,Z l,˘ Z Proof. The canonical morphisms as in (1.3.2.51) correspond to pushouts of extensions of B (resp. B ∨ ) by T˘ (resp. T˘∨ ) under the canonical homomorphism T˘ → T ˘ (resp. Y˘ ) to X (resp. Y ). Hence, (resp. T˘∨ → T ∨ ) induced by the restriction from X the realizations of the Hecke twists are compatible in the desired ways. (We omit the details for simplicity.) Suppose σ e ⊂ P+ is a top-dimensional nondegenerate rational polyhedral cone e Φ f H

e e in Σ, e and suppose σ in the cone decomposition Σ ˘ is the image of σ e ⊂ P+ e under Φf Φ H

the first morphism in (1.2.4.20). Then we have

f H

b˘ σ ˘⊥ = S Φc H

b=H e b , and we have the following: (see Definition 1.2.4.29) for any such σ ˘ , where H G

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Proposition 1.3.2.56. (Compare with Lemma 1.3.2.28 and Proposition 1.3.2.31.) The scheme   ∼ ˘ e e ⊕ ΨΦ˘ f,δ˘f (`) ΞΦ˘ f,δ˘f,˘σ = SpecO H

H

H

˘ σ⊥ `∈˘

e C ˘ ˘ ,δ Φ f H f H

H

e ˘ ˘ is a torsor under the split torus E e˘ ˘ ⊥ , which over C σ with character group σ ΦH ΦH f,˘ f,δH f b ˘ , which b ˘ with character group S is canonically isomorphic to the split torus E ΦH ΦH c c b b = (H ee ∩ H e e 0 )/H ee depends only on H (see Definition 1.2.4.53). We have P˘Z

b∨ b ∨ /S S ˘ ˘ Φ Φ 1

c H

P˘Z

PeZ

U2,eZ

∼ ˆ H b˘ b ˘ := S b ˘ (Z)/ b b , where S = U 2,Z Φb Φ1 U2,˘Z

G(ˆ Z)

is the kernel of the canonical

homomorphism SΦ˘ 1  SΦ e 1 (see Definition 1.2.4.29). e˘ ˘ e ˘ ˘ is universal for the additional strucThe torus torsor S := Ξ ΦH σ → CΦ f,δH f,˘ f,δH f H b b /U(n) b b -orbits of ´etale-locally-defined pairs (b tures (b τ b , τb∨ ), which are H τn , τbn∨ ), H

P˘Z

b H

where:

(1) τbn : 1 1 Y n

P˘Z

∼

˘ × X,S

⊗ −1 1 ˘)∗ PB is a trivialization of biextensions. → (˘ c∨ YS × c n|n ∼

⊗ −1 (2) τbn∨ : 1 1 Y˘ × X,S → (˘ c∨ ˘|XS )∗ PB is a trivialization of biextensions. n ×c n ∨ (3) τbn and τbn satisfy the analogues of the usual O-compatibility condition. (4) τbn and τbn∨ satisfy the symmetry condition that τbn |1Y × Y˘ ,S and τbn∨ |1Y˘ × Y,S coincide under the canonical isomorphism induced by the swapping isomor∼ phism 1Y × Y˘ ,S → 1Y˘ × Y,S and the symmetry automorphism of PB . (5) τbn |1 1 Y × X,S = τbn∨ |1 1 Y × X,S . n

n

b ˘ ˘ when we want to emphasize that (by Lemma e˘ ˘ We shall denote Ξ ΦH σ by ΞΦ c,δH c f,δH f,˘ H b e b and (Z ˘ b , δ˘ b ) (see Definition 1.2.4.17), ˘ b, Φ 1.3.2.25) it depends only on H = H G H H H but does not depend on the choice of σ ˘. b ˘ ˘ defines a homomorphism b ˘ -torsor structure of Ξ b˘ ˘ → C The E Φc Φ c,δ c Φ c,δ c H

H

H

H

H

˘ b ˘ → Pic(C b ˘ ˘ ) : `˘ 7→ Ψ b ˘ ˘ (`), S Φc Φ c,δ c Φ c,δ c H

H

H

H

H

˘ over C b ˘ an invertible sheaf Ψ b ˘ ˘ (`) b ˘ ˘ (up to isoassigning to each `˘ ∈ S ΦH ΦH ΦH c c,δH c c,δH c morphism), together with isomorphisms b ∗˘ ∆ Φ

˘

˘ ˘0

c,δH c,`,` H

˘ b ˘ ˘ (`) :Ψ Φ c,δ c H

H

OCb

⊗

∼ b b ˘ ˘ (`˘0 ) → Ψ ΨΦ˘ c,δ˘c (`˘ + `˘0 ) Φ c,δ c H

H

H

H

˘ ˘ ,δ Φ c H c H

˘ `˘0 ∈ S b ˘ , satisfying the necessary compatibilities with each other making for all `, ΦH c ˘ an O b b ˘ ˘ (`) ⊕ Ψ Φ c,δ c C ˘ ˘ -algebra, such that

˘ S b˘ `∈ Φ

H

H

Φ c,δ c H H

c H

b˘ ˘ ∼ Ξ Φ c,δ c = SpecO H

H

b C ˘ ˘ ,δ Φ c H c H



 ˘ b ΨΦ˘ c,δ˘c (`) .

⊕ ˘ S b˘ `∈ Φ

H

c H

H

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˘ such that either y ∈ Y or χ ∈ X, we When `˘ = [y ⊗ χ] for some y ∈ Y˘ and χ ∈ X have a canonical isomorphism ˘ ∼ b ˘ ˘ (`) Ψ c∨ (y), c˘(χ))∗ PB . = (˘ Φ c,δ c H

H

˘ n , then the canonical morphism ˘n and) Φ If we fix the choice of (Z b˘ ˘ → Ξ b˘ ˘ Ξ Φn ,δn Φ c,δ c H

(1.3.2.57)

H

b b 0 /U(n) b b 0 -torsor, and induces an isomorphism is an H P P ˘ Z

˘ Z

∼ b b b 0 /U(n) b b0 ) → b ˘ ˘ /(H ΞΦ˘ c,δ˘c Ξ Φn ,δn P P ˘ Z

(1.3.2.58)

H

H

˘ Z

(cf. Lemma 1.3.2.25). e ˘ ˘ as in Lemmas Proof. These follow from the corresponding properties of Ξ ΦH f,δH f 1.3.2.25 and 1.3.2.28, and Proposition 1.3.2.31, because the restriction from SΦ˘ f to H b ˘ (see Definition 1.2.4.29) corresponds to taking orbits of restrictithe subgroup S ΦH c

∼

⊗ −1 ons of τ˘n : 1 1 Y˘ × X,S → (˘ c∨ × c˘)∗ PB to 1 1 Y ˘ n n the pairs (b τn , τbn∨ ) as above.

˘ × X,S

and 1 1 Y˘ × X,S , which form n

For each rational polyhedral cone ρ˘ ⊂ (SΦ˘ f )∨ ˘ as a face, we have an R having σ H affine toroidal embedding   ˘ e e e ΞΦ˘ f,δ˘f,˘σ ,→ ΞΦ˘ f,δ˘f,˘σ (˘ ρ) := SpecO ⊕ ΨΦ˘ f,δ˘f (`) (1.3.2.59) H

H

H

H

e C ˘ Φ

˘ f,δH f H

H

˘ σ ⊥ ∩ ρ˘∨ `∈˘

H

as in (1.3.2.32). b ˘ )∨ , we have an affine In general, for each rational polyhedral cone ρb ⊂ (S ΦH c R toroidal embedding   ˘ b b b ρ) := SpecO (1.3.2.60) ⊕ ΨΦ˘ c,δ˘c (`) . ΞΦ˘ c,δ˘c ,→ ΞΦ˘ c,δ˘c (b H

H

H

H

b C ˘ ˘ ,δ Φ c H c H

H

˘ ρ∨ `∈b

H

By Proposition 1.3.2.56, (1.3.2.59) and (1.3.2.60) can be canonically identified b=H e b , when (Z ˘ b , δ˘ b ) is determined by (Z ˘ e , δ˘ e ) as in Definition ˘ b, Φ ˘ e, Φ when H G H H H H H H 1.2.4.17, and when ρb = pr(Sb ˘ )∨ (˘ ρ). (Hence, (1.3.2.59) depends only on these Φc R H

induced parameters.) b ˘ ˘ , where Ξ b ˘ ˘ (b Both sides of (1.3.2.60) are relative affine over C ρ) → ΦH ΦH c,δH c c,δH c b b CΦ˘ c,δ˘c is smooth when the cone ρb is smooth. The ρb-stratum of ΞΦ˘ c,δ˘c (b ρ) is H H H H   ˘ , b˘ ˘ b ˘ ˘ (`) ⊕ Ψ (1.3.2.61) Ξ Φ c,δ c,b ρ := SpecO Φ c,δ c H

H

b C ˘ ˘ ,δ Φ c H c H

˘ ρ˘⊥ `∈

H

H

e˘ ˘ which is canonically isomorphic to Ξ ΦH ˘ (cf. (1.3.2.33)). The affine morphism f,δH f,ρ ∼ ˘ b b b˘ ΞΦ˘ c,δ˘c,bρ → CΦ˘ c,δ˘c is a torsor under the torus E ΦH ρ = EΦ ˘ with character group c,b f,ρ H H H H H ⊥ ∼ ⊥ ρb = ρ˘ . (Note that these two instance of ⊥ are taken in different ambient spaces.)

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b ˘ as in b ˘ of P For each ΓΦ˘ c -admissible rational polyhedral cone decomposition Σ ΦH ΦH c c H Definition 1.2.4.40, we have (as in (1.3.2.34)) a toroidal embedding b ˘ ˘ ,→ Ξ b˘ ˘ = Ξ b˘ ˘ b , Ξ Φ c,δ c Φ c,δ c Φ c,δ c,Σ ˘ H

H

H

H

H

H

(1.3.2.62)

Φc H

b ˘ ˘ , with an open the right-hand side being only locally of finite type over C ΦH c,δH c covering b˘ ˘ = Ξ Φ c,δ c H

H

∪ b˘ ρ b∈Σ Φ

b ˘ ˘ (b Ξ Φ c,δ c ρ),

(1.3.2.63)

b˘ ˘ Ξ Φ c,δ c,b ρ

(1.3.2.64)

H

H

c H

(cf. (1.3.2.35)) inducing a stratification b˘ ˘ = Ξ Φ c,δ c H

a

H

H

b˘ ρ b∈Σ Φ

H

c H

` (cf. (1.3.2.36)). (The notation “ ” only means a set-theoretic disjoint union. The b ˘ ˘ .) Let algebro-geometric structure is still the one inherited from Ξ Φ c,δ c H

b˘ ˘ X Φ c,δ c,b ρ := H

H

b ˘ ˘ (b ρ))∧ (Ξ b˘ ΦH Ξ c,δH c Φ

H

(1.3.2.65)

˘ ,ρ c,δH c b H

b˘ ˘ , b ˘ ˘ (b ρ) along its ρb-stratum Ξ (cf. (1.3.2.39)), the formal completion of Ξ ΦH ρ ΦH c,δH c,b c,δH c e˘ ˘ e ˘ ˘ (˘ ρ , the formal completion of Ξ which is canonically isomorphic to X ΦH σ ), ΦH σ ,ρ˘ f,δH f,˘ f,δH f,˘ e˘ ˘ e ˘ ˘ (˘ e˘ ˘ (cf. (1.3.2.40)). ρ), along its ρ˘-stratum Ξ in Ξ the closure of Ξ ΦH σ f,δH f,˘

Also, let us define

ΦH ˘ f,δH f,ρ

ΦH f,δH f

(1.3.2.66)

b˘ ˘ b b˘ ˘ = X X Φ c,δ c Φ c,δ c,Σ ˘ H

H

H

Φc H

H

b˘ ˘ b (cf. Lemma 1.3.2.41) to be the formal completion of Ξ Φ c,δ c,Σ ˘ H

H

b+ . b˘ ˘ b ˘ and ρb ⊂ P the σ b-strata Ξ b∈ Σ ˘ Φc Φ c,δ c,b ρ for ρ Φ H

H

H

along the union of

Φc H

c H

Proposition 1.3.2.67. (Compare with Propositions 1.3.1.15, 1.3.2.24, 1.3.2.45, b 0 (A∞ ) on the collection {Ξ b˘ ˘ } b , and 1.3.2.55.) There is a Hecke action of P ˘ Φ c,δ c H b 0 Z H

H

P ˘ Z

realized by finite ´etale surjections pulling tautological objects back to Hecke b ˘ (A∞ ) on the collection twists, which is compatible with the Hecke action of G 1,Z b˘ ˘ } b b˘ ˘ → C b ˘ ˘ (with varying H) b {C under the canonical morphisms Ξ Φ c,δ c H b Φ c,δ c Φ c,δ c H

H

G

H

1,˘ Z

H

H

H

b 0 (A∞ ) → G b ˘ (A∞ ) = P b 0 (A∞ )/U b ˘ (A∞ ). and the canonical homomorphism P 1,Z 2,Z ˘ ˘ Z Z` b ˘ (A∞ ) on the collection { Ξ b ˘ ˘ } b , where There is also a Hecke action of P ΦH b c,δH c HP

Z

˘ Z

˘ b , δ˘ b )] sharing the same Z ˘ b , realized by ˘ b, Φ the disjoint unions are over classes [(Z H H H H finite ´etale surjections pulling tautological objects back to Hecke twists, which indub 0 (A∞ ) = P b ˘ (A∞ )/P b 0 (A∞ ) on the index sets {[(Z ˘ b , δ˘ b )]}, ˘ b, Φ ces an action of G Z ˘ H H H l,˘ Z Z b ˘ (A∞ )/U b ˘ (A∞ ) on the collection which is compatible with the Hecke action of P Z

2,Z

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`b { C ˘ c,δ˘c }H bb Φ H

H

b P˘ /HU Z 2,˘ Z

b

page 91

91

(with the same index sets and the same induced action of

b 0 (A∞ )) under the canonical morphisms Ξ b˘ ˘ → C b ˘ ˘ (with varying H) b and G ΦH ΦH l,˘ Z c,δH c c,δH c b ˘ (A∞ ) → P b ˘ (A∞ )/U b ˘ (A∞ ). the canonical homomorphism P Z

2,Z

Z

Any such Hecke action b ˘0 [b g] : Ξ Φ

,δ˘0c0 c0 H H

b˘ 0 covering [b g] : C Φ

,δ˘0c0 c0 H H

H

H

H

b˘ ˘ →Ξ Φ c,δ c

,δ˘0c0 c0 H H

H

b˘ 0 between torus torsors over C Φ

,δ˘0c0 c0 H H

c0 H

H

b ˘ ˘ induces a morphism →C Φ c,δ c b ˘0 Ξ Φ

b˘ 0 E Φ

b˘ ˘ →Ξ Φ c,δ c

H

× b˘ C Φ

b˘ 0 C Φ

,δ˘0c0 c0 H H

˘ c,δH c H

, which is equivariant with the morphism

b ˘0 b˘ → S b ˘ dual to the homomorphism S → E Φ Φc Φc H

˘ ⊗Q → X ˘ 0 ⊗ Q, f ˘ : X Y

of morphisms (fX˘

c0 H

H

∼

Z

induced by the pair ∼

: Y˘ 0 ⊗ Q → Y˘ ⊗ Q) defining

Z

Z

Z

˘ b , δ˘ b ) of cusp labels (which is the ˘ 0 , δ˘0 ) →gb (Z ˘ b, Φ ˘0b 0 , Φ the gb-assignment (Z b0 b0 H H H H H H ∞ e ˘ (A ) ∩ P e 0 (A∞ ) lifting gb ∈ P b ˘ (A∞ ) = ge-assignment for any element ge ∈ P Z

Z

e Z

e ˘ (A∞ ) ∩ P e 0 (A∞ ))/U e ˘ (A∞ ), which is nevertheless independent of the choice of (P Z 2,Z e Z ge; cf. Lemma 1.2.4.42 and [62, Def. 5.4.3.9]). ˘ b , δ˘ b , ρb) b ˘ (A∞ ) is as above and if (Φ ˘ 0 , δ˘0 , ρb0 ) is a gb-refinement of (Φ If gb ∈ P Z b0 b0 H H H H (cf. Lemma 1.2.4.42 and [62, Def. 6.4.3.1]), then there is a canonical morphism b ˘0 [b g] : Ξ Φ

,δ˘0c0 c0 H H

b ˘ ˘ (b (b ρ0 ) → Ξ Φ c,δ c ρ)

b˘ 0 (cf. (1.3.2.46)) covering [b g] : C Φ

,δ˘0c0 c0 H H

b ˘0 b ˘ ˘ , mapping Ξ Ξ Φ Φ c,δ c H

H

,δ˘0c0 ,b ρ0 c0 H H

H

(1.3.2.68)

H

b ˘ ˘ , extending [b b ˘0 → C g] : Ξ Φ c,δ c Φ H

,δ˘0c0 c0 H H

H

→

b ˘ ˘ , and inducing a canonical morphism to Ξ Φ c,δ c,b ρ H

H

b ˘0 [b g] : X Φ

,δ˘0c0 ,b ρ0 c0 H H

b˘ ˘ →X Φ c,δ c,b ρ H

(1.3.2.69)

H

b 0 0 ) is a gb-refinement b ˘ (A∞ ) is as above and if (Φ ˘ 0 , δ˘0 , Σ (cf. (1.3.2.47)). If gb ∈ P Z ˘ b0 H b0 Φ H c0 H

˘ b , δ˘ b , Σ b ˘ ) (cf. Lemma 1.2.4.42 and [62, Def. 6.4.3.2]), then morphisms like of (Φ ΦH H H c (1.3.2.68) patch together and define a canonical morphism b ˘0 [b g] : Ξ Φ

b0 ,δ˘0c0 ,Σ ˘0 c0 H Φ H c0 H

b˘ 0 (cf. (1.3.2.48)) covering [b g] : C Φ

,δ˘0c0 c0 H H

b˘ ˘ b →Ξ Φ c,δ c,Σ ˘ H

H

(1.3.2.70)

Φc H

b ˘ ˘ , extending [b b ˘0 → C g] : Ξ Φ c,δ c Φ H

,δ˘0c0 c0 H H

H

→

b ˘ ˘ , and inducing a canonical morphism Ξ Φ c,δ c H

H

b ˘0 [b g] : X Φ

b0 ,δ˘0c0 ,Σ ˘0 c0 H Φ H c0 H

b˘ ˘ b →X Φ c,δ c,Σ ˘ H

H

(1.3.2.71)

Φc H

(cf. (1.3.2.49)) compatible with each (1.3.2.69) as above (under canonical morphisms).

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Proof. By Proposition 1.3.2.56 (see in particular (1.3.2.58)), the assertions in the first three paragraphs are reduced to the ones for the principal levels, which `e then follow from the corresponding assertions for the collection { Ξ ˘ f,δ˘f }H e e0 Φ H

H

P ˘ Z

e ˘ (A∞ ) to P e ˘ (A∞ ) ∩ P e 0 (A∞ )), because the tauto(by restricting the action of P Z Z e Z b˘ ˘ = Ξ b˘ logical objects over Ξ are canonically induced by those over Φn ,δn Φb ,δ˘ b U (n)

e˘ ˘ = Ξ e˘ Ξ Φn ,δn Φe

. ˘e U (n) ,δU (n)

U (n)

The assertions in the last paragraph then follow from the

universal properties of toroidal embeddings (cf. [62, Prop. 6.2.5.11]). Lemma 1.3.2.72. (Compare with Lemma 1.3.2.50.) By comparing the universal properties, we obtain a canonical morphism e ˘ ˘ → ΞΦ ,δ Ξ H H Φ f,δ f H

(1.3.2.73)

H

covering (1.3.2.51), by sending τ˘He , which is an orbit of ´etale-locally-defined tri∼ ⊗ −1 → (˘ c∨ × c˘)∗ PB for some integer n ≥ 1 such that vializations τ˘n : 1 1 Y˘ × X,S ˘ n e e U(n) ⊂ H, to the orbit τH of ´etale-locally-defined trivializations τn = τ˘n |1 1 . n

Y × X,S

The morphisms (1.3.2.73) and (1.3.2.51) induce a canonical morphism e ˘ ˘ → ΞΦ ,δ Ξ H H Φ f,δ f H

H

× CΦH ,δH

e˘ ˘ C Φ f,δ f H

(1.3.2.74)

H

e ˘ ˘ , equivariant with the homomorphism E ˘ → between torus torsors over C ΦH ΦH f f,δH f EΦH dual to the canonical homomorphism SΦH → SΦ˘ f (see (1.2.4.18)). H Suppose the image of a rational polyhedral cone ρ˘ ⊂ (SΦ˘ f )∨ R under the (canoH nical) second morphism in (1.2.4.20) is contained in some rational polyhedral cone ρ ⊂ (SΦH )∨ R . Then there is a canonical morphism e ˘ ˘ (˘ Ξ Φ f,δ f ρ) → ΞΦH ,δH (ρ) H

(1.3.2.75)

H

e˘ ˘ covering (1.3.2.51) and extending (1.3.2.73), mapping Ξ ΦH ˘ to ΞΦH ,δH ,ρ and f,δH f,ρ inducing a canonical morphism e˘ ˘ X Φ f,δ f,ρ˘ → XΦH ,δH ,ρ . H

(1.3.2.76)

H

e ˘ and ΣΦ are cone decompositions of P ˘ and PΦ , respectively, such that If Σ H H ΦH ΦH f f e ˘ under the (canonical) second morphism in (1.2.4.20) the image of each ρ˘ in Σ ΦH f

is contained in some ρ ∈ ΣΦH , then morphisms like (1.3.2.75) patch together and define a canonical morphism e˘ ˘ e Ξ Φ f,δ f,Σ ˘ H

H

Φf H

→ ΞΦH ,δH ,ΣΦH

(1.3.2.77)

covering (1.3.2.51), extending (1.3.2.73), and inducing a canonical morphism e˘ ˘ e X Φ f,δ f,Σ ˘ H

H

Φf H

→ XΦH ,δH ,ΣΦH

compatible with each (1.3.2.76) as above (under canonical morphisms).

(1.3.2.78)

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Proof. The statements are self-explanatory. Lemma 1.3.2.79. (Compare with Lemmas 1.3.2.50 and 1.3.2.72.) By comparing the universal properties (cf. Proposition 1.3.2.56), we obtain a canonical morphism b ˘ ˘ → ΞΦ ,δ Ξ (1.3.2.80) ΦH c,δH c

covering (1.3.2.51), by sending the pair

H

H

∨ (b τHb , τbH b ),

which is an orbit of ´etale-locallye e to the orbit τH defined pairs for some integer n ≥ 1 such that U(n) ⊂ H, ∨ of ´etale-locally-defined τn = τbn |1 1 Y × X,S = τbn |1 1 Y × X,S , as in Proposition 1.3.2.56. (b τn , τbn∨ )

n

n

The morphisms (1.3.2.80) and (1.3.2.51) induce a canonical morphism b ˘ ˘ → ΞΦ ,δ b˘ ˘ Ξ × C (1.3.2.81) H H Φ c,δ c Φ c,δ c H

H

CΦH ,δH

H

H

b˘ ˘ = C e ˘ ˘ , equivariant with the surjective hobetween torus torsors over C ΦH ΦH c,δH c f,δH f b ˘ → EΦ (see Proposition 1.3.2.56) dual to the canonical injective momorphism E ΦH c

H

b ˘ (see Definition 1.2.4.29). homomorphism SΦH ,→ S Φc H

b ˘ )∨ under (1.2.4.37) is Suppose the image of a rational polyhedral cone ρb ⊂ (S ΦH f R contained in some rational polyhedral cone ρ ⊂ (SΦH )∨ . Then there is a canonical R morphism b ˘ ˘ (b ρ) → ΞΦ ,δ (ρ) (1.3.2.82) Ξ ΦH c,δH c

H

H

b˘ ˘ (cf. (1.3.2.75)) covering (1.3.2.51) and extending (1.3.2.80), mapping Ξ ΦH ρ to c,δH c,b ΞΦH ,δH ,ρ and inducing a canonical morphism b˘ ˘ X (1.3.2.83) Φ c,δ c,b ρ → XΦH ,δH ,ρ H

H

b ˘ and PΦ , reb ˘ and ΣΦ are cone decompositions of P (cf. (1.3.2.76)). If Σ H H ΦH ΦH c c b ˘ under (1.2.4.37) is contained in spectively, such that the image of each ρb in Σ ΦH c

some ρ ∈ ΣΦH , then morphisms like (1.3.2.82) patch together and define a canonical morphism b˘ ˘ b Ξ → ΞΦH ,δH ,ΣΦH (1.3.2.84) Φ c,δ c,Σ ˘ H

H

Φc H

(cf. (1.3.2.77)) covering (1.3.2.51), extending (1.3.2.80), and inducing a canonical morphism b˘ ˘ b X → XΦH ,δH ,ΣΦH (1.3.2.85) Φ c,δ c,Σ ˘ H

H

Φc H

(cf. (1.3.2.78)) compatible with each (1.3.2.83) as above (under canonical morphisms). By the same argument in [61, Sec. 3C], using the extended Kodaira–Spencer isomorphism as in [62, Prop. 6.2.5.18], the morphism (1.3.2.84) is log smooth and we have a canonical isomorphism 1

ΩΞb ˘

/ΞΦH ,δH ,ΣΦ ˘ ,Σ b Φ c,δ ˘ H c Φ H H c H

∼ b˘ ˘ b = (Ξ Φ c,δ c,Σ ˘ H

H

Φc H

(1.3.2.86)

e Lie∨ \ → CΦH ,δH ) HomO (X, G /CΦ ∗

H ,δH

),

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where

1

ΩΞb ˘

/ΞΦH ,δH ,ΣΦ ˘ ,Σ b Φ c,δ ˘ H c Φ H H c H

b˘ ˘ b ((Ξ Φ c,δ c,Σ ˘ H

Φc H

H

:= (Ω1b

ΞΦ ˘

/CΦH ,δH ˘ ,Σ b ˘ c,δH c Φ H c H

→ ΞΦH ,δH ,ΣΦH )∗ Ω1Ξ

ΦH ,δH ,ΣΦ

H

[d log ∞])/ [d log ∞])

/CΦH ,δH

is the sheaf of modules of relative log 1-differentials, and where G\ → CΦH ,δH is the tautological semi-abelian scheme as in Proposition 1.3.2.12. Moreover, the canonical morphism b˘ ˘ b˘ ˘ b × C (1.3.2.87) Ξ → ΞΦ ,δ ,Σ ΦH ˘ c,δH c,ΣΦ

H

H

ΦH

c H

ΦH c,δH c

CΦH ,δH

(induced by (1.3.2.51) and (1.3.2.84)) induces a canonical short exact sequence b˘ ˘ b b ˘ ˘ ) ∗ Ω1 0 → (Ξ →C b Φ c,δ c Φ c,δ c,Σ ˘ C /C H

H

H

Φc H

˘ ˘ ,δ Φ c H c H

H

ΦH ,δH

1

→ ΩΞb ˘

(1.3.2.88)

/ΞΦH ,δH ,ΣΦ ˘ ,Σ b Φ c,δ ˘ H c Φ H H c H

1

→ ΩΞb ˘

/(ΞΦH ,δH ,ΣΦ × ˘ ,Σ b Φ c,δ ˘ H C c Φ H H ΦH ,δH c H

where 1 ΩΞb ˘

/(ΞΦH ,δH ,ΣΦ × ˘ ,Σ b Φ c,δ ˘ H C c Φ H H ΦH ,δH c H

b˘ ˘ b ((Ξ Φ c,δ c,Σ ˘ H

H

Φc H

b˘ C Φ

˘ ) c,δH c H

b˘ C Φ

:= (Ω1b

→ ΞΦH ,δH ,ΣΦH )∗ Ω1Ξ

ΦH ,δH ,ΣΦ

˘ ) c,δH c H

ΞΦ ˘

H

→ 0,

/CΦ ˘ ˘ ,Σ ˘ ,δ b ˘ c H c c,δH c Φ H H c H

/CΦH ,δH

[d log ∞])/

b

[d log ∞])

is the sheaf of modules of relative log 1-differentials, which is exact and has locally free terms, and which can be canonically identified with the pullback under b˘ ˘ b Ξ → CΦH ,δH of the canonical short exact sequence Φ c,δ c,Σ ˘ H

H

Φc H

e Lie∨ 0 → HomO (X, B/CΦ

H ,δH

e Lie∨ \ ) → HomO (X, G /CΦ

H ,δH

)

(1.3.2.89) → →0 of locally free sheaves (compatible with (1.3.2.54)). Hence, (1.3.2.87) is also log smooth (by [45, 3.12]). e Lie∨ HomO (X, T /CΦH ,δH )

Proof. The statements are self-explanatory. Proposition 1.3.2.90. (Compare with Proposition 1.3.2.55.) Under the canonie and H), and under the canonical cal morphisms as in (1.3.2.73) (with varying H 0 ∞ 0 ∞ e e homomorphisms P˘Z (A ) → PZ (A ) and P˘Z,eZ (A∞ ) → PZ (A∞ ), the Hecke action e 0 (A∞ ) on the collection {Ξ e ˘ ˘ } e is compatible with the Hecke action of of P ˘ Z

ΦH e0 f,δH f HP

˘ Z

e ˘ e(A∞ ) on the P0Z (A∞ ) on the collection {ΞΦH ,δH }HP0 ; and the Hecke action of P Z,Z Z `e collection { ΞΦ˘ f,δ˘f }He e is compatible with the Hecke action of PZ (A∞ ) on the ` H H P˘Z,eZ collection { ΞΦH ,δH }HPZ , where the index sets are as in Proposition 1.3.2.45. These Hecke actions are all compatible with those in Proposition 1.3.2.55. They are also compatible with extensions to toroidal embeddings and their formal completions.

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Proof. As in the case of Proposition 1.3.2.55, the canonical morphisms as in (1.3.2.80) correspond to pushouts of extensions of B (resp. B ∨ ) by T˘ (resp. T˘∨ ) under the canonical homomorphism T˘ → T (resp. T˘∨ → T ∨ ) induced by the re˘ (resp. Y˘ ) to X (resp. Y ). Hence, the realizations of the Hecke striction from X twists are compatible in the desired ways. (We omit the details for simplicity.) Proposition 1.3.2.91. (Compare with Propositions 1.3.2.55 and 1.3.2.90.) Under b and H), and under the the canonical morphisms as in (1.3.2.80) (with varying H 0 ∞ 0 ∞ ∞ b b canonical homomorphisms P˘Z (A ) → PZ (A ) and P˘Z (A ) → PZ (A∞ ), the Hecke b 0 (A∞ ) on the collection {Ξ b ˘ ˘ } b is compatible with the Hecke action action of P ˘ Φ c,δ c H b 0 Z H

H

P ˘ Z

b ˘ (A∞ ) on on the collection {ΞΦH ,δH }HP0 ; and the Hecke action of P Z Z `b the collection { ΞΦ˘ c,δ˘c }Hb b is compatible with the Hecke action of PZ (A∞ ) on P˘ H H Z ` the collection { ΞΦH ,δH }HPZ , where the index sets are as in Proposition 1.3.2.45. These Hecke actions are all compatible with those in Proposition 1.3.2.55. They are also compatible with extensions to toroidal embeddings and their formal completions. of

P0Z (A∞ )

b ˘ (A∞ ) on Proof. As in the proof of Proposition 1.3.2.67, the Hecke action of P Z `b e the collection { ΞΦ˘ c,δ˘c }Hb b is induced by the Hecke action of P˘Z,eZ (A∞ ) on the P˘ H H Z `e collection { ΞΦ˘ f,δ˘f }He e 0 . Hence, these statements follow from the corresponding H

H

P ˘ Z

statements of Proposition 1.3.2.90. 1.3.3

Toroidal Compactifications of PEL-Type Kuga Families and Their Generalizations

For simplicity, in the remainder of this subsection, all morphisms between schemes or algebraic stacks over S0 = Spec(F0 ) will be defined over S0 , unless otherwise specified. Let Q be any O-lattice. Consider the abelian scheme GMH over MH in (1) of Theorem 1.3.1.3. By [62, Prop. 5.2.3.9], the group functor HomO (Q, GMH ) over MH is relatively representable by a proper smooth group scheme which is an extension of a finite ´etale group scheme, whose rank has no prime factors other than those of Disc = DiscO/Z [62, Def. 1.1.1.6], by an abelian scheme HomO (Q, GMH )◦ , the fiberwise geometric identity component of HomO (Q, GMH ) (see [62, Def. 5.2.3.8]). Example 1.3.3.1. If Q ∼ = O⊕s for some integer s ≥ 0, then HomO (Q, GMH )◦ = ×s HomO (Q, GMH ) ∼ = GMH is the s-fold fiber product of GMH over MH . Example 1.3.3.2. If O ∼ = Mk (OF ) and Q is of finite index in OF⊕k for some integer k ≥ 1, then the relative dimension of HomO (Q, GMH )◦ over MH is 1/k of the relative dimension of GMH over MH . Definition 1.3.3.3. (See [61, Def. 2.4].) A Kuga family over MH is an abelian scheme Ngrp → MH that is Q× -isogenous to HomO (Q, GMH )◦ for some O-lattice Q.

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Definition 1.3.3.4. A generalized Kuga family over MH is a torsor N → MH under some Kuga family Ngrp → MH as in Definition 1.3.3.3. Lemma 1.3.3.5. (See [61, Lem. 2.6].) The abelian scheme HomZ (Q∨ , G∨ MH ) is isomorphic to the dual abelian scheme of HomZ (Q, GMH ). Lemma 1.3.3.6. (See [61, Lem. 2.9].) Let jQ : Q∨ ,→ Q be as in Lemma 1.2.4.1. Then the isogeny λMH ,jQ ,Z : HomZ (Q, GMH ) → HomZ (Q∨ , G∨ MH ) induced canonically by jQ and λMH : GMH → G∨ MH is a polarization. Proposition 1.3.3.7. (See [61, Prop. 2.10 and Cor. 2.12].) The abelian scheme ◦ × ◦ HomO (Q∨ , G∨ MH ) is Q -isogenous to the dual abelian scheme of HomO (Q, GMH ) . ∨ Moreover, given any jQ : Q ,→ Q as in Lemma 1.2.4.1, the composition λMH ,jQ : HomO (Q, GMH )◦ ,→ HomZ (Q, GMH ) λMH ,jQ ,Z

→

◦ HomZ (Q∨ , G∨ MH )  (HomO (Q, GMH ) )

∨

(1.3.3.8)

induced canonically by jQ and the polarization λMH : GMH → G∨ MH is a polarization. Definition 1.3.3.9. Let N → MH be as in Definition 1.3.3.4. Then we define the dual N∨ → MH to be the dual abelian scheme Ngrp,∨ → MH of Ngrp → MH . Remark 1.3.3.10. By [92, XIII, Prop. 1.1], N∨ = Ngrp,∨ → MH is canonically isomorphic to Pic0 (N/MH ) → MH (which can be defined as in the case of abelian schemes; cf. [62, Def. 1.3.2.1]). Note that this is always a group scheme, with its identity section, even when N → MH is a nontrivial torsor of Ngrp → MH . Definition 1.3.3.11. By abuse of notation, we denote by LieN/MH (resp. Lie∨ N/MH , ∨ ∨ resp. LieN∨ /MH , resp. LieN∨ /MH ) the locally free sheaf LieNgrp /MH (resp. LieNgrp /MH , resp. LieNgrp,∨ /MH , resp. Lie∨ Ngrp,∨ /MH ) over MH , although N → MH might have no section. Lemma 1.3.3.12. We have: ∼ Lie∨ N/MH = HomOMH (LieN/MH , OMH ), ∼ Lie∨ N∨ /MH = HomOMH (LieN∨ /MH , OMH ), Ω1N/MH ∼ = (N → MH )∗ Lie∨ N/MH , 1 ∨ ∗ ∼ (N → MH ) Lie∨∨ ΩN∨ /MH = N /MH , ∼ (N → MH )∗ Ω1 , = Lie∨ N/MH

N/MH

∼ (N → = Lie∨ N∨ /MH , 1 R (N → MH )∗ ON ∼ = LieN∨ /MH , R1 (N∨ → MH )∗ ON∨ ∼ = LieN/MH . ∨

MH )∗ Ω1N∨ /MH

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The relative de Rham cohomology H idR (N/MH ) := Ri (N → MH )∗ (Ω•N/MH ) and its Hodge filtration and Gauss–Manin connection ∇ are canonically isomorphic to those of H idR (Ngrp /MH ). Proof. The first two follows from the definition and the corresponding statement ∨ for LieNgrp /MH , Lie∨ Ngrp /MH , LieNgrp,∨ /MH , and LieNgrp,∨ /MH . The remaining ones follow by ´etale descent from the corresponding ones for Ngrp → MH (cf. [62, Cor. 2.1.5.9 and Lem. 2.1.5.11]). Corollary 1.3.3.13. (Compare with [61, Cor. 2.13].) If a generalized Kuga family N → MH is a torsor under a Kuga family Ngrp → MH which is Q× -isogenous to HomO (Q, GMH )◦ for some O-lattice Q, then we have canonical isomorphisms of locally free sheaves over MH : LieN/MH ∼ = HomO (Q, LieGMH /MH ), LieN∨ /MH ∼ = HomO (Q∨ , LieG∨ /MH ), ∼ ), Lie∨ = Hom (Q∨ , Lie∨ N/MH

Lie∨ N∨ /MH

O

GMH /MH

∼ = HomO (Q, Lie∨ G∨ M

H

/MH ).

Remark 1.3.3.14. We do not need to choose a polarization Ngrp → Ngrp,∨ in the isomorphisms in Corollary 1.3.3.13. The algebraically constructed toroidal compactifications of Kuga families and their generalizations in characteristic zero can be described as follows: Theorem 1.3.3.15. (Compare with [61, Thm. 2.15].) Let Q be any O-lattice. Suppose that H is neat (see [89, 0.6] or [62, Def. 1.4.1.8]), and that Σ is as in Definition 1.2.2.13, so that the moduli problem MH is representable by a scheme tor quasi-projective over S0 , and so that (by Theorem 1.3.1.3) Mtor H = MH,Σ is an algebraic space proper and smooth over S0 . (By Theorem 1.3.1.10, if Σ is projective as in Definition 1.2.2.14, then Mtor H,Σ is projective over S0 .) Consider the ++ + ++ sets KQ,H ⊂ K+ ⊂ K and K Q,H,Σ ⊂ KQ,H,Σ ⊂ KQ,H,Σ as in Definitions Q,H Q,H 1.2.4.44 and 1.2.4.50, with compatible directed partial orders. These sets parameterize the following data: b Σ) b ∈ K++ , if Hκ := H bG (which is contained in H, so (1) For each κ = (H, Q,H that MHκ is a finite ´etale cover of MH ; see Definition 1.2.4.4), then there is a generalized Kuga family Nκ → MHκ (see Definition 1.3.3.4), which is a torsor under a Kuga family Ngrp → MHκ (see Definition 1.3.3.3) with a κ × Q -isogeny κisog : HomO (Q, GMHκ )◦ → Ngrp κ

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of abelian schemes over MHκ , together with an open dense immersion κtor : Nκ ,→ Ntor κ of schemes over S0 , such that the scheme Ntor κ is projective and smooth over S0 , and such that the complement of Nκ in Ntor (with its reduced κ structure) is a relative Cartier divisor E∞,κ with simple normal crossings. The scheme Ntor κ has a stratification by locally closed subschemes a b ˘ ˘ Ntor Z κ = [(Φ c,δ c,b τ )] , H

H

˘ c,δ˘c,b [(Φ τ )] H H

˘ b , δ˘ b , τb)] running through a complete set of equivalence classes of with [(Φ H H b + and τb ∈ Σ ˘ b , δ˘ b , τb) (as in Lemma 1.2.4.42) with τb ⊂ P b (Here b ˘ ∈ Σ. (Φ ˘c ΦH H H c Φ H ` ˘ b is suppressed in the notation by our convention. The notation “ ” only Z H means a set-theoretic disjoint union. The algebro-geometric structure is still b ˘ 0 ˘0 0 ˘ 0 , δ˘0 , τb0 )]-stratum Z that of Ntor .) In this stratification, the [(Φ κ

b H

[(Φ c,δ c,b τ )]

b H

H

H

b ˘ ˘ ˘ b , δ˘ b , τb)]-stratum Z lies in the closure of the [(Φ [(ΦH τ )] if and only if H H c,δH c,b 0 0 ˘0 ˘ ˘ ˘ [(ΦHb , δHb , τb)] is a face of [(ΦHb , δHb , τb )] as in Lemma 1.2.4.42. In particular, b[(0,0,{0})] is an open dense stratum in this stratification. Nκ = Z b ˘ ˘ ˘ b , δ˘ b , τb)]-stratum Z The [(Φ is smooth over S0 and isomorphic H

H

[(ΦH τ )] c,δH c,b

b˘ ˘ to the support of the formal scheme X ΦH τ (see (1.3.2.65)) for every c,δH c,b ˘ b , δ˘ b , τb) of [(Φ ˘ b , δ˘ b , τb)], which is the completion of an representative (Φ H H H H b ˘ ˘ ) of a b τ ) (along its τb-stratum Ξ affine toroidal embedding Ξ ˘ ˘ (b ΦH τ c,δH c,b

ΦH c,δH c

b ˘ ˘ over a finite b ˘ ˘ over an abelian scheme torsor C torus torsor Ξ Φ c,δ c Φ c,δ c H

H

H

H

˘

c c b ˘ZH b ΦH ´etale cover M b of the scheme MH b (quasi-projective over S0 ) in Lemma H 1.3.2.50 and Proposition 1.3.2.56. ∧ b ˘ ˘ The formal completion (Ntor of Ntor κ )Z κ along Z[(Φ b τ )] is canonic,δ c,b H

˘ ,τ ˘ ,δ [(Φ c H c b )] H

H

b ˘ ˘ ; and the formal completion (Ntor )∧ cally isomorphic to X κ ∪Z b Φ c,δ c,b τ H

H

, ˘ ,τ ˘ ,δ [(Φ c H c b )] H

b ˘ ˘ b ˘ ˘ b˘ , b∈ Σ where ∪ Z ΦH [(ΦH τ )] is the union of all strata Z[(Φ τ )] with τ c c,δH c,b c,δH c,b H b ˘ ˘ /Γ ˘ (cf. (5) of Theorem 1.3.1.3 and is canonically isomorphic to X ΦH ΦH c c,δH c Lemma 1.3.2.41). (Such isomorphisms can be induced by strata-preserving tor b ˘ ˘ isomorphisms between ´etale neighborhoods of points of Z [(ΦH τ )] in Nκ c,δH c,b b˘ ˘ b ˘ ˘ (b and ´etale neighborhoods of points of Ξ in Ξ τ ).) ΦH τ c,δH c,b

ΦH c,δH c

Each Ntor admits a canonical proper surjection Ntor → Mmin extending κ κ H the canonical proper surjection Nκ → MH , and the latter is the pullback of the former under the canonical morphism MH ,→ Mmin on the target H ˘ b , δ˘ b , τb)]-stratum (see Theorem 1.3.1.5). Such a morphism maps the [(Φ H H tor min b ˘ ˘ Z [(Φ c,δ c,b τ )] of Nκ to the [(ΦH , δH )]-stratum Z[(ΦH ,δH )] of MH if and only H

H

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˘ b , δ˘ b )] as in if the cusp label [(ΦH , δH )] is assigned to the cusp label [(Φ H H Lemma 1.2.4.15. If κ ∈ K+ Q,H , then Hκ = H and hence MHκ = MH . If κ ∈ KQ,H , then grp Nκ = Nκ → MHκ = MH is a Kuga family. For each relation κ0  κ in K++ etale surjection Q,H , there is a proper log ´ tor tor fκtor 0 ,κ : Nκ0 → Nκ ,

extending a canonical finite ´etale surjection fκ0 ,κ : Nκ0 → Nκ inducing a canonical finite ´etale surjection Nκ0 → Nκ × MHκ0 equivariant MHκ

with the canonical Q× -isogeny isog grp fκgrp ◦ ((κ0 )isog )−1 : Ngrp × MHκ0 , 0 ,κ := κ κ0 → Nκ MHκ

i

(fκtor 0 ,κ )∗ ONtor κ0

such that R = 0 for i > 0. These surjections are compatible with the canonical morphisms to Mmin H . (2) For each κ ∈ K++ , the structural morphism fκ : Nκ → MH extends Q,H,Σ tor tor (necessarily uniquely) to a surjection fκtor : Ntor κ → MH = MH,Σ , which is proper and log smooth (as in [45, 3.3] and [43, 1.6]) if we equip Ntor κ and Mtor H with the canonical (fine) log structures given respectively by the relative Cartier divisors with (simple) normal crossings E∞,κ and D∞,H (see (1) above and (3) of Theorem 1.3.1.3). Then we have the following commutative diagram:  +NCD / Ntor Nκ  κ proper smooth surjective

fκ

  MH 

proper log smooth surjective +NCD

fκtor

 / Mtor H

projective smooth

proper smooth

&/

S0

tor b ˘ ˘ ˘ b , δ˘ b , τb)]-stratum Z The morphism fκtor maps the [(Φ [(ΦH τ )] of Nκ to H H c,δH c,b the [(ΦH , δH , τ )]-stratum Z[(ΦH ,δH ,τ )] of Mtor H if and only if (the cusp label ˘ b , δ˘ b )] as in Lemma 1.2.4.15 and) [(ΦH , δH )] is assigned to the cusp label [(Φ H H b ˘ under (1.2.4.37) is contained in τ ∈ ΣΦ as in Condithe image of τb ∈ Σ H ΦH c b˘ ˘ tion 1.2.4.49. In this case, the compatible morphisms X → XΦ ,δ ,τ ΦH τ c,δH c,b

b˘ ˘ b and X Φ c,δ c,Σ ˘ H

H

Φc H

H

H

→ XΦH ,δH ,ΣΦH induced by fκtor (and the canonical isomor-

phisms in (1) above and in (5) of Theorem 1.3.1.3) coincide with the canonical morphisms as in (1.3.2.83) and (1.3.2.85). (These morphisms can be induced by compatible morphisms between ´etale neighborhoods of points of the supports of formal schemes in relevant ambient schemes as in (1) above, compatible with all stratifications.) If κ0  κ, then we have the compatibility fκtor = fκtor ◦ fκtor 0 0 ,κ .

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Compactifications with Ordinary Loci ++ (3) Suppose κ ∈ K+ Q,H,Σ (not just in KQ,H,Σ , so that Hκ = H). For simplicity, let us suppress the subscripts “κ” from the notation. (All canonical isomorphisms will be required to be compatible with the canonical isomorphisms + 0 defined by pulling back under fκtor 0 ,κ for each relation κ  κ in KQ,H,Σ .) Then the following are true:

(a) Let Ω1Ntor /S0 [d log ∞] and Ω1Mtor /S0 [d log ∞] denote the sheaves of moH dules of log 1-differentials over S0 given by the (respective) canonical log structures defined in (2). Let 1

:= (Ω1Ntor /S0 [d log ∞])/((f tor )∗ (Ω1Mtor /S0 [d log ∞])). ΩNtor /Mtor H H

Then there is a canonical isomorphism ∼ 1 (f tor )∗ (HomO (Q∨ , Lie∨ G/Mtor )) = ΩNtor /Mtor H

(1.3.3.16)

H

between locally free sheaves over Ntor , extending the composition of canonical isomorphisms f ∗ (HomO (Q∨ , Lie∨ GM

H

/MH ))

∼ ∼ 1 = f ∗ Lie∨ N/MH = ΩN/MH

(1.3.3.17)

over N (see Lemma 1.3.3.12). (b) For each integer b ≥ 0, there exist canonical isomorphisms a )∼ Rb f∗tor (ΩNtor /Mtor ))) = (∧b (HomO (Q∨ , LieG∨ /Mtor H H

⊗ (∧a (HomO (Q∨ , Lie∨ G/Mtor )))

OMtor

(1.3.3.18)

H

H

and a a Rb f∗tor (ΩNtor /Mtor ⊗ IE∞ ) ∼ ) ⊗ ID∞,H = Rb f∗tor (ΩNtor /Mtor H H ONtor

OMtor H

(1.3.3.19) of locally free sheaves over where IE∞ (resp. ID∞,H ) is the ONtor -ideal (resp. OMtor -ideal) defining the relative Cartier divisor H E∞ = E∞,κ (resp. D∞,H ) (with its reduced structure), compatible with cup products and exterior products, extending the canonical isomorphism over MH induced by the composition of canonical isomorphisms Mtor H ,

Rb f∗ (ON ) ∼ = ∧b (HomO (Q∨ , LieG∨M = ∧b LieN∨ /MH ∼

H

•

/MH )).

(1.3.3.20)

1

(c) Let ΩNtor /Mtor := ∧• ΩNtor /Mtor be the log de Rham complex associated H H tor tor with f : N → Mtor (with differentials inherited from Ω•N/MH ). Let H the (relative) log de Rham cohomology be defined by •

i tor H ilog-dR (Ntor /Mtor ). H ) := R f∗ (ΩNtor /Mtor H

Then the (relative) Hodge spectral sequence a

tor E1a,b := Rb f∗tor (ΩNtor /Mtor ) ⇒ H a+b /Mtor H ) log-dR (N H

(1.3.3.21)

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degenerates at E1 terms, and defines a Hodge filtration on H ilog-dR (Ntor /Mtor H ) with locally free graded pieces given by a Rb f∗tor (ΩNtor /Mtor ) for integers a+b = i, extending the canonical Hodge H i filtration on H dR (N/MH ). As a result, for each integer i ≥ 0, there is a canonical isomorphism ∼

i tor ∧i H 1log-dR (Ntor /Mtor /Mtor H ) → H log-dR (N H ),

compatible with the Hodge filtrations defined by (1.3.3.21), extending ∼ the canonical isomorphism ∧i H 1dR (N/MH ) → H idR (N/MH ) over MH (defined by cup product). (d) For each jQ : Q∨ ,→ Q as in Lemma 1.2.4.1, the Q× -polarization ∨

λMH ,jQ : HomO (Q, GMH )◦ → (HomO (Q, GMH )◦ ) in Proposition 1.3.3.7 induces a Q× -polarization λN,jQ : Ngrp → Ngrp,∨ ,

and hence defines canonically (as in [23, 1.5], by ´etale descent; see Lemma 1.3.3.12) a perfect pairing h · , · iλMH ,jQ : H 1dR (N/MH ) × H 1dR (N/MH ) → OMH (1). Then H 1log-dR (Ntor /Mtor H ) is (under the restriction morphism) canonically isomorphic to the unique subsheaf of 1 (MH ,→ Mtor H )∗ (H dR (N/MH ))

satisfying the following conditions: . i. H 1log-dR (Ntor /Mtor H ) is locally free of finite rank over OMtor H 1

ii. The sheaf f∗tor (ΩNtor /Mtor ) can be identified with the subsheaf of H 1 ) (f (Ω (MH ,→ Mtor ∗ ∗ H N/MH )) formed (locally) by sections that are 1 also sections of H log-dR (Ntor /Mtor H ). (Here we view all sheaves 1 canonically as subsheaves of (MH ,→ Mtor H )∗ (H dR (N/MH )).) 1 tor tor iii. H log-dR (N /MH ) is self-dual under the push-forward (MH ,→ Mtor H )∗ h · , · iλMH ,jQ . (e) The Gauss–Manin connection ∇ : H •dR (N/MH ) → H •dR (N/MH ) ⊗ Ω1MH /S0

(1.3.3.22)

OMH

extends to an integrable connection 1

• tor ∇ : H •log-dR (Ntor /Mtor /Mtor H ) → H log-dR (N H ) ⊗ ΩMtor H /S0 OMtor H

(1.3.3.23) with log poles along D∞,H , called the extended Gauss–Manin connection, satisfying the usual Griffiths transversality with the Hodge filtration defined by (1.3.3.21).

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b ∞ ) with image (4) (Hecke actions.) Suppose we have an element gb ∈ G(A ∞ ∞ b gh ∈ G(A ) under the canonical homomorphism G(A ) → G(A∞ ), and ˆ such suppose we have two neat open compact subgroups H and H0 of G(Z) −1 0 0 0 that H ⊂ gh Hgh . Suppose Σ = {ΣΦ0 0 }[(Φ0 0 ,δ0 0 )] is a compatible choice H H H of admissible smooth rational polyhedral cone decomposition data for MH0 , which is a gh -refinement of Σ as in [62, Def. 6.4.3.3]. Consider the sets ++ + ++ KQ,H0 ⊂ K+ Q,H0 ⊂ KQ,H0 and KQ,H0 ,Σ0 ⊂ KQ,H0 ,Σ0 ⊂ KQ,H0 ,Σ0 as in 0 0 Definitions 1.2.4.44 and 1.2.4.50 (for H and Σ ), with compatible directed partial orders, parameterizing generalized Kuga families and their compactifications with properties as in (1), (2), and (3) above. The sets K++ Q,H etc ++ and KQ,H0 etc (and the objects they parameterize) satisfy the compatibility with gb (and gh ) in the sense that the following are true: b Σ) b ∈ K++ (resp. K+ , resp. KQ,H ), and for each (a) For each κ = (H, Q,H Q,H ˆ such that H b 0 ⊂ G( b Z) b 0 ⊂ gbHb b g −1 (so that open compact subgroup H −1 0 b b Hκ = HG and Hκ0 = HG satisfy Hκ0 ⊂ gh Hκ gh ), there exists an b0 , Σ b 0 ) ∈ K++ 0 (resp. K+ 0 , resp. KQ,H0 ) such that element κ0 = (H Q,H

Q,H

there exists a (necessarily unique) finite ´etale surjection [b g ] : Nκ0 → Nκ

(1.3.3.24)

covering the compatible surjections [gh ] : MH0 → MH and [gh ] : MHκ0 → MHκ given by [62, Prop. 6.4.3.4] (see Proposition 1.3.1.15), inducing a finite ´etale surjection Nκ0 → Nκ × MHκ0 of abelian MHκ

scheme torsors equivariant with the isogeny (not just Q× -isogeny) grp × MHκ0 induced by (κ0 )isog , κisog , and the Q× -isogeny Ngrp κ0 → Nκ MHκ

GMH

κ0

→ GMHκ × MHκ0 realizing GMHκ × MHκ0 as a Hecke twist MHκ

MHκ

of GMH 0 by gh (which is the pullback of the Q× -isogeny GMH0 → κ GMH × MH0 realizing GMH × MH0 as a Hecke twist of GMH0 by gh ). MH

MH

(Here all the base changes from MH to MH0 and from MHκ to MHκ0 use the surjections denoted by [gh ].) b Σ) b and H b 0 as in (4a) such that κ ∈ K++ (resp. (b) For each κ = (H, Q,H b0 , Σ b 0 ) ∈ K++ 0 (resp. K+ , resp. KQ,H ), there is an element κ0 = (H Q,H

Q,H

K+ g ] is defined as in (4a) (see (1.3.3.24)), Q,H0 , resp. KQ,H0 ) such that [b 0 b b (cf. Lemma 1.2.4.42 and [62, and such that Σ is a gb-refinement of Σ Def. 6.4.3.3]), which extends to a (necessarily unique) proper log ´etale surjection [b g]

tor

tor : Ntor κ0 → Nκ

(1.3.3.25)

such that tor

Ri [b g ]∗ O(N0 0 )tor = 0 κ

for all i > 0.

(1.3.3.26)

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b ˘0 ˘ 0 , δ˘0 , τb0 )]-stratum Z Under (1.3.3.25), the [(Φ b0 H b0 [(Φ H

,δ˘0c0 ,b τ 0 )] c0 H H

of Ntor κ0

tor b ˘ ˘ ˘ b , δ˘ b , τb)]-stratum Z is mapped to the [(Φ if and [(ΦH τ )] of Nκ H H c,δH c,b 0 0 ˘ ˘ ˘ ˘ only if there are representatives (ΦHb , δHb , τb) and (ΦHb 0 , δHb 0 , τb0 ) of ˘ b , δ˘ b , τb)] and [(Φ ˘ 0 , δ˘0 , τb0 )], respectively, such that (Φ ˘ 0 , δ˘0 , τb0 ) [(Φ H

H

b0 H

b0 H

b0 H

b0 H

˘ b , δ˘ b , τb) (cf. Lemma 1.2.4.42 and [62, Def. is a gb-refinement of (Φ H H b ˘ 0 ˘0 0 → 6.4.3.1]). In this case, the compatible morphisms X Φ ,δ ,b τ c0 H

b˘ ˘ b ˘0 X Φ c,δ c,b τ and XΦ H

b0 ,δ˘0c0 ,Σ ˘0 c0 H Φ H c0 H

H

b˘ ˘ b →X Φ c,δ c,Σ ˘ H

H

c0 H

induced by (1.3.3.25) (and

Φc H

the canonical isomorphisms in (1) above) coincide with the canonical morphisms as in (1.3.2.69) and (1.3.2.71). + If κ ∈ K++ Q,H,Σ (resp. KQ,H,Σ , resp. KQ,H,Σ ), we may assume in the ++ 0 above that κ ∈ KQ,H0 ,Σ0 (resp. K+ Q,H0 ,Σ0 , resp. KQ,H0 ,Σ0 ), so that (1.3.3.25) covers the surjection [gh ]

tor

tor : Mtor H0 ,Σ0 → MH,Σ

given by [62, Prop. 6.4.3.4], tor 0 (c) Suppose [b g] is defined as in (4b) for some κ ∈ K+ Q,H,Σ and κ ∈ + ++ ++ KQ,H0 ,Σ0 (not just in KQ,H,Σ and KQ,H0 ,Σ0 ). Then there is a canonical isomorphism ([b g]

tor ∗

) : ([gh ]

∼

tor ∗

a+b tor tor tor tor ) H a+b log-dR (Nκ /MH,Σ ) → H log-dR (Nκ0 /MH0 ,Σ0 )

respecting the Hodge filtrations and compatible with the canonical isomorphisms tor ∗

([b g]

tor ∗

1

∼

1

) ΩNtor → ΩNtor0 /Mtor0 tor κ /MH,Σ

) : ([b g]

H ,Σ0

κ

([gh ]

tor ∗

) : ([gh ]

tor ∗

([gh ]

,

∼

tor ∗

) LieG∨ /Mtor → LieG∨ /Mtor0 H,Σ

H ,Σ0

∼

tor ∗

) : ([gh ]

∨ ) Lie∨ G/Mtor → LieG/Mtor0

H ,Σ0

H,Σ

,

,

tor and the canonical isomorphisms in (3) for Ntor κ and Nκ0 . 0 ∞ 0 b (d) If we have an element gb ∈ G(A ) with image gh ∈ G(A∞ ) under the b ∞ ) → G(A∞ ), with a similar setup such canonical homomorphism G(A tor 0 tor that [b g ] : Nκ00 → Nκ0 and [b g 0 ] : Ntor κ00 → Nκ0 are compatibly defined tor ++ 00 0 for some κ ∈ KQ,H00 ,Σ00 , then [b g gb] : Nκ00 → Nκ and [b g 0 gb] : Ntor κ00 → tor Nκ are also compatibly defined and satisfy the identities [b g 0 gb] = [b g] ◦ tor + + 0 0 tor 0 tor 0 [b g ] and [b g gb] = [b g ] ◦ [b g ] . If κ ∈ KQ,H,Σ , κ ∈ KQ,H0 ,Σ0 , and tor κ00 ∈ K+ , we also have [b g 0 gb]∗ = [b g 0 ]∗ ◦ [b g ]∗ and ([b g 0 gb] )∗ = 00 00 Q,H ,Σ tor tor ([b g 0 ] )∗ ◦ ([b g ] )∗ (in both applicable senses above).

(5) (Q× -isogenies.) Let gl be an element of GLO ⊗ A∞ (Q ⊗ A∞ ). Then the Z

Z

ˆ in Q ⊗ A∞ determines a unique O-lattice Q0 (up to submodule gl (Q ⊗ Z) Z

Z

isomorphism), together with a unique choice of an isomorphism ∼

[gl ]Q : Q ⊗ Q → Q0 ⊗ Q, Z

Z

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Compactifications with Ordinary Loci ∼ ˆ with inducing an isomorphism Q ⊗ A∞ → Q0 ⊗ A∞ matching gl (Q ⊗ Z) Z

Z

Z

ˆ and inducing a canonical Q× -isogeny Q0 ⊗ Z, Z

[gl ]∗Q : HomO (Q0 , GMH )◦ → HomO (Q, GMH )◦ ++ defined by [gl ]Q . Consider the sets KQ0 ,H ⊂ K+ Q0 ,H ⊂ KQ0 ,H and KQ0 ,H,Σ ⊂ + ++ KQ0 ,H,Σ ⊂ KQ0 ,H,Σ as in Definitions 1.2.4.44 and 1.2.4.50 (with Q replaced with Q0 ), with compatible directed partial orders, parameterizing generalized Kuga families and their compactifications with properties as in (1), (2), ++ and (3) above. The sets K++ Q,H etc and KQ0 ,H etc (and the objects they parameterize) satisfy the compatibility with gl in the sense that the following are true: b Σ) b ∈ K++ (resp. K+ , resp. KQ,H ), there is an (a) For each κ = (H, Q,H

Q,H

+ b0 , Σ b 0 ) ∈ K++ element κ0 = (H Q0 ,H,Σ (resp. KQ0 ,H,Σ , resp. KQ0 ,H,Σ ) such bG , such that the Q× -isogeny b 0 ⊂ Hκ = H that Hκ0 = H G

[gl ]∗,grp κ0 ,κ

grp := κisog ◦ [gl ]∗Q ◦ ((κ0 )isog )−1 : Ngrp × MHκ0 κ0 → Nκ MHκ

is an isogeny (not just a quasi-isogeny), and such that there is a (necessarily unique) finite ´etale surjection [gl ]∗κ0 ,κ : Nκ0 → Nκ inducing a finite ´etale surjection Nκ0 → Nκ × MHκ0 of abelian MHκ

scheme torsors equivariant with the isogeny [gl ]∗,grp κ0 ,κ . b Σ) b as in (5a), there is an element κ0 = (H b0 , Σ b 0) ∈ (b) For each κ = (H, ++ + KQ0 ,H (resp. KQ0 ,H , resp. KQ0 ,H ) such that [gl ]∗κ0 ,κ is defined as in (5a) and extends to a (necessarily unique) proper log ´etale surjection tor tor [gl ]∗,tor κ0 ,κ : Nκ0 → Nκ ,

(1.3.3.27)

Ri ([gl ]∗,tor =0 κ0 ,κ )∗ ONtor 0

(1.3.3.28)

such that κ

for all i > 0. + If κ ∈ K++ Q,H,Σ (resp. KQ,H,Σ , resp. KQ,H,Σ ), we may assume in ++ 0 the above that κ ∈ KQ0 ,H,Σ (resp. K+ Q0 ,H,Σ , resp. KQ0 ,H,Σ ). Then (1.3.3.27) is compatible with the canonical morphisms fκtor : Ntor κ → tor tor tor tor MH,Σ and fκ0 : Nκ0 → MH,Σ . + 0 (c) Suppose [gl ]κ∗,tor 0 ,κ is defined as in (5b) for some κ ∈ KQ,H,Σ and κ ∈ + ++ ++ KQ0 ,H,Σ (not just in KQ,H,Σ and KQ0 ,H,Σ ). Then, for each integer i ≥ 0, there is a canonical isomorphism ∼

i i tor tor tor tor ∗ ([gl ]∗,tor κ0 ,κ ) : H log-dR (Nκ /MH,Σ ) → H log-dR (Nκ0 /MH,Σ )

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extending the canonical isomorphism ∼

([gl ]∗κ0 ,κ )∗ : H idR (Nκ /MH ) → H idR (Nκ0 /MH ) induced by [gl ]Q , respecting the Hodge filtrations and inducing canonical isomorphisms a

a

∼

b tor ∗ b tor (ΩNtor0 /Mtor ) ([gl ]κ∗,tor (ΩNtor tor ) → R f∗ 0 ,κ ) : R f∗ κ /MH H κ

(for integers a + b = i) compatible (under the canonical isomorphisms tor in (3) for Ntor κ and Nκ0 ) with the canonical isomorphisms ∼

∨

∼

∨

([gl ]∗Q )∗ : HomO (Q∨ , LieG∨ /Mtor ) → HomO ((Q0 ) , LieG∨ /Mtor ) H H and ∨ 0 ([gl ]∗Q )∗ : HomO (Q∨ , Lie∨ G/Mtor ) → HomO ((Q ) , LieG/Mtor ). H

H

(d) If we have an element gl0 ∈ GLO ⊗ A∞ (Q ⊗ A∞ ) with a similar setup Z

Z

00 such that [gl0 ]∗κ00 ,κ0 and [gl0 ]∗,tor κ00 ,κ0 are compatibly defined for some κ ∈ ++ 0 ∗,tor 0 ∗ KQ00 ,H,Σ , then [gl gl ]κ00 ,κ and [gl gl ]κ00 ,κ are also compatibly defined and satisfy the identities [gl gl0 ]∗κ00 ,κ = [gl ]∗κ0 ,κ ◦ [gl0 ]∗κ00 ,κ0 and [gl gl0 ]∗,tor κ00 ,κ = ∗,tor + + + 0 00 0 ∗,tor [gl ]κ0 ,κ ◦ [gl ]κ00 ,κ0 . If κ ∈ KQ,H,Σ , κ ∈ KQ0 ,H,Σ , and κ ∈ KQ00 ,H,Σ , ∗ we also have ([gl gl0 ]∗κ00 ,κ )∗ = ([gl0 ]∗κ00 ,κ0 )∗ ◦ ([gl ]∗κ0 ,κ )∗ and ([gl gl0 ]∗,tor κ00 ,κ ) = ∗,tor ∗ ∗ ([gl0 ]κ∗,tor 00 ,κ0 ) ◦ ([gl ]κ0 ,κ ) .

Remark 1.3.3.29. The statements of Theorem 1.3.3.15 are more general than those in [61, Thm. 2.15], because we now consider not just Kuga families, but also generalized Kuga families over a finite ´etale cover MHκ of MH . Nevertheless, the proof of [61, Thm. 2.15] works almost verbatim for such generalizations. We will explain the necessary modifications in the next section. (We will only need the compactified Kuga families, i.e., those with κ ∈ KQ,H,Σ , for the construction of canonical and subcanonical extensions of automorphic bundles in Section 1.4.2, and for all applications we know. We included their generalizations in Theorem 1.3.3.15 only because it seems natural to do so.) Remark 1.3.3.30. The second, third, and fourth paragraphs of (1) of Theorem 1.3.3.15 follow from the construction of Nκ and Ntor κ using the toroidal boundary of e a larger PEL-type moduli problem MHe , and from (2) and (5) of Theorem 1.3.1.3, and from Lemma 1.3.2.41 (and from the justifications provided in Section 1.3.2 and to be provided in Section 1.3.4 below). The second paragraphs of (2) and tor (4b) of Theorem 1.3.3.15 follow from the construction of fκtor and [b g ] using the e tor = M e tor (given by (6) of Theorem universal property of certain suitably chosen M e eΣ e H H, 1.3.1.3), which is consistent with the construction of the canonical morphisms in Lemmas 1.3.2.41 and 1.3.2.79 and Proposition 1.3.2.67 using the various universal properties (all given in terms of degeneration data). These statements were not in [61, Thm. 2.15], but are implicit in the theory and can be deduced from other

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known statements. Similarly, the second last paragraph of statement (1) of Theorem 1.3.3.15 was not in [61, Thm. 2.15], but can be deduced from the other statements (in Theorems 1.3.1.3 and 1.3.1.5, Propositions 1.3.1.15 and 1.3.1.14, and Theorem 1.3.3.15). We omit the proof here because the proof of a subtler statement in mixed characteristics will be given for Theorem 7.1.4.1 (see Proposition 7.2.4.14 below). Remark 1.3.3.31. The isomorphism (1.3.3.19) (even just in the case of Kuga families) was not in the statement of [61, Thm. 2.15], although it is implicit in the arguments of the (rather lengthy) proof there. We included it here for the sake of completeness. The details of the proof are similar to those for (7.1.4.5) (to be given below in Section 7.3), and hence are omitted here. Remark 1.3.3.32. Statements (4) and (5) of Theorem 1.3.3.15 were not as explicitly stated in [61, Thm. 2.15], but follow from the same argument of the proof there (based on an analogue of Proposition 1.3.1.15). Remark 1.3.3.33. (Compare with Remarks 1.1.2.1 and 1.3.1.4.) By varying the e we can (in practice) allow choices of L and Q, and hence varying the choices of L, b in the parameter κ = (H, b Σ) b to be any open compact subgroup of G(A b ∞ ). the H Nevertheless, this can be achieved by varying the lattice Q alone, and hence is already incorporated in (5) of Theorem 1.3.3.15. 1.3.4

Justification for the Parameters for Kuga Families

For later constructions (in Chapter 7), and for some applications, we would like to spell out what fκ : Nκ → MH and κisog : HomO (Q, GMHκ )◦ → Ngrp κ are for each κ ∈ ++ tor tor tor tor b b ∈ K++ . Let KQ,H , and what fκ : Nκ → MH = MH,Σ is for each κ = (H, Σ) Q,H,Σ us review some of the materials in [61, Sec. 3]. (We will also take this opportunity to correct a mistake there. See Remark 1.3.4.5 below.) e etc be chosen as in Section 1.2.4. Let κ e h · , · ie, e e δ), e Σ, e σ Let (L, h0 ), (e Z, Φ, e = (H, e) ++ ++ e be any element in the set KQ,H as in Definition 1.2.4.11, and let κ = [e κ] ∈ KQ,H be as in Definition 1.2.4.44. e e as in ˆ define a moduli problem M e h · , · ie, e e ⊂ G( e Z) The data of O, (L, h0 ), and H H e is neat and Σ e is projective (and smooth), by Theorems 1.3.1.3 Section 1.1.2. Since H e tor = M e tor of M e e which is and 1.3.1.10, we have a toroidal compactification M e eΣ e H H H, e satisfies projective and smooth over S0 , as in the latter half of Section 1.3.2. Since H ef f ∼ e ZH ∼ e ΦH Condition 1.2.4.7, by Lemmas 1.3.2.1 and 1.3.2.5, we have M e = MHκ , e = MH H bG = GreZ−1 (H e e 0 ) = GreZ−1 (H e e ). By the construction of C ee e → where Hκ = H P Pe Φ f,δ f e

e Z

H

Z

H

f e ΦH M e in [62, Sec. 6.2.3–6.2.4] (see also the correction in Remark 1.3.1.6), it is a torsor H e grp under an abelian scheme C canonically Q× -isogenous to Hom (Q, GM )◦ . If

e

ef e f,δ Φ H H

O

H

b satisfies Condition 1.2.4.8, then we have Hκ = H and hence MH = MH . If H b H κ

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ee e = C e grp also satisfies Condition 1.2.4.9, then C e e → MHκ = MH is an abelian Φ f,δ f H

H

scheme, not just a torsor. (See Remark 1.3.1.6.)

ΦH f,δH f

ef f ∼ e ZH ∼ e ΦH Remark 1.3.4.1. The isomorphism M e = MH e = MHκ means we do not need to H ∼ ∼ consider nontrivial twisted objects (ϕ e−2,He , ϕ e0,He ) above (ϕ e−2,He , ϕ e0,He ) and ϕ e−1,He = αHκ . e

Since σ e ⊂ P+ is a top-dimensional nondegenerate rational polyhedral cone in e Φ f H

e by (2) of Theorem 1.3.1.3, the locally closed e e in Σ, the cone decomposition Σ ΦH f e tor is a zero-dimensional torus bundle e e e stratum Z e σ )] (not its closure) of MH [(ΦH f,e f,δH e e e ee e over MH . In other words, Z is over the abelian scheme torsor C ΦH f f,δH

[(ΦH σ )] f,δH f,e

κ

e e e ee e . Let us define Nκe to be this stratum Z canonically isomorphic to C ΦH [(ΦH σ )] , f,δH f f,δH f,e and denote the canonical morphism Nκe → MH by fκe (which is the composition of the canonical morphisms Nκe → MHκ and MHκ → MH ). Let us denote the e grp canonical Q× -isogeny HomO (Q, GMH )◦ → Ngrp := C by κ eisog . Note that κ e e e ΦH f,δH f

e e e ee e for every Σ e and every topNκe = Z [(ΦH σ )] is canonically isomorphic to CΦ f,δH f,e f,δH f H ee . dimensional cone σ e in Σ ΦH f

ee e (see (5) of Theorem 1.3.1.5 Lemma 1.3.4.2. The abelian scheme torsor C ΦH f,δH f e grp and Definition 1.2.1.15) and the canonical isogeny Hom (Q, GM )◦ → C of O

Hκ

ef e f,δ Φ H H

ef e ΦH M e H

∼ = MHκ depend (up to canonical isomorphism) only on ˆ (see Definitions 1.2.4.3 and 1.2.4.4) b=H e b of G( b Z) the open compact subgroup H G ˆ still e Moreover, if H e 0 is any open compact subgroup of G( e Z) determined by H. 0 b b e satisfying Condition 1.2.4.7 such that H = HG n H b under the isomorphism

abelian schemes over

b G

U

ˆ ∼ ˆ n U( ˆ induced by the splitting δe (cf. Condition 1.2.4.9), then we b Z) b Z) G( = G(Z) grp ∼ e e e grp . have CΦ =C ef0 = C e e f0 ,δ e e e H

H

ΦH f0 ,δH f0

ΦH f,δH f

Proof. These follow from the corresponding statements of Lemma 1.3.2.7. Consequently, Nκe and κ eisog depend (up to canonical isomorphism) only on the open ˆ determined by H b b Z) e (see Definitions 1.2.4.3 and 1.2.4.4). compact subgroup H of G( tor Let us take Nκe to be the schematic closure of the locally closed stratum e e e e tor . Then we obtain the canonical open dense immersion κ Z etor : eΣ e [(ΦH σ )] in MH, f,δH f,e b but also on the choices of Σ ee Nκe ,→ Ntor . Certainly, Ntor depends not only on H κ e

and σ e.

κ e

ΦH f

e is neat, the Lemma 1.3.4.3. (See [61, Lem. 3.1].) Under the assumption that H tor e closure of every stratum in M eΣ e has no self-intersection. H,

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e ++ , e Σ, e σ Corollary 1.3.4.4. (Compare with [61, Cor. 3.2].) For each κ e = (H, e) ∈ K Q,H e e e e tor is projective and smooth over S0 , and the closure Ntor of Nκe = Z in M κ e

[(ΦH σ )] f,δH f,e

eΣ e H,

the complement of Nκe in Ntor κ e (with its reduced structure) is a relative Cartier divisor with simple normal crossings. Proof. Combine Lemma 1.3.4.3, (3) of Theorem 1.3.1.3, and Theorem 1.3.1.10. Remark 1.3.4.5. In [61, Sec. 3], the κ e etc above were denoted κ etc without the tildes. However, the binary relation  introduced there is not a directed partial order. We take this opportunity to correct this mistake and (at the same time) provide a formulation better for the applications. The desired parameters should be given by equivalence classes κ = [e κ] of κ e, with the natural partial order  among them. (See Definitions 1.2.4.44 and 1.3.4.20, and see Proposition 1.3.4.19 below. Before then, we can not yet assert the second half of [61, Cor. 3.2].) e tor induces a stratification of Ntor . By (2) of Theorem The stratification of M e H ˘ e , δ˘ e , τ˘)] 1.3.1.3, the strata of Ntor are parameterized by equivalence classes [(Φ H H e e having [(Φ e , δ e , σ e)] as a face (as in Definition 1.2.2.19), spelled out in Section H

H

1.2.4. By Lemma 1.2.4.42, they can also be parameterized by the equivalence classes ˘ b , δ˘ b , τb)]. [(Φ H H Let σ ˘ be the image of σ e ⊂ P+ e under the first morphism in (1.2.4.20). Consider Φ e+ e˘ the sets Σ ˘ Φ f,˘ σ and ΣΦ H

f H

, and the groups ΓΦ˘ f,ΦH , ΓΦ˘ f,ΦH ,˘σ , ΓΦ˘ c , and ΓΦ˘ c,ΦH de,˘ σ H

H

f H

H

H

b ˘ and (S b ˘ )∨ defined in Definition fined in Definition 1.2.4.21; consider the sets S ΦH ΦH c c R b˘ 1.2.4.29; consider the Γ ˘ -admissible rational polyhedral cone decomposition Σ ΦH c

ΦH c

b ˘ defined in Corollary 1.2.4.40; consider the collection Σ b defined in Lemma of P ΦH c b Σ) b = [e 1.2.4.42; and consider the set KQ,H of equivalence classes κ = (H, κ] of ele0 0 0 e Q,H , with a directed partial order κ = (H b ,Σ b )  κ = (H, b Σ) b when ments κ e in K b0 ⊂ H b 0 and when Σ b 0 is a refinement of Σ, b as in Definition 1.2.4.44 and Lemma H 1.2.4.47. e ++ , consider the degenerating e Σ, e σ Construction 1.3.4.6. For each κ e = (H, e) in K Q,H family e ei, α e tor e λ, (G, e e) → M (1.3.4.7) H

eΣ e H,

e e as in Theorem 1.3.1.3. As in (1.3.2.18), let of type M H b bi) → Ntor b λ, (G,

(1.3.4.8) e e tor . denote the pullback of (1.3.4.7) to the closure of Nκe = Z[(Φ ef,e e f,δ eΣ e σ )] in MH, H H ee e because σ e is top-dimensional. Note that Nκe is canonically isomorphic to C ΦH f,δH f e e , by proceeding as in Construction 1.3.2.16, Although α e e is defined only over M κ e

Ntor κ e ,

H

H

we can define a (partial) pullback b bi, α e\ , ei\ , βe\ ) → Ntor b λ, e\ , λ (G, bHb ) := (G κ e e H

(1.3.4.9)

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of the degenerating family (1.3.4.7) to Ntor κ e , with the convention that (as in the case e ei, α e ei) is defined over all e λ, e λ, of (G, eHe ) itself) α bHb is defined only over Nκe , while (G, tor of Nκe as in (1.3.4.8). By construction, the pullback bN , biN , α bN , λ ee e , (G bHb ) → Nκe ∼ =C κ e κ e κ e Φ f,δ f H

(1.3.4.10)

H

e e , δe e ) of (1.3.4.9) to Nκe determines and is determined by (the prescribed (e ZHe , Φ H H and) the tautological object ee e ((A, λ, i, αHκ ), (e cHe , e c∨ e )) → CΦ H f,δ f H

(1.3.4.11)

H

e e ; i.e., elements of Γ e ). Here (up to isomorphisms inducing automorphisms of Φ Φf H H

f ∼ e eZH (A, λ, i, αHκ ) is the tautological object over M e = MHκ . As explained in Remark H 1.3.4.1, we do not need to consider nontrivial twisted objects (ϕ e∼ e∼ e, ϕ e ) above −2,H 0,H e e (ϕ e e, ϕ e e ) and ϕ e e = αH . With the fixed choice of (e Z, Φ, δ), the tautological

−2,H

0,H

−1,H

κ

b and hence so is the tuple (1.3.4.10). Thus, the object (1.3.4.11) depends only on H, notation α bHb is justified. b bi, α b λ, Construction 1.3.4.12. Let (G, bHb ) → Ntor κ e be as in (1.3.4.9) in Construction 1.3.4.6. Consider any morphism ξ : Spec(V ) → Ntor ¯ κ e centered at a geometric point s tor of Nκe such that V is a complete discrete valuation ring with fraction field K, and such that η := Spec(K) is mapped to the generic point of the irreducible component ˘ e , δ˘ e , ρ˘)]-stratum containing the image of s¯. Suppose the image of s¯ lies on the [(Φ H H e ˘ ˘ e tor , where [(Φ ˘ e , δ˘ e , ρ˘)] is represented by some (Φ ˘ e , δ˘ e , ρ˘) with Z of M e e [(ΦH , δ , ρ)] ˘ H H H H f H f H,Σ ˘ ˘ ˘ ˘ ˘ ˘ (Z e , Φ e = (X, Y , φ, ϕ˘ e , ϕ˘ e ), δ e ) representing some cusp label as in Section H

H

−2,H

0,H

H

˘ e , δ˘ e , τ˘) because the symbol τ 1.2.4. (We avoid using the more familiar notation (Φ H H will be used for another purpose below.) For simplicity, let us fix compatible choices eϕ e and (Z ˘ ϕ˘−2 , ϕ˘0 ), δ), ˘ e = (X, e Ye , φ, ˘ = (X, ˘ Y˘ , φ, ˘, Φ of representatives (e Z, Φ e−2 , ϕ e0 ), δ) e as in Section 1.2.4, in their H-orbits. e˘ ˘ Since X is formally smooth over S0 , there exists a complete regular local ΦH ˘ f,δH f,ρ

ring Ve and an ideal Ie ⊂ Ve such that Ve /Ie ∼ = V and such that the morphism e : Spf(Ve , I) e˘ ˘ e →X Spec(V ) → Ntor extends to a morphism ξ κ e ΦH ˘ which induces a f,δH f,ρ e˘ ˘ e e e dominant morphism from Spec(V ) to Spec(R), where R is the local ring of X

ΦH ˘ f,δH f,ρ

at the image of s¯. Let ‡ e‡ , ei‡ , α e‡ , λ e (G eH e ) → Spec(V )

(1.3.4.13)

denote the pullback of (1.3.4.7) under the composition of ξe with the canonical e tor , and let e˘ ˘ morphism X eΣ e Φ f,δ f,ρ˘ → MH, H

H

‡ b‡ , bi‡ , α b‡ , λ (G bH b ) → Spec(V )

denote the pullback of (1.3.4.9) under ξ.

(1.3.4.14)

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e As in (6) of Theorem 1.3.1.3, (1.3.4.13) defines an object of DEGPEL,M e f (V ), H which corresponds to an object \,‡ e ‡ , λ e‡ , i e‡ , X e ‡ , Ye ‡ , φe‡ , e (B c‡ , e c∨,‡ , τe‡ , [e αH e ]) B B \,‡ e in DDPEL,M αH e f (V ) under [62, Thm. 5.3.1.19], where [e e ] is represented by some H

\,‡ ‡,∼ ‡ ‡ e‡ , e e‡ , ϕ α eH e‡−1,He , ϕ e‡,∼ c∨,‡ eH e = (ZH e e−2,H e, ϕ e , δH e cH e, e e ,τ e) 0,H H \,‡ e ‡ , Ye ‡ , φe‡ , [e as in [62, Def. 5.3.1.14; see also the errata]. Note that (X αH e ]) determines ‡ e ‡ e‡ ˘ ˘ e some cusp label [(Z , Φ , δ )] equivalent to the cusp label [(Z e , Φ e , δ˘ e )] represene H

e H

H

e H

H

H

˘ introduced above (where the (ϕ e ˘ δ) ˘, Φ, ted by the H-orbit of the (Z e‡−2,He , ϕ e‡0,He ) in ‡,∼ ˘ e is induced by (ϕ Φ e‡,∼ e0, e, ϕ e ) as in the corrected [62, Def. 5.4.2.8] in the errata). H −2,H H For simplicity, we shall use entries in this last representative to replace their isomorphic (or equivalent) objects, and say in this case that (ϕ e‡,∼ e‡,∼ e, ϕ e ) induces −2,H 0,H (ϕ˘−2,He , ϕ˘0,He ). ˘ e e ‡ , λ e‡ , i e‡ , X, ˘ Y˘ , φ, By definition, the pullback of (B c‡ , e c∨,‡ ) to the subscheme B B b‡ , bi‡ ) → Spec(V ). Let us denote it by b‡ , λ Spec(V ) of Spec(Ve ) depends only on (G ˘ b b ‡ , λ b‡ , i b‡ , X, ˘ Y˘ , φ, (B c‡ , b c∨,‡ ). B B

eϕ e e e = (X, e Ye , φ, Note that the H-orbit (e ZHe , Φ e−2,He , ϕ e0,He ), δeHe ) is part of the data of H ˘ b , (ϕ˘ b , ϕ˘ b ), and δ˘ b , which κ e. By Lemma 1.2.4.16, it makes sense to consider Z H

−2,H

0,H

H

∼ ˘ ∼ ˆ Z(1)), ˆ ˆ and b ˘ ⊗ Z, ˘, (ϕ˘−2 : Gr˘Z−2 → are the H-orbits of Z HomZˆ (X ϕ˘0 : GrZ0 → Y˘ ⊗ Z), Z

Z

˘ respectively. Moreover, by extending restrictions to subgroups of L/n e L e (with δ, ‡ e ˘ Z−1,n replaced with its subgroup Z−1,n ) as in Construction 1.3.4.6, α eHe induces a ‡ ‡ ‡ b level-H structure ϕ e of (B , λ b‡ , i b‡ ) depending only on α b , which we denote e −1,H

B

B

b H

b by ϕ b‡−1,Hb . Then it also makes sense to consider the H-orbit (ϕ e‡,∼ e‡,∼ b, ϕ b ), which −2,H 0,H

‡,∼ ‡,∼ we denote by (ϕ b−2, b0, ˘−2,Hb , ϕ˘0,Hb ) × ϕ b‡−1,Hb which b, ϕ b ), which is a subscheme of (ϕ H H ˘ ZH c

b U(n)-orbits, b can be identified with a system of H/ where n ≥ 1 are integers such b e b ⊂ H, b which surjects under the two projections to the orbits that U(n) := U(n) G defining (ϕ˘−2,Hb , ϕ˘0,Hb ) and ϕ b‡−1,Hb . In this case, we say that (ϕ b‡,∼ b‡,∼ b, ϕ b ) induces −2,H 0,H b the H-orbit (ϕ˘ b , ϕ˘ b ). −2,H

0,H

e := Frac(Ve ) and ηe := Spec(K). e By [62, Lem. 4.2.1.7], the trivialization Let K ∼ ⊗ −1 ∨,‡ ‡ of biextensions τe : 1Y˘ × X,e c ×e c‡ )∗ PB determines a homomorphism e ι‡ : ˘ η → (e e ‡ ,e η e ‡ . It also determines a homomorphism e ˘ ηe → e \,‡ lifting e ι∨,‡ : X Y˘ηe → G c∨,‡ : Y˘ → B η e

e ∨,\,‡ lifting e ˘ →B e ∨,‡ , which is compatible with e G c‡ : X ι‡ under the homomorphisms η e e e‡ : B e\,‡ : G ˘ and λ e‡ → B e ∨,‡ (and the homomorphism λ e \,‡ → G e ∨,\,‡ φ˘ : Y˘ → X B determined by them), by symmetry of τe‡ . Let V 1 be the localization of Ve at the (prime) kernel Ie1 of Ve → V . By [62, Prop. 4.5.3.11 and Cor. 4.5.3.12], the restriction

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e \,‡1 (resp. e e ι‡ |Y (resp. e ι∨,‡ |X ) extends to a homomorphism e ι‡,1 : Y → G ι∨,‡,1 : X → V ∨,\,‡ e 1 ), whose pullback to the closed point η of Spec(V 1 ) is a homomorphism b G ι‡ : V \,‡ ∨,‡ ∨,\,‡ ‡,1 ∨,‡,1 b b Y → Gη (resp. b ι : X → Gη ). The two homomorphisms e ι and e ι (resp.

b ι‡ and b ι∨,‡ ) are compatible with each other under the homomorphisms φ : Y → X e\,‡ : G b\,‡ : G e \,‡ → G e ∨,\,‡ (resp. λ b \,‡ → G b ∨,\,‡ ). By the same argument as in and λ ‡ ∨,‡ the proof of [62, Lem. 4.2.1.7], the pair (b ι ,b ι ) determines a pair (b τ ‡ : 1Y

∼

˘ × X,η

⊗ −1 → (b c∨,‡ |Y × b c‡ )∗ PB b ‡ ,η , ∼

⊗ −1 τb∨,‡ : 1Y˘ × X,η → (b c∨,‡ × b c‡ |X )∗ PB b ‡ ,η )

(satisfying certain familiar compatibility conditions, which we omit for simplicity). ˆ there exists an ´etale covering e e ⊂ G( e Z), For each integer n ≥ 1 such that U(n) ⊂H ∼ e n )  ηe = Spec(K) e and an H e e 0 -orbit of liftings τe‡ : 1 1 ˘ ˘ ηen = Spec(K n ηn → PeZ n Y × X,e ‡ ∗ ⊗ −1 of τe‡ , which determines orbits of homomorphisms e ι‡n : 1 Y˘ηe → (c∨,‡ n ×c ) P ‡ n

B ,e ηn

n

e \,‡ and e ˘ ηe → G e ∨,\,‡ compatible with liftings cn∨,‡ : 1 Y˘ηe → B e ‡ and : n1 X G ι∨,‡ n n n η en η en η en n e e‡ : B e\,‡ : G ˘ ηe → B e ∨,‡ (and with φ˘n : 1 Y˘ → 1 X, ˘ λ e‡ → B e ∨,‡ , and λ e \,‡ → c‡n : 1 X n

η en

n

n

n

B

e ∨,\,‡ ). Let Ven be the normalization of Ve in K e n , let Ien1 := rad(Ie1 · Ven ), let Vn1 be the G localization of Ven at the multiplicative subset complement to Ie1 , and let Kn be the ee e reduction of Vn1 modulo Vn1 · Ien1 . By the construction of X ΦH σ as a completion f,δH f,e e of the affine toroidal embedding Ξ e e (e σ ) along its σ e-strata, we may choose Ven ΦH f,δH f

such that Kn is a finite ´etale K-algebra. Let ηn = Spec(Kn ). Let Vn be the normalization of V in Kn . By [62, Prop. 4.5.3.11 and Cor. 4.5.3.12], the restriction 1 e \,‡ e ι‡n | n1 Yηen (resp. e ιn∨,‡ | n1 Xηen ) extends to a homomorphism e ι‡,1 en → GV 1 (resp. n : n Yη n ∨,\,‡ 1 e : e ι∨,‡,1 X → G ), whose pullback to the dense subscheme η = Spec(K 1 n n) η e n n

Vn

n

b \,‡ b ∨,\,‡ of Spec(Vn ) is a homomorphism b ι‡n : n1 Yηn → G ιn∨,‡ : n1 Xηn → G ηn (resp. b ηn ). ∨,‡ ‡ ∨,‡,1 ‡,1 ιn ) are compatible with (resp. b ιn and b ιn These two homomorphisms e ιn and e e\,‡ : G e \,‡ → G e ∨,\,‡ each other under the homomorphisms φn : n1 Y → n1 X and λ ‡ 1 \,‡ \,‡ ∨,\,‡ ∨,‡,1 b b b e : n Y → BV 1 and (resp. λ : G → G ), and determine homomorphisms e cn n ∨,‡ 1 1 1 ‡ ∨,‡ ‡ ∨,‡ ‡,1 b e b b cn : X → Bη ). The H-orbit of cn : Y → Bη and b e cn : X → B 1 (resp. b n

Vn

n

n

n

n

ι‡n , b ιn∨,‡ ) is well defined (i.e., independent of the choice of the representative c∨,‡ (b c‡n , b n ,b ‡ ∨,‡ ‡ e e 0 -orbit) and descends (as compatible subschemes of schemes (e cn , e cn , τen ) in its H P e Z

of homomorphisms) to η. Such a descended object is independent of n, which we shall denote by (b c‡Hb , b c∨,‡ ι‡Hb , b ι∨,‡ b ,b b ). H H Then, as above, the pair (b ι‡Hb , b ι∨,‡ b ) determines a pair H ‡ ∨,‡ (b τH bH b, τ b )

(whose detailed definitions we omit for simplicity).

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b bi, α b λ, In summary, given the family (G, bHb ) → Ntor κ e as in Construction 1.3.4.6, tor each morphism ξ : Spec(V ) → Nκe as above determines a tuple \,‡ ˘ b b ‡ , λ b‡ , i b‡ , X, ˘ Y˘ , φ, (B c‡ , b c∨,‡ , τb‡ , τb∨,‡ , [b αH b ]), B B

(1.3.4.15)

\,‡ where [b αH b ] is an equivalence class of \,‡ ‡ ∨,‡ ˘b, b ˘ b, ϕ α bH b‡,∼ b‡−1,Hb , ϕ b‡,∼ c‡Hb , b c∨,‡ bH bH b = (ZH b, ϕ b , δH b ,τ b, τ b ) −2,H 0,H H

(1.3.4.16)

(whose precise definitions we omit for simplicity). Given a tuple as in (1.3.4.15), if we set b ‡ , λ b‡ , i b‡ , ϕ (B ‡ , λB ‡ , iB ‡ , ϕ−1,Hκ ) := (B b) B B b−1,H and (c‡ , c∨,‡ , τ ‡ ) := (b c‡ |X , b c∨,‡ |Y , τb‡ |1Y ×X,η ), \,‡ and define [αH ] using similar restrictions, then the tuple κ \,‡ (B ‡ , λB ‡ , iB ‡ , X, Y, φ, c‡ , c∨,‡ , τ ‡ , [αH ]) κ

(1.3.4.17)

defines an object of DDPEL,MHκ (V ). On the other hand, the pullback ‡ ‡ b‡ b‡ b (G , λ , i , α bHb ) → Spec(V ) is determined up to isomorphism by its generic fiber ‡ ‡ b‡ b‡ b bH,η (Gη , λη , iη , α b ) → Spec(K), which (up to isomorphism) determines and is de-

‡ ∨,‡ termined by a tuple ((G‡η , λ‡η , i‡η , αH ), (e c‡H,η cH,η e ,e e )) → Spec(K) parameterized by κ ,η ‡ ‡ ‡ ‡ ee e . The abelian part (G , λ , i , α ) extends to a degenerating family C η

ΦH f,δH f

η

η

Hκ ,η

‡ (G‡ , λ‡ , i‡ , αH ) κ

(1.3.4.18)

‡ of type MHκ over Spec(V ) (with αH still defined only over Spec(K)) which defines κ an object of DEGPEL,MHκ (V ). By the theory of two-step degenerations (see [28, Ch. III, Thm. 10.2] and [62, Sec. 4.5.6]), and by analyzing endomorphism structures and level structures as in [62, Sec. 5.1–5.3], under [62, Thm. 5.3.1.19], this last object (1.3.4.18) corresponds to the above object (1.3.4.17) in DDPEL,MHκ (V ). ¯ As for (e c‡H,η c∨,‡ e ,e e ), they are determined by their values on K-valued points, H,η ¯ is any fixed algebraic closure of K, which are orbits of compatible homowhere K 1 e ¯ : 1X e → B ∨,‡ (K) ¯ and e ¯ ¯ c∨,‡ Y → B ‡ (K). On the other morphisms e c‡n (K) n (K) : n

n

‡ ∨,‡ hand, by the same argument as in the proof of [62, Lem. 4.2.1.7], (b τH bH b, τ b ) is de1 ˘ ∨,\,‡ ¯ 0,‡ ¯ (K) and termined by orbits of compatible homomorphisms b ιn (K) : n X → G 1 ˘ \,‡ ¯ \,‡ ∨,\,‡ ‡ ¯ b ι∨,0,‡ ( K) : Y → G ( K), where G and G are determined by b c | X : X → n n ∨,‡ ∨,‡ ‡ ‡ ‡ 0,‡ ¯ b b B = B and b c |Y : Y → B = B , respectively. By definition, b ι (K) and ¯ are compatible with the homomorphism ι∨,‡ (K) ¯ : X → G∨,\,‡ (K) ¯ and b ι∨,0,‡ (K) ‡ ¯ \,‡ ¯ ‡ ι (K) : Y → G (K) defined by τ . Given the splitting δ, there exists a subgroup ˘ n (resp. Y˘n ) of 1 X ˘ (resp. 1 Y˘ ) containing X (resp. Y ) such that the admissible X n n e ˘ X e (resp. s ˘ : Y˘  Ye ) induces an isomorphism X ˘ n /X ∼ surjection s ˘ : X = 1X X

Y

n

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(resp. Y˘n /Y ∼ = n1 Ye ). Hence, by [62, Prop. 4.5.5.3], we can form equivariant quo¯ and ι‡ (K), ¯ and obtain (by restrictions) orbits of comtients by the images of ι∨,‡ (K) 1 e ∼ ˘ ∨,‡ ¯ ¯ ∼ ¯ (K)(X)) patible homomorphisms n X = Xn /( n1 X) → B ∨ (K) = (G∨,\,‡ (K))/(ι 1 e ∼ ˘ 1 \,‡ ¯ ‡ ¯ ∼ ¯ and n Y = Yn /( n Y ) → B(K) = (G (K))/(ι (K)(Y )). These coincide with the ¯ and e ¯ because of the following reasons: At level one, above orbits of e c‡n (K) cn∨,‡ (K) this follows from the theory of two-step degenerations (see [28, Ch. III, Sec. 10] and [62, Sec. 4.5.6]), because e c‡ and e c∨,‡ (which are defined over K) are induced by e →G e ∨,\ and Ye → G e\ , (reductions of extensions of) compatible homomorphisms X η e η e e = Spec(Frac Ve ) is some auxiliary choice as above. At higher where ηe = Spec(K) levels, this follows from the way we reconstruct level structures from its graded pieces using the various splittings. In brief, the tuple over Spec(V ) as in (1.3.4.15) determines and is determined by the tuple (1.3.4.14) (up to isomorphism, over Spec(K)). As in [62, (6.2.5.10)], the pair (b τ ‡ , τb∨,‡ ) defines compatible morphisms υτb‡ : ˘ → Z and υτb∨,‡ : Y˘ × X → Z (using the discrete valuation υ : Inv(V ) → Z of Y ×X V ), which define the same element b ˘ )∨ υτb‡ = υτb∨,‡ ∈ (S Φc R H

(see (1.2.4.29)). On the other hand, as in (6) of Theorem 1.3.1.3, τe‡ defines a ˘ → Z, which defines an element morphism υτe‡ : Y˘ × X υτe‡ ∈ ρ˘ ⊂ P+ ˘ , Φ f H

‡

∨,‡

where ρ˘ is as above. Since (b τ , τb see that

) is defined by extending restrictions of υτe‡ , we

υτb‡ = υτb∨,‡ ∈ ρb = pr(Sb ˘

∨ Φ c )R H

b˘ (˘ ρ) ⊂ P Φc H

(see (1.2.4.41)). If ρ˘ is replaced with another representative, then ρb is replaced with a translation under the action of ΓΦ˘ c . (This finishes Construction 1.3.4.12.) H

e Σ, e σ e0 , Σ e 0, σ Proposition 1.3.4.19. Suppose κ e = (H, e) and κ e 0 = (H e0 ) are elements ++ ++ 0 e in K κ0 ]  κ = [e κ] in KQ,H (see Definition 1.2.4.44). Let Q,H such that κ = [e tor 0 0 0 0 b bi, α b , bi , α b λ, b ,λ (G, b b) → N (resp. (G b ) → Ntor 0 ) denote the pullback of the deH

κ e

b0 H

κ e

0 e ei, α e tor (resp. (G e0 , ei0 , α e tor ) as in e λ, e0 , λ generating family (G, eHe ) → M eH eΣ e e 0 ) → MH e 0 ,Σ e0 H,

tor tor Construction 1.3.4.6. Then there is a canonical morphism fκetor 0 ,e κ : Nκ e 0 → Nκ e 0 tor b0 , bi0 , α b0 , λ such that (G bH b 0 ) → Nκ e0 is canonically isomorphic to the pullback of tor b bi, α b λ, (G, bHb ) → Ntor κ e under fκ e0 ,e κ. e ++ , the closure Ntor of Nκe = e Σ, e σ In particular, for each κ e = (H, e) ∈ K Q,H

κ e

e e e e tor and the open embedding κ Z etor : Nκe ,→ Ntor depend (up to eΣ e κ e [(ΦH σ )] in MH, f,δH f,e ++ b Σ) b in K canonical isomorphism) only on the pair κ = [e κ] = (H, . Q,H

tor tor The morphism fκetor etale locally given by equivariant morphisms 0 ,e κ : Nκ e 0 → Nκ e is ´ between toric schemes mapping strata to strata, which is log ´etale essentially by

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definition (see [45, Thm. 3.5]). Moreover, as in [28, Ch. V, Rem. 1.2(b)] and in the proof of [62, Lem. 7.1.1.4], we have Ri (fκetor = 0 for i > 0 by [50, Ch. I, 0 ,e κ )∗ ONtor κ e0 Sec. 3]. b (resp. H b 0 ) is determined by some H e (resp. H e 0 ) satisfying ConProof. Suppose H 0 0 b e b0 ∩ H b (resp. dition 1.2.4.7. By Lemma 1.3.4.2, we may replace H (resp. H ) with H 0 e ∩ H), e in which case we have a canonical (forgetful) morphism fκe0 ,eκ : Nκe0 ∼ H = ∨,0 0 0 0 0 0 ee e → C ee e ∼ N (by constructions). Suppose ((G , λ , i , α ), (e c , e c C = κe Hκ0 e0 H e 0 )) ΦH ΦH H f0 ,δH f0 f,δH f ee e (resp. (resp. ((G, λ, i, αH ), (e c e, e c∨ ))) is the tautological object over C H

κ

ΦH f0 ,δH f0

e H

ee e ), as in Construction 1.3.4.6, which determines and is determined by C ΦH f,δH f 0 b0 , bi0 , α bN , biN , α b0 , λ b ,λ (G bH bHb ) → Nκe ), the pullback of e0 (resp. (GNκ e κ e κ e Nκ Nκ Nκ b 0 ) → Nκ e0 e0 e0 tor 0 b0 b0 0 tor b b b b b b ) → N ) to Nκe0 (resp. Nκe ). Then fκe0 ,eκ is (G , λ , i , α b ) → N 0 (resp. (G, λ, i, α H

κ e

b0 H

κ e

ee e , such also the canonical morphism determined by the universal property of C ΦH f,δH f 0 ,e )) under f is canonically isomorphic that the pullback of ((G, λ, i, αHκ ), (e cHe , e c∨ κ e κ e H b c∨,0 )); or, ratc∨,0 )) of ((G0 , λ0 , i0 , α0 0 ), (e c0 , e c0 , e to the H-orbit ((G0 , λ0 , i0 , α0 ), (e Hκ

e H

H0

e H

e0 H

e0 H

bN , biN , α bN , λ her, such that the pullback of (G bHb ) → Nκe under fκe0 ,eκ is canonically κ e κ e κ e 0 0 0 0 0 b0 , bi0 , α b b0 , λ b b b bH isomorphic to the H-orbit (GNκe 0 , λNκe 0 , iNκe 0 , α bHb ) → Nκe0 of (G Nκ Nκ Nκ b0 ) → e0 e0 e0 Nκe0 . Since Ntor κ e0 is noetherian normal, by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, 0 b0 , bi0 , α b0 , λ Prop. 3.3.1.5], since (G bH e0 is canonically isomorphic to the Nκ Nκ Nκ b 0 ) → Nκ e0 e0 e0 b b b pullback of (GNκe , λNκe , iNκe , α bHb ) → Nκe under fκe0 ,eκ , as soon as fκe0 ,eκ : Nκe0 → Nκe tor tor b0 , bi0 , α b0 , λ , we know that (G b0 ) → Ntor extends to a morphism f 0 : Ntor 0 → N 0 is κ e ,e κ

κ e

κ e

κ e

b0 H

tor b bi, α b λ, canonically isomorphic to the pullback of (G, bHb ) → Ntor κ e under fκ e0 ,e κ . Such an tor extension fκe0 ,eκ is necessarily unique, because Nκe (resp. Nκe0 ) is dense in Ntor κ e (resp. 0 0 Ntor ). Hence, it suffices to show that f : N → N extends locally. κ e ,e κ κ e κ e κ e0 e ˘ ˘ 0 -stratum of M e tor , Let s¯ be any geometric point of Ntor κ e0 on the Z[(Φ e 0 ,Σ e0 ˘ )] f0 ,δH f0 ,ρ H H 0 0 ˘ ˘ ˘ ˘ ˘ ˘ where [(ΦHe 0 , δHe 0 , ρ˘ )] is represented by some (ΦHe 0 , δHe 0 , ρ˘ ) with (ZHe 0 , ΦHe 0 = ˘ ϕ˘ e 0 , ϕ˘ e 0 ), δ˘ e 0 ) representing some cusp label as in Section 1.2.4. ˘ Y˘ , φ, (X, −2,H

0,H

H

e = For simplicity, let us fix compatible choices of representatives (e Z, Φ eϕ e and (Z ˘ ϕ˘−2 , ϕ˘0 ), δ), ˘ as in Section 1.2.4, in their e Ye , φ, ˘ = (X, ˘ Y˘ , φ, ˘, Φ (X, e−2 , ϕ e0 ), δ) e 0 -orbits. As in Construction 1.3.4.12, each morphism ξ 0 : Spec(V ) → Ntor H 0 centeκ e

red at a geometric point s¯ of Ntor κ e0 , where V is a complete discrete valuation ring with fraction field K, and where η := Spec(K) is mapped to the generic point of the irreducible component containing the image of s¯, determines a tuple \,‡ ˘ b b ‡ , λ b‡ , i b‡ , X, ˘ Y˘ , φ, (B c‡ , b c∨,‡ , τb‡ , τb∨,‡ , [b αH b 0 ]) B B \,‡ as in (1.3.4.15), where [b αH b 0 ] is an equivalence class of \,‡ ‡,∼ ∨,‡ ‡ ∨,‡ ˘ b0 , b ˘ b0 , ϕ α bHb 0 = (Z b‡−1,Hb 0 , ϕ b‡,∼ c‡Hb 0 , b cH bHb 0 , τbH b0 , ϕ b 0 , δH b0 , τ b0 ) H b−2,H 0,H

as in (1.3.4.16), and the pair (b τ ‡ , τb∨,‡ ) defines an element υτb‡ = υτb∨,‡ in ρb0 for some + 0 b b ρb ⊂ PΦ˘ in ΣΦ˘ c0 . (We should have denoted all these entries with some extra 0 in c0 H

H

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0 b0 , bi0 , α b0 , λ their superscripts, because they are determined by the pullback of (G bH b0 ) → tor b N 0 . But we omit them for the sake of simplicity.) By forming H-orbits, we obtain κ e

a tuple \,‡ ˘ b b ‡ , λ b‡ , i b‡ , X, ˘ Y˘ , φ, (B c‡ , b c∨,‡ , τb‡ , τb∨,‡ , [b αH b ]), B B \,‡ where [b αH b ] is an equivalence class of ‡ ∨,‡ \,‡ ˘b, b ˘ b, ϕ b‡,∼ b‡−1,Hb , ϕ b‡,∼ c‡Hb , b c∨,‡ bH bH α bH b ,τ b, τ b ), b = (ZH b, ϕ b , δH −2,H 0,H H

and the pair (b τ ‡ , τb∨,‡ ) defines the same element υτb‡ = υτb∨,‡ in ρb0 . By assumption, 0 b˘ ∼ b ˘ , we b is a refinement of Σ. b Hence, under the canonical isomorphism P Σ =P ΦH ΦH c0 c + 0 b b ˘ , so that υτb‡ = υτb∨,‡ lies in ρb. have ρb ⊂ ρb for some cone ρb ⊂ P in Σ ˘c Φ H

ΦH c

˘f e ΦH M e H

b see Defini(which depends only on H; By the universal property of ‡,∼ ˘ ϕ˘ b , ϕ˘ b ), δ˘ b ), (ϕ ˘ b = (X, ˘ Y˘ , φ, ˘ b, Φ tion 1.2.1.15), the data (Z b−2,Hb , ϕ b‡,∼ b ), and H H −2,H 0,H H 0,H ‡ b ‡ , λ b‡ , i b‡ , ϕ (B b ) on the torus and abelian parts define a canonical morphism B B b−1,H ˘

˘

f f e ΦH e ΦH e˘ ˘ → M ξ1 : Spec(V ) → M e , the additional e . By the universal property of CΦ f,δH f H H H e˘ ˘ data (b c‡ , b c∨,‡ ) lifting (b c‡ , b c∨,‡ ) define a canonical morphism ξ0 : Spec(V ) → C

b H

ΦH f,δH f

b H

lifting ξ1 . By the construction of 

∼ e˘ ˘ Ξ Φ f,δ f,˘ σ = SpecO H

H

˘ e ˘ ˘ (`) ⊕ Ψ Φ f,δ f H

˘ σ⊥ `∈˘

e C ˘ ˘ ,δ Φ f H f H



H

e ˘ ˘ , which we can canonically identify as over C ΦH f,δH f   ˘ b˘ ˘ ∼ b ˘ ˘ (`) Spec Ξ Ψ ⊕ = Φ c,δ c Φ c,δ c O H

H

b C ˘ ˘ ,δ Φ c H c H

˘ S b˘ `∈ Φ

H

H

c H

b ˘ ˘ (see Proposition 1.3.2.56), it enjoys the universal property (similar to over C ΦH c,δH c ‡ ∨,‡ e ˘ ˘ ) such that the final part of the data (b e˘ ˘ → C τH bH that of Ξ b, τ b ) lifting ΦH ΦH f,δH f f,δH f e˘ ˘ (b τ ‡ , τb∨,‡ ) determines a canonical morphism ξ˜K : Spec(K) → Ξ lifting ξ0 ΦH σ f,δH f,˘

e ˘ ˘ . Since the element υτb‡ = υτb∨,‡ e˘ ˘ under the canonical morphism Ξ Φ f,δ f,˘ σ → CΦ f,δ f H

H

H

H

defined by (b τ ‡ , τb∨,‡ ) lies in ρb0 ⊂ ρb, by the construction of   ∼ ˘ e ˘ ˘ (˘ e ˘ ˘ (`) Ξ ρ ) Spec ⊕ Ψ = Φ f,δ f,˘ σ Φ f,δ f O H

H

e C ˘ Φ

˘ f,δH f H

H

˘ σ ⊥ ∩ ρ˘∨ `∈˘

(see [61, Sec. 3B]), which we can canonically identify as  b ˘ ˘ (b b˘ Ξ ρ) = Spec ⊕ Ψ ΦH c,δH c

OCb

˘ ˘ ,δ Φ c H c H

˘ ρ∨ `∈b

H

 ˘ ( `) Φ c,δ˘c H

H

b and on ρb∨ ∼ (see (1.3.2.60)), which depends only on H ˘ ⊥ ∩ ρ˘∨ , and by the same =σ argument as in the proof of [62, Prop. 6.2.5.11], the morphism ξ˜K extends to a

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e ˘ ˘ (˘ morphism ξ˜ : Spec(V ) → Ξ ΦH σ ρ) lifting ξ0 under the canonical morphism f,δH f,˘ e e ΞΦ˘ f,δ˘f,˘σ (˘ ρ) → CΦ˘ f,δ˘f , which maps the special point of Spec(V ) to the ρ˘-stratum H H H H e˘ ˘ e ˘ ˘ (˘ Ξ of Ξ ρ). (Alternatively, we can noncanonically lift υτb‡ = υτb∨,‡ to ΦH ˘ f,δH f,ρ

ΦH σ f,δH f,˘

e ˘ ˘ and Ξ e ˘ ˘ (˘ elements of ρ˘ ⊂ P+ ˘ , work with ΞΦ Φ f,δ f ρ) directly, and invoke the f,δ f Φ H

f H

H

H

H

original [62, Prop. 6.2.5.11].) Since V is complete, ξ˜ induces a morphism ξˆ from e˘ ˘ e ˘ ˘ (˘ Spf(V ) to X ˘-stratum ΦH σ ,ρ˘, the formal completion of ΞΦ σ ρ) along its ρ f,δH f,˘ f,δH f,˘ H e˘ ˘ e ˘ ˘ . Then the composition of ξˆ with the canonical morphism X Ξ → ΦH ˘ f,δH f,ρ

ΦH σ ,ρ˘ f,δH f,˘

tor Ntor κ e gives a canonical morphism ξ : Spec(V ) → Nκ e . As explained in Construction 1.3.4.12, ξη := ξ|η : η = Spec(K) → Nκe is de0 b0 , bi0 , α b 0 b0 b0 b0 ) → Ntor b 0η , λ termined by the pullback (G η η bH b0 κ e0 b 0 ,η ) → Spec(K) of (G , λ , i , α H 0 0 tor 0 b0 b0 0 b b under ξη := ξ |η : Spec(K) → N 0 , whose H-orbit (Gη , λη , iη , α b ) → Spec(K) κ e

b H,η

is (as explained in the first paragraph of this proof) isomorphic to the pullback b bi, α b λ, of (G, bHb ) → Ntor under the composition of ξη0 : Spec(K) → Nκe0 with κ e tor tor fκe0 ,eκ : Nκe0 → Nκe . Hence, ξη = fκe0 ,eκ ◦ ξη0 by the universal property of Nκe , and ξ : Spec(V ) → Ntor can be interpreted as a (necessarily unique) extension κ e of fκe0 ,eκ ◦ ξη0 : Spec(K) → Nκe . Since ξ 0 : Spec(V ) → Ntor ¯ (the prescribed center of ξ 0 ) are arbitrary, and κ e0 and s tor since Nκe0 is noetherian normal, this shows that fκe0 ,eκ extends to fκetor 0 ,e κ , as desired. By considering good algebraic models as in the paragraph preceding [61, Lem. tor tor 5.10], the morphism fκetor is ´etale locally given by the canoni0 ,e κ : Nκ e 0 → Nκ e 0 e e cal morphism Ξ ˘ ˘ ρ), because the tautological data (as in ρ ) → Ξ ˘ ˘ (˘ 0 (˘ ΦH σ f0 ,δH f0 ,˘

ΦH σ f,δH f,˘

e ˘ ˘ (˘ e˘ ˘ ρ0 ) is the pullback of the one over Ξ (1.3.4.15)) over Ξ ΦH σ ρ). By conΦH σ 0 (˘ f,δH f,˘ f0 ,δH f0 ,˘ 0 e˘ ˘ e ˘ ˘ (˘ struction, Ξ etale and equivariant with respect ρ) → Ξ ΦH ΦH σ ρ) is log ´ σ 0 (˘ f0 ,δH f0 ,˘ f,δH f,˘ e˘ e to the canonical homomorphism E between tori, which (by Pro0 → E˘ ΦH σ f0 ,˘

ΦH σ f,˘

position 1.3.2.56 again) can be canonically identified with the canonical log ´etale b ˘ ˘ (b b ˘ ˘ (b morphism Ξ ρ), equivariant with respect to the canonical hoρ0 ) → Ξ ΦH ΦH c0 ,δH c0 c,δH c b˘ b ˘ between tori (dual to the canonical homomorphism momorphism E → E ΦH c0

ΦH c

b˘ → S b ˘ ). The remainder of the proposition then follows. S Φc Φ c0 H

H

Thanks to Lemma 1.3.4.2 and Proposition 1.3.4.19, we can make the following: e ++ which defines κ = [e Definition 1.3.4.20. For κ e∈K κ] ∈ K++ Q,H Q,H (see Definition isog ◦ 1.2.4.44), we shall denote κ e : HomO (Q, GMHκ ) → Ngrp and κ etor : Nκe ,→ Ntor κ e κ e isog ◦ grp tor tor by κ : HomO (Q, GMHκ ) → Nκ and κ : Nκ ,→ Nκ , respectively. For κ e and e ++ such that κ0 = [e κ e0 in K κ0 ]  κ = [e κ] in K++ Q,H Q,H , we shall denote the canonical grp grp × MHκ0 , morphisms fκe0 ,eκ : Nκe0 → Nκe , fκe0 ,eκ := κ eisog ◦ ((e κ0 )isog )−1 : Ngrp κ e 0 → Nκ e MHκ

grp tor tor isog 0 0 and fκetor ◦ ((κ0 )isog )−1 : Ngrp 0 ,e κ0 → κ : Nκ e0 → Nκ e by fκ ,κ : Nκ → Nκ , fκ0 ,κ := κ grp tor tor tor Nκ × MHκ0 , and fκ0 ,κ : Nκ0 → Nκ , respectively. (That is, we drop the tildes in MHκ

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b ˘ ˘ ˘ b , δ˘ b , τb)]-stratum of Ntor all such notations.) We shall denote by Z κ , [(ΦH τ )] the [(ΦH H c,δH c,b tor ∼ ∼ e ˘ ˘ ˘ e , δ˘ e , τ˘)]-stratum Z e˘ ˘ b˘ ˘ which is the [(Φ of N under =Ξ =Ξ H

H

[(ΦH τ )] f,δH f,˘

ΦH τ f,δH f,˘

ΦH τ c,δH c,b

κ e

tor the canonical identification between Ntor (up to canonical isomorphism) κ and Nκ e ˘ ˘ ˘ ˘ (when (ΦHb , δHb , τb) is determined by (ΦHe , δHe , τ˘) as in Section 1.2.4).

Now the question is whether the structural morphism fκ : Nκ → MH extends tor tor (necessarily uniquely) to a (proper) morphism fκtor : Ntor κ → MH = MH,Σ between the compactifications. ++ Let K++ Q,H,Σ be the subset of KQ,H defined at the end of Section 1.2.4, which is ++ the subset of KQ,H consisting of elements κ satisfying Condition 1.2.4.49. The main b Σ) b = [e e Σ, e σ result of [61, Sec. 3B] is the following: For κ = (H, κ] = [(H, e)] ∈ K++ Q,H,Σ

e Σ, e σ (which means κ e = (H, e) satisfies Condition 1.2.4.48, for some and hence every representative κ e of κ), the structural morphism fκ : Nκ → MH extends to a (unique) tor morphism fκtor : Ntor etale locally given by morphisms between κ → MH , which is ´ toric schemes equivariant under (surjective) morphisms between tori. (The proof b to satisfy either Conditions 1.2.4.8 or 1.2.4.9.) In the remainder does not require H of [61, Sec. 3–5], it was shown that the collection of such extended morphisms satisfy the remaining requirements of Theorem 1.3.3.15. (The proofs of these used a particular representative κ e of κ = [e κ], which nevertheless suffices, by Proposition 1.3.4.19. Also, they assumed that κ ∈ KQ,H,Σ , in which case Nκ → MHκ = MH is an abelian scheme—but this is not really necessary: For log smoothness, in [61, e e e is Sec. 3C], the proof using the extended Kodaira–Spencer isomorphism over M H,Σ insensitive to whether Nκ → MH is an abelian scheme or not. We note that for the condition on equidimensionality, in [61, Sec. 3D], the proofs there are combinatorial in nature and also insensitive to whether Nκ → MH is an abelian scheme or not. For the statements (4) and (5) of Theorem 1.3.3.15, the proof using the Hecke action of e e e } e e are also insensitive to whether Nκ → MH is an e ∞ ) on the collection {M G(A H,Σ H,Σ abelian scheme or not. In [61, Sec. 4–5], the proof for statements in (3) of Theorem 1.3.3.15 can be verified by ´etale descent, and hence can be proved with the same methods even when we only assume κ ∈ K+ Q,H,Σ , in which case Nκ → MHκ = MH is only an abelian scheme torsor.) Hence, the same methods of the proof of [61, Thm. 2.15; see also the errata] work here for the slightly generalized Theorem 1.3.3.15. 1.4

1.4.1

Automorphic Bundles and Canonical Extensions in Characteristic Zero Automorphic Bundles

Suppose there exists a finite extension F00 of F0 in C such that there exists an O ⊗ F00 -module L0 such that L0 ⊗ C ∼ = V0 , where V0 is as in (1.1.1.4). Once 0 Z

F0

the choice of F00 is fixed, the choice of L0 is unique up to isomorphism because O ⊗ F00 -modules are uniquely determined by their multi-ranks. (See [62, Lem. 1.1.3.4 Z

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and Def. 1.1.3.5] for the notion of multi-ranks.) Let ∨ 0 h · , · ican. : (L0 ⊕ L∨ 0 (1)) ×(L0 ⊕ L0 (1)) → F0 (1)

be the alternating pairing defined by h(x1 , f1 ), (x2 , f2 )ican. := f2 (x1 ) − f1 (x2 ) (cf. [62, Lem. 1.1.4.13]). Definition 1.4.1.1. (See [61, Def. 6.2].) For each F00 -algebra R, set   ∨  (g, r) ∈ GLO ⊗ R ((L0 ⊕ L0 (1)) ⊗0 R) × Gm (R) :  Z F0 G0 (R) := , hgx, gyican. = rhx, yican. , ∀x, y ∈ (L0 ⊕ L∨ 0 (1)) ⊗ R F00

∨ P0 (R) := {(g, r) ∈ G0 (R) : g(L∨ 0 (1) ⊗ R) = L0 (1) ⊗ R}, F00

F00

M0 (R) := GLO ⊗ R (L∨ 0 (1) ⊗ R) × Gm (R), F00

Z

where we view M0 (R) canonically as a quotient of P0 (R) by P0 (R) → M0 (R) : (g, r) 7→ (g|L∨0 (1) ⊗ R , r). 0 F0

The assignments are functorial in R and define group functors G0 , P0 , and M0 over F00 . Lemma 1.4.1.2. (See [61, Lem. 6.3].) Suppose R is the algebraic closure of F00 in C. Then there is an isomorphism (L ⊗ R, h · , · i) ∼ = ((L0 ⊕ L∨ 0 (1)) ⊗ R, h · , · ican. ), F00

Z

which induces an isomorphism G ⊗ R ∼ = G0 ⊗ R over R. (Consequently, P0 (R) can F00

Z

be identified with a “parabolic” subgroup of G(R).) (In practice, it is not necessary to take R to be algebraically closed. Much smaller rings would suffice for the existence of isomorphisms as in Lemma 1.4.1.2.) In the remainder of this subsection, by abuse of notation, we shall replace MH etc with their base changes from Spec(F0 ) to Spec(F00 ), and replace S0 = Spec(F0 ) with Spec(F00 ). Definition 1.4.1.3. The principal G0 -bundle over MH is the relative scheme EG0 := IsomO ⊗ OM ((H dR 1 (GMH /MH ), h · , · iλ , OMH (1)), Z

((L0 ⊕ L∨ 0 (1))

H

⊗ OMH , h · , · ican. , OMH (1))),

F00

the sheaf of isomorphisms of OMH -sheaves of symplectic O-modules, over MH . (The group G0 acts as automorphisms on (L ⊗ OMH , h · , · iλ , OMH (1)) by definition. The Z

third entries in the tuples represent the values of the pairings.)

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Definition 1.4.1.4. The principal P0 -bundle over MH is the relative scheme ∨ EP0 := IsomO ⊗ OM ((H dR 1 (GMH /MH ), h · , · iλ , OMH (1), LieG∨ M Z

H

H

/MH (1)),

∨ ((L0 ⊕ L∨ 0 (1)) ⊗ OMH , h · , · ican. , OMH (1), L0 (1) ⊗ OMH )), F00

F00

the sheaf of isomorphisms of OMH -sheaves of symplectic O-modules with maximal totally isotropic O ⊗ F00 -submodules, over MH . (The group P0 acts as automorphisms Z

on (L ⊗ OMH , h · , · iλ , OMH (1), L∨ 0 (1) ⊗ OMH ) by definition. The third entries in the F00

Z

tuples represent the values of the pairings.) Definition 1.4.1.5. The principal M0 -bundle over MH is the relative scheme EM0 := IsomO ⊗ OM ((Lie∨ G∨ M Z

H

H

∨ /MH (1), OMH (1)), (L0 (1)

⊗ OMH , OMH (1))),

F00

the sheaf of isomorphisms of OMH -sheaves of O ⊗ F00 -modules, over MH .

(We

Z

view the second entries in the pairs as an additional structure, inherited from the corresponding objects for P0 . The group M0 acts as automorphisms on (L∨ 0 (1) ⊗ OMH , OMH (1)) by definition.) F00

Remark 1.4.1.6. The Tate twists on Lie∨ G∨ M

H

/MH (1)

in Definitions 1.4.1.4 and

1.4.1.5 have been omitted in most of this author’s writing so far (in, for example, [61], [59], and [70]), which unfortunately made it unclear whether the duality and LieG∨M /MH involves a Tate twist or not. For the sake of between Lie∨ G∨ MH /MH H clarity, we have reinstated such Tate twists, as explained in Remark 1.1.2.3. Lemma 1.4.1.7. The relative scheme EG0 (resp. EP0 , resp. EM0 ) over MH is an ´etale torsor under (the pullback of) the group scheme G0 (resp. P0 , resp. M0 ). Proof. The existence of sections over geometric points of MH is guaranteed by the determinantal condition for LieA/MH,1 . By the infinitesimal deformation theory explained in [62, Ch. 2] (based on well-known ideas due to Grothendieck, Mumford, and others), we have isomorphisms between ∨ (H dR 1 (A/MH ), h · , · iλ , OMH (1), LieA∨ /MH )

and ∨ ((L0 ⊕ L∨ 0 (1)) ⊗ OMH , h · , · ican. , OMH (1), L0 (1) ⊗ OMH ) F00

F00

over the formal completions of MH at points of finite type over S0 . Since the sheaves involved are all coherent, we can algebraize the isomorphisms over formal bases by Grothendieck’s formal existence theory [35, III-1, 5.1.2], and obtain sections of these functors over complete local rings (at points of finite type over S0 ). Since the base scheme S0 = Spec(F00 ) is a point, and since these functors are locally of finite presentation (because they are defined by morphisms between coherent sheaves over the scheme MH of finite type over S0 ), Artin’s approximation theory [3, Thm. 1.10 and Cor. 2.5] implies that they have sections ´etale locally over MH , as desired.

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Definition 1.4.1.8. For each F00 -algebra R, we denote by RepR (G0 ) (resp. RepR (P0 ), resp. RepR (M0 )) the category of R-modules with algebraic actions of G0 ⊗ R (resp. P0 ⊗ R, resp. M0 ⊗ R). F00

F00

F00

Definition 1.4.1.9. Let R be any F00 -algebra. For each W ∈ RepR (G0 ), we define G0 ⊗ R

EG0 ,R (W ) := (EG0 ⊗ R) F00

0 F0

×

W,

called the automorphic sheaf over MH ⊗ R associated with W . It is called an F00

automorphic bundle if W is locally free of finite rank over R. We define similarly EP0 ,R (W ) (resp. EM0 ,R (W )) for W ∈ RepR (P0 ) (resp. W ∈ RepR (M0 )) by replacing G0 with P0 (resp. M0 ) in the above expression. Lemma 1.4.1.10. Let R be any F00 -algebra. (1) The assignment EG0 ,R ( · ) (resp. EP0 ,R ( · ), resp. EM0 ,R ( · )) defines an exact functor from RepR (G0 ) (resp. RepR (P0 ), resp. RepR (M0 )) to the category of quasi-coherent sheaves over MH . (2) If we consider an object W ∈ RepR (G0 ) as an object of RepR (P0 ) by restriction to P0 , then we have a canonical isomorphism EG0 ,R (W ) ∼ = EP0 ,R (W ). (3) If we view an object W ∈ RepR (M0 ) as an object of RepR (P0 ) via the canonical homomorphism P0 → M0 , then we have a canonical isomorphism EP0 ,R (W ) ∼ = EM0 ,R (W ). (4) Suppose W ∈ RepR (P0 ) has a decreasing filtration by subobjects Fa (W ) ⊂ W in RepR (P0 ) such that each graded piece GraF (W ) := Fa (W )/Fa+1 (W ) can be identified with an object of RepR (M0 ). Then EP0 ,R (W ) has a filtration Fa (EP0 ,R (W )) := EP0 ,R (Fa (W )) with graded pieces EM0 ,R (GraF (W )). The proofs of these statements can be found in [70, Sec. 1.3]. Lemma 1.4.1.11. For any F00 -algebra R, the pullback of LieGM /MH (resp. H top Lie∨ Lie∨ GM /MH , resp. ωMH = ∧ GM /MH ; see (2) of Theorem 1.3.1.5) to MH ⊗ R H

H

is canonically isomorphic to EM0 ,R (W ) for W = L0 ⊗ R (resp. F00

L∨ 0

F00

⊗ R, resp.

F00

∧top L∨ 0 ⊗ R). F00

Proof. This follows from Definitions 1.4.1.5 and 1.4.1.9, and from Lemma 1.4.1.10.

1.4.2

Canonical Extensions

Now let us explain the construction of canonical extensions using Theorem 1.3.3.15. By taking Q = O, so that HomO (Q, GMH )◦ ∼ = GMH and so that there exists some Q× -isogeny κisog : GMH → N = Ngrp = Nκ for some κ ∈ KQ,H,Σ as in κ

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Theorem 1.3.3.15, the locally free sheaf H 1dR (N/MH ) ∼ = H 1dR (GMH /MH ) extends to 1 tor tor the locally free sheaf H log-dR (N /MH ) over OMtor . Let H 1 tor H log-dR /Mtor ). (Ntor /Mtor H ), OMtor H ) := HomOMtor (H log-dR (N 1 H H

Then this

H log-dR (Ntor /Mtor 1 H )

can qualifies as the H dR in the following: 1 (GMH /MH )

Proposition 1.4.2.1. (See [61, Prop. 6.9].) There exists a unique locally free can over OMtor satisfying the following properties: sheaf H dR 1 (GMH /MH ) H can (1) The sheaf H dR , canonically identified with a subsheaf of the 1 (GMH /MH ) dR quasi-coherent sheaf (MH ,→ Mtor H )∗ (H 1 (N/MH )), is self-dual under the tor pairing (MH ,→ MH )∗ h · , · iλ . We shall denote the induced pairing by h · , · ican λ . dR (1) as a subsheaf totally isotropic (2) H 1 (GMH /MH )can contains Lie∨ G∨ /Mtor H under the pairing h · , · ican . λ can (3) The quotient sheaf H dR can be canonically iden/Lie∨ 1 (GMH /MH ) G∨ /Mtor H tor tified with the subsheaf LieG/Mtor of (MH ,→ MH )∗ LieGM /MH . H H

∼

(4) The pairing h · , · ican induces an isomorphism LieG/Mtor → LieG∨ /Mtor λ H H which coincides with dλ. can (5) Let H 1dR (GMH /MH )can := HomOMtor (H dR , OMtor ). Then the 1 (GMH /MH ) H H

Gauss–Manin connection

∇ : H 1dR (GMH /MH ) → H 1dR (GMH /MH ) ⊗ Ω1MH /S0 OMH

extends to an integrable connection 1

∇ : H 1dR (GMH /MH )can → H 1dR (GMH /MH )can ⊗ ΩMtor H /S0 OMtor

(1.4.2.2)

H

with log poles along D∞,H , called the extended Gauss–Manin connection, such that the composition (ignoring Tate twists; see Remark 1.1.2.3) 1 can Lie∨ G/Mtor ,→ H dR (GMH /MH ) H

1

∇

→ H 1dR (GMH /MH )can ⊗ ΩMtor H /S0 OMtor H

(1.4.2.3)

1

 LieG∨ /Mtor ⊗ ΩMtor H /S0 H OMtor H

induces by duality the extended Kodaira–Spencer morphism 1

∨ Lie∨ G/Mtor ⊗ LieG∨ /Mtor → ΩMtor H /S0 H

OMtor

H

H

as in [62, Def. 4.6.3.44], which factors through KS (in Definition 1.3.1.2) and induces the extended Kodaira–Spencer isomorphism KSG/Mtor in (4) H /S0 of Theorem 1.3.1.3.

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can With these characterizing properties, we say (H dR , ∇) is the cano1 (GMH /MH ) dR nical extension of (H 1 (GMH /MH ), ∇).

Remark 1.4.2.4. The notion of canonical extensions is closely related to the notion of regular singularities of algebraic differential equations. See [61, Rem. 6.12] for a list of references to this notion. can Then the principal bundle EG0 extends canonically to a principal bundle EG 0 tor over MH by setting can can EG := IsomO ⊗ OMtor ((H dR , h · , · ican (1)), λ , OMtor 1 (GMH /MH ) 0 H Z

H

((L0 ⊕ L∨ , h · , · ican. , OMtor (1))), 0 (1)) ⊗ OMtor H H

(1.4.2.5)

F00

the principal bundle EP0 extends canonically to a principal bundle EPcan over Mtor H 0 by setting can , h · , · ican (1), Lie∨ EPcan := IsomO ⊗ OMtor ((H dR λ , OMtor 1 (GMH /MH ) G∨ /Mtor (1)), 0 H Z

H

H

((L0 ⊕ L∨ 0 (1))

⊗ OMtor ,h·, H

F00

· ican. , OMtor (1), L∨ 0 (1) H

⊗ OMtor )), H

F00

(1.4.2.6)

can and the principal bundle EM0 extends canonically to a principal bundle EM over 0 tor MH by setting can (1)), EM := IsomO ⊗ OMtor ((Lie∨ G∨ /Mtor (1), OMtor 0 H Z

(L∨ 0 (1)

H

H

(1.4.2.7)

⊗ OMtor , OMtor (1))). H H

F00

can can (resp. EPcan , resp. EM ) over Mtor Lemma 1.4.2.8. The relative scheme EG H is an 0 0 0 ´etale torsor under (the pullback of) the group scheme G0 (resp. P0 , resp. M0 ).

Proof. As in the proof Lemma 1.4.1.7, these define ´etale torsors by Artin’s approximation theory (cf. [3, Thm. 1.10 and Cor. 2.5]), because these schemes have sections over the formal completions of Mtor H at points of finite type over S0 (be∨ tor cause Lie∨ (1) ⊗ O can be compared using the Lie algebra conand L tor MH 0 G∨ /M H

F00

dition [62, Def. 1.3.4.1 and Lem. 1.2.5.11], and because the pairings h · , · ican and λ h · , · ican. can be compared using [62, Cor. 1.2.3.10]). Definition 1.4.2.9. Let R be any F00 -algebra. For each W ∈ RepR (G0 ), we define G0 ⊗ R can EG (W ) 0 ,R

:=

can (EG 0

⊗ R)

F00

0 F0

×

W,

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called the canonical extension of EG0 ,R (W ), and define sub can EG (W ) := EG (W ) ⊗ ID∞,H , 0 ,R 0 ,R OMtor H

called the subcanonical extension of EG0 ,R (W ), where ID∞,H is the OMtor -ideal H defining the relative Cartier divisor D∞,H (with its reduced structure) in (3) of can Theorem 1.3.1.3. We define similarly EPcan (W ) and EPsub (W ) (resp. EM (W ) 0 ,R 0 ,R 0 ,R sub and EM0 ,R (W )) with G0 and its principal bundle replaced accordingly with P0 (resp. M0 ) and its principal bundle. Then we have: Lemma 1.4.2.10. Lemma 1.4.1.10 remains true if we replace the automorphic sheaves with their canonical or subcanonical extensions. As remarked in [71, Sec. 4.2], the same proofs for Lemma 1.4.1.10 also work here. Lemma 1.4.2.11. (Compare with Lemma 1.4.1.11.) For any F00 -algebra R, the pullback of LieG/Mtor (resp. Lie∨ , resp. ωMtor = ∧top Lie∨ ; see (3) of TheG/Mtor G/Mtor H H H H tor can orem 1.3.1.5) to MH ⊗ R is canonically isomorphic to EM0 ,R (W ) for W = L0 ⊗ R (resp.

L∨ 0

F00 top

⊗ R, resp. ∧

F00

F00

L∨ 0

⊗ R).

F00

Proof. This follows from (1.4.2.7), Definition 1.4.2.9, and Lemma 1.4.2.10. 1.4.3

Hecke Actions

Proposition 1.4.3.1. (Compare with (4) of Theorem 1.3.3.15.) Let R be any F00 -algebra, and consider any W ∈ RepR (G0 ). Suppose we have an element g ∈ ˆ such G(A∞ ), and suppose we have two open compact subgroups H and H0 of G(Z) 0 −1 that H ⊂ gHg . Then there is (by abuse of notation) a canonical isomorphism ∼

[g]∗ : [g]∗ EP0 ,R (W ) → EP0 ,R (W )

(1.4.3.2)

of coherent sheaves over MH0 , where the first EP0 ,R (W ) is defined over MH , and where the second is defined over MH0 . Suppose Σ = {ΣΦH }[(ΦH ,δH )] and Σ0 = {Σ0Φ0 0 }[(Φ0 0 ,δ0 0 )] are compatible choices H H H of admissible smooth rational polyhedral cone decomposition data for MH and MH0 , respectively, such that Σ0 is a g-refinement of Σ as in [62, Def. 6.4.3.3], so that tor tor [g] : Mtor H0 ,Σ0 → MH,Σ is defined as in Proposition 1.3.1.15. Then there is (by abuse of notation) a canonical isomorphism tor ∗

([g]

∼ tor ∗ can ) EP0 ,R (W ) →

) : ([g]

Mtor H0 ,Σ0 ,

EPcan (W ) 0 ,R

(1.4.3.3)

of coherent sheaves over where the first is defined over Mtor H,Σ , tor and where the second is defined over MH0 ,Σ0 . There is also (by abuse of notation) a canonical morphism tor ∗

([g]

can EM (W ) 0 ,R

tor ∗ sub ) EP0 ,R (W )

) : ([g]

→ EPsub (W ) 0 ,R

(1.4.3.4)

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of coherent sheaves over Mtor H0 ,Σ0 (which is not an isomorphism in general). The canonical morphisms (1.4.3.2), (1.4.3.3), and (1.4.3.4) are compatible with each other. The same statements are true if we replace P0 with G0 or M0 . If g = g1 g2 , where g1 and g2 are elements of G(A∞ ), each having a setup similar tor tor tor to that of g, then we have [g]∗ = [g1 ]∗ ◦ [g2 ]∗ and ([g] )∗ = ([g1 ] )∗ ◦ ([g2 ] )∗ whenever the involved isomorphisms are defined. Proof. Thanks to the construction of EPcan based on the canonical extensions 0 dR can can 0 (G /M ) and H (G /M H dR ) in Proposition 1.4.2.1, which are in MH H MH0 H 1 1 turn based on the relative de Rham homology in Theorem 1.3.3.15, we have the isomorphisms (1.4.3.2) and (1.4.3.3) because of (4c) of Theorem 1.3.3.15, the latter of which inducing the morphism (1.4.3.4), and we have the last statement (for P0 ) because of (4d) of Theorem 1.3.3.15. By Lemmas 1.4.1.10 and 1.4.2.10, these statements for P0 imply the analogous statements for G0 and M0 . 1.5

Comparison with the Analytic Construction

min All algebraically constructed objects in this chapter (such as Mtor H,Σ , MH , EP0 ,R , etc) are naturally compatible with their analytically constructed (algebraic) analogues. More precisely, the canonical open and closed immersion (1.1.3.1) extends to stratapreserving open and closed immersions tor Shtor H,Σ ,→ [MH,Σ ]

(1.5.1)

min Shmin H ,→ MH

(1.5.2)

and

(over S0 = Spec(F0 )), and the same are true for other objects defined on them. (For this to make sense, we can only consider Σ that works both for [62] and for works such as [89].) For more details, see [59]. We will not really need these results in this work. For applications, it suffices to know that one can first compare the analytically constructed (algebraic) objects with the algebraically constructed objects in [62] in characteristic zero, as carried out in [59]. Then one can compare the algebraically constructed objects in characteristic zero with all new objects constructed in this work.

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Chapter 2

Flat Integral Models

From now on, let us fix a choice of a rational prime number p > 0. Let MH be as in Section 1.1. In this chapter, we explain some general constructions of noetherian normal flat integral models of MH and their compactifications reviewed in Section 1.3. Beyond some basic properties due to their constructions, our understanding of their refined local structures is limited. (Nevertheless, in certain special cases, we can deduce the normality of the characteristic p fiber at the bottom level at p—i.e., when H is of the form H = Hp Hp with Hp = G(Zp )—from results in the theory of local models. See, for example, [65, Sec. 14].) Thus, the reader should keep in mind that the schemes constructed in this chapter are only auxiliary in nature. Because of our applications in mind, these integral models will be constructed only over Z(p) , although one can also obtain the models over Z by essentially the same constructions. We will cite [62] for the constructions of the various auxiliary models in this subsection. It is tempting to cite only [28] for these constructions, and this is indeed feasible for most constructions in this section. But for the construction of Hecke actions of elements in G(A∞ ) in Section 2.2.3, this is no longer logically sufficient, because the construction in [28], by requiring that the cone decompositions are admissible for GLg (Z) in the case of Siegel moduli of principally polarized abelian ˆ schemes of relative dimension g, only allowed Hecke actions of elements in Gaux (Z). 2.1

Auxiliary Choices

2.1.1

Auxiliary Choices of Smooth Moduli Problems

Lemma 2.1.1.1. For each integer d ≥ 1, there exist integers a1 > 0 and a2 ≥ 0, and a positive definite symmetric bilinear pairing ( · , · )aux : Z⊕(a1 +a2 ) × Z⊕(a1 +a2 ) → Z

(2.1.1.2)

satisfying the following properties: (1) Suppose that [L# : L] = d2 . Then, under the canonical embedding L⊕(a1 +a2 ) ,→ Laux := L⊕ a1 ⊕(L# )⊕ a2 125

(2.1.1.3)

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induced by L ,→ L# , the alternating pairing h · , · i ⊗( · , · )aux on L⊕(a1 +a2 ) ∼ = L ⊗ Z⊕(a1 +a2 ) extends to an alternating pairing h · , · iaux on Z

Laux valued in Z(1) that is self-dual at p in the sense that p - [L# aux : Laux ]. (2) Let A be a (relative) abelian scheme over an algebraic stack S, and let ×(a1 +a2 ) λ : A → A∨ be a polarization such that deg(λ) = d2 . Let AM aux := A O × a1 ∨ × a2 and Aaux := A ×(A ) , which are fiber products over S; and let S

a1 O ∨ f := Id× × λ × a 2 : AM aux → Aaux . Then λ : A → A and the morphism A S

∼

( · , · )∗aux : Z⊕(a1 +a2 ) → Z⊕(a1 +a2 )

(2.1.1.4)

M M,∨ canonical induced by ( · , · )aux induce a polarization λM aux : Aaux → Aaux (cf. Lemmas 1.2.4.1, 1.3.3.5, and 1.3.3.6, or rather [61, Lem. 2.5, 2.6, and ∨ −1 O O,∨ 2.9, and their proofs]), and λO ◦ λM aux := (f ) aux ◦ f : Aaux → Aaux is a × polarization (not just a Q -polarization) of degree prime to p. Moreover, 2 deg(λO aux ) depends only on deg(λ) = d and the choices of (a1 , a2 ) and ( · , · )aux , but not on A and λ.

If p - d, then we can take (a1 , a2 ) = (1, 0) and take ( · , · )aux : Z × Z → Z to be the pairing sending (1, 1) to 1. Otherwise, we can take
(a1 , a2 ) = (4, 4), and take ( · , · )aux to be defined by some 2 × 2 matrix t1x dx2 over M4 (Z) such that t xx = d2 − 1. Proof. The statement is obvious when p - d. Otherwise, we can arrange that h · , · iaux is self-dual (at every prime)  by the proof ofZarhin’s trick (as in [104, Sec. 2] and [80, IX, 1.1]), by taking x = x21 + x22 + x23 + x24

2

x1 x2 x3 x4

−x2 x1 x4 −x3

−x3 −x4 x1 x2

−x4 x3 −x2 x1

for any integers x1 , x2 , x3 , x4

such that = d − 1, which exist by the fact (due to Lagrange) that every nonnegative integer can be written as the sum of four squares of integers. Lemma 2.1.1.5. Let (Z, λZ ) be any polarized abelian scheme over a scheme S. Given any integer d ≥ 1, let us fix the choices of (a1 , a2 ) and ( · , · )aux as in Lemma 2.1.1.1. Then the functor that assigns to each scheme T over S the set of isomorphism classes of polarized abelian schemes (A, λ) over T such that deg(λ) = d2 and O O O (Z, λZ ) × T ∼ = (AO aux , λaux ) over T , where (Aaux , λaux ) is defined by (A, λ) as in (2) S

of Lemma 2.1.1.1, is representable by a scheme finite over S. Proof. By [81, Sec. 16], deg(λZ ) = d2aux for some integer daux ≥ 1. The assertion to prove is trivially true unless the construction in (2) of Lemma 2.1.1.1 assigns O to each pair (A, λ) of genus g and polarization degree d2 a pair (AO aux , λaux ) of 2 genus gaux = (a1 + a2 )g and polarization degree daux . Hence, it suffices to treat the universal case, which we explain as follows. Consider the Siegel moduli Ag,d (resp. Agaux ,daux ) of genus g (resp. gaux ) and polarization degree d2 (resp. d2aux ), which is an algebraic stack separated and of finite type over Spec(Z) (see [80, VII, 4.3] or [17, Def. 1.1 and Rem. 1.2]). The

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O assignment of pairs (AO aux , λaux ) to pairs (A, λ) parameterized by Ag,d as in (2) of Lemma 2.1.1.1 is functorial, and defines (by universal property) a morphism

Ag,d → Agaux ,daux .

(2.1.1.6)

To prove the lemma, it suffices to show that (2.1.1.6) is finite. Suppose V is the spectrum of a discrete valuation ring V with fraction field K. Suppose (AK , λK ) is an object of Ag,d (Spec(K)), and suppose the corresponding O O O object (AO aux,K , λaux,K ) of Agaux ,daux (Spec(K)) extends to an object (Aaux,V , λaux,V ) of Agaux ,daux (Spec(V )). By the semistable reduction theorem (see, for example, [62, Thm. 3.3.2.4]), up to replacing K with a finite extension field and replacing V accordingly, we may assume that AK extends to a semi-abelian scheme AV over Spec(V ). By the theory of N´eron models (see [10]; cf. [92, IX, 1.4], [28, Ch. I, Prop. ×(a1 +a2 ) 2.7], or [62, Prop. 3.3.1.5]), the isogeny fK : AM → AO aux,K = AK aux,K extends ×(a +a )

to an isogeny AV 1 2 → AO aux,V , and (since a1 + a2 > 0) this is possible only when AV is an abelian scheme. Also, the polarization λK extends to a polarization λV of AV . Consequently, we have an object (AV , λV ) of Ag,d (Spec(V )), which O O O must correspond to the unique extension (AO aux,V , λaux,V ) of (Aaux,K , λaux,K ) (up to unique isomorphism, by the theory of N´eron models again, or by the separateness of Agaux ,daux ). Hence, (2.1.1.6) is proper by the valuative criterion (and the fact that Ag,d and Agaux ,daux are separated and of finite type over Spec(Z)). In order to show that (2.1.1.6) is finite, it suffices to show that the induced proper morphism Ag,d ⊗ Z[ n1 ] → Agaux ,daux ⊗ Z[ n1 ] Z

(2.1.1.7)

Z

is finite for at least two integers n prime to each other. For each n ≥ 3, the algebraic stack Agaux ,daux ⊗ Z[ n1 ] admits a finite ´etale cover by the quasiZ

projective scheme Agaux ,daux ,n , defined as in [83, Ch. 7], parameterizing isomor∼ phisms γaux,n : Z⊕ 2gaux → Aaux [n] for each object (Aaux , λaux ) of Agaux ,daux ⊗ Z[ n1 ]. Z

(In order to avoid confusion with our later terminologies, we refrain from calling such isomorphisms level structures, because they are not required to respect the pairings on both sides.) Similarly, the algebraic stack Ag,d ⊗ Z[ n1 ] admits a finite Z

´etale cover by the quasi-projective scheme Ag,d,n,n parameterizing isomorphisms ∼ ∼ γn : Z⊕ 2g → A[n] and γn∨ : Z⊕ 2g → A∨ [n] for each object (A, λ) of Ag,d . (This is even more naive—the two isomorphisms γn and γn∨ are not required to be related to each other under λ.) By assigning to each object (A, λ, γn , γn∨ ) of Ag,d,n,n the object O O × a1 (AO ×(γn∨ )× a2 ) of Agaux ,daux ,n , we obtain a proper morphism aux , λaux , γaux,n := γn Ag,d,n,n → Agaux ,daux ,n

(2.1.1.8)

lifting (2.1.1.7). Then it suffices to show that (2.1.1.8) is finite, or rather just quasi-affine, by [35, III-1, 4.4.2]. Let ωAg,d,n,n and ωAgaux ,daux ,n denote the Hodge invertible sheaves over Ag,d,n,n and Agaux ,daux ,n , respectively, defined by the top exterior powers of the duals of the

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relative Lie algebras of the tautological abelian schemes, which are ample by [80, IX, 3.1]. By [80, IX, 2.4] and by the construction of (2.1.1.8), the pullback of a positive power of ωAgaux ,daux ,n to Ag,d,n,n is isomorphic to a positive power of ωAg,d,n,n . By [35, II, 5.1.6], these show that (2.1.1.8) is quasi-affine, as desired. Consider any integral PEL datum (Oaux , ?aux , Laux , h · , · iaux , h0,aux ), where (Laux , h · , · iaux ) is as in Lemma 2.1.1.1, such that Oaux is a subring of O stabilized by ? , with an induced (positive) involution we denote by ?aux , and such that h0,aux is canonically induced by h0 by the isomorphism Laux ⊗ R ∼ = L⊕(a1 +a2 ) ⊗ R Z

Z

induced by (2.1.1.3). Suppose moreover that p is a good prime for the integral PEL datum (Oaux , ?aux , Laux , h · , · iaux , h0,aux ) (see Definition 1.1.1.6), which is possible because we already know that p - [L# aux : Laux ], and that the action of Oaux on 0 Laux extends to an action of a maximal order Oaux in Oaux ⊗ Q containing Oaux Z

(cf. Condition 1.2.1.1). These are possible, for example, by taking Oaux = Z with trivial involution ?aux . From now on, we shall fix the auxiliary choices of (a1 , a2 ), ( · , · )aux , and (Oaux , ?aux , Laux , h · , · iaux , h0,aux ). Lemma 2.1.1.9. With the assumptions as above, the assignment (g, r) 7→ (g × a1 ×(r t g −1 )× a2 , r) defines an injective homomorphism G → Gaux

(2.1.1.10)

of algebraic group functors over Spec(Z), where Gaux is the group functor over Spec(Z) defined by the order Oaux (with positive involution ?aux ), the lattice Laux , and the pairing h · , · iaux as in Definition 1.1.1.3, which is compatible with the simiˆ (which stabilizes L ⊗ Z) ˆ to a subgroup of Gaux (Z) ˆ litude characters and maps G(Z) Z

ˆ (which stabilizes Laux ⊗ Z). Z

Proof. The assignment is injective because a1 > 0, and defines a homomorphism as asserted because Oaux is a subring of O, because ?aux is the restriction of ? , and because hx, rg −1 yi = hgx, yi = hx, t gyi by the definition of ν(g). Lemma 2.1.1.11. The reflex field F0,aux defined by the integral PEL datum (Oaux , ?aux , Laux , h · , · iaux , h0,aux ) (see [53, p. 389] or [62, Def. 1.2.5.4]) is contained in F0 (as subfields of C). Proof. Since h0,aux is canonically induced by h0 by the isomorphism Laux ⊗ R ∼ = Z

(L⊕ a1 ⊕(L# )⊕ a2 ) ⊗ R induced by (2.1.1.3), we have a canonical isomorphism Z

⊕(a +a ) V0,aux ∼ = V0 1 2 as Oaux ⊗ C-modules. By [62, Cor. 1.2.5.6], F0 (resp. F0,aux ) Z

is the subfield of C generated over Q by the traces TrC (b|V0 ) for b ∈ O (resp. TrC (b|V0,aux ) for b ∈ Oaux ). Hence, F0,aux is contained in F0 , as desired.

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129

p ˆ (resp. Gaux (Z ˆ p )), For each open compact subgroup Haux (resp. Haux ) of Gaux (Z) let MHaux (resp. MHpaux ) denote the moduli problem defined by the integral PEL datum (Oaux , ?aux , Laux , h · , · iaux , h0,aux ) over S0,aux = Spec(F0,aux ) (resp. ~S0,aux = Spec(OF0,aux ,(p) )), which is an algebraic stack separated, smooth, and of finite type over S0,aux (resp. ~S0,aux ) by [62, Thm. 1.4.1.11]. Let [MHaux ] (resp. [MHpaux ]) denote the coarse moduli space associated with MHaux (resp. MHpaux ; see [62, Sec. A.7.5]), which is a scheme quasi-projective over S0,aux (resp. ~S0,aux ) by [62, Cor. 7.2.3.10]. Moreover, let [MHpaux ⊗ Q] denote the coarse moduli space associated with Z

MHpaux ⊗ Q, which is canonically isomorphic to [MHpaux ] ⊗ Q, because the association Z

Z

of coarse moduli spaces is compatible with flat base changes. p ˆ → When Haux is the image of Haux under the canonical homomorphism Gaux (Z) p ˆ Gaux (Z ), we have a canonical finite morphism MHaux → MHpaux ⊗ Q,

(2.1.1.12)

Z

by forgetting the level structure at p, which factors as a composition of canonical finite morphisms MHaux → MHpaux G(Zp ) → MHpaux ⊗ Q. Z

Remark 2.1.1.13. There is a subtle difference between MHpaux ⊗ Q and the moZ

duli problem MHpaux Gaux (Zp ) over S0,aux = Spec(F0,aux ), because the former is not equipped with a level structure at p. Nevertheless, the canonical morphism MHpaux Gaux (Zp ) → MHpaux ⊗ Q

(2.1.1.14)

Z

is finite ´etale, which is an isomorphism at least when Oaux ⊗ Q is simple, because Z

p is a good prime for MGaux (Zˆ p ) (by [62, Prop. 1.4.4.3] and [53, Sec. 8]). (In what follows, we will not need (2.1.1.14) to be an isomorphism.) Proposition 2.1.1.15. With assumptions as above, for any open compact subgroup ˆ such that H is mapped into Haux under the homomorphism G(Z) ˆ → Haux of Gaux (Z) ˆ Gaux (Z) given by (2.1.1.10), we can define a finite morphism MH → MHaux O O O (AO aux , λaux , iaux , αHaux )

over S0,aux such that the pullback over MHaux to MH satisfies the following properties:

(2.1.1.16) of the tautological object

× a1 (1) AO × (A∨ )× a2 for the same integers (a1 , a2 ) as in aux is isomorphic to A MH

Lemma 2.1.1.1, which is equipped with an isogeny ×(a1 +a2 ) f : AM → AO aux := A aux

induced by λ : A → A∨ . O O,∨ (2) The polarization λO aux : Aaux → Aaux coincides with the composition × M M M,∨ (f ∨ )−1 ◦ λM aux ◦ f (as Q -isogenies), where λaux : Aaux → Aaux is indu∨ ced by λ : A → A and ( · , · )aux as in (2) of Lemma 2.1.1.1.

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Compactifications with Ordinary Loci O (3) The isogeny f : AM aux → Aaux is compatible with the Oaux -actions defined by M the Oaux -structure iaux : Oaux → EndMH (AM aux ) induced by the restriction O of i : O → EndMH (A) to Oaux , and by iO : aux Oaux → EndMH (Aaux ). (4) At each geometric point s¯ of MH , the level structure αH induces an H-orbit ∼ ˆ → T As¯, which in turn induces an Haux -orbit of isomorphisms α ˆ s¯ : L ⊗ Z Z

of isomorphisms

⊕(a1 +a2 )

α ˆ s¯

∼

⊗ A∞ : Laux ⊗ A∞ → V AO aux,¯ s ˆ Z

Z

(which makes sense because H is mapped into Haux under the homomorˆ → Gaux (Z) ˆ given by (2.1.1.10)). On the other hand, the level phism G(Z) O structure αHaux induces an Haux -orbit of isomorphisms ∼

∞ α ˆ sO : Laux ⊗ A∞ → V AO ¯ ⊗A aux,¯ s. ˆ Z

Z

These two Haux -orbits of isomorphisms coincide. p ˆ → When Haux is the image of Haux under the canonical homomorphism Gaux (Z) p ˆ ), by composition with (2.1.1.12), we obtain a morphism Gaux (Z

MH → MHpaux ⊗ Q

(2.1.1.17)

Z

(over S0,aux ), which induces a finite morphism MH → [MHpaux ] ⊗ Q, such that the Z

O O O ) of the tautological object over MHpaux to MH satispullback (AO aux , λaux , iaux , αHp aux fies the same properties as above, with (4) replaced with the following:

(40 ) At each geometric point s¯ of MH , the level structure αH induces an H-orbit ∼ p ˆ → -orbit of isomorphisms α ˆ s¯ : L ⊗ Z T As¯, which in turn induces an Haux Z

of isomorphisms

⊕(a1 +a2 )

α ˆ s¯

∼

⊗ A∞,p : Laux ⊗ A∞,p → Vp AO aux,¯ s ˆ Z

Z

p (which makes sense because H is mapped into Haux Gaux (Zp ) under the hoˆ ˆ momorphism G(Z) → Gaux (Z) given by (2.1.1.10)). On the other hand, the O p -orbit of isomorphisms level structure αH induces an Haux p aux ∼

∞,p α ˆ sO,p : Laux ⊗ A∞,p → Vp AO ¯ ⊗A aux,¯ s. ˆp Z

Z

p These two Haux -orbits of isomorphisms coincide. 0 0 Suppose we replace Haux with an open compact subgroup Haux such that Haux ˆ ˆ still contains the image of H under the homomorphism G(Z) → Gaux (Z) given by (2.1.1.10). Then the morphism MH → MH0aux → MH0,p ⊗ Q thus obtained are comaux Z

patible with (2.1.1.16) and (2.1.1.17), and with the (compatible) canonical morphisms MH0aux → MHaux and MH0,p ⊗ Q → MHpaux ⊗ Q. aux Z

Z

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131

O M O Proof. Let AM aux , Aaux , λaux , λaux , and f be defined as in (2) of Lemma 2.1.1.1 (with S = MH there). Since Oaux ⊂ O and since the involution ?aux is the restriction of ? , the O-structure i : O → EndMH (A) of (A, λ) induces an Oaux -structure iM aux : M M Oaux → EndMH (AM ) of (A , λ ), which in turn induces an O ⊗ Q-structure aux aux aux aux Z

O O O iO aux : Oaux ⊗ Q → EndMH (Aaux ) ⊗ Q of (Aaux , λaux ) as in [62, Def. 1.3.3.1] by Z

Z

M −1 iO for each b ∈ Oaux . aux (b) := f ◦ iaux (b) ◦ f At each geometric point s¯ of MH , the level structure αH lifts to an ∼ ˆ → ˆ T As¯, which induces an isomorphism α ˆ s¯ : L ⊗ Z O ⊗ Z-equivariant Z

Z

ˆ isomorphism Oaux ⊗ Z-equivariant Z ⊕(a1 +a2 )

α ˆ sM ˆ s¯ ¯ := α

∼

ˆ → T AM : (L⊕(a1 +a2 ) ) ⊗ Z aux,¯ s Z

and an Oaux ⊗ A∞ -equivariant isomorphism Z ∼

∞ α ˆ sM : (L⊕(a1 +a2 ) ) ⊗ A∞ → V AM ¯ ⊗A aux,¯ s ˆ Z

Z

(all matching similitudes, implicitly). By [62, Lem. 1.3.5.2], under the isomorphism ∼ α ˆ s¯ ⊗ A∞ : L ⊗ A∞ → V As¯, the polarization λs¯ : As¯ → A∨ s¯ (as an O-equivariant ˆ Z

Z

ˆ of L ⊗ A∞ . Hence, the isogeny) corresponds to the open compact subgroup L# ⊗ Z Z

Z

∞ ˆ restriction of α ˆ sM induces an Oaux ⊗ Z-equivariant isomorphism ¯ ⊗A ˆ Z

Z

α ˆ sO ¯

∼ ˆ→ : Laux ⊗ Z T AO aux,¯ s. Z

Since the choices of s¯ and α ˆ s¯ are arbitrary, by [62, Lem. 1.3.5.2] again, the O Oaux ⊗ Q-structure iO : O aux ⊗ Q → EndMH (Aaux ) ⊗ Q induces an Oaux -structure aux Z

Z

Z

O O O iO aux : Oaux → EndMH (Aaux ) of (Aaux , λaux ). Moreover, by forgetting the factor at p O ˆ p, the α ˆ s¯ above induces an Oaux ⊗ Z -equivariant isomorphism Z

α ˆ sO,p ¯

∼ ˆp → : Laux ⊗ Z Tp A O aux,¯ s. Z

Since the H-orbit of α ˆ s¯ is π1 (MH , s¯)-invariant, and since H is mapped into p ˆ → Gaux (Z) ˆ (resp. G(Z) ˆ → Haux (resp. Haux ) under the homomorphism G(Z) p O O p ˆ Gaux (Z )) given by (2.1.1.10), the Haux -orbit [α ˆ s¯ ]Haux of α ˆ s¯ (resp. Haux -orbit O,p p [ˆ αsO,p ] of α ˆ ) is π (M , s ¯ )-invariant. By [62, Prop. 1.4.3.4], the tuple 1 H Haux s¯ ¯ O,p O O O O O O p (AO , λ , i , [ α ˆ ] ) (resp. (A , λ , i , [ˆ α ] )) defines an object Haux s¯ aux aux aux s¯ Haux aux aux aux O O O O O O O O (Aaux , λaux , iaux , αHaux ) (resp. (Aaux , λaux , iaux , αHpaux )) of MHaux (resp. MHpaux ) over MH , which satisfies the properties described in the proposition by its very construction. We would like to show that LieAOaux /MH with its O ⊗ Q-module structure given Z

by iO aux satisfies the determinantal condition given by (Laux ⊗ R, h · , · iaux , h0,aux ) Z

as in [62, Def. 1.3.4.1]. Since this condition is closed by definition, and is open in

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characteristic zero by [62, Lem. 1.2.5.11], it suffices to verify it at each C-point t O O of MH . Let (At , λt , it ) and (AO aux,t , λaux,t , iaux,t ) denote the respective pullbacks of O O O (A, λ, i) and (Aaux , λaux , iaux ) to such a C-point t. By [62, Lem. 1.2.5.11] again, since LieA/MH with its O ⊗ Q-module structure given by i satisfies the determinanZ ∼ V0 as O ⊗ C-modules, tal condition given by (L ⊗ R, h · , · i, h0 ), we have LieA = t

Z

Z

⊕(a +a ) ∼ = V0 1 2 ∼ = V0,aux as

∼ and it suffices to note that LieAOaux,t = Oaux ⊗ C-modules (cf. the proof of Lemma 2.1.1.11). a1 Lie⊕ At

a2 ⊕ Lie⊕ A∨ t

Z

Thus, we have obtained the desired morphisms (2.1.1.16) and (2.1.1.17) by the moduli interpretations of MHaux and MHpaux , which are compatible with each other is obtained from α ˆ sO under (2.1.1.12) because α ˆ sO,p ¯ ¯ by forgetting the factor at p. The morphisms (2.1.1.16) and (2.1.1.17) between algebraic stacks are schematic and finite by Lemma 2.1.1.5 (for the abelian schemes and polarizations), by [62, Prop. 1.3.3.7] (for the endomorphism structures), and by the fact that the level structures are defined by isomorphisms between finite ´etale group schemes. Lemma 2.1.1.18. With assumptions as above, suppose the image Hp of H under ˆ → G(Z ˆ p ) is neat (which means, a fortiori, that H is the canonical morphism G(Z) p ˆ p ) such ⊂ Gaux (Z also neat). Then there exists a neat open compact subgroup Haux p that H is mapped into Haux = Haux Gaux (Zp ) under the injective homomorphism ˆ → Gaux (Z) ˆ given by (2.1.1.10). (If we only assume that H is neat, then we G(Z) 0 ˆ such that H is mapped ⊂ Gaux (Z) can still find a neat open compact subgroup Haux 0 ˆ ˆ into Haux under G(Z) → Gaux (Z).) p Proof. Let n0 ≥ 3 be any integer prime to p such that U p (n0 ) ⊂ H, and let Haux p p be generated by Uaux (n0 ) and the image of H under the injective homomorphism ˆ p ) → Gaux (Z ˆ p ) given by (2.1.1.10). Then every element of Hp is congruent G(Z aux

modulo n0 to the image of some element of Hp , which is neat by assumption. Hence, p p and Haux = Haux Gaux (Zp ) are also neat, by definition (see [89, 0.6] or [62, Def. Haux 1.4.1.8]), and by Serre’s lemma that no nontrivial root of unity can be congruent to 1 modulo n if n ≥ 3. (This is the same argument used in the proof of Lemma 1.2.4.45. The parenthetical remark in the statement of the lemma follows from the method of the proof.) 2.1.2

Auxiliary Choices of Toroidal and Minimal Compactifications

Let us continue with the setting in Section 2.1.1. ˆ (see Definition 1.2.1.2) Each symplectic admissible filtration Z = {Z−i }i of L ⊗ Z Z

ˆ by setting induces a symplectic admissible filtration Zaux = {Zaux,−i }i of Laux ⊗ Z Z ⊕(a1 +a2 )

Zaux,−i := (Z−i

)⊗A ˆ Z


∞


ˆ ∩ Laux ⊗ Z Z

(2.1.2.1)

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as submodules of Laux ⊗ A∞ . If Z is fully symplectic (see Definition 1.2.1.3), which Z

means Z extends to a symplectic filtration ZA = {Z−i,A }i of L ⊗ A, then Zaux = Z

{Zaux,−i }i also extends to a filtration Zaux,A = {Zaux,−i,A }i on Laux ⊗ A, by setting Z

Zaux,−i,A :=

⊕(a +a ) Z−i,A1 2 .

These definitions are compatible with actions of G(A) and Gaux (A) (and with the homomorphism G(A) → Gaux (A) given by (2.1.1.10)), and are compatible with reductions modulo n for any integer n ≥ 1. Thus, there is a well-defined assignment Z 7→ Zaux .

(2.1.2.2)

If Φ = (X, Y, φ, ϕ−2 , ϕ0 ) is a torus argument of Z (see Definition 1.2.1.5), then we define Xaux := X ⊕ a1 ⊕ Y ⊕ a2 and Yaux := Y ⊕ a1 ⊕ X ⊕ a2 . Lemma 2.1.2.3. With the setting as above, there exist canonically induced morphisms φaux : Yaux ,→ Xaux , aux ∼ ˆ Z(1)), ˆ ϕaux,−2 : GrZ−2 → HomZˆ (Xaux ⊗ Z,

Z

and ∼ ˆ ϕaux,0 : GrZ0aux → Yaux ⊗ Z Z

making Φaux := (Xaux , Yaux , φaux , ϕaux,−2 , ϕaux,0 ) a torus argument of Zaux , and making the diagrams   IdY

⊕ a1

2) Y ⊕(a1 +a _

φ ⊗( · , · )∗ aux

⊕ φ⊕ a2



X ⊕(a1 +a2 ) o



⊕ a1

IdX

⊕ φ⊕ a2

⊕(a1 +a2 )

φaux

? _ Xaux

o ϕaux,−2

o



⊕(a1 +a2 ) ˆ Z(1))) ˆ (HomZˆ (X ⊗ Z, Z

(2.1.2.4)

/ / GrZaux −2

(GrZ−2 )⊕(a1 +a2 ) ϕ−2

/ Yaux _

⊕ a1

(IdX

 ˆ Z(1)) ˆ / / Hom ˆ (Xaux ⊗ Z, Z ∗

⊕ φ ⊕ a2 )

Z

(2.1.2.5)

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and (GrZ0 )⊕(a1 +a2 )  ⊕(a1 +a2 )

ϕ0



/ GrZaux

(2.1.2.6)

0

o ϕaux,0

o



ˆ ⊕(a1 +a2 ) (Y ⊗ Z) Z

 ⊕ a1

IdY

⊕ φ ⊕ a2

 ˆ / Yaux ⊗ Z Z

commutative, where ( · , · )∗aux is canonically induced by ( · , · )aux as in Lemma 2.1.1.1. Proof. These follow from Lemma 2.1.1.1 and from the construction of the filtration Zaux,−i in (2.1.2.1). ∼ ˆ then it induces a ˆ is a splitting of the filtration Z of L ⊗ Z, If δ : GrZ → L ⊗ Z Z

Z

splitting of the filtration ZA∞ of L ⊗ A∞ , and hence induces a splitting δaux of the Z

filtration Zaux of Laux ⊗ A∞ . Z

The above assignments are compatible with the formations of orbits. That is, ˆ → Gaux (Z) ˆ given by when H is mapped into Haux under the homomorphism G(Z) (2.1.1.10), we have a well-defined assignment of representatives of cusp labels (ZH , ΦH , δH ) 7→ (ZHaux , ΦHaux , δHaux ).

(2.1.2.7)

This assignment is also compatible with the equivalence relations among representatives of cusp labels, and induces a well-defined assignment of cusp labels [(ZH , ΦH , δH )] 7→ [(ZHaux , ΦHaux , δHaux )].

(2.1.2.8)

Moreover, by Lemma 2.1.2.3, tensor products with the symmetric bilinear pairing ( · , · )aux in Lemma 2.1.1.1 induce an embedding ∨ (SΦH )∨ Q ,→ (SΦHaux )Q : y 7→ y ⊗( · , · )aux

(2.1.2.9)

(by forgetting the compatibility of the pairings with O, but retaining only the compatibility of the pairings with Oaux ). Since ( · , · )aux is positive definite, the embedding ∨ (SΦH )∨ R ,→ (SΦHaux )R

(2.1.2.10)

+ induced by (2.1.2.9) maps PΦH (resp. P+ ΦH ) to PΦHaux (resp. PΦHaux ). By construction, the pullback of a nondegenerate rational polyhedral cone σaux in PΦHaux (resp. P+ ΦHaux ) under (2.1.2.10) is either empty or a nondegenerate rational polyhedral cone σ in PΦH (resp. P+ ΦH ). (However, σ might not be smooth when σaux is.) The dual of (2.1.2.9) gives a surjection

(SΦHaux )Q := SΦHaux ⊗ Q  (SΦH )Q := SΦH ⊗ Q, Z

(2.1.2.11)

Z

which induces a homomorphism SΦHaux → SΦH .

(2.1.2.12)

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p ˆ → Gaux (Z ˆ p ), When Haux is mapped to Haux under the homomorphism Gaux (Z) by suppressing the factors at p, we obtain compatible assignments

(ZHaux , ΦHaux , δHaux ) 7→ (ZHpaux , ΦHpaux , δHpaux )

(2.1.2.13)

and [(ZHaux , ΦHaux , δHaux )] 7→ [(ZHpaux , ΦHpaux , δHpaux )],

(2.1.2.14)

together with the canonical homomorphism SΦHp

aux

→ SΦHaux ,

(2.1.2.15)

which induces the canonical isomorphisms ∼

(SΦHp )Q := SΦHp aux

aux

⊗ Q → (SΦHaux )Q ,

(2.1.2.16)

Z

∼

∨ (SΦHaux )∨ Q → (SΦHp )Q , aux

(2.1.2.17)

and ∼

∨ (SΦHaux )∨ R → (SΦHp )R . aux

(2.1.2.18)

By composing (2.1.2.7), (2.1.2.8), (2.1.2.15), (2.1.2.16), (2.1.2.9), and (2.1.2.10) with (2.1.2.13), (2.1.2.14), (2.1.2.12), (2.1.2.11), (2.1.2.17), and (2.1.2.18), respectively, we obtain (ZH , ΦH , δH ) 7→ (ZHpaux , ΦHpaux , δHpaux ),

(2.1.2.19)

[(ZH , ΦH , δH )] 7→ [(ZHpaux , ΦHpaux , δHpaux )],

(2.1.2.20)

SΦHp

aux

→ SΦH ,

(2.1.2.21)

(SΦHp )Q  (SΦH )Q ,

(2.1.2.22)

∨ (SΦH )∨ Q ,→ (SΦHp )Q ,

(2.1.2.23)

∨ (SΦH )∨ R ,→ (SΦHp )R ,

(2.1.2.24)

aux

aux

and aux

respectively. Definition 2.1.2.25. Let Σ (resp. Σaux , resp. Σpaux ) be a compatible choice of admissible smooth rational polyhedral cone decomposition data for MH (resp. MHaux , resp. MHpaux ). We say that Σ and Σaux (resp. Σpaux ) are compatible with each other if, for each representative (ZH , ΦH , δH ) of cusp labels of MH with assigned representative (ZHaux , ΦHaux , δHaux ) (resp. (ZHpaux , ΦHpaux , δHpaux )) of cusp labels of MHaux (resp. MHpaux ) as in (2.1.2.7) (resp. (2.1.2.19)), the image of each σ ∈ ΣΦH under the embedding (2.1.2.10) (resp. (2.1.2.24)) is contained in some cone σaux ∈ ΣΦHaux p p (resp. σaux ) is assig∈ ΣΦHp ). In this case, we say that (ΦHpaux , δHpaux , σaux aux ned to (ΦH , δH , σ), and (since this is compatible with the equivalence relations)

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p )] is assigned to [(ΦH , δH , σ)]. We say that we also say that [(ΦHpaux , δHpaux , σaux p Σaux and Σaux are compatible (resp. Σpaux induces Σaux ) if, for each representative (ZHaux , ΦHaux , δHaux ) of cusp label of MHaux with assigned representative (ZHpaux , ΦHpaux , δHpaux ) of cusp label of MHpaux as in (2.1.2.13), the image of each σaux ∈ ΣΦH under the isomorphism (2.1.2.18) is contained in some cone in p ΣΦHp (resp. is exactly some cone σaux in ΣΦHp ). In this case, we say that aux aux p p ) is (ΦHpaux , δHpaux , σaux ) is assigned to (ΦHaux , δHaux , σaux ) (resp. (ΦHpaux , δHpaux , σaux induced by (ΦHaux , δHaux , σaux )), and (since this is compatible with the equivalence p relations) we also say that [(ΦHpaux , δHpaux , σaux )] is assigned to [(ΦHaux , δHaux , σaux )] p p p (resp. [(ΦHaux , δHaux , σaux )] is induced by [(ΦHaux , δHaux , σaux )])

Lemma 2.1.2.26. If Σ and Σaux are compatible, and if Σaux and Σpaux are compatible, then Σ and Σpaux are also compatible. If Σ and Σpaux are compatible, and if Σpaux induces Σaux , then Σ and Σaux are also compatible. Proof. These follow immediately from the definitions. Lemma 2.1.2.27. Suppose Σaux and Σpaux are compatible. Then the morphism (2.1.1.12) canonically extends to a morphism tor p Mtor ⊗ Q, Haux ,Σaux → MHp aux ,Σaux

(2.1.2.28)

Z

tor where Mtor are as in Theorem 1.3.1.3 and [62, Thm. 6.4.1.1], p Haux ,Σaux and MHp aux ,Σaux such that the tautological tuple (Gaux , λaux , iaux , αHaux ) over Mtor Haux ,Σaux induces (by forgetting the factor at p of αHaux ) the pullback of the tautological tuple tor (Gaux , λaux , iaux , αHpaux ) over Mtor (denoted similarly, by p Haux ,Σaux over MHp aux ,Σaux abuse of notation), mapping the [(ΦHaux , δHaux , σaux )]-stratum Z[(ΦHaux ,δHaux ,σaux )] p tor p p p of Mtor p ,δ p ,σaux )] of MHp Haux ,Σaux to the [(ΦHaux , δHaux , σaux )]-stratum Z[(ΦHp aux ,Σaux aux Haux p p p when [(ΦHaux , δHaux , σaux )] is assigned to [(ΦHaux , δHaux , σaux )].

Proof. This follows by comparing the universal properties of Mtor Haux ,Σaux and Mtor , as in (6) of Theorem 1.3.1.3 and [62, Thm. 6.4.1.1]. p p Haux ,Σaux Proposition 2.1.2.29. With assumptions as in Proposition 2.1.1.15, there exist compatible choices Σ, Σaux , and Σpaux of admissible smooth rational polyhedral cone decomposition data for MH , MHaux , and MHpaux , respectively, such that Σ, Σaux , and Σpaux are compatible with each other as in Definition 2.1.2.25, and such that the morphism (2.1.1.17) canonically extends to a morphism tor Mtor H,Σ → MHaux ,Σaux ,

(2.1.2.30)

which induces by composition with (2.1.2.28) a morphism tor p Mtor ⊗ Q. H,Σ → MHp aux ,Σaux

(2.1.2.31)

Z

The morphism (2.1.2.30) (resp. (2.1.2.31)) maps the [(ΦH , δH , σ)]-stratum Z[(ΦH ,δH ,σ)] of Mtor H,Σ to the [(ΦHaux , δHaux , σaux )]-stratum Z[(ΦHaux ,δHaux ,σaux )] of

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p tor p p p Mtor ) p ,δ p ,σaux )] of MHp Haux ,Σaux (resp. [(ΦHaux , δHaux , σaux )]-stratum Z[(ΦHp aux ,Σaux aux Haux p when (ΦHaux , δHaux , σaux ) (resp. (ΦHpaux , δHpaux , σaux )) is assigned to (ΦH , δH , σ) (see Definition 2.1.2.25). Let (G, λ, i, αH ) (resp. (Gaux , λaux , iaux , αHaux ), resp. (Gaux , λaux , iaux , αHpaux )) denote the degenerating family of type MH (resp. MHaux , tor tor , denoted similarly by resp. MHpaux ) over Mtor p H,Σ (resp. MHaux ,Σaux ; resp. MHp aux ,Σaux abuse of notation) as in Theorem 1.3.1.3 (or rather [62, Thm. 6.4.1.1]). Then the tor pullback of Gaux (from either Mtor ) to Mtor p Haux ,Σaux or MHp H,Σ is isomorphic to aux ,Σaux × a1 ∨ × a2 G × (G ) , and satisfies analogues of the characterizing properties in ProMtor H,Σ

position 2.1.1.15. (In fact, by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5], the pullbacks of (Gaux , λaux , iaux , αHaux ) and (Gaux , λaux , iaux , αHpaux ) are determined up to unique isomorphisms by their restrictions to MH , which are then characterized by the properties stated in Proposition 2.1.1.15.) Proof. As in (2) of Lemma 2.1.1.1 and as in the proof of Proposition 2.1.1.15, × a1 ∨ ×(a1 +a2 ) ×(a1 +a2 ) × (G∨ )× a2 , and , GO , GM,∨ let GM aux := G aux := (G ) aux := G GO,∨ aux

× a2

∨ × a1

× G

:= (G )

, which are fiber products over

Mtor H,Σ

Mtor H,Σ tor MH,Σ , whose

pullbacks

O,∨ O M,∨ to MH can be canonically identified with AM aux , Aaux , Aaux , and Aaux , respectively. a1 a1 a2 M O ∨ a2 O,∨ M,∨ Let f := IdG × λ : Gaux → Gaux and f := IdG∨ × λ : Gaux → Gaux , Mtor H,Σ

Mtor H,Σ dual isogenies of each other. Let λM aux be defined by λ and : O in Lemma 2.1.1.1, and let iM (GM aux → EndMtor aux aux ) H,Σ

whose pullbacks to MH are the morphism ( · , · )∗aux as be induced by the restriction of i to Oaux . By (2) of Lemma 2.1.1.1, and by [92, IX, O ∨ −1 ◦ λM 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5], λO aux ◦ f : Gaux → aux := (f ) O,∨ × Gaux is an isogeny (not just a Q -isogeny) of degree prime to p whose pullback (GO to MH is a polarization, and we have an iO aux ) uniquely aux : Oaux → EndMtor H,Σ O O extending its pullback to MH . Together with the αHaux (resp. αH ) over MH p aux constructed in the proof of Proposition 2.1.1.15, we obtain a degenerating family O O O O O O O )) of type MHaux (resp. MHpaux ) (GO aux , λaux , iaux , αHaux ) (resp. (Gaux , λaux , iaux , αHp aux tor over MH,Σ . O O O tor To show that (GO aux , λaux , iaux , αHaux ) → MH,Σ is canonically isomorphic to the pullback of (Gaux , λaux , iaux , αHaux ) → Mtor Haux ,Σaux under a canonically determined morphism (2.1.2.30), we need to verify the condition as in [62, Thm. 6.4.1.1(6)] (cf. (6) of Theorem 1.3.1.3). In the association of degeneration data, over any Spec(V ) → Mtor H,Σ such that V is a complete discrete valuation ring with algebraically closed residue field k and valuation υ : Inv(V ) → Z, and such that Spec(Frac(V )) is mapped to a point s ∼ ˆ → of MH , and for any lifting α ˆ s¯ : L ⊗ Z T Gs¯ at a geometric point s¯ above s, the Z

(noncanonical) filtration Z is defined to be the pullback of the geometric filtration 0 ⊂ T Ts¯ ⊂ T G\s¯ ⊂ T Gs¯, whose H-orbit ZH is uniquely determined by αH . If O ˆ ∼ we define α ˆ sO ˆ s¯ as in the proof of Proposition 2.1.1.15, ¯ : Laux ⊗ Z → T Gaux,¯ s by α Z

then the filtration Zaux defined by Z as in (2.1.2.1) agrees with the pullback of the

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O,\ O O geometric filtration 0 ⊂ T Taux,¯ s ⊂ T Gaux,¯ s , because this last filtration s ⊂ T Gaux,¯ O O O on T Gaux,¯s is induced by the filtration 0 ⊂ V Taux,¯s ⊂ V GO,\ aux,¯ s ⊂ V Gaux,¯ s on ∼ O M O V Gaux,¯s , whose pullback under the isomorphism V(f ) : V Gaux,¯s → V Gaux,¯s agrees M,\ M M M with the filtration 0 ⊂ V Taux,¯ s ⊂ V Gaux,¯ s on V Gaux,¯ s (which naturally s ⊂ V Gaux,¯ \ agrees with the filtration induced by 0 ⊂ V Ts¯ ⊂ V Gs¯ ⊂ V Gs¯ on V Gs¯). Suppose, under the equivalence of categories in [62, Thm. 5.3.1.19], \ (B, λB , iB , X, Y, φ, c, c∨ , τ, [αH ])

(2.1.2.32)

is the object of DDPEL,MH (V ) associated with the object of DEGPEL,MH (V ) defined by the pullback of the degenerating family (G, λ, i, αH ) → Mtor H,Σ under Spec(V ) → Mtor , and suppose H,Σ \ (Baux , λBaux , iBaux , Xaux , Yaux , caux , c∨ aux , τaux , [αHaux ])

(2.1.2.33)

is the object of DDPEL,MHaux (V ) associated with the object of DEGPEL,MHaux (V ) O tor O O defined by the pullback of the degenerating family (GO aux , λaux , iaux , αHaux ) → MH,Σ tor under Spec(V ) → MH,Σ . Then (2.1.2.33) is induced by (2.1.2.32) in a sense that can be made precise, which implies in particular the following: Under the assignment (2.1.2.8), the cusp label [(ZH , ΦH = (X, Y, φ, ϕ−2,H , ϕ0,H ), δH )] determined by (2.1.2.32) gives the cusp label [(ZHaux , ΦHaux , δHaux )] determined by (2.1.2.33). If we fix a representative (ZH , ΦH , δH ) of [(ZH , ΦH , δH )], then the assignment (2.1.2.7) gives a representative (ZHaux , ΦHaux , δHaux ) of [(ZHaux , ΦHaux , δHaux )]. With such choices of (ZH , ΦH , δH ) and (ZHaux , ΦHaux , δHaux ), if B : SΦH → Inv(V ) and Baux : SΦHaux → Inv(V ) are determined by (2.1.2.32) and (2.1.2.33), respectively, then (2.1.2.9) maps υ ◦ B : SΦH → Z ,→ Q to υ ◦ Baux : SΦHaux → Z ,→ Q because λM aux is induced by λ and ( · , · )aux . If υ ◦ B defines an element of σ ∈ ΣΦH , and if the image of σ under (2.1.2.24) is contained in some σaux ∈ ΣΦHaux , then υ ◦ Baux defines an element of σaux . Thus, if Σ and Σaux are compatible with each other as in Definition 2.1.2.25, by considering all morphisms Spec(V ) → Mtor H,Σ as above, we see that O O O O (Gaux , λaux , iaux , αHaux ) satisfies the condition as in [62, Thm. 6.4.1.1(6)] (cf. (6) of Theorem 1.3.1.3), as desired. O O O The case for (GO ) → Mtor aux , λaux , iaux , αHp H,Σ and (2.1.2.31) is similar, by supaux pressing the factors at p in the above argument (and hence the obtained (2.1.2.31) is tautologically compatible with (2.1.2.30)). (Or one may just apply Lemma 2.1.2.27.) Consider the invertible sheaves top ∗ 1 ωMtor := ∧top Lie∨ eG ΩG/Mtor G/Mtor = ∧ H,Σ H,Σ

over

H,Σ

Mtor H,Σ , ωMtor H

aux ,Σaux

:= ∧top Lie∨ Gaux /Mtor

Haux ,Σaux

= ∧top e∗Gaux Ω1Gaux /Mtor

Haux ,Σaux

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over Mtor Haux ,Σaux , and ωMtorp

p Haux ,Σaux

:= ∧top Lie∨ Gaux /Mtorp

p Haux ,Σaux

= ∧top e∗Gaux Ω1Gaux /Mtorp

p Haux ,Σaux

Mtor . p Hp aux ,Σaux

over , resp. ωMtorp ) to MH (resp. We shall denote the pullback of ωMtor (resp. ωMtor H,Σ H aux

Haux

MHaux , resp. MHpaux ) by ωMH (resp. ωMHaux , resp. ωMHp ), which is independent of aux the choice of Σ (resp. Σaux , resp. Σpaux ). Lemma 2.1.2.34. The pullback of ωMtorp

p Haux ,Σaux

canonically isomorphic to ωMtor H

aux ,Σaux

to Mtor Haux ,Σaux under (2.1.2.28) is

.

Proof. This follows from Lemma 2.1.2.27 and the definitions of ωMtor and Haux ,Σaux ωMtorp p . Haux ,Σaux

Lemma 2.1.2.35. There exists an integer 1 ≤ a0 ≤ 2 such that the pullback ⊗ a0 ⊗ a0 of ωM (resp. ωM ) to Mtor tor tor H,Σ under the morphism (2.1.2.30) (resp. p p Haux ,Σaux

Haux ,Σaux

⊗a (2.1.2.31)) is isomorphic to ωM tor , where a := a0 (a1 + a2 ). We may take a0 = 1 H,Σ when a2 is even.

We shall henceforth fix a choice of a0 . Proof of Lemma 2.1.2.35. Consider also the invertible sheaf 0 top ∗ ωM := ∧top Lie∨ eG∨ Ω1G∨ /Mtor . tor G∨ /Mtor = ∧ H,Σ

H,Σ

H,Σ

By Proposition 2.1.2.29, the pullback of ωMtor H

aux ,Σaux

(resp. ωMtorp

p Haux ,Σaux

) to Mtor H,Σ is

canonically isomorphic to ⊗ a1 ωM tor

H,Σ

0 ⊗ a2 ⊗ (ωM . tor )

Mtor H,Σ

H,Σ

(This is consistent with Lemma 2.1.2.34.) By [80, IX, 2.4, and its proof], there exists an integer 1 ≤ a0 ≤ 2 such that ⊗ a0 ∼ 0 ⊗ a0 ωM . = (ωM tor ) tor H,Σ

H,Σ

Hence, up to replacing a0 with 1 when a2 is even, the lemma follows. min Let Mmin ) denote the minimal compactification of MHaux (resp. Haux (resp. MHp aux MHpaux ), which is by construction a projective variety over S0,aux = Spec(F0,aux ) (resp. ~S0,aux = Spec(OF0,aux ,(p) )) containing the coarse moduli space [MHaux ] of MHaux (resp. [MHpaux ] of MHpaux ) as an open subscheme. By [62, Thm. 7.2.4.1], p p there exists an integer N1 ≥ 1 (depending on Haux , which is 1 when Haux is neat) ⊗ N1 min such that ωMtorp p descends to an ample invertible sheaf over MHpaux , which we Haux ,Σaux

⊗ N1 denote by ωM by abuse of notation. In this case, by Lemma 2.1.2.34 and by the min p Haux

universal property of the projective spectra  ⊗k tor ∼ Mmin Haux = Proj ⊕ Γ(MHaux ,Σaux , ωMtor k≥0

Haux ,Σaux

 )

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and  ⊗k ∼ p Mmin Proj ⊕ Γ(Mtor , ωM = tor Hp Hp aux aux ,Σaux p

p Haux ,Σaux

k≥0

⊗ N1 (see (3) of Theorem 1.3.1.5), ωM tor Haux ,Σaux ⊗ N1 min over MHaux , which we denote by ωM min Haux

 )

also descends to an ample invertible sheaf by abuse of notation, and the morphism

(2.1.2.28) induces a morphism min Mmin ⊗Q Haux → MHp aux

(2.1.2.36)

Z

⊗ N1 under which the pullback of ωM min

p Haux

⊗ N1 is canonically isomorphic to ωM min . Haux

On the other hand, since H is neat, ωMtor descends to an ample invertible sheaf H,Σ min ωMmin over MH . H Proposition 2.1.2.37. With assumptions as in Proposition 2.1.1.15, there exists a morphism min Mmin H → MHaux

(2.1.2.38)

extending (2.1.1.16) and compatible with (2.1.2.30), which induces by composition with (2.1.2.36) a morphism min Mmin ⊗Q H → MHp aux

(2.1.2.39)

Z

extending (2.1.1.17) and compatible with (2.1.2.31). The morphism (2.1.2.38) to the (resp. (2.1.2.39)) maps the [(ΦH , δH )]-stratum Z[(ΦH ,δH )] of Mmin H p p [(ΦHaux , δHaux )]-stratum Z[(ΦHaux ,δHaux )] of Mmin Haux (resp. [(ΦHaux , δHaux )]-stratum Z[(ΦHp ,δHp )] of Mmin ) when [(ΦHaux , δHaux )] (resp. [(ΦHpaux , δHpaux )]) is assigned Hp aux aux aux to [(ΦH , δH )] as in (2.1.2.20) (with the filtrations ZH , ZHaux , and ZHpaux suppressed in the notation). If N1 ≥ 1 is as above, and if a0 ≥ 1 and a ≥ 1 are integers as in ⊗ a0 N1 ⊗ a0 N1 Lemma 2.1.2.35, then the pullback of ωM (resp. ωM ) to Mmin min min H is canonically Haux

⊗ aN1 isomorphic to ωM min . H Consequently, Mmin H

p Haux

min is the normalization of Mmin ⊗ Q) in MH Haux (resp. MHp aux Z

min under the morphism MH → Mmin ⊗ Q) induced by (2.1.1.16) Haux (resp. MH → MHp aux Z

p (resp. (2.1.1.17)) and the canonical morphism MHaux → Mmin Haux (resp. MHaux ⊗ Q →

Z

⊗ Q). Mmin Hp aux Z

Proof. The first paragraph follows from Proposition 2.1.2.29, from Lemma 2.1.2.35, and from the universal properties of the projective spectrum   ⊗k tor ∼ Mmin H = Proj ⊕ Γ(MH,Σ , ωMtor ) k≥0

H,Σ

min (and the ones for Mmin above). Haux and MHp aux ⊗ a0 N1 ⊗ a0 N1 ⊗ aN1 min Since ωMmin (resp. ωMmin ) is ample over Mmin ), since ωM is min Haux (resp. MHp aux Haux

ample over

Mmin H ,

p Haux

H

and since the pullback of the former is canonically isomorphic to

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min the latter, the canonical morphism from Mmin H to the normalization of MHaux (resp. min MHpaux ⊗ Q) in MH is finite (see [35, II, 5.1.6, and III-1, 4.4.2]). Since both the Z

source and target of this finite morphism are normal, and since they share an open dense subscheme MH , the second paragraph follows from Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11]), as desired. 2.2 2.2.1

Flat Integral Models as Normalizations and Blow-Ups Flat Integral Models for Minimal Compactifications

~ H denote the normalization of M Proposition 2.2.1.1. Let M ˆ p ) in MH under Gaux (Z p ˆ p ) there). p the morphism MH → MHaux induced by (2.1.1.17) (with Haux = Gaux (Z ~ ~ Then MH is a normal algebraic stack flat over S0 := Spec(OF0 ,(p) ) equipped with ~ H × S0 ∼ ~H → a canonical isomorphism M = MH over S0 , and with a morphism M ~ S0

MHpaux = MGaux (Zˆ p ) extending (2.1.1.17). The tautological tuple (A, λ, i, αH ) over MH extends to a degenerating family ~ H (see [62, Def. 5.3.2.1] and Definition 1.3.1.1), ~ ~λ,~i, α (A, ~ H ) of type MH over M ~ ~ where (A, λ) is a polarized abelian scheme with an O-structure ~i such that LieA/ ~H ~ M ~ with its O ⊗ Z(p) -module structure given naturally by i satisfies the determinantal Z

condition in [62, Def. 1.3.4.1] given by (L ⊗ R, h · , · i, h0 ), and where α ~ H is defiZ

~ aux , ~λaux ,~iaux , α ned only over MH . If we denote by (A ~ Gaux (Zˆ p ) ) the pullback of the tautological tuple (Aaux , λaux , iaux , αGaux (Zˆ p ) ) over MGaux (Zˆ p ) under the morphism ~ H → MHp induced by (2.1.1.17), then (A ~ aux , ~λaux ) is isomorphic to the polariM aux 0 0 ~ aux , ~λaux ) defined by (A, ~ ~λ) as in (2) of Lemma 2.1.1.1, ~i is zed abelian scheme (A ~ H (by [92, IX, the unique extension of i over the noetherian normal base scheme M 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5]), and α ~ Gaux (Zˆ p ) is determined by αH in the sense that its further pullback to MH is determined by αH as in Propop ˆ p ) there). Then ωM extends to the invertible = Gaux (Z sition 2.1.1.15 (with Haux H sheaf top ωM Lie∨ = ∧top e∗A~ Ω1A/ ~ H := ∧ ~ ~ M ~ ~ M A/ H

H

~ H , which is ample if H is neat. If a0 ≥ 1 and a ≥ 1 are integers as in over M ⊗ a0 a Lemma 2.1.2.35, then ω ⊗ ~ H is canonically isomorphic to the pullback of ωMGaux (ˆZp ) M ~H→M under the morphism M ˆ p induced by (2.1.1.17). Gaux (Z )

~ H ] of M ~ H is canonically isomorphic to the normaliThe coarse moduli space [M zation of [MGaux (Zˆ p ) ] in [MH ] under the morphism [MH ] → [MGaux (Zˆ p ) ] induced by (2.1.1.17), which is a normal scheme quasi-projective and flat over ~S0 equipped with ∼ [MH ] over S0 . In particular, if H is neat, ~ H ] × S0 = a canonical isomorphism [M ~ S0

~H∼ ~ H ] is a scheme. then M = [M

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~ H (up to canonical isomorphism) satisfying We obtain the same normalization M ˆ p ) with any open compact subgroup the analogous properties if we replace Gaux (Z p ˆ p ) such that Hp Gaux (Zp ) still contains the image of H under the Haux of Gaux (Z aux ˆ → Gaux (Z) ˆ given by (2.1.1.10). homomorphism G(Z) ~ H and hence [M ~ H ] depend only on the linear Up to canonical isomorphism, M algebraic data defining MH , but not on the auxiliary choices in Section 2.1 defining MGaux (Zˆ p ) or MHpaux . Proof. The first paragraph is self-explanatory. As for the second paragraph, except for the ampleness of ωM ~ H when H is neat, it suffices to show that the tautological ~ H . (Once ~ ~λ) over M (A, λ) over MH extends to some polarized abelian scheme (A, this is shown, the remainder of the paragraph will follow from the uniqueness of extensions by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5].) Since the genus of A and the polarization degree of λ is determined by the level structure αH , the tautological (A, λ) over MH defines (by forgetting the additional structures) a morphism from MH to the Siegel moduli Ag,d of genus g = 21 rkZ (L) and polarization degree d2 = [L# : L], which induces a finite morphism MH → Ag,d ⊗ Q Z

by [62, Prop. 1.3.3.7, Cor. 2.2.2.8, and Prop. 2.2.2.9]. Similarly, the tautological (Aaux , λaux ) defines a morphism from MGaux (Zˆ p ) to the Siegel moduli Agaux ,daux of genus gaux = 12 rkZ (Laux ) and polarization degree d2aux = [L# aux : Laux ], which induces a finite morphism MGaux (Zˆ p ) → Agaux ,daux ⊗ Z(p) . As explained in the Z

proof of Lemma 2.1.1.5, the construction as in (2) of Lemma 2.1.1.1 defines a finite morphism Ag,d → Agaux ,daux . By comparing the universal properties, the composition MH → MGaux (Zˆ p ) ⊗ Q → Agaux ,daux ⊗ Q of finite morphisms coincides Z

Z

with the composition MH → Ag,d ⊗ Q → Agaux ,daux ⊗ Q of finite morphisms. Since Z

Z

~H Ag,d → Agaux ,daux and MGaux (Zˆ p ) → Agaux ,daux ⊗ Z(p) are finite, it follows that M Z

is canonically isomorphic to the normalization of Ag,d ⊗ Z(p) under the canonical Z

morphism MH → Ag,d ⊗ Z(p) . In particular, the tautological object (A, λ) over MH Z

~ H → Ag,d . ~ ~λ) parameterized by the canonical morphism M extends to an object (A, This also shows, as in the last paragraph of the statement of the proposition, that ~ H is canonical and independent of the auxiliary choices. M ~ H ] of M ~ H is canonically isomorphic to the normalizaThe coarse moduli space [M tion of [MGaux (Zˆ p ) ] in [MH ] by the universal property of coarse moduli spaces, and by Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11], or the formulation in [62, Prop. ~ H ] over ~S0 , and 7.2.3.4] for algebraic spaces). Except for the quasi-projectivity of [M for the ampleness of ωM ~ H when H is neat, both of which will follow from Proposition 2.2.1.2 below, the remaining statements of the proposition are self-explanatory. Although Proposition 2.2.1.1 is stated without any reference to compactificati~ H ] over ~S0 , and the ampleons, the easiest way to show the quasi-projectivity of [M

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ness of ωM ~ H when H is neat, is to introduce the minimal compactifications. (This is a natural consideration because this is what the minimal compactifications in [5] did over C.) ~ min denote the normalization of Mmin Proposition 2.2.1.2. Let M H G

in Mmin H ˆ p) = Gaux (Z

ˆp )

aux (Z

p min under the morphism Mmin H → MGaux (Z ˆ p ) induced by (2.1.2.39) (with Haux ~ min is a normal scheme projective and flat over ~S0 = Spec(OF ,(p) ) there). Then M H 0 ~ min × S0 ∼ equipped with a canonical isomorphism M = Mmin over S0 . H

H

~ S0

~ H ] is an open dense subscheme of M ~ min , because [M By construction, [M ˆp ) ] H Gaux (Z min is an open dense subscheme of MG (Zˆ p ) (by [62, Thm. 7.2.4.1]). aux p If N1 ≥ 1 is as in the paragraph preceding Proposition 2.1.2.37 (for Haux = p ˆ )), and if a0 ≥ 1 and a ≥ 1 are integers as in Lemma 2.1.2.35, then ω ⊗ aN1 Gaux (Z MH ⊗ aN1 ~ min , which we compatibly extend to an ample invertible sheaf over M and ωM min H H ~ H is denote by ω ⊗ aN1 by abuse of notation, such that the pullback of ω ⊗ aN1 to M ~ min M H

canonically isomorphic to canonically isomorphic to

~ min M H ⊗ a0 N1 ⊗ aN1 ~ min to M , such that the pullback of ωMmin ω~ H MH Gaux (ˆ Zp ) ⊗ aN1 ω ~ min , and so that there is a canonical isomorphism MH

is

  ~ min ∼ ~ min ⊗ aN1 k ) . M H = Proj ⊕ Γ(MH , ωM min ~ k≥0

H

~ min M H

(up to canonical isomorphism) if we We obtain the same normalization ˆ p ) with any open compact subgroup Hp of Gaux (Z ˆ p ) such that replace Gaux (Z aux p ˆ → Haux Gaux (Zp ) still contains the image of H under the homomorphism G(Z) ˆ Gaux (Z) given by (2.1.1.10), in which case we might reduce the size of N1 in the above statements. By Lemma 2.1.1.18, if the image of H under the canonical hoˆ → G(Z ˆ p ) is neat, then we can choose Hp to be neat, so that momorphism G(Z) aux N1 = 1. ~ min (up to canonical isomorphism) if we replace Mmin We also obtain the same M H H min with [MH ], and if we replace the morphism Mmin H → MG ˆ p ) induced by (2.1.2.39) (Z aux

with the morphism [MH ] → Mmin ˆ p ) induced by (2.1.1.17) (cf. the second paragraph Gaux (Z of Proposition 2.1.2.37). ~ H in Proposition 2.2.1.1, it is also true that, up to canonical As in the case of M min ~ isomorphism, MH depends only on the linear algebraic data defining MH , but not on the auxiliary choices in Section 2.1 defining MGaux (Zˆ p ) or MHpaux . However, the proof of this is somewhat indirect and will be postponed until Corollary 2.2.1.15 below. Proof of Proposition 2.2.1.2. By construction as a normalization, we know that ⊗ a0 N1 ~ min is normal, and that the morphism M ~ min → Mmin M H H ˆ p ) is finite. Since ωMmin G (Z aux

is ample over Mmin G

aux

Gaux (ˆ Zp )

~ min ˆ p ) , its pullback to MH is also ample, which we define as the (Z

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aN1 ⊗ aN1 ⊗ aN1 common extension ω ⊗ and ωM min . (This is consistent with Lemma ~ min of ωMH MH

H

~ min is projective 2.1.2.35 and Proposition 2.1.2.37.) This shows in particular that M H ~ min is normal and hence has no p-torsion, over ~S0 . Since the structural sheaf of M H ⊗ a0 N1 to MGaux (Zˆ p ) is canonically it is also flat over ~S0 . Since the pullback of ωM min Gaux (ˆ Zp )

⊗ a0 N1 ~ H , which is canonically isomorphic to ωM , its further pullback to M Gaux (ˆ Zp ) aN1 to the pullback of ω ⊗ ~ min by construction, is canonically isomorphic to M H

isomorphic aN1 ω⊗ (by ~ MH

the part of Proposition 2.2.1.1 we have proved). The remaining statements of the proposition are self-explanatory. Now the proof of Proposition 2.2.1.1 is also complete. Remark 2.2.1.3. In our constructions (including ones to be given below), taking normalizations will never introduce pathologies, either because we are talking integral closures in (products of) separable field extensions (see [77, Sec. 33, Lem. 1]), or because the schemes in questions are all excellent (being a localization of a scheme of finite type over Z; see [76, Sec. 31–34] for more discussions). For each stratum Z[(ΦH ,δH )] as in (4) of Theorem 1.3.1.5, consider its closure ~ min . Then we define a locally and its closure ~Z[(ΦH ,δH )] in M Z[(ΦH ,δH )] in Mmin H H closed subscheme ~Z 0 0 ~Z := ~Z − ∪ (2.2.1.4) [(ΦH ,δH )]

[(ΦH ,δH )]

Z[(ΦH ,δH )] *Z[(Φ0

H

,δ 0 )] H

[(ΦH ,δH )]

~ min . By definition, we have the following: of M H 0 )] of Z[(Φ0 ,δ 0 )] , Lemma 2.2.1.5. If Z[(ΦH ,δH )] is contained in the closure Z[(Φ0H ,δH H H ~ 0 )] , and the latter agrees with the closure of then ~Z[(ΦH ,δH )] is contained in Z[(Φ0H ,δH ~Z[(Φ0 ,δ0 )] . H

H

Remark 2.2.1.6. It is nontrivial that the collection {~Z[(ΦH ,δH )] }[(ΦH ,δH )] does de~ min (see [65, Sec. 12]). Nevertheless, without actually using fine a stratification of M H this fact, we shall still call ~Z[(ΦH ,δH )] the [(ΦH , δH )]-stratum, by abuse of language. ~ min is canonical and independent of the auxiliary choices, For showing that M H and for many applications, it is desirable to know the following: Proposition 2.2.1.7. The image of the canonical morphism ~ H ⊗ Fp → M ~ min ⊗ Fp M H Z

(2.2.1.8)

Z

(see Proposition 2.2.1.2) is an open and dense subset. ~ min . Consider any morphism ξ : Spec(R) → M ~ min , Proof. Let s be any point of M H H where R is a complete discrete valuation ring with fraction field K of characteristic

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zero and with algebraically closed residue field k of characteristic p, such that the special point Spec(k) is mapped to s, and such that the restriction of ξ to the generic point Spec(K) factors as the composition of a morphism ξK : Spec(K) → MH with ~ min . (Such morphisms ξ and ξK exist because the canonical morphism MH → M H min ~ ~ MH and M H are of finite type over S0 , and because the image [MH ] of MH is open ~ min .) By the semistable reduction theorem (see, for example, [28, Ch. dense in M H I, Thm. 2.6] or [62, Thm. 3.3.2.4]), and by the theory of N´eron models (see [10]; cf. [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5]), up to replacing K with a finite extension field and replacing R accordingly, the pullback under ξK of the tautological tuple (A, λ, i, αH ) over MH extends to a degenerating family † † (G† , λ† , i† , αH ) of type MH over Spec(R), where αH is defined only over the generic point Spec(K). By applying the construction of elevators as in the proof of [58, Thm. 3.1] to ˜ ˜i) of type (PE, O) (see [58, Def. ˜ λ, (G† , λ† , i† ), there exists a degenerating family (G, ˜ where R ˜ is a noetherian 2.1]; see also Definition 4.1.3.2 below) over S := Spec(R), integral domain over R which is complete with respect to some ideal I˜ such that ˜ = I, ˜ satisfying the following properties: rad(I) (1) There exists a morphism Spec(R) → S under which (G† , λ† , i† ) is isomor˜ ˜i). ˜ λ, phic to the pullback of (G, ˜ is an abelian (2) There exists an open dense subscheme S1 of S over which G scheme, such that S1 ⊗ Fp is nonempty and dense in S ⊗ Fp . (This is beZ

Z

cause, in the proof of [58, Thm. 3.1] in [58, Sec. 3], the scheme Ξ◦ is smooth over S and where Ξ◦ ⊗ Fp is nonempty and dense in Ξ◦ (σ) ⊗ Fp .) Z

Z

(3) For any morphism Spec(V ) → S as in (6) of Theorem 1.3.1.3, centered at the geometric point Spec(k) → S induced by the morphism ‡ Spec(R) → S above, there exist some (Z‡H , Φ‡H , δH ) and σ ‡ such that σ ‡ + is a one-dimensional cone in P ‡ . (This is because the cone σ in the proof ΦH

of [58, Thm. 3.1] in [58, Sec. 3] can be taken to be one-dimensional.) ¯ be an algebraic closure of K. Then there exists an affine integral scheme Let K ¯ → S 0 liffinite ´etale over S1,Q , together with a morphism η¯ := Spec(K) 1,Q ˜ S 0 , ˜iS 0 ) satisfies the ˜ S0 , λ ting the above morphism η → S1,Q , such that (G 1,Q 1,Q 1,Q determinantal condition in [62, Def. 1.3.4.1] given by (L ⊗ R, h · , · i, h0 ) and is

0 S1,Q

Z

ˆ h · , · i) as in [62, Def. 0 equipped with a level-H structure α ˜ H,S1,Q of type (L ⊗ Z, Z

0 ˜ S 0 , ˜iS 0 , α ˜ S0 , λ 0 1.3.7.6], and such that the pullbacks of (G ˜ H,S1,Q ) → S1,Q and 1,Q 1,Q 1,Q † (G† , λ† , i† , αH ) → Spec(R) to η¯ are isomorphic to each other. By the universal property of MH , there is a canonical morphism 0 S1,Q → MH

(2.2.1.9)

˜ S 0 , ˜iS 0 , α ˜ S0 , λ 0 under which (G ˜ H,S1,Q ) is isomorphic to the pullback of the tauto1,Q 1,Q 1,Q 0 logical tuple (A, λ, i, αH ) over MH . By construction, the compositions η¯ → S1,Q →

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MH and η¯ → η → MH coincide with each other (cf. the proof of [58, Thm. 4.1]). Let S 0 and S10 denote the normalizations of S and S1 under the canonical mor0 0 ˜ S 0 , ˜iS 0 , α ˜ S0 , λ 0 phisms S1,Q → S and S1,Q → S1 , respectively. Then (G ˜ H,S1,Q ) 1,Q 1,Q 1,Q 0 ˜ ˜ ˜ canonically extends to degenerating families (GS 0 , λS 0 , iS 0 , α ˜ H,S 0 ) → S and 0 ˜ S 0 , ˜iS 0 , α ˜ S 0 , ˜iS 0 ) and (G ˜ S 0 , ˜iS 0 ) ˜ S0 , λ ˜ S0 , λ ˜ S0 , λ 0) → S (G ˜ of type M , where ( G H H,S 1 1 1 1 1 1 1 1 ˜ ˜i) from S to S 0 and S 0 , respectively, and where ˜ λ, are just the pullbacks of (G, 1 0 α ˜ H,S 0 and α ˜ H,S10 are defined only over S1,Q . ~ H and M ~ min are the normalizations of MHp and Mmin By definition, M in H aux Hp aux p MH , as in Propositions 2.2.1.1 and 2.2.1.2, for some open compact subgroup Haux ˆ p ). By the same argument as in the proof of Proposition 2.1.1.15, by of Gaux (Z ˜ S 0 , ˜iS 0 , α ˜ S0 , λ forgetting the factor of α ˜ H,S10 at p, the tuple (G ˜ H,S10 ) induces a tuple 1 1 1 parameterized by MHpaux , and induces a morphism (2.2.1.10) S10 → MHpaux 0 p by the universal property of MHaux , whose restriction to S1,Q coincides with the composition of (2.2.1.9) with the morphism MH → MHpaux induced by (2.1.1.17) (as ~ H as a normalization, in Proposition 2.2.1.1). Consequently, by the definition of M (2.2.1.10) induces a morphism ~H S10 → M (2.2.1.11) extending (2.2.1.9). By the same argument as in the proof of Proposition 2.1.2.29, the degenera˜ S 0 , ˜iS 0 , α ˜ S0 , λ ting family (G ˜ H,S 0 ) → S 0 of type MH induces a degenerating family O O O O ˜ ˜ ˜ (G ˜ Hpaux ,S 0 ) → S 0 of type MHpaux , which defines a morphism aux,S 0 , λaux,S 0 , iaux,S 0 , α p S 0 → Mtor (2.2.1.12) Hp aux ,Σaux p for any compatible choice of Σaux for MHpaux as in [62, Def. 6.3.3.4] (cf. Definition 1.2.2.13), because the property (3) above ensures that the degenerating fa0 O ˜ O 0 , ˜iO 0 , α ˜O 0 , λ mily (G 0 ) → S satisfies the condition as in [62, Thm. aux,S ˜ Hp aux,S aux,S aux ,S 6.4.1.1(6)] (cf. Hthe proof of [62, Prop. 6.3.3.17].) By composition with the canonical morphism Hpaux : Mtor → Mmin as in [62, Thm. 7.2.4.1(3)] (cf. (3) of p Hp Hp aux ,Σaux aux Theorem 1.3.1.5), (2.2.1.12) induces a morphism S 0 → Mmin , (2.2.1.13) Hp aux whose restriction to S10 is the composition of (2.2.1.11) with the canonical morphism ~H→M ~ min . Consequently, by the definition of M ~ min as a normalization, (2.2.1.13) M H H induces a morphism ~ min S0 → M (2.2.1.14) H extending (2.2.1.11). Since the geometric point s¯ → S lifts to some geometric point s¯ → S 0 by the finiteness of S 0 → S, and since S10 ⊗ Fp is nonempty and dense in S 0 ⊗ Fp because

Z

Z

S1 ⊗ Fp is nonempty and dense in S ⊗ Fp (by the property (2) above), the image s of Z

Z

~ min ⊗ Fp is the specialization of some point of M ~ H ⊗ Fp . Since s is arbitrary, s¯ in M Z

Z

this shows that the open image of (2.2.1.8) is a dense subset, as desired.

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~ min constructed Corollary 2.2.1.15. Up to canonical isomorphism, the scheme M H in Proposition 2.2.1.2 depends only on the linear algebraic data defining MH , but not on the auxiliary choices defining MGaux (Zˆ p ) or MHpaux . ~ min is flat over Z(p) and is noetherian Proof. By Propositions 2.2.1.2 and 2.2.1.7, M H min ~ ~ min is of codimension at least two. normal, and the complement of [MH ] ∪ MH in M H Hence, the canonical restriction morphism ~ min , ω ⊗ aN1 k ) → Γ([M ~ H ] ∪ Mmin , (ω ⊗ aN1 k )| ~ Γ(M H H ~ min ~ min [MH ] ∪ Mmin ) MH

MH

(2.2.1.16)

H

is an isomorphism for each k ≥ 0. By Propositions 2.2.1.1 and 2.2.1.2, the righthand side of (2.2.1.16) depends onlyon the linear algebraic data defining MH . Since  min ~ min ∼ ~ M Proj ⊕ Γ( M , ω ⊗ aN1 k ) , the corollary follows, as desired. = H

2.2.2

H

k≥0

~ min M H

Flat Integral Models for Projective Toroidal Compactifications

Proposition 2.2.2.1. Let H, Σ, pol, H,dpol , and JH,dpol be as in Theorem 1.3.1.10 (for each integer d ≥ 1). (In particular, H is neat and Σ is projective.) For each d ≥ 1, let J~H,dpol be the coherent O ~ min -ideal defining the schematic closure in MH

~ min of the closed subscheme of Mmin defined by the coherent O min -ideal JH,dpol . M MH H H Suppose d0 ≥ 1 is any integer such that the statement in Theorem 1.3.1.10 is true. Let ~ tor M H,d0 pol := NBlJ~H,d

0 pol

~ min ). (M H

~ tor ~ Then M H,d0 pol is a normal scheme projective and flat over S0 = Spec(OF0 ,(p) ) tor tor ∼ ~ equipped with a canonical isomorphism M H,d0 pol × S0 = MH,Σ over S0 = Spec(F0 ). ~ S0 H min The canonical morphism H : Mtor extends to a canonical morphism H,Σ → MH H~ tor min ~ ~ :M →M . Moreover, the canonical morphisms H

H,d0 pol

H

H~ OM ~ min → H,∗ OM ~ tor H

(2.2.2.2)

H,d0 pol

~ min defined by J~H,dpol necessarily is an isomorphism. Since closed subscheme of M H H ~ H in M ~ min , the pullback of ~ under the canonical lies in the closed complement of M H H ~H →M ~ min is an isomorphism, which canonically identifies M ~ H as an morphism M H tor ~ open dense subscheme of MH,d0 pol . aN1 ⊗ aN1 ~ tor By abuse of notation, we denote the pullback of ω ⊗ ~ min to MH,d0 pol by ω ~ tor MH

(cf. Proposition 2.2.1.2). Then

aN1 k ω⊗ ~ tor M

0 pol

is generated by global sections for suffi-

H,d0 pol

ciently large k ≥ 1, and we have a canonical isomorphism   ⊗ aN1 k ~ min ∼ ~ tor M ) . H = Proj ⊕ Γ(MH,d0 pol , ωM ~ tor k≥0

MH,d

H,d0 pol

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Proof. The statements in the first paragraph are all self-explanatory. The statement in the second paragraph is true because, for each k ≥ 0, ⊗ aN1 k ~ min , ω ⊗ aN1 k ) by the projection formula [35, 0I , ~ tor ) ∼ Γ(M = Γ(M H H,d0 pol , ω ~ tor ~ min MH,d

MH

0 pol

5.4.10.1], and because

aN1 ω⊗ ~ min M H

~ min (see Proposition 2.2.1.2). is ample over M H

Proposition 2.2.2.3. With the assumptions in Proposition 2.2.2.1, suppose H0 , Σ0 , and pol0 are as in Proposition 1.3.1.11. Then we define ~ min )∗ J~H,d pol ~ min J~H0 ,d pol0 := (M 0 → M H

H

0

0

and ~ tor0 M H ,d0 pol0 := NBlJ~

H0 ,d0 pol0

~ min (M H0 ).

~ tor0 ~ tor Then M H ,d0 pol0 enjoys analogues of properties of MH,d0 pol in Proposition 2.2.2.1, and we have a canonical morphism ~ tor0 ~ tor M (2.2.2.4) H ,d0 pol0 → MH,d0 pol ~ tor0 which is finite. Moreover, M H ,d0 pol0 is canonically isomorphic to the normalization tor ~ of MH,d0 pol in MH0 under the composition of canonical morphisms MH0 → MH ,→ ~ tor Mtor H,Σ → MH,d0 pol . 0 ~ tor0 If Σ is smooth, then M 0 can also be constructed as in Proposition H ,d0 pol

2.2.2.1, and the schemes we obtain in the two constructions are canonically isomorphic. Proof. As in the proof of Proposition 1.3.1.11, since J~H0 ,d0 pol0 is the pullback of ~ min ~ min J~H,d0 pol under the finite morphism M H0 → MH , the morphism (2.2.2.4) exists and ~ tor0 is finite, by the universal property of the normalization of blow-up. Since M H ,d0 pol0 tor ~ is normal, it is canonically isomorphic to the normalization of M in MH0 by H,d0 pol

Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11]). If Σ0 is smooth, then the J~H0 ,d0 pol0 defined as a pullback is canonically isomorphic ~ min ~ tor to the J~H0 ,d0 pol0 defined on M H0 itself, and hence the two constructions of MH0 ,d0 pol0 by normalizations of blow-ups give canonically isomorphic schemes. ~ tor Remark 2.2.2.5. We introduce the scheme M H,d0 pol in Proposition 2.2.2.1 mainly for technical reasons, and for the sake of completeness. This is even more so for ~ tor0 the scheme M H ,d0 pol0 in Proposition 2.2.2.3. (We will need special cases of them in Sections 5.2.3 and 6.1.1 below.) For each stratum Z[(ΦH ,δH ,σ)] as in (2) of Theorem 1.3.1.3, consider its closure ~ ~ tor Z[(ΦH ,δH ,σ)] in Mtor H,Σ and its closure Z[(ΦH ,δH ,σ)],d0 pol in MH,d0 pol . Then we define a locally closed subscheme ~Z 0 0 ~Z − ∪ := ~Z [(ΦH ,δH ,σ)],d0 pol

[(ΦH ,δH ,σ)],d0 pol

Z[(ΦH ,δH ,σ)] *Z[(Φ0

H

,δ 0 ,τ )] H

[(ΦH ,δH ,τ )],d0 pol

(2.2.2.6)

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~ tor of M H,d0 pol . By definition, we have the following: 0 ,τ )] of Lemma 2.2.2.7. If Z[(ΦH ,δH ,σ)] is contained in the closure Z[(Φ0H ,δH ~ 0 ,τ )] , then ~ 0 ,τ )],d pol , and the latter Z[(Φ0H ,δH Z[(ΦH ,δH ,σ)],d0 pol is contained in Z[(Φ0H ,δH 0 tor ~ 0 ,τ )],d pol in M agrees with the closure of ~Z[(Φ0H ,δH . Moreover, the canonical 0 H,d0 pol H~ tor min ~ ~ morphism :M →M maps each ~Z to (an open subscheme

H

H,d0 pol

[(ΦH ,δH ,σ)],d0 pol

H

of) ~Z[(ΦH ,δH )] . Remark 2.2.2.8. It is not clear whether the collection {~Z[(ΦH ,δH ,σ)],d0 pol }[(ΦH ,δH ,σ)] ~ tor defines a stratification of M H,d0 pol (cf. Remark 2.2.1.6). However, we shall still call ~Z[(Φ ,δ ,σ)],d pol the [(ΦH , δH , σ)]-stratum, by abuse of language. 0 H H 2.2.3

Hecke Actions

Proposition 2.2.3.1. (Compare with Proposition 1.3.1.14.) Suppose that g = (g0 , gp ) ∈ G(A∞,p ) × G(Zp ) ⊂ G(A∞ ) and that H and H0 are two open compact ˆ such that H0 ⊂ gHg −1 . Then there is a canonical finite surjection subgroups of G(Z) ~ :M ~ H0 → M ~H [g] (over ~S0 = Spec(OF0 ,(p) )) extending the canonical finite surjection [g] : MH0 → MH k ~ (over S0 = Spec(F0 )) defined by the Hecke action of g, such that ω ⊗ ~ H over MH is M ~ H0 (up to canonical isomorphism) whenever k is divisible pulled back to ω ⊗ k over M ~ 0 M H

by the integer a in Lemma 2.1.2.35. Moreover, there is a canonical finite surjection ~ min : M ~ min ~ min [g] H0 → MH (over ~S0 ) extending the canonical finite surjection [[g]] : [MH0 ] → [MH ] (over S0 ) k ~ min ~ min is pulled back to ω ⊗ k over M induced by [g], such that ω ⊗ H0 (up ~ min ~ min over MH MH0

MH

to canonical isomorphism, compatible with the previous one) whenever the former is defined. (This canonical morphism is compatible with the canonical isomorphism ∼ min ⊗k ⊗k ([g] )∗ ωM By restriction, the surjection min → ωMmin in Proposition 1.3.1.14.) H

H0

~ ~ min induces the surjection [[g]] ~ : [M ~ H0 ] → [M ~ H ] induced by [g]. [g] min ~ The surjection [g] maps the [(Φ0 0 , δ 0 0 )]-stratum ~Z 0 H ~ min M H

H

0 )] [(ΦH0 ,δH 0

~ min of M to H0

the [(ΦH , δH )]-stratum ~Z[(ΦH ,δH )] of if and only if there are representati0 0 0 ves (ΦH , δH ) and (ΦH0 , δH0 ) of [(ΦH , δH )] and [(Φ0H0 , δH 0 )], respectively, such that 0 0 (ΦH , δH ) is g-assigned to (ΦH0 , δH0 ) as in [62, Def. 5.4.3.9]. If g = g1 g2 , where g1 = (g1,0 , g1,p ) and g2 = (g2,0 , g2,p ) are elements of G(A∞,p ) × G(Zp ) ⊂ G(A∞ ), each having a setup similar to that of g, then we ~ = [g~2 ] ◦ [g~1 ], [[g]] ~ = [[g~2 ]] ◦ [[g~1 ]], and [g] ~ min = [g~2 ]min ◦ [g~1 ]min . have [g] Proof. Since H0 ⊂ gHg −1 , by considering their images under the canonical homoˆ → G(Z ˆ p ) and G(Z) ˆ → G(Zp ) (and the canonical homomorphisms morphisms G(Z)

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ˆ p ) → Gaux (Z ˆ p ) and G(Zp ) → Gaux (Zp ) given by (2.1.1.10)), there exist an G(Z 0,p ˆ p ) contained in g0 Gaux (Z ˆ p )g −1 such that open compact subgroup Haux of Gaux (Z 0 0,p 0 ˆ → Gaux (Z) ˆ Haux Gaux (Zp ) contains the image of H under the homomorphism G(Z) given by (2.1.1.10). Since gp ∈ G(Zp ), the Hecke twists of tautological objects over MH0 and 0 MH0,p ⊗ Q are realized by compatible Z× (p) -isogenies, and hence [g] : MH → MH aux Z

and [g0 ] ⊗ Q : MH0,p ⊗ Q → MGaux (Zˆ p ) ⊗ Q are compatible (see (2) of Lemma 2.1.1.1 aux Z

Z

Z

and Proposition 2.1.1.15). Then we have a commutative diagram ~ H0 /M

MH0

~ [g]

[g]

'

(2.1.1.17)

%

MH

~H /M

(2.1.1.17)



 / M 0,p Haux

MH0,p ⊗Q aux Z

[g0 ] [g0 ] ⊗ Q

&  MGaux (Zˆ p ) ⊗ Q

Z

$  /M ˆp ) Gaux (Z

Z

of solid arrows, in which all unnamed morphisms are canonical morphisms, inducing the desired (compatible) dotted arrow. By taking normalizations and by Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11]), we obtain a commutative diagram ~ min /M 0

Mmin H0

H

~ min [g]

min

[g]

' (2.1.2.39)

$

Mmin H

~ min /M H

(2.1.2.39)

 / Mmin H0,p



Mmin ⊗Q H0,p aux

aux

Z

[g0 ]min [g0 ]min ⊗ Q Z

&  Mmin ˆp ) ⊗ Q G (Z aux

Z

#  / Mmin G

ˆp aux (Z )

of solid arrows compatible with the previous one, in which all unnamed morphisms are canonical morphisms, inducing the desired dotted arrow (compatible with all the other arrows in both diagrams). The remaining statements in the proposition then follow from the known statements (including those in Proposition 1.3.1.14) and from the various universal properties.

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Corollary 2.2.3.2. (Compare with [62, Cor. 7.2.5.2].) Suppose we have two open ˆ such that H0 is a normal subgroup of H. Then compact subgroups H and H0 of G(Z) the canonical morphisms defined in Proposition 2.2.3.1 induce an action of the finite ~ min : M ~ min ~ min ~ min defined by group H/H0 on M H0 . The canonical surjection [1] H0  MH min ~ Proposition 2.2.3.1 can be identified with the quotient of M 0 by this action. H

~ min Proof. The existence of such an action is clear. Since M H0 is projective over min 0 ~ 0 /(H/H ) exists as a scheme (cf. [25, V, 4.1]). S0 and normal, the quotient M H Then it follows from Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11]) that the 0 ~ min ~ min (with noetherian normal target) is an induced morphism M H0 /(H/H ) → MH ~ min , by [62, Cor. 7.2.5.2]—in fact, isomorphism, because it is generically so (over M H the proof here is part of that of [62, Cor. 7.2.5.2]). For later references, let us define: Definition 2.2.3.3. For each integer i ≥ 0, we define S0,i := Spec(F0 [ζpi ]) and ~S0,i := Spec(OF ,(p) [ζpi ]). 0 Definition 2.2.3.4. For each integer i ≥ 0, we define MH,i (resp. Mmin H,i , resp. tor min tor MH,Σ,i ) to be the base change MH × S0,i (resp. MH × S0,i , resp. MH,Σ × S0,i ) over S0

S0

S0

as in (4) of Theorem S0,i . For each locally closed subscheme Z[(ΦH ,δH )] of Mmin H 1.3.1.5, we denote by Z[(ΦH ,δH )],i its pullback under the canonical morphism Mmin H,i → tor min MH . For each locally closed subalgebraic stack Z[(ΦH ,δH ,σ)] of MH,Σ as in (2) of Theorem 1.3.1.3, we denote by Z[(ΦH ,δH ,σ)],i its pullback under the canonical tor morphism Mtor H,Σ,i → MH,Σ . ~ H,i (resp. M ~ min ) to be the Definition 2.2.3.5. For each integer i ≥ 0, we define M H,i min ~ ~ ~ ~ normalization of MH × S0,i (resp. MH × S0,i ). For each locally closed subscheme ~ S0

~ S0

~Z[(Φ ,δ )] of M ~ min as in (2.2.1.4), we denote by ~Z[(Φ ,δ )],i its pullback under the H H H H H ~ min . ~ min → M canonical morphism M H H,i For each integer i ≥ 0, and for each H, Σ, pol, and d0 as in Proposition 2.2.2.1 ~ tor ~ tor such that M H,d0 pol is defined, we also define MH,d0 pol,i to be the normalization of ~ tor ~ ~ ~ tor M H,d0 pol × S0,i . For each locally closed subscheme Z[(ΦH ,δH ,σ)],d0 pol of MH,d0 pol as ~ S0

in (2.2.2.6), we denote by ~Z[(ΦH ,δH ,σ)],d0 pol,i its pullback under the canonical morH~ ~ tor ~ tor ~ tor ~ phism M × ~S0,i : M H,d0 pol,i → MH,d0 pol . Then the base change H,d0 pol × S0,i → H ~ S

~ S

0 0 H ~ min × ~S0,i induces a canonical morphism ~ ~ tor ~ min , mapping each M : M → M H H,i H,d0 pol,i H,i

~ S0

~Z[(Φ ,δ ,σ)],d pol,i as above to ~Z[(Φ ,δ )],i . We naturally extend these definitions to 0 H H H H ~ tor0 the schemes M constructed in Proposition 2.2.2.3. 0 H ,d0 pol ~ min : M ~ min ~ min is defined as in PropoFor all integers i0 ≥ i ≥ 0, if [g] H0 → MH ~ min ~ min sition 2.2.3.1, then we denote the canonically induced morphism M 0 0 → M H ,i

H,i

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~ min (compatible with ~S0,i0 → ~S0,i ) by [g] i0 ,i . 2.2.4

The Case When p is a Good Prime

Suppose p is a good prime (for the integral PEL datum (O, ?, L, h · , · i, h0 )) as in ˆ is an open compact subgroup. By considering Definition 1.1.1.6. Suppose H ⊂ G(Z) ˆ → G(Z ˆ p ), we know that there the image of H under the canonical morphism G(Z) p p ˆ ) such that H ⊂ H0 := Hp G(Zp ). exists some open compact subgroup H ⊂ G(Z By [62, Prop. 1.4.4.3], there is a canonical open and closed immersion MH0 ,→ MHp × S0 .

(2.2.4.1)

~ S0

(By [53, Sec. 8], this is an isomorphism at least when O ⊗ Q is simple, but we do Z

not need to know that.) Lemma 2.2.4.2. With assumptions as above, there is a canonical open and closed immersion min ~ min M H0 ,→ MHp

(2.2.4.3)

inducing a canonical open and closed immersion ~ H0 ] ,→ [MHp ], [M

(2.2.4.4)

min ~ ~ min so that M H0 (resp. [MH0 ]) is the scheme closure of [MH0 ] in MHp (resp. [MHp ]) under the canonical morphism induced by (2.2.4.1). ~ ~ min In particular, the construction of M H0 (resp. [MH0 ]) is independent of the auxi~ min (resp. liary choice of (Oaux , ?aux , Laux , h · , · iaux , h0,aux ). The same is true for M H 0 ~ H ]), regardless of the choice of H . [M

Proof. With the setting in Proposition 2.1.1.15, under the additional assumption in this lemma that p is a good prime, we can arrange that the canonical morphism (2.1.1.17) extends to a composition MH0 → MHp → MGaux (Zˆ p )

(2.2.4.5)

of canonical morphisms, the latter one being finite by the same argument as in the proof of Proposition 2.1.1.15. (In fact, by Lemma 2.1.1.1, we can take MGaux (Zˆ p ) to be MG(Zˆ p ) in this case.) Then (2.2.4.5) induces a composition min MH0 → Mmin Hp → MG

ˆp )

aux (Z

(2.2.4.6)

of canonical morphisms, the latter one being finite by the same argument as in the proofs of Proposition 2.1.2.29 and Corollary 2.1.2.37. (Again by Lemma 2.1.1.1, we min min can take MGaux (Zˆ p ) to be MG(Zˆ p ) , in which case Mmin ˆ p ) is MG(Z ˆ p ) .) Since MHp Gaux (Z ~ H0 ] and M ~ min is normal, by definition of [M (see Propositions 2.2.1.1 and 2.2.1.2) 0 H

and by Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11]), the open and closed

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immersion (2.2.4.1) induces the desired open and closed immersions (2.2.4.4) and (2.2.4.3). ~ min and [M ~ H ]) then follows, because the canoThe last assertion (concerning M H min nical morphism [MH ] → MG (Zˆ p ) induced by (2.1.1.17) (see Proposition 2.2.1.2) aux

min factors as a composition [MH ] → [MH0 ] → Mmin Hp → MGaux (Z ˆ p ) of canonical mormin ~ ~ H ]) is the normalization of M ~ min phisms (see (2.2.4.6)), and hence M (resp. [M H H0 ~ H0 ]) under the canonical morphism [MH ] → M ~ min ~ H0 ]). (resp. [M (resp. [MH ] → [M 0 H

This does not depend on the choice of H0 because replacing H0 with a finite index subgroup only results in finite morphisms between normal schemes, and the ~ min ~ construction of M H0 (resp. [MH0 ]) by normalization is insensitive to such morphisms.

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Chapter 3

Ordinary Loci

In this chapter, we introduce the notions of ordinary (semi-)abelian schemes and ordinary level structures, define the moduli problems parameterizing them, and construct the ordinary loci by normalizing these moduli problems after suitable base changes. The terminology of ordinary loci is, admittedly, an abuse of language. Nevertheless, these ordinary loci can be embedded into (normalizations of suitable base changes) of the flat integral models constructed in Chapter 2 (which we view as the total models). The main point is that, while we cannot describe the local structures of the total models in detail, we can describe the local structures of these ordinary loci rather precisely, because these ordinary loci are constructed as normalizations of moduli problems with explicit and mild singularities. 3.1 3.1.1

Ordinary Semi-Abelian Schemes and Serre’s Construction Ordinary Abelian Schemes and Semi-Abelian Schemes

Definition 3.1.1.1. Let U be a scheme. We say that a quasi-finite flat commutative group scheme H of finite presentation over U is of ´ etale-multiplicative type if it is ´etale locally an extension of a (commutative) ´etale group scheme by a finite flat group scheme of multiplicative type. (For simplicity, we shall often suppress the modifiers such as being commutative or being of finite presentation when we mention group schemes of ´etale-multiplicative type.) Definition 3.1.1.2. Let U be a scheme. We say that a semi-abelian scheme Z → U is ordinary if, for every integer m ≥ 1, the (commutative) quasi-finite flat group scheme Z[m] (of finite presentation over U ) is of ´etale-multiplicative type. We say an abelian scheme Z → U is ordinary if it is ordinary as a semi-abelian scheme. Remark 3.1.1.3. It suffices to verify this condition over strict local rings of U for rational prime numbers m > 1 that are residue characteristics of U . Remark 3.1.1.4. Suppose U is the spectrum of a strict local ring of residue characteristic p > 0. Then it suffices to verify that, at a geometric point above the special point, the connected part of the p-divisible group of the pullback of the 155

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semi-abelian scheme is of multiplicative type. This is a condition for the slopes in the Dieudonn´e–Manin classification of isogeny classes of p-divisible groups over an algebraically closed field of characteristic p (see [75]). If U is a scheme over Spec(Q), then every semi-abelian scheme Z → U is ordinary. If U is the spectrum of an algebraic closed field of characteristic p, then a semi-abelian scheme Z → U , which is an extension of an abelian scheme Z ab by a torus Z tor , is ordinary if and only if Z ab is an ordinary abelian variety. Lemma 3.1.1.5. If Z → Z 0 is an isogeny between semi-abelian schemes over U , then Z is ordinary if and only if Z 0 is. Proof. This follows from Remarks 3.1.1.3 and 3.1.1.4. For later reference, let us define: Definition 3.1.1.6. For each scheme S over Spec(Z) and any integer n ≥ 1, we define a functor on the category of ´etale sheaves of Z/nZ-modules over S by setting ( · )mult := HomS (HomS ( · , ((Z/nZ)(1))S ), µn,S ) ∼ = HomS (HomS ( · , ((Z/nZ)(1))S ), Gm,S ). . By funcFor constant sheaves of Z/nZ-modules, we denote (( · )S )mult by ( · )mult S toriality, ´etale sheaves carrying O-actions are sent to finite flat group schemes of multiplicative type also carrying O-actions. Definition 3.1.1.7. When S is a scheme of characteristic p, we extend such a definition to the category of ´etale sheaves of Zp -modules over S by setting ( · )mult := lim( · /(pr · ))mult , −→ r which is a p-divisible group of multiplicative type. Example 3.1.1.8. Let S be a scheme of characteristic p. Let (Qp /Zp )S = lim(( p1r Zp )/Zp )S denote the split rank-one ´etale p-divisible group over S. (For −→ r such constant objects, if S is a geometric point, and if the context is clear, we shall often suppress S from the notation.) Let µp∞ ,S = lim µpr ,S = lim Gm,S [pr ] denote −→ −→ r r the split rank-one p-divisible group of multiplicative type over S. Then we have ∼ lim µ r = µ ∞ , (Zp (1))mult = lim((Z/pr Z)(1))mult = p ,S −→ −→ p ,S r

which is the Serre dual of (Qp /Zp )S .

r

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157

Serre’s Construction for Ordinary Abelian Schemes

The following definition has been alluded to when we cited [62, Prop. 5.2.3.9] in Section 1.3.3: Definition 3.1.2.1. (See [62, Def. 5.2.3.6].) Let U be a scheme, and let N be an ´etale sheaf of left O-modules that becomes a constant finitely generated O-module N over a finite ´etale covering of U . Let (Z, λZ ) be a polarized abelian scheme over U with a left O-action given by some iZ : O → EndU (Z). Then we denote by HomO (N , Z) the (commutative) group functor of O-equivariant group homomorphisms from the group functor N to the group functor Z. The aim of this subsection is to further generalize [62, Prop. 5.2.3.9], to include a treatment of the fiberwise geometric identity components and group schemes of fiberwise geometric connected components when the base scheme U has residue characteristics ramified in O and when the abelian scheme Z → U in question is ordinary (see Definition 3.1.1.2). Lemma 3.1.2.2. Suppose that W is a commutative proper group scheme of finite presentation over U . Suppose that W0 is an abelian subscheme of W (i.e. a subgroup scheme that is an abelian scheme), and that there is an integer m ≥ 1 such that multiplication by m defines a homomorphism [m] : W → W with schematic image a (closed) subscheme of W0 and such that W [m], the m-torsion subgroup scheme of W , is finite flat over U . Then, for every geometric point s¯ → U , the fiber (W0 )s¯ is the reduced subscheme of the connected component of Ws¯ containing the identity section. Moreover, W is flat and the quotient group functor W/W0 is representable by a commutative finite flat group scheme E. The group π0 (Ws¯) of connected components can be canonically identified with the s¯-valued points of Es¯. Proof. Since W is commutative and since W0 is (fppf locally) m-divisible as an abelian scheme, the condition that [m] sends W to W0 shows that W/W0 can be identified with the quotient of W [m] by W0 [m] = W0 ∩ W [m]. Since W0 is an abelian scheme, W0 [m] is finite flat over U and the quotient W/W0 is representable (by [25, V, 4.1]). Since W [m] is finite flat, the quotient W [m]/W0 [m] is also finite flat (by [25, VIA , 3.2 and 5.4]). (All of these statements are over U .) The statements on the identity components and group of connected components of geometric fibers are obvious. Definition 3.1.2.3. (Compare with [62, Def. 5.2.3.8].) Suppose W0 is an abelian subscheme of a proper group scheme W of finite presentation over a base scheme U , such that for every geometric point s¯ → U , the fiber (W0 )s¯ is the reduced subscheme of the connected component of Ws¯ containing the identity section. Then we say (for simplicity) that W0 is the fiberwise geometric identity component of W (without emphasizing that it is reduced), and denote it by W ◦ . (By [35, IV-2,

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4.5.13], it is also correct to say that W0 is the fiberwise identity component, without the term geometric.) Suppose the quotient group functor W/W0 is representable by a finite group scheme E. Then we say that E is the group scheme of fiberwise geometric connected components, and denote it by π 0 (W/U ). By Lemma 3.1.2.2, the finite group scheme π 0 (W/U ) is defined and is finite flat over U if W is commutative and if there is an integer m ≥ 1 such that multiplication by m defines a homomorphism [m] : W → W with schematic image a (closed) abelian subscheme of W0 and such that W [m] is finite flat over U . Now we can state our (slight) generalization of [62, Prop. 5.2.3.9]: Proposition 3.1.2.4. With the setting as in Definition 3.1.2.1, suppose N is constant with value some finitely generated O-module N . Then the following are true: (1) The group functor HomO (N, Z) is representable by a proper subgroup scheme of an abelian scheme over U . (2) Suppose that N is torsion of order annihilated by some integers m ≥ 1, and that Z[m] is a finite flat group scheme of ´etale-multiplicative type over U . Then HomO (N, Z) is also finite flat of ´etale-multiplicative type over U . (3) If N is projective as an O-module, then HomO (N, Z) is representable by an abelian scheme. (4) Suppose that N is an O-lattice, and that Z is an ordinary abelian scheme over U (see Definition 3.1.1.2). Then HomO (N, Z) is representable by a proper flat group scheme which is an extension of a (commutative) finite flat group scheme of ´etale-multiplicative type, whose rank has no prime factors other than those of the discriminant of Disc = DiscO/Z [62, Def. 1.1.1.6], by an abelian scheme over U . Following Definition 3.1.2.3, we shall say that HomO (N, Z) is the extension of the finite flat group scheme π 0 (HomO (N, Z)/U ) of ´etale-multiplicative type by the abelian scheme HomO (N, Z)◦ . We shall still call this Serre’s construction (as in [62, Prop. 5.2.3.9]). Proof. (This is essentially the same proof of [62, Prop. 5.2.3.9].) Since O is (left) noetherian (see, for example, [93, Cor. 2.10]), and since N is finitely generated, there is a free resolution O⊕ r1 → O⊕ r0 → N → 0 for some integers r0 , r1 ≥ 0. By taking HomO ( · , Z), we obtain an exact sequence 0 → HomO (N, Z) → Z r0 → Z r1

(3.1.2.5)

(of fppf sheaves) over U , where Z r0 (resp. Z r1 ) stands for the fiber products of r0 (resp. r1 ) copies of Z over U , which shows that HomO (N, Z) is representable because it is the kernel of the homomorphism Z r0 → Z r1 between abelian schemes in (3.1.2.5).

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To show that HomO (N, Z) is proper over U , note that the first homomorphism in (3.1.2.5) is a closed immersion because Z s is separated over U , and every closed subscheme of Z r is proper over U . This proves (1) of Proposition 3.1.2.4. Suppose that N is torsion of order annihilated by some integers m ≥ 1, and that Z[m] is a finite flat group scheme of ´etale-multiplicative type over U . Then HomO (N, Z) is isomorphic to the closed subscheme HomO (N, Z[m]) of the finite flat group scheme HomZ (N, Z[m]) of ´etale-multiplicative type over U . Over an ´etale covering of U over which Z[m] admits an O-equivariant filtration by finite flat subgroup schemes whose graded pieces are either constant group schemes or dual to constant group schemes (i.e., split multiplicative), the condition of compatibilities with O-actions on the constant schemes involved is both open and closed. This implies that HomO (N, Z) is also finite flat of ´etale-multiplicative type over U . This proves (2) of Proposition 3.1.2.4. If N is projective, then it is flat by [93, Cor. 2.16]. This is the same for its dual ˜ defined by an ideal I (right) O-module N ∨ . Hence, for every embedding U ,→ U 2 ˜ such that I = 0, the surjectivity of the morphism Z(U ) → Z(U ) of O-modules implies the surjectivity of the morphism ˜) ∼ ˜ ) → (N ∨ ⊗ Z)(U ) ∼ (N ∨ ⊗ Z)(U = N ∨ ⊗ Z(U = N ∨ ⊗ Z(U ). O

O

O

O

This shows that HomO (N, Z) → U is formally smooth, and hence smooth because it is (locally) of finite presentation (see [35, IV-4, 17.3.1 and 17.5.2]). Moreover, since N is projective, there exists some projective O-module N 0 such that N ⊕ N 0 ∼ = O⊕ r for some r ≥ 0. Then we have HomO (N, Z) × HomO (N 0 , Z) ∼ = Z r , which shows U

that the geometric fibers of HomO (N, Z) → U are connected. Hence, by definition, HomO (N, Z) is an abelian scheme over U . This proves (3) of Proposition 3.1.2.4. Finally, suppose that N is an O-lattice, and that Z and hence Z ∨ are ordinary abelian schemes over U (see Lemma 3.1.1.5). Let O0 be any maximal order in O ⊗ Q containing O. By [62, Prop. 1.1.1.21], Z

there exists an integer m ≥ 1, with no prime factors other than those of Disc, such ∨ that mO0 ⊂ O. Consider the intersection K of the kernels of [b] : Z ∨ [m] → Z ∨ [m] 0 for all b ∈ mO . By working over an ´etale covering over which Z[m] and Z ∨ [m] admit O-equivariant filtrations by finite flat subgroup schemes whose graded pieces are either constant group schemes or split multiplicative (as in the proof of (2) of Proposition 3.1.2.4 above), we see that K is a finite flat subgroup scheme of Z ∨ [m], and hence so is its orthogonal complement K ⊥ in Z[m] (with respect to the canonical pairing eZ[m] : Z[m] × Z ∨ [m] → µm,U ). By construction, this K ⊥ is the smallest subgroup scheme of Z[m] containing the images of [b] : Z[m] → Z[m] for all b ∈ mO0 . Therefore, by forming the isogeny Z  Z 0 := Z/K ⊥ , the action of O on Z induces an action of O0 on Z 0 . In this case, there is also a canonical isogeny Z 0  Z whose pre- and post-compositions with the previous isogeny Z  Z 0 are multiplications by m on Z and Z 0 , respectively. Let N 0 be the O0 -span of N in N ⊗ Q. Since N 0 is the O0 -space of N , the Z

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canonical isogeny Z  Z 0 induces a canonical homomorphism HomO (N, Z) → HomO0 (N 0 , Z 0 ). On the other hand, the canonical isogeny Z 0  Z above induces a canonical homomorphism HomO0 (N 0 , Z 0 ) → HomO (N, Z), whose pre- and post-composition with the previous canonical homomorphism is nothing but the multiplications by m on HomO (N, Z) and HomO0 (N 0 , Z 0 ), respectively. As usual, we denote by [m] all such multiplications by m. Since O0 is maximal, N 0 is projective as an O0 -module by [62, Prop. 1.1.1.23]. By (3) of Proposition 3.1.2.4 proved above, we know that HomO0 (N 0 , Z 0 ) is an abelian scheme. Since [m] : HomO (N, Z) → HomO (N, Z) factors as the composition of canonical homomorphisms HomO (N, Z) → HomO0 (N 0 , Z 0 ) → HomO (N, Z), this shows that the schematic image of [m] : HomO (N, Z) → HomO (N, Z) is an abelian scheme. On the other hand, by working over an ´etale covering over which Z[m] admits an O-equivariant filtration by finite flat subgroup schemes whose graded pieces are either constant group schemes or split multiplicative (as above), we see that HomO (N, Z)[m] ∼ = HomO (N, Z[m]) is finite flat of ´etale-multiplicative type (of rank dividing a power of m) over U . Hence, by Lemma 3.1.2.2, we see that both π 0 (HomO (N, Z)/U ) and HomO (N, Z)◦ are defined with the desired properties. Remark 3.1.2.6. The materials in this subsection generalize naturally to the case when U is an algebraic stack (which is Deligne–Mumford by our convention). 3.1.3

Extensibility of Isogenies

Suppose that U is noetherian scheme and that U is a noetherian normal scheme containing U as an open dense subscheme. Let f : Z → Z 0 be an isogeny of semiabelian schemes over U (cf. Lemma 3.1.1.5), and let Z → U be a semi-abelian scheme extending Z → U (cf. Definition 3.1.1.2), in the sense that Z U = Z × U U

is isomorphic to U as a group scheme. (Then Z is determined up to canonical isomorphism by Z, by noetherian normality of U and by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5].) Let K := ker(f ), which is quasi-finite flat and of finite presentation over U . Let K denote the schematic closure of K in Z. Let N be an integer such that K is a closed subgroup scheme of Z[N ], which exists because U is noetherian. Then K is a closed subgroup scheme of Z[N ] (by the universal property of schematic closures), which is quasi-finite over U . Lemma 3.1.3.1. With assumptions as above, suppose U is Dedekind (i.e., noetherian normal and of dimension at most one). Then K is a group scheme quasi-finite 0 flat over U , and there exists a semi-abelian scheme Z → U , unique up to unique isomorphism, such that the isogeny f : Z → Z 0 over U uniquely extends to an 0 isogeny f : Z → Z over U .

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Proof. Since U is one-dimensional, as explained in [10, Sec. 10.1, Prop. 4 and 7; see also the middle of paragraph 2 on p. 310], the (locally of finite type) N´eron models of Z and Z 0 over U (uniquely) exist, and the (fiberwise) identity compo0 nents of them are group schemes Z and Z over U (the former being up to canonical isomorphism the same Z as above) which are commutative, separated, smooth, and 0 of finite type, and have geometrically connected fibers. (Since Z → U admits the identity section, by [35, IV-2, 4.5.13], its connected fibers are also geometrically connected.) Moreover, by the universal property of N´eron models and by the definition of identity components, the homomorphism f : Z → Z 0 uniquely extends to 0 a homomorphism f : Z → Z over U . Since K = ker(f ) because the latter is the 0 (closed) preimage of the identity section of the separated group scheme Z → U , if N is any integer as above such that K = ker(f ) ⊂ Z[N ], then K = ker(f ) ⊂ Z[N ]. This forces f to be quasi-finite and hence surjective (because it is between schemes with geometrically connected fibers that are separated, smooth, of finite type, and of the same dimension). Therefore, K = ker(f ) is quasi-finite and flat over 0 U (see [62, Lem. 1.3.1.11]), and Z → U is also semi-abelian (begin an isogenous quotient of Z → U , whose fibers are still extensions of abelian varieties by tori), as desired. Lemma 3.1.3.2. With assumptions as above, suppose moreover that Z → U is an ordinary semi-abelian scheme (but suppose no longer that U is one-dimensional). Then K is a group scheme quasi-finite flat over U (of fiber degrees dividing those of 0 K), and there exists a semi-abelian scheme Z → U such that the isogeny f : Z → Z 0 0 0 over U extends to an isogeny f : Z → Z over U . (Then Z and f are determined up to unique isomorphism by Z and f , by noetherian normality of U and by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5].) Proof. Since U is noetherian normal, by [92, XI, 1.13], Z is locally quasi-projective. Hence, by [80, IV, 7.1.2] (see also [62, Lem. 3.4.3.1]), over any open subscheme U 0 of U over which the quasi-finite group scheme K is flat, the quotient (Z × U 0 )/(K × U 0 ) U

U

is representable by a semi-abelian scheme over U 0 , which is also ordinary by Lemma 0 3.1.1.5. Hence, the flatness of K and the constructibility of Z and f are equivalent conditions over open subschemes of U . If K were not flat, then it must be so at some point u, and we may take this point to be maximally so. Therefore, to show that K is flat, we may replace U with its strict localization at an arbitrary point u (see [35, 0I , (6.6.3), and IV-4, 18.8.8(iii)]), and we may enlarge U and assume that U is the full open complement of u in U . By Lemma 3.1.3.1, we may assume that u has codimension at least two. Let N be any integer as above such that K = ker(f ) ⊂ H := Z[N ]. By the assumption that Z → U is ordinary, and by arguing as in [62, Sec. 3.4.1], H := Z[N ] admits a filtration mult f 0⊂H ⊂ H ⊂ H, mult

f

where H is finite flat of multiplicative type, where H is the maximal finite flat subgroup scheme of H over U (which is open and closed in H, because U is

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f

mult

Henselian), where the quotient H := H /H is finite ´etale, and where the f quotient H/H is quasi-finite ´etale (whose special fiber over u consists of only the mult f f,´ et × U , H f := H × U , H f,´et := H × U , identity section). Let H mult := H U

U

U

K mult := K ∩ H mult , and K f := K ∩ H f . Note that K f is (finite and) flat over U because H f is open and closed in H. Let us denote Cartier duals by superscripts “D”. Note that K mult is the kernel of the composition of morphisms H mult ,→ H f  H f /K f between finite flat group schemes. (The quotient H f /K f is defined because K f is finite flat.) The Cartier dual of this morphism is (H f /K f )D → (H mult )D . Since (H mult )D is ´etale over U , its fibers over U are disjoint unions of closed points. Hence, by the fiberwise criterion of flatness as in [35, IV-3, 11.3.10 a)⇒b)], the image of (H f /K f )D → (H mult )D is finite flat. Then the cokernel of (H f /K f )D → (H mult )D is defined and also finite flat of finite presentation; and its Cartier dual is K mult and is also finite flat of finite presentation. Thus, we can define K f,´et := K f /K mult , which is a finite ´etale subgroup scheme of H f,´et . mult f,´ et Let K (resp. K ) be the schematic closure of K mult (resp. K f,´et ) in H mult mult f,´ et (resp. H f,´et ). Since U is strict local and normal, H (resp. H ) is dual to (resp. mult f,´ et is) a constant group scheme over U . Hence, K (resp. K ) is also finite flat of multiplicative (resp. finite ´etale). + f f,´ et f,´ et + Let K := (H  H )−1 (K ) and let K + := K × U . Given K f as an U

extension of K f,´et by K mult , the natural inclusion H mult ,→ K + induces an isomorphism H mult /K mult ∼ = K + /K f , and induces a surjection K +  H mult /K mult . Note that this surjection determines K f in the sense that its kernel is a finite flat subgroup scheme of H f that is an extension of K f,´et by K mult , which is just K f . + mult mult Since U is normal and K and H /K are finite flat, the above surjection extends to a surjection K

+

H

mult

/K

mult

, f

f,´ et

whose kernel defines a finite flat subgroup scheme of H that is an extension of K mult f , which must coincide with the closure of K f in H , which is nothing but by K f f f f K := K ∩ H . Hence, K is finite flat over U . Since K − K has an empty fiber f over u and coincides with K − K over U = U − {u}, we see that K is flat over all of U , as desired. Remark 3.1.3.3. Lemma 3.1.3.2 is incorrect if we do not assume that Z → U is ordinary. See [18, Sec. 6] for an example even when U is regular. (One might as well introduce conditions on U as in [101] to ensure that it is healthy regular . However, such conditions tend to impose restrictions on the ramification of the universal base ring, which conflicts with our goal.)

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Proposition 3.1.3.4. Let (Z, λZ ) → U be as in Definition 3.1.2.1. Let N be an O-lattice. Suppose Z is ordinary, so that HomO (N, Z) is defined and representable by a proper flat subgroup scheme over U , which is an extension of a finite flat group scheme π 0 (HomO (N, Z)/U ) of ´etale-multiplicative type by the abelian scheme HomO (N, Z)◦ as in Proposition 3.1.2.4. Suppose that U is noetherian normal and that Z → U is an ordinary semi-abelian scheme extending Z → U . Then HomO (N, Z) is defined and representable over U by an extension of a quasifinite flat group scheme π 0 (HomO (N, Z)/U ) of ´etale-multiplicative type by a semiabelian scheme HomO (N, Z)◦ . The restriction of this extension to U is the extension HomO (N, Z) of the finite flat group scheme π 0 (HomO (N, Z)/U ) of ´etalemultiplicative type by an abelian scheme HomO (N, Z)◦ in (4) of Proposition 3.1.2.4. Proof. The same argument of the proof of (1) of Proposition 3.1.2.4 shows that HomO (N, Z) is representable by a (closed) subgroup scheme of a semi-abelian scheme over U . The same argument of the proof of (3) of Proposition 3.1.2.4 shows that, when N is projective as an O-module, there exists some projective O-module N 0 such ×r for some integer r, which implies that that HomO (N, Z) × HomO (N 0 , Z) ∼ = Z U

HomO (N, Z) is a semi-abelian scheme over U , because it is commutative, separated, smooth, and of finite type, and because its geometric fibers are all connected with trivial unipotent radicals (see [10, Sec. 7.3, paragraph following Lem. 1]). Let m ≥ 1, O0 , Z  Z 0 , Z 0  Z, and N 0 be as in the proof of (4) of Proposition 3.1.2.4. Since U is noetherian normal, by Lemma 3.1.3.2, there exist a 0 0 0 semi-abelian scheme Z and two isogenies Z  Z and Z  Z over U extending the isogenies Z  Z 0 and Z 0  Z over U . By the previous paragraph, we 0 know that HomO0 (N 0 , Z ) is representable by a semi-abelian scheme over U . Moreover, the same argument of the proof of (2) of Proposition 3.1.2.4 shows that HomO (N, Z)[m] is quasi-finite flat of ´etale-multiplicative type over U for every integer m ≥ 1. Hence, the same argument of the proof of (4) of Proposition 3.1.2.4 shows that the schematic-image of [m] : HomO (N, Z) → HomO (N, Z) is an semi0 abelian scheme over U , which is an isogenous quotient of HomO0 (N 0 , Z ) (by Lemma 3.1.3.2 again). Let us denote this semi-abelian scheme by HomO (N, Z)◦ . As in the proof of Lemma 3.1.2.2, since HomO (N, Z)◦ is (fppf locally) m-divisible (as a semi-abelian scheme) over U , the quotient HomO (N, Z)/HomO (N, Z)◦ can be identified (as a fppf sheaf over U ) with HomO (N, Z)[m]/HomO (N, Z)◦ [m], which is representable by a quasi-finite flat scheme π 0 (HomO (N, Z)/U ) over U because HomO (N, Z)◦ [m] is closed in HomO (N, Z)[m] (by [52, II, 6.16], an algebraic space quasi-finite and separated over a scheme is a scheme). Thus, HomO (N, Z) is an extension of a quasi-finite flat group scheme π 0 (HomO (N, Z)/U ) of ´etale-multiplicative type by a semi-abelian scheme HomO (N, Z)◦ over U . By its very construction, this extension extends the corresponding extension in (4) of Proposition 3.1.2.4, as desired.

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Linear Algebraic Data for Ordinary Loci Necessary Data for Ordinary Reductions

Suppose there exists an abelian scheme A over S = Spec(R), where R is a complete noetherian normal domain with fraction field K of characteristic zero and algebraically closed residue field k of characteristic p > 0, and suppose there exist ~ H (see Proposition 2.2.1.1), which induces a field homomora morphism ξ : S → M ¯ be phism F0 → K factoring through (F0 )υ → K for a place υ of F0 above p. Let K ¯ an algebraic closure of K. Let s := Spec(k), η := Spec(K), and η¯ := Spec(K). For simplicity, let us assume that H is neat. Then the restriction ξ0 : η → MH of ξ defines an object (Aη , λη , iη , αH,η ) of MH (η). By [92, IX, 1.4], [28, Ch. I, Prop. ∨ 2.7], or [62, Prop. 3.3.1.5], λη : Aη → A∨ η extends to an isogeny λ : A → A over S, and iη : O → Endη (Aη ) extends to a homomorphism i : O → EndS (A). By [92, XI, 1.16], the symmetric invertible sheaf (IdA , λ)∗ PA is ample because its restriction to η is. Therefore, λ is also a polarization (by definition; cf. [23, 1.2, 1.3, 1.4] and [62, Prop. 1.3.2.15 and Def. 1.3.2.16]). Note that the extension i satisfies the Rosati condition defined by λ, because it already does over η. Hence, i is an O-endomorphism structure as in [62, Def. 1.3.3.1]. Moreover, LieA/S with its O ⊗ Z(p) -module structure given naturally by i satisfies the determinantal condition Z

in [62, Def. 1.3.4.1] given by (L ⊗ R, h · , · i, h0 ), because the condition is given by Z

an identity of polynomial functions, and because η is dense in S. For each p-divisible group G, we denote by G mult (resp. G conn , resp. G ´et ) the multiplicative-type (resp. connected, resp. ´etale) part of G, whose formation is functorial and compatible with all automorphisms of G. Let As [p∞ ] = lim As [pr ] (resp. −→ r ∞ ∨ r [p ] = lim [p ]) denote the p-divisible group attached to the abelian variety A∨ A s s −→ r As (resp. A∨ s ). The canonical perfect duality ∞ eAs [p∞ ] : As [p∞ ] × A∨ s [p ] → µp∞ ,s

induces canonical perfect dualities ∞ ´ et As [p∞ ]mult × A∨ s [p ] → µp∞ ,s

and ∞ mult As [p∞ ]´et × A∨ → µp∞ ,s , s [p ]

which induce canonical isomorphisms ∼

∞ ´ et As [p∞ ]mult → Homs (A∨ s [p ] , µp∞ ,s )

and ∼

∞ mult As [p∞ ]´et → Homs (A∨ , µp∞ ,s ), s [p ]

respectively, compatible with their O-actions induced by i.

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Proposition 3.2.1.1. With the setting as above, suppose moreover that the abelian scheme A over S is ordinary as in Definition 3.1.1.2. Let us fix the choice of a ∼ system of compatible isomorphisms {ζpr ,¯η : (Z/pr Z)(1) → µpr ,¯η }r≥0 , which exists because K is of characteristic zero. Then the following are true: (1) As is an ordinary abelian variety in the usual sense. (2) The physical Tate modules Tp As and Tp A∨ s are free Zp -modules of rank dims As (when their O-module structures are ignored). ∼ ˆ → T Aη¯ be any lifting of αH (whose H orbit determines (3) Let (ˆ α, νˆ) : L ⊗ Z Z

αH ) (as in [62, Lem. 1.3.6.5]). Then the factor ∼

α ˆ p : L ⊗ Zp → Tp Aη¯ Z

of α ˆ at p and the canonical O ⊗ Zp -equivariant extension Z ∞ mult

0 → As [p ]

→ As [p∞ ] → As [p∞ ]´et → 0

define a filtration D1 = 0 ⊂ D0 ⊂ D−1 = L ⊗ Zp

(3.2.1.2)

Z

:= Let us set Gr−1 D

of O ⊗ Zp -modules, whose H-orbit is canonical. Z 0

D−1 /D and Gr0D := D0 /D1 as usual. Then there exists canonically induced O ⊗ Zp -equivariant isomorphisms Z

∼ (Gr0D )mult = As [p∞ ]mult s (see Definition 3.1.1.7), ∼ Gr−1 D = Tp A s , and ∞ ´ et ∼ Gr−1 D ⊗ (Qp /Zp ) = As [p ] . Zp

(4) By duality, we have an analogous filtration D#,1 = 0 ⊂ D#,0 ⊂ D#,−1 = L# ⊗ Zp

(3.2.1.3)

Z

of O ⊗ Zp -modules, together with canonically induced O ⊗ Zp -equivariant Z

Z

isomorphisms ∞ mult ∼ (Gr0D# )mult , = A∨ s s [p ]

∼ T A∨ , Gr−1 D# = p s and ∞ ´ et Gr−1 ⊗ (Qp /Zp ) ∼ = A∨ s [p ] , D# Zp

where Gr−1 := D#,−1 /D#,0 and Gr0D# := D#,0 /D#,1 as usual. D#

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(5) The canonical (O-equivariant) embedding L ,→ L# and the polarization λ : A → A∨ induce canonical O ⊗ Zp -equivariant morphisms Z

: (Gr0D )mult → (Gr0D# )mult s s

(φ0D )mult s and

−1 −1 φ−1 D : GrD → GrD# ,

and φ−1 such that (φ0D )mult D ⊗ (Qp /Zp ) are canonically dual to each other, s Zp

making the diagrams

ν ˆord (φ0D )mult s



∼

(Gr0D# )mult s Gr−1 D

/ As [p∞ ]mult

∼

(Gr0D )mult s

λ

∞ mult / A∨ s [p ]

/ Tp A s

∼

φ−1 D

 Gr−1 D#





λ

/ Tp A ∨ s

∼

and Gr−1 D ⊗ (Qp /Zp )

∼

/ As [p∞ ]´et

∼

 ∞ ´ et / A∨ s [p ]

Zp

⊗ (Qp /Zp ) φ−1 D

λ

Zp

 ⊗ (Q Gr−1 p /Zp ) # D Zp

commutative for some canonically determined νˆord ∈ Z× p. (6) Let us consider LieA/S as an O ⊗ R-module using the O-structure i. Then Z

there are canonical isomorphisms

, Zp ) ⊗ R ∼ HomZp (Gr−1 = LieA/S D# Zp

and ∼ HomZp (Gr−1 D , Zp ) ⊗ R = LieA∨ /S Zp

of O ⊗ R-modules making the diagram Z

HomZp (Gr−1 , Zp ) ⊗ R D#

∼

/ LieA/S

Zp

t

(φ−1 D )

dλ



HomZp (Gr−1 D , Zp ) ⊗ R Zp

∼



/ LieA∨ /S

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commutative. These are dual to canonical isomorphisms ∼

−1 Lie∨ A/S → GrD# ⊗ R Zp

and ∼

−1 Lie∨ A∨ /S → GrD ⊗ R Zp

of O ⊗ R-modules making the diagram Z

Lie∨ A∨ /S

∼

/ Gr−1 D ⊗R Zp

λ

∗



Lie∨ A/S

∼



φ−1 D

/ Gr−1 ⊗R D# Zp

commutative. ¯ (7) Let F00 and L0 be as in Section 1.4. Then, for each homomorphism F00 → K of extension fields of F0 , there is an isomorphism ∼ LieA/S ⊗ K ¯ = ¯ L0 ⊗ K F00

R

¯ of O ⊗ K-modules. (This follows from the fact that (Aη , λη , iη , αH,η ), as an Z

object of MH (η), satisfies the Lie algebra condition in [62, Def. 1.3.4.1].) Consequently, we have an isomorphism ∼ HomZ (Gr−1 ¯ ¯ = , Zp ) ⊗ K (3.2.1.4) L0 ⊗ K p D# F00

Zp

¯ of O ⊗ K-modules. Z

(8) The canonical homomorphism Qp → (F0 )υ is an isomorphism. Proof. Statements (1) and (2) are clear form the definitions (see Remark 3.1.1.4). Statements (3), (4), and (5) follow from the rigidity of groups of multiplicative type (see [26, IX, 3.6 and 3.6 bis]), by (uniquely) lifting the torsion subgroup schemes r mult As [pr ]mult and A∨ to subgroup schemes A[pr ]mult and A∨ [pr ]mult of multis [p ] r plicative type of A[p ] and A∨ [pr ], respectively, for each r ≥ 0, so that α ˆ induces ∼ ∞ mult → A [p ] , extending to an isomorvia {ζpr ,¯η }r≥0 an isomorphism (Gr0D )mult η¯ η¯ ¯ ¯ phism over the normalization R of R in K and inducing the desired isomorphism ∼ ∼ (Gr0D )mult → As [p∞ ]mult ; so that the dual of α ˆ induces an isomorphism (Gr0D# )mult = s η¯ 0 mult ∼ ∨ ∞ mult ∞ mult Aη¯ [p ] , compatible with the above isomorphism (GrD )η¯ → Aη¯[p ] up to a scalar νˆord ∈ Z× ν ord mod pr ) ζpr ,¯η = νˆ mod pr for every r ≥ 0, exp such that (ˆ ¯ as above and inducing the desired isomorphism tending to an isomorphism over R 0 mult ∼ ∞ mult (GrD )s → As [p ] ; and so that the rest of the assertions follow by various canonical identifications. Statement (6) follows from [47, 3.4], and statement (7) is ¯ generated self-explanatory. Because of the isomorphism (3.2.1.4), the subfield of K ¯ by traces TrC (b|L0 ⊗ K) for b ∈ O is contained in Qp . Since this is true for every F00

¯ statement (8) follows. field homomorphism F00 → K,

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3.2.2

Maximal Totally Isotropic Submodules at p

Now we will turn the observations made in Proposition 3.2.1.1 into formal definitions. Let p > 0 be any rational prime number. Lemma 3.2.2.1. Consider the following two kinds of data: (1) A filtration D1Qp = 0 ⊂ D0Qp ⊂ D−1 Qp = L ⊗ Qp

(3.2.2.2)

Z

of O ⊗ Qp -modules such that D0Qp is (maximal) totally isotropic under the Z

pairing induced by h · , · i and such that D0Qp is its own annihilator under the pairing. For simplicity, we shall call D0Qp a maximal totally isotropic submodule of L ⊗ Qp , without mentioning the O ⊗ Qp -submodule Z

structure. (2) A filtration

Z

D1 = 0 ⊂ D0 ⊂ D−1 = L ⊗ Zp

(3.2.2.3)

Z

of O ⊗ Zp -modules such that D0 is totally isotropic under the pairing induced Z

−1 /D0 is torsionby h · , · i, such that the quotient O ⊗ Zp -module Gr−1 D =D Z

free (as a Zp -module), and such that D0 is its own annihilator under the pairing. For simplicity, we shall call D0 a maximal totally isotropic submodule of L ⊗ Zp , without mentioning the O ⊗ Zp -submodule strucZ

ture.

Z

These two kinds of data determine each other in the following way: D0Qp is the Qp -span of D0 in L ⊗ Qp , while D0 is the intersection of D0Qp with L ⊗ Zp . Z

Z

Proof. The statements are self-explanatory. Lemma 3.2.2.4. Each choice of a filtration (3.2.2.2) (or equivalently a filtration (3.2.2.3); see Lemma 3.2.2.1) determines the following list of data: (1) A filtration D#,1 = 0 ⊂ D#,0 ⊂ D#,−1 = L# ⊗ Zp

(3.2.2.5)

Z #,0 such that D#,0 = DQ ∩(L# ⊗ Zp ) in L ⊗ Qp , so that Gr−1 = D#,−1 /D#,0 is D# p Z

Z

torsion-free (as a Zp -module). (2) A perfect duality Gr0D × Gr−1 → Zp (1) D# induced by h · , · i.

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169

(3) A perfect duality Gr0D# × Gr−1 D → Zp (1) induced by h · , · i. (4) An canonical inclusion φ0D : Gr0D ,→ Gr0D# (with finite cokernel) dual to a canonical inclusion −1 −1 φ−1 D : GrD ,→ GrD#

(with finite cokernel). (5) For each integer r ≥ 0, we have Gr0D,pr := D0pr := D0 /pr D0 , −1 −1 −1 r Gr0D# ,pr := Dp#,0 := D#,0 /pr D#,0 , Gr−1 r D,pr := GrD /p GrD , and GrD# ,pr := 0 0 −1 −1 0 r GrD# /p GrD# , together with the morphisms φD,pr : GrD,pr → GrD# ,pr and −1 −1 −1 0 φ−1 D,pr : GrD,pr → GrD# ,pr induced by φD and φD , respectively. Proof. The statements are self-explanatory. Lemma 3.2.2.6. Under the assumption that L satisfies Condition 1.2.1.1, any filtration (3.2.2.3) as in Lemma 3.2.2.1 (noncanonically) splits. The filtration (3.2.2.5) it determines as in Lemma 3.2.2.4 also splits. (The splittings might not be compatible with each other under the canonical morphisms induced by L ,→ L# and λ : A → A∨ .) Proof. By Condition 1.2.1.1, the action of O on L extends to an action of some maximal order O0 in O ⊗ Q containing O. By Lemma 3.2.2.1, D0 is the intersection Z

of D0Qp with L ⊗ Zp . Hence, the action of O on the submodule D0 of L ⊗ Zp extends Z

Z

to an action of O0 (compatible with those on D0Qp and L ⊗ Zp ), and the filtration D of Z

L ⊗ Zp is O0 ⊗ Zp -equivariant. A similar argument shows that the filtration D# on Z

Z

L# ⊗ Zp is also O0 ⊗ Zp -equivariant. Since O0 is maximal, O0 ⊗ Zp is also maximal, Z

Z

Z

which is hereditary in the sense that all O0 ⊗ Zp -lattices (namely, finitely generated Z

O0 ⊗ Zp -modules with no p-torsion) are projective O0 ⊗ Zp -modules (cf. [62, Prop. Z

Z

1.1.1.12 and 1.1.1.23]). Hence, the filtrations (3.2.2.3) and (3.2.2.5) split because −1 Gr−1 D and GrD# are torsion-free (as Zp -modules), as desired. Definition 3.2.2.7. Let O, (L, h · , · i) be given as above. We define for each Zp -algebra R   (g, r) ∈ GLO ⊗ R (L ⊗ R) × Gm (R) : Z Z Pord , D (R) :=  (g, r) ∈ G(R), g(D ⊗ R) = D ⊗ R  Zp

Zp

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Mord D (R) :=

  (g, r) ∈ GLO ⊗ R (GrD ⊗ R) × Gm (R) : Z

Zp

 hgx, gyi = rhx, yi, ∀x, y ∈ GrD ⊗ R 

,

Zp

ord ord Uord D (R) := ker(GrD : PD (R) → MD (R)),

and i Uord,i (R) := ker(GriD : Pord D D (R) → GLO ⊗ R (GrD ⊗ R)) Z

Zp

for each i. These assignments are functorial in R and define group functors Pord D , ord,i ord ord Mord , U , and U over Spec(Z ). By definition, P is a subgroup of G ⊗ Zp , p D D D D Z

and (by Lemma 3.2.2.6) there is an exact sequence 1 → Uord → Pord → Mord → 1. D D D As in the case of G, the projections to the second factor (g, r) 7→ r define homomorphisms ν : Pord → Gm ⊗ Zp , ν : Mord → Gm ⊗ Zp , and ν : Uord,i → Gm ⊗ Zp , D D D Z

Z

Z

which we call the similitude characters. For simplicity, we shall often denote elements (g, r) by simply g, and denote by ν(g) the value of r when we need it. Definition 3.2.2.8. For all integers 0 ≤ r and 0 ≤ r1 ≤ r0 , we set: Up (pr ) := ker(G(Zp ) → G(Z/pr Z)), r Up,0 (pr ) := (G(Zp ) → G(Z/pr Z))−1 (Pord D (Z/p Z)),

Up,1 (pr ) := (G(Zp ) → G(Z/pr Z))−1 (UDord,−1 (Z/pr Z)), bal r r Up,1 (p ) := (G(Zp ) → G(Z/pr Z))−1 (Uord D (Z/p Z)),

Up,1,0 (pr1 , pr0 ) := Up,1 (pr1 ) ∩ Up,0 (pr0 ), bal bal r1 Up,1,0 (pr1 , pr0 ) := Up,1 (p ) ∩ Up,0 (pr0 ), ord r U ord (pr ) := ker(Mord D (Zp ) → MD (Z/p Z)).

Definition 3.2.2.9. We say that an open compact subgroup Hp ⊂ G(Zp ) is of standard form with respect to D if there exists an integer r ≥ 0 such that bal r Up,1 (p ) ⊂ Hp ⊂ Up,0 (pr ).

In this case, we say that r is the depth of Hp , and write r = depthD (Hp ). ˆ is of standard form with We say that an open compact subgroup H ⊂ G(Z) p ˆ p ) and Hp ⊂ G(Zp ), respect to D if it is of the form H = H Hp , where Hp ⊂ G(Z such that Hp is of standard form with respect to D. In this case, we set depthD (H) := depthD (Hp ). We say that two open compact subgroups Hp and Hp0 of G(Zp ) (resp. H and H0 ˆ of standard form with respect to D are equally deep if depth (Hp ) = of G(Z)) D depthD (Hp0 ) (resp. depthD (H) = depthD (H0 )). We shall suppress the term “with respect to D” when the choice of D is clear from the context.

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By Proposition 3.2.1.1, for the filtration D = {Di }i of O ⊗ Zp -modules of L ⊗ Zp Z

Z

as in Lemma 3.2.2.1 to be useful for our purpose of defining and studying the ordinary loci in mixed characteristics, we need the following: Assumption 3.2.2.10. There exists a place υ of F0 above p such that the canonical homomorphism Qp → (F0 )υ is an isomorphism, and there exists an extension field K of (F0 )υ (and F0 ), together with a homomorphism F00 → K of fields over F0 , such that (3.2.2.11) L0 ⊗ K ∼ = HomZ (Gr−1 # , Zp ) ⊗ K p

F00

D

Zp

as O ⊗ K-modules. Z

This assumption will be made when we define moduli problems for ordinary level structures. 3.2.3

Compatibility with Cusp Labels

ˆ be of standard form (with respect to D) as in Definition 3.2.2.9, so that Let H ⊂ G(Z) p bal r H = H Hp and Up,1 (p ) ⊂ Hp ⊂ Up,0 (pr ) for r = depthD (H). Let [(ZH , ΦH , δH )] be a cusp label at level H for the PEL-type O-lattice (L, h · , · i, h0 ). By definition, ZH is an H-orbit of strongly symplectic admissible filtrations Z on ˆ This includes, in particular, the datum of an Hp /U(pr )-orbit of symplectic L ⊗ Z. Z

admissible filtrations Z ⊗ Zp = {Z−i ⊗ Zp }i on L ⊗ Zp . ˆ Z

ˆ Z

Z

Definition 3.2.3.1. We say that the cusp label [(ZH , ΦH , δH )] is compatible with the filtration D if there exists at least one representative Z in the H-orbit ZH such that we have Z−2 ⊗ Zp ⊂ D0 ⊂ Z−1 ⊗ Zp , ˆ Z

(3.2.3.2)

ˆ Z

which induces a filtration D−1 = {Di−1 }i on GrZ−1 ⊗ Zp given by ˆ Z

Z D1−1 := 0 ⊂ D0−1 := D0 /(Z−2 ⊗ Zp ) ⊂ D−1 −1 := Gr−1 ⊗ Zp ˆ Z

ˆ Z

(serving the same purpose as the filtration D does for L ⊗ Zp ). Z r

By taking reduction modulo p , we have the compatibility Z−2,pr ⊂ D0pr ⊂ Z−1,pr ,

(3.2.3.3)

which induces a filtration D−1,pr = {Di−1,pr }i on GrZ−1,pr given by Z D1−1,pr := 0 ⊂ D0−1,pr := D0pr /Z−2,pr ⊂ D−1 −1,pr := Gr−1,pr

(3.2.3.4)

r

(serving the same purpose as the filtration Dpr does for L/p L). Similarly, we have the compatibility #,0 # Z# −2,pr ⊂ Dpr ⊂ Z−1,pr ,

(3.2.3.5)

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#,i Z which induces a filtration D# −1,pr = {D−1,pr }i on Gr−1,pr given by #

#,0 #,0 # #,−1 Z D#,1 −1,pr := 0 ⊂ D−1,pr := Dpr /Z−2,pr ⊂ D−1,pr := Gr−1,pr .

(3.2.3.6)

bal r Remark 3.2.3.7. Since Up,1 (p ) ⊂ Hp ⊂ Up,0 (pr ), and since the action of Up,0 (pr ) r 0 stabilizes Dpr as an O ⊗ (Z/p Z)-submodule of L/pr L, the compatibilities (3.2.3.2), Z

(3.2.3.3), and (3.2.3.5) are independent of the choice of Z. Hence, it is justified to have the following: Definition 3.2.3.8. We say that a cusp label [(ZH , ΦH , δH )] at level H is D-ordinary if it is compatible with the filtration D as in Definition 3.2.3.1. We shall simply say that [(ZH , ΦH , δH )] is ordinary, or an ordinary cusp label, if the choice of D is clear in the context. For later references, let us define: Definition 3.2.3.9. (Compare with Definition 1.2.1.11.) Suppose Z is compatible with D as in (3.2.3.2). For each Zp -algebra R, we define the following quotients of subgroups of PZ (R) (see Definitions 1.2.1.10 and 1.2.1.11): ord (1) Pord Z,D (R) := PZ (R) ∩ PD (R). (2) Because of the compatibility (3.2.3.2), ZZ (R) ∩ Pord D (R) = ZZ (R) does not define a new group. This is similar for UZ (R), U2,Z (R), U1,Z (R) , Gl,Z (R), and G0l,Z (R). ord ord (3) Pord h,Z,D (R) := PZ,D (R)/ZZ,D (R) is the subgroup of elements of Gh,Z (R) preserving the filtration D−1 induced by D on GrZ−1 ⊗ Zp as in Definition 3.2.3.1. ˆ Z

ord 0 (4) Pord,0 Z,D (R) := PZ (R) ∩ PD (R) is the kernel of the canonical homomorphism Z Z 0 (ν −1 Gr−2 , Gr0 ) : Pord Z,D (R) → Gl,Z (R). ord,0 ord (5) P1,Z,D (R) := PZ,D (R)/U2,Z (R), which is (under any splitting δ above) isoord morphic to (Pord h,Z,D n U1,Z )(R) := Ph,Z,D (R) n U1,Z (R). ord ord ∼ ∼ ord,0 ∼ ord (6) Pord,0 h,Z,D (R) := P1,Z,D (R)/U1,Z (R) = PZ,D (R)/UZ (R) = PZ,D (R)/ZZ,D (R) = ord Ph,Z,D (R).

3.3 3.3.1

Level Structures Level Structures Away from p

ˆ p ) is an open compact subgroup. Suppose n0 ≥ 1 is an integer Suppose Hp ⊂ G(Z prime to p such that U p (n0 ) ⊂ Hp . Let Hn0 := Hp /U p (n0 ). Definition 3.3.1.1. (Compare with [62, Def. 1.3.6.1].) Let S be a scheme over Spec(Z(p) ). Let A be an abelian scheme over S, with a polarization λ : A → A∨ and an O-endomorphism structure i : O ,→ EndS (A) as in [62, Def. 1.3.3.1]. Let H, n0 , and Hn0 be as above. A naive principal level-n0 structure of (A, λ, i) of type (L/n0 L, h · , · i) is a pair (αn0 , νn0 ), where

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173

∼

(1) αn0 : (L/n0 L)S → A[n0 ] is an O-equivariant isomorphism of (´etale) group schemes over S. ∼ (2) νn0 : ((Z/n0 Z)(1))S → µn0 ,S is an isomorphism of group schemes over S making the diagram h·,·i

(L/n0 L)S ×(L/n0 L)S S

/ ((Z/n0 Z)(1))S o νn0

αn0 ×αn0 o

 A[n0 ] × A[n0 ]

e

S

λ

/ µn ,S 0

commutative, where eλ is the λ-Weil pairing. By abuse of notation, we often denote such a symplectic isomorphism by (αn0 , νn0 ) : ∼ ∼ (L/n0 L)S → A[n0 ], or simply by αn0 : (L/n0 L)S → A[n0 ], and denote νn0 by ν(αn0 ) (although αn0 does not always determine νn0 ). Definition 3.3.1.2. (Compare with [62, Def. 1.3.6.2].) We say a naive principal level-n0 structure (αn0 , νn0 ) of (A, λ, i) of type (L/n0 L, h · , · i) in Definition 3.3.1.1 ˆ p , h · , · i) if it satisfies the following is a principal level-n0 structure of type (L ⊗ Z Z

symplectic-liftability condition: There exists (noncanonically) a tower (Sm0  S)n0 |m0 ,p-m0 of finite ´etale coverings such that we have the following: (1) Sn0 = S. (2) For each l0 such that n0 |l0 and l0 |m0 , there is a finite ´etale covering Sm0  Sl0 whose composition with Sl0 → S is the finite ´etale covering Sm0 → S. (3) There is a naive principal level-m0 structure (αm0 ,Sm0 , νm0 ,Sm0 ) of (A, λ, i) × Sm0 of type (L/m0 L, h · , · i) over each Sm0 . S

(4) For each l0 such that n0 |l0 and l0 |m0 , the pullback of (αl,Sl0 , νl0 ,Sl0 ) to Sm0 is the reduction modulo l0 of (αm0 ,Sm0 , νm0 ,Sm0 ). Definition 3.3.1.3. (Compare with [62, Def. 1.3.7.3].) Let S and (A, λ, i) be as in Definition 3.3.1.1. Let Hp , n0 , and Hn0 be as above. A naive level-Hn0 structure of (A, λ, i) of type (L/n0 L, h · , · i) is an Hn0 -orbit αHn0 of naive principal level-n0 ∼ structures (L/n0 L)S → A[n0 ], namely a (finite ´etale) subscheme αHn0 of the finite ´etale scheme IsomS ((L/n0 L)S , A[n0 ]) × IsomS (((Z/n0 Z)(1))S , µn0 ,S ) S

over S that becomes the disjoint union of elements in some Hn0 -orbit of naive principal level-n0 structures of type (L/n0 L, h · , · i) after a finite ´etale surjective base change in S. In this case, we denote by ν(αHn0 ) the projection of αHn0 to IsomS (((Z/n0 Z)(1))S , µn0 ,S ), which is a ν(Hn0 )-orbit of ´etale-locally-defined isomorphisms with its natural interpretation.

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Definition 3.3.1.4. (Compare with [62, Def. 1.3.7.6].) Let S and (A, λ, i) be as in Definition 3.3.1.1. Let Hp be as above. For each integer n0 ≥ 1 such that p - n0 and U p (n0 ) ⊂ Hp , set Hn0 := Hp /U p (n0 ) as above. Then a level-Hp structure of ˆ p , h · , · i) is a collection αHp = {αH }n labeled by integers (A, λ, i) of type (L ⊗ Z n0 0 Z

n0 ≥ 1 such that p - n0 and U p (n0 ) ⊂ Hp , with elements αHn0 described as follows: (1) For each index n0 , the element αHn0 is a naive level structure of (A, λ, i) of type (L/n0 L, h · , · i) and level Hn0 as in Definition 3.3.1.3. (2) For all indices n0 and m0 such that n0 |m0 , the Hn0 -orbit αHn0 is the schematic image of the Hm0 -orbit αHm0 under the canonical (finite ´etale) morphism IsomS ((L/m0 L)S , A[m0 ]) × IsomS (((Z/m0 Z)(1))S , µm0 ,S ) S

→ IsomS ((L/n0 L)S , A[n0 ]) × IsomS (((Z/n0 Z)(1))S , µn0 ,S ), S

which is equivalent to the formation of U p (n0 )/U p (m0 )-orbits (see [62, Lem. 1.3.7.5]). Remark 3.3.1.5. In these definitions, unlike in [62, Sec. 1.3.6], we no longer assume that the polarization λ has degree prime to p. Hence, these level structures away from p do not detect the polarization type of λ. Lemma 3.3.1.6. Let S and (A, λ, i) be as in Definition 3.3.1.1. Let Hp be as above, ˆ p , h · , · i) as Definition and let αHp be a level-Hp structure of (A, λ, i) of type (L ⊗ Z Z

3.3.1.4. Let s¯ be any geometric point of S. Then there exists an O-equivariant symplectic isomorphism ∼ ˆp → (ˆ αsp¯ , νˆs¯p ) : L ⊗ Z Tp As¯

(3.3.1.7)

Z

such that, for each integer n0 ≥ 1 such that p - n0 and U p (n0 ) ⊂ Hp , the Hn0 -orbit of ∼ the reduction of (αn0 ,¯s , νn0 ,¯s ) : L/n0 L → A[n0 ]s¯ of (α ˆ sp¯ , νˆs¯p ) modulo n0 coincides with the pullback of αHn0 from S to s¯. We say that this (ˆ αsp¯ , α ˆ sp¯ ), or for simplicity just α ˆ sp¯ , is a lifting of αHp at s¯. The Hp -orbit [ˆ αsp¯ ]Hp of α ˆ sp¯ is unique (i.e., independent p of the choice of α ˆ s¯ ). If S is locally noetherian, then the Hp -orbit [ˆ αsp¯ ]Hp is π1 (S, s¯)-invariant. Moreover, we can recover the collection αHp = {αHn0 }n0 over the connected component of s¯ on S from the π1 (S, s¯)-invariant Hp -orbit [ˆ αsp¯ ]Hp . Proof. The pullback of the compatible collection αHn0 to s¯ allows us to choose ∼ a compatible collection of O-equivariant isomorphisms {(αn0 ,¯s , νn0 ,¯s ) : L/n0 L → A[n0 ]s¯}n0 , such that the Hn0 -orbit of (αn0 ,¯s , νn0 ,¯s ) is αHn0 for each n0 , which is equivalent to the desired (ˆ αs¯, νˆs¯) in (3.3.1.7). When S is locally noetherian, the Hp -orbit of (α ˆ sp¯ , νˆs¯p ) is invariant under the action of π1 (S, s¯) because the Hn0 -orbit of (αn0 ,¯s , νn0 ,¯s ), or rather the pullback αHn0 ,¯s of αHn0 to s¯, is invariant under π1 (S, s¯) by definition of αHn0 (see Definition 3.3.1.3) and by definition of π1 (S, s¯).

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175

Hecke Twists Defined by Level Structures Away from p

Suppose g0 ∈ G(A∞,p ), and suppose we have two open compact subgroups Hp and ˆ p ) such that H0,p ⊂ g0 Hp g −1 . Let S and (A, λ, i) be as in Definition H0,p of G(Z 0

0 3.3.1.1, and let αH0,p = {αHm }m0 be a level-H0,p structure of (A, λ, i) of type 0 ˆ p , h · , · i) as in Definition 3.3.1.4, indexed by integers m0 ≥ 1 such that p - m0 (L ⊗ Z

Z

0 and U p (m0 ) ⊂ H0,p , defining Hm := H0,p /U p (m0 ) for each such m0 . 0

Proposition 3.3.2.1. With assumptions as above, there exists a tuple 0 (A0 , λ0 , i0 , αH p ) (over S, unique up to isomorphism), called the Hecke twist of −1 0 (A, λ, i, αH0,p ) by g0 , equipped with a Z× (p) -isogeny [g0 ] : A → A of abelian sche0 mes (whose formal inverse we denote by [g0 ] : A → A), satisfying the following characterizing conditions: ∨

(1) λ0 : A0 → A0,∨ is a polarization defined by λ0 = r0 [g0 ] ◦ λ ◦ [g0 ] (as positive Q× -isogenies), where r0 is the unique number in Z× (p),>0 such that p p ˆ ˆ r0 ν(g0 )Z = Z .

(2) i0 : O → EndS (A0 ) is an O-structure of (A0 , λ0 ) making [g0−1 ] an O-equivariant Z× (p) -isogeny. 0 ˆ p , h · , · i). (3) α p is a level-Hp structure of (A0 , λ0 , i0 ) of type (L ⊗ Z H

Z

∼ ˆp → (4) At each geometric point s¯, there exist a lifting (α ˆ sp¯ , νˆs¯p ) : L ⊗ Z Tp As¯ Z

∼

ˆ p → Tp A0 ) of αH0,p (resp. α0 p ) as in Lemma (resp. (ˆ αsp,0 ˆs¯p,0 ) : L ⊗ Z ¯ ,ν s¯ H Z

∼

3.3.1.6, such that the induced morphisms α ˆ sp¯,A∞,p : L ⊗ A∞,p → Vp As¯, Z

∼

∼

∞,p α ˆ sp,0 → Vp A0s¯, and Vp ([gp ]) : Vp As¯ → Vp A0s¯ satisfy ¯,A∞,p : L ⊗ A Z

p −1 α ˆ sp,0 ˆ sp¯,A∞,p ◦ g0 , and such that νˆs¯p,0 = νˆs¯p ◦ (r0 ν(g0 )) ¯,A∞,p = V ([g0 ]) ◦ α ∼ ˆ p (1) → as isomorphisms Z Tp Gm,¯s , where r0 as in (1) above. In this case, p 0,p the H -orbit [ˆ αs¯ ]H0,p determines a (g0 Hp g0−1 )-orbit [ˆ αsp¯ ]g0 Hp g−1 because

H0,p ⊂ g0 Hp g0−1 , and hence induces an Hp -orbit [ˆ αsp,0 ¯ ]Hp .

0

We consider such a Hecke twist to be away from p because the two tuples are −1 0 related by a canonical Z× (p) -isogeny [g0 ] : A → A which induces an isomorphism A[pr ] → A0 [pr ] for each integer r ≥ 1. ˆ p ), each having a setup If g0 = g1,0 g2,0 , where g1,0 and g2,0 are elements of G(Z analogous to that of g0 , then the Hecke twists by g0 can be constructed in two steps −1 −1 using Hecke twists by g1,0 and g2,0 , such that [g0−1 ] = [g2,0 ] ◦ [g1,0 ] (or, equivalently, [g0 ] = [g1,0 ] ◦ [g2,0 ]). Proof. For [g0−1 ] to exist, at each geometric point s¯ of S and for each lifting ∼ α ˆ s¯ of αH0,p , the induced isomorphism Vp ([g0−1 ]s¯) : Vp As¯ → Vp A0s¯ must map ˆ p )) to Tp A0 = α ˆ p ). (Since Hp and H0,p are subgroups of G(Z ˆ p ), α ˆ s¯(g0 (L ⊗ Z ˆ s0¯(L ⊗ Z s¯ Z

Z

this condition is independent of the choice of α ˆ s¯.)

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Let us construct [g0−1 ] as follows. (When S is locally noetherian, the construction can be much simpler using π1 (S, s¯)-modules, as in [62, Sec. 6.4.3]. But we spell out the details in a more general context, because later we will encounter some analogue of this construction, for which the techniques of π1 (S, s¯)-modules do not work.) Let n0 , m0 , and N0 be positive integers prime to p such that n0 |m0 , U p (n0 ) ⊂ p H , U p (m0 ) ⊂ H0,p , and ˆ p ⊂ N −1 g0 (L ⊗ Z ˆ p ) ⊂ n−1 N −1 g0 (L ⊗ Z ˆ p ) ⊂ m−1 (L ⊗ Z ˆ p ). L⊗Z 0 0 0 0 Z

Z

Z

(3.3.2.2)

Z

(Such integers always exist.) Then we have in particular an O-submodule ˆ p ) ⊂ (m−1 (L ⊗ Z ˆ p ))/(L ⊗ Z ˆ p) ∼ ˆ p ))/(L ⊗ Z (N0−1 g0 (L ⊗ Z = L/m0 L. 0 Z

Z

Z

(3.3.2.3)

Z

By (3.3.2.2), we have U p (m0 ) ⊂ g0 U p (n0 )g0−1 , and the inclusion H0,p ,→ g0 Hp g0−1 0 induces a homomorphism Hm = H0,p /U p (m0 ) → Hn0 = Hp /U p (n0 ). 0 0 0 , which is an By definition of αH0,p = {αHm }m0 , over the scheme S˜ = αHm 0 0 0 Hm0 -torsor (finite ´etale) over S, there is a tautological principal level-m0 structure ∼ (αm0 , νm0 ), where αm0 : (L/m0 L)S˜ → AS˜ [m0 ] is an O-equivariant isomorphism ∼ of (´etale) group schemes over S, and where νm0 : ((Z/m0 Z)(1))S˜ → µm0 ,S˜ is an ˜ satisfying the usual symplectic and liftability isomorphism of group schemes over S, conditions defining a (principal) level structure. ˆ p ))/(L ⊗ Z ˆ p ) (see (3.3.2.3)) Let KS˜ be the schematic image of (N0−1 g0 (L ⊗ Z Z

Z

under αm0 , which is an O-invariant subgroup scheme of AS˜ [m0 ], which is finite 0 ´etale over Sm0 . Since the tautological action of Hm on S˜ → S is compatible with 0 the isomorphism αm0 , we can descend KS˜ to a finite ´etale subgroup scheme K of A[m0 ], and define an isogeny A → A0 := A/K.

(3.3.2.4)

Z× (p) -isogeny

Then we define the [g0−1 ] : A → A0 (see [62, Def. 1.3.1.17]) to be the composition of (3.3.2.4) with [N0 ]−1 , and denote the isogeny (3.3.2.4) as [N0 g0−1 ]. (Note that [N0 g0−1 ] = N0 [g0−1 ], and [g0−1 ] = [N0 ]−1 if g0 = N0 Id.) The Z× (p) -isogeny −1 [g0 ] is independent of the choice of m0 and N0 . (When S is locally noetherian, we can reformulate the definition of level structures away from p using the language of π1 (S, s¯)-modules, for a geometric point s¯ on each connected component of S. Then we can construct [g0−1 ] as in [62, Sec. 6.4.3].) 2 ˆp ˆp Let r0 ∈ Z× (p),>0 be such that r0 ν(g0 )Z = Z . Note that N0 r0 ∈ Z>0 because −1 ˆ p ⊂ N g0 (L ⊗ Z ˆ p ). Then we define a Q× -isogeny λ0 : A0 → A0,∨ by setting L⊗Z Z

0

Z

∨

∨

λ := r0 [g0 ] ◦ λ ◦ [g0 ] = (N02 r0 )([N0 g0−1 ] )−1 ◦ λ ◦ [N0 g0−1 ]−1 . This Q× -isogeny λ0 is a Q× -polarization by [62, Cor. 1.3.2.18 and 1.3.2.21]. It is an isogeny (and hence a polarization) because we have the inclusions 0

ˆ p ⊂ N −1 g0 (L ⊗ Z ˆ p ) ⊂ N −1 g0 (L# ⊗ Z ˆ p) L⊗Z 0 0 Z

=

Z −2 −1 −1 ˆ p ))# N0 r0 (N0 g0 (L ⊗ Z Z

⊂

Z −2 −1 N0 r0 (L#

ˆ p ), ⊗Z Z

(3.3.2.5)

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which by the descent construction of [N0 g0−1 ] corresponds to a factorization of N02 r0 λ : A → A∨ as a composition of isogenies A

[N0 g0−1 ]

→

∨

λ0

A0 → A0,∨

[N0 g0−1 ]

→

A∨ , ∼

in which [N0 g0−1 ] induces an isomorphism ker(N02 r0 λ)[pr ] → ker(λ0 )[pr ] for each integer r ≥ 1. The above constructions of [g0−1 ] : A → A0 and λ0 are both compatible with the actions of O. Hence, we obtain an induced O-structure i0 : O ,→ EndS (A0 ). By construction, and by (3.3.2.2), the isomorphism αm0 ◦ g0 induces an isomorphism α0 : L/n0 L ∼ = (n−1 N −1 L)/(N −1 L) n0

0

0

0

αm0

g0 ∼

∼ −1 −1 0 ˆp ˆp → (n−1 ˜ [n0 ], 0 N0 g0 (L ⊗ Z ))/(N0 g0 (L ⊗ Z )) → AS Z

Z

together with an isomorphism ∼

νn0 0 : ((Z/n0 Z)(1))Sn0 → µn0 ,S˜ 0 induced by restricting νm0 ◦ (r0 ν(g0 )). The homomorphism Hm → Hn0 induced 0 −1 0,p p by H ,→ g0 H g0 (i.e., conjugation by g0 ) induces a well-defined Hn0 -orbit of 0 0 0 ˜ which descends to a naive level structure α0 (αn0 0 , νn0 0 ) over S, Hn0 of (A , λ , i ) of type (L/n0 L, h · , · i) and level Hn0 as in Definition 3.3.1.3. Since we can repeat the above procedure for each integer n0 ≥ 1 such that p - n0 and U p (n0 ) ⊂ Hp , we 0 0 ˆ p , h · , · i) obtain a level-Hp structure αH }n0 of (A0 , λ0 , i0 ) of type (L ⊗ Z p = {αH n 0

Z

as in Definition 3.3.1.4. 0 This finishes the construction of the Hecke twist (A0 , λ0 , i0 , αH p ). Since Hecke twists (in this proposition) are constructed using prime-to-p isogenies (and their formal inverses), which are uniquely determined by their behaviors on geometric fibers of torsion subgroup schemes of abelian schemes over S of ranks prime to p (which are finite ´etale group schemes over S), the last statement of the proposition follows from the characterizing conditions preceding it. 3.3.3

Ordinary Level Structures at p

Let us fix a choice of a filtration D as in Lemma 3.2.2.1. For each integer r ≥ 0, the perfect dualities in Lemma 3.2.2.4 induce canonical isomorphisms ∼ ∼ (Gr0D,pr )mult , Gm,S ) and (Gr0D# ,pr )mult = HomS (Gr−1 = HomS (Gr−1 D,pr , Gm,S ) (cf. S S D# ,pr 0 0 0 Definition 3.1.1.6), and a morphism φD,pr : GrD,pr → GrD# ,pr . Let Hp ⊂ G(Zp ) be an open compact subgroup of standard form as in Definition bal r (p ) ⊂ Hp ⊂ Up,0 (pr ) for r = depthD (Hp ), where Up,0 (pr ) and 3.2.2.9, so that Up,1 bal r bal r Up,1 (p ) are as in Definition 3.2.2.8. Let Hpord := Hp /Up,1 (p ), which is a subgroup r ord r of MD (Z/p Z). Definition 3.3.3.1. Let S be a scheme over Spec(Z). Let A be an abelian scheme over S, with a polarization λ : A → A∨ and an O-endomorphism structure i : O ,→

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0 0 EndS (A) as in [62, Def. 1.3.3.1]. Let L ⊗ Zp , D, Hp , Hpord r , and φD,pr : GrD,pr → Z

Gr0D# ,pr be as above. A naive principal ordinary level-pr structure of (A, λ, i) of type (L/pr L, h · , · i, Dpr ), or rather of type φ0D,pr : Gr0D,pr → Gr0D# ,pr , is a triple ord,0 , νpord , αpord,#,0 αpord r r ), r = (αpr

where the first two entries are O-equivariant homomorphisms : (Gr0D,pr )mult → A[pr ] αpord,0 r S and αpord,#,0 : (Gr0D# ,pr )mult → A∨ [pr ] r S that are closed immersions, and where the third entry νpord is a section of r × (Z/pr Z)S ∼ = IsomS (((Z/pr Z)(1))S , ((Z/pr Z)(1))S ) ∼ = Isom (µ r , µ r ) S

p ,S

p ,S

which are symplectic in the sense that the two homomorphisms are compatible with the homomorphisms 0 mult → (Gr0D# ,pr )mult νpord : (Gr0D,pr )mult r ◦ (φD,pr )S S S

and λ : A[pr ] → A∨ [pr ], namely that the following diagram αord,0 pr

(Gr0D,pr )mult S 0 mult νpord r ◦(φD,pr )S

/ A[pr ] λ



(Gr0D# ,pr )mult S

αord,#,0 pr

 / A∨ [pr ]

is commutative, or equivalently that the following diagram (Gr0D,pr )mult S (φ0D,pr )mult S

αord,0 pr

λ



(Gr0D# ,pr )mult S

/ A[pr ]

ord,#,0 νpord r ◦αpr

 / A∨ [pr ]

is commutative, and that the schematic images of the two homomorphisms αpord,0 r and αpord,#,0 are annihilators of each other under the canonical pairing eA[pr ] : r A[pr ] × A∨ [pr ] → µpr ,S . We shall denote νpord by ν(αpord r r ). We shall also denote ord,0 ord,#,0 the schematic image of αpr (resp. αpr ), which is a closed subgroup scheme of A (resp. A∨ ), by image(αpord,0 ) (resp. image(αpord,#,0 )). r r

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= Definition 3.3.3.2. We say a naive principal ordinary level-pr structure αpord r r (αpord,0 , αpord,#,0 , νpord r r r ) of (A, λ, i) of type (L/p L, h · , · i, Dpr ) in Definition 3.3.3.1 is a principal ordinary level-pr structure of type (L ⊗ Zp , h · , · i, D) if it satisfies Z

the following symplectic-liftability condition: There exists (noncanonically) a tower (Spr0  S)r0 ≥r of quasi-finite ´etale coverings such that we have the following: (1) Spr = S. (2) For each r00 such that r0 ≥ r00 ≥ r, there is a quasi-finite ´etale covering Spr0  Spr00 whose composition with Spr00 → S is the quasi-finite ´etale covering Spr0 → S. 0 (3) There is a naive principal ordinary level structure αpord r 0 ,S 0 of (A, λ, i) × Spr S

pr

r0

of type (L/p L, h · , · i, Dp ) over each Sp . (4) For each r00 such that r0 ≥ r00 ≥ r, the pullback of αpord r 00 ,S r0

r0

pr

00

reduction modulo pr of αpord r 0 ,S

pr

0

00

to Spr0 is the

.

Definition 3.3.3.3. Let S and (A, λ, i) be as in Definition 3.3.3.1, and let Hpord r be as above. A naive ordinary level-Hpord structure of (A, λ, i) of type r ord etale-locally-defined (naive) princi(L/pr L, h · , · i, Dpr ) is an Hpord r -orbit α ord of ´ H r p

pal ordinary level structures of type (L ⊗ Zp , h · , · i, D) and level pr , namely a (finite Z

ord ´etale) subscheme αH etale scheme ord of the quasi-finite ´ pr

× 0 mult r HomS (GrD,pr )S , A[p ] × HomS (Gr0D# ,pr )mult , A∨ [pr ] × (Z/pr Z)S S S

S

over S that becomes the (scheme-theoretic) Hpord r -orbit of some naive principal ordinary level structures of type (L ⊗ Zp , h · , · i, D) and level pr (see Definition 3.3.3.2) Z

after a finite ´etale surjective base change in S. ord,0 ord,#,0 ord ord We shall denote by αH , resp. νH ord = ν(αH ord )) the schematic ord (resp. αH ord pr pr pr pr

0 0 ord mult r mult image of αH in Hom (Gr ) , A[p ] (resp. Hom (Gr , A∨ [pr ] , r ord D,p S S S D# ,pr )S r p

×

resp. (Z/pr Z)S ). Since the action of Hpord does not modify the schematic image of αpord,0 (resp. r r ord,#,0 ) in an orbit, it makes sense (by ´etale descent, and by abuse of language) to αpr ord,0 ord,#,0 consider the common schematic image of αH ), which is a closed ord (resp. αH ord pr

∨

subgroup scheme of A (resp. A ), by

ord,0 image(αH ord ) pr

pr

ord,#,0 (resp. image(αH )). ord pr

Definition 3.3.3.4. Let S and (A, λ, i) be as in Definition 3.3.3.1, and let Hpord be r as above. Consider the open compact subgroup can. 
−1 ord can. ∼ ord r r bal r Hpord := Mord (Hpr ) (3.3.3.5) D (Zp ) → MD (Z/p Z) → Up,0 (p )/Up,1 (p ) ord ord r ∼ bal r of Mord (p ) = Hpord = Hp /Up,1 (p ). Let S and (A, λ, i) r D (Zp ), so that Hp /U be as in Definition 3.3.3.1. For each integer r0 such that r0 ≥ r, set Hpord := r0

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0

0

bal r Hpord /U ord (pr ), which is then viewed as a subgroup of Up,0 (pr )/Up,1 (p ). Then ord an ordinary level-Hp structure αHp of (A, λ, i) of type (L ⊗ Zp , h · , · i, D) is a Z

ord r naive ordinary level-Hpord structure αH r ord of (A, λ, i) of type (L/p L, h · , · i, Dpr ) that r p

satisfies the following symplectic-liftability condition: There exists (noncanonically) a tower (Spr0  S)r0 ≥r of quasi-finite ´etale coverings such that we have the following: (1) Spr = S. (2) For each r00 such that r0 ≥ r00 ≥ r, there is a quasi-finite ´etale covering Spr0  Spr00 whose composition with Spr00 → S is the quasi-finite ´etale covering Spr0 → S. ord (3) There is a naive ordinary level-Hpord structure αH ord of (A, λ, i) of type r0 0 pr

r0

(L/p L, h · , · i, Dpr0 ) over each Spr0 . ord (4) For each r00 such that r0 ≥ r00 ≥ r, the pullback of αH ord ,S pr

00

ord is the reduction modulo pr of αH ord ,S pr

ord the schematic image of αH ord ,S pr

0

pr

0

0

pr

0

00

pr

to Spr0

00

ord , in the sense that αH ord ,S pr

00

00 pr

is

under the canonical (quasi-finite ´etale)

morphism HomS

pr


r0 (Gr0D,pr0 )mult S r0 , A[p ]

0

p

× HomS

0 pr

Spr0

→ HomS

pr

00




0

×

× (Z/pr Z)S

pr

Spr0

0


r 00 (Gr0D,pr00 )mult S r00 , A[p ] p

∨ r 00 (Gr0D# ,pr00 )mult Spr00 , A [p ]

00

00 pr

×

× (Z/pr Z)S




× HomS

Spr00

∨ r0 (Gr0D# ,pr0 )mult Spr0 , A [p ]

pr

Spr00

00

00

defined by restriction to the pr -torsion in the sources. ord,0 ord,#,0 ord We shall denote αH (resp. αH , resp. νH ord ord ord r pr

pr

ord ν(αH resp. ord ), r

=

p

p

ord,0 ord,#,0 ord,0 ord,#,0 ord ord image(αH )) by αH (resp. αH , resp. νH = ν(αH ), ord ), resp. image(αH ord p p p p pr

pr

ord,0 ord,#,0 resp. image(αH ), resp. image(αH )). p p

Remark 3.3.3.6. Even when r = 0, the existence of a principal ordinary level-pr structure of (A, λ, i) of type (L ⊗ Zp , h · , · i, D) in Definition 3.3.3.2 forces A to be Z

ordinary (see Definition 3.1.1.2). The same is true for the existence of an ordinary level-Hp structure of (A, λ, i) of type (L ⊗ Zp , h · , · i, D) in Definition 3.3.3.4. Z

Lemma 3.3.3.7. Let S and (A, λ, i) be as in Definition 3.3.3.1. Suppose moreover i that S is a scheme over Spec(Fp ). For each integer i ≥ 0, let A(p ) := A × S S,FiS

(resp. A

∨,(pi )

:= A

∨

∨

× S) denote the pullback of A (resp. A ) under the i-th S,FiS

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iteration FiS : S → S of the absolute Frobenius morphism FS : S → S, and let i i (i) (i) FA/S : A → A(p ) (resp. FA∨ /S : A∨ → A∨,(p ) ) denote the relative Frobenius i

i

morphism induced by the universal property of A(p ) (resp. A∨,(p ) ) as a fiber product. i i i Then A∨,(p ) is the dual abelian scheme of A(p ) , with polarization λ(p ) := λ × S. S,FiS

Let

ord αH p

=

ord,0 ord,#,0 ord (αH , αH , νH ) p p p

be any ordinary level-Hp structure of (A, λ, i)

ord,0 i of type (L ⊗ Zp , h · , · i, D) as in Definition 3.3.3.4. Then we have image(αH ord )[p ] = (i) ker(FA/S )

lar,

pr

Z

(i) ord,#,0 and image(αH )[pi ] = ker(FA∨ /S ), for each 0 ≤ i ≤ r. (In particuord pr (i) (i) )[pi ] = ker(FA∨ /S ) )[pi ] = ker(FA/S ) and image(αpord,#,0 we have image(αpord,0 r r , νpord , αpord,#,0 = (αpord,0 a principal ordinary level-pr structure αpord r r r ) of type r

for (L ⊗ Zp , h · , · i, D) as in Definition 3.3.3.2.) Z

Proof. The first paragraph is nothing but definitions. For the second paragraph, since it is about comparison of finite flat group schemes of finite presentation over S, we may reduce to the case that S is Henselian local, and hence to the case that S is the spectrum of an algebraically closed field of characteristic p > 0. Then ord,0 i the assertions follow from the fact that, given their ranks, both image(αH ord )[p ] pr

(resp.

ord,#,0 image(αH )[pi ]) ord pr

and

(i) ker(FA/S )

(resp.

(i) ker(FA∨ /S ))

are the unique maximal

subgroup scheme of multiplicative type of the ordinary abelian variety A[pi ] (resp. A∨ [pi ]) over S (see Remark 3.3.3.6). Corollary 3.3.3.8. In Definitions 3.3.3.2 and 3.3.3.4, if S is a scheme over Spec(Fp ), then we may assume that the tower (Spr0  S)r0 ≥r of quasi-finite ´etale coverings is finite ´ etale. ord Proof. In this case, for each r0 ≥ r, we know before constructing αH = ord pr

ord,0 ord,#,0 ord (αH , νH ord ) ord , αH ord r0 r0 r0 p

that we must have

p

p

ord,0 image(αH ord ) r0

=

(r 0 ) ker(FA/S )

0

and

p

(i)

ord,#,0 ord image(αH ) = ker(FA∨ /S ), and hence the desired αH ord (tautologically) exists ord pr

0

pr

0

over some open and closed subscheme Spr0 of the finite ´etale scheme (r 0 )
IsomS (Gr0D,pr0 )mult , ker(FA/S ) S


× 0 (r 0 ) × IsomS (Gr0D# ,pr0 )mult , ker(FA∨ /S ) × (Z/pr Z)S , S S

S

which is finite ´etale over S (and we can compatibly form a tower of such subschemes such that the necessary compatibility conditions between Spr0 and Spr00 , when r00 ≥ r0 ≥ r, are satisfied). Lemma 3.3.3.9. Let S and (A, λ, i) be as in Definition 3.3.3.1. Let Hp and Hpord r ord be as above, and let αH be an ordinary level-H structure of (A, λ, i) of type p p

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(L ⊗ Zp , h · , · i, D) as Definition 3.3.3.4. Let s¯ be any geometric point of S. Then Z

there exists a triple , νˆs¯ord ) ,α ˆ sord,#,0 α ˆ sord = (ˆ αsord,0 ¯ ¯ ¯

(3.3.3.10)

where the first two entries are injective O-equivariant homomorphisms : (Gr0D )mult α ˆ sord,0 → As¯[p∞ ] ¯ s¯ and ∞ : (Gr0D# )mult α ˆ sord,#,0 → A∨ ¯ s¯ s¯ [p ],

(see Definition 3.1.1.7) and where the third entry νˆs¯ord is a section of ∼ ∼ (Z× s , µp∞ ,¯ s ), p )s¯ = Isoms¯(((Zp )(1))s¯, ((Zp )(1))s¯) = Isoms¯(µp∞ ,¯ satisfying the symplectic condition as in Definition 3.3.3.1 which we shall spell out ord to the pr -torsion in ˆ sord below, such that the Hpord r -orbit of the restriction αpr ,¯ ¯ s of α ord the sources coincides with the pullback of αH ord from S to s¯. We say that this α ˆ sord is ¯ r p

ord is unique (i.e., independent of α ˆ sord αsord at s¯. The Hpord -orbit [ˆ a lifting of αH ¯ ¯ ]Hord p p ord of the choice of α ˆ s¯ ). = (ˆ αsord,0 The symplectic condition for α ˆ sord ,α ˆ sord,#,0 , νˆs¯ord ) is that the two homo¯ ¯ ¯ ord,0 ord,#,0 morphisms α ˆ s¯ and α ˆ s¯ are compatible with the homomorphisms

→ (Gr0D# )mult νˆs¯ord ◦ (φ0D )mult : (Gr0D )mult s¯ s¯ s¯ and ∞ λs¯ : As¯[p∞ ] → A∨ s¯ [p ],

namely that the following diagram (Gr0D )mult s¯ ν ˆs¯ord ◦(φ0D )mult s ¯

α ˆ ord,0 s ¯

/ As¯[p∞ ] λs¯



(Gr0D# )mult s¯

α ˆ ord,#,0 s ¯

 ∞ / A∨ s¯ [p ]

is commutative, or equivalently that the following diagram (Gr0D )mult s¯ (φ0D )mult s ¯

α ˆ ord,0 s ¯

λs¯



(Gr0D# )mult s¯

/ As¯[p∞ ]

ν ˆs¯ord ◦α ˆ ord,#,0 s ¯

 ∞ / A∨ s¯ [p ]

is commutative, and that the images of the two homomorphisms α ˆ sord,0 and ¯ ord,#,0 α ˆ s¯ are annihilators of each other under the canonical pairing eAs¯[p∞ ] : ∞ As¯[p∞ ] × A∨ s. s¯ [p ] → µp∞ ,¯

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Proof. As in the proof of Lemma 3.3.1.6, the pullback of the compatible tower ord (αH ¯ of S (with a ord )r 0 ≥r defined over (Spr 0  S)r 0 ≥r to the geometric point s pr

0

compatible choice of liftings to each Spr0 → S) allows us to choose a compatible ord 0 tower (αpord r 0 )r ≥r of principal ordinary level structures, such that the Hpr 0 -orbit ord 0 of each αpord ≥ r, which is equivalent to the desired α ˆ sord in r 0 is αH ord for each r ¯ pr

0

(3.3.3.10). The symplectic condition for α ˆ sord follows from that of αpord r 0 over each ¯ Spr0 → S. 3.3.4

Hecke Twists Defined by Ordinary Level Structures at p

ord For each element gp of Pord = (gp,0 , gp,−1 ) the D (Qp ) ⊂ G(Qp ), we denote by gp 0 −1 −1 ∼ action of gp on the graded pieces GrD = GrD ⊕ GrD , and by gp,#,0 = t gp,−1 , where 0 t gp,−1 is the induced action on GrD# by transposition (with respect to the perfect −1 gp,0 pairing Gr0D# × Gr−1 D → Zp (1); see Lemma 3.2.2.4). Note that gp,#,0 and ν(gp ) 0 induce the same element in GLO ⊗ Qp (DQp ). Z

Although a more general theory might exist, we shall construct ordinary Hecke twists defined by ordinary level structures at p under a list of conditions. Nevertheless, such ordinary Hecke twists are sufficient for the applications we know. Suppose we have two open compact subgroups Hp and Hp0 of G(Qp ) of standard form as in Definition 3.2.2.9 such that r = depthD (Hp ) ≤ r0 = depthD (Hp0 ). Suppose that gp ∈ Pord D (Qp ) satisfies the following conditions: (1) Hp0 ⊂ gp Hp gp−1 . (2) There exist integers r0 and r0 ≥ r00 ≥ r such that Gr0D ⊂ p−r0 gp,0 (Gr0D ) ⊂ p−r

00

−r0

0

gp,0 (Gr0D ) ⊂ p−r Gr0D

(3.3.4.1)

and 00

Gr0D# ⊂ pr0 gp,#,0 (Gr0D# ) ⊂ p−r+r0 gp,#,0 (Gr0D# ) ⊂ p−r Gr0D#

(3.3.4.2)

Note that (3.3.4.2) is equivalent (by duality) to 00

−1 r−r0 −r0 pr Gr−1 gp,−1 (Gr−1 gp,−1 (Gr−1 D ⊂p D )⊂p D ) ⊂ GrD .

The relations (3.3.4.1) and (3.3.4.2) define O-submodules 0 ∼ Gr0 (p−r0 gp,0 (Gr0 ))/ Gr0 ⊂ (p−r Gr0 )/ Gr0 = D

D

D

D

D,pr0

(3.3.4.3)

and 00 (pr0 gp,#,0 (Gr0D# ))/ Gr0D# ⊂ (p−r Gr0D# )/ Gr0D# ∼ (3.3.4.4) = Gr0D# ,pr00 respectively. Since these conditions are complicated, we include some basic examples:

ord 0 Example 3.3.4.5 (elements in Pord D (Zp )). Suppose gp ∈ PD (Zp ) and Hp ⊂ 0 −1 gp Hp gp . Then the remaining conditions above are automatic, because gp,0 (GrD ) = Gr0D , gp,#,0 (Gr0D# ) = Gr0D# , and r0 ≥ r, and because we can take r0 = 0 and take any r0 ≥ r00 ≥ r for (3.3.4.1) and (3.3.4.4) to hold. (We will continue in Example 3.3.4.18 below.)

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Example 3.3.4.6 (multiplication by powers of p). Suppose gp ∈ Pord D (Qp ) ord r0 r 0 , p 0 Id by g = (g , g ) = (p Id acts on GrDQp ∼ = Gr0DQp ⊕ Gr−1 p,0 p,−1 Gr p DQp Gr−1 ) D D

for some integer r0 . Suppose Hp0 ⊂ gp Hp gp−1 = Hp . Then gp,0 (Gr0D ) = pr0 Gr0D , gp,#,0 (Gr0D# ) = p−r0 Gr0D# , r0 ≥ r, and the remaining conditions are automatic, because we can take r0 as it is and take any r0 ≥ r00 ≥ r for (3.3.4.1) and (3.3.4.4) to hold. (We will continue in Example 3.3.4.19 below.) ∼ Example 3.3.4.7 (Up operator). Suppose gp ∈ Pord D (Qp ) acts on GrDQp = −1 0 0 ord −1 = (gp,0 , gp,−1 ) = (p IdGr0D , IdGr−1 ). Suppose Hp ⊂ GrDQp ⊕ GrDQp by gp D

gp Hp gp−1 . Then gp,0 (Gr0D ) = p−1 Gr0D , gp,#,0 (Gr0D# ) = Gr0D# , and r0 > r. Then the remaining conditions are automatic, because we can take r0 = 0 and take any r0 > r00 ≥ r for (3.3.4.1) and (3.3.4.4) to hold. (We will continue in Example 3.3.4.20 below.) Example 3.3.4.8 (generalized Up operator). Ignoring the O-module structures, suppose L = Z⊕ 2n for some integer n ≥ 0, with D defined by D0 = −1 0 ∼ Zp⊕ n ⊂ L ⊗ Zp . Suppose gp ∈ Pord D (Qp ) acts on GrDQp = GrDQp ⊕ GrDQp by Z

gpord = (gp,0 , gp,−1 ) = (diag(p−r1 , p−r2 , . . . , p−rn ), diag(p−rn+1 , p−rn+2 , . . . , p−r2n )) for some integers r1 ≥ r2 ≥ · · · ≥ rn . Since ri + rn+i (satisfying p−ri −rn+i = ν(gp )) is a constant independent of 1 ≤ i ≤ n, this forces rn+1 ≤ rn+2 ≤ · · · ≤ r2n . Suppose Hp0 ⊂ gp Hp gp−1 . Suppose rn ≥ r2n and r0 − r ≥ r1 − rn+1 . Then the remaining conditions hold if we take any r0 such that rn ≥ −r0 ≥ r2n and take any r00 such that r0 − (r1 + r0 ) ≥ r00 ≥ r − (rn+1 + r0 ) for (3.3.4.1) and (3.3.4.4) to hold. (This rather elaborate example includes both Examples 3.3.4.6 and 3.3.4.7 as special cases. However, we will not continue this example as in Examples 3.3.4.19 and 3.3.4.20 below.) 0 ord Let S and (A, λ, i) be as in Definition 3.3.3.1, and let αH 0 be an ordinary level-Hp p

structure of (A, λ, i) of type (L ⊗ Zp , h · , · i, D) as in Definition 3.3.3.4, with Hp0,ord Z

00 (resp. Hp0,ord ≥ r0 ) defined by Hp0 as Hpord (resp. Hpord r 00 , for each r 00 , for each integer r 00 integer r ≥ r) is defined by Hp .

Proposition 3.3.4.9. With assumptions as above, there exists a tuple ord,0 (A0 , λ0 , i0 , αH ) (over S, up to isomorphism), called the ordinary Hecke twist p ord of (A, λ, i, αH0p ) by gp , equipped with a Q× -isogeny [gp−1 ]ord : A → A0 (whose formal inverse we denote by [gp ]ord : A0 → A) satisfying the following characterizing conditions: −1 ord (1) [gp−1 ]ord is the composition ([p−r0 gp,−1 ]ord )−1 ◦ [pr0 gp,0 ] ◦ [pr0 ]−1 , where r0 r0 [p ] : A → A is the multiplication by p on A when r0 ≥ 0, or the formal inverse of the multiplication by p−r0 on A when r0 < 0; and where −1 ord [pr0 gp,0 ] : A → A00 and [p−r0 gp,−1 ]ord : A0 → A00 are isogenies of p-power

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degrees (whose formal inverses we denote by [p−r0 gp,0 ]ord : A00 → A and −1 ord [pr0 gp,−1 ] : A00 → A0 , respectively). ∨ (2) λ0 : A0 → A0,∨ is a polarization defined by λ0 = rp ([gp ]ord ) ◦ λ ◦ [gp ]ord (as positive Q× -isogenies), where rp is the unique power of p such that rp ν(gp )Zp = Zp . (3) i0 : O → EndS (A0 ) is an O-structures of (A0 , λ0 ) making [gp ]ord an O-equivariant Q× -isogeny. ord,0 (4) αH is an ordinary level-Hp structure of (A0 , λ0 , i0 ) of type p (L ⊗ Zp , h · , · i, D). Z

, νˆs¯ord ) ,α ˆ sord,#,0 (5) At each geometric point s¯, there exist a lifting α ˆ sord = (ˆ αsord,0 ¯ ¯ ¯ ord,0 ord,0,0 ord,#,0,0 ord,0 ord,0 ord (resp. α ˆ s¯ = (ˆ αs¯ ,α ˆ s¯ , νˆs¯ )) of αH0p (resp. αHp ) as in Lemma ord 3.3.3.9, such that α ˆ sord,0 is compatible with α ˆ sord under the Q× -isogeny ¯ ¯ ◦ gp r0 r0 −1 ord [p ] : A → A and the isogenies [p gp,0 ] : A → A00 and [p−r0 gp,−1 ]ord : A0 → A00 , in the following sense: −1 ord (a) ker([pr0 gp,0 ] )s¯ is the schematic image of the submodule

((p−r0 gp,0 (Gr0D ))/ Gr0D )mult s¯ → (see (3.3.4.3)) under α ˆ sord,0 : (Gr0D )mult ⊂ (Gr0D )mult of (Gr0D,pr0 )mult ¯ s¯ s¯ s¯ ∞ As¯[p ]. Then α ˆ sord,−1 : As¯[p∞ ] → Gr−1 ¯ D ⊗ (Qp /Zp ) Zp

∞ → A∨ (which is the Serre dual of α ˆ sord,#,0 : (Gr0D# )mult ¯ s¯ s¯ [p ]) satisfies

ker(α ˆ sord,−1 ) = image(ˆ αsord,0 ), ¯ ¯ and induces an injection −1 ord α ˆ sord,0,00 := [pr0 gp,0 ]s¯ ◦ α ˆ sord,0 ◦ (p−r0 gp,0 ) : (Gr0D )mult → A00s¯ [p∞ ] ¯ ¯ s¯

and a surjection α ˆ sord,−1,00 : A00s¯ [p∞ ]  Gr−1 ¯ D ⊗ (Qp /Zp ) Zp

satisfying ker(α ˆ sord,−1,00 ) = image(ˆ αsord,0,00 ). ¯ ¯ Consequently, ∞ α ˆ sord,#,0 : (Gr0D# )mult → A∨ ¯ s¯ s¯ [p ]

is liftable to an injection ∞ α ˆ sord,#,0,00 : (Gr0D# )mult → As00,∨ ¯ ¯ [p ]. s¯

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(b) The isogeny [p−r0 gp,−1 ]ord : A0 → A00 is dual to an isogeny −1 −1 [p−r0 gp,#,0 ]ord : A00,∨ → A0,∨ , and ker([p−r0 gp,#,0 ]ord )s¯ is the schematic image of the submodule ((pr0 gp,#,0 (Gr0D# ))/ Gr0D# )mult s¯ of (Gr0D# ,pr00 )mult ⊂ (Gr0D# )mult . Then (see (3.3.4.4)) under α ˆ sord,#,0,00 ¯ s¯ s¯ ∞ : (Gr0D# )mult α ˆ sord,#,0,0 → A0,∨ ¯ s¯ [p ] s¯

agrees with the composition −1 [p−r0 gp,#,0 ◦ (pr0 gp,#,0 ). ]s¯ ◦ α ˆ sord,#,0,00 ¯

(c) The kernel of the dual surjection −1 ∞ : A00,∨ α ˆ sord,#,−1,00 s¯ [p ] → GrD# ⊗ (Qp /Zp ) ¯ Zp

of α ˆ sord,0,00 is the schematic image of α ˆ sord,#,0,00 , which ¯ ¯ −1 ker([pr0 gp,#,0 ])s¯. Hence, α ˆ sord,#,−1,00 induces a surjection ¯

contains

−1 ∞ A0,∨ s¯ [p ] → GrD# ⊗ (Qp /Zp ), Zp

which is dual to an injection (Gr0D )mult → A0s¯[p∞ ] s¯ lifting α ˆ sord,0,00 . This injection coincides with α ˆ sord,0,0 . ¯ ¯ ∼ (d) νˆs¯ord,0 : µp∞ ,¯s → µp∞ ,¯s is induced by νˆs¯ord ◦ (rp ν(gp )), where rp is as in (2) above. By abuse of language, we say that α ˆ sord,0 = [gp−1 ]ord ◦α ˆ sord ◦ gpord . In ¯ s¯ ¯ ord ord ord ord −1 0,ord ˆ s¯ ]H0,ord determines a (gp Hp (gp ) )-orbit this case, the Hp -orbit [α p 0,ord ord ⊂ gpord Hpord (gpord )−1 , and hence induces [α ˆ s¯ ]gpord Hord ord −1 because Hp p (gp ) αsord,0 an Hpord -orbit [ˆ ]Hord . ¯ p If gp = g1,p g2,p , where g1,p and g2,p are elements of Pord D (Qp ), each having a setup similar to that of gp , then the ordinary Hecke twist by gp can be constructed in two −1 ord steps using ordinary Hecke twists by g1,p and g2,p , such that [gp−1 ]ord = [g2,p ] ◦ −1 ord ord ord ord [g1,p ] (or, equivalently, [gp ] = [g1,p ] ◦ [g2,p ] ). Proof. Since ordinary Hecke twists (in this proposition) are constructed using p-power isogenies (and their formal inverses), which are uniquely determined by their behaviors on geometric fibers of p-power torsion subgroup schemes of (ordinary) abelian schemes over S (which are finite flat group schemes of ´etalemultiplicative type over S), the last statement of the proposition follows from the characterizing conditions preceding it. Therefore, we can construct the desired ordinary Hecke twist by gp in two steps using the ordinary Hecke twists by p−r0 gp and by pr0 Id (the latter being given by [p−r0 ] = [pr0 ]−1 , which is defined for all r0 ∈ Z

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ord and does not alter the additional structures λ, i, and αH 0 ; see Example 3.3.4.19 p −1 ord below). Hence, our main tasks are to construct [pr0 gp,0 ] and [p−r0 gp,−1 ]ord , so r0 −1 ord −r0 ord −1 r0 −1 ord that [p gp ] = ([p gp,−1 ] ) ◦ [p gp,0 ] , and to construct the additional ord,0 ord,0 structures λ0 , i0 , and αH (the main focus being on αH ). p p ord ord By definition of α 0 , over the scheme S˜ = α 0,ord , which is an Hp0,ord -torsor r Hp

H

pr

0

0

(finite ´etale) over S, there is a tautological principal ordinary level-pr structure ord,0 αpord , αpord,#,0 , νpord r 0 = (αpr 0 r 0 ) as in Definition 3.3.3.2, where r0 0

αpord,0 : (Gr0D,pr0 )mult → AS˜ [pr ] ˜ r0 S and 0

r αpord,#,0 : (Gr0D# ,pr0 )mult → A∨ ˜ ˜ [p ] r0 S S

are closed immersions, and where the third entry νpord ˜ → µpr0 ,S ˜ r 0 : µpr 0 ,S ×

0

is a section of (Z/pr Z)S˜ , satisfying the usual symplectic and liftability conditions defining a (principal) ordinary level structure. The schematic image of the submodule ((p−r0 gp,0 (Gr0D ))/ Gr0D )mult ˜ S of (Gr0D,pr0 )mult (see (3.3.4.3)) under the closed immersion αpord,0 defines a subgroup r0 ˜ S 0

, scheme K0,S˜ of AS˜ [pr ]. Since K0,S˜ is isomorphic to ((p−r0 gp,0 (Gr0D ))/ Gr0D )mult ˜ S ˜ Since the it is finite flat, of finite presentation, and of multiplicative type over S. on S˜ → S is compatible with the homomorphism αpord,0 , tautological action of Hp0,ord r r0 0

we can descend K0,S˜ to a subgroup scheme K0 of A[pr ], which is also finite flat, of finite presentation, and of multiplicative type over S. (This is a question of descending the finite ´etale character group scheme of K0,S˜ .) Hence, we can define the isogeny −1 ord [pr0 gp,0 ] : A → A00 := A/K0 ,

(3.3.4.10)

0

inducing an embedding A[pr ]/K0 ,→ A00 . By (3.3.4.1), we know that 00

0

A00 [pr ] ⊂ A[pr ]/K0

(3.3.4.11)

(as a closed subgroup scheme). ˜ the homomorphism αord,0 Over S, induces a homomorphism pr 0   ord,#,0 r0 −1 ord −r0 αpord,0,00 := [p g ] ◦ α ◦ (p g ) : 00 0 p,0 |(Gr0 00 )mult ˜ r p,0 S pr ˜ D,pr

(Gr0D,pr00 )Smult ˜

→

S

00 A00S˜ [pr ]

−1 ord that is a closed immersion by (3.3.4.1). Since the kernel K0,S˜ of [pr0 gp,0 ]S˜ : AS˜ → 00 AS˜ is contained in the kernel of the surjection 0

αpord,−1 : AS˜ [pr ] → (Gr−1 )˜ r0 D,pr0 S

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dual to αpord,#,0 , the surjection αpord,−1 induces a surjection r0 r0 0

AS˜ [pr ]/K0,S˜ → (Gr−1 ) ˜. D,pr0 S 00

By restriction to A00S˜ [pr ] (see (3.3.4.11)), we obtain an induced surjection 00

αpord,−1,00 : A00S˜ [pr ] → (Gr−1 ) ˜, r 00 D,pr00 S which is dual to a homomorphism 00

r αpord,#,0,00 : (Gr0D# ,pr00 )mult → A00,∨ ˜ r 00 ˜ [p ] S S

that is a closed immersion, lifting the homomorphism 00

r αpord,#,0 : (Gr0D# ,pr00 )mult → A∨ ˜ ˜ [p ] r 00 S S

induced by αpord,#,0 . r0 The schematic image of the submodule ((pr0 gp,#,0 (Gr0D# ))/ Gr0D# )mult ˜ S of (Gr0D# ,pr00 )mult (see (3.3.4.4)) under the closed immersion αpord,#,0,00 defir 00 ˜ S 00

r nes a subgroup scheme K0,#,S˜ of A00,∨ Since K0,#,S˜ is isomorphic to ˜ [p ]. S 0 mult 0 r0 ((p gp,#,0 (GrD# ))/ GrD# )S˜ , it is finite flat, of finite presentation, and of multipli˜ Since the tautological action of Hp0,ord on S˜ → S is compatible cative type over S. r

with the homomorphism αpord,#,0,00 (and the other homomorphisms involved in its r 00 00

definition), we can descend K0,#,S˜ to a subgroup scheme K0,# of A00,∨ [pr ], which is also finite flat, of finite presentation, and of multiplicative type over S. (This is a question of descending the finite ´etale character group scheme of K0,#,S˜ .) Hence, we can define the isogeny −1 [p−r0 gp,#,0 ]ord : A00,∨ → A0,∨ := A00,∨ /K0,# ,

(3.3.4.12)

and define the isogeny [p−r0 gp,−1 ]ord : A0 → A00

(3.3.4.13)

to be the dual of [p−r0 gp,#,0 ]ord , with kernel isomorphic to the Cartier dual of K0,# . By (3.3.4.2), we know that 00

A0,∨ [pr ] ⊂ A00,∨ [pr ]/K0,#

(3.3.4.14)

(as a closed subgroup scheme). Over S˜ again, the homomorphism αpord,#,0,00 induces a homomorphism r 00   ord,#,0,00 −1 r0 αpord,#,0,0 := [p−r0 gp,#,0 ]ord ◦ α ◦ (p g ) : r 00 p,#,0 |(Gr0# r )mult ˜ S pr ˜ S D ,p (3.3.4.15) r (Gr0D# ,pr )mult → A0,∨ ˜ ˜ [p ] S S −1 that is a closed immersion by (3.3.4.2). Since the kernel K0,#,S˜ of [p−r0 gp,#,0 ]ord ˜ : S 00,∨ 0,∨ AS˜ → AS˜ is contained in the kernel of the surjection 00

−1 r αpord,#,−1,00 : A00,∨ ˜ r 00 ˜ [p ] → (GrD# ,pr00 )S S

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dual to αpord,0,00 , the surjection αpord,#,−1,00 induces a surjection r 00 r 00 00

) ˜. A00,∨ [pr ]/K0,# → (Gr−1 D# ,pr00 S r By restriction to A0,∨ ˜ [p ] (see (3.3.4.14)), we obtain an induced surjection S −1 r αpord,#,−1,0 : A0,∨ r ˜, ˜ [p ] → (GrD# ,pr )S S

which is dual to a homomorphism → A0S˜ [pr ] αpord,0,0 : (Gr0D,pr )mult r ˜ S

(3.3.4.16)

that is a closed immersion, lifting the homomorphism : (Gr0D,pr00 )mult → A00S˜ [pr ] αpord,0,00 r ˜ S induced by αpord,0,00 . r 00 −1 ˜ Since r0 ≥ r by assumption, the section νpord r 0 ◦ (rp ν(gp )) over S, where rp is as in the statement of the proposition, induces a section ∼

: µpr ,S˜ → µpr ,S˜ νpord,0 r

(3.3.4.17)

×

of (Z/pr Z)S˜ . Before moving on, let us first justify the construction of λ0 and i0 outlined in the statements of the proposition. The only part that is not clear is that the Q× -polarization λ0 is indeed a polarization. This is only a statement of checking whether a Q× -isogeny is an isogeny, which can be verified after pulled back to ge˜ Hence, it ometric points of S, which we can always lift to geometric points of S. r0 −1 ord −r0 −1 ord follows from the above construction of [p gp,0 ] , [p gp,#,0 ] , and [p−r0 gp,−1 ]ord (see (3.3.4.10), (3.3.4.12), and (3.3.4.13)), and the relations among the various (O ⊗ Zp )-submodules of Gr0D and Gr0D# . Z

Since all the above constructions over S˜ are compatible with the action of the indutautological action of Hp0,ord on S˜ → S, the homomorphism Hp0,ord → Hpord r r r0 ced by Hp0,ord ,→ gpord Hpord (gpord )−1 (i.e., conjugation by gpord ), or rather by Hp0 ,→ ord,0 gp Hp gp−1 , induces a well-defined Hpord = (αpord,0,0 , αpord,#,0,0 , νpord,0 ) r r r r -orbit of αpr ˜ over S (see (3.3.4.16), (3.3.4.15), and (3.3.4.17)), which descends to a naive orord,0 0 0 0 r dinary level-Hpord structure αH r ord of (A , λ , i ) of type (L/p L, h · , · i, Dpr ) as in pr

Definition 3.3.3.3. Since we can repeat the above procedure when the objects involved are (´etale locally) liftable to higher levels, we obtain an ordinary level-Hp ord,0 structure αH of (A0 , λ0 , i0 ) of type (L ⊗ Zp , h · , · i, D) as in Definition 3.3.3.4. By p Z

ord,0 construction, at each geometric point s¯ of S, there exist liftings of αH satisfying p the characterizations in (5) of the proposition. ord,0 This finishes the construction of the ordinary Hecke twist (A0 , λ0 , i0 , αH ). p

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Example 3.3.4.18 (elements in Pord D (Zp )). (This is a continuation of Example ord,0 ord 3.3.4.5.) In this case, the ordinary Hecke twist (A0 , λ0 , i0 , αH ) of (A, λ, i, αH 0 ) p p −1 0 by gp can be described as follows: The isogeny [gp ] : A → A is an isomorphism allowing us to identify (A0 , λ0 , i0 ) with (A, λ, i). Over the scheme ord r0 S˜ = αH structure 0,ord , where there is a tautological principal ordinary level-p pr

0

ord ord αpord = (αpord,0 , αpord,#,0 , νpord := (αpord,0 ◦ r0 r 0 ), we have a twisted triple αpr 0 ◦ gp r0 r0 r0 ord,0 r gp,0 , αpord,#,0 ◦ gp,#,0 , νpord . r 0 ◦ ν(gp )), whose reduction modulo p defines a triple αpr r0 ord,0 ord,0 The Hpord descends to S and agrees with αH . r -orbit of αpr p

Example 3.3.4.19 (multiplication by powers of p). (This is a continuation of ord,0 Example 3.3.4.6.) In this case, the ordinary Hecke twist (A0 , λ0 , i0 , αH ) of p ord 0 0 0 (A, λ, i, αH0p ) by gp can be described as follows: The triple (A , λ , i ) can be identified with (A, λ, i), so that the Q× -isogeny [gp−1 ] : A → A0 is identified with the Q× -isogeny [p−r0 ] = [pr0 ]−1 : A → A, as in (1) of Proposition 3.3.4.9. Over the ord r0 scheme S˜ = αH struc0,ord , where there is a tautological principal ordinary level-p 0 pr

ture r

αpord r0

p of the

ord,0 = (αpord,0 , αpord,#,0 , νpord to be the r 0 ), we can take αpr r0 r0 ord,0 ord ord triple αpr0 . The Hpr -orbit of αpr descends to S and

reduction modulo ord,0 agrees with αH . p

Example 3.3.4.20 (Up operator and relative Frobenius). (This is a continuord,0 ation of Example 3.3.4.7.) In this case, the ordinary Hecke twist (A0 , λ0 , i0 , αH ) of p ord,0 ord (A, λ, i, αH )[p], the 0 ) by gp can be described as follows: Consider K0 := image(αH0 p p ord,0 p-torsion subgroup scheme of image(αH ). Then [gp−1 ]ord : A → A0 can be identi0 p fied with the quotient A → A/K0 . If S is a scheme over Spec(Fp ), we can identify A0 with the pullback A(p) of A by the absolute Frobenius morphism FS : S → S, and identify [gp−1 ] : A → A0 with the relative Frobenius morphism FA/S : A → A(p) ; and, accordingly, we can also identify λ0 and i0 with the pullbacks λ(p) and i(p) by FS , reord spectively. Over the scheme S˜ = αH 0,ord (but no longer assuming that S is a scheme pr

0

0

over Spec(Fp )), where there is a tautological principal ordinary level-pr structure ord,0 ord,0 αpord , αpord,#,0 , νpord = (αpord,0,0 , αpord,#,0,0 , νpord,0 r r r 0 = (αpr 0 r 0 ), we can take αpr r 0 ) such r0 that αpord,0,0 is obtained from αpord,0 by first taking the quotient of the source and r r0 target by the p-torsion subgroup and K0 , respectively, and restrict the induced morphism (which has image in A0S˜ ) to the pr -torsion subgroup; such that αpord,#,0,0 r

to the pr -torsion subgroup (whose image in A∨ is the restriction of αpord,#,0 r0 ˜ canoS 0,∨ nically lifts to a subgroup scheme of A under the pullback to S˜ of the ´etale dual morphism ([gp ]ord )

∨

˜ S

: A0,∨ → A∨ of [gp ]ord ); and such that νpord,0 is induced by r0

ord,0 ord,0 −1 νpord ν(gp )). The Hpord descends to S and agrees with αH . r -orbit of αpr r 0 ◦ (p p

Proposition 3.3.4.21.

Suppose that g = (g0 , gp ) ∈ G(A∞,p ) × Pord D (Qp ) ⊂

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G(A∞ ) (see Definition 3.2.2.7), and that H and H0 are two open compact subˆ such that H0 ⊂ gHg −1 , and such that H and H0 are of standard groups of G(Z) form as in Definition 3.2.2.9. Suppose moreover that gp satisfies the conditions given in Section 3.3.4. Let S be a scheme over Spec(Z(p) ), let A be an abelian scheme over S, let λ : A → A∨ be a polarization, let i : O ,→ EndS (A) be an O-endomorphism structure as in [62, Def. 1.3.3.1], let αH0,p = {αHn0 0 }n0 be a level-H0,p structure of ˆ p , h · , · i) as in Definition 3.3.1.4, and let αord0 be an ordinary (A, λ, i) of type (L ⊗ Z level-Hp0

Hp

Z

structure of (A, λ, i) of type (L ⊗ Zp , h · , · i, D) as in Definition 3.3.3.4. Z

Under these assumptions, the constructions in Propositions 3.3.2.1 and 3.3.4.9 are both applicable and are compatible with each other. By Proposition 3.3.2.1, the tuple (A, λ, i, αH0,p ) admits a Hecke twist by g0 , which × −1 00 00 is a tuple (A00 , λ00 , i00 , αH p ) equipped with a Z (p) -isogeny [g0 ] : A → A compati∼

ble with all other structures. Since [g0−1 ] induces an isomorphism A[pr ] → A00 [pr ] ord,00 for each r ≥ 0, we have a canonically induced ordinary level structure αH on 0 p 00 00 00 00 (A , λ , i , αHp ). ord,00 By Proposition 3.3.4.9, the tuple (A00 , λ00 , i00 , αH ) admits an ordinary Hecke 0 p ord,0 twists by gp , which is a tuple (A0 , λ0 , i0 , αH ) equipped with a Q× -isogeny [gp−1 ]ord : p −1 ord 00 0 −r0 A → A , which is the composition ([p gp,−1 ]ord )−1 ◦ [pr0 gp,0 ] ◦ [pr0 ]−1 of isogenies of p-power degrees or their formal inverses. Since [gp−1 ]ord induces an iso∼ morphism A00 [n0 ] → A0 [n0 ] for each integer n0 ≥ 1 such that p - n0 , we have a 0 0 0 0 canonically induced level structure αH p on (A , λ , i ). ord,0 0 0 0 0 ), which we call the ordiThus, we have obtained a tuple (A , λ , i , αHp , αH p ord nary Hecke twist of (A, λ, i, αH0,p , αH 0 ) by g = (g0 , gp ), which is equipped with a p

Q× -isogeny [g −1 ]ord : A → A0 defined by the composition [g −1 ]ord := [gp−1 ]ord ◦ [g0−1 ], whose formal inverse we denote by [g]ord : A0 → A. By construction, we have ∨ λ0 = r([g]ord ) ◦ λ ◦ [g]ord (as positive Q× -isogenies), where r is the unique number ˆ ˆ in Q× >0 such that rν(g)Z = Z. If g = g1 g2 , where g1 = (g1,0 , g1,p ) and g2 = (g2,0 , g2,p ) are elements of G(A∞,p ) × Pord D (Qp ), each having a setup similar to that of g, then the ordinary Hecke twists by g can be constructed in two steps using ordinary Hecke twists by g1 and g2 , such that [g −1 ]ord = [g2−1 ]ord ◦ [g1−1 ]ord (or, equivalently, [g]ord = [g1 ]ord ◦ [g2 ]ord ). Proof. The statements are self-explanatory. (Since the constructions of isogenies in Propositions 3.3.2.1 and 3.3.4.9 are achieved by quotients by torsion subgroup schemes of prime-to-p and p-power ranks, respectively, and since the quotients by two torsion subgroup schemes of ranks relative prime to each other can be performed in any order, in order to construct the Hecke twists by g = (g0 , gp ), we might as well form the ordinary Hecke twist by gp first, and form the Hecke twist by g0 second, so that [g −1 ]ord = [gp−1 ]ord ◦ [g0−1 ] = [g0−1 ] ◦ [gp−1 ]ord , by abuse of notation.

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For the same reason, the last statement of the proposition follows from the last statements of Propositions 3.3.2.1 and 3.3.4.9, because [g −1 ]ord = [gp−1 ]ord ◦ [g0−1 ] = −1 ord −1 ord −1 −1 −1 ord −1 −1 ord −1 [g2,p ] ◦[g1,p ] ◦[g2,0 ]◦[g1,0 ] = [g2,p ] ◦[g2,0 ]◦[g1,p ] ◦[g1,0 ] = [g2−1 ]ord ◦[g1−1 ]ord , by a similar abuse of notation.) 3.3.5

Comparison with Level Structures in Characteristic Zero

ˆ be of standard form (with respect to D) as in Definition 3.2.2.9, so Let H ⊂ G(Z) × bal r rν that H = Hp Hp , Up,1 (p ) ⊂ Hp ⊂ Up,0 (pr ), and ν(Hp ) = ker(Z× p → (Z/p Z) ) for some integer rν ≤ r = depthD (H). Over each scheme S over Spec(Q[ζprν ]), ∼ there exists a canonical isomorphism ζprν ,S : ((Z/prν Z)(1))S → µprν ,S (which is the pullback of the canonical ζprν over Spec(Q[ζprν ])). Proposition 3.3.5.1. Let S be a scheme over S0,rν = Spec(F0 [ζprν ]) (see Definition 2.2.3.3), and let (A, λ, i, αH ) be an object of MH (S) (see Section 1.1.2). Then the level-H structure αH = {αHn }n (labeled by integers n ≥ 1 such that U(n) ⊂ H) determines the following data: bal r (1) Since Up (pr ) ⊂ Up,1 (p ) ⊂ Hp , for each integer n0 ≥ 1 such that p - n0 and p p U (n0 ) ⊂ H , we have U(n0 pr ) ⊂ H. Let αHn0 be the schematic image of αHn0 pr under the canonical (reduction modulo n0 ) morphism

IsomS ((L/n0 pr L)S , A[n0 pr ]) × IsomS (((Z/n0 pr Z)(1))S , µn0 pr ,S ) S

→ IsomS ((L/n0 L)S , A[n0 ]) × IsomS (((Z/n0 Z)(1))S , µn0 ,S ).

(3.3.5.2)

S

Then the collection αHp = {αHn0 }n0 labeled by integers n0 ≥ 1 such that p - n0 and U p (n0 ) ⊂ Hp defines a level-Hp structure αHp of (A, λ, i) of type ˆ p , h · , · i) (see Definition 3.3.1.4). (L ⊗ Z Z

(2) Let αHpr be the schematic image of αHn0 pr under the canonical (reduction modulo pr ) morphism IsomS ((L/n0 pr L)S , A[n0 pr ]) × IsomS (((Z/n0 pr Z)(1))S , µn0 pr ,S ) S

→ IsomS ((L/pr L)S , A[pr ]) × IsomS (((Z/pr Z)(1))S , µpr ,S ).

(3.3.5.3)

S

Over some finite flat covering S 0 → S of finite presentation, suppose that ∼ there exists some isomorphism ζpr ,S 0 : ((Z/pr Z)(1))S 0 → µpr ,S 0 lifting the ∼ pullback ζprν ,S 0 of ζprν ,S : ((Z/prν Z)(1))S → µprν ,S to S 0 . Consider the canonical morphism IsomS 0 ((L/pr L)S 0 , AS 0 [pr ]) × IsomS 0 (((Z/pr Z)(1))S 0 , µpr ,S 0 )

→

S0 0 mult r ∨ r HomS 0 ((GrD,pr )S 0 , AS 0 [p ]) × HomS 0 ((Gr0D# ,pr )mult S 0 , AS 0 [p ]) S0 × × (Z/pr Z)S 0 S0

(3.3.5.4)

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over S 0 defined by sending ∼

∼

(αpr : (L/pr L)S 0 → AS 0 [pr ], νpr : ((Z/pr Z)(1))S 0 → µpr ,S 0 ) in the source to ord,0 , νpord , αpord,#,0 αpord r r ) r = (αpr

in the target, where: ∼

0 mult : (Gr0D,pr )mult (a) αpord,0 → AS 0 [pr ] is composition of ζp−1 → r r ,S 0 : (GrD,pr )S 0 S0 0 0 (GrD,pr )S 0 with the restriction of αpr to (GrD,pr )S 0 . #,mult r : (Gr0D# ,pr )mult (b) αpord,#,0 → A∨ to r S0 S 0 [p ] is the restriction of αpr ∼ #,mult 0 mult # r # mult ∨ r (GrD# ,pr )S 0 , where αpr : (L /p L )S 0 → AS 0 [p ] is the inverse of the Cartier dual of αpr over S 0 . × is a section of (Z/pr Z)S 0 defined by the composition ζp−1 (c) νpord r r ,S 0 ◦ νpr : ∼ r r (Z/p Z)(1))S 0 → ((Z/p Z)(1))S 0 .

Then the schematic image of αHpr × S 0 under this canonical morphism is S

independent of the lifting ζpr ,S 0 of ζprν ,S 0 over S 0 , and defines by descent ord (see [33, VIII, 1.9, 1.11, 5.5]) a subscheme αH ord of r p

× , A[pr ]) × HomS ((Gr0D# ,pr )mult , A∨ [pr ]) × (Z/pr Z)S HomS ((Gr0D,pr )mult S S S S ord that defines an ordinary level-Hp structure αH of (A, λ, i) of type p (L ⊗ Zp , h · , · i, D) (see Definition 3.3.3.4). Z

This assignment ord αH 7→ (αHp , αH ) p

(3.3.5.5)

induces an injection from the set of level-H structures of (A, λ, i) of type ˆ h · , · i) as in [62, Def. 1.3.7.6] to the set of pairs (αHp , αord ), where αHp (L ⊗ Z, Hp Z

ˆ p , h · , · i) (see Definition is a level-Hp structure αHp of (A, λ, i) of type (L ⊗ Z Z

ord ord 3.3.1.4), and where αH is an ordinary level-Hp structure αH of (A, λ, i) of type p p (L ⊗ Zp , h · , · i, D) (see Definition 3.3.3.4). Z

∼

ˆ → T As¯ is a lifting of αH as in Lemma 3.3.1.6 (or Suppose α ˆ s¯ : L ⊗ Z Z

rather in [62, Sec. 1.3.7]).

Then α ˆ s¯ induces an O ⊗ A∞,p -equivariant isomorZ

∼

ˆ p → Tp As¯, which is a lifting of αHp as in Lemma 3.3.1.6, bephism α ˆ sp¯ : L ⊗ Z Z

cause the assignment of αHp to αH in (1) above is defined by the canonical morphisms as in (3.3.5.2). Similarly, α ˆ s¯ induces an O ⊗ Zp -equivariant isomorphism ∼

Z

α ˆ s¯,p : L ⊗ Zp → Tp As¯, which induces by taking graded pieces and by duality a triple Z

ord α ˆ sord = (ˆ αsord,0 ,α ˆ sord,#,0 , νˆs¯ord ), which is a lifting of αH as in Lemma 3.3.3.9, because ¯ ¯ ¯ p ord the assignment of αHp to αH in (2) above is defined by the canonical morphisms as in (3.3.5.3) and (3.3.5.4).

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ord Proof. It is clear that the assignments define αHp and αH as naive level structures. p The symplectic-liftability conditions are verified because αH itself is symplecticliftable, and its symplectic liftings to higher levels induce the desired symplectic ord liftings of αHp and αH . p As for the injectivity of (3.3.5.5), first note that if two geometric points of

IsomS ((L/n0 pr L)S , A[n0 pr ]) × IsomS (((Z/n0 pr Z)(1))S , µn0 pr ,S ) S

have the same images under both the morphisms (3.3.5.2) and (3.3.5.3), then they must be the same. Second note that if two geometric points of IsomS 0 ((L/pr L)S 0 , AS 0 [pr ]) × IsomS 0 (((Z/pr Z)(1))S 0 , µpr ,S 0 ) S0

are mapped to the same point under the morphism (3.3.5.4), then they are in the bal r = Hp /Up,1 (p ) by definition (see the beginning of same Up,0 (pr )-orbit. Since Hpord r Section 3.3.3), the injectivity of (3.3.5.5) follows. The last paragraph of the proposition is self-explanatory. Proposition 3.3.5.6. Let g, H, and H0 be as in Proposition 3.3.4.21. Suppose × rν 0 ν(Hp0 ) = ker(Z× Z) ). (Then rν 0 ≥ rν because Hp0 ⊂ Hp .) Let S p → (Z/p be a scheme over S0,rν 0 = Spec(F0 [ζprν 0 ]), and let (A, λ, i, αH0 ) be an object of ord MH0 (S). Let αH0,p and αH 0 be determined by αH0 as in Proposition 3.3.5.1. Let p 0 0 0 0 (A , λ , i , αH ) be the Hecke twist of (A, λ, i, αH0 ) by g as in [62, Sec. 6.4.3], equipped ord,0 0 0 be determined by αH with a Q× -isogeny [g] : A → A0 , and let αH p and αH p ord,00 00 as in Proposition 3.3.5.1. Let (A00 , λ00 , i00 , αH ) be the ordinary Hecke twist p , αH p ord of (A, λ, i, αH0,p , αH0p ) by g as in Proposition 3.3.4.21. Then there is a canonical ord,00 ord,0 00 0 ) matching ) and (A00 , λ00 , i00 , αH isomorphism between (A0 , λ0 , i0 , αH p , αH p , αH p p −1 0 −1 ord 00 [g ] : A → A and [g ] : A → A .

Proof. Since we are in characteristic zero, all kernels of isogenies involved are finite ´etale group schemes, and all level structures are defined by homomorphisms between ´etale group schemes. Hence, the statements of this proposition can be verified after pulled back to geometric points of S. At each geometric point s¯ of S, the validity of the corresponding statements follows from the last paragraph of Proposition 3.3.5.1, and from the descriptions of the effects of Hecke twists over geometric points in (4) of Proposition 3.3.2.1 and (5) of Proposition 3.3.4.9 (and in the analogue of (4) of Proposition 3.3.2.1 for usual Hecke twists defined by g in characteristic zero). 3.3.6

Valuative Criteria

Definition 3.3.6.1. We say an element gp ∈ Pord D (Qp ) is of Up type if it acts ord on GrDQp ∼ by g = (g , g ) = (p−a IdGr0D , IdGr−1 ) for some = Gr0DQp ⊕ Gr−1 p,0 p,−1 DQp p D integer a > 0, and if for one (and hence every) splitting of the filtration (3.2.2.3) (resp. (3.2.2.5)), gp (resp. gp,# = t gp−1 ) maps the image of Gr−1 (resp. Gr−1 ) to D D# L ⊗ Zp (resp. L# ⊗ Zp ). Z

Z

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We say an element gp ∈ Pord D (Qp ) is of twisted Up type if it acts on ord by g = (gp,0 , gp,−1 ) such that gp,0 (Gr0D ) = p−a Gr0D and GrDQp ∼ = Gr0DQp ⊕ Gr−1 p DQp 0 0 0 −b t −1 b gp,−1 (Gr−1 Gr−1 D ) = p D , so that gp,#,0 (GrD# ) = gp,−1 (GrD# ) = p GrD# , for some integers a ≥ b. In this case, we define depthD (gp ) := a − b. Z

≥0 Z Remark 3.3.6.2. Suppose gp ∈ Pord Pord D (Qp ) is of Up type, then p gp D (Zp ) is ord a subsemigroup of PD (Qp ), whose elements are all of twisted Up type. (Thus, the elements of twisted Up type includes Examples 3.3.4.5, 3.3.4.7, and 3.3.4.7 as special cases.)

Remark 3.3.6.3. Suppose that Hp , Hp0 , and gp ∈ Pord D (Qp ) satisfies the conditions 0 given in Section 3.3.4, with r := depthD (Hp ) and r := depthD (Hp0 ), and that gp is of twisted Up type as in Definition 3.3.6.1, such that r0 − depthD (gp ) = r. Then, by setting r0 := −b as in Definition 3.3.6.1 and r00 := r, we have Gr0D ⊂ p−r0 gp,0 (Gr0D ) ⊂ p−r

00

−r0

0

gp,0 (Gr0D ) = p−r Gr0D

(3.3.6.4)

and 00

Gr0D# = pr0 gp,#,0 (Gr0D# ) ⊂ p−r+r0 gp,#,0 (Gr0D# ) = p−r Gr0D#

(3.3.6.5)

(cf. (3.3.4.1) and (3.3.4.2)). 0 00 Lemma 3.3.6.6. Suppose that Hp , Hp0 , gp ∈ Pord D (Qp ), r0 , r, r , r are as in Remark 3.3.6.3, and that r > 0. Suppose R is a discrete valuation ring over Z(p) , with ~ := Spec(R) fraction field Frac(R) and residue field k of characteristic p > 0. Let S ord,0 ord,#,0 ord ord , νH and S := Spec(Frac(R)). Suppose that (A, λ, i, αH0p ) = (αH0p , αH 0 ) 0 p p is defined over S as in Proposition 3.3.4.9, so that the ordinary Hecke twist ord,0 ord,0,0 ord,#,0,0 ord,0 ord (A0 , λ0 , i0 , αH = (αH , αH , νH )) of (A, λ, i, αH 0 ) by gp is defined, equipp p p p p

ped with a Q× -isogeny [gp−1 ]ord : A → A0 over S. ~ 0 over S, ~ so that Moreover, suppose that A0 extends to a semi-abelian scheme A 0,∨ 0 0 ~ A is defined (as in [62, Thm. 3.4.3.2]), and so that λ and i also (uniquely) ~0 → A ~ 0,∨ and ~i0 : O → End ~ (A ~ 0 ) by [92, IX, 1.4], [28, extend to some ~λ0 : A S ord,0,0 Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5]; and suppose that image(αH ) (resp. p ord,#,0,0 image(αH )) extends to a finite flat subgroup scheme K 0 (resp. K #,0 ) of mulp ~ 0 [pr ] (resp. A ~ 0,∨ [pr ]), so that αord,0 (uniquely) extends to some tiplicative type of A Hp ord,0 ord,0,0 ord,#,0,0 ord,0 bal r α ~H = (~ αH ,α ~H , ~νH ), which is ´etale locally over S an Hp /Up,1 (p )-orbit p p p p ord,0 ord,0,0 ord,#,0,0 ord,0 ord,0,0 0 mult ~ 0 [pr ], of some α ~ pr = (~ α pr ,α ~ pr , ~νpr ), where α ~ pr : (GrD,pr ) → A ×

~ S

ord,#,0,0 ~ 0,∨ [pr ], and ~νpord,0 α ~H : (Gr0D# ,pr )mult →A ∈ (Z/pr Z)S~ satisfy the same comr ~ p S patibility conditions as in Definitions 3.3.3.1, 3.3.3.2, 3.3.3.3, and 3.3.3.4; and so ~→S ~ is ordinary, by Remark 3.3.3.6. that A ord ~ ~ ~ ~ ord0 ) over S, ~ Then (A, λ, i, αH 0 ) also extends to an analogous tuple (A, λ, i, α Hp p ~ → A ~ 0 , extending [gp−1 ]ord : A → A0 , equipped with a Q× -isogeny [gp−1 ]ord : A ~ 0 is an abelian scheme, then A ~ is between ordinary semi-abelian schemes. If A

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ord,0 ~ ~λ,~i) of also an abelian scheme; α ~H is an ordinary level-Hp0 structure of (A, p ord,0 ~ 0 , ~λ0 ,~i0 , α type (L ⊗ Zp , h · , · i, D) as in Definition 3.3.3.4; and (A ~H ) is the ordinary p Z

ord ~ ~λ,~i, α Hecke twist of (A, ~H 0 ) by gp as in Proposition 3.3.4.9. p

Proof. By Lemma 3.1.3.1, sufficiently divisible multiples of the formal inverse [gp ]ord : A0 → A of [gp−1 ]ord extends to sufficiently divisible multiples of the formal ~0 → A ~ of [gp−1 ]ord , where A ~ is determined by any such extensions inverse [gp ]ord : A by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5]; and λ and i also (uni~→A ~ ∨ and ~i : O → End ~ (A) ~ (by the same references). quely) extends to some ~λ : A S ~ ~ ~0 → S ~ is. Moreover, by Lemma 3.1.1.5, A → S is ordinary because A Hence, the question is whether the schematic closure K (resp. K # ) of ord,0 ord,0 ~ ~∨ image(αH ord ) (resp. image(αH ord )) in A (resp. A ), which is quasi-finite flat and pr

pr

~ because A ~ →S ~ is ordinary, is of multiplicative of ´etale-multiplicative type over S type. (The rest of the lemma will be self-explanatory.) By (3.3.6.4) and (3.3.6.5) in Remark 3.3.6.3, using the crucial condition that 0 r − depthD (gp ) = r, we have the following exact sequences of finite flat group schemes ord,0,0 ord,0 ord,0 ) → image(αH )→0 ) → image(αH 0 → pr image(αH 0 0 p p p

and ord,#,0,0 ord,#,0 ord,#,0 ) → image(αH ) ) → image(αH 0 → pr image(αH 0 0 p p p

of multiplicative type over S. By definition of the ordinary level structures, both are ´etale locally (after forgetting their O-module structures) filtered by subobjects whose graded pieces are isomorphic to 0

0

0 → (pr Z/pr Z)mult → (Z/pr Z)mult → (Z/pr Z)mult . S S S ~ and A ~ ∨ , respectively, correspond to the exact sequences Their closures in A 0 → pr K → K → K 0 → 0 and 0 → pr K # → K # → K #,0 → 0. By taking normalizations over the ´etale cover, both are fppf locally (after forgetting their O-module structures) filtered by subobjects whose graded pieces are exact sequences of the form 0 → pr C → C → (Z/pr Z)mult → 0, ~ S 0

where C is cyclic of order pr , and where r > 0 by assumption. Hence, C must also be of multiplicative type, by the same argument in the proof of [49, Thm. 6.7.11(2)]; and so are K and K # , as desired.

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Remark 3.3.6.7. The proof of Lemma 3.3.6.6 shows that, in order to have similar valuative criteria for more general elements gp ∈ Pord D (Qp ), such as those generalized Up operators in Example 3.3.4.8 (which are, in general, not of twisted Up type as in Definition 3.3.6.1), we should consider not just the group Up,0 (pr ) stabilizing a maximal totally isotropic subgroup D0pr of L ⊗ (Z/pr ) (which can be called the Z

canonical subgroup), but also a sequence of isotropic subgroups Dipri ⊂ L ⊗ (Z/pri ) Z

such that ri ≥ rj and Djpri = (Djprj /pri Djprj ) ⊂ Dipri whenever j ≥ i (which can be considered a sequence of partial canonical subgroups of increasing depth). This is technically possible, but we omit its treatment because it introduces complications that we do not immediate need (in an already lengthy work). Lemma 3.3.6.8. In Lemma 3.3.6.6, if the characteristic of Frac(R) is p > 0, then the same conclusion holds without the assumptions that r0 − depthD (gp ) = r in Remark 3.3.6.3 and that r > 0 in Lemma 3.3.6.6. ~ is a scheme over Spec(Fp ), we have the idenProof. By Lemma 3.3.3.7, since S (r 0 ) (r 0 ) ord,0 ord,#,0 tities image(αH0p ) = ker(FA/S ) and image(αH ) = ker(FA∨ /S ) over S, which 0 p 0

0

(r ) (r ) # ~ necessarily extends to the identities K = ker(FA/ = ker(FA~ ∨ /S~ ) over S, ~ S ~ ) and K

where K and K # are as in the proof of Lemma 3.3.6.6. Hence, K and K # are of multiplicative type without the assumption that r0 − depthD (gp ) = r > 0. 3.4

Ordinary Loci

From now on, we fix the choice of D as in Lemma 3.2.2.1 that satisfies Assumption 3.2.2.10. 3.4.1

Naive Moduli Problems with Ordinary Level Structures

ˆ be of standard form as in Definition 3.2.2.9, so that H = Hp Hp and Let H ⊂ G(Z) bal r bal r := Hp /Up,1 (p ) and let Up,1 (p ) ⊂ Hp ⊂ Up,0 (pr ) for r = depthD (H). Let Hpord r ord Hp be defined as in (3.3.3.5). Definition 3.4.1.1. Let H, Hp , Hp , r, and Hpord be as above. The moduli problem ...ord MH is defined as the category fibered in groupoids over (Sch / Spec(Z(p) )) whose ...ord fiber over each scheme S is the groupoid MH (S) described as follows: The objects ...ord ord of MH (S) are tuples (A, λ, i, αHp , αH ), where: p (1) (2) (3) (4)

A is an abelian scheme over S. λ : A → A∨ is a polarization. i : O ,→ EndS (A) is an O-endomorphism structure as in [62, Def. 1.3.3.1]. ˆ p , h · , · i) (see DefiαHp is a level-Hp structure of (A, λ, i) of type (L ⊗ Z Z

nition 3.3.1.4).

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Compactifications with Ordinary Loci ord (5) αH is an ordinary level-Hp structure of (A, λ, i) of type p (L ⊗ Zp , h · , · i, D) (see Definition 3.3.3.4). (This forces A to be ordinary. Z

See Remark 3.3.3.6.) ...ord The morphisms of MH (S) are naive isomorphisms (between the abelian schemes, matching all additional structures). ...ord ...ord ...ord bal r If Hp = U p (n0 ) and Hp = Up,1 (p ), then we denote MH by Mn0 pr = Mn , and we denote by αn0 (resp. αpord r ) the unique principal level-n0 structure (resp. principal ord ordinary level-pr structure) determined by αHp (resp. αH ). p As always, the symplectic isomorphisms carry the additional data of isomorphisms between values of pairings, which can be called the similitudes of the symplectic isomorphisms. If p is a good prime, then an argument similar to that in [62, Ch. 2] shows that ...ord the moduli problem MH is an algebraic stack separated and of finite type over Spec(Z(p) ). (See the proof of Theorem 3.4.1.9 below.) However, the argument there used the crucial technical result [62, Thm. B.3.11] (due to Artin) to suppress the technical condition [62, Sec. 2.3.4, Cond. 4] in the verification of Artin’s criterion, which requires the infinitesimal deformation rings to be noetherian and normal. As we will see below, when p is not a good prime, the infinitesimal deformation rings might not even be geometrically unibranched (and hence not geometrically normal either), a situation to which no variant of [62, Thm. B.3.11] along the lines of [3, Thm. 3.9] seems to apply. To circumvent this difficulty, we shall again introduce some auxiliary moduli problems. (For later references, in this subsection, we will develop more about these auxiliary moduli problems than we need for the proof of representability.) Construction 3.4.1.2. Let (Oaux , ?aux , Laux , h · , · iaux , h0,aux ) be chosen as in the paragraph preceding Lemma 2.1.1.9, where (Laux = L⊕ a1 ⊕(L# )⊕ a2 , h · , · iaux ) is as in Lemma 2.1.1.1. Then the filtration D on L ⊗ Zp induces a filtration Z

D1aux = 0 ⊂ D0aux := (D0 )⊕ a1 ⊕(D#,0 )⊕ a2 ⊂ D−1 aux = Laux ⊗ Zp . Z

Since D0aux,Qp := D0aux ⊗ Qp is mapped to (D0 )⊕(a1 +a2 ) ⊗ Qp under the canonical isoZp

Zp

morphism Laux ⊗ Qp ∼ = L⊕(a1 +a2 ) ⊗ Qp , its submodule D0aux = D0aux,Qp ∩(Laux ⊗ Zp ) Z

Z

Z

is maximal totally isotropic under the pairing on Laux ⊗ Zp induced by h · , · iaux Z

(cf. Lemma 3.2.2.1). Since h · , · iaux is self-dual at p, the pairing h · , · iaux indu∼ # ces an isomorphism Laux ⊗ Zp → L# aux ⊗ Zp matching Daux with Daux (cf. Lemma Z

Z

∼

3.2.2.4), and induces an isomorphism φ0Daux : Gr0Daux → Gr0D# . Hence, we have a aux

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commutative diagram: 1 +a2 ) (Gr0D )⊕(a _

φ0D ⊗( · , · )∗ aux



/ Gr0D aux

(3.4.1.3)

o φ0Daux



 ? _ Gr0#

(Gr0D# )⊕(a1 +a2 ) o

Daux

(This finishes Construction 3.4.1.2.) ˆ p ) and Hp = U bal (pr ) for some integer Suppose H = Hp Hp such that Hp ⊂ G(Z p,1 p ˆ r ≥ 0. Let Haux ⊂ Gaux (Z) be of the form Haux = Haux Haux,p such that Hp,aux = bal Up,1,aux (pr ), where Up,1,aux (pr ) is defined by the filtration Daux on Laux ⊗ Zp as in Z

Definition 3.2.2.8, and such that H is mapped into Haux under the homomorphism ˆ → Gaux (Z) ˆ given by (2.1.1.10). G(Z) Lemma 3.4.1.4. Let H and Haux be as above. Then there is a morphism ...ord ...ord (3.4.1.5) MH → MHaux compatible with (2.1.1.17). Proof. The construction of (3.4.1.5) is similar to the construction of (2.1.1.17), but let us still spell out the details in steps where they slightly differ. ...ord ord Suppose (A, λ, i, αHp , αH ) is the tautological tuple over MH . Then we obtain p × a1 ∨ × a2 ×(a1 +a2 ) , the canoand AO the abelian schemes AM aux := A aux := A ...× (A ) M ord H M O M nical morphism f : Aaux → Aaux , the polarization λaux and the Oaux -structure iM aux , O and the polarization λO of degree prime to p and the O ⊗ Q-structure i aux aux aux as Z

in (2) of Lemma 2.1.1.1 and in the proof of Proposition 2.1.1.15. O The kernel K of the canonical isogeny f : AM aux → Aaux decomposes canonically p p as a fiber product K ...× Kp , where K is a finite ´etale group scheme of rank primeM ord H

...ord ∼ ˆp → to-p such that, at each geometric point s¯ of MH , any lifting α ˆ sp¯ : L ⊗ Z Tp As¯ Z

of αHp (as in Lemma 3.3.1.6) defines an isomorphism ∼ ˆp → (L# /L)⊕ a2 ⊗ Z Ks¯p ; Z

and where Kp is a finite flat group scheme of p-primary rank such that, at each ...ord ord geometric point s¯ of MH , any lifting α ˆ sord = (ˆ αsord,0 ,α ˆ sord,#,0 , νˆs¯ord ) of αH (as in ¯ ¯ ¯ p Lemma 3.3.3.9) defines a short exact sequence ⊕ a2 0 → ((Gr0D# / Gr0D )⊕ a2 )mult → Kp,¯s → ((Gr−1 / Gr−1 )s¯ → 0. s¯ D ) D#

(Although [62, Lem. 1.3.5.2] is not applicable to isogenies of degree not prime to p, we can use such short exact sequences to study quasi-isogenies formed by isogenies whose kernels are finite flat group schemes of ´etale-multiplicative type.) Therefore,

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... (AO the O ⊗ Q-structure iO aux : Oaux ⊗ Q → End M ord aux ) ⊗ Q induces an O-structure H Z Z Z O O ... (AO iO ) of (A , λ ). aux : Oaux → End M ord aux aux aux H ...ord Moreover, away from p, at each geometric point s¯ of MH , the isomorphism ∼ p O M O Vp (f ) : Vp AM aux,¯ s → V Aaux,¯ s (which can be defined even though f : Aaux → Aaux ˆ p -equivariant isomoris not prime to p) and the lifting α ˆ sp¯ above induces an Oaux ⊗ Z Z

∼

ˆ p → Tp A O : Laux ⊗ Z phism α ˆ sO,p ¯ aux,¯ s (matching similitudes, implicitly). Since the Z ...ord p under the Hp -orbit of α ˆ sp¯ is π1 (MH , s¯)-invariant, and since Hp is mapped into Haux p p p ˆ ˆ homomorphism G(Z ) → Gaux (Z ) given by (2.1.1.10), the Haux -orbit [ˆ αsO,p ¯ ]Hp aux ... ord O,p of α ˆ s¯ is invariant under π1 (MH , s¯). This allows us to construct a level structure O O lifts αH αH (away from p) as in the proof of [62, Prop. 1.4.3.4] such that α ˆ sO,p p p ¯ aux aux (as in Lemma 3.3.1.6). bal r bal Finally, at p, since Hp = Up,1 (p ) and Haux,p = Up,1,aux (pr ) for some integer r ≥ 0, we can start with the unique triple ord,0 αpord , αpord,#,0 , νpord r r = (αpr r ) ord,O ord determined by αH , and define αH by defining p aux,p

αpord,O := (αpord,0,O , αpord,#,0,O , νpord,O ), r r r r where: )× a2 as homomorphisms from (1) αpord,0,O := (αpord,0,O )× a1 ...× (αpord,#,0,O r r r M ord H

...ord (Gr0Daux ,pr )mult M

H M ord H ord,0,O ...ord )−1 ◦ ((φ0Daux )mult := λO aux ◦ αpr MH 0 r ...ord to AO,∨ phisms from (GrD# ,pr )mult [p ]. aux MH aux × ord,O ord r ... νpr := νpr as a section of (Z/p Z) M ord . H H

(2) (3)

r ∼ ...ord )× a1 ...× ((Gr0D# ,pr )mult ...ord )× a2 to AO = ((Gr0D,pr )mult aux [p ]. M M

αpord,#,0,O r

H

(cf. (3.4.1.3)) as homomor-

...ord ord Thus, we assigned to the tautological tuple (A, λ, i, αHp , αH ) over MH a tuple p ...ord ord,O O O O , αH ) parameterized by the moduli problem MHaux , which (AO aux , λaux , iaux , αHp aux,p aux ...ord then induces the desired morphism (3.4.1.5) by the universal property of MHaux . By ...ord O O O ) their very constructions, the pullback to MH of the tuple (AO aux , λaux , iaux , αHp aux constructed in the proof of Proposition 2.1.1.15 gives, via Proposition 3.3.5.1, the ...ord ord,O O O O pullback to MH of the tuple (AO , αH ) constructed here. (It aux , λaux , iaux , αHp aux,p aux O O is nevertheless an abuse of notation when we use the notation AO aux , λaux , and iaux in both of them.) Hence, (3.4.1.5) is compatible with (2.1.1.17), as desired. bal r Assuming no longer that Hp = Up,1 (p ), we still have:

ˆ and (resp. Haux = Hp Haux,p ⊂ Lemma 3.4.1.6. Suppose H = Hp Hp ⊂ G(Z) aux ˆ Gaux (Z)) is an open compact subgroup such that there exists integers r ≥ r0 such

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0

bal r bal r bal that Up,1 (p ) ⊂ Hp ⊂ Up,0 (p ) and Up,1,aux (pr ) ⊂ Haux,p ⊂ Up,0,aux (pr ), and such ˆ → Gaux (Z) ˆ given by that H is mapped into Haux under the homomorphism G(Z) (2.1.1.10). In this case, there is a morphism ...ord ...ord (3.4.1.7) MH → MHaux

compatible with (2.1.1.17) (and with (3.4.1.5)). Proof. The operations of taking orbits of level structures on both sides of (3.4.1.5) are compatible with each other. Hence, the morphism (3.4.1.5) at any sufficiently high level induces the morphism (3.4.1.7) by forgetting part of the structures. Lemma 3.4.1.8. The morphism (3.4.1.7) is schematic, separated, and quasi-finite. Proof. As in the proof of Proposition 2.1.1.15, this follows from Lemma 2.1.1.5 (for the abelian schemes and polarizations), from [62, Prop. 1.3.3.7] (for the endomorphism structures), from the fact that the level structures away from p are defined by isomorphisms between finite ´etale group schemes, and from the fact that the ordinary level structures are defined by morphisms between finite flat group schemes of ´etale-multiplicative type. ...ord Theorem 3.4.1.9. The moduli problem MH is an algebraic stack separated and of finite type over Spec(Z(p) ). It is representable by an algebraic space if the objects it parameterizes have no nontrivial automorphism, which is, in particular, the case −1 when Hp is neat. Its local structures can be described as follows: Let φ−1 D : GrD → be as in Lemma 3.2.2.4, and consider the finitely generated Zp -module Gr−1 D#   −1 0 0 d ⊗ φ−1 D (d ) − d ⊗ φD (d) ~SD := (Gr−1 ⊗ Gr−1 )/ . (3.4.1.10) D D# d,d0 ∈Gr−1 (bd) ⊗ d# − d ⊗ (b? d# ) Zp D , d# ∈Gr−1 # ,b∈O D

Let ~SD,Z be any noncanonical choice of a finitely generated Z-module such that ~SD,Z ⊗ Zp ∼ = ~SD and such that the maximal torsion submodule ~SD,Z,tor of ~SD,Z is Z

isomorphic to the maximal torsion submodule ~SD,tor of ~SD (as Z-modules). (This latter condition is not necessary for our purpose, but we impose it for simplicity ...ord of later exposition.) Then the completions of strict local rings of MH at geometric points of characteristic p are isomorphic to the completions of strict local rings of the group scheme ED,Z of multiplicative type of finite type over Spec(Z(p) ) with character group the finitely generated commutative group ~SD,Z . ...ord Proof. By Lemma 3.4.1.8, MH is schematic, separated, and quasi-finite over ...ord ...ord MHaux . Since p is a good prime for the moduli problem MHaux , we can show that ...ord MHaux is an algebraic stack separated and of finite type over Spec(Z(p) ) by an ar...ord gument similar to that in [62, Ch. 2]: The moduli problem MHaux is an algebraic stack (quasi-separated and) locally of finite type over Spec(Z(p) ) by the theory of

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infinitesimal deformations and Artin’s criterion. Note that, since the infinitesimal deformation rings are smooth (by [62, Prop. 2.2.4.9] and by the fact that αHpaux and ord αH are both parameterized by ´etale objects), and since the moduli problem aux,p ...ord MHaux can be extended to a moduli problem over a Dedekind ring having infinitely many residue characteristics, we can suppress the technical condition [62, Sec. 2.3.4, Cond. 4] in Artin’s criterion by [62, Thm. B.3.11]. The diagonal 1-morphism ...ord ...ord ...ord ∆... : MHaux → MHaux × MHaux is finite by the theory of N´eron models M ord H aux

Spec(Z(p) )

(applied to the abelian schemes and the additional structures—the ordinary level structures, once they exist, have finite automorphism group schemes). This shows ...ord ...ord that MHaux is separated over Spec(Z(p) ) (by definition). The algebraic stack MHaux is quasi-compact (and hence of finite type over Spec(Z(p) )) because it can be covered by a quasi-compact scheme parameterizing the sections of ample invertible sheaves, the endomorphism structures, and the ordinary level structures. (See [62, Ch. 2] ...ord ...ord for more details.) Thus, we have shown that MHaux and hence MH are algebraic stacks separated and of finite type over Spec(Z(p) ). If Hp is neat, then the tuple (A, λ, i, αHp ) admits no nontrivial automorphism ...ord ord at p), in which case MH is representable by (regardless of the level structure αH p an algebraic space. The existence of ordinary level structures in Definition 3.4.1.1 forces the abelian ...ord varieties parameterized by geometric points of characteristic p of MH to be ordinary (see Remark 3.3.3.6). Hence, we can describe the completion of strict local rings ...ord of MH at geometric points of characteristic p using the Serre–Tate deformation theory of ordinary abelian varieties (as in, for example, [47]). By [47, Thm. 2.1, 1) and 2)], the formal moduli of any ordinary abelian variety A over s¯ := Spec(k), where k is an algebraically closed field of characteristic p, is canonically isomorphic ˆ m ), where Tp A and Tp A∨ are the physical to formal torus HomZp (Tp A ⊗ Tp A∨ , G Zp

Tate modules of A and A∨ , which are free Zp -modules of rank dim(A) = dim(A∨ ). ...ord ord ord ) of MH (¯ If A is part of an object (A, λ, i, αHp , αH s), then any lifting αsord ¯ of αHp as p −1 ∨ ∼ in Lemma 3.3.3.9 defines isomorphisms Tp A ∼ = Gr−1 D and Tp A = GrD# compatible −1 with each other under φD and λ (cf. Proposition 3.2.1.1). Hence, the formal moduli −1 ˆ of A itself is canonically isomorphic to the formal torus HomZp (Gr−1 D ⊗ GrD# , Gm ). Zp

By [47, Thm. 2.1, 3) and 4)], the formal submoduli for liftings of A also carrying ord liftings of the additional structures λ, i, αHp , and αH is the formal subgroup p ~ ˆ scheme of multiplicative type HomZ (SD , Gm ), where the condition for λ is dual to p

−1 −1 0 0 0 the relations d ⊗ φ−1 D (d ) − d ⊗ φD (d) for all d, d ∈ GrD ; where the condition for i is dual to the relations (bd) ⊗ d# − d ⊗ (b? d# ) for all d ∈ Gr−1 and b ∈ O; D and where no conditions are needed for the level structures, because they are given by morphisms between finite ´etale group schemes, or by morphisms between finite flat groups schemes of multiplicative type, which always uniquely lift. Hence, the

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...ord completion of strict local rings of MH can be described as in the statement of the theorem, as desired. (We emphasize that the elegant argument in [47, Sec. 1] requires the prorepresentability as an input. Therefore, it is not true that one can avoid the references as in [62, Ch. 2] to the original theory of deformation of abelian schemes developed by Grothendieck, Mumford, and others.) 3.4.2

Ordinary Loci as Normalizations

...ord Let H be as at the beginning of Section 3.4.1, so that MH is defined as in Definition × rν 3.4.1.1. Suppose that ν(Hp ) = ker(Z× p → (Z/p Z) ) (where rν ≤ r) (cf. also Section 3.3.5). Definition 3.4.2.1. Let ~SD be as in Theorem 3.4.1.9. Then we define rD ≥ 0 to be the smallest nonnegative integer such that prD annihilates all (p-primary) torsion elements in ~SD , and we define rH := max(rD , rν ). Remark 3.4.2.2. If p is a good prime as in Definition 1.1.1.6, then rD = 0 by [62, Prop. 1.2.2.3], and hence rH = rν . Lemma 3.4.2.3. The canonical morphism ...ord MH,rν → MH ⊗ F0 [ζprν ]

(3.4.2.4)

Z

over S0,rν = Spec(F0 [ζprν ]) induced by the assignment (3.3.5.5) is an open and closed immersion. Proof. The morphism (3.4.2.4) is an open and closed immersion because over F0 the Lie algebra condition given by (L ⊗ R, h · , · i, h0 ) (in [62, Def. 1.3.4.1]) is defined Z

and is an open and closed condition by [62, Prop. 2.2.2.9], because all level structures involved are defined by homomorphisms between finite ´etale group schemes, and because (3.3.5.5) is injective. Theorem 3.4.2.5. Let Mord denote the open and closed subalgebraic stack of H ...ord r MH ⊗ F0 [ζp H ] over S0,rH = Spec(F0 [ζprH ]) (see Definition 2.2.3.3) given by Z

the image of the induced canonical open and closed immersion MH,rH ,→ ...ord ...ord ord ~ ord of M MH ⊗ F0 [ζprH ]. Then the normalization M H in MH under the canonical H Z ...ord morphism Mord H → MH is a regular algebraic stack which is separated, smooth, and of finite type over ~S0,rH = Spec(OF0 ,(p) [ζprH ]) (see Definition 2.2.3.3). (In parti~ ord → Spec(Z(p) ) all comes cular, the nonsmoothness of the structural morphism M H ~ from that of the finite flat morphism S0,rH = Spec(OF0 ,(p) [ζprH ]) → Spec(Z(p) ).) ~ ord is representable by an algebraic space when the moduli The algebraic stack M H ...ord problem MH is (see Theorem 3.4.1.9), which is the case when Hp is neat. Let ~SD,Z be noncanonically chosen as in Theorem 3.4.1.9, and let ~SD,Z,free be the ~ ord are maximal free quotient of ~SD,Z . Then the completions of strict local rings of M H

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isomorphic to the completions of strict local rings of the torus ED,Z,free over ~S0,rH with character group ~SD,Z,free . bal r ~ ord by M ~ ord r = M ~ ord , If Hp = U p (n0 ) and Hp = Up,1 (p ), then we denote M n0 p n H r ord where n = n0 p , and we denote by αn0 (resp. αpr ) the unique principal level-n0 ord structure (resp. principal ordinary level-pr structure) determined by αHp (resp. αH ). p Proof. The statements in the first paragraph are self-explanatory. Since all objects involved are excellent (see [35, IV-2, 7.8.3]), the operations of taking formal completions and taking normalizations are interchangeable. Hence, by Theorem 3.4.1.9, it suffices to study the normalization of ED,Z in ED,Z ⊗ F0 [ζprH ]. With ~SD,Z , ~SD,Z,tor , Z(p)

~SD,Z,free , ED,Z , and ED,Z,free as in the statements of Theorem 3.4.1.9 and this theorem, let ED,Z,tor denote the group scheme of multiplicative type over Spec(Z(p) ) with character group ~SD,Z,tor . Then we have a short exact sequence 0 → ED,Z,free → ED,Z → ED,Z,tor → 0

(3.4.2.6)

of group schemes of multiplicative type over Spec(Z(p) ). Since rH ≥ rD , we have a canonical isomorphism ζprD ,~S0,r

H

: ((Z/prD Z)(1))~S0,r

∼

H

→ µprD ,~S0,r

H

over S0,rH = Spec(F0 [ζprH ]), inducing a canonical isomorphism ED,Z,tor ⊗ F0 [ζprH ] ∼ = HomS0,rH ((SD,tor )S0,rH , µprD ,S0,r ) H

Z(p)

ζ −1 S0,r prD ,~ ∼

→

H

Hom(SD,tor , (Z/prD Z)(1))S0,rH

(under the assumption that the torsion submodule ED,Z,tor is p-primary in Theorem 3.4.1.9), which shows that the underlying scheme of ED,Z,tor ⊗ F0 [ζprH ] is Z(p)

isomorphic to a disjoint union of duplicates of the base scheme S0,rH . Therefore, the scheme ED,Z ⊗ F0 [ζprH ] over S0,rH is a disjoint union of duplicaZ(p)

tes of the scheme ED,Z,free ⊗ F0 [ζprH ] over S0,rH , which admit smooth models Z(p)

ED,Z,free ⊗ OF0 ,(p) [ζ

p rH

] over ~S0,rH = Spec(OF0 ,(p) [ζprH ]). This shows that the

Z(p)

normalization of ED,Z in ED,Z ⊗ F0 [ζprH ] is a disjoint union of duplicates of the Z(p)

smooth scheme ED,Z,free ⊗ OF0 ,(p) [ζprH ] over ~S0,rH = Spec(OF0 ,(p) [ζprH ]), which is Z(p)

regular and satisfies the descriptions of the completions of strict local rings in the proposition, as desired. ~ ord ⊗ Fp is nonempty. Nevertheless, in Remark 3.4.2.7. It is not obvious that M H Z

~ ord ⊗ Fp by constructing its some special cases, we can show the nonemptiness of M H Z

partial toroidal compactifications. See Section 6.3.3 for more discussions.

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Remark 3.4.2.8. (Compare with Remark 1.1.2.1.) As in Remark 1.1.2.1, if we have chosen another PEL-type O-lattice L0 in L ⊗ Q which satisfies L ⊗ Z(p) = L0 ⊗ Z(p) , Z

Z

Z

then (by [62, Prop. 1.4.3.4 and Cor. 1.4.3.8]) we have an Z× (p) -isogeny between the tautological abelian schemes over MH (matching their additional structures), and ~ ord hence also between those over Mord H (in characteristic zero). Since MH is noetherian ord ~ normal, and since the tautological abelian scheme A → MH is ordinary, by Lemma ord 3.1.3.2, the Z× (p) -isogeny between tautological abelian schemes over MH extends 0 to a Z× (p) -isogeny A → A relating the corresponding tautological abelian schemes ~ ord (which can be identified with the isomorphic moduli problem defined over M using L0 ), and their additional structures are automatically matched by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5]. Hence, the Z× (p) -isogenous class of the tautological object depends only on the choice of L ⊗ Z(p) . Then we can Z

~ ord }H indexed by H of the form Hp Hp , with Hp an arbitrary define a collection {M H ˆ p )), and with Hp satisfying open compact subgroup of G(A∞,p ) (not just one of G(Z bal r r Up,1 (p ) ⊂ Hp ⊂ Up,0 (p ) for some integer r ≥ 0 (carrying a Hecke action as in Proposition 3.4.4.1 below). The choice of L ⊗ Zp and its filtration D, however, are Z

more substantial. Modifying the choice of L ⊗ Zp will inevitably incur isogenies of Z

degree divisible by p, which can still be done (because Lemma 3.1.3.2 also works) but will make the theory much more complicated. Convention 3.4.2.9. To facilitate the language, for S a scheme over ~S0,rH = ...ord ord ) of MH (S) is parameSpec(OF0 ,(p) [ζprH ]), we say that an object (A, λ, i, αHp , αH p ...ord ~ ord if the tautological morphism S → M terized by M H determined by the universal H ~ ord . Then it also makes sense to consider the pullproperty factors through S → M H ...ord ~ ord . back of the tautological tuple over MH as the tautological tuple over M H Definition 3.4.2.10. (Compare with [62, Def. 5.3.2.1] and Definition 1.3.1.1.) Let S be a normal locally noetherian algebraic stack over ~S0,rH . A tuple ord ~ ord , or (G, λ, i, αHp , αH ) over S is called a degenerating family of type M H p simply a degenerating family when the context is clear, if there exists a dense subalgebraic stack S1 of S, such that we have the following: (1) By viewing group schemes as relative schemes (cf. [37]), G is a semi-abelian scheme over S whose restriction GS1 to S1 is an abelian scheme. In this case, the dual semi-abelian scheme G∨ exists (up to unique isomorphism; cf. [80, IV, 7.1] or [62, Thm. 3.4.3.2]), whose restriction G∨ S1 to S1 is the dual abelian scheme of GS1 . (2) λ : G → G∨ is a group homomorphism that induces by restriction a polarization λS1 of GS1 . (3) i : O → EndS (G) is a homomorphism that defines by restriction an O-structure iS1 : O → EndS1 (GS1 ) of (GS1 , λS1 ).

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ord ~ ord (see (4) (GS1 , λS1 , iS1 , αHp , αH ) → S1 defines a tuple parameterized by M H p Convention 3.4.2.9). ord (5) The ordinary level structure αH , which is an orbit of ´etale-locally-defined p ord,0 → GS1 [pr ] and : (Gr0D,pr )mult , νpord , αpord,#,0 triples (αpord,0 r r r ), where αpr S1 r ord αpord,#,0 : (Gr0D# ,pr )mult → G∨ r S1 S1 [p ] are closed immersions and where νpr × is a section of (Z/pr Z)S1 , extend to an orbit of ´etale-locally-defined triples ord,#,0 ord,0 0 mult → G[pr ] and αpord,#,0 : (αpord,0 , νpord r ,S ), where αpr ,S : (GrD,pr )S r ,S r ,S , αpr ,S 0 mult ∨ r ord (GrD# ,pr )S → G [p ] are closed immersions and where νpr ,S is a section × of (Z/pr Z)S , that is also symplectic-liftable as in Definition 3.3.3.4. By ord abuse of language, we say in this case that αH extends to an ordinary level p ord structure of (G, λ, i) over S. Since αHp is determined by its restriction to every dense subalgebraic stack, by abuse of notation, we shall use the same notation for every such restriction.

Remark 3.4.2.11. The extensibility condition (5) in Definition 3.4.2.10 is nontrivial even when G is an abelian scheme, because it forces G to be an ordinary abelian scheme over S. If S1 is merely (of characteristic zero) defined over S0,rν , then GS1 is always ordinary over S1 , but G is not an ordinary abelian scheme over S in general. Remark 3.4.2.12. Conditions (2), (3), and (4) are closed conditions for structures on abelian schemes defined over ~S0,rH . Hence, the rather weak condition for S1 in Definition 3.4.2.10 is justified because S1 can always be replaced with the largest subalgebraic stack of S over ~S0,rH (which is open dense in S) such that GS1 is an abelian scheme. (Conditions (2) and (3) are closed by [62, Lem. 4.2.1.6] and by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5]. Condition (4) is closed thanks to condition (5); see Remark 3.4.2.11.) 3.4.3

Properties of Kodaira–Spencer Morphisms

Definition 3.4.3.1. (Compare with Definitions 1.1.2.8 and 1.3.1.2.) Let ord (G, λ, i, αHp , αH )→S p

~ ord (as in Definition 3.4.2.10). Then we define be a degenerating family of type M H the OS -module KS = KS(G,λ,i)/S = KS(G,λ,i,αHp ,αord H )/S p

by setting KS :=

(Lie∨ G/S

⊗

OS

Lie∨ G∨ /S )/



λ∗ (y) ⊗ z − λ∗ (z) ⊗ y (b? x) ⊗ y − x ⊗(by)

 x∈Lie∨ G/S , y,z∈Lie∨ , G∨ /S b∈O

We also define the OS -module KSfree = KS(G,λ,i)/S,free = KS(G,λ,i,αHp ,αord H )/S,free p

.

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to be the quotient of KS defined as the image of the canonical morphism KS → KS ⊗ Q Z

of OS -modules. Remark 3.4.3.2. By definition, the sheaf KS(G,λ,i)/S,free contains no p-torsion and hence is flat over ~S0,rH = Spec(OF0 ,(p) [ζprH ]) (although it can be pathologically different from KS if S is not flat over ~S0,r ). (G,λ,i)/S

H

Proposition 3.4.3.3. (Compare with [62, Prop. 2.3.5.2].) Let ord (A, λ, i, αHp , αH )→S p

~ ord , with tautological morphism f : S → M ~ ord . Suppose be a tuple parameterized by M H H ~ that S is smooth over S0,rH , which implies that S is flat over Spec(Z(p) ), and that the OS -module Ω1S/~S is locally free. Let KS(A,λ,i)/S and KS(A,λ,i)/S,free be defined 0,rH

ord by (A, λ, i, αHp , αH ) → S as in Definition 3.4.3.1 (with G = A and S1 = S; cf. p Definition 1.1.2.8). Then the Kodaira–Spencer morphism

KS = KSA/S/~S0,r

H

∨ 1 : Lie∨ A/S ⊗ LieA∨ /S → ΩS/~ S OS

0,rH

(see [62, Def. 2.1.7.9]) canonically induces a morphism KS : KS(A,λ,i)/S → Ω1S/~S

0,rH

,

(3.4.3.4)

which factors through KS : KS(A,λ,i)/S,free → Ω1S/~S

0,rH

.

(3.4.3.5)

Moreover, the morphism f is ´ etale if and only if it is flat and (3.4.3.5) is an isomorphism. Proof. The canonical morphism (3.4.3.4) exist by the same argument in the proof of [62, Prop. 2.3.5.2], because only the properties of (A, λ, i) are used there. Since Ω1S/~S is locally free over the scheme S flat over Spec(Z(p) ), it is also torsion-free. 0,rH

Hence, (3.4.3.4) factors through (3.4.3.5). Suppose the morphism f is ´etale. To show that (3.4.3.5) is an isomorphism over ~ ord , or rather over the completions of the S, it suffice to show it (universally) over M H ord ~ strict local rings of M H at its geometric points. Let us replace S with the spectrum b1 of any such complete local rings, and replace Ω1S/~S with its completion Ω S/~ S 0,rH

0,rH

(with respect to the topology of the complete local ring). At geometric points of residue characteristic zero, we can conclude the proof by citing [62, Prop. 2.3.5.2]. Hence, we only need to consider geometric points of residue characteristic p. By ~ ord (see Theorem 3.4.2.5), the scheme S is the normalization the construction of M H ...ord of the spectrum S 0 of the completion of a strict local ring of MH ⊗ OF0 ,(p) [ζprH ], Z(p)

and we may assume that (A, λ, i) → S is the pullback of some (A0 , λ0 , i0 ) → S 0 . Let

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b1 Ω S 0 /~ S

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0,rH

denote the similar completion of Ω1S 0 /~S

. (These are the correct targets

0,rH

of the Kodaira–Spencer morphisms over completed bases which are not necessarily of finite type over ~S0,rH ; cf. the proof of [62, Prop. 2.3.5.2].) The canonical morphism b1 0 (S → S 0 )∗ Ω S /~ S

0,rH

b1 →Ω S/~ S

0,rH

can be identified with the pullback (from Spec(Z(p) ) to S) of the canonical morphism ∨ Lie∨ ED,Z / Spec(Z(p) ) → LieED,Z,free / Spec(Z(p) )

induced by (3.4.2.6), with kernel the maximal torsion subsheaf of Lie∨ ED,Z / Spec(Z(p) ) . This identification is compatible with the Kodaira–Spencer isomorphism ∼ b1 KS(A0 ,λ0 ,i0 )/S 0 → Ω S 0 /~ S

(3.4.3.6)

0,rH

induced by the Kodaira–Spencer morphism in the Serre–Tate deformation theory (see [47]) (which is compatible with the usual Kodaira–Spencer morphism defined over smooth schemes), and with the compatible canonical isomorphisms from 0 KS(A0 ,λ0 ,i0 )/S 0 and KS(A,λ,i)/S to pullbacks of Lie∨ ED,Z / Spec(Z(p) ) to S and S, respectively, induced by (6) of Proposition 3.2.1.1. Thus, we see that the pullback of the isomorphism (3.4.3.6) (from S 0 to S) induces the desired isomorphism. Conversely, suppose (3.4.3.5) is an isomorphism. By the previous paragraph, we ∼ 1 , where by abuse of nohave an isomorphism KS : KS(A,λ,i)/M ~ ord ,free → ΩM ~ ord /~ S H

H

0,rH

~ ord . Since tation we have also used (A, λ, i) to denote the tautological objects over M H the construction of KS(A,λ,i)/S,free commutes with flat base change (between schemes smooth over ~S0,r ), and since the association of Kodaira–Spencer morphisms H

is functorial, the first morphism in the exact sequence f ∗ Ω1M ~

~

H,rH /S0,rH

→ Ω1S/~S

0,rH

→ Ω1S/M ~

H,rH

→0

is an isomorphism. This shows that f is unramified, and hence ´etale because it is flat by assumption. 3.4.4

Hecke Actions

Proposition 3.4.4.1. (Compare with Propositions 1.3.1.14 and 2.2.3.1.) Suppose ∞ we have an element g = (g0 , gp ) ∈ G(A∞,p ) × Pord D (Qp ) ⊂ G(A ) (see Definition ˆ such 3.2.2.7), and suppose we have two open compact subgroups H and H0 of G(Z) 0 −1 0 that H ⊂ gHg , and such that H and H are of standard form as in Definition ...ord ...ord 3.2.2.9 (so that MH and MH0 are defined as in Definition 3.4.1.1). Suppose moreover that gp satisfies the conditions given in Section 3.3.4. Then the assignment in Proposition 3.3.4.21 of ordinary Hecke twists by g induces a canonical quasi-finite surjection ... ord ...ord ...ord [g] : MH0 → MH

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...ord (over Spec(Z(p) )), such that the tautological tuple over MH is pulled back to the ... ord ...ord ordinary Hecke twist of the tautological tuple over MH0 by g. The surjection [g] is finite if the levels Hp and Hp0 at p are equally deep as in Definition 3.2.2.9, or if gp is of twisted Up type as in Definition 3.3.6.1 and depthD (Hp0 ) − depthD (gp ) = depthD (Hp ) > 0. ... ord ...ord ...ord By Proposition 3.3.5.6, the surjection [g] : MH0 → MH is compatible with the surjection [g] : MH0 → MH over Mord H0 , and induces surjective quasi-finite flat morphisms ord [g]ord : Mord H0 → MH

(compatible with S0,rH0 → S0,rH ) and ~ ord : M ~ ord0 → M ~ ord [g] H H (compatible with ~S0,rH0 → ~S0,rH ) compatible with each other. If gp ∈ Pord D (Zp ), then the induced morphisms ord ord [g]ord × ~S0,rH0 rH0 : MH0 → MH ~ S0,rH

and ~ ord : M ~ ord0 → M ~ ord × ~S0,r 0 [g] H H rH0 H ~ S0,rH

... ord are quasi-finite ´etale. These morphisms induced by [g] are characterized by the property that the pullback of the tautological tuple is the ordinary Hecke twist of the ... ord tautological tuple by g. They are finite if [g] is finite (see the previous paragraph). If g = g1 g2 , where g1 = (g1,0 , g1,p ) and g2 = (g2,0 , g2,p ) are elements of G(A∞,p ) × Pord D (Qp ), each having a setup similar to that of g, then we have ... ord ... ord ... ord ~ ord = [g~2 ]ord ◦ [g~1 ]ord . [g] = [g2 ] ◦ [g1 ] , inducing [g] ... ord ...ord ...ord Proof. The morphism [g] : MH0 → MH (uniquely) exists by the definition ...ord of ordinary Hecke twists, and by the definition of MH as a moduli problem. Its surjectivity is a consequence of the liftability conditions in the definitions of level structures. Its quasi-finiteness, and its finiteness when the levels Hp and Hp0 at p are equally deep, follow from the definition of ordinary level structures as orbits of ´etale-locally-defined principal ordinary level structures (see Definitions 3.3.3.3 and 3.3.3.4). When gp is of twisted Up type as in Definition 3.3.6.1 and depthD (Hp0 ) − ... ord depthD (gp ) = depthD (Hp ) > 0, since [g] is (of finite presentation and) quasifinite, its finiteness follows from Lemma 3.3.6.6 (which verifies its properness by ~ ord0 and M ~ ord are regular the valuative criterion; cf. [35, IV-3, 8.11.1]). Since M H H and equidimensional of the same dimension (see Theorem 3.4.2.5), the morphism ~ ord0 → M ~ ord is automatically flat (by [35, IV-3, 15.4.2 e0 )⇒b)]; cf. [62, Lem. [g]ord : M H H

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~ ord : M ~ ord0 → M ~ ord × ~S0,r 0 is ´etale because 6.3.1.11]). The induced morphism [g] H H rH 0 H ~ S0,rH

it induces an isomorphism between the completions of strict local rings (again see Theorem 3.4.2.5). The last statement of this proposition follows from the last statement of Proposition 3.3.4.21. The remaining statements of this proposition are self-explanatory. ~ H, Definition 3.4.4.2. For each algebraic stack or scheme over Spec(Z), such as M we denote in Fraktur its formal completion along its fiber over Spec(Fp ) → Spec(Z), ~ H , and consider it a formal algebraic stack over Spf(Zp ), with support an such as M algebraic stack over Spec(Fp ). By abuse of notation, we denote the formal comple~ H ] as [M ~ H ]. tion of [M Corollary 3.4.4.3. With the setting as in Proposition 3.4.4.1, the morphism ~ ord : M ~ ord0 → M ~ ord [g] H H ~ ord : M ~ ord0 → M ~ ord is finite flat. Hence, if gp ∈ Pord (Zp ), then the induced by [g] D H H induced morphism ~ ord : M ~ 0,r 0 ~ ord0 → M ~ ord × S [g] H H rH0 H ~ 0,r S H

is finite ´etale (because it is quasi-finite ´etale by Proposition 3.4.4.1). ~ ord : M ~ ord : M ~ ord and [g] ~ ord0 → ~ ord0 → M Proof. The quasi-finite flat morphisms [g] H H H rH0 ~ 0,r 0 are finite because the induced morphism ~ ord × S M H

~ 0,r S H

H

~ ord : M ~ ord0 ⊗ Fp → M ~ ord ⊗ Fp [g] H H Z

Z

is proper by Lemma 3.3.6.8 (cf. [35, IV-3, 8.11.5, or IV-4, 8.12.6]). Corollary 3.4.4.4. With the setting as in Proposition 3.4.4.1, if g = (g0 , gp ) ∈ ˆ p ) × Pord (Zp ) (cf. Examples 3.3.4.5 and 3.3.4.18), if H0,p = g0 Hp g −1 in G(Z ˆ p ), G(Z D 0 ord 0 ord −1 ord in MD (Zp ) (see (3.3.3.5)), then (rH0 = rH and) the and if Hp = (gp Hp gp ) ord ~ ~ ord0 → M ~ ord is an isomorphism. (These conditions are induced morphism [g] :M H

H

true, in particular, when g = 1 and when H = Hp Hp and H0 = H0,p Hp0 satisfy ord H0,p = Hp and Hp0 = Hpord .) Proof....By assumption (and by the moduli interpretations), the canonical mor... ... ... ¯ phism [g] : MH0 → MH between the moduli problems induces a bijection MH0 (F p) → ... ¯ ¯ p -valued points. Hence, by the description of the local strucbetween their F MH (Fp )... ... tures of MH0 and MH in Theorem 3.4.2.5 (which asserts that the completions of strict ¯ p -points are isomorphic to the completions of local rings of both of them at their F strict local rings of the same group of multiplicative type), and by the definition of

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~ ord : M ~ ord0 and M ~ ord as their respective normalizations, we see that [g] ~ ord0 → M ~ ord M H H H H ord ¯ ord ¯ ~ ~ ¯ also induces a bijection MH0 (Fp ) → MH (Fp ) between their Fp -valued points. ~ ord : M ~ ord0 → M ~ ord (being Hence, by Corollary 3.4.4.3, the induced morphism [g] H H ¯ p -valued points) is an isomorphism. finite ´etale and a bijection on F ord

= (gp Hp gp−1 )ord is Example 3.4.4.5. The condition in Corollary 3.4.4.4 that Hp0 0 r 1 r0 0 also true, for example, when gp = 1, Hp = Up,1,0 (p , p ), and Hp = Up,1,0 (pr1 , pr0 ) for some r1 ≤ r0 ≤ r00 . Corollary 3.4.4.6 (elements of Up type). Suppose in Proposition 3.4.4.1 that g0 = 1 and gp is of Up type as in Definition 3.3.6.1 (so that it is of twisted Up type and depthD (gp ) = 1). Then the induced morphism ~ ord : M ~ ord ⊗ Fp ~ ord0 ⊗ Fp → M [g] H H Z

(3.4.4.7)

Z

is finite flat and coincides with the composition of the (finite flat) absolute Frobenius morphism ~ ord ~ ord FM ~ ord ⊗ Fp : MH0 ⊗ Fp → MH0 ⊗ Fp H0

Z

Z

Z

with the canonical finite flat morphism ~ ord : M ~ ord0 ⊗ Fp → M ~ ord ⊗ Fp [1] H H Z

(3.4.4.8)

Z

(see Corollary 3.4.4.3). ord = Hpord as open compact subgroups of Mord If Hp0 D (Zp ) (see (3.3.3.5)), then (rH0 = rH and) the canonical morphism (3.4.4.8) is an isomorphism by Corollary 3.4.4.4, and the composition ord −1

~ ([1]

~ ord M H

⊗ Fp

∼

)

→

~

ord

~ ord0 ⊗ Fp [g]→ M ~ ord ⊗ Fp M H H

Z

Z

Z

coincides with the (finite flat) absolute Frobenius morphism ~ ord ~ ord FM ~ ord ⊗ Fp : MH ⊗ Fp → MH ⊗ Fp . H

Z

Z

Z

0 ord Proof. Since g0 = 1, the ordinary Hecke twist (A0 , λ0 , i0 , αH p , αH ) of the tautolop ord ord ~ 0 ⊗ Fp by g = (g0 , gp ) is defined essentially gical object (A, λ, i, αH0,p , α 0 ) over M Hp

H

Z

only by gp , equipped with the morphism [gp−1 ] : A → A0 which is nothing but 0 ∼ (p) the relative Frobenius morphism FA/M , with the additi~ ord ⊗ Fp : A → A = A 0 H

Z

onal structures naturally induced, as explained in Examples 3.3.4.7 and 3.3.4.20. 0 ord Hence, (A0 , λ0 , i0 , αH p , αH ) coincides with the object naturally induced by the pulp ord lback of (A, λ, i, αH0,p , αH 0 ) by the absolute Frobenius F ~ ord M ⊗ Fp . Hence, the first p H0

Z

paragraph of the corollary follows. The second paragraph of the corollary is selfexplanatory.

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Remark 3.4.4.9. By Kunz’s theorem [54] (cf. [76, Sec. 42, Thm. 107]), the absolute Frobenius morphisms FM ~ ord ⊗ Fp and FM ~ ord ⊗ Fp in Corollary 3.4.4.6 are flat because H0

Z

H

Z

~ ord0 ⊗ Fp and M ~ ord ⊗ Fp are regular (by smoothness of M ~ ord0 and M ~ ord over ~S0,r 0 M H H H H H Z

Z

and ~S0,rH , respectively; see Theorem 3.4.2.5). 3.4.5

The Case When p is a Good Prime

Let H, Hp , and Hp be as in Section 3.4.1. When p is a good prime (for the integral PEL datum (O, ?, L, h · , · i, h0 ); see ~ ord directly without taking the normalization of Definition 1.1.1.6), we can define M H an object in characteristic zero. In this case, the pairing h · , · i is self-dual after base change to Zp , and hence we can define MHp over ~S0 = Spec(OF0 ,(p) ) as in [62, Def. 1.4.1.4]. On the other hand, consider Up,0 (p0 ) = G(Zp ). Then rD = 0 by [62, Prop. 1.2.2.3] and by the assumption that p is a good prime, and ν(G(Zp )) = Z× p implies that rHp G(Zp ) = rν(G(Zp )) = 0 (see Definition 3.4.2.1). Let H0 := Hp G(Zp ). Then ∼ ~ ord0 over ~S0 as in Theorem 3.4.2.5, such that M ~ ord0 ⊗ Q ∼ we can define M = Mord H H H0 = MH0 Z

over S0 . Lemma 3.4.5.1. There is a canonical open immersion ~ ord0 ,→ MHp M H

(3.4.5.2)

such that the pullback of the tautological object (A, λ, i, αHp ) over MHp is part of ~ ord0 (cf. Convention 3.4.2.9). the tautological object over M H Proof. Consider the open immersion (cf. [35, IV-4, 17.9.1, and IV-2, 6.15.3]) Mord Hp ,→ MHp

(3.4.5.3)

ord representing the ordinary level structure αG(Z over MHp (which is unique up to p) ...ord isomorphism if it exists). The two moduli problems Mord Hp and MH0 (cf. Definition 3.4.1.1) are almost identical, except that the former requires the Lie algebra condition in [62, Def. 1.3.4.1] given by (L ⊗ R, h · , · i, h0 ) (and hence has to be defined Z

over the finite extension OF0 ,(p) of Z(p) ). Therefore, we have a canonical finite morphism ...ord Mord (3.4.5.4) Hp → MH0 . ~ ord0 satisfies the Lie algebra On the other hand, the tautological tuple over M H condition, because the condition is given by an identity of polynomials (which is a ∼ closed condition), and because it is already satisfied over the generic fiber Mord H0 = MH0 . Therefore, by Proposition 3.4.3.3 and [62, Prop. 2.3.5.2], and by the valuative criterion of properness, there is a canonical finite ´etale morphism ~ ord0 → Mordp M H H

(3.4.5.5)

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by the universal property of Mord Hp , whose composition with (3.4.5.4) is the canonical ...ord ~ ord0 to M Comparing this with the open and closed finite morphism from M 0 . H

H

immersion (2.2.4.1), we see that (3.4.5.5) is also an open and closed immersion, and that the composition of (3.4.5.5) with (3.4.5.3) gives the desired open immersion (3.4.5.2). By the liftability of level structures, we have a canonical finite ´etale surjection ...ord ...ord MH  MH0 and a canonical quasi-finite ´etale surjection MH  MH0 , which induces a canonical quasi-finite ´etale morphism ~ ord  M ~ ord0 × ~S0,r . M H H H

(3.4.5.6)

~ S0

(Here rH = rν because rD = 0.) Alternatively, we have: Proposition 3.4.5.7. With assumptions as above, we can construct (3.4.5.6), or rather the canonical quasi-finite ´etale morphism ~ ord → MHp × ~S0,r M H H

(3.4.5.8)

~ S0

(which is the composition of (3.4.5.6) with (3.4.5.2)), as a canonical open and closed subalgebraic stack, given by taking the schematic closure of Mord H , in a relatively representable functor of ordinary level-Hp structures of type (L ⊗ Zp , h · , · i, D). Z

Proof. The relative representability, quasi-finiteness, and ´etaleness are all clear form the definitions. 3.4.6

Quasi-Projectivity of Coarse Moduli

In this subsection, we no longer assume that p is a good prime (for the integral PEL datum (O, ?, L, h · , · i, h0 )). Our ultimate source of projectivity or quasi-projectivity is [80, IX, 2.1], or its reformulation in [28, Ch. V, Prop. 2.1] and [62, Prop. 7.2.1.1]. For this purpose, we need a semi-abelian scheme over a proper algebraic stack, and we shall resort to the auxiliary moduli problems (again) and their compactifications (also). (This seems to be our only source of projectivity, since geometric invariant theory as in [83] is not known to be applicable to the construction of minimal compactifications.) Lemma 3.4.6.1. Let H and Haux be as in Lemma 3.4.1.6. Let top top ∗ 1 eA ΩA/M ωM Lie∨ ~ ord := ∧ ~ ord = ∧ ~ ord , A/M H

H

H

and ωM ~ ord

Haux

:= ∧top Lie∨ A

~ ord aux /MHaux

= ∧top e∗Aaux Ω1A

~ ord aux /MHaux

,

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~ ord (resp. Aaux → M ~ ord ) is the tautological abelian scheme with where A → M Haux ~ ord → A (resp. eA : M ~ ord → Aaux ). Then the morphism identity section eA : M aux Haux (3.4.1.7) canonically induces a quasi-finite morphism ~ ord → M ~ ord M (3.4.6.2) H

Haux

a0 compatible with (2.1.1.17), such that the pullback of ω ⊗ ~ ord

MHaux

is canonically isomor-

a phic to ω ⊗ ~ ord , where a0 ≥ 1 and a ≥ 1 are integers as in Lemma 2.1.2.35. MH

Proof. These follow from the constructions of the morphisms (3.4.1.5) and (3.4.1.7) (see the proofs of Lemmas 3.4.1.4 and 3.4.1.6) and the two normalizations Mord H and Mord , and from Lemma 3.4.1.8. Haux Proposition 3.4.6.3. With the setting as in Proposition 3.4.2.5, there is a canonical morphism ~ ord → M ~H M (3.4.6.4) H (see Proposition 2.2.1.1) inducing an open immersion ~ ord ] ,→ [M ~ H,r ] [M

(3.4.6.5) H H ~ ~λ,~i) (see Definition 2.2.3.5). Under (3.4.6.4), the pullback of the tautological (A, (see Proposition 2.2.1.1) is canonically isomorphic to the tautological (A, λ, i) over ~ H,r is a scheme quasi~ ord . If Hp is neat, then (H = Hp Hp is also neat and) M M H H ~ ~ ord (which, a projective over S0,rH by Proposition 2.2.1.1, and the algebraic stack M H priori, is an algebraic space by Theorem 3.4.2.5) is a scheme quasi-projective over ~S0,r and is canonically embedded as an open subscheme of M ~ H,r . H H Proof. By Lemma 3.4.6.1, we have a quasi-finite morphism ~ ord → M ~ ord ˆ M H

Gaux (Z)

(cf. (3.4.6.2)). By composition with the canonical open immersion ~ ord ˆ ,→ M M ˆp ) Gaux (Z G (Z) aux

as in (3.4.5.2), we obtain a quasi-finite morphism ~ ord → M M ˆp ) , H Gaux (Z ~ H as a norwhich induces the morphism (3.4.6.4) by the universal property of M malization, and induces the open immersion (3.4.6.5) by Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11], and the formulation in [62, Prop. 7.2.3.4] for algebraic spaces). Since the morphism (3.4.6.4) extends the canonical morphism Mord ∼ = MH,r → MH , H

H

~ ~λ,~i) to Mord is canonically isomorphic to the pullback of the the pullback of (A, H ~ ord to Mord . Since M ~ ord is noetherian and normal, tautological (A, λ, i) over M H H H ord ord ~ , by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, and since MH is dense in M H ~ ~λ,~i) under (3.4.6.4) is canonically Prop. 3.3.1.5], it follows that the pullback of (A, ord ~ isomorphic to the tautological (A, λ, i) over MH , as desired.

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Chapter 4

Degeneration Data and Boundary Charts

In this chapter, we explain how to incorporate the considerations of ordinary level structures into the theory of degeneration data and the boundary construction in [62]. (We no longer assume as in Section 3.4.5 that p is a good prime.) This is the technical heart of the whole work. Readers are encouraged to read this chapter only after mastering the earlier results in [82], [28, Ch. II–IV], and [62, Ch. 4–6]. Although the notation is quite heavy, it is designed to be as close as possible to the one in [62, Ch. 4–6], so that readers who are already familiar with the arguments there can easily see what the new considerations here are. 4.1

Theory of Degeneration Data

4.1.1

Degenerating Families of Type (PE, O)

Let O be as above; that is, as in Definition 1.1.1.1, O is an order in a finitedimensional semisimple Q-algebra with a positive involution ? stabilizing O. Definition 4.1.1.1. Let S be any normal locally noetherian scheme over Spec(Z). A degenerating family of type (PE, O) is a tuple (G, λ, i) over S such that we have the following: (1) G is a semi-abelian scheme over S. (2) There exists an open dense subscheme S1 of S such that GS1 is an abelian scheme. In this case, there is a unique semi-abelian scheme G∨ (up to unique isomorphism), called the dual semi-abelian scheme of G (cf. cf. [80, IV, 7.1] or [62, Thm. 3.4.3.2]), such that G∨ S1 is the dual abelian scheme of GS1 . (3) λ : G → G∨ is a group homomorphism that induces by restriction a polarization λS1 of GS1 . (4) i : O → EndS (G) is a map that defines by restriction an O-structure iS1 : O → EndS1 (GS1 ) of (GS1 , λS1 ). (See [62, Def. 1.3.3.1].) Remark 4.1.1.2. In Definition 4.1.1.1, we allow S to be an arbitrary scheme over Spec(Z) (without any reference to the reflex field F0 , as opposed to the case in 215

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Definition 1.3.1.1). 4.1.2

Common Setting for the Theory of Degeneration

Let R be a noetherian normal domain complete with respect to an ideal I, with rad(I) = I for convenience. Let S := Spec(R), K := Frac(R), η := Spec(K) the generic point of S, and S0 := Spec(R/I). We shall denote the pullbacks to η or S0 with subscripts “η” or “0”, respectively. 4.1.3

Degeneration Data for Polarized Abelian Schemes with Endomorphism Structures

Definition 4.1.3.1. (See [62, Def. 5.1.1.2].) With notation and assumptions as in Section 4.1.2, the category DEGPE,O (R, I) has objects consisting of degenerating families (G, λ, i) of type (PE, O) (over S = Spec(R)) such that G0 is an extension of an abelian scheme B0 by a isotrivial torus T0 (see [62, Def. 3.1.1.5]) over S0 . Definition 4.1.3.2. (See [62, Def. 5.1.1.3].) With notation and assumptions as in Section 4.1.2, the category DDPE,O (R, I) has objects of the form (B, λB , iB , X, Y , φ, c, c∨ , τ ), with entries described as follows: (1) B is an abelian scheme over S, λB : B → B ∨ is a polarization of B, and iB : O ,→ EndS (B) is an O-endomorphism structure of (B, λB ). (2) X and Y are two ´etale sheaves (of O-lattices) canonically dual to two isotrivial tori T and T ∨ , respectively, carrying O-actions over S, together with an O-equivariant embedding φ : Y → X with finite cokernel. (We shall denote the actions of an element b ∈ O on X and Y by iX (b) and iY (b), respectively. When the context is clear, we shall simply denote the actions by b.) (3) c : X → B ∨ and c∨ : Y → B are two O-equivariant morphisms of group schemes over S, satisfying the compatibility cφ = λB c∨ . ∼ ⊗ −1 (4) τ : 1Y × X,η → (c∨ × c)∗ PB,η is a trivialization of biextensions with symmetric pullback under IdY × φ : Y × Y → Y × X, satisfying the following conditions: (a) (Compatibility with O-actions:) For each b ∈ O, we have a canonical identification of sections (iY (b) × IdX )∗ τ = (IdY ×iX (b? ))∗ τ ∼ under the canonical isomorphism (iB (b) × IdB ∨ )∗ PB = ∨ ∗ (IdB ×(iB (b)) ) PB . (b) (Positivity:) After a finite ´etale surjective base change in S if necessary, let us assume that X and Y are constant with values in X and Y , respectively. For each y ∈ Y and χ ∈ X, the trivialization τ (y, χ) defines an isomorphism of invertible sheaves from (c∨ (y), c(χ))∗ PB,η to 1η . Under this isomorphism (which we again denote by τ (y, χ)), the canonical R-integral structure (c∨ (y), c(χ))∗ PB of (c∨ (y), c(χ))∗ PB,η

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determines an invertible R-submodule Iy,χ of K. Then the positivity condition is that Iy,φ(y) ⊂ I for all nonzero y in Y . (Clearly, I0,0 = R.) By the theory of degeneration data for polarized abelian varieties in [28, Ch. II and III] (explained in [62, Ch. 4]), generalized by functoriality for polarized abelian varieties with endomorphism structures (see [62, Sec. 5.1.1]), the so-called Mumford’s construction induces an equivalence of categories MPE,O (R, I) : DDPE,O (R, I) → DEGPE,O (R, I) : (B, λB , iB , X, Y , φ, c, c∨ , τ ) 7→ (G, λ, i)

(4.1.3.3)

realizing (G, λ, i) (up to isomorphism) as the image of an object of DDPE,O (R, I). We say that (B, λB , iB , X, Y , φ, c, c∨ , τ ) is the degeneration datum of (G, λ, i). The theory works even when X and Y are zero. (Then, by [28, Ch. I, 2.8], G ∼ =B is an abelian scheme over S, and the positivity condition for τ is trivially verified.) We shall suppress I from the notation when it is clear from the context. (This is the case, for example, when R is a discrete valuation ring.) 4.1.4

Degeneration Data for Principal Ordinary Level Structures

Let S = Spec(R) be as in Section 4.1.2. Let (G, λ, i) be a degenerating family of type (PE, O) over S as in Definition 4.1.3.1, with degeneration datum (B, λB , iB , X, Y , φ, c, c∨ , τ ) given by (4.1.3.3). Suppose (Gη , λη , iη ) is equipped with some naive principal ordinary level-pr r = (αpord,0 , αpord,#,0 , νpord structure αpord r r r ) of type (L/p L, h · , · i, Dpr ) (see Definition r 3.3.3.1). Explicitly, the first two entries are O-equivariant homomorphisms → G[pr ]η αpord,0 : (Gr0Dpr )mult r η and : (Gr0D# )mult → G∨ [pr ]η αpord,#,0 r η pr

that are closed immersions, which, together with the third entry ∼

νpord : µpr ,η → µpr ,η , r satisfy the symplectic condition ord,#,0 λη ◦ αpord,0 = νpord ◦ (φ0D,pr )mult r r ◦ αpr η

in Definition 3.3.3.2. Assume in addition that it satisfies the following: Condition 4.1.4.1. The schematic image of αpord,0 (resp. αpord,#,0 ) contains the r r r ∨ r r ∨ r subscheme T [p ]η (resp. T [p ]η ) of G[p ]η (resp. G [p ]η ). Definition 4.1.4.2. If the principal ordinary level-pr structure αpord satisfies Conr dition 4.1.4.1, then we say that αpord is compatible with the degeneration. r

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satisfies Condition One immediate consequence of the assumption that αpord r 4.1.4.1 (i.e., is compatible with degeneration) is that the pullbacks of the two subgroup schemes of multiplicative type ) ((Gr0Dpr )mult ) = αpord,0 T [pr ]η ⊂ image(αpord,0 r r η and ((Gr0D# )mult ) T ∨ [pr ]η ⊂ image(αpord,#,0 ) = αpord,#,0 r r η pr

determine, respectively, two totally isotropic and αpord,#,0 under αpord,0 r r O-submodules of Gr0D,pr and Gr0D# ,pr which are compatible with φ0D,pr : Gr0D,pr → Gr0D# ,pr in the sense that φ0D,pr maps the first submodule to the second submodule. This is equivalent to the determination of a symplectic filtration Z−3,pr = 0 ⊂ Z−2,pr ⊂ Z−1,pr ⊂ Z0,pr = L/pr L

(4.1.4.3)

of O-submodules, with dual filtration # # # # r # Z# −3,pr = 0 ⊂ Z−2,pr ⊂ Z−1,pr ⊂ Z0,pr = L /p L

(4.1.4.4)

with respect to the pairing h · , · i (which implies that canonical morphism L/pr L → L# /pr L# induces morphisms Z−i,pr → Z# −i,pr for each i), satisfying the compatibilities Z−2,pr ⊂ D0pr ⊂ Z−1,pr

(4.1.4.5)

#,0 # Z# −2,pr ⊂ Dpr ⊂ Z−1,pr

(4.1.4.6)

and

(cf. (3.2.3.3) and (3.2.3.5)) such that the above two submodules of Gr0D,pr and Gr0D# ,pr are respectively the two submodules GrZ−2,pr ⊂ Gr0D,pr

(4.1.4.7)

and #

GrZ−2,pr ⊂ Gr0D# ,pr ,

(4.1.4.8)

together with two isomorphisms ∼

(ϕ−2,pr )mult : (GrZ−2,pr )mult → T [pr ]η η η

(4.1.4.9)

and #

∼

mult (ϕ# : (GrZ−2,pr )mult → T ∨ [pr ]η η −2,pr )η

determined by mult νpord := αpord,0 |(GrZ−2,pr )mult r r ◦ (ϕ−2,pr )η η

and mult (ϕ# := αpord,#,0 |(GrZ# r −2,pr )η

)mult −2,pr η

.

(4.1.4.10)

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mult are equivalent to the two isoThese two morphisms (ϕ−2,pr )mult and (ϕ# η −2,pr )η morphisms ∼

ϕ−2,pr : GrZ−2,pr → Hom((X/pr X)η , (Z/pr Z)(1))

(4.1.4.11)

and #

∼

Z r r ϕ# −2,pr : Gr−2,pr → Hom((Y /p Y )η , (Z/p Z)(1)),

(4.1.4.12)

respectively. (Thus far, by abuse of notation, we have used notation such as GrZ−2,pr #

#

and GrZ−2,pr for the constant sheaves (GrZ−2,pr )η and (GrZ−2,pr )η over the point η. We will adopt a similar abuse of notation in what follows.) By the perfect duality # h · , · i02 : GrZ0,pr × GrZ−2,pr → (Z/pr Z)(1) induced by h · , · i (by the definition of (3.2.3.3) and (3.2.3.5)), this last isomorphism ϕ# −2,pr is canonically equivalent to an isomorphism ∼

ϕ0,pr : GrZ0,pr → (Y /pr Y )η .

(4.1.4.13)

The compatibility (4.1.4.5) induces a filtration D−1,pr = {Di−1,pr }i on GrZ−1,pr given by Z D1−1,pr := 0 ⊂ D0−1,pr := D0pr /Z−2,pr ⊂ D−1 −1,pr := Gr−1,pr

(4.1.4.14)

(serving the same purpose as the filtration Dpr does for L/pr L). Similarly, the #,i Z# compatibility (4.1.4.6) induces a filtration D# −1,pr = {D−1,pr }i on Gr−1,pr given by #

# #,−1 #,1 #,0 #,0 Z D−1,p r := 0 ⊂ D−1,pr := Dpr /Z−2,pr ⊂ D−1,pr := Gr−1,pr .

(4.1.4.15)

The filtrations (4.1.4.14) and (4.1.4.15) are dual to each other with respect to the # pairing h · , · i11 : GrZ−1,pr × GrZ−1,pr → (Z/pr Z)(1) induced by h · , · i. Then we have Gr0D−1,pr = Gr0D,pr / GrZ−2,pr

(4.1.4.16)

and Gr0D#

−1,pr

#

= Gr0D# ,pr / GrZ−2,pr ,

(4.1.4.17)

and we have a morphism φ0D−1,pr : Gr0D−1,pr → Gr0D#

−1,pr

.

(4.1.4.18)

Lemma 4.1.4.19. With the setting as above, if the homomorphisms αpord,0 and r ord,#,0 ord,#,0 ord,0 0 mult r : αpr extend to homomorphisms αpr ,S : (GrDpr )S → G[p ] and αpr ,S ∨ r (Gr0D# )mult → G [p ] over S, respectively, then these extensions are closed imS pr

mersions with schematic images contained in G\ [pr ] and G∨,\ [pr ], respectively, and ord,0 αpord , αpord,#,0 , νpord r r = (αpr r ) satisfies Condition 4.1.4.1.

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ord,#,0 Proof. The extension αpord,0 ) is a closed immersion over S because r ,S (resp. αpr ,S it is so over η, and the closure of the identity section over η is the identity section over S because G (resp. G∨ ) is separated over S. Since G\ [pr ] (resp. G∨,\ [pr ]) is the maximal finite flat closed subgroup scheme of G[pr ] (resp. G∨ [pr ]) (see [62, Sec. 3.4.2], or rather [34, IX, 7.3]), this shows that the schematic image of αpord,0 r ,S (resp. ord,#,0 \ r ∨,\ r αpr ,S ) is contained in G [p ] (resp. G [p ]). Since the schematic images of αpord,0 r are annihilators of each other under the canonical pairing eG[pr ]η : are αpord,#,0 r G[pr ]η × G∨ [pr ]η → µpr ,η , and since T [pr ]η (resp. T ∨ [pr ]η ) is the annihilator of G∨,\ [pr ]η (resp. G\ [pr ]η ) under this canonical pairing, we see by duality that αpord r satisfies Condition 4.1.4.1, as desired. ord,0 , νpord , αpord,#,0 Lemma 4.1.4.20. With the setting as above, suppose αpord r r ) r = (αpr satisfies Condition 4.1.4.1. Then there exist homomorphisms

: (Gr0Dpr )mult → G\ [pr ]η αpord,0,\ r η and : (Gr0D# )mult → G∨,\ [pr ]η αpord,#,0,\ r η pr

αpord,0 r

over η such that (resp. αpord,#,0 ) is the composition of αpord,0,\ (resp. αpord,#,0,\ ) r r r with the canonical morphism G\ [pr ]η ,→ G[pr ]η (resp. G∨,\ [pr ]η ,→ G∨ [pr ]η ). The homomorphism αpord,0 (resp. αpord,#,0 ) extends to a homomorphism αpord,0 r r r ,S : ord,#,0 0 0 mult r mult ∨ r (GrDpr )S → G[p ] (resp. αpr ,S : (GrD# )S → G [p ]) over S if and only pr ord,0,\ ord,#,0,\ if the homomorphism αpr (resp. αpr ) also extends to a homomorphism ord,0,\ ord,#,0,\ 0 mult \ r αpr ,S : (GrDpr )S → G [p ] (resp. αpr ,S : (Gr0D# )mult → G∨,\ [pr ]) over S. r S p

) contains the subscheme (resp. αpord,#,0 Proof. Since the schematic image of αpord,0 r r r ∨ r r ∨ r T [p ]η (resp. T [p ]η ) of G[p ]η (resp. G [p ]η ), by duality as in the proof of Lemma 4.1.4.19, the homomorphisms αpord,0 and αpord,#,0 have schematic images contained r r in G\ [pr ]η and G∨,\ [pr ]η , respectively, and induce the two homomorphisms αpord,0,\ r and αpord,#,0,\ . The assertion about extensions over S is obvious. r Proposition 4.1.4.21. Let S = Spec(R), η = Spec(K), and (G, λ, i) be as at the beginning of this Section 4.1.4. Let (B, λB , iB , X, Y , φ, c, c∨ , τ ) be the degeneration datum associated with (G, λ, i) under the equivalence (4.1.3.3). A naive principal ordinary level structure ord,0 αpord , αpord,#,0 , νpord r r = (αpr r )

of type (L ⊗ Zp , h · , · i, D) and level pr on (Gη , λη , iη ) (see Definition 3.3.3.1) saZ

tisfying Condition 4.1.4.1 determines (up to isomorphism) a tuple (Zpr , (ϕ−2,pr , ϕ0,pr ), D−1,pr , ϕord −1,pr ) with entries described as follows:

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(1) A symplectic admissible filtration Zpr = {Z−i,pr }i of O-submodules on L/pr L (see (4.1.4.5) and (4.1.4.6)) satisfying the compatibilities (3.2.3.3) and (3.2.3.5) and determining the O-submodules GrZ−2,pr ⊂ Gr0D,pr and #

GrZ−2,pr ⊂ Gr0D# ,pr (see (4.1.4.7) and (4.1.4.8)). (2) A pair of isomorphisms (ϕ−2,pr , ϕ# −2,pr ), or equivalently a pair of isomorphisms (ϕ−2,pr , ϕ0,pr ) (see (4.1.4.11), (4.1.4.12), and (4.1.4.13)), inducing isomorphisms ∼

(ϕ−2,pr )mult : (GrZ−2,pr )mult → T [pr ] S S and ∼

#

mult (ϕ# : (GrZ−2,pr )mult → T ∨ [pr ] S −2,pr )S

(uniquely extending (4.1.4.9) and (4.1.4.10)), the latter of which is equivalent to an isomorphism ∼

ϕ0,pr ,S : (GrZ0,pr )S → (Y /pr Y )S (uniquely extending (4.1.4.11)). (3) A filtration D−1,pr = {Di−1,pr }i on GrZ−1,pr (see (4.1.4.14)) that satisfies the analogous conditions as the filtration Dpr does for L/pr L (see Lemma 3.2.2.1). (4) A (naive) principle ordinary level structure ord,0 ord,#,0 ord ϕord −1,pr = (ϕ−1,pr , ϕ−1,pr , ν−1,pr )

of type φ0D−1,pr : Gr0D−1,pr → Gr0D#

−1,pr

(see (4.1.4.18)) on (Bη , λB,η , iB,η ) as

in Definition 3.3.3.2 such that ∼

ord ν−1,p r : µpr ,η → µpr ,η

is equal to νpord r . Explicitly, the first two entries are O-equivariant homomorphisms ord,0 0 ϕ−1,p )mult → B[pr ]η r : (GrD −1,pr η

and 0 ϕord,#,0 −1,pr : (GrD#

−1,pr

)mult → B ∨ [pr ]η η

that are closed immersions, satisfying the symplectic condition ord,0 ord,#,0 ord λB,η ◦ ϕ−1,p ◦ (φ0D−1,pr )mult r = ν−1,pr ◦ ϕ−1,pr η

as in Definition 3.3.3.2. The homomorphism αpord,0 (resp. αpord,#,0 , resp. νpord r r r ) extends to a homomorphism ord,0 ord,#,0 0 mult r αpr ,S : (GrDpr )S → G[p ] (resp. αpr ,S : (Gr0D# )mult → G∨ [pr ], resp. νpord r ,S : S ∼

pr

ord,#,0 µpr ,S → µpr ,S ) over S if and only if the homomorphism ϕord,0 −1,pr (resp. ϕ−1,pr ,

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ord,0 0 ord )mult → B[pr ] resp. ν−1,p r ) also extends to a homomorphism ϕ−1,pr ,S : (GrD −1,pr S ∼

0 (resp. ϕord,#,0 −1,pr ,S : (GrD#

−1,pr

ord )mult → B ∨ [pr ], resp. ν−1,p r ,S : µpr ,S → µpr ,S ) over S. S

If αpord is a principal ordinary level-pr structure of type (L ⊗ Zp , h · , · i, D) such r Z 0

that, for each integer r0 ≥ r, there exists some lifting to level pr (over some extension of η) compatible with degeneration (i.e., satisfying the analogue of Condition 0 are compatibly 4.1.4.1 at level pr ), then all of the above data determined by αpord r liftable to their analogues over Zp . Proof. Only statement (4) of the proposition requires some explanation: The two ord,#,0 homomorphisms ϕord,0 are induced, respectively, by the two homo−1,pr and ϕ−1,pr ord,#,0,\ ord,0,\ in Lemma 4.1.4.20. (The statements on extensions and αpr morphisms αpr over S also follow from there. The statements on liftability are obvious.) However, not all tuples (Zpr , (ϕ−2,pr , ϕ0,pr ), D−1,pr , ϕord −1,pr ) as in Proposition 4.1.4.21 come from (naive) principal ordinary level structures. To formulate the additional condition needed, we introduce splittings both for the constant side, na# mely the filtrations 0 ⊂ GrZ−2,pr ⊂ Gr0D,pr and 0 ⊂ GrZ−2,pr ⊂ Gr0D# ,pr (see (4.1.4.7) and (4.1.4.8)), and for the geometric side, namely the filtrations 0 ⊂ T [pr ]η ⊂ ) of group schemes of ) and 0 ⊂ T ∨ [pr ]η ⊂ αpord,#,0 ((Gr0D# )mult αpord,0 ((Gr0Dpr )mult r r η η pr

multiplicative type over η. #

Lemma 4.1.4.22. Let GrZ−2,pr ⊂ Gr0D,pr , GrZ−2,pr ⊂ Gr0D# ,pr , Gr0D−1,pr , and Gr0D#

−1,pr

= (αpord,0 , αpord,#,0 , νpord be determined by αpord r r r ) as above (see (4.1.4.7), (4.1.4.8), r (4.1.4.16), and (4.1.4.17)). Suppose that Condition 1.2.1.1 holds, and that, for each 0 integer r0 ≥ r, there exists some lifting to level pr (over some extension of η) satisfying the analogue of Condition 4.1.4.1. Then there are splittings ∼

δpord,0 : GrZ−2,pr ⊕ Gr0D−1,pr → Gr0D,pr r

(4.1.4.23)

and ∼

#

: GrZ−2,pr ⊕ Gr0D# δpord,#,0 r

−1,pr

→ Gr0D# ,pr ,

(4.1.4.24)

of O-modules, which are compatible with φ0D,pr : Gr0D,pr → Gr0D# ,pr only in the sense #

that φ0D,pr (GrZ−2,pr ) ⊂ GrZ−2,pr (inducing the expected morphisms φ∗ : GrZ−2,pr → #

GrZ−2,pr and φ0D−1,pr : Gr0D−1,pr → Gr0D#

−1,pr

φ0D,pr (δpord,0 (Gr0D−1,pr )) r

⊂

δpord,#,0 (Gr0D# r

−1,pr

between the subquotients), but not that

). These splittings are liftable to splittings

over Zp with analogous compatibility properties. Proof. By Condition 1.2.1.1, the action of O on L extends to an action of some maximal order O0 in O ⊗ Q containing O, and Lemma 3.2.2.6 and its proof Z

show that the filtrations D and Z ⊗ Zp on L ⊗ Zp and the filtrations D# and ˆ Z

Z

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Z# ⊗ Zp on L# ⊗ Zp are all O0 ⊗ Zp -equivariant, whose graded pieces are projective ˆ Z

Z

Z

O0 ⊗ Zp -modules. Hence, there exist (noncanonical) splittings of the short exact Z

#

sequences 0 → GrZ−2 → Gr0D → Gr0D−1 → 0 and 0 → GrZ−2 → Gr0D# → Gr0D# → 0 −1

of O0 ⊗ Zp -lattices, which induce the desired splittings (4.1.4.23) and (4.1.4.24) by Z

reduction modulo pr . Lemma 4.1.4.25. Choices of the splittings δpord,0 and δpord,#,0 as in (4.1.4.23) and r r (4.1.4.24) define a morphism #

Gr0D−1,pr → GrZ−2,pr of O-modules by sending d ∈ Gr0D−1,pr to prGrZ#

−2,pr


# ◦(δpord,#,0 )−1 ◦ φ0D,pr ◦ δpord,0 (0, d) ∈ GrZ−2,pr , r r #

which (by the perfect duality h · , · i02 : GrZ0,pr × GrZ−2,pr → (Z/pr Z)(1) induced by h · , · i) is equivalent to a pairing 0 Z r h · , · iord 10,pr : GrD−1,pr × Gr0,pr → (Z/p Z)(1)

(4.1.4.26)

? ord satisfying hb · , · iord 10,pr = h · , b · i10,pr for every b ∈ O.

In other words, the pairing h · , · iord 10,pr measures the failure of the condition that (Gr0D# (Gr0D−1,pr )) ⊂ δpord,#,0 φ0D,pr (δpord,0 r r

−1,pr

).

On the other hand, we would like to have splittings of the filtrations ) 0 ⊂ T [pr ]η ⊂ αpord,0 ((Gr0Dpr )mult r η and 0 ⊂ T ∨ [pr ]η ⊂ αpord,#,0 ((Gr0D# )mult ) r η pr

of group schemes of multiplicative type, which are O-equivariant isomorphisms ∼

0 mult : T [pr ]η ⊕ ϕord,0 ) → αpord,0 ((Gr0Dpr )mult ) ςpord,0 r r η −1,pr ((GrD−1,pr )η

(4.1.4.27)

and 0 ςpord,#,0 : T ∨ [pr ]η ⊕ ϕord,#,0 r −1,pr ((GrD#

∼

−1,pr

)mult ) → αpord,#,0 ((Gr0D# )mult ) r η η pr

(4.1.4.28)

respecting the subgroup schemes T [pr ]η and T ∨ [pr ]η , respectively. In particular, these splittings correspond, respectively, to O-equivariant homomorphisms 0 mult ϕ˜ord,0 ,→ G\ [pr ]η ,→ G\η −1,pr : (GrD−1,pr )η

(4.1.4.29)

and 0 ϕ˜ord,#,0 −1,pr : (GrD#

−1,pr

)mult ,→ G∨,\ [pr ]η ,→ G∨,\ η η

(4.1.4.30)

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ord,#,0 ord,0 that are closed immersions lifting ϕ−1,p r and ϕ−1,pr , respectively. (In this last step we are not asserting any compatibilities between (4.1.4.29) and (4.1.4.30) other than ord,#,0 those between ϕord,0 −1,pr and ϕ−1,pr .) By abuse of language, let us define the canonical isogenies ord,0 ord,0 0 ord )mult ) Bη  Bη,p r := Bη /image(ϕ−1,pr ) = Bη /ϕ−1,pr ((GrD −1,pr η

(4.1.4.31)

and ∨,ord ord,#,0 0 ∨ Bη∨  Bη,p := Bη∨ /image(ϕord,#,0 r −1,pr ) = Bη /ϕ−1,pr ((GrD#

−1,pr

)mult ). η

(4.1.4.32)

(These definitions depend on the principal ordinary level structure ϕord −1,pr .) 0 mult r ) → B[p ] of finite flat group schemes The close immersion ϕord,0 : (Gr r η D−1,pr η −1,p −1 ∨ r is dual to the surjection B [p ]η → (GrD# )η with kernel the schematic image of −1,pr

ϕord,#,0 −1,pr

:

(Gr0D# )mult η −1,pr

´etale subgroup scheme of (separable) isogeny

→ B [p ]η . Therefore, (Gr−1 D# ∨

r

−1,pr

∨,ord Bη,p r ,

)η is embedded as an

and the canonical isogeny (4.1.4.31) is dual to the

∨,ord ∨,ord −1 Bη,p  Bη,p r r /(Gr # D

−1,pr

)η ∼ = Bη∨ /B ∨ [pr ]η ∼ = Bη∨ .

(4.1.4.33)

Similarly, the canonical isogeny (4.1.4.32) is dual to the (separable) isogeny −1 ord ord ) ∼ Bη,p r  Bη,pr /(GrD = Bη /B[pr ]η ∼ = Bη . −1,pr η

(4.1.4.34)

Let us record our observations as follows: ∨,ord ord over η are canonically Lemma 4.1.4.35. The abelian schemes Bη,p r and Bη,pr dual to each other. The canonical isogenies (4.1.4.31) and (4.1.4.33) (resp. (4.1.4.32) and (4.1.4.34)) are dual to each other.

Lemma 4.1.4.36. (1) The embeddings ϕ˜ord,0 −1,pr as in (4.1.4.29) correspond to liftings of cη : X η → ∨ Bη to cord pr :

1 pr X η

∨,ord → Bη,p r ,

∨,ord are the canonical ones where the morphisms X η ,→ p1r X η and Bη∨  Bη,p r (see (4.1.4.32)). ∨ (2) The embeddings ϕ˜ord,#,0 −1,pr as in (4.1.4.30) correspond to liftings of cη : Y η → Bη to

c∨,ord : pr where the morphisms Y η ,→ (see (4.1.4.31)).

1 pr Y η

1 pr Y η

ord → Bη,p r,

ord and Bη  Bη,p r are the canonical ones

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Proof. The proof is similar to that of [62, Lem. 5.2.3.1]. Let us explain only the proof of the first part, because the proof for the second part is essentially the same. An embedding ϕ˜ord,0 −1,pr as in (4.1.4.29) defines in particular an isogeny 0

ord,0 0 )mult ). G\η → G\η := G\η /ϕ˜−1,p r ((GrD −1,pr η

The subgroup scheme Tη of G\η embeds into a subgroup scheme Tη0 of G\η because 0 mult ) the pullback to Tη = ker(G\η → Bη ) of the schematic image ϕord,0 −1,pr ((GrD−1,pr )η \ in Gη is trivial. Hence, we have a commutative diagram: / Tη

0

/ G\η

o

 / G\η 0

 / Tη0

0

/ Bη

ord,0 0 mult ) (4.1.4.31) mod ϕ−1,pr ((GrD−1,pr )η

We can complete this into a diagram / Tη / G\η 0 o

 / G\η 0

 / Tη0

0

mod Tη0 [pr ]

 / G\η

 / Tη

0

/0

 ord / Bη,p r

/0

/ Bη

/0

ord,0 0 mult ) (4.1.4.31) mod ϕ−1,pr ((GrD−1,pr )η

 ord / Bη,p r

/0

−1 (4.1.4.34) mod (GrD−1,pr )η

 / Bη

/0

in which every composition of two vertical arrows is the multiplication by pr . Therefore, finding an embedding of the form (4.1.4.29) is equivalent to finding an isogeny 0 G\η  G\η of the form: 0

/ Tη0

mod Tη0 [pr ]

0

 / Tη

/ G\η 0  / G\η

ord / Bη,p r

/0

−1 (4.1.4.34) mod(GrD−1,pr )S

 / Bη

/0

Since the surjection Tη0  Tη is the dual of the inclusion X η ,→ p1r X η , and since the isogeny (4.1.4.34) is dual to the isogeny (4.1.4.32) by Lemma 4.1.4.35, by [62, Prop. 0 3.1.5.1], isogenies G\η → G\η of the above form are equivalent to liftings cord pr : ∨,ord 1 ∨ over η of the homomorphism c : X → B defining the extension pr X η → B pr structure of 0 → T → G\ → B → 0. Since all the homomorphisms we consider above are O-equivariant, the lifting cord pr is also O-equivariant by functoriality of [62, Prop. 3.1.5.1]. Lemma 4.1.4.37. Choices of the embeddings ϕ˜ord,0 ˜ord,#,0 as in (4.1.4.29) −1,pr and ϕ −1,pr ord,0 ord,#,0 and (4.1.4.30) define splittings ςpr and ςpr as in (4.1.4.27) and (4.1.4.28), respectively, and define an O-equivariant homomorphism 0 mult ϕord,0 ) → T ∨ [pr ]η −1,pr ((GrD−1,pr )η

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0 mult ) to of group schemes of multiplicative type by sending a ∈ ϕord,0 −1,pr ((GrD−1,pr )η ord,0 −1
(a) ∈ T ∨ [pr ]η , )−1 ◦ λη ◦ ϕ˜ord,0 prT ∨ [pr ]η ◦(ςpord,#,0 r −1,pr ◦ (ϕ−1,pr )

which (by the perfect duality (Y /pr Y )η × T ∨ [pr ]η → µpr ,η ) is equivalent to a pairing ord,0 0 mult eord ) ×(Y /pr Y )η → µpr ,η 10,pr : ϕ−1,pr ((GrD−1,pr )η

(4.1.4.38)

? ord satisfying eord 10,pr (b · , · ) = e10,pr ( · , b · ) for every b ∈ O.

In other words, the pairing eord 10,pr measures the failure of the condition that 0 0 mult (ϕord,#,0 ))) ⊂ ςpord,#,0 (ϕord,0 λ(ςpord,0 r r −1,pr ((GrD# −1,pr ((GrD−1,pr )η

−1,pr

)mult )). η

There is (a priori) a second pairing analogous to (4.1.4.38): Since λB,η : Bη → ord,#,0 0 0 mult ), it induces a polarization )mult ) to ϕ−1,p Bη∨ maps ϕord,0 r ((Gr # η −1,pr ((GrD−1,pr )η D −1,pr

∨,ord ord λord B,η,pr : Bη,pr → Bη,pr

(4.1.4.39)

compatible with the two isogenies (4.1.4.31) and (4.1.4.32). Let us extend φ : Y → X naturally to φpr : p1r Y → p1r X. Since λB c∨ = cφ, for every section y of Y η , we see that ∨,ord 1 (λord − cord B,η,pr cpr pr φpr ,η )( pr y)

defines a section of ∨,ord  Bη∨ ) ∼ ker(Bη,p r = (Gr−1 D#

−1,pr

)η

(see (4.1.4.33) and Lemma 4.1.4.35). Therefore: Lemma 4.1.4.40. Choices of the embeddings ϕ˜ord,0 ˜ord,#,0 as in (4.1.4.29) −1,pr and ϕ −1,pr and (4.1.4.30) define by Lemma 4.1.4.36 an O-equivariant homomorphism ∨,ord −1 r λord − cord B,η,pr cpr pr φpr ,η : (Y /p Y )η → (GrD#

−1,pr

which (by the perfect duality (Gr0D−1,pr )mult ×(Gr−1 η D#

−1,pr

)η ,

)η → µpr ,η ; cf. Lemma

3.2.2.4) is equivalent to a pairing ord,0 0 mult dord ) ×(Y /pr Y )η → µpr ,η 10,pr : ϕ−1,pr ((GrD−1,pr )η

(4.1.4.41)

ord ? satisfying dord 10,pr (b · , · ) = d10,pr ( · , b · ) for every b ∈ O.

The two pairings in (4.1.4.38) and (4.1.4.41) are, without surprise, related: Proposition 4.1.4.42. With the same setting as in Lemmas 4.1.4.37 and 4.1.4.40, ord we have dord 10,pr = e10,pr .

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ord Proof. Since dord 10,pr = e10,pr is a statement about equalities, we may perform an injective continuous base change from R to a noetherian complete local ring, so that (by [76, 31.C, Cor. 2]) the integral closure of R in any finite extension of K = Frac(R) is finite over R. (This base change is unnecessary when char(K) = 0 or when R is excellent.) By replacing R with a finite ´etale extension, we may assume that X and Y are constant with values in X and Y , respectively. Consider the canonical morphisms ∨,ord HomZ ( p1r X, Bη∨ )  HomZ ( p1r X, Bη,p r )

(4.1.4.43)

ord HomZ ( p1r Y, Bη )  HomZ ( p1r Y, Bη,p r)

(4.1.4.44)

and

induced by (4.1.4.32) and (4.1.4.31), respectively. These are isogenies between abelian schemes, because X and Y are Z-lattices. Consider also the canonical homomorphism HomZ ( p1r Y, G\η ) → HomZ (Y, G\η ),

(4.1.4.45)

which factors through HomZ ( p1r Y, G\η ) → HomZ (Y, G\η )

× HomZ (Y,Bη )

ord HomZ ( p1r Y, Bη,p r)

(4.1.4.46)

ord qy(where the canonical morphism HomZ ( p1r Y, G\η ) → HomZ ( p1r Y, Bη,p r ) is the 1 1 \ composition of the canonical morphism HomZ ( pr Y, Gη ) → HomZ ( pr Y, Bη ) and (4.1.4.44)). These are isogenies with finite kernels between semi-abelian schemes. (The kernel of (4.1.4.45) is canonically isomorphic to HomZ ( p1r Y /Y, G\ [pr ]η ). The

((Gr0Dpr )mult )); kernel of (4.1.4.46) is canonically isomorphic to HomZ ( p1r Y /Y, αpord,0,\ r η see Lemma 4.1.4.20.) ∨,ord 1 ord : p1r Y → Bη,p and c∨,ord The two homomorphisms cord r pr pr : pr Xη → Bpr ∨,ord (with their O-equivariance ignored) define η-valued points of HomZ (X, Bη,p r ) and ord ord HomZ (Y, Bη,pr ), respectively. Moreover, the pair (ι, cpr ), where ι : Y → G\η is the ∼ ⊗ −1 homomorphism determined by τ : 1Y × X,η → (c∨ × c)∗ PB,η (which is compatible ∨ with c by definition), determines a η-valued point of the target of (4.1.4.46). Since (4.1.4.32) and (4.1.4.46) are isogenies with finite kernels between semi-abelian varie˜ of K such that, with η˜ := Spec(K), ˜ the ties over η, there exists a finite extension K η-valued points of the targets of (4.1.4.43) and (4.1.4.46) lift to η˜-valued points of the ˜ of R in the finite extension K ˜ of K = Frac(R) sources. Since the integral closure R ˜ ˜ is finite over R, we may replace R (resp. K) with R (resp. K) and assume that cord pr and cp∨,ord lift to homomorphisms r

c pr :

1 pr X

→ Bη∨

(4.1.4.47)

c∨ pr :

1 pr Y

→ Bη ,

(4.1.4.48)

and

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respectively, and that ι lifts to a homomorphism ι pr :

1 pr Y

→ G\η

(4.1.4.49)

c∨ pr

G\η

(by composition with the canonical morphism → Bη ), which compatible with ∼ ⊗ −1 ∨ ∗ is equivalent to a lifting τpr : 1 p1r Y × X,η → (cpr , cη ) PB,η of τ . By [62, Prop. 5.2.3.3], and by comparing its proof with that of Lemma 4.1.4.36, ∨ the lifting (cpr , c∨ pr , ιpr ) of (c, c , ι) as in (4.1.4.47), (4.1.4.48), and (4.1.4.49) defines a splitting ∼

ςpr : T [pr ]η ⊕ B[pr ]η ⊕( p1r Y /Y )η → G[pr ]η which is not necessarily O-equivariant, which nevertheless induces the two O-equivariant splittings (4.1.4.27) and (4.1.4.28) by taking graded pieces with re), and by duality. spect to the filtration defined by image(αpord,0 ) = αpord,0 ((Gr0Dpr )mult r r η By [62, Thm. 5.2.3.14], we have 1 d10,pr (a, p1r y) = eB[pr ]η (a, (λB,η c∨ pr − cpr φpr )( pr y))

= e10,pr (a, p1r y) = eλη (ςpr (0, a, 0), ςpr (0, 0, p1r y))

(4.1.4.50)

∈ ( p1r Y /Y )η . If we consider only a ∈ image(ϕord,0 −1,pr ) =

for all a ∈ B[pr ]η and

1 pr y

0 mult ϕord,0 ), −1,pr ((GrD−1,pr )η

then the canonical pairings eB[pr ]η : B[pr ]η × B ∨ [pr ]η → µpr ,η

and eλη : G[pr ]η × G[pr ]η → µpr ,η induce the canonical pairings 0 mult eimage(ϕord,0r ) : ϕord,0 ) ×(Gr−1 −1,pr ((GrD−1,pr )η D# −1,p

−1,pr

)η → µpr ,η ,

and e

λη image(αord,0 ) pr

: αpord,0 ((Gr0Dpr )mult ) ×(Gr−1 r η Dpr )η → µpr ,η ,

respectively, and the relation (4.1.4.50) implies that ord ∨,ord 1 1 dord − cord 10,pr (a, pr y) = eimage(ϕord,0r ) (a, (λB,η cpr pr φpr )( pr y)) −1,p

=

(4.1.4.51)

λ (ςpord,0 (0, a), ςpord,−1 ( p1r y, 0)) e η r r image(αord,0 ) r p

for all a ∈ image(ϕord,0 −1,pr ) and where

1 pr y

∈ ( p1r Y /Y )η , where ςpord,0 is as in (4.1.4.27), and r ∼

−1 ςpord,−1 : ( p1r Y /Y )η ⊕(Gr−1 r D−1,pr )η → (GrDpr )η

is canonically dual to the inverse of the ςpord,#,0 as in (4.1.4.28). By comparison r with the definition in Lemma 4.1.4.37, we have λη (ς ord,0 (0, a), ςpord,−1 ( p1r y, 0)). r r image(αord,0 ) p pr

1 eord 10,pr (a, pr y) = e

Thus, the proposition follows from (4.1.4.51), as desired.

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Now it is natural to compare the pairing (4.1.4.26) with the pairing (4.1.4.41). Proposition 4.1.4.52. (This is a continuation of Proposition 4.1.4.21.) With the same setting as in Proposition 4.1.4.21, the tuple (Zpr , (ϕ−2,pr , ϕ0,pr ), D−1,pr , ϕord −1,pr ) there satisfies the following condition: ) as in Lemma 4.1.4.22, which , δpord,#,0 := (δpord,0 For each pair of splittings δpord r r r determines a pairing Z 0 r h · , · iord 10,pr : GrD−1,pr × Gr0,pr → (Z/p Z)(1)

as in Lemma 4.1.4.25, and accordingly a pairing mult (h · , · iord : (Gr0D−1,pr )mult ×(GrZ0,pr )η → µpr ,η , 10,pr )η η

there exists a (necessarily unique) pair of liftings (cord pr :

1 pr X η

∨,ord ∨,ord → Bη,p : r , cpr

1 pr Y η

ord → Bη,p r)

of (c : X → B ∨ , c∨ : Y → B), which by Lemma 4.1.4.36 is equivalent to a pair of embeddings (ϕ˜ord,0 ˜ord,#,0 −1,pr , ϕ −1,pr ) as in (4.1.4.29) and (4.1.4.30), which determines a pairing ord,0 0 mult dord ) ×(Y /pr Y )η → µpr ,η 10,pr : ϕ−1,pr ((GrD−1,pr )η

as in Lemma 4.1.4.40, such that ∗ ord ord ord mult (ϕord,0 . −1,pr , ϕ0,pr ) (d10,pr ) = ν−1,pr ◦ (h · , · i10,pr )η

This condition is independent of the choice of (δpord,0 , δpord,#,0 ). r r ord Conversely, each tuple (Zpr , (ϕ−2,pr , ϕ0,pr ), D−1,pr , ϕ−1,pr ) that satisfies this = condition comes from some naive principal ordinary level-pr structure αpord r ord,#,0 r ord r ) of type (L/p L, h · , · i, D ) on (G , λ , i ) (see Definition (αpord,0 , α , ν r p η η η pr pr 3.3.3.2) as in Proposition 4.1.4.21. If the tuple is liftable (satisfying the analogous conditions for the liftings), then it comes from some principal ordinary level-pr of type (L ⊗ Zp , h · , · i, D). structure αpord r Z

ord,0 , αpord,#,0 , νpord Explicitly, we can recover the triple αpord r r ) from the tuple r = (αpr ord (Zpr , (ϕ−2,pr , ϕ0,pr ), D−1,pr , ϕ−1,pr ) as follows: Suppose (δpord,0 , δpord,#,0 ) and (ϕ˜ord,0 ˜ord,#,0 r r −1,pr , ϕ −1,pr ) have been chosen such that the above condition is satisfied, the latter of which determines a pair of splittings (ςpord,0 , ςpord,#,0 ) as in (4.1.4.27) and (4.1.4.28). Then we have the following defir r ning relations:

(1) The homomorphism αpord,0 : (Gr0D,pr )mult → G[pr ]η is defined as follows: r η The sum of the composition ∼

can.

ord mult ν−1,p : (GrZ−2,pr )mult → T [pr ]η ,→ G\ [pr ]η r ◦ (ϕ−2,pr )η η

and 0 mult ϕ˜ord,0 → G\ [pr ]η −1,pr : (GrD−1,pr )η

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defines a homomorphism (GrZ−2,pr )mult ⊕(Gr0D−1,pr )mult → G\ [pr ]η η η that is a closed embedding, and the pre-composition of this homomorphism with ∼

⊕(Gr0D−1,pr )mult → (GrZ−2,pr )mult ((δpord,0 )mult )−1 : (Gr0D,pr )mult r η η η η gives the homomorphism : (Gr0D,pr )mult → G\ [pr ]η , αpord,0,\ r η with the canonical embedding is the composition of αpord,0,\ and αpord,0 r r G\ [pr ]η ,→ G[pr ]η . → G∨ [pr ]η is defined as fol: (Gr0D# ,pr )mult (2) The homomorphism αpord,#,0 r η lows: The sum of the composition can.

∼

#

mult (ϕ# : (GrZ−2,pr )mult → T ∨ [pr ]η ,→ G∨,\ [pr ]η η −2,pr )η ord (note that we do not compose with ν−1,p r here) and 0 ϕ˜ord,#,0 −1,pr : (GrD#

−1,pr

)mult → G∨,\ [pr ]η η

defines a homomorphism #

⊕(Gr0D# (GrZ−2,pr )mult η

−1,pr

)mult → G\ [pr ]η η

that is a closed embedding, and the pre-composition of this homomorphism with ∼

#

⊕(Gr0D# ((δpord,#,0 )mult )−1 : (Gr0D# ,pr )mult → (GrZ−2,pr )mult r η η η

−1,pr

)mult η

gives the homomorphism → G∨,\ [pr ]η , αpord,#,0,\ : (Gr0D# ,pr )mult r η and αpord,#,0 is the composition of αpord,#,0,\ with the canonical embedding r r ∨,\ r G [p ]η ,→ G∨ [pr ]η . ∼ ord ord (3) The isomorphism νpord : µpr ,η → µpr ,η is equal to ν−1,p r r , where ν−1,pr : ∼ ord µpr ,η → µpr ,η is part of the data of ϕ−1,pr . Proof. The statements are self-explanatory. Remark 4.1.4.53. In the theory of degeneration for naive principal ordinary level structures, there is no need to consider some lifting τpord of τ as in the theory for r principal level structures (away from p) as in [62, Ch. 5].

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Definition 4.1.4.54. (See [62, Prop. 5.2.2.1].) Let φ : Y → X be an O-equivariant embedding with finite cokernel between ´etale sheaves of O-lattices, which is dual to an O-equivariant isogeny T  T ∨ of tori with O-actions. Then we define for each integer m the pairing eφm : T [m]η ×(Y /mY )η → µm,η to be the canonical pairing can. ∼

T [m]η ×(Y /mY )η → (X/mX)mult ×(Y /mY )η η Id ×φ

can.

→ (X/mX)mult ×(X/mX)η → µm,η η

with the sign convention that eφm (t, y) = t(φ(y)) = (φ(y))(t) for every section t of T [m]η and y of (Y /mY )η . Definition 4.1.4.55. (Compare with [62, Def. 5.2.7.8].) With the setting as in Section 4.1.2, suppose we are given a tuple (B, λB , iB , X, Y , φ, c, c∨ , τ ) in DDPE,O (R, I). A naive ordinary pre-level-n structure datum of type (L/nL, h · , · i, Dpr ) over η is a tuple ord ord ∨,ord ord αn\,ord := (Zn , ϕ−2,n , ϕord , τn ) −1,n , ϕ0,n , δn , cn , cn

consists of the following data: (1) A symplectic admissible filtration Zn on L/nL. This determines a dual sym# # plectic admissible filtration Z# n on L /nL , with induced perfect pairings #

h · , · iij,n : GrZ−i,n × GrZj,n → (Z/nZ)(1), for i + j = 2, inducing (possibly nonperfect) pairings h · , · iij,n : GrZ−i,n × GrZ−j,n → (Z/nZ)(1) denoted by the same symbols (by abuse of notation). By reduction modulo n0 (resp. pr ), we also obtain an admissible symplectic filtration and pairings between the graded pieces, with n replaced with n0 (resp. pr ) in the above notation. The filtration Zpr is compatible with Dpr in the sense that Z−2,pr ⊂ Dpr ⊂ Z−1,pr , and hence induces a filtration D−1,pr = {Di−1,pr }i on GrZ−1,pr (see (4.1.4.14)) that satisfies the analogous conditions as the filtration Dpr does for L/pr L (see Lemma 3.2.2.1). ord (2) A pair ϕord −1,n = (ϕ−1,n0 , ϕ−1,pr ) consisting of: (a) A (naive) principal level-n0 structure ∼

ϕ−1,n0 : GrZ−1,n0 → B[n0 ]η of (Bη , λB,η , iB,η ) of type (GrZ−1,n0 , h · , · i11,n0 ) over η, equipped with an isomorphism ∼

ν−1,n0 : (Z/n0 Z)(1) → µn0 ,η , as in Definition 3.3.1.2.

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(b) A (naive) principle ordinary level structure ord,0 ord,#,0 ord ϕord −1,pr = (ϕ−1,pr , ϕ−1,pr , ν−1,pr )

of (Bη , λB,η , iB,η ) of type φ0D−1,pr : Gr0D−1,pr → Gr0D#

−1,pr

(see (4.1.4.18))

as in Definition 3.3.3.2 (see also (4) of Proposition 4.1.4.21). (3) A pair of O-equivariant isomorphisms ∼

ϕ−2,n : GrZ−2,n → Homη ((X/nX)η , (Z/nZ)(1)) and ∼

ϕ0,n : GrZ0,n → (Y /nY )η satisfying ((ϕ−2,n )mult × ϕ0,n )∗ eφn = ((h · , · i20,n )mult , η η where ∼

: (GrZ−2,n )mult (ϕ−2,n )mult → T [n]η η η and (h · , · i20,n )mult : (GrZ−2,n )mult ×(GrZ0,n )η → µn,η η η are canonically induced by ϕ−2,n and h · , · i20,n . By reduction modulo n0 , we obtain isomorphisms and pairings with n replaced with n0 in the notation. By reduction modulo pr , we obtain isomorphisms over η that are equivalent to the pullbacks to η of the isomorphisms ∼

(ϕ−2,pr )mult : (GrZ−2,pr )mult → T [pr ] S S and ∼

#

mult (ϕ# : (GrZ−2,pr )mult → T ∨ [pr ] S −2,pr )S

over S. (4) A pair δnord = (δn0 , δpord r ) consisting of: ∼

(a) A splitting δn0 : GrZn0 → L/n0 L of O-modules, which determines the pairings h · , · i10,n0 : GrZ−1,n0 × GrZ0,n0 → (Z/n0 Z)(1) and h · , · i00,n0 : GrZ0,n0 × GrZ0,n0 → (Z/n0 Z)(1). (b) A pair δpord = (δpord,0 , δpord,#,0 ) of splittings r r r ∼

δpord,0 : GrZ−2,pr ⊕ Gr0D−1,pr → Gr0D,pr r and #

δpord,#,0 : GrZ−2,pr ⊕ Gr0D# r

−1,pr

∼

→ Gr0D# ,pr

of O-modules as in Lemma 4.1.4.22, which determines a pairing 0 Z r h · , · iord 10,pr : GrD−1,pr × Gr0,pr → (Z/p Z)(1)

as in Lemma 4.1.4.25, and accordingly a pairing mult ×(GrZ0,pr )η → µpr ,η . (h · , · iord : (Gr0D−1,pr )mult 10,pr )η η

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(5) Liftings cord n :

1 n Xη

cn∨,ord :

∨,ord 0 := Bη∨ /ϕord,#,0 → Bη,p r −1,pr ((GrD#

1 nY η

−1,pr

)mult ), η

ord,0 0 ord )mult ), → Bη,p r := Bη /ϕ−1,pr ((GrD −1,pr η

and τnord := τn0 : 1 n1

0

∼

Y × X,η S

∗ ⊗ −1 → (c∨ n0 , cη ) PB,η

of c : X → B ∨ , c∨ : Y → B and τ : 1Y

∼

× X,η S

⊗ −1 → (c∨ , cη )∗ PB,η over η,

respectively. The liftings cord and c∨,ord determine and are determined by liftings cn0 : n n ∨,ord ∨,ord 1 1 1 ∨ ∨ : p1r Y η → X → B , c : Y → B, cord n0 pr : pr X η → Bη,pr , and cpr n0 n0 ∨ ∨ ∨ ∨ ord Bη,pr of c : X → B , c : Y → B, c : X → B , and c : Y → B over η, respectively. By [62, Lem. 5.2.3.12], the liftings cn0 , c∨ n0 , and τn0 defines two pairings d10,n0 : B[n0 ]η ×(Y /n0 Y )η → µn0 ,η and d00,n0 : (Y /n0 Y )η ×(Y /n0 Y )η → µn0 ,η by setting 1 d10,n0 (a, n10 y) := eB[n0 ] (a, (λB c∨ n0 − cn0 φn0 )( n0 y) ∈ µn0 (η) 1 1 n0 y of n0 Y , and by setting d00,n0 ( n10 y, n10 y 0 ) := τn0 ( n10 y, φ(y 0 ))τn0 ( n10 y 0 , φ(y))−1 ∈ sections n10 y and n10 y 0 of n10 Y . ∨,ord Lemmas 4.1.4.36 and 4.1.4.40, the liftings cord pr and cpr

for sections a of B[n0 ]η and for

By ring

µm (η) define a pai-

ord,0 0 mult dord ) ×(Y /pr Y )η → µpr ,η . 10,pr : ϕ−1,pr ((GrD−1,pr )η

We say that the naive ordinary pre-level-n structure datum αn\,ord is symplectic, and call it a naive ordinary level-n structure datum of type (L/nL, h · , · i, Dpr ) over η, if the following conditions are satisfied: (ϕ−1,n0 × ϕ0,n0 )∗ (d10,n0 ) = ν−1,n0 ◦ h · , · i10,n0 , (ϕ0,n0 × ϕ0,n0 )∗ (d00,n0 ) = ν−1,n0 ◦ h · , · i00,n0 , and ∗ ord ord ord mult (ϕord,0 . −1,pr , ϕ0,pr ) (d10,pr ) = ν−1,pr ◦ (h · , · i10,pr )η

We remove “ naive” from the above terminologies, and call them ordinary preˆ h · , · i, D) and ordinary level-n struclevel-n structure datum of type (L ⊗ Z, Z

ˆ h · , · i, D), when the data are compatibly liftable to data ture datum of type (L ⊗ Z, Z

(at all higher levels) satisfying the analogous conditions.

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Proposition 4.1.4.56. With the setting in Propositions 4.1.4.21 and 4.1.4.52, suppose moreover that η is a scheme over Spec(Z(p) ). Then each ordinary level-n ˆ h · , · i, D) (in Definition 4.1.4.55) defines an structure datum αn\,ord of type (L ⊗ Z, Z ...ord extension of the triple (Gη , λη , iη ) to an object (Gη , λη , iη , αn0 , αpord r ) of Mn (η) (see Definition 3.4.1.1). Moreover, each such pair (αn0 , αpord r ) of level structures comes \,ord from some αn in this way. Proof. The only thing not explained yet is the statements concerning the principal level-n0 structure αn0 . In [62, Sec. 5.2], it was assumed that the polarization degree is prime to the residue characteristics, and that the generic point is defined over the rings of integers of the reflex field, but these assumptions were not really used in a substantial way. (They were only used to ensure that one obtains an object over η parameterized by the PEL-type moduli problem.) Therefore, the arguments there (with the liftability condition away from p) still work here, and allows us to construct αn0 , as desired. \,ord is not one to one. (The freedom However, the assignment of (αn0 , αpord r ) to αn comes from the choice of various splittings.) We can define as in [62, Def. 5.2.7.11] a notion of equivalence classes [αn\,ord ] of objects like αn\,ord . (The definition can be made precise, but we will not record the details here.)

Definition 4.1.4.57. With the setting as in Section 4.1.2, suppose moreover that (R, I) has objects of the η is a scheme over Spec(Z(p) ). The category DEGPEL,... M ord n ord form (G, λ, i, αn0 , αpr ) (over S), where: (1) (G, λ, i) defines an object of DEGPE,O (R, I) (see Definition 4.1.3.1). ...ord (2) (Gη , λη , iη , αn0 , αpord r ) defines an object of Mn (η) (see Definition 3.4.1.1). r (3) αpord r is a principal ordinary level-p structure of type (L ⊗ Zp , h · , · i, D) such Z

0

that, for each integer r0 ≥ r, there exists some lifting to level pr (over some ´etale extension of η) compatible with degeneration (i.e., satisfying the 0 analogue of Condition 4.1.4.1 at level pr ). Definition 4.1.4.58. With the setting as in Section 4.1.2, suppose moreover that η is a scheme over Spec(Z(p) ). The category DDPEL,... (R, I) has objects of the M ord n form (B, λB , iB , X, Y , φ, c, c∨ , τ, [αn\,ord ]), where: (1) (B, λB , iB , X, Y , φ, c, c∨ , τ ) defines an object of DDPE,O (R, I) (see Definition 4.1.3.2). (2) [αn\,ord ] is an equivalence class of ordinary level-n structure data αn\,ord of ˆ h · , · i, D) defined over η (see Definition 4.1.4.55). type (L ⊗ Z, Z

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Now it follows from Propositions 4.1.4.21, 4.1.4.52, and 4.1.4.56 that we have the following: Theorem 4.1.4.59. There is an equivalence of categories MPEL,... (R, I) : DDPEL,... (R, I) → DEGPEL,... (R, I) : M ord M ord M ord n n n (B, λB , iB , X, Y , φ, c, c∨ , τ, [αn\,ord ]) 7→ (G, λ, i, αn0 , αpord r ).

4.1.5

Degeneration Data for General Ordinary Level Structures

ˆ Definition 4.1.5.1. (See [62, Def. 1.3.7.1].) For each Z-algebra R, set ˆ → G(R)). Gess (R) := image(G(Z) Definition 4.1.5.2. (See [62, Def. 5.3.1.4].) Let Zn be a symplectic filtration on L/nL. Then we define the following subgroups or quotients of subgroups of Gess (Z/nZ): ess −1 Pess Zn := {gn ∈ G (Z/nZ) : gn (Zn ) = Zn }, Zn ess Zess Zn := {gn ∈ PZn : Gr−1 (gn ) = IdGrZn and ν(gn ) = 1}, −1

Zn ess Uess Zn := {gn ∈ PZn : Gr (gn ) = IdGrZn and ν(gn ) = 1},   n (g−1,n , rn ) ∈ GLO (GrZ−1 ) × Gm (Z/nZ) : ess Gh,Zn := , Zn ∃gn ∈ Pess Zn s.t. Gr−1 (gn ) = g−1,n and ν(gn ) = rn   n (g−2,n , g0,n ) ∈ GLO (GrZ−2 ) × GLO (GrZ0n ) : Gess := , Zn Zn l,Zn ∃gn ∈ Zess Zn s.t. Gr−2 (gn ) = g−2,n and Gr0 (gn ) = g0,n ( ) n g20,n ∈ HomO (GrZ0n , GrZ−2 ):   1 g20,n Uess , 2,Zn := −1 ∃gn ∈ Uess 1 Zn s.t. δn ◦ gn ◦ δn = 1   Zn Zn Zn n , Gr  (g21,n , g10,n ) ∈ HomO (GrZ−1 O (Gr0 , Gr−1 ) :  −2 ) × Hom  1 g21,n g20,n Uess . 1,Zn := −1 1 g10,n ∃gn ∈ Uess , for some g20,n  Zn s.t. δn ◦ gn ◦ δn = 1

We define similar subgroups and quotients of subgroups with n replaced with either n0 or pr . n Remark 4.1.5.3. Since ν(GrZ−1 (gn )) = ν(gn ) by definition, the condition ν(gn ) = 1 n in the definition of Zess is redundant if we interpret GrZ−1 (gn ) = IdGrZn as an Zn −1 identity of symplectic isomorphisms (which are required to preserve the similitude isomorphisms; see [62, Def. 1.1.4.8]).

Lemma 4.1.5.4. (See [62, Lem. 5.3.1.6].) By definition, there are natural inclusions ess ess ess ess Uess 2,Zn ⊂ UZn ⊂ ZZn ⊂ PZn ⊂ GZn ,

(4.1.5.5)

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and natural exact sequences: ess ess 1 → Zess Zn → PZn → Gh,Zn → 1, 1→

ess ess Uess Zn → ZZn → Gl,Zn → 1, ess ess Uess 2,Zn → UZn → U1,Zn → 1.

1→ We have similar statements with n replaced with either n0 or pr .

(4.1.5.6) (4.1.5.7) (4.1.5.8)

Definition 4.1.5.9. (See [62, Def. 5.3.1.11].) Let Hn be a subgroup of Gess (Z/nZ). For each of the subgroups > in (4.1.5.5), we define Hn,> := Hn ∩ >. For each of the quotients of two groups > = >1 />2 in (4.1.5.5), (4.1.5.6), (4.1.5.7), or (4.1.5.8), we define Hn,> := Hn,>1 /Hn,>2 . Thus, we have defined the groups Hn,Pess , Hn,Zess , Zn Zn ess ess ess ess , H , H , H , and H , so that we have the natural Hn,Uess n,Gh,Zn n,Gl,Zn n,U2,Zn n,U1,Zn Zn inclusions Hn,Uess ⊂ Hn,Uess ⊂ Hn,Zess ⊂ Hn,Pess ⊂ Hn Zn Zn Zn 2,Zn and natural exact sequences 1 → Hn,Zess → Hn,Pess → Hn,Gess → 1, Zn Zn h,Zn → 1, 1 → Hn,Uess → Hn,Zess → Hn,Gess Zn Zn l,Zn 1 → Hn,Uess → Hn,Uess → Hn,Uess → 1. Zn 2,Zn 1,Zn We define similar subgroups with n replaced with either n0 or pr . ˆ p ) be an open compact subgroup such that Proposition 4.1.5.10. Let Hp ⊂ G(Z p p U (n0 ) ⊂ H for some integer n0 ≥ 1 is an integer prime to p. Let Hn0 := Hp /U p (n0 ). Let S = Spec(R) be as in Section 4.1.2. Let (G, λ, i) be a degenerating family of type (PE, O) over S as in Definition 4.1.3.1, with degeneration datum (B, λB , iB , X, Y , φ, c, c∨ , τ ) given by (4.1.3.3). Suppose moreover that η is a scheme over Spec(Z(p) ). Let Zn be a symplectic-liftable filtration on L/nL. Consider compositions of finite ´etale morphisms of schemes ∼ ∼ τHn0  (cHn0 , c∨ Hn0 )  (ϕ−2,Hn0 , ϕ0,Hn0 ) (4.1.5.11) ∼  ϕ−1,Hn0  δHn0 → ZHn0  η such that we have the following: (1) ZHn0  η is an Hn0 -orbit of ´etale-locally-defined filtrations Zn0 , which is isomorphic to (the pullback of) the constant scheme Hn0 ,Pess \Hn0 over Zn 0 some finite ´etale extension of η. ∼ (2) δHn0 → ZHn0 is an isomorphism giving choices of splittings of filtrations parameterized by ZHn0 . (3) ϕ−1,Hn0  δHn0 is an Hn0 ,Gess -torsor realized as a finite ´etale subscheme h,Zn 0 of the pullback to δHn0 of the finite ´etale scheme Isomη ((L/n0 L)η , B[n0 ]η ) × Isomη (((Z/n0 Z)(1))η , µn0 ,η ) η

over η, which is an Hn0 ,Gess -torsor giving an Hn0 ,Gess -orbit of ´etaleh,Zn h,Zn 0 0 locally-defined ϕ−1,n0 . (This defines some naive level structure on the pullback of (Bη , λB,η , iB,η ) to δHn0 ; cf. Definition 3.3.1.3.)

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∼ -torsor giving an (4) (ϕ∼  ϕ−1,Hn0 is an Hn0 ,Gess −2,Hn0 , ϕ0,Hn0 ) l,Zn 0 -orbit of ´etale-locally-defined pairs (ϕ−2,n0 , ϕ0,n0 ). Hn0 ,Gess l,Zn 0 ∨ ∼ ess (5) (cHn0 , cHn )  (ϕ∼ −2,Hn0 , ϕ0,Hn0 ) is an Hn0 ,U1,Zn0 -torsor giving an 0 ∨ ess Hn0 ,U1,Zn -orbit of ´etale-locally-defined pairs (cn0 , cn0 ). 0 ess ess (6) τHn0  (cHn0 , c∨ Hn0 ) is an Hn0 ,U2,Zn0 -torsor giving an Hn0 ,U2,Zn0 -orbit of ´etale-locally-defined τn0 .

(Each of the datum or pairs of data is built on top of the earlier ones.) Then each such scheme τHn0  η determines a naive level-Hn0 structure αHn0 of (Gη , λη , iη ) of type (L/n0 L, h · , · i) (see Definition 3.3.1.3). If these schemes are orbits of liftable objects, then they determine a level-Hp structure αHp of (Gη , λη , iη ) of type ˆ p , h · , · i) (see Definition 3.3.1.4). (L ⊗ Z Z

ˆ p , h · , · i) Conversely, each level-Hp structure αHp of (Gη , λη , iη ) of type (L ⊗ Z Z

arises this way for some (noncanonical choice of) symplectic-liftable Zn0 . (The Hn0 -orbit of Zn0 , or rather the Hp -orbit of the lifting Zp , is nevertheless canonically determined by αHp .) Proof. This follows from the same arguments as in [62, Sec. 5.3]. Definition 4.1.5.12. With the setting in Definition 4.1.5.2, let Zpr := Zn /pr Zn , and assume moreover that Zpr satisfies the compatibility Z−2,pr ⊂ D0pr ⊂ Z−1,pr (see (3.2.3.3)). Then we define the following subgroups or quotients of subgroups of Gess (Z/pr Z): ess r −1 0 0 r r ∼ Pess Dpr := {gpr ∈ G (Z/p Z) : gpr (Dpr ) = Dpr } = Up,0 (p )/Up (p ), ess −1 Pess Zpr ,Dpr := {gpr ∈ PDpr : gpr (Zpr ) = Zpr }, Z

r

p ess Zess Zpr ,Dpr := {gpr ∈ PZpr ,Dpr : Gr−1 (gpr ) = IdGrZpr and ν(gpr ) = 1}, −1

Uess Zpr ,Dpr Gess l,Zpr ,Dpr

Uess 2,Zpr ,Dpr

Uess 1,Zpr ,Dpr

Pess Zpr ,Dpr

Zpr

:= {gpr ∈ : Gr (gpr ) = IdGrZpr and ν(gpr ) = 1},   Zpr Zpr  (g−2,pr , g0,pr ) ∈ GLO (Gr−2 ) × GLO (Gr0 ) :  ∃gpr ∈ Zess := , Zpr ,Dpr s.t.   Zpr  GrZpr (g r ) = g  r r r and Gr (g ) = g p −2,p p 0,p −2 0 ( ) Zpr Zpr g20,pr ∈ HomO (Gr0 , Gr−2 ) :  1 g20,pr := , −1 ∃gpr ∈ Uess 1 Zpr ,Dpr s.t. δpr ◦ gpr ◦ δpr = 1   Zpr Zpr HomO (Gr−1 , Gr−2 )     r r (g21,p , g10,p ) ∈  Z r Zpr :    × HomO (Gr0p , Gr−1 )   1 g21,pr g20,pr := . −1 ess  1 g10,pr ,  ∃gpr ∈ UZpr ,Dpr s.t. δpr ◦ gpr ◦ δpr =    1     for some g20,pr

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ess ess ess ess ess Note that Zess Zpr ,Dpr = ZZpr , UZpr ,Dpr = UZpr , and Gl,Zpr ,Dpr = Gl,Zpr . Let us consider bal

bal r U p,1 (pr ) := Up,1 (p )/Up (pr )

as a subgroup of Gess Dpr . Then we also define the following subgroups or quotients of subgroups of Gess (Z/pr Z): bal

r ∼ r bal r := Pess MZess,ord Dpr /U p,1 (p ) = Up,0 (p )/Up,1 (p ), pr ,Dpr can.

ess,ord ess ess Pess,ord Zpr ,Dpr := image of PZpr ,Dpr under PDpr → MZpr ,Dpr , can.

ess,ord ess ess ZZess,ord := image of Zess Zpr ,Dpr = ZZpr under PDpr → MZpr ,Dpr , pr ,Dpr can.

ess,ord ess ess := image of Uess UZess,ord Zpr ,Dpr = UZpr under PDpr → MZpr ,Dpr , pr ,Dpr ess,ord ess,ord Gess,ord h,Zpr ,Dpr := PZpr ,Dpr /ZZpr ,Dpr , ess,ord ess,ord ∼ ess ess ess Gl,Z := Zess,ord Zpr ,Dpr /UZpr ,Dpr = ZZpr /UZpr = Gl,Zpr , pr ,Dpr

Uess,ord 2,Zpr ,Dpr := {Id}, ess,ord U1,Z := UZess,ord . pr ,Dpr pr ,Dpr

Lemma 4.1.5.13. By definition, there are natural inclusions ess,ord ess,ord ess,ord ess,ord U2,Z ⊂ Uess,ord Zpr ,Dpr ⊂ ZZpr ,Dpr ⊂ PZpr ,Dpr ⊂ MZpr ,Dpr , pr ,Dpr

(4.1.5.14)

and natural exact sequences: ess,ord ess,ord 1 → Zess,ord Zpr ,Dpr → PZpr ,Dpr → Gh,Zpr ,Dpr → 1,

1→

Uess,ord Zpr ,Dpr

1→

Uess,ord 2,Zpr ,Dpr

→

Zess,ord Zpr ,Dpr

→

→

Uess,ord Zpr ,Dpr

Gess,ord l,Zpr ,Dpr

→

(4.1.5.15)

→ 1,

Uess,ord 1,Zpr ,Dpr

(4.1.5.16)

→ 1.

(4.1.5.17)

Definition 4.1.5.18. Let Hp be an open compact subgroup of G(Zp ) such that bal r Up,1 (p ) ⊂ Hp ⊂ Up,0 (pr ), which defines a subgroup Hpr := Hp /Up (pr ) of bal r ess := Hp /Up,1 (p ) of MZess,ord PDpr , and defines a subgroup Hpord . For each of r pr ,Dpr ord ord the subgroups > in (4.1.5.14), we define Hpr ,> := Hpr ∩ >. For each of the quotients of two groups > = >1 />2 in (4.1.5.14), (4.1.5.15), (4.1.5.16), or ord ord (4.1.5.17), we define Hpord r ,> := Hpr ,> /Hpr ,> . Thus, we have defined the groups 1 2 ord ord ord , Hpr ,Pess,ord , Hpr ,Zess,ord , Hpr ,Uess,ord , Hpord , Hpord , Hpord r ,Gess,ord r ,Gess,ord r ,Uess,ord Zpr ,Dpr

Zpr ,Dpr

and Hpord r ,Uess,ord

Zpr ,Dpr

h,Zpr ,Dpr

l,Zpr ,Dpr

, so that we have the natural inclusions

1,Zpr ,Dpr

Hpord r ,Uess,ord

2,Zpr ,Dpr

ord ord ord ⊂ Hpord r ,Uess,ord ⊂ Hpr ,Zess,ord ⊂ Hpr ,Pess,ord ⊂ Hpr Zpr ,Dpr

Zpr ,Dpr

Zpr ,Dpr

and natural exact sequences ord ord 1 → Hpord r ,Zess,ord → Hpr ,Pess,ord → Hpr ,Gess,ord

→ 1,

ord ord 1 → Hpord r ,Uess,ord → Hpr ,Zess,ord → Hpr ,Gess,ord

→ 1,

Zpr ,Dpr

Zpr ,Dpr

1 → Hpord r ,Uess,ord

2,Zpr ,Dpr

Zpr ,Dpr

Zpr ,Dpr

h,Zpr ,Dpr

l,Zpr ,Dpr

ord → Hpord r ,Uess,ord → Hpr ,Uess,ord Zpr ,Dpr

1,Zpr ,Dpr

→ 1.

2,Zpr ,Dpr

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∼ under the = Hpr ,Gess l,Z r

Remark 4.1.5.19. By definition, we have Hpord r ,Gess,ord

p

l,Zpr ,Dpr

canonical isomorphism

Gess,ord l,Zpr ,Dpr

page 239

∼ = Gess l,Zpr .

Proposition 4.1.5.20. Let Hpord be as in Definition 4.1.5.18. Let S = Spec(R) be r as in Section 4.1.2. Let (G, λ, i) be a degenerating family of type (PE, O) over S as in Definition 4.1.3.1, with degeneration datum (B, λB , iB , X, Y , φ, c, c∨ , τ ) given by (4.1.3.3). Let Zpr be a symplectic filtration on L/pr L, and assume moreover that Zpr satisfies the compatibility Z−2,pr ⊂ D0pr ⊂ Z−1,pr (see (3.2.3.3)). Consider compositions of finite ´etale morphisms of schemes ∼

∨,ord ord ord ord ord τH ord → (cH ord , c ord )  (ϕ−2,H ord , ϕ0,H ord ) H r r r r p

pr

p

p

p

(4.1.5.21)

∼

ord  δH  ϕord ord → ZH ord  η, −1,H ord r r r p

p

p

such that we have the following: (1) ZH ord etale η is an Hpord r -orbit (or equivalently an Hpr -orbit) of ´ pr locally-defined filtrations Zpr that are compatible with Dpr in the sense that Z−2,pr ⊂ Dpr ⊂ Z−1,pr , which is isomorphic to (the pullback of) the conover some finite ´etale extension of η. This \Hpord stant scheme Hpord r r ,Pess,ord Zpr ,Dpr

etale-locally-defined filtrations conjugate also determines an Hpord r -orbit of ´ to D−1,pr = {Di−1,pr }i on GrZ−1,pr . ord,0 ord,#,0 ∼ ord (2) δH ) → ZH ord gives choices of pairs of splittings δpord = r ord = (δ ord , δ ord r H H r pr

p

p

pr

, δpord,#,0 ). (δpord,0 r r ord ord (3) ϕ−1,H ord  δH ord is an Hpord r ,Gess,ord r r p

p

-torsor realized as a (finite ´etale) sub-

h,Zpr ,Dpr

scheme of the quasi-finite ´etale scheme )mult , B[pr ]δordord Homδordord (Gr0D,H ord r p

H r p

× Homδordord

δ ord ord

H r p

H r p




H r p

)mult , B ∨ [pr ]δordord (Gr0D# ,H ord pr H pr

ord ord over δH ord , which is an H r p ,Gess,ord r p




×

× (Z/pr Z)δord

H ord pr

δ ord ord H r p

-orbit of ´etale-locally-defined triples

h,Zpr ,Dpr

ord,0 ord,#,0 ord ϕord −1,pr = (ϕ−1,pr , ϕ−1,pr , ν−1,pr ). ord ord , ϕ0,H ord )  ϕ−1,H ord -torsor giving an (4) (ϕ−2,H ord is an Hpord r ,Gess,ord r r r p

p

p

Hpord r ,Gess,ord l,Z r ,D

l,Zpr ,Dpr

-orbit of ´etale-locally-defined pairs (ϕ−2,pr , ϕ0,pr ).

pr

p

(5) (cord , c∨,ord )  (ϕord , ϕord ) is an Hpord r ,Uess,ord H ord −2,H ord 0,H ord H ord r r r pr

p

Hpord r ,Uess,ord

p

1,Zpr ,Dpr

p

∨,ord -orbit of ´etale-locally-defined pairs (cord ). pr , cpr

∼

∨,ord ord ord ord (6) τH ord → (cH ord , c ord ) is an H r H p ,Uess,ord r r p

-torsor giving an

1,Zpr ,Dpr

p

pr

giving no new structure.

2,Zpr ,Dpr

-torsor; i.e., it is an isomorphism

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(Each of the datum or pairs of data is built on top of the earlier ones.) Then each ord ord ord such scheme τH structure αH ord  η determines a naive ordinary level-Hpr ord of r r p

p

(Gη , λη , iη ) of type (L/pr L, h · , · i, Dpr ) (see Definition 3.3.3.3). If these schemes are ord orbits of liftable objects, then they determine an ordinary level-Hp structure αH of p (Gη , λη , iη ) of type (L ⊗ Zp , h · , · i, D) (see Definition 3.3.3.4). (This forces G to be Z

ordinary. See Remark 3.3.3.6.) ord Conversely, each ordinary level-Hp structure αH of (Gη , λη , iη ) of type p (L ⊗ Zp , h · , · i, D) arises this way for some (noncanonical choice of) symplecticZ

liftable Zpr . (The (Hp /Up (pr ))-orbit of Zpr , or rather the Hp -orbit of the lifting to a symplectic filtration on L ⊗ Zp , is nevertheless well defined.) Z

Proof. This follows from arguments similar to those in [62, Sec. 5.3]. Definition 4.1.5.22. (Compare with [62, Def. 5.3.1.12].) With the setting as in Section 4.1.2, suppose we are given a tuple (B, λB , iB , X, Y , φ, c, c∨ , τ ) in DDPE,O (R, I). Let H, Hp , Hp , and r be as in Definition 3.4.1.1. Let n0 and be as in Definition Hn0 := Hp /U p (n0 ) be as in Proposition 4.1.5.10, and let Hpord r 4.1.5.18 (and hence in Proposition 4.1.5.20). Let n = n0 pr , and let Hn := H/U(n). By an Hn -orbit of ´ etale-locally-defined naive ordinary level-n structure data of type (L/nL, h · , · i, Dpr ), we mean a scheme \,ord ord ord ∨,ord ord αH = (ZHn , ϕ−2,Hn , ϕord −1,Hn , ϕ0,Hn , δHn , cHn , cHn , τHn ) n ord ) finite ´etale over η, which is a composition of schemes (or rather just τH n ∼

∨,ord ord ord ord ord ord τH  (cord Hn , cHn )  (ϕ−2,Hn , ϕ0,Hn )  ϕ−1,Hn  δHn → ZHn  η n

where we have ord ord ∼ (1) τH = τHn0 × τH ord , n r η

p

∨,ord ∼ ∨,ord ord ∨ , cH (2) (cord ord ), Hn , cHn ) = (cHn0 , cHn ) ×(cH ord r 0

(3)

ord (ϕord −2,Hn , ϕ0,Hn )

η

pr

p

∼ ord ∼ , ϕord ), = (ϕ∼ −2,Hn0 , ϕ0,Hn0 ) ×(ϕ−2,H ord 0,H ord r r η

p

p

ord ∼ (4) ϕord , −1,Hn = ϕ−1,Hn0 × ϕ−1,H ord r η

p

ord ∼ ord (5) δH = δHn0 × δH ord , and n pr η (6) ZHn ∼ , = ZHn0 × ZH ord r η

p

with objects at the right-hand sides in some composition (4.1.5.11) in Proposition 4.1.5.10 and some composition (4.1.5.21) in Proposition 4.1.5.20. We use the same \,ord terminology Hn -orbit of ´ etale-locally-defined for each of the entries in αH . n We remove “ naive” from the terminology, and call it an Hn -orbit of ´ etaleˆ h · , · i, D), locally-defined ordinary level-n structure data of type (L ⊗ Z, Z

when the data are compatibly liftable to data (at all higher levels) satisfying the analogous conditions.

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As in [62, Def. 5.3.1.13], the equivalence relations among ordinary level-n strucˆ h · , · i, D) over η then induce equivalence relations among ture data of type (L ⊗ Z, their Hn -orbits.

Z

Definition 4.1.5.23. (Compare with Definition [62, Def. 5.3.1.14].) With the setting as in Section 4.1.2, suppose we are given a tuple (B, λB , iB , X, Y , φ, c, c∨ , τ ) in DDPE,O (R, I). Let H, Hp , Hp , and r be as in Definition 3.4.1.1. For each integer n0 ≥ 1 such that p - n and U p (n0 ) ⊂ Hp , set Hn0 := Hp /U p (n) and Hn0 pr := H/U(n0 pr ) as usual. Then an ordinary level-H structure datum of ˆ h · , · i, D) over η is a collection α\,ord = {α\,ord }n indexed by intetype (L ⊗ Z, 0 H Hn pr 0

Z

\,ord gers n0 ≥ 1 such that p - n0 and U (n0 ) ⊂ H , with elements αH described as n0 pr follows: p

p

\,ord is an Hn0 pr -orbit of ´etale-locally(1) For each index n0 , the element αH n 0 pr ˆ h · , · i, D) as in Defidefined ordinary level-n structure data of type (L ⊗ Z, Z

nition 4.1.5.22. \,ord (2) For all indices n0 and m0 such that n0 |m0 , the Hn0 pr -orbit αH is den pr 0

\,ord termined by the Hm0 pr -orbit αH by reduction modulo n0 pr . m pr 0

\,ord It is customary to denote αH by a tuple \,ord ord ord ∨,ord ord αH = (ZH , ϕ−2,H , ϕord , τH ), −1,H , ϕ0,H , δH , cH , cH \,ord each subtuple or entry being a collection indexed by n0 as αH is, and to denote ord ∼ ord ord by ιH the collection corresponding to τH . For convenience, we also write τH = ord τHp × τHp etc as in Definition 4.1.5.22. η

As in [62, Def. 5.3.1.13], the equivalence relations among naive ordinary level-n ˆ h · , · i, D) over η then induce equivalence relations structure data of type (L ⊗ Z, among their Hn -orbits.

Z

Convention 4.1.5.24. (Compare with [62, Conv. 5.3.1.15].) To facilitate the lan\,ord guage, we shall call αH an H-orbit, with similar usages applied to other objects with subscripts “H”. If we have two open compact subgroups H0 ⊂ H for which \,ord ordinary level structures at those levels make sense, and if we have an object αH 0 \,ord 0 at level H , then there is a natural meaning of the object αH at level H determined \,ord \,ord \,ord by αH . We say in this case that αH is the H-orbit of αH . 0 0 As in [62, Def. 5.3.1.16], and as above, the equivalence relations among naive ordinary level-n structure data then also induce equivalent relations among ordinary level-H structure data. Definition 4.1.5.25. (Compare with Definition 4.1.4.57.) With the setting as in Section 4.1.2, suppose moreover that η is a scheme over Spec(Z(p) ). The category ord DEGPEL,... (R, I) has objects of the form (G, λ, i, αHp , αH ) (over S), where: M ord p H

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(1) (G, λ, i) defines an object of DEGPE,O (R, I) (see Definition 4.1.3.1). ...ord ord (2) (Gη , λη , iη , αHp , αH ) defines an object of MH (η) (see Definition 3.4.1.1). p ord (3) αH is defined by a (scheme-theoretic) Hpord r -orbit of some principal ordip nary level-pr structure of type (L ⊗ Zp , h · , · i, D) such that, for each integer Z

0

r0 ≥ r, there exists some lifting to level pr (over some ´etale extension of η) compatible with degeneration (i.e., satisfying the analogue of Condition 0 4.1.4.1 at level pr ). Definition 4.1.5.26. (Compare with Definition 4.1.4.58.) With the setting as in Section 4.1.2, suppose moreover that η is a scheme over Spec(Z(p) ). The category DDPEL,... (R, I) has objects of the form M ord H \,ord ]), (B, λB , iB , X, Y , φ, c, c∨ , τ, [αH

where: (1) (B, λB , iB , X, Y , φ, c, c∨ , τ ) defines an object of DDPE,O (R, I) (see Definition 4.1.3.2). \,ord \,ord (2) [αH ] is an equivalence class of ordinary level-n structure data αH of ˆ h · , · i, D) defined over η. (See Definition 4.1.5.23.) type (L ⊗ Z, Z

Now it follows from Propositions 4.1.4.21, 4.1.4.52, and 4.1.4.56 that we have the following: Theorem 4.1.5.27. There is an equivalence of categories ...ord (R, I) : DD ...ord (R, I) → DEG ...ord (R, I) : M PEL, M H

PEL, M H ∨

(B, λB , iB , X, Y , φ, c, c

4.1.6

\,ord , τ, [αH ])

PEL, M H

ord 7→ (G, λ, i, αHp , αH ). p

Comparison with Degeneration Data for Level Structures in Characteristic Zero

Let H, Hp , Hp , r, and rν be as in beginning of Section 3.3.5. Let n0 be some integer prime to p such that U p (n0 ) ⊂ Hp . Let Hn0 := Hp /U p (n0 ) and Hpr := Hp /Up (pr ), bal r := Hp /Up,1 (p ) as in Section 3.3.3. Let n := n0 pr . Then U(n) ⊂ H and let Hpord r and we set Hn := H/U(n). Let S = Spec(R) be as in Section 4.1.2. Assume moreover that the generic point η of S is a point over S0,prν = Spec(F0 [ζprν ]) (see Proposition 3.3.5.1). By normality of S, this forces S to be a scheme over Spec(Z[ζprν ]), so that there exists a canonical ∼ isomorphism ζprν ,S 0 : ((Z/prν Z)(1))S 0 → µprν ,S 0 for each scheme S 0 → S (which is the pullback of the canonical ζprν over Spec(Z[ζprν ])). Let (G, λ, i) be a degenerating family of type (PE, O) over S as in Definition 4.1.3.1, with degeneration datum (B, λB , iB , X, Y , φ, c, c∨ , τ ) given by (4.1.3.3). For simplicity, let us assume (until we finish the proof of Proposition 4.1.6.1) that X and Y are constant with values X and Y , respectively.

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Let [(ZH , ΦH , δH )] be an ordinary cusp label at level H for the PEL-type O-lattice (L, h · , · i, h0 ) (see Definition 3.2.3.8). Let \ ∼ ∨ αH = (ZH , ϕ∼ −2,H , ϕ−1,H , ϕ0,H , δH , cH , cH , τH )

ˆ h · , · i), as in [62, Def. 5.3.1.14], be a level-H structure datum of type (L ⊗ Z, Z

∼ with (ϕ∼ −2,H , ϕ0,H ) inducing the (ϕ−2,H , ϕ0,H ) in a representative (ZH , ΦH = (X, Y, φ, ϕ−2,H , ϕ0,H ), δH ) of [(ZH , ΦH , δH )], as in the corrected [62, Def. 5.4.2.8] in the errata. Then ∼ ∨ (ZH , (X, Y, φ, ϕ∼ −2,H , ϕ0,H ), (B, λB , iB , ϕ−1,H ), δH , (cH , cH , τH ))

is an object of DDfil.-spl. PEL,MH (R, I), as in [62, Lem. 5.4.2.10; see also the errata]. This means we have a composition of finite ´etale morphisms ∼

∼ ∼ τHn  (cHn , c∨ Hn )  (ϕ−2,Hn , ϕ0,Hn )  ϕ−1,Hn  δHn → ZHn  η

as in [62, Sec. 5.3]. By taking reduction modulo n0 or restrictions of various objects, we obtain an induced composition as in (4.1.5.11) in Proposition 4.1.5.10. We claim that we can also obtain an induced composition as in (4.1.5.21) in Proposition 4.1.5.20. By definition, ZH is an H-orbit of strongly symplectic admissible filtrations Z ˆ This includes, in particular, the datum of an Hp -orbit of a symplectic on L ⊗ Z. Z

admissible filtration Z ⊗ Zp = {Z−i ⊗ Zp }i on L ⊗ Zp . By Definition 3.2.3.1, and ˆ Z

ˆ Z

Z

by replacing Z with another representative in the H-orbit ZH if necessary, we shall assume that Z−2 ⊗ Zp ⊂ D0 ⊂ Z−1 ⊗ Zp ˆ Z

ˆ Z

(see (3.2.3.2)), which induces a filtration D−1 = {Di−1 }i on GrZ−1 ⊗ Zp . By taking ˆ Z

reduction modulo pr , we have the compatibility (3.2.3.3) (resp. (3.2.3.5)), which induces a filtration D−1,pr = {Di−1,pr }i on GrZ−1,pr given by (3.2.3.4) (resp. (3.2.3.6)). Note that (3.2.3.3) (resp. (3.2.3.4), resp. (3.2.3.5), resp. (3.2.3.6)) is a special case of (4.1.4.5) (resp. (4.1.4.14), resp. (4.1.4.6), resp. (4.1.4.15)) in the sense that we did not assume that the latter comes from some symplectic-liftable admissible filtration. bal r Since Up,1 (p ) ⊂ Hp ⊂ Up,0 (pr ), and since the action of Up,0 (pr ) stabilizes 0 Dpr as an O ⊗ (Z/pr Z)-submodule of L/pr L, the compatibility (3.2.3.3) is indeZ

pendent of the choice of Z, once Z exists (cf. Remark 3.2.3.7). Moreover, the Hpord r -orbit ZH ord (which is equivalently an Hpr -orbit) of Zpr = {Z−i,pr }i determines r p

#

Z 0 Z 0 the Hpord r -orbits of the two O-submodules Gr−2,pr ⊂ GrD,pr and Gr−2,pr ⊂ GrD# ,pr 0 0 Z as in (4.1.4.7) and (4.1.4.8), which in turn define GrD−1,pr = GrD,pr / Gr−2,pr and

Gr0D#

−1,pr

#

= Gr0D# ,pr / GrZ−2,pr as in (4.1.4.16) and (4.1.4.17). These are all indepen-

dent of the choice of Z. Thus, we have obtained a well-defined assignment of ZH ord to ZH . r p

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The datum δH in the cusp label [(ZH , ΦH , δH )] is by definition the H-orbit of some splitting δ, which includes in particular the datum of the Hpr -orbit of some splitting ∼

δpr : GrZ−2,pr ⊕ GrZ−1,pr ⊕ GrZ0,pr → L/pr L. Since we have the compatibility (3.2.3.3), we have splittings ∼

: GrZ−2,pr ⊕ Gr0D−1,pr → Gr0D,pr δpord,0 r and #

δpord,#,0 : GrZ−2,pr ⊕ Gr0D# r

−1,pr

∼

→ Gr0D# ,pr

as in (4.1.4.23) and (4.1.4.24). Note that the Hpr -orbit of δpr determines and is determined by the Hpord r -orbit of the pair δpord := (δpord,0 , δpord,#,0 ). r r r (Since the splitting δpr does not respect pairings, the two splittings δpord,0 and r δpord,#,0 are compatible with φ0D,pr : Gr0D,pr → Gr0D# ,pr only in the sense that r #

φ0D,pr (GrZ−2,pr ) ⊂ GrZ−2,pr ; cf. Lemma 4.1.4.22.) By abuse of notation, let us deord,0 ord,#,0 ord ord ) by δH , δpord,#,0 ). note the Hpord = (δpord,0 r r r -orbit of the pair δpr ord = (δ ord , δ ord H H r p

pr

pr

ord This is the same δH ord as in Proposition 4.1.5.20. r p

ord Thus, we have obtained a well-defined assignment of δH ord to δH . r p

The assignment of ϕord as in Proposition 4.1.5.20 to ϕ−1,H or rather some −1,Hpord r ...ord ...ord ϕ−1,Hpr is as in Proposition 3.3.5.1 for MHh and an obvious analogue MHh of MH (cf. Definitions 1.2.1.15 and 3.4.1.9). This is where we need η to be defined over S0,prν = Spec(F0 [ζprν ]). The datum ΦH in the cusp label [(ZH , ΦH , δH )] is by definition the H-orbit ˆ does not modify of some tuple (X, Y, φ, ϕ−2 , ϕ0 ) (where the action of H ⊂ G(Z) the first three entries), which includes in particular the datum of an Hpr -orbit, or equivalently an Hpord r -orbit, of pairs of isomorphisms (ϕ−2,pr , ϕ0,pr ), or equivalently pairs of isomorphisms (ϕ−2,pr , ϕ# −2,pr ), as in (4.1.4.11), (4.1.4.12), and (4.1.4.13). The two isomorphisms ϕ−2,pr and ϕ# −2,pr induce respectively the two isomorphisms ∼

(ϕ−2,pr )mult : (GrZ−2,pr )mult → T [pr ] S S and #

∼

mult (ϕ# : (GrZ−2,pr )mult → T ∨ [pr ] S −2,pr )S

as in (2) of Proposition 4.1.4.21. By abuse of notation, let us denote ord mult the (scheme-theoretic) Hpr -orbit of the pair ((ϕ−2,pr )mult , (ϕ# ) by −2,pr )S S # mult mult ((ϕ−2,H ord ) , (ϕ ) ), which determines and is determined by a scheme S r −2,H ord S p

pr

(ϕ−2,H ord , ϕ0,H ord ) over η. r r p

p

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If we also include ϕ−1,Hpr and ϕord into consideration, then we have a sub−1,H ord r p

∼ scheme (ϕ∼ −2,Hpr , ϕ0,Hpr ) of (ϕ−2,Hpr , ϕ0,Hpr ) × ϕ−1,Hpr which is an Hpr -orbit ZH ord pr

of ´etale-locally-defined ((ϕ−2,pr , ϕ0,pr ), ϕ−1,pr ) and surjects under the two projections to (ϕ−2,Hpr , ϕ0,Hpr ) and ϕ−1,Hpr . Moreover, it induces a subscheme , )mult , (ϕ# )mult ) × ϕ−1,H ord ((ϕord )mult , (ϕord,# )mult ) of ((ϕ−2,H ord S S r r −2,H ord −2,H ord S −2,H ord S r p

pr

p

pr

p

ZH ord pr

, ϕord ) of which determines and is determined by a subscheme (ϕord −2,H ord 0,H ord r r p

p

etale, both subschemes being Hpord ) × ϕ−1,H ord , ϕ0,H ord (ϕ−2,H ord r -orbits of ´ r r r p

p

p

ZH ord pr

locally-defined objects inducing surjections under the two projections. ) to , ϕ0,H ord Thus, we have obtained well-defined assignments of (ϕ−2,H ord r r p

p

ΦH , or rather just to (ϕ−2,H , ϕ0,H ); and of ((ϕord )mult , (ϕord,# )mult ) and S −2,H ord −2,H ord S r pr

p

∼ (ϕord , ϕord ) to (ϕ∼ −2,H , ϕ0,H ), which are compatible with each other. −2,H ord 0,H ord r r p

p

∼ ∼ ess The pair (cHn , c∨ Hn ) as a scheme over (ϕ−2,Hn , ϕ0,Hn ) is an Hn,U1,Zn -torsor giving 1 ∨ an Hn,Uess -orbit of ´etale-locally-defined pairs (cn , cn ). Each cn : n X → Bη∨ deter1,Zn mines by restriction to p1r X and by composition with (4.1.4.32) a homomorphism

cord pr :

1 nX

∨,ord 0 → Bη∨ → Bη,p = Bη∨ /ϕord,#,0 r −1,pr ((GrD#

−1,pr

)mult ) η

1 lifting c : X → B ∨ , while each c∨ n : n Y → Bη determines by restriction to by composition with (4.1.4.31) a homomorphism

: cp∨,ord r

1 pr Y

1 pr Y

and

ord,0 0 ord → Bη,p )mult ) r = Bη /ϕ−1,pr ((GrD −1,pr η

lifting c∨ : Y → B. Since the actions of Hn,Uess and Hpord r ,Uess,ord 1,Zn

respect pairings

1,Zpr ,Dpr

and are compatible, the Hpord r ,Uess,ord

1,Zpr ,Dpr

∨,ord -orbit of (cord ) is independent of choices, pr , cpr

and defines a scheme (cord , c∨,ord ) over (ϕord , ϕord ). H ord −2,H ord 0,H ord H ord r r r pr

p

p

p

∨,ord ∨ Thus, we have a well-defined assignment of (cord , cH ord ) to (cH , cH ). H ord r p

pr

ess Finally, the scheme τHn  (cHn , c∨ Hn ) is an Hn,U2,Zn -torsor giving an Hn0 ,Uess -orbit of ´ e tale-locally-defined τ , which induces the (trivial) n 2,Zn

Hpord r ,Uess,ord

2,Zpr ,Dpr

∼

∨,ord ord ord ord -torsor τH ord → (cH ord , c ord ). (Note that the group H r H p ,Uess,ord r r p

p

pr

is

2,Zpr ,Dpr

trivial.) This is consistent with the convention that τnord := τn0 does not see any information at p. Thus, we have verified our claim that we can also obtain an induced composition as in (4.1.5.21) in Proposition 4.1.5.20. Proposition 4.1.6.1. With the assumptions as above, there is a commutative dia-

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gram of finite ´etale morphisms: ord / τHord ∼ τ × τH ord n = Hn0 r

τHn

η

n





2,Zpr ,Dpr

∨,ord ∼ ∨ ord / (cord , c∨,ord ) Hn , cHn ) = (cHn0 , cHn0 ) ×(cH ord H ord r

(cHn , c∨ Hn )

η

mod Hn,Uess 1,Z

∼ (ϕ∼ −2,Hn , ϕ0,Hn )

n0



× H ord ess,ord pr ,U

1,Zpr ,Dpr

ord ∼ ord ∼ ∼ / (ϕord , ϕord ) −2,Hn , ϕ0,Hn ) = (ϕ−2,Hn0 , ϕ0,Hn0 ) ×(ϕ−2,H ord 0,H ord r r η

mod

mod Hn ,Gess 0 l,Z

n0





p

p

Hn ,Gess × H ord ess,ord 0 l,Zn pr ,G 0 l,Zpr ,Dpr

ord ∼ / ϕord −1,Hn = ϕ−1,Hn0 × ϕ−1,H ord r

ϕ−1,Hn

η

mod Hn,Gess h,Z



pr

p

mod Hn ,Uess 0 1,Z

n



p

mod Hn ,Uess × H ord ess,ord 0 2,Zn pr ,U 0

mod Hn,Uess 2,Z

n

δH n

p

mod Hn ,Gess 0 h,Z

n0



× H ord ess,ord pr ,G

h,Zpr ,Dpr

ord ord ∼ / δH × δH δ ord n = Hn0 r

∼

η

o

p

o





ZHn ∼ = ZHn0 × ZH ord r

ZHn

p

η





η

η

In this diagram, the objects at the right-hand sides form an Hn -orbit of ´etalelocally-defined naive ordinary level-n structure data (see Definition 4.1.5.22), and all horizontal morphisms over ZHn are torsors of the expected constant finite groups: (1) The induced morphism τHn → (cHn , c∨ Hn )

× ∨,ord (cord Hn ,cHn )

ord τH n

is a torsor under ess,ord ess ∼ Uess 2,Zpr ,Dpr = ker(U2,Zpr ,Dpr  U2,Zpr ,Dpr ) ∼  Hpord = ker(Hpr ,Uess r ,Uess,ord 2,Z r p

).

2,Zpr ,Dpr

(2) The induced morphism ∼ ∼ (cHn , c∨ Hn ) → (ϕ−2,Hn , ϕ0,Hn )

∨,ord (cord Hn , cHn )

×

ord (ϕord −2,Hn ,ϕ0,Hn )

is a torsor under ess,ord ∼ ker(Uess 1,Zpr ,Dpr  U1,Zpr ,Dpr ) = ker(Hpr ,Uess 1,Z

pr

 Hpord r ,Uess,ord

1,Zpr ,Dpr

).

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(3) The induced morphism ∼ (ϕ∼ −2,Hn , ϕ0,Hn ) → ϕ−1,Hn

247

ord × (ϕord −2,Hn , ϕ0,Hn )

ϕord −1,Hn

∼ is an isomorphism, because Hpr ,Gess = Hpord r ,Gess,ord l,Z r p

h,Zpr ,Dpr

under the canonical

l,Zpr ,Dpr

∼ ess,ord isomorphism Gess l,Zpr ,Dpr = Gl,Zpr ,Dpr . (4) The morphism ϕ−1,Hn → ϕord −1,Hn is a torsor under ker(Pess r r  Gess,ord ) ∼ = ker(Hpr ,Gess h,Zp ,Dp

page 247

h,Zpr

 Hpord r ,Gess,ord

).

h,Zpr ,Dpr

In particular, all horizontal morphisms in the above commutative diagram and all the induced morphisms are ´etale and surjective. Proof. Since η is a point over S0,prν = Spec(F0 [ζprν ]), the assignment to the data on the right-hand sides determines the data on the left-hand side. (However, this does not imply that all data as in (4.1.5.21) in Proposition 4.1.5.20 comes from such an assignment.) The statements on the induced morphisms follow from their definitions as forgetful functors. The isomorphisms between various kernels follow from the very definitions of the groups (see Definition 4.1.5.12). Theorem 4.1.6.2. With notation and assumptions as in the first two paragraphs of this subsection, let DDPEL,Mord (R, I) be the full subcategory of DDPEL,MH (R, I) H formed by objects each of whose underlying ZH is ordinary. (That is, each such ZH is compatible with the filtration D in the sense that, over an ´etale extension of η over which ZH becomes split and becomes part of a representative of a cusp label, it is compatible with D as in Definition 3.2.3.1.) Let DEGPEL,Mord (R, I) be the essential H image of DDPEL,Mord (R, I) under the equivalence of categories H MPEL,MH (R,I) : DDPEL,MH (R, I) → DEGPEL,MH (R, I) (4.1.6.3) in [62, Thm. 5.3.1.19], which induces an equivalence of categories MPEL,Mord : DDPEL,Mord (R, I) → DEGPEL,Mord (R, I). (4.1.6.4) H (R,I) H H Then there is a commutative diagram / DD ... (R, I) DDPEL,Mord (R, I) DDPEL,MH (R, I) o PEL, M ord H H MPEL,MH (R,I) o



MPEL,Mord (R,I) o H

DEGPEL,MH (R, I) o



DEGPEL,Mord (R, I) H

(R,I) o MPEL,... M ord H

 ...ord (R, I)

/ DEG PEL, M H

where MPEL,... (R, I) is the equivalence of categories as in Theorem 4.1.5.27. All M ord H horizontal morphisms in this commutative diagram are fully faithful. Proof. This follows from Proposition 4.1.6.1 by ´etale descent (which allows us to reduce to the case where X and Y are constant), and from the proof of Proposition 4.1.4.42 (which shows that the pairing conditions for DDPEL,MH (R, I) and for DDPEL,... (R, I) are compatible with each other at principal levels). M ord H

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In this section, let us continue with the settings in Section 4.1.6. Let rH be as in Definition 3.4.2.1. Let us fix a representative (ZH , ΦH , δH ) of an ordinary cusp label at level H for the PEL-type O-lattice (L, h · , · i, h0 ) (see Definition 3.2.3.8). Let us fix the choice of a representative (Zn , Φn , δn ) in the Hn -orbit (ZHn , ΦHn , δHn ). 4.2.1

Constructions with Level Structures but without Positivity Conditions ...ord ...ord,Zn The moduli problem Mn has a boundary version Mn , with tautological object (B, λB , iB , ϕ−1,n0 , ϕord −1,pr ) giving the abelian parts of degenerations. Then we have isogenies ord,0 0 mult ... B  Bpord := B/image(ϕord,0 r −1,pr ) = B/ϕ−1,pr ((GrD−1,pr ) M ord,Zn )

(4.2.1.1)

n

and ord,#,0 0 ∨ := B ∨ /image(ϕord,#,0 B ∨  Bp∨,ord r −1,pr ) = B /ϕ−1,pr ((GrD#

−1,pr

...ord,Zn ) (4.2.1.2) )mult M n

as in (4.1.4.31) and (4.1.4.32), respectively, together with isogenies −1 Bpord  Bpord )...ord,Zn ∼ r r /(GrD =B = B/B[pr ] ∼ −1,pr M n

(4.2.1.3)

∼ ∨ ∨ r ∼ ∨ )... n = B /B [p ] = B , M ord,Z n

(4.2.1.4)

and /(Gr−1  Bp∨,ord Bp∨,ord r r D#

−1,pr

as in (4.1.4.34) and (4.1.4.33), respectively. Consider the canonical homomorphism HomO ( n1 X, Bp∨,ord ) → HomO (X, B ∨ ) r defined by pre-composition with X

∼ → n1 X

(4.2.1.4) Bp∨,ord  r

(4.2.1.5)

and by post-composition with [n0 ]

B∨ → B∨.

(4.2.1.6)

(We can compare these with the following: The canonical morphism HomO ( n1 X, B ∨ ) → HomO (X, B ∨ ) induced by restriction to X can be defined ∼ alternatively by pre-composition with X → n1 X and by post-composition with [n] : B ∨  B ∨ .) Similarly, consider the canonical homomorphism HomO ( n1 Y, Bpord r ) → HomO (Y, B) ∼ 1 nY

defined by pre-composition with Y → Bpord r

(4.2.1.3)



(4.2.1.7)

and by post-composition with [n0 ]

B → B.

(4.2.1.8) ∨

∨

Consider also the canonical homomorphisms HomO (X, B ) → HomO (Y, B ) (resp. HomO (Y, B) → HomO (Y, B ∨ )) defined by pre-composition with the morphism φ : Y → X (resp. by post-composition with λB : B → B ∨ ). All these group functors

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defined by HomO ( · , · ) are relatively representable by Proposition 3.1.2.4, because the abelian schemes involved are all ordinary (see Definition 3.1.1.2). Lemma 4.2.1.9. The kernels of (4.2.1.5) and (4.2.1.7) are finite ´etale group ...ord,Zn . The kernels of HomO (X, B ∨ ) → HomO (Y, B ∨ ) and schemes over Mn HomO (Y, B) → HomO (Y, B ∨ ) are finite flat group schemes of ´etale-multiplicative ...ord,Zn . type over Mn Proof. Since ker((4.2.1.4)) ∼ = (Gr−1 D#

−1,pr

∨ )... → B∨) ∼ = n and ker([n0 ] : B M ord,Z n

B ∨ [n0 ] are finite ´etale, ker((4.2.1.6)) is also finite ´etale. Hence, the kernel of HomO ( n1 X, Bp∨,ord ) → HomO (X, B ∨ ) is finite ´etale because it is a finite flat subr group scheme of HomZ ( n1 X/X, ker((4.2.1.6))). Similarly, since ker((4.2.1.3)) ∼ = −1 ∼ ... (GrD−1,pr ) M nord,Zn and ker([n0 ] : B → B) = B[n0 ] are finite ´etale, the kernel of HomO ( n1 Y, Bpord etale. r ) → HomO (Y, B) is also finite ´ Since B (resp. B ∨ ) is ordinary, its (commutative) finite flat subgroup schemes are all of ´etale-multiplicative type. Hence, the kernel of HomO (X, B ∨ ) → HomO (Y, B ∨ ) (resp. HomO (Y, B) → HomO (Y, B ∨ )) is a finite flat group scheme of ...ord,Zn because it is a finite flat subgroup scheme of ´etale-multiplicative type over Mn HomZ (X/Y, B ∨ ) (resp. HomZ (Y, ker(λB ))). ...ord,Zn ...ord representing Let C Φn be the (relative) proper flat group scheme over Mn the fiber product HomO ( n1 X, Bp∨,ord ) r

× HomO (Y,B ∨ )

HomO ( n1 Y, Bpord r ).

(4.2.1.10)

...ord Then C Φn carries a tautological pair (cord n :

1 nX

→ Bp∨,ord , cn∨,ord : r

→ Bpord r )

1 nY

of liftings of (c : X → B ∨ , c∨ : Y → B) satisfying the compatibility λB c∨ = cφ, which is equivalent to two tautological pairs (cn0 :

1 n0 X

→ B ∨ , c∨ n0 :

1 n0 Y

→ B)

and (cord pr :

1 pr X

, cp∨,ord : → Bp∨,ord r r

of liftings of (c : X → B ∨ , c∨ : Y → B). Let us extend φ : Y → X naturally to φn :

1 nY

1 pr Y

→

→ Bpord r ) 1 n X.

Proposition 4.2.1.11. ◦ be the (reduced) fiberwise geometric identity compo(1) Let HomO ( n1 X, Bpord r ) 1 nent of HomO ( n X, Bpord r ) (see (4) of Proposition 3.1.2.4). Then the canonical homomorphism ∨,ord ◦ 1 HomO ( n1 X, Bpord ) r ) → HomO ( X, Bpr n

× HomO (Y,B ∨ )

HomO ( n1 Y, Bpord r )

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...ord over Mn has kernel the finite flat group scheme ord ◦ 1 HomO ( n1 X/φn ( n1 Y ), ker(λord B,pr )) ∩ HomO ( n X, Bpr )

(see (4.1.4.39)) of ´etale-multiplicative type (see Definition 3.1.1.1) and sche...ord,◦ ...ord matic image an abelian subscheme C Φn of C Φn . (See Lemma 3.1.2.2 and Definition 3.1.2.3.) ...ord (2) There exists an integer m ≥ 1 such that multiplication by m maps C Φn ...ord,◦ scheme-theoretically to a subscheme of C Φn , so that the group scheme ...ord ...ord ...ord ...ord π 0 (C Φn /Mn ) of fiberwise connected components of C Φn over Mn is defined and is of ´etale-multiplicative type. (See Lemma 3.1.2.2 and Definition ...ord ...ord 3.1.2.3.) Moreover, the rank of π 0 (C Φn /Mn ) has no prime factors other than those of Disc, n, [X : φ(Y )], and the rank of ker(λB ) (or rather ...ord ...ord ker(λord B,pr )). (This implies that the rank of π 0 ( C Φn /Mn ) does not contain prime factors other than those of Disc, n and [L# : L].) Proof. The first claim of the lemma is clear, because the finite flat group scheme HomO ( n1 X/φn ( n1 Y ), ker(λord B,pr )) ord 1 1 = HomO ( n1 X, ker(λord B,pr )) ∩ HomO ( n X/φn ( n Y ), Bpr )

is the kernel of ∨,ord 1 HomO ( n1 X, Bpord ) r ) → HomO ( X, Bpr n

× HomO (Y,B ∨ )

HomO ( n1 Y, Bpord r ),

and because B and hence Bpord are ordinary (by Lemma 3.1.1.5). r ◦ )◦ , HomO ( n1 Y, Bpord and For the second claim, let HomO ( n1 X, Bp∨,ord r r ) , ∨ ◦ HomO (Y, B ) denote respectively the fiberwise geometric identity components of ...ord,◦◦◦ ∨ ◦ HomO ( n1 X, Bp∨,ord denote ), HomO ( n1 Y, Bpord r r ), and HomO (Y, B ) , and let C Φ n the proper smooth group scheme representing the fiber product HomO ( n1 X, Bp∨,ord )◦ r

× HomO (Y,B ∨ )◦

◦ HomO ( n1 Y, Bpord r ) .

...ord ...ord (4) of Proposition 3.1.2.4, the group schemes π 0 (C Φn /Mn ) and ...ord,◦◦◦ ...ord π 0 (C Φn /Mn ) are defined, and their ranks differ up to multiplication by numbers having only prime factors of those of Disc. Therefore, it suffices to show that ...ord,◦◦◦ ...ord the rank of π 0 (C Φn /Mn ) has no prime factors other than those of n, [X : φ(Y )], and the rank of ker(λB ) (or rather ker(λord B,pr )). ...ord,◦◦◦ The kernel Kn of the canonical homomorphism C Φn → HomO (Y, B ∨ )◦ is given by a fiber product Kn,1 × Kn,2 , where By

MZnn

Kn,1 := HomO ( n1 X/φ(Y ), Bp∨,ord ) ∩ HomO ( n1 X, Bp∨,ord )◦ r r and ord ◦ 1 Kn,2 := HomO ( n1 Y, ker(λord B,pr )) ∩ HomO ( n Y, Bpr ) .

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...ord,◦◦◦ ...ord Since HomO (Y, B ∨ )◦ is an abelian scheme, the group π 0 (C Φn /Mn ) can be identified with a quotient of Kn . Since the rank of Kn is the product of the ranks of Kn,1 and of Kn,2 , it has no prime factors other than those of n, X/φ(Y ), and the rank of ker(λord B,pr ), as desired. Let us consider the finitely generated commutative group (cf. [62, (6.2.3.5)])   ... ...ord y ⊗ φ(y 0 ) − y 0 ⊗ φ(y) . S Φn := S Φn0 := (( n10 Y ) ⊗ X)/ (b n10 y) ⊗ χ − ( n10 y) ⊗ (b? χ) y,y0 ∈Y, Z χ∈X,b∈O

(4.2.1.12) As in [62, Sec. 6.2.2–6.2.3], the formal properties of the pullbacks of the Poincar´e biextension (as in [62, Lem. 6.2.2.5]) allow us to assign to each X ...ord `= [( n10 yi ) ⊗ χi ] ∈ S Φn 1≤i≤k

a well-defined rigidified invertible sheaf Ψord n (`) :=

⊗

O... ,1≤i≤k C ord

∗ 1 (c∨ n0 ( n0 yi ), c(χi )) PB

(4.2.1.13)

Φn

...ord over C Φn , together with canonical isomorphisms ord ∆ord,∗ n,`,`0 : Ψn (`)

∼

0 ord 0 ⊗ Ψord n (` ) → Ψn (` + ` ) O... ord C Φn

...ord for all `, `0 ∈ S Φn , satisfying the necessary compatibilities with each other making ... -algebra, so that we can define ⊕ Ψord n (`) an O C ord ... Φn ord

`∈ S Φn

  ...ord ord ⊕ Ψ (`) . Ξ Φn := SpecO...ord ...ord n C `∈ S Φn

Φn

...ord ...ord If we denote by E Φn = HomZ ( S Φn , Gm ) the group of multiplicative type of finite ...ord ...ord ...ord type with character group S Φn over Spec(Z), then Ξ Φn is an E Φn -torsor, and we have tautological trivializations τnord = τn0 : 1( n1

0

∼

Y )×X

∗ ⊗ −1 → (c∨ n0 × c) PB

...ord over Ξ Φn , which corresponds to a tautological homomorphism ιord n = ι n0 : ∼

1 n0 Y

→ G\ .

⊗ −1 Let τ : 1Y ×X → (c∨ × c)∗ PB be the restriction of τnord to 1Y ×X , which corresponds to a tautological homomorphism ι : Y → G\ . ...ord ...ord ...ord Let S Φn ,tor denote the torsion subgroup of S Φn , and let S Φn ,free denote the ...ord ...ord ...ord quotient of S Φn by S Φn ,tor , namely the free commutative quotient group of S Φn . ...ord ...ord ...ord ...ord Let E Φn ,tor := HomZ ( S Φn ,tor , Gm ) (resp. E Φn ,free := HomZ ( S Φn ,free , Gm )) be

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...ord the group of multiplicative type of finite type with character group S Φn ,tor (resp. ...ord S Φn ,free ) over Spec(Z). Then the exact sequence ...ord ...ord ...ord 0 → S Φn ,tor → S Φn → S Φn ,free → 0 induces an exact sequence ...ord ...ord ...ord 0 → E Φn ,free → E Φn → E Φn ,tor → 0 ...ord in the reversed direction. Since S Φn ,free is a finitely generated free commutative ...ord group, E Φn ,free is by definition a torus (cf. [62, Def. 3.1.1.5]). As in Definition 4.1.5.9, the choice of Zn in the Hn -orbit ZHn determines the groups Hn,Uess ⊂ Hn,Uess ⊂ Hn,Zess ⊂ Hn,Pess ⊂ Hn , Zn Zn Zn 2,Zn with short exact sequences 1 → Hn,Zess → Hn,Pess → Hn,Gess → 1, Zn Zn h,Zn 1 → Hn,Uess → Hn,Zess → Hn,Gess → 1, Zn Zn l,Zn 1 → Hn,Uess → Hn,Uess → Hn,Uess → 1, Zn 2,Zn 1,Zn together with similar subgroups or quotients of subgroups when n is replaced with n0 or pr . Note that the quotient Hn,Pess \Hn describes elements in the orbit ZHn . The Zn 0 fiber of ΦHn → ZHn at Zn is naturally an orbit under the image Hn,G ess of Hn,Pess in Zn l,Zn ess ess ess Gl,Zn . By viewing the semidirect product Gh,Zn n UZn as a subgroup of Gess (Z/nZ) ess ess using the splitting δn , and by viewing Gess h,Zn n U1,Zn as its quotient by U2,Zn , we can ess and Hn,Gess nUess , fitting define as in Definition 4.1.5.9 the groups Hn,Gess 1,Zn h,Zn nUZn h,Zn into short exact sequences 0 ess → Hn,Pess → H 1 → Hn,Gess →1 n,Gess Zn h,Zn nUZn l,Z n

and ess → Hn,Gess nUess 1 → Hn,Uess → Hn,Gess → 1. 2,Zn 1,Zn h,Zn nUZn h,Zn

0 ess Let Hn,G denote the canonical image of Hn,Gess in Gess ess h,Zn , so that we have h,Zn nU1,Zn h,Zn an exact sequence 0 ess 1 → Hn,Uess → Hn,Gess → Hn,G → 1. ess 1,Zn h,Zn nU1,Zn h,Z n

Let us also define similar subgroups or quotients of subgroups when n is replaced with n0 or pr . As in Definition 4.1.5.18, the compatibility of Zn with Dpr allows us to define subgroups Hpord r ,Uess,ord

2,Zpr ,Dpr

ord ord ord ⊂ Hpord r ,Uess,ord ⊂ Hpr ,Zess,ord ⊂ Hpr ,Pess,ord ⊂ Hpr , Zpr ,Dpr

Zpr ,Dpr

Zpr ,Dpr

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with short exact sequences ord ord 1 → Hpord r ,Zess,ord → Hpr ,Pess,ord → Hpr ,Gess,ord

→ 1,

ord ord 1 → Hpord r ,Uess,ord → Hpr ,Zess,ord → Hpr ,Gess,ord

→ 1,

Zpr ,Dpr

Zpr ,Dpr

Zpr ,Dpr

h,Zpr ,Dpr

Zpr ,Dpr

1 → Hpord r ,Uess,ord

l,Zpr ,Dpr

ord → Hpord r ,Uess,ord → Hpr ,Uess,ord Zpr ,Dpr

2,Zpr ,Dpr

→ 1.

1,Zpr ,Dpr

As in Remark 4.1.5.19, note that ∼ . = Hpr ,Gess l,Z r

Hpord r ,Gess,ord

p

l,Zpr ,Dpr

Let H ord,0 ess,ord r p ,Gl,Z

pr ,Dpr

under the canonical homomordenote the image of Hpord r ,Pess,ord Zpr ,Dpr

ess,ord phism Pess,ord Zpr ,Dpr → Gl,Zpr ,Dpr . By definition, we have

∼ = Hp0 r ,Gess l,Z r

H ord,0 ess,ord r p ,Gl,Z

p

pr ,Dpr

∼ ess under the canonical isomorphism Gess,ord l,Zpr ,Dpr = Gl,Zpr . ord Let δpr be induced by δpr as in Section 4.1.6. Then we can view the semidirect ess,ord ess,ord ord product Gess,ord h,Zpr ,Dpr n UZpr ,Dpr as a subgroup of MZpr ,Dpr using δpr , which coinciess ess ess des with the image of PZpr ,Dpr ∩(Gh,Zn n UZn ) under the canonical homomorphism ess,ord ess,ord ess,ord Pess Dpr → MZpr ,Dpr . Note that Gh,Zpr ,Dpr n U1,Zpr ,Dpr is the (isomorphic) quotient of ess,ord ess,ord Gh,Z n Uess,ord Zpr ,Dpr by U2,Zpr ,Dpr = {Id}. Hence, we can define as in Definition pr ,Dpr ord 4.1.5.18 the groups Hpr ,Gess,ord nUess,ord and Hpord , fitting into r ,Gess,ord nU1,ess,ord Zpr ,Dpr

h,Zpr ,Dpr

Zpr ,Dpr

h,Zpr ,Dpr

short exact sequences 1 → Hpord r ,Gess,ord

h,Zpr ,Dpr

nUess,ord Z r ,D r p

p

ord,0 → Hpord ess,ord r ,Pess,ord → H r

→1

p ,Gl,Z

Zpr ,Dpr

pr ,Dpr

and 1 → Hpord r ,Uess,ord

2,Zpr ,Dpr

→ Hpord r ,Gess,ord

h,Zpr ,Dpr

(Certainly, Hpord r ,Uess,ord

2,Zpr ,Dpr

Hpord r ,Gess,ord

h,Zpr ,Dpr

ess,ord nU1,Z r ,D p

pr

nUess,ord Z r ,D r p

p

→ Hpord r ,Gess,ord

h,Zpr ,Dpr

= {Id}, so that Hpord r ,Gess,ord

h,Zpr ,Dpr

nUess,ord 1,Z r ,D p

nUess,ord Z r ,D r p

→ 1.

pr

is isomorphic to

p

via the last short exact sequence above.) Let H ord,0 ess,ord r p ,Gh,Z

denote the canonical image of

Hpord r ,Gess,ord h,Z r ,D p

pr

nUess,ord 1,Z r ,D p

in pr

Gess,ord h,Zpr ,Dpr ,

so that we have

an exact sequence 1 → Hpord r ,Uess,ord

1,Zpr ,Dpr

→ Hpord r ,Gess,ord

h,Zpr ,Dpr

nUess,ord 1,Z r ,D p

pr

→ H ord,0 ess,ord r p ,Gh,Z

pr ,Dpr

pr ,Dpr

→ 1.

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Then we define the following groups: Hnord ord Hn,Pess Zn ord Hn,U ess Zn

:= Hn0 × Hpord r , := Hn0 ,Pess × Hpord r ,Pess,ord , Zn 0

:= Hn,Uess Zn

Zpr ,Dpr

× Hpord r ,Uess,ord Z r ,D r p

,

p

ord ord , Hn,Z ess := Hn0 ,Zess × H r Zn p ,Zess,ord Z n

ord Hn,G ess h,Zn

0

Zpr ,Dpr

× Hpord r ,Gess,ord h,Z r ,D

:= Hn0 ,Gess h,Zn

0

ord,0 := Hn0 0 ,Gess Hn,G ess h,Z h,Zn

p

, pr

× H ord,0 ess,ord r

,

p ,Gh,Z

n0

pr ,Dpr

ord , × Hpord Hn,G ess := Hn0 ,Gess r ,Gess,ord l,Zn l,Z n

ord,0 Hn,G ess l,Zn

0

l,Zpr ,Dpr

× H ord,0 pr ,Gess,ord l,Z r ,D

,

ord Hn,U := Hn0 ,Uess × Hpord ess r ,Uess,ord 1,Zn 1,Z

,

ord Hn,U := Hn0 ,Uess × Hpord ess r ,Uess,ord 2,Zn 2,Z

,

:=

Hn0 0 ,Gess l,Zn

0

n

0

n

ord Hn,G ess h,Z

n

0

2,Zpr ,Dpr

× Hpord r ,Gess,ord

:= Hn0 ,Gess h,Zn

0

nUess Zn

nUess 1,Zn

:= Hn0 ,Gess h,Zn

0

nUess 1,Zn

Note that some of these are not new: H ord ess ∼ = Hn ,Gess n,Gl,Zn

ord,0 Hn,G ess l,Zn

0

pr

1,Zpr ,Dpr

nUess Zn

n

ord Hn,G ess h,Z

p

l,Zn 0

∼ = Hn0 0 ,Gess l,Z

n0

h,Zpr ,Dpr

× Hpord r ,Gess,ord

nUess,ord Z r ,D r p

h,Zpr ,Dpr

,

p

nUess,ord 1,Z r ,D p

.

pr

∼ × Hpr ,Gess , = Hn,Gess l,Z r l,Zn p

× Hp0 r ,Gess l,Z

pr

0 ∼ ess , = Hn,G l,Z n

ord ∼ Hn,U . ess = Hn0 ,Uess 2,Zn 2,Z n

0

Consider the following commutative diagram, in which every square is Cartesian: ...ord ord ord ord ord ord ord / / ... / / ... / / ... ess ess ess nUess Ξ Φn Ξ Φn /Hn,U Ξ Φn /Hn,U Ξ Φn /Hn,G Zn Zn 2,Z h,Z n

% ...  ord C Φn

n

 ...ord ord // C /H Φn n,Uess 1,Z

n

' ...  ord,Zn Mn

 ...ord ord // C /H Φn n,Gess h,Z

n

nUess 1,Zn

 ... n ord,0 / / Mord,Z /Hn,G ess n h,Z

n

(4.2.1.14) Now we consider the equivariant quotient of a ...ord a ...ord

ord ord → ess ess Ξ Φn /Hn,Gess C Φn /Hn,Gess h,Zn nUZn h,Zn nU1,Zn a ...ord,Zn
ord,0 → /Hn,G ess Mn h,Zn

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ord,0 0 ∼ by Hn,G ess , where the disjoint unions are over elements Φn (with the = Hn,G ess l,Zn l,Zn same (X, Y, φ)) in the fiber of ΦHn → ZHn above Zn , which is a torsor under ord,0 0 ∼ Hn,G ess . This is (up to canonical isomorphisms) the same as the equi= Hn,G ess l,Zn l,Zn variant quotient of a ...ord a ...ord a ...ord,Zn Ξ Φn → C Φn → Mn bal r by Hn (or rather by Hnord , since the kernel Up,1 (p ) of Hpr → Hpord acts trivir ally), where the disjoint unions are over representatives (Zn , Φn , δn ) (with the same (X, Y, φ)) in the Hn -orbit (ZH , ΦH , δH ). Let us denote this equivariant quotient (up to canonical isomorphisms) by ...ord,ΦH ...ord ...ord , (4.2.1.15) Ξ ΦH ,n → C ΦH ,n → MH

whose terms carry compatible actions of ΓΦH (see Definition 1.2.2.3). (We keep the ...ord ...ord subscripts “n” in the notation because Ξ ΦH ,n and C ΦH ,n depend on the choice of n.) ...ord By construction, Ξ ΦH ,n is universal for tuples ord (ZH , (X,Y, φ, ϕord −2,H , ϕ0,H ), ord ord ∨,ord ord (B, λB , iB , ϕ−1,Hp , ϕord , τH )), −1,Hp ), δH , (cH , cH

up to automorphism by ΓΦH , describing degeneration data without positivity conord ord dition, where (ϕord −2,H , ϕ0,H ) (resp. δH ) induces the same (ϕ−2,H , ϕ0,H ) (resp. δH ) in the representative (ZH , ΦH = (X, Y, φ, ϕ−2,H , ϕ0,H ), δH ) we have fixed. The ca...ord,ΦH ...ord ...ord ...ord ord forget the data τH nonical morphisms Ξ ΦH ,n → C ΦH ,n and C ΦH ,n → MH ∨,ord ), respectively. and (cord H , cH Lemma 4.2.1.16. The canonical morphism ...ord,Zn ...ord,ΦH ord,0 /Hn,G → MH ess Mn h,Z

(4.2.1.17)

is an isomorphism, and hence the canonical morphisms ...ord,ΦH ...ord,ZH ...ord,ΦH ...ord → MH := MH /ΓΦH → MHh , MH

(4.2.1.18)

n

(cf. Definition 1.2.1.15 and Lemmas 1.3.2.1 and 1.3.2.5) are finite ´etale. If, for some (and hence every) choice of a representative (Zn , Φn , δn ) in (ZH , ΦH , δH ), the ess ess ess image Hn,Pess in Gess h,Zn × Gl,Zn is the direct product Hn,Gh,Zn × Hn,Gl,Zn , then we Zn ord,0 ord,0 ord ord have Hn,G = Hn,G and Hn,G ess ess ess = Hn,Gess , and the canonical morphisms in h,Zn l,Zn h,Zn l,Zn (4.2.1.18) are isomorphisms.

...ord,Zn ord,0 Proof. The first statement is true because Mn /Hn,G is finite ´etale over ess h,Zn ...ord ∼ ...ord,Zn ord,0 ord ord /Hn,G (since Hn,G and Hn,G act by forming oress ess ess MHh = Mn h,Zn h,Zn h,Zn bits of ordinary level structures), because the index set of the disjoint union

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` ...ord,Zn ord,0 ord,0 0 ∼ is a torsor under Hn,G /Hn,G ess , and because ΓΦH acts = Hn,G ess ess Mn l,Zn l,Zn h,Zn ...ord,ΦH on MH via the canonical homomorphisms ord,0 ord,0 ord ∼ ord ΓΦH → Hn,G ess /Hn,Gess = Hn,Gess /Hn,Gess . l,Z h,Z h,Z l,Z n

n

The second statement follows from the definitions of

n

n

ord,0 Hn,G ess h,Zn

ord and Hn,G ess . h,Zn

Lemma 4.2.1.19. (1) The canonical morphism ...ord ord ...ord → ess C Φn /Hn,Gess C nU ΦH ,n 1,Z h,Zn n

(4.2.1.20)

is an isomorphism, compatible with (4.2.1.17). (2) Suppose that, for some (and hence every) choice of a representative (Zn , Φn , δn ) in (ZH , ΦH , δH ), the splitting of the canonical homomorphism ess ess Gess h,Zn n U1,Zn  Gh,Zn defined by δn induces a splitting of the canoni0 ess cal homomorphism Hn,Gess  Hn,G ess , and hence an isomorphism h,Zn nU1,Zn h,Zn 0 ∼ ess Hn,Gess n Hn,Uess . In this case, the splitting of the ca= Hn,Gess 1,Zn h,Zn nU1,Zn h,Zn nonical homomorphism ess,ord ess,ord ess,ord ess ess (Gess h,Zn0 × Gh,Zpr ,Dpr ) n (U1,Zn0 × U1,Zpr ,Dpr )  (Gh,Zn0 × Gh,Zpr ,Dpr ) (4.2.1.21) ord ord defined by δn = (δn0 , δpr ) induces a splitting of the canonical hoord,0 ord momorphism Hn,G  Hn,G ess nUess ess , and hence an isomorphism 1,Zn h,Zn h,Zn ...ord ord,0 ord ord ∼ . Under this assumption, C ΦH ,n is H n H Hn,G ess nUess = n,Gess n,Uess 1,Zn 1,Zn h,Zn h,Zn ...ord,ΦH a proper flat group scheme over MH such that there exists an inte...ord ger m ≥ 1 such that multiplication by m maps C ΦH ,n scheme-theoretically ...ord,ΦH ...ord,◦ , so to a subgroup scheme C ΦH ,n that is an abelian scheme over MH ...ord ...ord,ΦH ) of fiberwise connected compothat the group scheme π 0 (C ΦH ,n /MH ...ord,ΦH ...ord nents of C ΦH ,n over MH is defined and is of ´etale-multiplicative type. (See Lemma 3.1.2.2 and Definition 3.1.2.3; cf. Proposition 4.2.1.11.) The ...ord ...ord,ΦH ) has no prime factors other than those of Disc, rank of π 0 (C ΦH ,n /MH n, [X : φ(Y )], and the rank of ker(λB ). (This implies that the rank of ...ord ...ord,ΦH π 0 (C ΦH ,n /MH ) does not contain prime factors other than those of # Disc, n and [L : L].) (3) In general (no longer making the assumption on splittings as in state...ord,ΦH ...ord ment (2)), the morphism C ΦH ,n → MH is a torsor under a pro...ord,ΦH ...ord,grp per flat group scheme C ΦH ,n → MH satisfying the properties as ...ord,grp ...ord,ΦH ) is defined. Then, the in statement (2), for which π 0 (C ΦH ,n /MH ...ord,ΦH ...ord Stein factorization (see [35, III-1, 4.3.3]) of C ΦH ,n → MH , which ...ord,ΦH ...ord ...ord,ΦH ...ord ) → MH , is the we denote abusively as C ΦH ,n → π 0 (C ΦH ,n /MH

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...ord ...ord ...ord,ΦH composition of an abelian scheme torsor C ΦH ,n → π 0 (C ΦH ,n /MH ) ...ord,ΦH ...ord ...ord,ΦH with a torsor π 0 (C ΦH ,n /MH ) → MH under the group scheme ...ord,ΦH ...ord,grp ...ord,ΦH . ) → MH π 0 (C ΦH ,n /MH Proof. Statement (1) is true because the common index set of the disjoint uni

` ...ord ord ` ...ord,Zn ord,0 ord,0 ∼ ons and /Hn,G is a torsor under Hn,G ess ess ess = C Φn /Hn,Gess Mn h,Zn nU1,Zn h,Zn l,Zn ...ord,ΦH ...ord 0 Then the morphism C ΦH ,n → MH can be canonically identiHn,G ess . l,Zn ...ord,Zn ...ord ord ord,0 of /Hn,G fied with the equivariant quotient C Φn /Hn,Gess nUess → ess M n 1,Zn h,Zn h,Zn ...ord,Zn ...ord . C Φn → Mn Via the splitting of (4.2.1.21) defined by δnord = (δn0 , δpord r ), the equivariant ess,ord ess action of Gh,Zn × Gh,Zpr ,Dpr is compatible with the group scheme structure of ...ord,Zn0 ...ord ...ord ord . Since the action of Hn,U on C Φn (modifying the tautologiess C Φn → Mn 1,Zn cal object (cord n :

1 nX

, cn∨,ord : → Bp∨,ord r

1 nY

→ Bpord r )) is realized by the translation ...ord,Zn ...ord , the quoaction of a (commutative) finite flat subgroup scheme of C Φn → Mn ... ...ord ord ord,Zn is a proper flat group scheme. Under the assumptions tient C Φn /Hn,Uess → Mn 1,Z n

ord on the splittings as in statement (2), the equivariant action of Hn,G is compatible ess h,Zn ...ord,Zn ...ord ord , and hence the equivawith the group scheme structure of C Φn /Hn,Uess → Mn 1,Z ...ord,Zn nord,0 ...ord ord → Mn /Hn,Gess (see the diagram (4.2.1.14), riant quotient C Φn /Hn,Gess nUess 1,Zn h,Zn h,Zn in which every square is Cartesian) is again a proper flat group scheme, which in...ord,Zn ...ord (see Proposition 4.2.1.11) the properties described as herits from C Φn → Mn in statement (2). In general, without the assumptions on the splittings as in statement ord,0 ord ord (2), Hn,G and Hn,G n Hn,U are two different subgroups of ess nUess ess ess 1,Z 1,Z h,Zn

n

h,Zn

n

ess,ord ess,ord ess (Gess h,Zn0 × Gh,Zpr ,Dpr ) n (U1,Zn0 × U1,Zpr ,Dpr ). By the same reasoning as in the pre...ord,Zn ...ord ord,0 ord,0 ord vious paragraph, the quotient C Φn /(Hn,G n Hn,U /Hn,G ess ) → Mn ess ess 1,Zn h,Zn h,Zn is a proper flat group scheme, with properties described as in statement (2). ...ord,Zn ...ord ord The group scheme structure of C Φn /Hn,U → Mn might not descend to ess 1,Zn ...ord,Zn ...ord ord ord,0 → Mn /Hn,Gess , but nevertheless makes the latter a toress C Φn /Hn,Gess h,Zn nU1,Zn h,Zn ...ord,Zn ...ord ord,0 ord,0 ord sor under C Φn /(Hn,Gess n Hn,Uess ) → Mn /Hn,G ess . Hence, statement (3) 1,Zn h,Zn
` ...ordh,Zn ord,0 ...ord,grp ord follows if we define C ΦH ,n to be the quotient of n Hn,U by ess ) C Φn /(Hn,Gess 1,Zn h,Zn ord,0 0 ∼ H ess , where the disjoint unions are over elements Φn in the fiber of Hn,Gess = n,Gl,Zn l,Zn ...ord,ΦH ...ord ΦHn → ZHn above Zn , as in the definition of C ΦH ,n → MH .

...ord ord Lemma 4.2.1.22. The quotient by the action of Hn,U on E Φn is realized by ess 2,Zn the translation action of a (commutative) finite flat subgroup scheme, so that the

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...ord ...ord ord quotient E ΦH ,n := E Φn /Hn,U is a group scheme of multiplicative type of finite ess ...ord2,Zn ...ord ...ord type with character group S ΦH ,n a subgroup of S Φn . Let S ΦH ,n,tor be the torsion ...ord ...ord ...ord subgroup of S ΦH ,n , and let S ΦH ,n,free be the free quotient of S ΦH ,n . Then the canonical short exact sequence ...ord ...ord ...ord 0 → S ΦH ,n,tor → S ΦH ,n → S ΦH ,n,free → 0 induces a short exact sequence ...ord ...ord ...ord 0 → E ΦH ,n,free → E ΦH ,n → E ΦH ,n,tor → 0. ...ord ...ord Then we have canonical isomorphisms S ΦH ,n,free ∼ = SΦH and E ΦH ,n,free ∼ = EΦH ∼ = HomZ (SΦH , Gm ), where SΦH and EΦH are defined as in [62, Lem. 6.2.4.4]. Proof. These statements are about group schemes of multiplicative type of finite type over Spec(Z) with constant character groups, and hence they can be verified after base change to Spec(Q) (or any base ring of residue characteristics prime to ...ord the order of S ΦH ,n,tor ). Then they all follow from [62, Lem. 6.2.4.4]. Lemma 4.2.1.23. (1) The canonical morphism ...ord ord ...ord ess → Ξ Φ ,n Ξ Φn /Hn,Gess nU H Z h,Zn n

(4.2.1.24)

is an isomorphism, compatible with (4.2.1.17) and (4.2.1.20). ...ord ...ord ...ord (2) The morphism Ξ ΦH ,n → C ΦH ,n is a torsor under the pullback of E ΦH ,n , ...ord ...ord ...ord which factors as a composition Ξ ΦH ,n → Ξ ΦH ,n,tor → C ΦH ,n , in which ...ord ...ord the morphism Ξ ΦH ,n → Ξ ΦH ,n,tor is a torsor under the pullback of the ...ord torus E ΦH ,n,free ∼ = EΦH ∼ = HomZ (SΦH , Gm ), and in which the morphism ...ord ...ord Ξ ΦH ,n,tor → C ΦH ,n is a torsor under the pullback of the finite flat group ...ord scheme E ΦH ,n,tor of multiplicative type. Proof. Statement (1) is true because the common index set of the disjoint uni


` ...ord ord ` ...ord ord ` ...ord,Zn ord,0 ons , and /Hn,G ess ess , ess Ξ Φn /Hn,Gess C Φn /Hn,Gess Mn h,Zn nU1,Zn h,Zn nUZn h,Zn ...ord ...ord ord,0 0 ∼ is a torsor under Hn,G Then the morphism Ξ ΦH ,n → C ΦH ,n ess . = Hn,G ess l,Zn l,Zn ...ord ord can be canonically identified with the equivariant quotient Ξ Φn /Hn,G ess nUess → Zn h,Zn ...ord ord ...ord ...ord ord /H of → . Since the action of H (modifying ess C Φn Ξ Φn C Φn n,Gess n,Uess 2,Z h,Z nU1,Z n

n

n

∼

∗ ⊗ −1 → (c∨ ) is realized as n0 × c) PB 0 ...ord the same action of a finite flat subgroup scheme of E Φn as in Lemma 4.2.1.22, ...ord ord ...ord ...ord we see that Ξ Φn /Hn,U → C Φn is a torsor under the pullback of E ΦH ,n , and ess 2,Zn ...ord ord ...ord ord hence so is Ξ Φn /Hn,G after equivariant quotient ess nUess → C Φ /Hn,Gess nUess n Z 1,Z

the tautological object τnord = τn0 : 1( n1

h,Zn

ord by Hn,G ess

ess . h,Zn nU1,Zn

n

Y )×X

h,Zn

n

(See the diagram (4.2.1.14), in which every square is Cartesian.) ...ord ...ord Then the factorization of Ξ ΦH ,n → C ΦH ,n follows.

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By [62, Prop. 6.2.4.7; see also the errata], there is an algebraic stack ΞΦH ,δH separated, smooth, and schematic over MZHH over S0 = Spec(F0 ), whose quotient by ΓΦH is universal for tuples ∼ ∨ (ZH , (X, Y, φ, ϕ∼ −2,H , ϕ0,H ), (B, λB , iB , ϕ−1,H ), δH , (cH , cH , τH ))

up to automorphism by ΓΦH , describing degeneration data without positivity condi∼ ∼ tion, such that (ZH , Φ∼ H = (X, Y, φ, ϕ−2,H , ϕ0,H ), δH ) induces the same representative (ZH , ΦH = (X, Y, φ, ϕ−2,H , ϕ0,H ), δH ) we have fixed, as in [62, Lem. 5.4.2.10; see also the errata]. The structural morphism ΞΦH ,δH → MZHH factorizes canonically as H the composition ΞΦH ,δH → CΦH ,δH → MΦ → MZHH , where ΞΦH ,δH → CΦH ,δH H is a torsor under the pullback of the torus EΦH ∼ = HomZ (SΦH , Gm ), where H H CΦH ,δH → MΦ is an abelian scheme torsor, and where MΦ → MZHH is finite H H ΦH ´etale. The morphisms ΞΦH ,δH → CΦH ,δH and CΦH ,δH → MH forget the data τH and (cH , c∨ H ), respectively. By the construction in [62, Sec. 6.2.4; see also ZH H the errata], ΞΦH ,δH , CΦH ,δH , MΦ H , and MH , are, respectively, quotients of objects at principal level n by suitable subgroups of H/U(n), which are subgroups of ˆ p ) × Up,0 (pr ))/(U p (n0 ) × Up (pr )). (See also the descriptions and characteriza(G(Z tions of these objects in Section 1.3.2.) ...ord,ZH H Let Mord,Z be the open subalgebraic stack of MH ⊗ F0 [ζprH ] given by the H Z

image of the canonical open immersion

...ord,ZH H MZH,r := MZHH × S0,rH = MZHH ⊗ F0 [prH ] ,→ MH ⊗ F0 [ζprH ] H S0

F0

Z

...ord,ZH H ~ ord,ZH be the normalization of M (cf. Theorem 3.4.2.5), and let M in Mord,Z H H H under the canonical morphism ...ord,ZH H Mord,Z → MH H ~ ord in Theorem 3.4.2.5). (with properties analogous to those of M H

Let ord,ZH Ξord , ΦH ,δH := ΞΦH ,δH × MH

(4.2.1.25)

H CΦord := CΦH ,δH × Mord,Z , H H ,δH

(4.2.1.26)

H H H Mord,Φ := MΦ . × Mord,Z H H H

(4.2.1.27)

Z

MHH

Z

MHH

and Z

MHH

...ord ...ord By Theorem 4.1.6.2, they carry tuples parameterized by Ξ ΦH ,n , C ΦH ,n , and ...ord,ΦH , respectively, and induce (by universal properties) canonical morphisms MH ...ord Ξord ΦH ,δH → Ξ ΦH ,n , ...ord CΦord → C ΦH ,n , H ,δH

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and ...ord,ΦH H Mord,Φ → MH H ord,ZH and compatible with each other, and with their canonical morphisms to MH ...ord,ZH ...ord ord,ΦH ord ord ~ ~ ~ ) be the normalization of Ξ ΦH ,n . Let ΞΦH ,δH (resp. CΦH ,δH , resp. MH MH ...ord,ΦH ...ord H ord ord (resp. C ΦH ,n , resp. MH ) in ΞΦH ,δH (resp. CΦH ,δH , resp. Mord,Φ ) under the H canonical morphism. Then we obtain the following commutative diagram

Ξord ΦH ,δH

~ ord /Ξ ΦH ,δH

ord / ... Ξ Φ ,n

 CΦord H ,δH

 ~ ord /C

 ...ord /C ΦH ,n

 H Mord,Φ H

 ~ ord,ΦH /M

 ... H / Mord,Φ H

 H Mord,Z H

 ~ ord,ZH /M H

 ... H / Mord,Z H

 S0,rH = Spec(F0 [ζprH ])

 / ~S0,r = Spec(OF ,(p) [ζprH ]) H 0

 / Spec(Z(p) )

H

ΦH ,δH

H

(4.2.1.28)

of canonically induced morphisms (which is generally not Cartesian). The objects in this diagram all carry compatible canonical actions of ΓΦH . (The actions on those in the bottom two rows are all trivial.) ~ ord,ΦH → M ~ ord,ZH Proposition 4.2.1.29. In (4.2.1.28), the canonical morphism M H H ∼ ~ ord,ΦH /ΓΦ → M ~ ord,ZH . Moreover, is finite ´etale, which induces an isomorphism M H H ...ord,ΦH H ~ ord,ΦH → M ~ ord,ZH is an isomorphism. the canonical morphism M × M H H H ...ord,ZH MH

If the condition in the second statement of Lemma 4.2.1.16 is satisfied, then the ~ ord,ΦH → M ~ ord,ZH is an isomorphism. canonical morphism M H H ...ord,ΦH ord,ZH H Proof. The morphism Mord,Φ → MH × MH is an isomorphism esH ...ord,Z MH

H

...ord,ΦH ...ord,ZH sentially by construction. Since the canonical morphism MH → MH is ~ ord,ΦH is defined by normalization, the finite ´etale by Lemma 4.2.1.16, and since M H remainder of the first two statements follows. The last statement also follows from Lemma 4.2.1.16. ~ ord,ΦH is ~ ord Proposition 4.2.1.30. In (4.2.1.28), the morphism C ΦH ,δH → MH ~ ord an abelian scheme torsor. Moreover, the canonical morphism C → ΦH ,δH

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ord,ΦH MH

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261

~ ord,ΦH is a closed immersion. If the condition in (2) of Lemma M H

~ ord,ΦH is an abe~ ord 4.2.1.19 is satisfied, then the abelian scheme torsor C ΦH ,δH → MH lian scheme. Proof. By construction, we have the following commutative diagram, in which the vertical columns in the diagram are Stein factorizations, by Lemma 4.2.1.19: /

ord CΦ H ,δH

/

...ord ord,ΦH × M C Φ ,n H ...ord,ΦH H MH



...ord,ΦH ... ord,ΦH π 0 ( C ord ) × M ΦH ,n / M H ...ord,ΦH H MH

/

...ord ~ ord,ΦH × M C Φ ,n H ...ord,ΦH H MH



...ord,ΦH ... ~ ord,ΦH M π 0 ( C ord ) × ΦH ,n / M H ...ord,ΦH H MH

 H / M~ ord,Φ H

 , Mord,ΦH H

H Since CΦord → Mord,Φ is an abelian scheme torsor, the canonical morphism H H ,δH ... ord ord,ΦH ord CΦH ,δH → C ΦH ,n ... × MH induces (by considering their Stein factorizations ord,ΦH

MH

over

H Mord,Φ ) H

a section

H Mord,Φ H

...ord ...ord,ΦH → π 0 (C ΦH ,n /MH ) ... ×

ord,ΦH MH

H Mord,Φ . H

(4.2.1.31)

...ord,ΦH ...ord ...ord,ΦH is a torsor under the group scheme ) → MH Since π 0 (C ΦH ,n /MH ...ord,ΦH ...ord,grp ...ord,ΦH in Lemma 4.2.1.19, which is (finite flat, of fiπ 0 (C ΦH ,n /MH ) → MH ~ ord,ΦH is normal nite presentation, and) of ´etale-multiplicative type, and since M H (by definition), the schematic closure of the image of (4.2.1.31) defines a section ...ord ...ord,ΦH ~ ord,ΦH → π (C ~ ord,ΦH . M ) ... × M (4.2.1.32) ΦH ,n /MH 0 H H ord,ΦH MH

...ord,ΦH ...ord,grp ...ord is a torsor under the proper flat group scheme C ΦH ,n → Since C ΦH ,n → MH ...ord,ΦH ...ord,grp ...ord,ΦH in Lemma 4.2.1.19, which is the extension of π 0 (C ΦH ,n /MH ) by an MH ...ord,grp,◦ abelian scheme C ΦH ,n , the pullback of the morphism ...ord ...ord ...ord,ΦH ~ ord,ΦH → π (C ~ ord,ΦH (4.2.1.33) ) ... × M C ΦH ,n ... × M ΦH ,n /MH 0 H H ord,ΦH ord,ΦH MH

MH

~ ord,ΦH under the abelian scheme under the section (4.2.1.32) is a torsor C → M H ...ord,grp,◦ ord,ΦH ~ , which is in particular separated, smooth, and of finite C ΦH ,n ... × MH ord,ΦH

MH

~ ord ~ ord type. By the construction of C ΦH ,δH and C, the canonical morphism CΦH ,δH → ...ord ~ ord,ΦH factors through a canonical morphism C ~ ord C ΦH ,n ... × M ΦH ,δH → C, which H ord,ΦH

MH

~ ord must be an isomorphism because C ΦH ,δH is defined by normalization. This also

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...ord ~ ord shows that C ΦH ,δH → C ΦH ,n ... ×

ord,ΦH MH

~ ord,ΦH is a closed immersion (because the M H

~ ord,ΦH is an abelian scheme torsor. ~ ord section (4.2.1.32) is), and that C ΦH ,δH → MH H If the condition in (2) of Lemma 4.2.1.19 is satisfied, then CΦH ,δH → MΦ H is an abelian scheme (see [62, Prop. 6.2.4.7; see also the errata]), and so is its pullback H H CΦord → Mord,Φ . By taking the closure of the identity section Mord,Φ → H H H ,δH ord,ΦH ord ord ~ ~ → CΦH ,δH , which shows that the abelian CΦH ,δH , we obtain a section MH ord,ΦH ord ~ ~ scheme torsor CΦH ,δH → MH is also an abelian scheme, as desired. ~ ord,Φ1 , and M ~ ord,Z1 denote the analogues of ~ ord , M Proposition 4.2.1.34. Let C 1 1 Φ1 ,δ1 ~ ord,ZH at principal level 1 (i.e., with H replaced with ~ ord,ΦH , and M ~ ord , M C ΦH ,δH H H bal 0 ~ ord,Φ1 = M ~ ord,Z1 (by definition), and ˆ U(1) = U p (1)Up,1 (p ) = G(Z)). Then M 1 1 ~ ord,ΦH is finite ´etale. (Since ~ ord ~ ord the canonical morphism C → C × M ΦH ,δH

Φ1 ,δ1

H

~ ord,Φ1 M 1

~ ord,ΦH → M ~ ord,ZH is finite ´etale by Proposition 4.2.1.29, the canonical morphism M H H ~ ord,ZH is also finite ´etale.) ~ ord ~ ord C → C × M ΦH ,δH Φ1 ,δ1 H ~ ord,Z1 M 1

bal r Proof. By Lemma 4.2.1.9, for any n = n0 pr such that U p (n0 ) ⊂ Hp and Up,1 (p ) ⊂ ... ... ... ord,Z n ord ord r is unramified, Hp ⊂ Up,0 (p ), the canonical morphism C Φn → C Φ1 ... × Mn ord,Z1

M1

because it is a homomorphism with a finite ´etale kernel. Since this morphism factorizes as a composition of canonical morphisms ...ord,Zn ...ord,Zn ...ord ...ord ...ord , (4.2.1.35) → C Φ1 ... × Mn C Φn → C ΦH ,n ... × Mn ord,Z1

ord,ΦH

M1

MH

in which the first one is surjective by definition (see (4.2.1.15) and (1) of Lemma ...ord 4.2.1.19), we see (by ´etale descent) that the canonical morphism C ΦH ,n → ...ord,ΦH ...ord ord,0 , which is the equivariant finite ´etale quotient by Hn,G of ess C Φ1 ... × MH h,Zn ord,Z1 M1

the second morphism in (4.2.1.35), is also unramified. Consequently, the pul...ord ...ord ~ ord,ΦH → C ~ ord,ΦH is unramified. By Propolback C ΦH ,n ... × M M Φ1 ... × H H ord,ΦH

ord,Z1

M1

MH

...ord ~ ord sition 4.2.1.30, the canonical morphisms C ΦH ,δH → C ΦH ,n ... × ~ ord C Φ1 ,δ1

× ~ ord,Φ1 M 1

...ord ~ ord,ΦH → C M Φ1 ... × H

~ ord ~ ord morphism C ΦH ,δH → CΦ1 ,δ1

ord,ΦH MH

ord,Z1 M1

× ~ ord,Φ1 M 1

~ ord,ΦH and M H

~ ord,ΦH are closed immersions. Hence, the M H

~ ord,ΦH (compatible with the above two closed M H

immersions) is also unramified. Since this is a morphism between abelian scheme torsors equivariant with a homomorphism of abelian schemes of the same relative dimension (which is automatically surjective), by [35, IV-3, 11.3.10 a)⇒b) and 15.4.2 e0 )⇒b)] (cf. the proof of [62, Lem. 1.3.1.11]), it is automatically flat, and hence finite ´etale, as desired.

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Corollary 4.2.1.36. Suppose H0 ⊂ H and suppose (with similar assumptions at ~ ord,ΦH0 is also defined. Then the canonical morphism ~ ord level H0 ) C ΦH0 ,δH0 → MH0 H0 ~ ord,Φ ~ ord ~ ord C →C × M is finite ´etale. 0 ΦH0 ,δH0

ΦH ,δH

H

~ ord,ΦH M H

Proof. By Proposition 4.2.1.34, the composition of canonical morphisms H0 H0 ~ ord,Φ ~ ord,Φ ~ ord ~ ord ~ ord C × M →C × M is finite ´etale. Since ΦH0 ,δH0 → CΦH ,δH Φ1 ,δ1 H0 H0 ~ ord,ΦH M H

~ ord,Φ1 M 1

these are morphisms between abelian scheme torsors equivariant under homomorphisms of abelian schemes of the same relative dimension, they are both surjective. Hence, they are both unramified, and (as in the proof of Proposition 4.2.1.34) finite ´etale. ~ ord ~ ord Proposition 4.2.1.37. In (4.2.1.28), the morphism Ξ ΦH ,δH → CΦH ,δH is a torsor ord ∼ ~ under the pullback to C ΦH ,δH of the torus EΦH = HomZ (SΦH , Gm ) (see Lemma ...ord ~ ord ~ ord × C 4.2.1.22). Moreover, the canonical morphism Ξ ΦH ,δH is a ΦH ,δH → Ξ ΦH ,n ...ord C ΦH ,n

closed immersion. Proof. By construction, we have the following commutative diagram: Ξord ΦH ,δH

ord / ... Ξ Φ ,n H

× ...ord

C ΦH ,n

 ...ord Ξ ΦH ,n,tor ... ×

C ord ΦH ,n

*

ord / ... Ξ Φ ,n

CΦord H ,δH

H

CΦord H ,δH

× ...ord

C ΦH ,n

ord / ... Ξ Φ ,n,tor H

~ ord C ΦH ,δH

 × ...ord

C ΦH ,n

~ ord C ΦH ,δH

 ~ ord /C ΦH ,δH

 CΦord H ,δH

ord ord Since Ξord ΦH ,δH → CΦH ,δH is a torsor under the pullback to CΦH ,δH of ord the torus EΦH ∼ = HomZ (SΦH , Gm ), the canonical morphism ΞΦH ,δH → ...ord ord Ξ ΦH ,n,tor ... × CΦH ,δH induces a section ord C ΦH ,n

...ord Ξord ΦH ,δH → Ξ ΦH ,n,tor ... ×

C ord ΦH ,n

CΦord . H ,δH

(4.2.1.38)

...ord (It suffices to show that the schematic image of Ξord ΦH ,δH → Ξ ΦH ,n,tor ... ×

C ord ΦH ,n

...ord is isomorphic to CΦord under the structural morphism Ξ ΦH ,n,tor ... × H ,δH

C ord ΦH ,n

CΦord H ,δH

CΦord → H ,δH

CΦord , which can be verified after making an ´etale localization that trivializes H ,δH ...ord ...ord ~ ord ~ ord the torsors.) Since Ξ ΦH ,n,tor ... × C ΦH ,δH → CΦH ,δH is a torsor under E ΦH ,n,tor , C ord ΦH ,n

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~ ord which is finite flat and of multiplicative type, and since C ΦH ,δH is normal (by definition), the schematic closure of the image of (4.2.1.38) defines (as above) a section ord ~ Φord,δ → ... ~ Φord,δ . C (4.2.1.39) Ξ ΦH ,n,tor ... × C H H H H ord C ΦH ,n

...ord ...ord Since the morphism Ξ ΦH ,n → Ξ ΦH ,n,tor is a torsor under the pullback to ...ord Ξ ΦH ,n,tor of the torus EΦH (see Lemma 4.2.1.23), the pullback of the morphism ...ord ord ~ Φord,δ → ... ~ Φord,δ (4.2.1.40) Ξ ΦH ,n ... × C Ξ ΦH ,n,tor ... × C H H H H ord ord C ΦH ,n

C ΦH ,n

~ ord under the section (4.2.1.39) is a torsor Ξ → C ΦH ,δH under the pullback of EΦH to ord ~ CΦH ,δH , which is in particular separated, smooth, and of finite type. By the con...ord ~ ord ~ ord ~ ord × C struction of Ξ ΦH ,δH and Ξ, the canonical morphism ΞΦH ,δH → Ξ ΦH ,n ...ord ΦH ,δH C ΦH ,n

~ ord factors through a canonical morphism Ξ ΦH ,δH → Ξ, which must be an isomorphism ord ~ ~ ord ~ ord because Ξ ΦH ,δH is defined by normalization. This shows that ΞΦH ,δH → CΦH ,δH ~ ord . This also shows is a torsor under the pullback of the torus EΦH to C ΦH ,δH ... ord ~ ord ~ ord × C that Ξ ΦH ,δH is a closed immersion (because the section ΦH ,δH → Ξ ΦH ,n ...ord C ΦH ,n

(4.2.1.39) is), as desired. ~ ord denote the analogue of Ξ ~ ord Proposition 4.2.1.41. Let Ξ Φ1 ,δ1 ΦH ,δH at principal level p bal 0 ˆ 1 (i.e., with H replaced with U(1) = U (1)Up,1 (p ) = G(Z)). Then the canonical ~ ord ~ ord ~ ord morphism Ξ × C etale. ΦH ,δH → ΞΦ1 ,δ1 ΦH ,δH is finite ´ ~ ord C Φ ,δ 1

1

Proof. By definition (see (4.2.1.12) and (4.2.1.15)), for any n = n0 pr such that ...ord bal r U p (n0 ) ⊂ Hp and Up,1 (p ) ⊂ Hp ⊂ Up,0 (pr ), the canonical morphism Ξ Φn → ...ord ...ord Ξ Φ1 ...× C Φn is unramified, because (as a morphism between torus torsors) it is C ord Φ1

...ord ...ord ... ´etale locally the canonical homomorphism E Φn = HomZ ( S Φn , Gm ) = E Φn0 = ... ...ord ... ... ...ord HomZ ( S Φn0 , Gm ) → E Φ1 = HomZ ( S Φ1 , Gm ) = E Φ1 = HomZ ( S Φ1 , Gm ) with a finite ´etale kernel. Since this morphism factorizes as a composition of canonical morphisms ...ord ...ord ...ord ...ord ...ord (4.2.1.42) Ξ Φn → Ξ ΦH ,n ... × C Φn → Ξ Φ1 ...× C Φn , ord ord C Φ1

C ΦH ,n

in which the first one is surjective by definition (see (4.2.1.15) and (1) of Lemma ...ord 4.2.1.23), we see (by ´etale descent) that the canonical morphism Ξ ΦH ,n → ...ord ...ord ord ess nUess Ξ Φ1 ...× C ΦH ,n , which is the equivariant finite ´etale quotient by Hn,G 1,Z h,Zn

C ord Φ1

n

of the second morphism in (4.2.1.42), is also unramified. Consequently, the ...ord ...ord ~ ord ~ ord pullback Ξ ΦH ,n ... × C By PropoΦH ,δH → Ξ Φ1 ...× CΦH ,δH is unramified. C ord ΦH ,n

C ord Φ1

...ord ~ ord sition 4.2.1.37, the canonical morphisms Ξ ΦH ,δH → Ξ ΦH ,n ... ×

C ord ΦH ,n

~ ord C ΦH ,δH and

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~ ord Ξ Φ1 ,δ1

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...ord ~ ord ~ ord × C ΦH ,δH → Ξ Φ1 ...× CΦH ,δH are closed immersions. Hence, the mor-

~ ord C Φ ,δ 1

phism

10374-main

C ord Φ1

1

~ ord Ξ ΦH ,δH

→

~ ord Ξ Φ1 ,δ1

~ ord × C ΦH ,δH (compatible with the above two closed immer-

~ ord C Φ ,δ 1

1

sions) is also unramified. Since this is a morphism between torus torsors equivariant with a homomorphism of tori of the same relative dimension (which is automatically surjective), by [35, IV-3, 11.3.10 a)⇒b) and 15.4.2 e0 )⇒b)] (cf. the proof of [62, Lem. 1.3.1.11]), it is automatically flat, and hence finite ´etale, as desired. Corollary 4.2.1.43. Suppose H0 ⊂ H and suppose (with similar assumptions at ~ ord ~ ord level H0 ) Ξ ΦH0 ,δH0 → CΦH0 ,δH0 are also defined. Then the canonical morphism ~ ord ~ ord ~ ord is finite ´etale. Ξ →Ξ × C ΦH0 ,δH0

ΦH ,δH

~ ord C Φ ,δ H

ΦH0 ,δH0

H

Proof. By Proposition 4.2.1.41, the composition of canonical morphisms ~ ord ~ ord ~ ord ~ ord ~ ord × C etale. Since Ξ × C ΦH0 ,δH0 → ΞΦ1 ,δ1 ΦH0 ,δH0 is finite ´ ΦH0 ,δH0 → ΞΦH ,δH ~ ord C Φ ,δ H

H

~ ord C Φ ,δ 1

1

these are homomorphisms between torus torsors (equivariant with homomorphisms between tori) of the same relative dimension, they are both surjective, and hence finite ´etale, as desired. ~ ord,ΦH , and M ~ ord,ZH ~ ord , C ~ ord , M Corollary 4.2.1.44. The algebraic stacks Ξ ΦH ,δH ΦH ,δH H H are all separated, smooth, and of finite type over ~S0,rH . If Hp is neat, they are all quasi-projective over ~S0,rH . Proof. These follow from Propositions 3.4.6.3, 4.2.1.30, and 4.2.1.37. Convention 4.2.1.45. (Compare with Convention 3.4.2.9.) We say that an object ...ord ~ ord over a scheme S over S0,rH parameterized by Ξ ΦH ,n is parameterized by Ξ ΦH ,δH ...ord if the tautological morphism S → Ξ ΦH ,n determined by the universal property fac~ ord . Then it also makes sense to consider the tautological tors through S → Ξ ΦH ,δH ord ~ ord,ΦH , and ~ ~ ord , M tuple over Ξ . We shall adopt the same convention for C ΦH ,δH ΦH ,δH H ~ ord,ZH . M H It follows from the constructions above that we have the following proposition, ~ ord in which Ξ ΦH ,δH etc are explicitly realized as normalizations in the paragraph preceding (4.2.1.28) (and hence they are, up to canonical isomorphisms, independent of the auxiliary choice of n): Proposition 4.2.1.46. (Compare with [62, Prop. 6.2.4.7; see also the errata].) Let H, Hp , and Hp be as in beginning of Section 3.3.5, and let rH be as in Definition 3.4.2.1. Let us fix a representative (ZH , ΦH , δH ) of an ordinary cusp label at level H (see Definition 3.2.3.8), where ΦH = (X, Y, φ, ϕ−2,H , ϕ0,H ), which defines a finite ~ ord,ΦH of an algebraic stack M ~ ord,ZH over ~S0,r = Spec(OF ,(p) [ζprH ]). ´etale cover M H 0 H H

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~ ord,ZH as in ~ ord,ΦH and M (Then we can talk about the tautological tuples over M H H Convention 4.2.1.45.) Let us consider the category fibered in groupoids over the category of locally ~ ord,ZH that are flat over ~S0,r whose fiber over noetherian normal schemes over M H H each scheme S (with the conditions just described) has objects the tuples ord (ZH , (X,Y, φ, ϕord −2,H , ϕ0,H ), ord ord ∨,ord ord (B, λB , iB , ϕ−1,Hp , ϕord , τH )) −1,Hp ), δH , (cH , cH

(4.2.1.47)

describing degeneration data without positivity condition over S such that ~ ord,ZH , (B, λB , iB , ϕ−1,Hp , ϕord −1,Hp ) is the pullback of the tautological tuple over MH ord ord such that (ϕord −2,H , ϕ0,H ) (resp. δH ) induces the (ϕ−2,H , ϕ0,H ) in ΦH (resp. δH ), and such that the pullbacks of each tuple as in (4.2.1.47) to maximal points (see [36, 0, 2.1.2]) of S are induced as in Section 4.1.6 by the pullbacks of the corresponding tuple ∼ ∨ (ZH , (X, Y, φ, ϕ∼ −2,H , ϕ0,H ), (B, λB , iB , ϕ−1,H ), δH , (cH , cH , τH ))

(4.2.1.48)

parameterized by ΞΦH ,δH (see [62, Prop. 6.2.4.7; see also the errata]). ~ ord Then there is an algebraic stack Ξ ΦH ,δH separated, smooth, and schematic over ord,ZH ~ ~ ord , MH , together with a tautological tuple and a natural action of ΓΦH on Ξ ΦH ,δH ~ ord /ΓΦ is isomorphic to the category described above such that the quotient Ξ H ΦH ,δH ~ ord,ZH ). Equivalently, for each tuple as in (as categories fibered in groupoids over M H ord,Z H ~ (4.2.1.47) over a scheme S over M (with properties described above), there is a H ~ ord,ZH ), which is unique after we fix an isomorphism ~ ord morphism S → Ξ ΦH ,δH (over MH ∼ ∼ (fY : Y → Y, fX : X → X) in ΓΦH , such that the tuple over S is the pullback of the ~ ord tautological tuple over Ξ ΦH ,δH if we identify X by fX and Y by fY . ~ ord,ZH factorizes as the composition ~ ord The structural morphism Ξ ΦH ,δH → MH ~ ord,ΦH → M ~ ord,ZH compatible with the natural actions of ~ ord ~ ord Ξ ΦH ,δH → CΦH ,δH → MH H ~ ord,ZH ), where Ξ ~ ord ~ ord ΓΦH (trivial on M ΦH ,δH → CΦH ,δH is a torsor under the torus H ~ ord,ΦH is an abelian scheme torsor, ~ ord EΦH ∼ = HomZ (SΦH , Gm ); where C ΦH ,δH → MH which is an abelian scheme when the condition in (2) of Lemma 4.2.1.19 is satisfied; ~ ord,ΦH → M ~ ord,ZH is as above (and is finite ´etale), which is an isomorand where M H H phism when the condition in the second statement of Lemma 4.2.1.16 is satisfied, ∼ ~ ord,ZH ~ ord,ΦH /ΓΦ → inducing an isomorphism M MH . The EΦH -torsor structure of H H ord ~ ΞΦH ,δH defines a canonical homomorphism ~ Φord,δ ) : ` 7→ Ψ ~ ord SΦH → Pic(C ΦH ,δH (`), H H

(4.2.1.49)

~ ord (`) over C ~ ord assigning to each ` ∈ SΦH an invertible sheaf Ψ ΦH ,δH ΦH ,δH (up to isomorphism), together with isomorphisms ~ ord,∗ ~ ord ∆ ΦH ,δH ,`,`0 : ΨΦH ,δH (`)

⊗

OC ~ ord

ΦH ,δH

∼ ~ ord ~ ord (`0 ) → Ψ ΨΦH ,δH (` + `0 ) ΦH ,δH

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for all `, `0 ∈ SΦH , satisfying the necessary compatibilities with each other making ~ ord (`) an O ~ ord -algebra, such that ⊕ Ψ ΦH ,δH C ΦH ,δH

`∈SΦH

∼ ~ ord Ξ ΦH ,δH = SpecO

 ~ ord C ΦH ,δH

⊕ `∈SΦH

 ~ ord Ψ ΦH ,δH (`) .

Remark 4.2.1.50. The condition that (B, λB , iB , ϕord −1,H ) is the pullback of the ord,ZH ~ tautological tuple over MH means in particular that ϕord −1,H extends over all of S, not just at a maximal point (see [36, 0, 2.1.2]) over S0,prH (cf. condition (5) in Definition 3.4.2.10). In general, this is a nontrivial condition even when the cusp label [(ZH , ΦH , δH )] has O-multi-rank zero [62, Def. 5.4.2.7], which implies that B is an ordinary abelian scheme over S (cf. Remark 3.4.2.11). (The condition of being ordinary is irrelevant over the maximal points over S0,prH because they are of characteristic zero.) 4.2.2

Toroidal Embeddings, Positivity Conditions, and Mumford Families

As in [62, Sec. 6.2.5], let S∨ ΦH := HomZ (SΦH , Z) be the Z-dual of SΦH , and let ∨ ∼ (SΦH )∨ := S ⊗ R Hom = Z (SΦH , R). By definition of SΦH (in [62, Lem. 6.2.4.4]), ΦH R Z

the R-vector space (SΦH )∨ R is isomorphic to the R-vector space of Hermitian pairings (| · , · |) : (Y ⊗ R) × (Y ⊗ R) → O ⊗ R, by sending a Hermitian pairing (| · , · |) to the Z

Z

Z

0 0 element in (SΦH )∨ R induced by the assignment y ⊗ φ(y ) 7→ Tr(O ⊗ R)/R (|y, y |). (See Z

∨ [62, Lem. 1.1.4.5].) Let PΦH (resp. P+ ΦH ) be the subset of (SΦH )R corresponding to positive semi-definite (resp. positive definite) Hermitian pairings with admissible ∨ radicals (see [62, Def. 6.2.5.6]). Then both PΦH and P+ ΦH are cones in (SΦH )R . Let ΣΦH be a ΓΦH -admissible smooth rational polyhedral cone decomposition of ∨ PΦH with respect to the integral structure given by S∨ ΦH in (SΦH )R (see Definition 1.2.2.6). For each σ ∈ ΣΦH , consider the affine toroidal embedding   ~ ord ~ ord ⊕ Ψ (4.2.2.1) Ξ ΦH ,δH (`) , ΦH ,δH (σ) := SpecO ~ ord C ΦH ,δH

`∈σ ∨ ⊂SΦH

with the σ-stratum defined by 

~ ord Ξ ΦH ,δH ,σ := SpecO

~ ord C ΦH ,δH

⊕ `∈σ ⊥ ⊂SΦH

 ~ ord (`) . Ψ ΦH ,δH

(4.2.2.2)

Then we have canonical morphisms ~ ord (τ ) → Ξ ~ ord (σ) Ξ ΦH ,δH ΦH ,δH

(4.2.2.3)

~ ord (σ) when τ ⊂ σ, which is an open immersion when τ is a face of σ. By gluing Ξ ΦH ,δH over cones σ in ΣΦH using such open immersions along the faces, we obtain the toroidal embedding ord

ord

~ ord ~ ~ Ξ ΦH ,δH ,→ ΞΦH ,δH = ΞΦH ,δH ,ΣΦ

H

(4.2.2.4)

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defined by ΣΦH (as in [62, Def. 6.1.2.3]). Proposition 4.2.2.5. (Compare with [28, Ch. IV, p. 102] and [62, Prop. 6.2.5.8].) ord

~ Φ ,δ has the properties described in [62, Thm. 6.1.2.8], with the By construction, Ξ H H following additional ones: (1) There are constructible ΓΦH -equivariant ´etale constructible sheaves (of ord

~ O-lattices) X and Y on Ξ ΦH ,δH , together with an (O-equivariant) embedding φ : Y ,→ X, which are defined as follows: Each admissible surjection X  X 0 of O-lattices (see Definition 1.2.1.2 and [62, Def. 1.2.6.7]) determines a surjection from (ZH , ΦH , δH ) to some 0 representative (Z0H , Φ0H , δH ) of a cusp label at level H by [62, Lem. 5.4.2.11], which is also compatible with the filtration D as in Definition 3.2.3.1 and hence also ordinary as in Definition 3.2.3.8, where Z0H and Φ0H = (X 0 , Y 0 , φ0 , ϕ0−2,H , ϕ00,H ) are uniquely determined by the construction. Consequently, it makes sense to define PΦ0H and an embedding PΦ0H ,→ PΦH for each admissible surjection X  X 0 . ~ ord Over the locally closed stratum Ξ ΦH ,δH ,σj , the sheaf X is the constant quotient sheaf Xσj of X, with the quotient X  Xσj an admissible surjection defining a pair (ZH,σj , ΦH,σj = (Xσj , Yσj , φσj , ϕ−2,H,σj , ϕ0,H,σj )) such that σj is contained in the image of the embedding P+ φσj ,→ PΦH . We shall interpret this as having a sheaf version of ΦH , written as ΦH = (X, Y , φ, ϕ−2,H , ϕ0,H ). (2) The formation of SΦH from ΦH applies to ΦH and defines a sheaf SΦH . ord

~ (3) There is a tautological homomorphism B : SΦH → Inv(Ξ ΦH ,δH ) of constructible sheaves of groups (see [62, Def. 4.2.4.1]) which sends the class ~ ord (`) on ⊗ of ` ∈ SΦH,σj to the sheaf of ideals OΞ Ψ ~ ord ΦH ,δH (σj ) ΦH ,δH

OC ~ ord

ΦH ,δH

~ ord (σj ), such that we have the following: Ξ ΦH ,δH (a) This homomorphism B is ΓΦH -equivariant (because it is compatible with twists of the identification of ΦH ) and EΦH -invariant (because ~ ord (`) corresponds to a weight subsheaf of the O ~ ord -algebra Ψ ΦH ,δH C OΞ ~ ord

ΦH ,δH

ΦH ,δH

under the action of EΦH ), and is trivial on the open subord

~ ~ ord scheme Ξ ΦH ,δH of ΞΦH ,δH . (b) For each local section y of Y , the support of B(y ⊗ φ(y)) is effective, and is the same as the support of y. This is because σ(y, φ(y)) ≥ 0 for all σ ⊂ PΦH and y ∈ Y , and σ(y, φ(y)) > 0 when σ ⊂ P+ ΦH,σ and 0 6= y ∈ Yσ . ord

~ ord (σ) of Ξ ~ Φ ,δ enjoys the folloMoreover, each open subalgebraic stack Ξ ΦH ,δH H H wing universal property as in [62, Prop. 6.2.5.11]:

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Let R be a noetherian normal domain with fraction field K, and suppose ~ ord we have a morphism tR : Spec(R) → C ΦH ,δH that is liftable over Spec(K) to ord ~ a morphism t˜K : Spec(K) → Ξ . By abuse of notation, let us denote by ΦH ,δH ord ~ ΨΦH ,δH (`)R the R-invertible module defined by the pullback under tR of the inver~ ord (`) over C ~ ord , and denote Ψ ~ ord (`)R ⊗ K by Ψ ~ ord (`)K . tible sheaf Ψ ΦH ,δH ΦH ,δH ΦH ,δH ΦH ,δH R   ∼ ~ ord (`) , the morphism t˜K defines iso~ ord ⊕ Ψ Since Ξ ΦH ,δH ΦH ,δH = SpecO ~ ord C ΦH ,δH

`∈SΦH

∼ ~ ord (`)R as an morphisms → K, which defines an embedding of Ψ ΦH ,δH R-invertible submodule I` of K. Therefore, the pullback of the homomorphism (4.2.1.49) in Proposition 4.2.1.46 determines a homomorphism

~ ord (`)K Ψ ΦH ,δH

B : SΦH → Inv(R) : ` 7→ I`

(4.2.2.6)

~ ord (`) ∼ (see [62, Def. 4.2.4.1]). If ` = [y ⊗ χ] for some y ∈ Y and χ ∈ X, then Ψ = ΦH ,δH ∨ ∗ (c (y), c(χ)) PB by construction, and hence I` = Iy,χ as R-invertible submodules of K (see [62, Def. 4.2.4.6]; cf. (4b) of Definition 4.1.3.2). For each discrete valuation υ : K × → Z of K, since I` is locally principal for every `, it makes sense to consider the composition υ ◦ B : SΦH → Z : ` 7→ υ(I` ),

(4.2.2.7)

which is an element in S∨ ΦH (cf. [62, (6.2.5.9) and (6.2.5.10)]). Proposition 4.2.2.8. (Compare with [62, Prop. 6.2.5.11].) With assumptions and ~ ord (σ) is as follows: The morphism notation as above, the universal property of Ξ ΦH ,δH ord ~ ~ ord (σ) if t˜K : Spec(K) → Ξ extends to a morphism t˜R : Spec(R) → Ξ ΦH ,δH ΦH ,δH and only if, for every discrete valuation υ : K × → Z of K such that υ(R) ≥ 0, the corresponding homomorphism υ ◦ B : SΦH → Z as in (4.2.2.7) (or rather its composition with Z ,→ R) lies in the closure σ of σ in (SΦH )∨ R.   ~ ord (`) is relatively affine over ~ ord (σ) ∼ Proof. Since Ξ ⊕ Ψ = SpecO ΦH ,δH ΦH ,δH ~ ord C ΦH ,δH

`∈σ ∨

~ ord , the morphism t˜K extends to a morphism t˜R : Spec(R) → Ξ ~ ord (σ) if I` ⊂ C ΦH ,δH ΦH ,δH ∨ R for every ` ∈ σ . Since R is noetherian and normal, this is true if (υ ◦ B)(`) ≥ 0 for every discrete valuation υ of K such that υ(R) ≥ 0 and for every ` ∈ σ ∨ , or equivalently if υ ◦ B pairs nonnegatively with σ ∨ under the canonical pairing ∨ between S∨ ΦH and (SΦH )R , or equivalently if υ ◦ B lies in σ, as desired. Remark 4.2.2.9. If t˜K extends to t˜R , then the homomorphism B : SΦH → Inv(R) ord

~ ˜ agrees with the pullback of the homomorphism B : SΦH → Inv(Ξ ΦH ,δH ) under tR . (Thus, the notation is consistent when B and B can be compared over R.) ~ ord ~ ord Remark 4.2.2.10. Recall that the σ-stratum Ξ ΦH ,δH ,σ of ΞΦH ,δH (σ) is defined (see ord ord ∼ ∼ ~ [62, Def. 6.1.2.7]) by the sheaf of ideals IΦH ,δH ,σ = ⊕ ΨΦH ,δH (`) in OΞ ~ ord (σ) = `∈σ0∨

ΦH ,δH

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~ ord (`) (cf. [62, Conv. 6.2.3.20]). Since σ ⊂ PΦ is positive semidefinite, ⊕ Ψ H ΦH ,δH

`∈σ ∨

we have σ(`) ≥ 0 for every ` of the form [y ⊗ φ(y)]. As a result, the trivialization τ (y, φ(y)) : OΞ ~ ord

∼ ~ ord Ψ ~ ord ΦH ,δH (y ⊗ φ(y)) → OΞ

⊗

ΦH ,δH

OC ~ ord

ΦH ,δH

ΦH ,δH

~ ord over Ξ ΦH ,δH extends to a section τ (y, φ(y)) : OΞ ~ ord

ΦH ,δH (σ)

~ ord Ψ ~ ord ΦH ,δH (y ⊗ φ(y)) → OΞ

⊗

(4.2.2.11)

ΦH ,δH (σ)

OC ~ ord

ΦH ,δH

~ ord (σ). If σ ⊂ P+ , then by [62, Lem. 6.2.5.7], we have σ(y ⊗ φ(y)) > 0 over Ξ ΦH ,δH ΦH ~ ord (σ) as in (4.2.2.11) for every y 6= 0. In this case, the section τ (y, φ(y)) over Ξ ΦH ,δH ord has image contained in IΦH ,δH ,σ . This is almost the positivity condition, except that the base scheme is not completed along IΦord . H ,δH ,σ ord

~ ord ~ ord ~ Let X ΦH ,δH = XΦH ,δH ,ΣΦH be the formal completion of ΞΦH ,δH along the union + + ord ~ ord ~ of the σ-strata Ξ ΦH ,δH ,σ for σ ⊂ PΦH . For each σ ⊂ PΦH , let XΦH ,δH ,σ be the formal ~ ord (σ) along the σ-stratum Ξ ~ ord completion of Ξ . Then, using the language ΦH ,δH

ΦH ,δH ,σ

of relative schemes over formal algebraic stacks (see [37]), there are tautological tuples of the form ord (ZH , (X,Y, φ, ϕord −2,H , ϕ0,H ),

(4.2.2.12)

ord ord ∨,ord ord (B, λB , iB , ϕ−1,Hp , ϕord , τH )). −1,Hp ), δH , (cH , cH

~ ord ~ ord over both the formal algebraic stacks X ΦH ,δH and XΦH ,δH ,σ , the one on the latter being the pullback of the one on the former under the canonical morphism ~ ord ~ ord ~ ord X ΦH ,δH ,σ → XΦH ,δH . Moreover, this tautological tuple over XΦH ,δH satisfies the positivity condition in the following sense: We have a functorial assignment that, to each connected affine formal scheme U with an ´etale (i.e., formally ´etale and ~ ord , assigns a tuple of the of finite type; see [35, I, 10.13.3]) morphism U → X ΦH ,δH form (4.2.2.12) (with positivity condition) over the (normal) scheme Spec(Γ(U, OU )) (smooth over ~S0,rH ). Let R = Γ(U, OU ), and let I be (the radical of) its ideal of definition. Then R and I satisfies the requirement in Section 4.1.2, and we obtain a tuple defining an object of DDPEL,Mord (R, I). H By Theorems 4.1.5.27 and 4.1.6.2, Mumford’s construction defines an object ( ♥ G,

♥

λ,

♥

i,

♥

αHp ,

♥ ord αHp )

→ Spec(R)

(4.2.2.13)

in DEGPEL,Mord (R, I), which comes from an object H ( ♥ G,

♥

λ,

♥

i,

♥

αH ) → Spec(R) ♥

♥ ord αHp )

(4.2.2.14) ♥

in DEGPEL,MH (R, I) in the sense that ( αHp , is assigned to αH under (3.3.5.5) over the generic point Spec(Frac(R)) of Spec(R). Since the tuple (4.2.2.14) is an object of DEGPEL,MH (R, I), it is a degenerating family of type MH as in Definition 1.3.1.1. Since ϕord −1,H is defined over all of R, the tuple (4.2.2.13) is a

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~ ord as in Definition 3.4.2.10. Moreover, the torus degenerating family of type M H ♥ part of each fiber of G over the support of U is split with character group X. If we have an ´etale (i.e., formally ´etale and of finite type; see [35, I, 10.13.3]) morphism Spf(R1 ) → Spf(R2 ) and if the degeneration datum over Spec(R2 ) pulls back to the degeneration datum over Spec(R1 ), then the family constructed by Mumford’s construction over Spec(R1 ) pulls back to a family over Spec(R2 ) with the same degeneration datum as the datum over Spec(R2 ). The functoriality in [62, Thm. 4.4.16] over Spec(R2 ) then assures that this pullback family agrees with the family constructed from the datum over Spec(R2 ). In particular, we see that the ord assignment of ( ♥ G, ♥ λ, ♥ i, ♥ αHp , ♥ αH ) → Spec(Γ(U, OU )) to U is functorial. p Hence, the assignment defines a (relative) degenerating family ( ♥ G,

♥

♥

λ,

♥

i,

αHp ,

♥ ord αHp )

~ ord , →X ΦH ,δH

(4.2.2.15)

which comes from a degenerating family ( ♥ G,

♥

λ,

♥

i,

♥

~ ord αH ) → X ΦH ,δH

(4.2.2.16)

in the sense that it is so over each affine formal scheme U as above. Since the cone decomposition ΣΦH is ΓΦH -admissible, the group ΓΦH acts naturally on all the objects involved in the degeneration data, and hence by functoriality on the ord ~ ord . )→X degenerating family ( ♥ G, ♥ λ, ♥ i, ♥ αHp , ♥ αH ΦH ,δH p For each σ ⊂ PΦH , let ΓΦH ,σ be defined as in [62, Def. 6.2.5.23], namely the subgroup of ΓΦH consisting of elements that maps σ to itself under the natural action of ΓΦH on PΦH . Then we have similarly the degenerating family ( ♥ G,

♥

λ,

♥

i,

♥

αHp ,

♥ ord αHp )

~ ord →X ΦH ,δH ,σ ,

(4.2.2.17)

together with an equivariant action of ΓΦH ,σ , which comes from a degenerating family ( ♥ G,

♥

λ,

♥

i,

♥

~ ord αH ) → X ΦH ,δH ,σ

(4.2.2.18)

(in a sense analogous to that of (4.2.2.15) and (4.2.2.16)), together with a compatible equivariant action of ΓΦH ,σ . By [62, Rem. 6.2.5.26 and Lem. 6.2.5.27], if the cone decomposition ΣΦH is chosen such that [62, Cond. 6.2.5.25] is satisfied, then each ΓΦH ,σ is finite and ~ ord X ΦH ,δH ,σ /ΓΦH ,σ is a formal (Deligne–Mumford) algebraic stack. Moreover, if H is neat (which is the case, for example, when Hp is neat), then ΓΦH ,σ is trivial and ~ ord ~ ord X ΦH ,δH ,σ /ΓΦH ,σ = XΦH ,δH ,σ is a formal algebraic space. From now on, as always, let us assume that the cone decomposition ΣΦH is chosen such that [62, Cond. 6.2.5.25] is satisfied. This is possible by refining any given cone decomposition ΣΦH . Then the compatible equivariant action of ΓΦH ,σ ord ♥ ♥ ♥ ♥ ~ ord ~ ord on ( ♥ G, ♥ λ, ♥ i, ♥ αHp , ♥ αH ) → X ΦH ,δH ,σ and ( G, λ, i, αH ) → XΦH ,δH ,σ p imply that we have a descended family ( ♥ G,

♥

λ,

♥

i,

♥

αHp ,

♥ ord αHp )

~ ord →X ΦH ,δH ,σ /ΓΦH ,σ ,

(4.2.2.19)

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which comes from a descended family ( ♥ G,

♥

♥

λ,

i,

♥

~ ord αH ) → X ΦH ,δH ,σ /ΓΦH ,σ

(4.2.2.20)

(in a sense analogous to that of (4.2.2.15) and (4.2.2.16)). Definition 4.2.2.21. (Compare with [62, Def. 6.2.5.28].) All the degenerating families (4.2.2.15), (4.2.2.16), (4.2.2.17), (4.2.2.18), (4.2.2.19), and (4.2.2.20) constructed above are called Mumford families. Remark 4.2.2.22. (Compare with [62, Rem. 6.2.5.29].) By abuse of notation, we ord will use the same notation ( ♥ G, ♥ λ, ♥ i, ♥ αHp , ♥ αH ) and ( ♥ G, ♥ λ, ♥ i, ♥ αH ) for p the Mumford families over various bases. Remark 4.2.2.23. The analogues of [62, Rem. 6.2.5.30 and 6.2.5.31] are also true in this context. 4.2.3

Extended Kodaira–Spencer Morphisms and Induced Isomorphisms

Since we have the tautological presence of G\ and ι (defined by the tautological ord

~ Φ ,δ separated, smooth, and tuple (B, X, Y , c, c∨ , τ )) over the algebraic stack Ξ H H locally of finite type over ~S0,rH , we can define (by ´etale descent if necessary) as in [62, Sec. 4.6.2] the extended Kodaira–Spencer morphism KS

ord

~ (G\ ,ι)/Ξ Φ

Lie∨ \

H ,δH

H ,δH

O~ ord ΞΦ

: Lie∨ ∨,\

⊗

ord

~ G /Ξ Φ

/~ S0,rH

G

→ Ω1 ord

ord

~ /Ξ Φ

H ,δH

H ,δH

~ Ξ Φ

H ,δH

/~ S0,rH

[d log ∞].

(4.2.3.1)

Let λ\ : G\ → G∨,\ be the homomorphism defined by the tautological data λB : B → B ∨ and φ : Y → X. Then λ\ induces an O-equivariant morphism → Lie∨ \ ~ ord . Let i\ : O → End~ ord (G\ ) denote the (λ\ )∗ : Lie∨ ∨,\ ~ ord G

/ΞΦ

G /ΞΦ

H ,δH

ΞΦ

H ,δH

H ,δH

tautological O-action morphism on G\ . Definition 4.2.3.2. (Compare with [62, Def. 2.3.5.1] and Definitions 1.1.2.8, 1.3.1.2, and 3.4.3.1.) The O~ord -module ΞΦH ,δH

KS = KS

ord

~ (G\ ,λ\ ,i\ )/Ξ Φ

H ,δH

is the quotient of Lie∨ \

ord

~ G /Ξ Φ

by the O~ ord ΞΦ

H ,δH

⊗

O~ ord ΞΦ

Lie∨ ∨,\ G

ord

~ /Ξ Φ

H ,δH

-submodule spanned by

H ,δH

(λ\ )∗ (y) ⊗ z − (λ\ )∗ (z) ⊗ y

H ,δH

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and ∨

(i\ (b))∗ (x) ⊗ y − x ⊗ (i\ (b) )∗ (y), for x ∈ Lie∨ \

ord

~ G /Ξ Φ

H ,δH

, y, z ∈ Lie∨ ∨,\ G

, and b ∈ O.

ord

~ /Ξ Φ

H ,δH

Remark 4.2.3.3. Unlike in the good reduction case (see [62, Rem. 6.2.5.17]), the formation of KS here may produce torsion elements. Therefore, we also introduce the following: Definition 4.2.3.4. (Compare with [62, Def. 6.2.5.16] and Definition 3.4.3.1.) The O~ord -module ΞΦH ,δH

KSfree = KS

ord

~ (G\ ,λ\ ,i\ )/Ξ Φ

is the quotient of KS

KS

H ,δH

→ KS

ord

~ (G\ ,λ\ ,i\ )/Ξ Φ

of O~ ord

,free

defined as the image of the canonical morphism

ord

~ (G\ ,λ\ ,i\ )/Ξ Φ

H ,δH

⊗Q

ord

~ (G\ ,λ\ ,i\ )/Ξ Φ

H ,δH

H ,δH

Z

-modules.

ΞΦ

H ,δH ,free

By definition, the sheaf KSfree = KS

ord

~ (G\ ,λ\ ,i\ )/Ξ Φ

H ,δH

,free

contains no p-torsion and

hence is flat over ~S0,rH = Spec(OF0 ,(p) [ζprH ]). Proposition 4.2.3.5. (Compare with [62, Prop. 6.2.5.18].) The Kodaira–Spencer morphism (4.2.3.1) factors through the sheaf KSfree defined in Definition 4.2.3.4, and induces an isomorphism ∼

KSfree → Ω1 ord ~ Ξ Φ

H ,δH

In particular, the O~ord

ΞΦH ,δH

/~ S0,rH

[d log ∞].

(4.2.3.6)

-module KSfree is locally free of finite rank. ord

~ Φ ,δ → ~S0,r as a composition Proof. Let us analyze the structural morphism Ξ H H H of smooth morphisms: ord

π0 ~ ord π ~ ord,ΦH π2 ~ ~ Φ ,δ → Ξ CΦH ,δH →1 M → S0,rH . H H H

For simplicity, let us denote the composition π1 ◦ π0 by π10 . Ω1 ord [d log ∞] has an increasing filtration ~ Ξ Φ

H ,δH

/~ S0,rH

∗ 0 ⊂ π01 Ω1~ ord,ΦH MH

/~ S0,rH

⊂ π0∗ Ω1C~ ord

∗ with graded pieces given by π01

Ω1 ord ~ Ξ Φ

H ,δH

Then

~ ord /C Φ ,δ H

~

ΦH ,δH /S0,rH

Ω1~ ord,ΦH MH

⊂ Ω1 ord

/~ S0,rH

~ Ξ Φ

H ,δH

, π0∗

/~ S0,rH

[d log ∞],

Ω1~ ord CΦ

H ,δH

[d log ∞], all of which are locally free of finite rank. H

ord,ΦH

~ /M H

, and

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On the other hand, the sheaf KSfree = KS

ord

~ (G\ ,λ\ ,i\ )/Ξ Φ

,free

H ,δH

has an increasing

∗ filtration given by π01 KS(B,λ ,i )/M ~ ord,ΦH ,free , the pullback (under π0 ) of the free B B H quotient KS(B,c,c∨ ,λ\ ,i\ )/C~ ord ,free of the quotient KS(B,c,c∨ ,λ\ ,i\ )/C~ ord of ΦH ,δH ΦH ,δH
Lie∨ ⊗ Lie∨ ~ ord ~ ord G \ /C B ∨ /C ΦH ,δH

OC ~ ord

ΦH ,δH

ΦH ,δH

+ Lie∨ ~ ord B/C

ΦH ,δH

Lie∨ ~ ord G∨,\ /C

⊗

OC ~ ord




ΦH ,δH

ΦH ,δH

(as an OC~ ord

ΦH ,δH

-submodule of Lie∨ ~ ord G \ /C

Lie∨ ~ ord G∨,\ /C

⊗

ΦH ,δH

OC ~ ord

) by relations as

ΦH ,δH

ΦH ,δH

in Definition 4.2.3.4, and the whole sheaf KSfree . Hence, it suffices to show that the morphism (4.2.3.1) respects the filtrations and matches isomorphically the (free quotients of) the graded pieces. ~ ord ~ ord,ΦH is ´etale over the base change M By Proposition 3.4.3.3, since M Hh ,rH of H ... ord ~ ord (defined by MH and M ~ M Hh as in Theorem 3.4.2.5) to S0,rH , the Kodaira– h Hh ord,ΦH ~ for B over M induces an isomorphism Spencer morphism KS ~ ord,ΦH ~ B/MH

H

/S0,rH

∼

KS(B,λ

~ ord,ΦH ,free B ,iB )/MH

→

Ω1~ ord,ΦH ~ M /S0,r H

,

(4.2.3.7)

H

and hence the same remains true after pulled back by π01 . Since the Kodaira– ord ~ Φ ,δ = π ∗ KS ~ ord,ΦH ~ for B over Ξ Spencer morphism KS ord ~ B/Ξ Φ

H ,δH

01

/~ S0,rH

B/MH

/S0,rH

H

H

is the restriction of the Kodaira–Spencer morphism KS in (4.2.3.1) (cf. the compatibility statements in [62, Sec. 4.6.2]), we see that the first filtered pieces are respected. By the deformation-theoretic interpretation of the Kodaira–Spencer morphisms KS(B,c)/C~ ord /~S0,r and KS(B ∨ ,c∨ )/C~ ord /~S0,r in [62, Sec. 4.6.1] (see in parH

ΦH ,δH

H

ΦH ,δH

ticular [62, Def. 4.6.1.2]), we see that the restrictions of both of them to Lie∨ ⊗ agree with KSB/C~ ord /~S0,r , which induces a Lie∨ ~ ord ~ ord B/C B ∨ /C ΦH ,δH

OC ~ ord

H

ΦH ,δH

ΦH ,δH

ΦH ,δH

surjection onto

π1∗

Ω1~ ord,ΦH ~ . MH /S0,rH

Lie∨ ~ ord G \ /C

Hence, they define a morphism
Lie∨ ~ ord B ∨ /C

⊗

ΦH ,δH

OC ~ ord

ΦH ,δH

ΦH ,δH

+ Lie∨ ~ ord B/C

Lie∨ ~ ord G∨,\ /C

⊗

ΦH ,δH

OC ~ ord




ΦH ,δH

ΦH ,δH

ΦH ,δH

compatible with the pullback of KSB/M ~ ord,ΦH /~ S H

the source of (4.2.3.8) with its quotient by

(4.2.3.8)

→ Ω1C~ ord

0,rH

/~ S0,rH

, which induces (after replacing

Lie∨ ~ ord B/C Φ ,δ H

⊗

H

OC ~ ord

Lie∨ ~ ord B ∨ /C

) a

ΦH ,δH

ΦH ,δH

morphism Lie∨ ~ ord T /C

Lie∨ ~ ord B ∨ /C

⊗

ΦH ,δH

OC ~ ord




ΦH ,δH

ΦH ,δH

+

Lie∨ ~ ord B/C Φ ,δ H

H

⊗

OC ~ ord

ΦH ,δH

Lie∨ ~ ord T ∨ /C

ΦH ,δH




→ Ω1~ ord CΦ

H ,δH

ord,ΦH

~ /M H

.

(4.2.3.9)

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Since c and c∨ satisfies the compatibility cφ = λB c∨ , the morphism (4.2.3.8) is compatible with quotients by relations as in Definition 4.2.3.4, and induces a morphism KS(B,c,c∨ ,λ\ ,i\ )/C~ ord

ΦH ,δH

→ Ω1C~ ord

(4.2.3.10)

~

ΦH ,δH /S0,rH

compatible with the pullback of the isomorphism (4.2.3.7). Then the morphism (4.2.3.9) induces a morphism
KS(B,c,c∨ ,λ\ ,i\ )/C~ ord /KS(B,λB ,iB )/C~ ord (4.2.3.11) → Ω1~ ord ~ ord,ΦH , free ΦH ,δH

where KS(B,λB ,iB )/C~ ord

CΦ

ΦH ,δH

ΦH ,δH

∼ =

π1∗

H ,δH

KS(B,λ

~ ord,ΦH B ,iB )/MH

/M H

. By Propositions 4.2.1.29 and

4.2.1.34, the morphism (4.2.3.11) is the pullback of the corresponding morphism
(4.2.3.12) KS(B,c,c∨ ,λ\ ,i\ )/C~ ord /KS(B,λB ,iB )/C~ ord free → Ω1C~ ord /M ~ ord,Z1 Φ1 ,δ1

Φ1 ,δ1

1

Φ1 ,δ1

...ord ...ord at principal level 1. Since C Φ1 is the universal space for (c, c∨ ) over M1 (satisfying the compatibility cφ = λB c∨ ), the source of (4.2.3.12) can be canonically identified with the pullback of the free quotient Ω1...ord ~ ord,Z1 )/M ~ ord,Z1 ,free M (C Φ × 1 1 1 ...ord,Z 1 M1 ...ord ~ ord,Z1 . By the con~ ord → C of Ω1...ord under C M Φ1 ... × 1 Φ1 ,δ1 ~ ord,Z1 )/M ~ ord,Z1 (C Φ × M ord,Z1 1 1 1 ...ord,Z M1 1 M 1

~ ord,Z1 as the pullback of (4.2.1.33) under the section ~ ord → M struction of C 1 Φ1 ,δ1 (4.2.1.32) in the proof of Proposition 4.2.1.30, the canonical morphism ...ord ~ ord,Z1 )∗ Ω1...ord ~ ord → C (C M Φ1 ... × Φ1 ,δ1 1 ~ ord,Z1 )/M ~ ord,Z1 ,free M (C Φ × ord,Z1 1 1 1 ...ord,Z M1 1 M 1

→ Ω1C~ ord

~ ord,Z1 Φ1 ,δ1 /M1

is an isomorphism. Thus, we see that the morphism (4.2.3.12) is an isomorphism, and hence that the morphism (4.2.3.11) is an isomorphism and induces (by a simple diagram chasing) the composition of canonical isomorphisms KS(B,c,c∨ ,λ\ ,i\ )/C~ ord

ΦH ,δH ,free

/KS(B,λB ,iB )/C~ ord

∼

→ KS(B,c,c∨ ,λ\ ,i\ )/C~ ord

ΦH ,δH

∼

→ Ω1~ ord CΦ

H ,δH

ord,ΦH

~ /M H

ΦH ,δH ,free

/KS(B,λB ,iB )/C~ ord




ΦH ,δH

(4.2.3.13)

free

.

This shows that the morphism (4.2.3.10) also induces an isomorphism KS(B,c,c∨ ,λ\ ,i\ )/C~ ord

ΦH ,δH ,free

∼

→ Ω1C~ ord

~

ΦH ,δH /S0,rH

.

(4.2.3.14) ord

~ Φ ,δ is induced by Since the pullback of this isomorphism (4.2.3.14) (under π0 ) to Ξ H H the restriction of the extended Kodaira–Spencer morphism KS in (4.2.3.1) (cf. [62, Rem. 4.6.2.7]), we see that the second filtered pieces are also respected, with an induced isomorphism between the second graded pieces.

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Finally, we arrive at the top filtered pieces, and the question is about the induced morphism Lie∨

Lie∨ ∨

⊗

ord

~ T /Ξ Φ

H ,δH

ΞΦ

H ,δH

→ Ω1 ord

ord

~ T /Ξ Φ

O~ ord

~ Ξ Φ

H ,δH

H ,δH

~ ord /C Φ ,δ H

[d log ∞]

between the top graded pieces. Let us denote by KS(T,λT ,iT )/~S0,r quotient of

Lie∨ ⊗ T /~ S0,rH O ~ S

Lie∨ T ∨ /~ S0,rH

(4.2.3.15)

H

H

,free

the free

by relations as in Definition 4.2.3.2, and

0,rH

ord

by KS

ord

~ (T,λT ,iT )/Ξ Φ

H ,δH

,free

and KS(T,λT ,iT )/Ξ ~ ord

ΦH ,δH

,free

~ their pullbacks to Ξ ΦH ,δH and

~ ord , respectively. Ξ ΦH ,δH Let us first consider the restriction Lie∨ ~ ord T /Ξ

Lie∨ ~ ord T ∨ /Ξ

⊗

O~ Ξord

ΦH ,δH

ΦH ,δH

→ Ω1Ξ ~ ord

~ ord ΦH ,δH /CΦH ,δH

(4.2.3.16)

ΦH ,δH

~ ord , which is induced by the Kodaira–Spencer morphism of (4.2.3.15) to Ξ ΦH ,δH KS(G\ ,ι)/Ξ ~ ord /~ S0,r defined deformation-theoretically as in [62, Def. 4.6.2.6]. Since H

ΦH ,δH

τ is compatible with the polarizations and O-endomorphism structures, the morphism (4.2.3.16) induces a morphism KS(T,λT ,iT )/Ξ ~ ord

ΦH ,δH ,free

→ Ω1Ξ ~ ord

~ ord ΦH ,δH /CΦH ,δH

.

(4.2.3.17)

By Proposition 4.2.1.41, the morphism (4.2.3.17) is the pullback of the corresponding morphism KS(T,λT ,iT )/Ξ ~ ord

Φ1 ,δ1 ,free

→ Ω1Ξ ~ ord

~ ord Φ1 ,δ1 /CΦ1 ,δ1

.

(4.2.3.18)

...ord ...ord Since Ξ Φ1 is the universal space for ι over C Φ1 (with symmetry condition, but without positivity condition), the source of (4.2.3.18) can be canonically identified with the pullback of the free quotient Ω1(...ord × C~ ord )/C~ ord ,free of Ξ Φ1 ... Φ1 ,δ1 Φ1 ,δ1 C ord Φ1 ... ~ ord → Ξ ord ~ ord Ω1(...ord × C~ ord )/C~ ord under Ξ Φ1 ...× CΦ1 ,δ1 . By the construction Φ1 ,δ1 Ξ Φ1 ... ord Φ ,δ Φ ,δ C ord Φ1

1

1

1

C Φ1

1

~ ord → C ~ ord as the pullback of (4.2.1.40) under the section (4.2.1.39) in the of Ξ Φ1 ,δ1 Φ1 ,δ1 proof of Proposition 4.2.1.37, the canonical morphism ord ~ ord → ... ~ ord )∗ Ω1...ord (Ξ Ξ Φ1 ...× C Φ1 ,δ1 Φ1 ,δ1 ~ ord )/C ~ ord ,free (ΞΦ × C ord Φ1 ,δ1 Φ1 ,δ1 1 ... C Φ1

C ord Φ1

→ Ω1Ξ ~ ord

~ ord ΦH ,δH /CΦH ,δH

is an isomorphism. Thus, we see that the morphism (4.2.3.18) is an isomorphism, and hence that the morphism (4.2.3.17) is an isomorphism KS(T,λT ,iT )/Ξ ~ ord

ΦH ,δH ,free

∼

→ Ω1Ξ ~ ord

~ ord ΦH ,δH /CΦH ,δH

.

(4.2.3.19)

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ord

~ Φ ,δ , then the morphism (4.2.3.15) is induced by the exIf we work over Ξ H H tended Kodaira–Spencer morphism KS \ ~ ord defined as in [62, Def. ~ (G ,ι)/ΞΦ

H ,δH

4.6.2.12]. Since its image in Ω1 ord ~ Ξ Φ

H ,δH

~ ord /C Φ ,δ H

/S0,rH

~ ord (`)) for [d log ∞] contains d log(Ψ ΦH ,δH H

all ` ∈ SΦH , which are exactly the generators, we see that (4.2.3.15) induces an isomorphism KS

∼

ord

~ (T,λT ,iT )/Ξ Φ

,free H ,δH

→ Ω1 ord ~ Ξ Φ

H ,δH

~ ord /C Φ ,δ H

[d log ∞]

H

and (by a simple diagram chasing) the composition of canonical isomorphisms KSfree /π0∗ KS(B,c,c∨ ,λ\ ,i\ )/C~ ord

ΦH ,δH ,free

∼

→ KS/π0∗ KS(B,c,c∨ ,λ\ ,i\ )/C~ ord

ΦH ,δH

∼

→

Ω1 ord ~ Ξ

~ ord ΦH ,δH /CΦH ,δH


free

∼ = KS

ord

~ (T,λT ,iT )/Ξ Φ

H ,δH

,free

[d log ∞]

between the top graded pieces. Hence, (4.2.3.6) is an isomorphism, as desired.

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Chapter 5

Partial Toroidal Compactifications

The goal of this chapter is to construct the partial toroidal compactifications for the ordinary loci defined in Chapter 3, based on the theory of degeneration and the construction of boundary charts given in Chapter 4. 5.1

Approximation and Gluing Along the Ordinary Loci

In this section, let us continue with the setting in Sections 4.1.6 and 4.2.1, including the choices of a level H = Hp Hp and an integer rH determined by H as in Definition 3.4.2.1. The materials in this section follows those of [62, Sec. 6.3] very closely. However, we spell out the precise statements to make sure that the definitions and constructions can indeed be generalized after some subtle modifications. 5.1.1

Ordinary Good Formal Models

Construction 5.1.1.1. Let S be an excellent normal algebraic stack that is flat ord over ~S0,rH = Spec(OF0 ,(p) [ζprH ]), and let (G, λ, i, αHp , αH ) be a degenerating fap ord ~ mily of type M over S (see Definition 3.4.2.10). (As remarked in [62, ConH

str. 6.3.1.1], the excellence assumption on S might be removed by direct limit arguments, but we do not need this generality for our purpose.) By definiord tion (see Condition 4 of Definition 3.4.2.10), (G, λ, i, αHp , αH ) → S defines a p ord ~ tuple parameterized by M H (see Convention 3.4.2.9). By the construction of ~ ord (see Theorem 3.4.2.5), this implies that there is a level-H structure αH on M H

ord (G, λ, i) ⊗ Q → S ⊗ Q such that (αHp , αH ) ⊗ Q is assigned to αH under (3.3.5.5) p Z

Z

Z

over S ⊗ Q. Then (G, λ, i, αH ) → S qualifies as a degenerating family of type Z

MH over S (by choosing S1 to be an open dense subscheme of S ⊗ Q in DeZ

finition 1.3.1.1), and [62, Constr. 6.3.1.1] applies and defines the sheaf objects ΦH = (X(G), Y (G), φ(G), ϕ−2,H (G), ϕ0,H (G)), SΦH (G) , and B(G) : SΦH (G) → Inv(S). (This finishes Construction 5.1.1.1.) 279

(5.1.1.2)

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Compactifications with Ordinary Loci

Let (ZH , ΦH , δH ) be a representative of an ordinary cusp label at level H (see Deord finition 3.2.3.8). Then we have the Mumford families ( ♥ G, ♥ λ, ♥ i, ♥ αHp , ♥ αH )→ p ord ♥ ♥ ♥ ♥ ord ~ ~ X /ΓΦ ,σ and ( G, λ, i, αH ) → X /ΓΦ ,σ as in (4.2.2.19) and ΦH ,δH ,σ

H

ΦH ,δH ,σ

H

(4.2.2.20). Let us summarize their properties as follows: Proposition 5.1.1.3. (Compare with [62, Prop. 6.3.1.6].) Let Sfor = Spf(R, I) be an affine formal scheme, with an ´etale (i.e., formally ´etale and of finite type; ~ ord see [35, I, 10.13.3]) morphism fˆ : Sfor → X ΦH ,δH ,σ /ΓΦH ,σ inducing a morphism ord ~ f : S = Spec(R) → ΞΦH ,δH (σ)/ΓΦH ,σ mapping the support Spec(R/I) of Sfor ~ ord ~ ord to the σ-stratum Ξ ΦH ,δH ,σ /ΓΦH ,σ of ΞΦH ,δH (σ)/ΓΦH ,σ . (In this case, the subscheme Spec(R/I) of S is the scheme-theoretic preimage of its image under f .) ord Let ( ♦ G, ♦ λ, ♦ i, ♦ αHp , ♦ αH ) → S = Spec(R) (resp. ( ♦ G, ♦ λ, ♦ i, ♦ αH ) → p ~ ord S) be the pullback of ( ♥ G, ♥ λ, ♥ i, ♥ αHp , ♥ αord ) → X /ΓΦ ,σ (resp. Hp

ΦH ,δH ,σ

H

ˆ ~ ord ( G, λ, i, αH ) → X ΦH ,δH ,σ /ΓΦH ,σ ) to Sfor under f (by abuse of language). ♦ ♦ ♦ ord In this case, ( αHp , αHp ) is assigned to αH under (3.3.5.5) over the generic point η = Spec(K) of Spec(R), where K is the fraction field of R. Then R is an I-adically complete excellent ring, which is formally smooth over the abelian scheme ~ ord , and hence also formally smooth over ~S0,r = Spec(OF ,(p) [ζprH ]). C H 0 ΦH ,δH ♥

♥

♥

♥

~ ord (σ)/ΓΦ ,σ determines a stratification of S = (1) The stratification of Ξ H ΦH ,δH Spec(R) parameterized by {faces τ of σ}/ΓΦH ,σ such that each stratum of S (with its reduced structure, namely, its structure as an open subscheme in a closed subscheme with reduced structure) is the scheme-theoretic preimage ~ ord (σ)/ΓΦ ,σ under f . of the corresponding stratum of Ξ H ΦH ,δH ♦ (2) The formal completion of G along the preimage of Spec(R/I) is canonically isomorphic to the pullback of G\ under fˆ (as a formal algebraic stack, rather than a relative scheme). (3) The ´etale sheaf X( ♦ G) (see [62, Thm. 3.3.1.9]) is the quotient sheaf of the constant sheaf X such that, over the (τ mod ΓΦH ,σ )-stratum, the sheaf X( ♦ G) is a constant quotient X(τ mod ΓΦH ,σ ) of X, with an admissible surjection X  X(τ mod ΓΦH ,σ ) inducing a torus argument ΦH,(τ mod ΓΦH ,σ ) from ΦH as in [62, Lem. 5.4.2.11], such that τ is contained in the ΓΦH -orbit of the image of the induced embedding P+ ,→ PΦH . (We know ΦH,(τ mod Γ ) ΦH ,σ

the surjection is admissible because of the existence of level-H structures; see [62, Lem. 5.2.2.2 and 5.2.2.4].) This produces a sheaf version ΦH ( ♦ G) of ΦH over S. The formation of SΦH from ΦH applies to ΦH ( ♦ G) and defines a sheaf SΦH ( ♦ G) (cf. Proposition 4.2.2.5). Then ΦH ( ♦ G) is equivalent (see Definition 1.2.1.6 and [62, Def. 5.4.2.2]) ~ ord (σ)/ΓΦ ,σ (see Proposition to the pullback of the tautological ΦH on Ξ H ΦH ,δH 4.2.2.5) under f . (4) Under the equivalence between ΦH ( ♦ G) and the pullback of ΦH above, the

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pullback f ∗ (B) : SΦH ( ♦ G) → Inv(S) of the tautological homomorphism B ~ ord (σ)/ΓΦ ,σ (see Proposition 4.2.2.5) under f agrees with the over Ξ H ΦH ,δH ord homomorphism B( ♦ G) defined by ( ♦ G, ♦ λ, ♦ i, ♦ αHp , ♦ αH ) → S (or p ♦ ♦ ♦ ♦ rather by ( G, λ, i, αH ) → S) as in Construction 5.1.1.1 (or rather as in [62, Constr. 6.3.1.1]). (5) Let KS( ♦ G, ♦ λ, ♦ i)/S,free be the OS -module (flat over ~S0,rH ) defined by ord ( ♦ G, ♦ λ, ♦ i, ♦ αHp , ♦ αH ) → S as in Definition 3.4.3.1. As in [62, p 1 b Sec. 4.6.3], let Ω denote the completion of Ω1 with respect to S/~ S0,rH

S/~ S0,rH

the topology of R defined by I, which is locally free of finite rank over b1 OS (cf. [35, 0IV , 20.4.9]), and let Ω [d log ∞] be the subsheaf of S/~ S 0,rH

b1 (η ,→ S)∗ (η ,→ S)∗ Ω S/~ S

0,rH

b1 generated locally by Ω S/~ S

and those d log q

0,rH

where q is a local generator of an irreducible component of the normal crossings divisor of Spec(R) induced by the corresponding normal crossings di~ ord (σ) (cf. [62, Thm. 6.1.2.8(5)]). Then the extended Kodaira– visor of Ξ ΦH ,δH Spencer morphism (see [62, Def. 4.6.3.44]) defines an isomorphism ∼ b1 →Ω [d log ∞]. KS ♦ : KS ♦ ♦ ♦ ~ G/S/S0,rH

(

G,

λ,

i)/S,free

S/~ S0,rH

~ ord (6) The morphism fˆ : Sfor = Spf(R, I) → X ΦH ,δH ,σ /ΓΦH ,σ , or rather the morord ~ phism f : S = Spec(R) → ΞΦH ,δH (σ)/ΓΦH ,σ , is tautological with respect to ~ ord (σ)/ΓΦ ,σ , in the following sense: the universal property of Ξ ΦH ,δH

H

The setting is as follows: The base ring R and the ideal I satisfy the setting ord ) be a degenerating family of type of Section 4.1.6. Let (G, λ, i, αHp , αH p ord ~ MH over S that defines an object in the essential image of the canonical ...ord (R, I) in Theorem 4.1.6.2. morphism DEGPEL,Mord (R, I) → DEG H

PEL, M H

By Theorem 4.1.6.2, the family determines an object in the essential image of DDPEL,Mord (R, I) → DDPEL,... (R, I), which determines an object of M ord H H DDPEL,Mord (R, I) up to isomorphism, which is by definition an object of H DDPEL,MH (R, I), the latter determining, in particular, an ordinary cusp label. Suppose (ZH , ΦH , δH ) is a representative of this ordinary cusp label. By [62, Lem. 5.4.2.10; see also the errata], there exists a tuple \ (B, λB , iB , X, Y, φ, c, c∨ , τ, [αH ])

defining the above object of DDPEL,MH (R, I), together with a representative \ ∼ ∨ αH = (ZH , ϕ∼ −2,H , ϕ−1,H , ϕ0,H , δH , cH , cH , τH ) \ ∼ of [αH ], such that (ϕ∼ −2,H , ϕ0,H ) induces the (ϕ−2,H , ϕ0,H ) in ΦH , as in the corrected [62, Def. 5.4.2.8] in the errata. By Theorem 4.1.6.2, this tuple is an object of DDPEL,Mord (R, I), and determines an object H \,ord ]) (B, λB , iB , X, Y , φ, c, c∨ , τ, [αH \,ord in DDPEL,... (R, I), with [αH ] represented by the tuple M ord H \,ord ord ord ord ord ord ∨,ord ord αH = (ZH , ϕord , τH ) −2,H , ϕ−1,H = (ϕ−1,Hp , ϕ−1,Hp ), ϕ0,H , δH , cH , cH

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Compactifications with Ordinary Loci \ ord ord determined by αH as in Section 4.1.6, so that (ϕord −2,H , ϕ0,H ) (resp. δH ) induces the same (ϕ−2,H , ϕ0,H ) in ΦH (resp. δH ). By Proposition 4.2.1.46, this tuple without its positivity condition defines a morphism Spec(K) → ~ ord Ξ ΦH ,δH that is unique up to an action of ΓΦH on the identification of ~ ord ~ ord ΦH , whose composition with Ξ ΦH ,δH → CΦH ,δH extends to a morphism ~ ord . Let B(G) : S Spec(R) → C ΦH (G) → Inv(S) be the homomorphism ΦH ,δH defined as in Construction 5.1.1.1. Then the universal property is as follows: Suppose there exists an identification of ΦH such that, for each discrete valuation υ : Inv(S) → Z defined by a height-one prime of R, the composition υ ◦ B(G) : SΦH (G) → Z defines an element in the closure σ of σ in (SΦH )∨ R . Such an identification of ΦH is ~ ord unique up to an element in ΓΦH ,σ , and all morphisms Spec(K) → Ξ ΦH ,δH ord ~ as above induce the same morphism Spec(K) → Ξ /Γ if they reΦH ,σ ΦH ,δH spect such identifications of ΦH . Then this morphism extends to a (neces~ ord (σ)/ΓΦ ,σ , sending the sarily unique) morphism f : S = Spec(R) → Ξ H ΦH ,δH ord ~ subscheme Spec(R/I) to the σ-stratum of Ξ (σ)/Γ ΦH ,σ and hence inΦH ,δH ord ˆ ~ ducing a morphism f : Sfor = Spf(R, I) → XΦH ,δH ,σ /ΓΦH ,σ between formal ord ) → S is isomorphic to the pulalgebraic stacks, such that (G, λ, i, αHp , αH p ♦ ♦ ♦ ♦ ♦ ord ord lback ( G, λ, i, αHp , αHp ) → S of ( ♥ G, ♥ λ, ♥ i, ♥ αHp , ♥ αH ) → p ord ˆ ~ X /ΓΦ ,σ under f (and so that (G, λ, i, αH ) → S is isomorΦH ,δH ,σ

H

phic to the pullback ( ♦ G, ♦ λ, ♦ i, ♦ αH ) → S of ( ♥ G, ♥ λ, ♥ i, ♥ αH ) → ˆ ~ ord X ΦH ,δH ,σ /ΓΦH ,σ under f , by the injectivity of the assignment (3.3.5.5)). Proof. The proof of [62, Prop. 6.3.1.6] based on [62, Thm. 4.6.3.16] works here if we replace the reference to [62, Prop. 6.2.5.18] there with an analogous reference to Proposition 4.2.3.5. As a byproduct of our usage of Proposition 4.2.3.5 in the proof: Corollary 5.1.1.4. (Compare with [62, Cor. 6.3.1.8].) Suppose fˆ : Sfor = ~ ord Spf(R, I) → X ΦH ,δH ,σ /ΓΦH ,σ is a morphism between noetherian formal schemes formally smooth over ~S0,rH , with induced morphism f : S = Spec(R) → ~ ord (σ)/ΓΦ ,σ such that the support Spec(R/I) of Sfor is the scheme-theoretic Ξ H ΦH ,δH ~ ord preimage under f of some subalgebraic stack Z of the σ-stratum Ξ ΦH ,δH ,σ /ΓΦH ,σ ord ~ of Ξ (σ)/Γ . Suppose moreover that the pullback of the stratification of ΦH ,σ ΦH ,δH ord ~ ΞΦH ,δH (σ)/ΓΦH ,σ induces a stratification of S = Spec(R) such that each stratum of S = Spec(R) (with its reduced structure, as in (1) of Proposition 5.1.1.3) is the ~ ord (σ)/ΓΦ ,σ . Under these assumpscheme-theoretic preimage of a stratum of Ξ H ΦH ,δH tions, we have an induced morphism f0 : Spec(R/I) → Z, and we can define (as in [62, Thm. 4.6.3.16] and (5) of Proposition 5.1.1.3) the extended Kodaira–Spencer morphism b1 KS ♦ : KS ♦ ♦ ♦ →Ω [d log ∞], (5.1.1.5) ~ G/S/S0,rH

(

G,

λ,

i)/S,free

S/~ S0,rH

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where KS( ♦ G, ♦ λ, ♦ i)/S,free is the sheaf defined as in Definition 3.4.3.1 by the ord ord pullback ( ♦ G, ♦ λ, ♦ i, ♦ αHp , ♦ αH ) → S of ( ♥ G, ♥ λ, ♥ i, ♥ αHp , ♥ αH ) → p p ord ˆ ~ X /Γ in the sense of relative schemes. Then the morphism f is forΦH ,σ ΦH ,δH ,σ mally ´ etale if and only if it satisfies the conditions that f is flat, that f0 is formally smooth, and that the morphism KS ♦ G/S/~S0,r in (5.1.1.5) is surjective. H

Proof. The proof of [62, Cor. 6.3.1.8] works here if we replace the reference to [62, Prop. 6.2.5.18] there with an analogous reference to Proposition 4.2.3.5. Corollary 5.1.1.6. (Compare with [62, Cor. 6.3.1.14].) In the context of Corollary 5.1.1.4, suppose that R is a strict local ring with (separably closed) residue field k, so ˜ mapping Spec(k) that the morphism f induces a morphism f˜ : Spec(R) → Spec(R) ord ˜ ˜ ~ to Spec(k), where R is a strict local ring of ΞΦH ,δH (σ)/ΓΦH ,σ with (separably closed) ˜ and suppose that k is of finite type over k. ˜ Then f˜ is formally ´etale residue field k, if and only if R and f satisfy the conditions that R is equidimensional and has ~ ord (σ)/ΓΦ ,σ , and that the induced canonical morphism the same dimension as Ξ H ΦH ,δH (5.1.1.5) is surjective. (This last condition forces the induced homomorphism k˜ → k to be an isomorphism.) Proof. The proof of [62, Cor. 6.3.1.14] also works here. Definition 5.1.1.7. (Compare with [28, Ch. IV, Sec. 3] and [62, Def. 6.3.1.15].) Let (ΦH , δH ) be a representative of an ordinary cusp label at level H (see Definition 3.2.3.8), and let σ ⊂ P+ ΦH be a nondegenerate smooth rational polyhedral cone. An ordinary good formal (ΦH , δH , σ)-model is a degenerating family ord ~ ord over Spec(R) (see Definition 3.4.2.10) ) of type M ( ♦ G, ♦ λ, ♦ i, ♦ αHp , ♦ αH H p where we have the following: (1) R is a strict local ring that is complete with respect to an ideal I = rad(I), together with a stratification of Spec(R) with strata parameterized by ΓΦH ,σ -orbits of faces of σ. ~ ord (σ)/ΓΦ ,σ such that (2) There exists a morphism f : Spec(R) → Ξ H ΦH ,δH Spec(R/I) is the scheme-theoretic preimage of the σ-stratum under f , satisfying the following properties: (a) The morphism f makes R isomorphic to the completion of a strict ~ ord (σ)/ΓΦ ,σ with respect to the ideal defining the local ring of Ξ H ΦH ,δH σ-stratum. (b) The stratification of Spec(R) is strictly compatible with that of ~ ord (σ)/ΓΦ ,σ in the sense that each stratum of Spec(R) (with Ξ H ΦH ,δH its reduced structure, as in (1) of Proposition 5.1.1.3) is the scheme~ ord (σ)/ΓΦ ,σ . theoretic preimage of the corresponding stratum of Ξ H ΦH ,δH ♦ ♦ ♦ ♦ ♦ ord (c) The degenerating family ( G, λ, i, αHp , αHp ) defines an object of DEGPEL,... (R, I) in the essential image of the canonical morM ord H phism DEGPEL,Mord (R, I) → DEGPEL,... (R, I) in Theorem 4.1.6.2, M ord H H

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Compactifications with Ordinary Loci ord and (by abuse of language) ( ♦ G, ♦ λ, ♦ i, ♦ αHp , ♦ αH ) → Spec(R) p ♥ ♥ ♥ ♥ ord is the pullback of the Mumford family ( G, λ, i, αHp , ♥ αH )→ p ord ord ˆ ~ ~ X /ΓΦ ,σ under the morphism f : Spf(R, I) → X /ΓΦ ,σ ΦH ,δH ,σ

ΦH ,δH ,σ

H

H

induced by f . Remark 5.1.1.8. (Compare with [62, Rem. 6.3.1.16].) As in Proposition 5.1.1.3, ~ ord the morphism fˆ : Spf(R, I) → X ΦH ,δH ,σ /ΓΦH ,σ in Definition 5.1.1.7 (making ♦ ♦ ♦ ♦ ♦ ord p ( G, λ, i, αH , αHp ) → Spec(R) the pullback of the Mumford family ~ ord ( ♥ G, ♥ λ, ♥ i, ♥ αHp , ♥ αord ) → X /ΓΦ ,σ ) is necessarily unique. Hp

ΦH ,δH ,σ

H

Remark 5.1.1.9. (Compare with [62, Rem. 6.3.1.17].) By the universal property ~ ord (σ)/ΓΦ ,σ (see Proposition 4.2.2.8 and (6) of Proposition 5.1.1.3), the of Ξ H ΦH ,δH ~ ord (σ)/ΓΦ ,σ in Definition 5.1.1.7 (with the desired morphism f : Spec(R) → Ξ H ΦH ,δH ~ ord (σ)/ΓΦ ,σ properties) is tautological for the induced morphism Spec(R) → C H ΦH ,δH ♦ and the homomorphism B( G) : SΦH ( ♦ G) → Inv(Spec(R)). Corollary 5.1.1.10. (Compare with [62, Cor. 6.3.1.18].) Suppose R is a regular strict local ring complete with respect to an ideal I = rad(I), together with ~ ord a morphism fˆ : Sfor := Spf(R, I) → X ΦH ,δH ,σ /ΓΦH ,σ inducing a morphism ord ~ f : S := Spec(R) → ΞΦH ,δH (σ)/ΓΦH ,σ such that Spec(R/I) is the scheme-theoretic preimage of the σ-stratum under f , and inducing an isomorphism between separable closures of residue fields. Then we can verify the statement that f makes R isomor~ ord (σ)/ΓΦ ,σ with respect to the phic to the completion of a strict local ring of Ξ H ΦH ,δH ideal defining the σ-stratum by verifying the following conditions: ~ ord (σ)/ΓΦ ,σ . (1) The scheme S has the same dimension as Ξ H ΦH ,δH ~ ord (σ)/ΓΦ ,σ induces a stratification of Spec(R) (2) The stratification of Ξ H ΦH ,δH ~ ord (σ)/ΓΦ ,σ in the sense that that is strictly compatible with that of Ξ H ΦH ,δH each stratum of S (with its reduced structure, as in (1) of Proposition 5.1.1.3) is the scheme-theoretic preimage of the corresponding stratum of ~ ord (σ)/ΓΦ ,σ . Ξ H ΦH ,δH (3) The extended Kodaira–Spencer morphism (see [62, Def. 4.6.3.44]) induces an isomorphism KS ♦ G/S/~S0,r where ( ♦ G, ♥

♦

♥

λ,

♦

♥

i,

∼

H

♦

♥

b1 : KS( ♦ G, ♦ λ, ♦ i)/S,free → Ω S/~ S

0,rH

[d log ∞],

♦ ord αHp ) → S is the pullback of the Mumford fa♥ ord ˆ ~ ord αHp ) → X ΦH ,δH ,σ /ΓΦH ,σ under f (by abuse of

αHp ,

mily ( G, λ, i, αHp , language), and where KS( ♦ G, ♦ λ, ♦ i)/S,free is defined as in Definition 3.4.3.1.

Remark 5.1.1.11. (Compare with [62, Rem. 6.3.1.19].) The various morphisms ~ ord from Spec(R/I) to the support of the formal algebraic stack X ΦH ,δH ,σ /ΓΦH σ , for the various ordinary good formal (ΦH , δH , σ)-models, cover the whole σ-stratum.

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Remark 5.1.1.12. (Compare with [62, Rem. 6.3.1.20].) An ordinary good for0 mal (ΦH , δH , σ)-model is an ordinary good formal (Φ0H , δH , σ 0 )-model if and only if 0 0 0 (ΦH , δH , σ) is equivalent to (ΦH , δH , σ ) (see Definition 1.2.2.10). Remark 5.1.1.13. (Compare with [62, Rem. 6.3.1.21].) For two smooth ratio0 nal polyhedral cones σ, σ 0 ∈ P+ ΦH such that σ ⊂ σ , an ordinary good formal (ΦH , δH , σ)-model is not necessarily an ordinary good formal (ΦH , δH , σ 0 )-model (cf. [62, Rem. 6.2.5.31] and Remark 4.2.2.23). 5.1.2

Ordinary Good Algebraic Models

Proposition 5.1.2.1. (Compare with [62, Prop. 6.3.2.1].) Let (ΦH , δH ) be a representative of an ordinary cusp label at level H (see Definition 3.2.3.8), and let σ ⊂ P+ ΦH be a nondegenerate smooth rational polyhedral cone. Let R be the strict local ~ ord (σ)/ΓΦ ,σ for some σ ⊂ P+ , ring of a geometric point x ¯ of the σ-stratum of Ξ H ΦH ,δH ΦH ∧ let R be the completion of R with respect to the ideal I defining the σ-stratum, and ord let I ∧ := I · R∧ ⊂ R∧ . Suppose ( ♦ G, ♦ λ, ♦ i, ♦ αHp , ♦ αH ) → Spec(R∧ ) defines p ∧ an ordinary good formal (ΦH , δH , σ)-model over S := Spec(R∧ ). Then we can find ord ~ ord over S := (noncanonically) a degenerating family (G, λ, i, αHp , αH ) of type M H p ord Spec(R) as in Definition 3.4.2.10, which approximates ( ♦ G, ♦ λ, ♦ i, ♦ αHp , ♦ αH ) p in the following sense: (1) Over Spec(R/I), we have ( ♦ G,

♦

λ,

♦

i) ⊗ (R/I) ∼ = (G, λ, i) ⊗ (R/I). (We R

R

ord do not compare ( ♦ αHp , ♦ αHp ) and (αHp , αH ) here, because they are not p defined over Spec(R/I).) (2) Under the canonical homomorphism R ,→ R∧ , the pullbacks of the objects ΦH (G), SΦ(G) , and B(G) defined as in Construction 5.1.1.1 are isomorphic to the objects ΦH ( ♦ G), SΦ( ♦ G) , and B( ♦ G) defined as in Proposition 5.1.1.3, respectively. ord (3) The pullback of (G, λ, i, αHp , αH ) → S under the canonical homop ∧ morphism R ,→ R defines an ordinary good formal (ΦH , δH , σ)-model ord (G, λ, i, αHp , αH ) ⊗ R∧ → S ∧ , and can be realized as the pullback of p R

~ ord → X ΦH ,δH ,σ /ΓΦH ,σ via ∧ ord ~ a canonically defined morphism Spf(R , I ) → XΦH ,δH ,σ /ΓΦH ,σ . Comparing this isomorphism with the original morphism Spf(R∧ , I ∧ ) → ~ ord X ΦH ,δH ,σ /ΓΦH ,σ making the ordinary good formal (ΦH , δH , σ)-model ♦ ord ( G, ♦ λ, ♦ i, ♦ αHp , ♦ αH ) → S ∧ a pullback of the Mumford family, we p see that they are approximate in the sense that the induced morphisms from ~ ord Spec(R/I) to the σ-stratum of X ΦH ,δH ,σ /ΓΦH ,σ (between the supports of the formal schemes) coincide. (4) The extended Kodaira–Spencer morphism (see [62, Def. 4.6.3.44]) for the ord above pullback (G, λ, i, αHp , αH ) ⊗ R∧ → S ∧ induces (cf. the proof of [62, p the Mumford family ( ♥ G,

♥

λ,

♥

i,

R

♥

αHp ,

♥ ord αHp ) ∧

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Thm. 4.6.3.43]) an isomorphism KSG/S/~S0,r

H

∼ e1 : KS(G,λ,i)/S,free → Ω S/~ S

0,rH

[d log ∞],

where KS(G,λ,i)/S,free is defined as in Definition 3.4.3.1, and where e1 e1 Ω [d log ∞] is defined by Ω , the coherent sheaf associated with S/~ S S/~ S 0,rH

0,rH

e1 the module of universal finite differentials Ω R/OF0 ,(p) [ζprH ] (see [55, Sec. 11– 12]), and by the normal crossings divisor of S = Spec(R) induced by the ~ ord (σ)/ΓΦ ,σ (as in (5) of Proposition 5.1.1.3, with Ω b1 one of Ξ H ΦH ,δH S/~ S 0,rH

e1 there replaced with Ω S/~ S

here). 0,rH

Proof. The proof of [62, Prop. 6.3.2.1] using Artin’s approximation theory (cf. [2, Thm. 1.10] and [62, Prop. 6.3.2.2]) and [28, Ch. IV, Lem. 4.2] also works here. Definition 5.1.2.2. (Compare with [62, Def. 6.3.2.5].) Let (ΦH , δH ) be a representative of an ordinary cusp label at level H (see Definition 3.2.3.8), and let σ ⊂ P+ ΦH be a nondegenerate smooth rational polyhedral cone. An ordinary good algebraic (ΦH , δH , σ)-model consists of the following data: (1) An affine scheme S = Spec(Ralg ), together with a stratification of S with strata parameterized by ΓΦH ,σ -orbits of faces of σ. ~ ord (σ)/ΓΦ ,σ making S an (2) A strata-preserving morphism S → Ξ H ΦH ,δH ~ ord (σ)/ΓΦ ,σ at the ´etale neighborhood of some geometric point x ¯ of Ξ H ΦH ,δH σ-stratum. ~ ord (σ)/ΓΦ ,σ at x Let R∧ be the completion of the strict local ring of Ξ ¯ H ΦH ,δH with respect to the ideal defining the σ-stratum. Then there is a “natural inclusion” ınat : Ralg ,→ R∧ . ord ~ ord over S as in Defini) of type M (3) A degenerating family (G, λ, i, αHp , αH H p tion 3.4.2.10, together with an embedding ıalg : Ralg ,→ R∧ , such that we have the following: (a) There are isomorphisms between the objects ΦH (G), SΦ(G) , and B(G) (see Construction 5.1.1.1), and the pullbacks of the tautological objects ~ ord (σ)/ΓΦ ,σ (see Proposition 4.2.2.5) under ΦH , S, and B over Ξ H ΦH ,δH ord ~ S → ΞΦH ,δH (σ)/ΓΦH ,σ . (b) The embedding ıalg : Ralg ,→ R∧ is close to the natural inclusion ınat in the sense that the following two morphisms Spf(R∧ , I) → ~ ord X ΦH ,δH ,σ /ΓΦH ,σ coincide over the σ-stratum: ord i. The pullback (G, λ, i, αHp , αH ) p

⊗

R∧ → S ∧ := Spec(R∧ )

Ralg ,ınat

defines an ordinary good formal (ΦH , δH )-model by the isomorphisms in (3a) above, and hence defines a canonical morphism ~ ord Spf(R∧ , I) → X ΦH ,δH ,σ /ΓΦH ,σ .

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ii. The embedding ıalg : Ralg ,→ R∧ defines a composition S∧

Spec(ıalg )

→

~ ord S→Ξ ΦH ,δH (σ)/ΓΦH ,σ ,

~ ord inducing a morphism Spf(R∧ , I) → X ΦH ,δH ,σ /ΓΦH ,σ . (c) The extended Kodaira–Spencer morphism (see [62, Def. 4.6.3.44]) induces an isomorphism KSG/S/~S0,r

∼

H

: KS(G,λ,i)/S,free → Ω1S/~S

0,rH

[d log ∞],

where KS(G,λ,i)/S,free is defined as in Definition 3.4.3.1. Proposition 5.1.2.3. (Compare with [62, Prop. 6.3.2.6].) There exist ordinary good algebraic (ΦH , δH , σ)-models such that the morphisms from them to ~ ord (σ)/ΓΦ ,σ cover the σ-stratum Ξ ~ ord ~ ord Ξ H ΦH ,δH ΦH ,δH ,σ /ΓΦH ,σ of ΞΦH ,δH (σ)/ΓΦH ,σ . Proof. The proof of [62, Prop. 6.3.2.6] works verbatim here. Remark 5.1.2.4. (Compare with [62, Rem. 6.3.2.7].) What is implicit behind Proposition 5.1.2.3 is that, although we need to approximate the (possibly infinitely many) good formal models at all geometric points of the σ-stratum, we only need ~ ord (σ). finitely many good algebraic models to cover it, by quasi-compactness of Ξ ΦH ,δH Remark 5.1.2.5. (Compare with [62, Rem. 6.3.2.8].) An ordinary good algebraic 0 (ΦH , δH , σ)-model is an ordinary good algebraic (Φ0H , δH , σ 0 )-model if and only if 0 0 0 (ΦH , δH , σ) is equivalent to (ΦH , δH , σ ) (see Remark 5.1.1.12). Remark 5.1.2.6. (Compare with [62, Rem. 6.3.2.9].) For two smooth rational 0 polyhedral cones σ, σ 0 ∈ P+ ΦH such that σ ⊂ σ , an ordinary good algebraic (ΦH , δH , σ)-model is not necessarily an ordinary good algebraic (ΦH , δH , σ 0 )-model (see [62, Rem. 6.2.5.31] and Remark 5.1.1.13). Proposition 5.1.2.7. (Compare with [62, Prop. 6.3.2.10].) Suppose x ¯ is any geometric point in the (τ mod ΓΦH ,H,σ )-stratum of an ordinary good algebraic ord ~ ord (σ)/ΓΦ ,σ , where τ (ΦH , δH , σ)-model (G, λ, i, αHp , αH ) → Spec(Ralg ) → Ξ H ΦH ,δH p ∧ is a face of σ. By pulling back to the completion Rx¯ of the strict local ring of Ralg at x ¯ with respect to the ideal defining the (τ mod ΓΦH ,H,σ )-stratum, we obtain a 0 good formal (Φ0H , δH , τ 0 )-model, where: ¯, which comes (1) Φ0H = (X 0 , Y 0 , φ0 , ϕ0−2,H , ϕ00,H ) is the pullback of ΦH to x equipped with a surjection (sX : X  X 0 , sY : Y  Y 0 ) (as in Definition 1.2.1.17) by definition of ΦH . 0 0 (2) δH is any splitting that makes (Φ0H , δH ) a representative of a cusp label. 0 0 Then there is a surjection (ΦH , δH )  (Φ0H , δH ) (the actual choice of δH does not matter). (3) τ 0 ⊂ P+ Φ0H is any nondegenerate smooth rational polyhedral cone whose image under the embedding PΦ0H ,→ PΦH induced by the surjection (sX , sY ) is the translation of τ by an element of ΓΦH .

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(This is the so-called openness of versality.) Proof. The proof of [62, Prop. 6.3.2.10] also works here (with the ingredients there replaced with their analogues above). 0 Remark 5.1.2.8. (Compare with [62, Rem. 6.3.2.15].) Suppose (Φ0H , δH , σ 0 ) is 0 a face of (ΦH , δH , σ), so that σ is identified with some face τ of σ under some 0 surjection (sX : X  X 0 , sY : Y  Y 0 ) : (ΦH , δH )  (Φ0H , δH ). Then there always exists some ordinary good algebraic (ΦH , δH , σ)-model that has a nonempty (τ mod ΓΦH ,σ )-stratum on the base scheme.

For later reference, we shall also make the following definition: Definition 5.1.2.9. (Compare with [62, Def. 6.3.2.16].) Let (ΦH , δH ) be a representative of an ordinary cusp label at level H (see Definition 3.2.3.8), and let 0 0 0 σ ⊂ P+ ΦH be a nondegenerate smooth rational polyhedral cone. Suppose (ΦH , δH , σ ) 0 is a face of (ΦH , δH , σ) such that the image of σ under the embedding PΦ0H ,→ PΦH 0 ) is induced by some surjection (sX : X  X 0 , sY : Y  Y 0 ) : (ΦH , δH )  (Φ0H , δH a ΓΦH -translation of a face τ of σ (which can be σ itself). Then we shall call the ~ ord (σ)/ΓΦ ,σ the [(Φ0 , δ 0 , τ 0 )]-stratum. (In this (τ mod ΓΦH ,σ )-stratum of Ξ H H H ΦH ,δH 0 0 0 case, [(ΦH , δH , τ )] is a face of [(ΦH , δH , σ)]; see Definition 1.2.2.19.) We shall also call the induced (τ mod ΓΦH ,σ )-strata of ordinary good formal (ΦH , δH , σ)-models and ordinary good algebraic good (ΦH , δH , σ)-models (see Definitions 5.1.1.7 and 0 5.1.2.2) their [(Φ0H , δH , τ 0 )]-strata. 5.1.3

´ Gluing in the Etale Topology

Definition 5.1.3.1. (Compare with Definition 1.2.2.13.) A compatible choice of admissible smooth rational polyhedral cone decomposition data for ~ ord is a complete set Σord = {ΣΦ }[(Φ ,δ )] of compatible choices of ΣΦ as M H H H H H in Definition 1.2.2.13, but with ΣΦH defined only for representatives (ZH , ΦH , δH ) of ordinary cusp labels (see Definition 3.2.3.8). Proposition 5.1.3.2. (Compare with [62, Prop. 6.3.3.5] and Proposition 1.2.2.17.) A compatible choice Σord of admissible smooth rational polyhedral cone decomposi~ ord exists. Moreover, each Σord for M ~ ord extends to some Σ for tion data for M H H MH (as in Definition 1.2.2.13), and we may assume that Σ is a refinement of any given collection Σ0 also inducing Σord . The same is true if we allow varying levels or twists by Hecke actions (see [62, Def. 6.4.2.8 and 6.4.3.2]). We may also assume that Σord or Σ is invariant under any choice of an open compact subgroup H0 of G(A∞ ) normalizing H. Conversely, each Σ for MH induces (by restriction to ~ ord . ordinary cusp labels) a valid Σord for M H Proof. By considering only ordinary cusp labels, which makes sense because ordinary cusp labels only surject to ordinary cusp labels (by definition; see Definition

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3.2.3.8), the same argument of the proof of [62, Prop. 6.3.3.5] (by induction on the magnitude of cusp labels) works here and shows that some Σord exists. That is, assertions in Proposition 1.2.2.17 are true when we only consider the ordinary cusps. The same argument also shows that, by starting with cone decompositions in each ~ ord and by extending them to cone decompositions at other cusp given Σord for M H labels (which may surject to either ordinary or nonordinary cusp labels), we can extend Σord to some Σ for MH , which can be a refinement of any given collection Σ0 . The last statement follows immediately from the definitions. Definition 5.1.3.3. (Compare with Definition 1.2.2.14.) A compatible choice Σord = {ΣΦH }[(ΦH ,δH )] of admissible smooth rational polyhedral cone decompo~ ord (see Definition 5.1.3.1) is projective if there is a colsition data for M H ord lection pol = {polΦH }[(ΦH ,δH )] of polarization functions labeled by representatives (ΦH , δH ) as in Definition 1.2.2.14, but with (ΣΦH and) polΦH defined only for representatives (ZH , ΦH , δH ) of ordinary cusp labels (see Definition 3.2.3.8). Proposition 5.1.3.4. (Compare with [62, Prop. 7.3.1.4] and Propositions 1.2.2.17 and 5.1.3.2.) There exists a compatible choice Σord = {ΣΦH }[(ΦH ,δH )] of admissible ~ ord (see Definition 5.1.3.1) smooth rational polyhedral cone decomposition data for M H that is projective, carrying a compatible collection of polarization functions polord as in Definition 5.1.3.3. Moreover, each such (Σord , polord ) extends to some (Σ, pol) for MH (as in Definition 1.2.2.14), and we may assume that Σ is a refinement of any given collection Σ0 also inducing Σord . The same is true if we allow varying levels or twists by Hecke actions (see [62, Def. 6.4.2.8 and 6.4.3.2]). We may also assume that Σord and polord , or Σ and pol, are invariant under any choice of an open compact subgroup H0 of G(A∞ ) normalizing H. Conversely, each (Σ, pol) for ~ ord . MH induces (by restriction to ordinary cusp labels) a valid (Σord , polord ) for M H Proof. As in the proof of Proposition 5.1.3.2, by considering only ordinary cusp labels, the same argument of the proofs of [62, Prop. 6.3.3.5 and 7.3.1.4] (by induction on the magnitude of cusp labels) works here and shows that some Σord and polord exist. The same argument also shows that, by starting with cone de~ ord and by compositions and polarization functions in each given (Σord , polord ) for M H extending them to ones at other cusp labels (which may surject to either ordinary or nonordinary cusp labels), we can extend (Σord , polord ) to some (Σ, pol) for MH , where Σ can be a refinement of any given collection Σ0 . And we may assume that these satisfy the additional requirements as in the proposition. The last statement follows immediately from the definitions. Let Σord = {ΣΦH }[(ΦH ,δH )] be any compatible choice of admissible smooth ratio~ ord . To construct the desired M ~ ord,tor nal polyhedral cone decomposition data for M H H,Σord ord,tor ord ~ ~ as an algebraic stack, it suffices to give an ´etale presentation UH  MH such ~ ord × U ~ ord is ´etale over U ~ ord via the two projections (see [62, that ~Rord := U H

H

~ ord,tor M ord H,Σ

H

H

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~ ord and Prop. A.7.1.1 and Def. A.7.1.3]). Equivalently, it suffices to construct the U H ord,tor ord ~R that satisfy the required groupoid relations, which then realizes M ~ as the H H,Σord ord ord ord ord ~ ~ ~ ~ quotient of U by R . Let us first explain our choices of U and R , then H

H

H

H

show that they have the desired properties. Construction 5.1.3.5. (Compare with [62, Constr. 6.3.3.1 and 6.3.3.9].) We shall ~ ord and a stratification on it as follows: construct U H (1) Choose a complete set of (mutually inequivalent) representatives (ΦH , δH ) of ordinary cusp labels at level H. This is a finite set because there is already a finite set of representatives for all cusp labels (including nonordinary ones, when we constructed toroidal compactifications for MH ; see the explanation in [62, Constr. 6.3.3.1]). (2) For each (ΦH , δH ) chosen above, choose a complete set of (mutually inequivalent) representatives σ in ΣΦH /ΓΦH , where ΣΦH is the ΓΦH -admissible smooth rational polyhedral cone decomposition chosen in Σord . This gives a complete set of representatives (ΦH , δH , σ) of equivalence classes [(ΦH , δH , σ)] defined in Definition 1.2.2.10 such that the cusp label [(ΦH , δH )] is ordinary. This is a finite set by the ΓΦH -admissibility (see Definition 1.2.2.4) of each ΣΦH . (3) For each representative (ΦH , δH , σ) above that satisfies moreover σ ⊂ P+ ΦH , choose finitely many ordinary good algebraic (ΦH , δH , σ)-models Spec(Ralg ) (see Definition 5.1.2.2) such that the corresponding ´etale mor~ ord phisms from the various Spec(Ralg /I)’s to the σ-stratum Ξ ΦH ,δH ,σ /ΓΦH ,σ of ord ~ ΞΦH ,δH (σ)/ΓΦH ,σ , where I denotes the ideal of Ralg defining the σ-stratum of Spec(Ralg ), cover the whole σ-stratum (see Proposition 5.1.2.3 and Remark 5.1.2.4). ~ ord This is possible by the quasi-compactness of Ξ ΦH ,δH ,σ /ΓΦH ,σ , because ΓΦH ,σ is finite (by [62, Rem. 6.2.5.26 and Lem. 6.2.5.27], because the cone decomposition ΣΦH is chosen such that [62, Cond. 6.2.5.25] is satisfied), ~ ord and because Ξ ΦH ,δH ,σ is a torus torsor over an abelian scheme torsor over a ~ ord,ZH separated and of finite type finite ´etale cover of the algebraic stack M H over ~S0,rH = Spec(OF0 ,(p) [ζprH ]) (see Section 4.2 and Theorem 3.4.2.5). (4) Let us form the scheme   disjoint union of the (finitely many) ~ ord = ordinary good algebraic (ΦH , δH , σ)-models , U H Spec(Ralg ) chosen above (smooth over ~S0,rH = Spec(OF0 ,(p) [ζprH ])) which comes equipped with a natural stratification labeled as follows: On an ordinary good algebraic (ΦH , δH , σ)-model Spec(Ralg ) used ~ ord above, its stratification inherited from in the construction of U H ord 0 ~ ΞΦH ,δH (σ)/ΓΦH ,σ can be relabeled using equivalence classes [(Φ0H , δH , τ 0 )]

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(see Definition 1.2.2.10) following the recipe in Definition 5.1.2.9, which are faces of [(ΦH , δH , σ)] (see Definition 1.2.2.19). Then we define the strati~ ord to be induced by those on the ordinary fication on the disjoint union U H good algebraic models. By the compatibility of the choice of Σord (see Definitions 1.2.2.13 and 0 5.1.3.1), we know that in each representative (Φ0H , δH , τ 0 ) of each face 0 0 0 0 [(ΦH , δH , τ )] of [(ΦH , δH , σ)], the cone τ is in the cone decomposition ΣΦ0H we have in Σord . Hence, we may label all the strata by the equivalence clas~ ord . For simplicity, we ses of triples we have taken in the construction of U H ~ ord , which we denote by call the [(0, 0, {0})]-stratum the [0]-stratum of U H ~ ord,[0] . U H (This finishes Construction 5.1.3.5.) ord The ordinary good algebraic models (G, λ, i, αHp , αH ) over the various p ord ) of Spec(Ralg )’s define (by taking union) a degenerating family (G, λ, i, αHp , αH p ord ord ~ ~ type MH over UH as in Definition 3.4.2.10, whose restriction to the [0]-stratum ~ ord,[0] is a tuple (G[0] , λ[0] , i[0] , αHp , αord ) parameterized by M ~ ord (see ConvenU Hp

H

H

~ ord,[0] → M ~ ord . This mortion 3.4.2.9). This determines a canonical morphism U H H ord,[0] ~ phism is ´etale because UH is locally of finite presentation, and the morphism ~ ord,[0] by the calcu~ ord,[0] → M ~ ord is formally ´etale at every geometric point of U U H H H lation of Kodaira–Spencer morphisms (using (3c) of Definition 5.1.2.2, [62, Thm. ~ ord,[0] → M ~ ord (sur4.6.3.16], and Proposition 3.4.3.3). As a result, the morphism U H H ord ~ ord ~ . This identifies M jective by definition) defines an ´etale presentation of M H H ~ ord,[0] by the ´etale groupoid ~Rord,[0] over U ~ ord,[0] defined by with the quotient of U H H H the representable functor ~Rord,[0] := Isom~ ord,[0] H U H

× ~ S0,r

∗ [0] ord , λ[0] , i[0] , αHp , αH ), ord,[0] ( pr1 (G ~ p UH

H

(5.1.3.6)

ord pr∗2 (G[0] , λ[0] , i[0] , αHp , αH )), p

~ ord,[0] × U ~ ord,[0] → U ~ ord,[0] denote, respectively, the two prowhere pr1 , pr2 : U H H H ~ S0,rH

jections. Proposition 5.1.3.7. (Compare with [62, Prop. 6.3.3.11].) Suppose R is a noetherian normal complete local domain with fraction field K and algebraically closed residue field k. Assume that Spec(R) is flat over ~S0,rH , and that we have a degenerating ‡ ord,‡ ~ ord over Spec(R) as in Definition 3.4.2.10. family (G‡ , λ‡ , i‡ , αH ) of type M p , αH H p Then the following conditions are equivalent: ~ ord sending the generic point (1) There exists a morphism Spec(R) → U H ‡ ord,‡ Spec(K) to the [0]-stratum such that (G‡ , λ‡ , i‡ , αH ) → Spec(R) p , αH p ord ord ~ is the pullback of (G, λ, i, αHp , α ) → U . Hp

H

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Compactifications with Ordinary Loci ‡ ord,‡ (2) The degenerating family (G‡ , λ‡ , i‡ , αH ) → Spec(R) is the pullback p , αH p ♥ ♥ ♥ ♥ ♥ ord ~ ord of the Mumford family ( G, λ, i, αHp , αH ) → X ΦH ,δH ,σ /ΓΦH ,σ p ord ~ via a morphism Spf(R) → XΦH ,δH ,σ /ΓΦH ,σ , or equivalently a morphism ~ ord (σ)/ΓΦ ,σ , for some (ΦH , δH , σ) (which can be assumed Spec(R) → Ξ H ΦH ,δH ~ ord ). to be a triple used in the construction of U H

‡ ord,‡ (3) The degenerating family (G‡ , λ‡ , i‡ , αH ) over Spec(R) defines an obp , αH p ject of the essential image of DEGPEL,Mord (R) → DEGPEL,... (R), which M ord H H corresponds to a tuple \,ord,‡ ]) (B ‡ , λB ‡ , iB ‡ , X ‡ , Y ‡ , φ‡ , c‡ , c∨,‡ , τ ‡ , [αH

in the essential image of DDPEL,Mord (R) → DDPEL,... (R) under (4.1.6.4) M ord H H in Theorem 4.1.6.2. Then we have a fully symplectic-liftable admissible \,ord,‡ ], which is compatible with the filtration filtration Z‡H determined by [αH D as in Definition 3.2.3.1. Moreover, the ´etale sheaves X ‡ and Y ‡ are necessarily constant, because the base scheme R is strict local. Hence, it makes sense to say that we also have a uniquely determined torus argument Φ‡H at level H for Z‡H . On the other hand, we have objects ΦH (G‡ ), SΦH (G‡ ) , and B(G‡ ), which

define objects Φ‡H , SΦ‡ , and in particular, B ‡ : SΦ‡ → Inv(R) over the H H special fiber. If υ : K × → Z is any discrete valuation defined by a height-one prime of . R, then υ ◦ B ‡ : SΦ‡ → Z makes sense and defines an element of S∨ Φ‡ H

H

‡ Then the condition is that, for some (and hence every) choice of δH ma‡ ‡ ‡ king (ZH , ΦH , δH ) a representative of an ordinary cusp label (see Definition 3.2.3.8), there is a cone σ ‡ in the cone decomposition ΣΦ‡ of PΦ‡ (given H

H

by the choice of Σord ; cf. Definition 5.1.3.1) such that the closure σ ‡ of σ ‡ ‡ in (SΦ‡ )∨ R contains all υ ◦ B obtained in this way. H

Proof. The implication from (1) to (2) is clear, as the morphism from Spec(R) to ~ ord necessarily factors through the completion of some strict local ring of U ~ ord . U H H The implication from (2) to (3) is analogous to Proposition 5.1.1.3. For the implication from (3) to (1), suppose there exists a cone σ ‡ in the cone decomposition ΣΦ‡ of PΦ‡ such that σ ‡ contains all the υ ◦ B ‡ ’s. Up to replacing H

H

σ ‡ with another cone in ΣΦ‡ , let us assume that σ ‡ is a minimal one. Then H

some linear combination with positive coefficients of the υ ◦ B ‡ ’s lie in σ ‡ , the interior of σ ‡ . On the other hand, by the positivity condition of τ ‡ , such a linear combination with positive coefficients must be positive definite on Y ‡ . Hence, σ ‡ ⊂ ~ ord P+‡ . Then there exists a unique triple (ΦH , δH , σ) chosen in the construction U H ΦH

‡ (see Construction 5.1.3.5) such that (Φ‡H , δH , σ ‡ ) and (ΦH , δH , σ) are equivalent (see Definition 1.2.2.10).

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~ ord ~ ord ~ ord,ZH is finite ´etale over the base change M Since M Hh ,rH of MHh (defined by H ...ord MHh and MHh as in Theorem 3.4.2.5) to ~S0,rH , since B ‡ is defined over R, and since R is noetherian, normal, and flat over ~S0,rH , as pointed out in Remark 3.4.2.12, ord,‡ ord,‡ the tuple (B ‡ , λB ‡ , iB ‡ , ϕ‡−1,Hp , ϕord,‡ −1,Hp ) and the ΓΦ‡H -orbit of (ϕ−2,H , ϕ0,H ) de~ ord,ZH as soon as its restriction to Spec(K) defines fine a morphism Spec(R) → M H ord,Z H ~ . By Proposition 4.2.1.46, the degeneration daa morphism Spec(K) → M H

‡ ord,‡ tum (without the positivity condition) associated with (G‡ , λ‡ , i‡ , αH ) → p , αH p ord ord ~ ~ Spec(R) determines (by the universal properties of Ξ ΦH ,δH and CΦH ,δH ) a morphism ord ~ Spec(K) → ΞΦH ,δH , whose composition with the (relatively affine) structural mor~ ord ~ ord ~ ord phism Ξ ΦH ,δH → CΦH ,δH extends to a morphism Spec(R) → CΦH ,δH . By Proposi~ ord tion 4.2.2.8 and the assumption on the υ ◦ B ‡ ’s, the morphism Spec(K) → Ξ ΦH ,δH ord ‡ ~ extends to a morphism Spec(R) → Ξ (σ), which identifies B(G ) with the ΦH ,δH

pullback of B under an identification of ΦH (G‡ ) with the pullback of ΦH . The ambiguity of the identifications can be removed (or rather intrinsically incorporated) ~ ord (σ)/ΓΦ ,σ . Hence, we have a uniquely determined if we form the quotient Ξ H ΦH ,δH ~ ord (σ)/ΓΦ ,σ , which is independent of strata-preserving morphism Spec(R) → Ξ H ΦH ,δH ‡ the identification of ΦH with ΦH we have chosen. This determines a morphism ~ ord Spf(R) → X ΦH ,δH ,σ /ΓΦH ,σ as in (2). Let us denote the image of the closed point of Spec(R) by x, which necessarily lies in the σ-stratum (thanks to the minimality of the choice of σ ‡ ). By construction, there is some ordinary good algebraic (ΦH , δH , σ)-model ~ ord such that (G, λ, i, αHp , αHp ) → Spec(Ralg ) used in the construction of U H ord ~ the image of the structural morphism Spec(Ralg ) → ΞΦH ,δH (σ)/ΓΦH ,σ contains x. Let Ralg ∧ be the completion of Ralg with respect to the ideal defining the σ-stratum of Spec(Ralg ), and let I ∧ be the induced ideal of definition. Then the ~ ord (σ)/ΓΦ ,σ induces a formally ´etale mor´etale morphism Spec(Ralg ) → Ξ H ΦH ,δH ∧ ∧ ~ ord phism Spf(Ralg , I ) → X /Γ . By formal ´etaleness, the morphism ΦH ,σ ΦH ,δH ,σ ord ~ Spf(R) → X ΦH ,δH ,σ /ΓΦH ,σ can be uniquely lifted to a morphism Spf(R) → Spf(Ralg ∧ , I ∧ ). The underlying morphism Spec(R) → Spec(Ralg ∧ ) identifies the ‡ ord,‡ degeneration datum associated with (G‡ , λ‡ , i‡ , αH ) → Spec(R) with the p , αH p ord degeneration datum associated with the pullback of (G, λ, i, αHp , αH ) ⊗ Ralg ∧ → p Ralg

Spec(Ralg ∧ ). Hence, the morphism Spec(R) → Spec(Ralg ∧ ) → Spec(Ralg ) identi‡ ord,‡ ord fies (G‡ , λ‡ , i‡ , αH ) → Spec(R) with the pullback of (G, λ, i, αHp , αH ) → p , αH p p Spec(Ralg ), as desired. The key to the gluing process is the following: Proposition 5.1.3.8. (Compare with [62, Prop. 6.3.3.13].) The two projections ~ ord × U ~ ord to U ~ ord are ´ from ~Rord etale. H := UH H H ~ ord,tor M ord H,Σ

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Proof. The same argument of the proof of [62, Prop. 6.3.3.13] works here, with good formal and algebraic models replaced with ordinary good formal and algebraic models, with algebraic stacks such as ΞΦH ,δH (σ)/ΓΦH ,σ replaced with ~ ord (σ)/ΓΦ ,σ , with the openness of versality provided by Proposition 5.1.2.7 Ξ H ΦH ,δH (instead of [62, Prop. 6.3.2.10]), and with the theory of degeneration provided by Theorems 4.1.5.27 and 4.1.6.2 (instead of [62, Thm. 5.3.1.19]). Corollary 5.1.3.9. (Compare with [62, Cor. 6.3.3.14].) The scheme ~Rord H over ~ ord defines an ´etale groupoid space (see [62, Def. A.5.1.2]), which extends the ´etale U H ord,[0] ~ ord,[0] . The scheme ~Rord is finite over U ~ ord × U ~ ord , groupoid space ~RH over U H H H H ~ S0,rH

~ ord /~Rord defines an algebraic stack separated and hence (by [62, Lem. A.7.2.9]) U H H ~ over S0,rH = Spec(OF0 ,(p) [ζprH ]). Proof. The same argument of the proof of [62, Cor. 6.3.3.14] works here, except ord,[0] ~ ord we need not only the property ~ ord × U over U that for the finiteness of ~RH H H ~ S0,rH

of the Isom functor of abelian schemes as in [62, Sec. 2.3.4, Cond. 2], but also the fact that the abelian schemes are ordinary (so that the ordinary level structures are defined by isomorphisms between finite ´etale group schemes, or rather their duals). Definition 5.1.3.10. (Compare with [62, Def. 6.3.3.15].) The separated alge~ ord /~Rord (see [62, Prop. A.7.1.1 and Def. A.7.1.3]) will be denoted braic stack U H H ord,tor ~ ~ ord,tor by MH (or M , to emphasize its dependence on the compatible choice H,Σord Σord = {ΣΦH }[(ΦH ,δH )] of cone decompositions). Corollary 5.1.3.11. (Compare with [62, Cor. 6.3.3.16].) Both the degenerating ord ~ ord and the stratification over U ~ ord (see Construction family (G, λ, i, αHp , αH )→U H H p ord,tor ~ 5.1.3.5) descend to M ord , which we again denote by the same notation. This H,Σ

~ ord as the [0]-stratum in the stratification, and identifies the restriction of realizes M H ord ~ ord with the tautological tuple over M ~ ord . (G, λ, i, αHp , αH ) to M H H p Proof. The same argument of the proof of [62, Cor. 6.3.3.16] works here. ~ ord is smooth and Remark 5.1.3.12. (Compare with [62, Prop. 6.3.3.17].) Since U H of finite type over ~S0,rH = Spec(OF0 ,(p) [ζprH ]), by Proposition 5.1.3.8 and Corollary ~ ord,tor 5.1.3.9, the algebraic stack M is separated, smooth, and of finite type over H,Σord ~S0,r . But M ~ ord,tor ~ (See Proposition 6.3.2.2 ord is almost never proper over S0,r . H,Σ

H

H

below.) 5.2

Partial Toroidal Compactifications of Ordinary Loci

In this section, let H, Hp , Hp , and r be as in beginning of Section 3.3.5, and let rH be as in Definition 3.4.2.1.

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Main Statements

~ ord can be described as follows: The partial toroidal compactifications of M H Theorem 5.2.1.1. (Compare with [62, Thm. 6.4.1.1] and Theorem 1.3.1.3.) With settings as above, to each compatible choice Σord = {ΣΦH }[(ΦH ,δH )] of admissible smooth rational polyhedral cone decomposition data as in Definition 5.1.3.1, there ~ ord,tor = M ~ ord,tor is associated an algebraic stack M separated, smooth, and of finite H H,Σord ~ type over S0,rH = Spec(OF0 ,(p) [ζprH ]) (see Definition 2.2.3.3), which is an alge~ ord braic space when Hp is neat (see [89, 0.6] or [62, Def. 1.4.1.8]), containing M H

as an open fiberwise dense subalgebraic stack, together with a degenerating family ord ~ ord over M ~ ord,tor (as in Definition 3.4.2.10) such that (G, λ, i, αHp , αH ) of type M H H p we have the following: ord (1) The restriction (GM ~ ord , λM ~ ord , iM ~ ord , αHp , αHp ) of the degenerating family H H H ~ ord is the tautological tuple over M ~ ord (see Conven(G, λ, i, αHp , αord ) to M Hp

H

H

tion 3.4.2.9). ~ ord,tor has a stratification by locally closed subalgebraic stacks (2) M H a ~Zord ~ ord,tor = M [(ΦH ,δH ,σ)] , H [(ΦH ,δH ,σ)]

with [(ΦH , δH , σ)] running through a complete set of equivalence classes of (ΦH , δH , σ) (as in Definition 1.2.2.10) with [(ΦH , δH )] an ordinary cusp laord bel (as in Definition 3.2.3.8) and with σ ⊂ P+ . (Here ΦH and σ ∈ ΣΦH ∈ Σ ` ZH is suppressed in the notation by our convention. The notation “ ” only means a set-theoretic disjoint union. The algebro-geometric structure is still ~ ord,tor .) that of M H 0 , σ 0 )]-stratum ~Zord In this stratification, the [(Φ0H , δH 0 ,σ 0 )] lies in the clo[(Φ0H ,δH ord sure of the [(ΦH , δH , σ)]-stratum ~Z if and only if [(ΦH , δH , σ)] is [(ΦH ,δH ,σ)]

0 a face of [(Φ0H , δH , σ 0 )] as in Definition 1.2.2.19 (see also Remark 5.1.2.8). The analogous assertion holds after pulled back to fibers over ~S0,rH . ~ The [(ΦH , δH , σ)]-stratum ~Zord [(ΦH ,δH ,σ)] is smooth over S0,rH and isomorphic ~ ord to the support of the formal algebraic stack X /ΓΦ ,σ for every reΦH ,δH ,σ

H

presentative (ΦH , δH , σ) of [(ΦH , δH , σ)], where the formal algebraic stack ~ ord X ΦH ,δH ,σ (before quotient by ΓΦH ,σ , the subgroup of ΓΦH formed by elements mapping σ to itself; see [62, Def. 6.2.5.23]) admits a canonical struc~ ord (σ) (along its ture as the completion of an affine toroidal embedding Ξ ΦH ,δH ~ ord ~ ord σ-stratum Ξ ΦH ,δH ,σ ) of a torus torsor ΞΦH ,δH over an abelian scheme tor~ ord,ΦH of the regular algebraic stack ~ ord sor C etale cover M ΦH ,δH over a finite ´ H ~ ord,ZH separated, smooth, and of finite type over ~S0,r (as in Propositions M H H 4.2.1.29, 4.2.1.30, and 4.2.1.37). (Note that ZH and the isomorphism class ~ ord,ZH depend only on the class [(ΦH , δH , σ)], but not on the choice of of M H the representative (ΦH , δH , σ).)

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~ ord is an open fiberwise dense stratum in this stratification. In particular, M H ~ ord in M ~ ord,tor (with its reduced structure) is a relative (3) The complement of M H H ~ ord with normal crossings, such that each irreducible comCartier divisor D ∞,H ~ ord,tor − M ~ ord is open dense in an intersection of irponent of a stratum of M H H ~ ord (including possible self-intersections). When reducible components of D ∞,H ~ ord have no self-intersections Hp is neat, the irreducible components of D ∞,H (cf. Condition 1.2.2.9, [62, Rem. 6.2.5.26], and [28, Ch. IV, Rem. 5.8(a)]). (4) The extended Kodaira–Spencer morphism [62, Def. 4.6.3.44] for G → ~ ord,tor induces an isomorphism M H KSG/M ~ ord,tor /~ S0,r H

∼

H

1 : KS(G,λ,i)/M ~ ord,tor ,free → ΩM ~ ord,tor /~ S H

H

(see Definition 3.4.3.1). Here the sheaf Ω1M ~ ord,tor /~ S H

0,rH

[d log ∞],

[d log ∞] is the sheaf 0,rH

~ ord,tor over ~S0,r , with respect to the of modules of log 1-differentials on M H H ord ~ relative Cartier divisor D ∞,H with normal crossings. (5) For every representative (ΦH , δH , σ) of [(ΦH , δH , σ)], the formal completion ~ ord,tor )∧ ~ ord,tor along the [(ΦH , δH , σ)]-stratum ~Zord (M of M H H ~ [(ΦH ,δH ,σ)] Zord [(ΦH ,δH ,σ)]

~ ord is canonically isomorphic to the formal algebraic stack X ΦH ,δH ,σ /ΓΦH ,σ . This isomorphism respects stratifications in the sense that, given any ´etale (i.e., formally ´etale and of finite type; see [35, I, ~ ord 10.13.3]) morphism Spf(R, I) → X ΦH ,δH ,σ /ΓΦH ,σ inducing a morphism ord ~ Spec(R) → ΞΦH ,δH (σ)/ΓΦH ,σ , the stratification of Spec(R) inherited from ~ ord (σ)/ΓΦ ,σ (see Proposition 5.1.1.3 and Definition 5.1.2.9) makes Ξ H ΦH ,δH ~ ord,tor a strata-preserving morphism. the induced morphism Spec(R) → M H ord ~ ord,tor ) over M The pullback of the degenerating family (G, λ, i, αHp , αH H p ord,tor ∧ ♥ ♥ ♥ ♥ ♥ ord ~ p to (M ) is the Mumford family ( G, λ, i, αH , α ) H

Hp

~ Zord [(Φ

H ,δH ,σ)]

~ ord over X ΦH ,δH ,σ /ΓΦH ,σ (see Definition 4.2.2.21) after we identify the bases ord using the isomorphism. (Here both the pullback of (G, λ, i, αHp , αH ) and p ♥ ♥ ♥ ♥ ♥ ord the Mumford family ( G, λ, i, αHp , αHp ) are considered as relative schemes with additional structures; cf. [37].) (6) Let S be an irreducible noetherian normal scheme flat over ~S0,rH , and † ord,† suppose that we have a degenerating family (G† , λ† , i† , αH ) of type p , αH p † ord,† ord † † † ~ M over S as in Definition 3.4.2.10. Then (G , λ , i , α p , α )→S H

H

Hp

ord ~ ord,tor via a (necessarily unique) is the pullback of (G, λ, i, αHp , αH )→M H p ~ ord,tor (over ~Sord ) if and only if the following condition is morphism S → M H

0

satisfied at each geometric point s¯ of S: Consider any dominant morphism Spec(V ) → S centered at s¯, where V is a complete discrete valuation ring with fraction field K, algebraically closed ‡ ord,‡ residue field k, and discrete valuation υ. Let (G‡ , λ‡ , i‡ , αH ) → p , αH p † ord,† Spec(V ) be the pullback of (G† , λ† , i† , αH ) → S. This pullback p , αH p

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family defines an object in the essential image of DEGPEL,Mord (V ) → H ... DEGPEL, M ord (V ), which corresponds to a tuple H \,ord,‡ (B ‡ , λB ‡ , iB ‡ , X ‡ , Y ‡ , φ‡ , c‡ , c∨,‡ , τ ‡ , [αH ])

(V ) → DDPEL,... (V ) under (4.1.6.4) in the essential image of DDPEL,Mord M ord H H in Theorem 4.1.6.2. Then we have a fully symplectic-liftable admissible \,ord,‡ filtration Z‡H determined by [αH ]. Moreover, the ´etale sheaves X ‡ and ‡ Y are necessarily constant, because the base ring R is strict local. Hence, it makes sense to say we also have a uniquely determined torus argument Φ‡H at level H for Z‡H . On the other hand, we have objects ΦH (G‡ ), SΦH (G‡ ) , and B(G‡ ) (see [62, Constr. 6.3.1.1]), which define objects Φ‡H , SΦ‡ , and in particular H

B ‡ : SΦ‡ → Inv(V ) over the special fiber. Then υ ◦ B ‡ : SΦ‡ → Z defines H H , where υ : Inv(V ) → Z is the homomorphism induced an element of S∨ Φ‡ H

by the discrete valuation of V . Then the condition is that, for each Spec(V ) → S as above (centered at ‡ s¯), and for some (and hence every) choice of δH , there is a cone σ ‡ in the cone decomposition ΣΦ‡ of PΦ‡ (given by the choice of Σord ; cf. Definition H

H

5.1.3.1) such that σ ‡ contains all υ ◦ B ‡ obtained in this way. (7) If Σord extends to a compatible choice Σ of admissible smooth rational polyhedral cone decomposition data for MH (cf. Proposition 5.1.3.2), then there is a canonical open immersion ~ ord,tor M ⊗ Q ,→ Mtor H,Σ,rH H,Σord

(5.2.1.2)

Z

(see Definition 2.2.3.4) over S0,rH extending the canonical isomorphism ord ∼ Mord H = MH,rH over S0,rH (see the definition of MH in Theorem 3.4.2.5), ord,tor ord ~ is canonically detersuch that the pullback of (G, λ, i, αHp , αHp ) → M H tor mined by the pullback of (G, λ, i, αH ) → MH,Σ (see Theorem 1.3.1.3) in the sense that the triples (G, λ, i) are isomorphic over S0,rH , and in the sense ord that the pullback of αH determines the pullback of (αHp , αH ) ⊗ Q as in p Z

Proposition 3.3.5.1. The open immersion (5.2.1.2) induces isomorphisms ∼ ~Zord [(ΦH ,δH ,σ)] ⊗ Q → Z[(ΦH ,δH ,σ)],rH

(5.2.1.3)

Z

(see Definition 2.2.3.4) when the cusp label [(ΦH , δH )] is ordinary; otherwise, the pullback of Z[(ΦH ,δH ,σ)],rH under (5.2.1.2) is empty. Remark 5.2.1.4. Although statement (6) resembles a valuative criterion, it does ~ ord,tor is proper as in the proof of [62, Prop. 6.3.3.17], because the not imply that M H ~ ord requires the condition (5) in condition of being a degenerating family of type M H Definition 3.4.2.10, which does not hold in general (cf. Remark 3.4.2.11).

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Proof of Theorem 5.2.1.1. The proof is almost identical to that of [62, Thm. 6.4.1.1]. However, since this is one of the most important theorems in this work, we repeat the arguments here for the sake of certainty. ~ ord and ~Rord be constructed (noncanonically) as in Section 5.1.3, and let us Let U H H ~ ord,tor = M ~ ord,tor take the separated algebraic stack M to be the groupoid quotient H H,Σord ord ord ~ /~R U (see [62, Prop. A.7.1.1 and Def. A.7.1.3]) as in Definition 5.1.3.10. H

H

Statements (1) and (2) follow from Corollaries 5.1.3.9 and 5.1.3.11. Statements (3) and (4) are ´etale local in nature, and hence are inherited from the ´etale presen~ ord of M ~ ord,tor (with descent data over ~Rord ) by construction. tation U H H H Let us prove statement (6) by explaining why it is essentially a restatement of Proposition 5.1.3.7. Suppose we have a degenerating family † ord,† † † † (G , λ , i , αHp , αHp ) → S as in the statement. Then there is an open dense subscheme S1 of S such that the restriction of the family defines an object parame~ ord (cf. Convention 3.4.2.9), together with a morphism S1 → M ~ ord . terized by M H H ord,tor ~ . The question is whether this morphism extends to a morphism S → M H By [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5], if this is the case, † ord,† then (G† , λ† , i† , αH ) → S is isomorphic to the pullback of the tautological p , αH p ord ~ ord,tor under this morphism, and the condition in the tuple (G, λ, i, αHp , α ) → M Hp

H

statement certainly holds. Conversely, assume that the condition holds. Since all objects involved are locally of finite presentation, we can apply [62, Thm. 1.3.1.3] and assume that S is excellent. Since extendability is a local question (because ~ ord,tor is separated over ~S0,r ), we can work with U ~ ord and apply Proposition M H H H † ord,† † † † 5.1.3.7 (to pullbacks of (G , λ , i , αHp , αHp ) → S to the completions of local rings of S). Next, let us prove statement (5). By statement (6) we have just proved, we ~ ord,tor . (More ~ ord know that there is a unique morphism from X ΦH ,δH ,σ /ΓΦH ,σ to MH ~ ord precisely, we apply statement (6) to an ´etale covering of X ΦH ,δH ,σ /ΓΦH ,σ by affine formal schemes with descent data.) This induces a canonical morphism ~ ord,tor )∧ ~ ord X . For an inverse morphism, note that by ΦH ,δH ,σ /ΓΦH ,σ → (MH ~ Zord [(ΦH ,δH ,σ)]

~ ord alconstruction there is a canonical morphism from the formal completion of U H ord ~ ong its [(ΦH , δH , σ)]-stratum to XΦH ,δH ,σ /ΓΦH ,σ . Since this canonical morphism is determined by the degeneration data associated with the pullback of the tautologiord cal tuple (G, λ, i, αHp , αH ) to the completion, and since the two pullbacks of the p tautological tuple to ~Rord are tautologically isomorphic by definition of ~Rord H H , we see ord ~ that the morphism from the completion of UH along its [(ΦH , δH , σ)]-stratum to ~ ord,tor )∧ ~ ord ~ ord X /ΓΦ ,σ descends to a morphism (M →X /ΓΦ ,σ . ΦH ,δH ,σ

H

H

~ Zord [(Φ

H ,δH ,σ)]

ΦH ,δH ,σ

H

Then it follows from the constructions that these two canonical morphisms are inverses of each other. ~ ord,tor , the Now, let us prove statement (7). In every step of our construction of M H characteristic zero fiber of the boundary charts we have used are the pullback from S0 = Spec(F0 ) to S0,rH = Spec(F0 [ζprH ]) of the corresponding boundary charts of

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~ ord,tor are also compatible with the corresponding Mtor H,Σ , and the gluing process for MH tor gluing process for MH,Σ . (See Sections 4.2 and 5.1.) The only difference is that we ~ ord,tor , while we need to only consider ordinary cusp labels in the construction for M H tor consider all cusp labels in the construction for MH,Σ . Hence, we have a canonical ~ ord,tor open immersion M ⊗ Q ,→ Mtor H,Σ,rH respecting their natural stratifications. H,Σord Z

(Without matching the stratifications explicitly, it still follows from (6) of Theorem ~ ord,tor 1.3.1.3 and statement (6) of this theorem that M ⊗ Q is canonically isomorphic H,Σord Z

to the open subalgebraic stack of Mtor H,Σ,rH formed by its strata associated with ordinary cusp labels (with cones), because they enjoy the same universal properties. However, the proofs of these universal properties are also based on the corresponding boundary chart constructions and gluing processes, from which we can directly deduce the full statement we need.) ~ ord Finally, suppose that Hp is neat. Then H = Hp Hp is neat, and X ΦH ,δH ,σ is a formal algebraic space by [62, Lem. 6.2.5.27] because we have assumed in Definition 5.1.3.1 that each cone decomposition ΣΦH in Σord satisfies Condition 1.2.2.9. By ~ ord,tor have no nontrivial automorstatements (2) and (5), it follows that points of M H ~ ord,tor ~ ord,tor × M ~ ord,tor → M phisms. Since the diagonal 1-morphism ∆M ~ ord,tor : MH H H H

~ S0,rH

~ ord,tor is an is finite (by Corollary 5.1.3.9), it must be a closed immersion. Hence, M H algebraic space when Hp is neat, as desired. Remark 5.2.1.5. (Compare with Remarks 1.1.2.1, 1.3.1.4, and 3.4.2.8.) Suppose we have chosen another lattice L0 in L ⊗ Q which nevertheless satisfies L ⊗ Z(p) = Z

Z

~ ord carries the corresponding abelian scheme A0 (with additional L0 ⊗ Z(p) , so that M H Z

0 ~ ord,tor structures) as in Remark 3.4.2.8, with a Z× (p) -isogeny f : A → A . Since MH,Σord ~ ord,tor is noetherian normal, since the tautological semi-abelian scheme G → M ord is H,Σ

ordinary, and since A = GM ~ ord , by Lemma 3.1.3.2 and by [92, IX, 1.4], [28, Ch. I, H

ext Prop. 2.7], or [62, Prop. 3.3.1.5], f extends to a Z× : G → G0 , and the (p) -isogeny f ord additional structures λ, i, αHp , and αH p of G naturally induce the additional strucord,0 0 0 0 0 tures λ , i , αHp , and αHp of G , which extend those of A0 . Hence, the Z× (p) -isogeny ord,tor ord ~ ~ , and carries well-defined addiclass of G over M ord extends that of A over M H

H,Σ

ord,0 0 ~ ord,tor tional structures. (It can be verified that (G0 , λ0 , i0 , αH )→M satisfies p , αH H,Σord p 0 the corresponding universal property defined by L and the corresponding collection of cone decompositions as in (6) of Theorem 5.2.1.1, so that the theory does not really depend on the choice of L within L ⊗ Z(p) . Then we can define a collection Z

~ ord,tor {M } indexed by H as in Remark 3.4.2.8 and collections Σord for the correH,Σord H ~ ord , carrying a Hecke action as in Proposition 5.2.2.2 below). However, sponding M H

as mentioned in Remark 3.4.2.8, modifying the choice of L ⊗ Zp and its filtration D will make the theory much more complicated.

Z

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5.2.2

Hecke Actions

Definition 5.2.2.1. (Compare with [62, Def. 6.4.3.3].) Suppose we have an element ∞ g = (g0 , gp ) ∈ G(A∞,p ) × Pord D (Qp ) ⊂ G(A ) (see Definition 3.2.2.7), and suppose ˆ such that H0 ⊂ gHg −1 , and we have two open compact subgroups H and H0 of G(Z) 0 such that H and H are of standard form as in Definition 3.2.2.9. Suppose moreover ~ ord : M ~ ord0 → M ~ ord that gp satisfies the conditions given in Section 3.3.4, so that [g] H

H

is defined (see Proposition 3.4.4.1). Let Σord = {ΣΦH }[(ΦH ,δH )] and Σord,0 = {Σ0Φ0 0 }[(Φ0 0 ,δ0 0 )] be compatible choices of admissible smooth rational polyhedral H H H ~ ord and M ~ ord0 , respectively. We say that Σord,0 is a cone decomposition data for M H

H

0 g-refinement of Σord if, for each g-assignment (fX , fY ) : (ΦH0 , δH0 ) →g (Φ0H , δH ) 0 0 of a representative (ΦH , δH ) of cusp label at level H to a representative (ΦH0 , δH0 ) of cusp label at level H0 as in [62, Def. 5.4.3.9], the first cusp label (and hence 0 0 both of them) being ordinary as in Definition 3.2.3.8, the triple (Φ0H0 , δH 0 , ΣΦ0 ) H0 is a g-refinement of (ΦH , δH , ΣΦH ) (under the pair of isomorphisms (fX , fY )) as in [62, Def. 6.4.3.2]; namely, the cone decomposition Σ0Φ0 0 of PΦ0H0 is a refinement H of the cone decomposition ΣΦH of PΦH under the identification between PΦ0H0 and PΦH defined by (fX , fY ). We say that Σord,0 is g-induced by Σord if each Σ0Φ0 0 H above is induced by ΣΦH under the identification between PΦ0H0 and PΦH defined by (fX , fY ). (This might not be possible because the running assumptions on cone decompositions, such as smoothness, might be incompatibly defined.)

Proposition 5.2.2.2. (Compare with Propositions 1.3.1.15 and 3.4.4.1.) Let g = (g0 , gp ), H, H0 , Σord = {ΣΦH }[(ΦH ,δH )] , and Σord,0 = {Σ0Φ0 0 }[(Φ0 0 ,δ0 0 )] be H

H

H

as in Definition 5.2.2.1, such that Σord,0 is a g-refinement of Σord . Then the ord ~ ord,tor by g (defined ordinary Hecke twist of the family (G, λ, i, αH0,p , αH 0 ) → M 0 H ,Σord,0 p ord )→ by Proposition 3.3.4.21 and Lemma 3.1.3.2) is the pullback of (G, λ, i, αHp , αH p ord,tor ~ via a (unique) surjection M H,Σord

~ ord,tor : M ~ ord,tor ~ ord,tor [g] →M . H0 ,Σord,0 H,Σord ~ ord,tor to M ~ ord (on the target) coincides with the surjection The pullback of [g] H ~ ord : M ~ ord0 → M ~ ord [g] H H ~ ord,tor is quasi-finite flat if Σord,0 is defined in Proposition 3.4.4.1. The morphism [g] g-induced by Σord as in Definition 5.2.2.1. (Here the flatness follows automatically from the quasi-finiteness, by [35, IV-3, 15.4.2 e0 )⇒b)]; cf. [62, Lem. 6.3.1.11].) ~ ord,tor is proper if the levels Hp and H0 at p are equally deep The surjection [g] p

as in Definition 3.2.2.9, or if gp is of twisted Up type as in Definition 3.3.6.1 and ~ ord,tor is finite if it is both depth (H0 )−depth (gp ) = depth (Hp ) > 0. (Note that [g] D

p

D

D

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quasi-finite and proper, by [35, IV-3, 8.11.1].) If gp ∈ Pord D (Zp ), then the induced morphism ~ ord,tor : M ~ ord,tor ~ ord,tor [g] →M × ~S0,rH0 rH 0 H0 ,Σord,0 H,Σord ~ S0,rH

is log ´etale, which is (quasi-finite) ´etale if Σord,0 is g-induced by Σord as in Definition 5.2.2.1. In particular, when g = 1 and H0 = H, we have proper log ´etale surjections ~ ord,tor : M ~ ord,tor ~ ord,tor [1] →M H,Σord,0 H,Σord when Σord,0 is a refinement of Σord . ~ ord,tor maps the [(Φ0 0 , δ 0 0 , σ 0 )]-stratum ~Zord0 0 0 Moreover, the surjection [g] H H [(ΦH0 ,δH0 ,σ )] ord,tor ord,tor ord ~ ~ ~ of M 0 ord,0 to the [(ΦH , δH , σ)]-stratum Z of M ord if and only H ,Σ

[(ΦH ,δH ,σ)]

H,Σ

0 0 if there are representatives (ΦH , δH , σ) and (Φ0H0 , δH 0 , σ ) of [(ΦH , δH , σ)] and 0 0 0 0 0 0 [(ΦH0 , δH0 , σ )], respectively, such that (ΦH0 , δH0 , σ ) is a g-refinement of (ΦH , δH , σ) as in [62, Def. 6.4.3.1]. If g = g1 g2 , where g1 = (g1,0 , g1,p ) and g2 = (g2,0 , g2,p ) are elements of G(A∞,p ) × Pord D (Qp ), each having a setup similar to that of g, then we have ord,tor ord,tor ord,tor ~ ~ ord = [g~2 ]ord ◦ [g~1 ]ord [g] = [g~2 ] ◦ [g~1 ] , extending the identity [g]

in Proposition 3.4.4.1. Finally, there exists Σ and Σ0 extending Σord and Σord,0 , respectively, as in Proposition 5.1.3.2, such that Σ0 is a g-refinement of Σ, and so that we have tor tor the canonical surjection [g] : Mtor H0 ,Σ0 → MH,Σ as in Proposition 1.3.1.15. Let tor tor [g]rH0 ,rH : Mtor H0 ,Σ0 ,rH0 → MH,Σ,rH denote the canonically induced morphism. Then tor tor ~ ~ ord,tor [g] ⊗ Q can be identified with the pullback of [g] to M ord ⊗ Q (on the tarZ

rH0 ,rH

H,Σ Z tor

~ get) under (5.2.1.2) in (7) of Theorem 5.2.1.1. In particular, [g] ´etale.

⊗ Q is proper log Z

~ ord,tor follows from a combination of [62, Prop. 5.4.3.8] Proof. The existence of [g] and (6) of Theorem 5.2.1.1: ~ ord,tor Since M is noetherian normal, and since the extensibility condition (5) H0 ,Σord,0 in Definition 3.4.2.10 implies that G is an ordinary semi-abelian scheme as in Definition 3.1.1.2, by Lemma 3.1.3.2, any isogenous quotient of GM ~ ord extends uniH0

quely (up to isomorphism) to an isogenous quotient of G. Hence, the usual ordi~ ord0 extends to the ordinary nary Hecke twist defined by Proposition 3.3.4.21 over M H ord,0 ord,tor 0 0 0 0 ord ~ ~ ord,tor Hecke twist (G , λ , i , αHp , αHp ) → MH0 ,Σord,0 of (G, λ, i, αH0,p , αH 0 ) → M 0 H ,Σord,0 p by g = (g0 , gp ). ord,0 0 ~ ord By construction, the restriction of (G0 , λ0 , i0 , αH ) → Mtor p , αH H0 ,Σord,0 to MH0 p ~ ord : M ~ ord0  M ~ ord defined in Proposidetermines the canonical surjection [g] H H tion 3.4.4.1. By [62, Prop. 5.4.3.8] and (6) of Theorem 5.2.1.1, the restriction ord,0 0 ~ ord,tor ~ ord,tor of (G0 , λ0 , i0 , αH ) → M to ´etale local charts of M admit p , αH H0 ,Σord,0 H0 ,Σord,0 p

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ord ~ ord,tor unique morphisms to (G, λ, i, αHp , αH )→M , by our assumption that Σord,0 H,Σord p is a g-refinement of Σord . (More precisely, the cones containing pairings of the form υ ◦ B 0 : Y 0 × X 0 → Z are carried to cones containing pairings of the form υ ◦ B : Y × X → Z under the identification between PΦ0H0 and PΦH defined by ∼

∼

(fX : X 0 ⊗ Z(p) → X ⊗ Z(p) , fY : Y ⊗ Z(p) → Y 0 ⊗ Z(p) ), when we have the objects Z

Z

Z

Z

as in the context of Definition 5.2.2.1.) These morphisms patch uniquely, and ~ ord,tor : ~ ord,tor hence descend to M 0 ord,0 . Therefore, there exists a unique morphism [g] H ,Σ

~ ord , which pulls (G, λ, i, αHp , αord ) → M ~ ord,tor ~ ord,tor ~ ord,tor M → M extending [g] Hp H0 ,Σord,0 H,Σord H,Σord ord,0 ord,tor ord,tor 0 0 0 0 ord ~ ~ ~ back to (G , λ , i , α p , α ) → M 0 ord,0 . Since M is dense in M ord , and H

Hp

H

H ,Σ

H,Σ

since the condition in (6) of Theorem 5.2.1.1 does not involve level structures, starting with a degeneration over a complete discrete valuation ring V centered ~ ord,tor at an arbitrary geometric point s¯ of M , we can construct degenerations over H,Σord 0 a complete discrete valuation ring V finite flat over V centered at a geometric ~ ord,tor point of M as soon as we can lift the level structures on the generic points. H0 ,Σord,0 ~ ord is surjective (by Proposition 3.4.4.1—in fact, this surjectivity Therefore, since [g] is essentially a consequence of the liftability conditions in the definitions of level ~ ord,tor is also surjective. structures), the morphism [g] ~ ord Moreover, if the levels Hp and Hp0 at p are equally deep, in which case [g] ~ ord,tor is proper, is finite by Proposition 3.4.4.1, then the (separated) morphism [g] because in this case the above argument also verifies the valuative criterion. On the other hand, if gp is of twisted Up type as in Definition 3.3.6.1 and depthD (Hp0 ) − depthD (gp ) = depthD (Hp ) > 0, then the desired valuative criterion follows from Lemma 3.3.6.6. If g = g1 g2 as in the last statement of this proposition, then we can also construct ord,0 ord 0 ) of (G, λ, i, αH0,p , αH the ordinary Hecke twist (G0 , λ0 , i0 , αH 0 ) in two steps, p , αH p p as in the last statement of Proposition 3.3.4.21. Hence, the induced morphisms ~ ord,tor = between partial toroidal compactifications satisfy the desired identity [g] ord,tor ord,tor ~ ord = [g~2 ]ord ◦ [g~1 ]ord in Proposition [g~2 ] ◦ [g~1 ] , extending the identity [g] 3.4.4.1. The assertions on the local structures can be verified by comparing the ´etale local structures, which then follows from the gluing construction of the partial toroidal compactifications (cf. Proposition 5.1.3.7). As for the last paragraph, the existence of Σ and Σ0 follows from Proposition tor 5.1.3.2. Since [g] is constructed in [62, Prop. 6.4.3.4] using Hecke twists of the tor tautological object, and since the tautological objects over Mtor H,Σ,rH and MH0 ,Σ0 ,rH0 ~ ord,tor ~ ord,tor induce the tautological objects over M ord ⊗ Q and M 0 ord,0 ⊗ Q, respectively, we H,Σ

tor

~ see that [g]

Z

H ,Σ

⊗ Q can be identified with the restriction of Z

Z tor [g]rH0 ,rH

~ ord,tor to M ⊗ Q. H0 ,Σord,0 Z

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tor ~ ord,tor This coincides with the pullback of [g]rH0 ,rH to M ⊗ Q (on the target) under H,Σord Z

~ ord,tor ~ ord,tor (5.2.1.2) because M ⊗ Q is the preimage of M ⊗ Q, by the statements H0 ,Σord,0 H,Σord Z

Z

concerning the strata in (7) of Theorem 5.2.1.1 and in Proposition 1.3.1.15. The remaining assertions of the proposition are self-explanatory. Corollary 5.2.2.3. (Compare with Corollary 3.4.4.3.) With the setting as in Proposition 5.2.2.2, the morphism ~ ord,tor : M ~ ord,tor ~ ord,tor [g] →M H0 ,Σord,0 H,Σord ~ ord,tor : M ~ ord,tor ~ ord,tor (cf. Definition 3.4.4.2) induced by [g] →M is proper, which H0 ,Σord,0 H,Σord ord,0 ord is finite flat if Σ is g-induced by Σ as in Definition 5.2.2.1. If gp ∈ Pord D (Zp ), then the induced morphism ~ ord,tor : M ~ ord,tor ~ ord,tor [g] →M rH 0 H0 ,Σord,0 H,Σord

~ 0,r 0 × S H

~ 0,r S H

is proper log ´etale (because it is log ´etale by Proposition 5.2.2.2). If moreover ~ ord,tor : M ~ ord,tor Σord,0 is g-induced by Σord as in Definition 5.2.2.1, then [g] → rH 0 H0 ,Σord,0 ord,tor ~ ~ M ord × S0,r 0 is finite ´etale (because it is quasi-finite ´etale by Proposition H,Σ

~ 0,r S H

H

5.2.2.2; cf. [35, IV-3, 8.11.5, or IV-4, 8.12.6]). Proof. These follow from the fact that the induced morphism ~ ord,tor : M ~ ord,tor ~ ord,tor ⊗ Fp ⊗ Fp → M [g] H,Σord H0 ,Σord,0 Z

Z

over Spec(Fp ) is proper by Lemma 3.3.6.8. Corollary 5.2.2.4. (Compare with Corollary 3.4.4.4 and Example 3.4.4.5.) With ˆ p ) × Pord (Zp ), if H0,p = the setting as in Proposition 5.2.2.2, if g = (g0 , gp ) ∈ G(Z D p −1 p 0 ord −1 ord ord ˆ g0 H g0 in G(Z ), if Hp = (gp Hp gp ) in MD (Zp ) (see (3.3.3.5)), and if Σord,0 is g-induced by Σord as in Definition 5.2.2.1, then (rH0 = rH and) the induced ~ ord,tor : M ~ ord,tor ~ ord,tor morphism [g] ord is an isomorphism. (See the remark at 0 ord,0 → M H,Σ

H ,Σ

the end of Corollary 3.4.4.4.) ~ ord,tor : M ~ ord,tor Proof. By Corollary 5.2.2.3, the induced morphism [g] → H0 ,Σord,0 ~ ord,tor to M ~ ord,tor ~ ord0 is an M is finite ´etale. By Corollary 3.4.4.4, the restriction of [g] H H,Σord ~ ord,tor ~ ord,tor isomorphism. Since M 0 ord,0 ⊗ Fp and M ord ⊗ Fp are regular (by smoothness of H ,Σ

H,Σ

Z

Z

~ ord,tor ~ ord,tor M and M over ~S0,rH0 and ~S0,rH , respectively; see Theorem 5.2.1.1), by H0 ,Σord,0 H,Σord Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11]), the induced finite morphism ~ ord,tor : M ~ ord,tor ~ ord,tor [g] 0 ord,0 ⊗ Fp → M ord ⊗ Fp is necessarily an isomorphism. Hence, H ,Σ

ord,tor

Z

H,Σ

Z

~ [g] : → (being finite ´etale and an isomorphism between the fibers over Spec(Fp )) is an isomorphism. ~ ord,tor M H0 ,Σord,0

~ ord,tor M H,Σord

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Corollary 5.2.2.5 (elements of Up type). (Compare with Corollary 3.4.4.6.) Suppose in Proposition 5.2.2.2 that g0 = 1 and gp is of Up type as in Definition 3.3.6.1 (so that it is of twisted Up type and depthD (gp ) = 1). Then Σord,0 is also a 1-refinement of Σord , and the induced morphism ~ ord,tor : M ~ ord,tor ~ ord,tor [g] ⊗ Fp → M ⊗ Fp H0 ,Σord,0 H,Σord Z

(5.2.2.6)

Z

is proper and coincides with the composition of the absolute Frobenius morphism FM ~ ord,tor

⊗ Fp H0 ,Σord,0 Z

~ ord,tor ~ ord,tor :M ⊗ Fp → M ⊗ Fp H0 ,Σord,0 H0 ,Σord,0 Z

Z

with the canonical proper morphism ~ ord,tor : M ~ ord,tor ~ ord,tor [1] ⊗ Fp → M ⊗ Fp . H0 ,Σord,0 H,Σord Z

(5.2.2.7)

Z

Suppose moreover that Σord,0 is g-induced by Σord as in Definition 5.2.2.1. Then Σ is also 1-induced by Σord , and the above morphisms (5.2.2.6) and (5.2.2.7) are both finite flat. ord If Hp0 = Hpord as open compact subgroups of Mord D (Zp ) (see (3.3.3.5)), then we ord,0 ord can take Σ to be g-induced by Σ as in Definition 5.2.2.1, so that (rH0 = rH and) the canonical morphism (5.2.2.7) is an isomorphism by Corollary 5.2.2.4, and so that the composition ord,0

ord,tor −1

~ ([1]

~ ord,tor M H,Σord

⊗ Fp

∼

)

~ ord,tor M ⊗ Fp H0 ,Σord,0

→

Z

Z

~ ord,tor [g]

→

~ ord,tor M ⊗ Fp H,Σord Z

coincides with the (finite flat) absolute Frobenius morphism FM ~ ord,tor

⊗ Fp

H,Σord Z

~ ord,tor ~ ord,tor ⊗ Fp . :M ⊗ Fp → M H,Σord H,Σord Z

Z

Proof. First note that gp acts by scalars on Gr0DQp and Gr−1 DQp , and hence g = (1, gp ) preserves any filtration Z ⊗ A∞ of L ⊗ A∞ satisfying Z−2 ⊗ Qp ⊂ DQp ⊂ ˆ Z

ˆ Z

Z

Z0 ⊗ Qp . Then any identification between PΦ0H0 and PΦH as in Definition 5.2.2.1 ˆ Z

is just some (positive) scalar multiplication, and hence being a g-refinement and a 1-refinement (or, being g-induced and 1-induced) are exactly the same notion. This explains all relevant statements in this corollary. Since the ordinary Hecke twist in Proposition 5.2.2.2 is realized as the relative Frobenius morphism (with naturally ~ ord0 ⊗ Fp as in Corollary induced additional structures) over the dense subscheme M H Z

~ ord,tor 3.4.4.6 (or rather its proof), it must be so over the whole M ⊗ Fp . Hence, H0 ,Σord,0 Z

the first paragraph of the corollary follows. The second paragraph of the corollary follows from the first paragraph and from Corollary 5.2.2.3. The third paragraph of the corollary follows from the second paragraph and from Corollary 5.2.2.4.

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Remark 5.2.2.8. (Compare with Remark 3.4.4.9.) By Kunz’s theorem [54] (cf. [76, Sec. 42, Thm. 107]), the absolute Frobenius morphisms FM ~ ord,tor ⊗ Fp and H0 ,Σord,0 Z

FM ~ ord,tor

⊗ Fp

H,Σord Z

~ ord,tor ~ ord,tor in Corollary 5.2.2.5 are flat because M ⊗ Fp and M ⊗ Fp H0 ,Σord,0 H,Σord Z

Z

~ ord,tor ~ ord,tor are regular (by smoothness of M and M over ~S0,rH0 and ~S0,rH , respectiH0 ,Σord,0 H,Σord vely; see Theorem 5.2.1.1). 5.2.3

The Case When p is a Good Prime

As in Section 3.4.5, suppose that p is a good prime (see Definition 1.1.1.6). Then we ~ ord,tor can also construct M using the toroidal compactifications already constructed H,Σord in [62, Thm. 6.4.1.1] in mixed characteristics, provided that Σord = {ΣΦH }[(ΦH ,δH )] is induced (in a natural sense similar to that in Definition 2.1.2.25) by a compatible choice Σ0 of admissible smooth rational polyhedral cone decomposition data for MH0 as in Definition 1.2.2.13, where H0 = Hp G(Zp ), as in [62, Constr. 7.3.1.6]. If Σ0 is projective with a collection pol0 of polarization functions as in Definitions 1.2.2.7 and 1.2.2.14, then Σord is also projective with an induced collection polord of polarization functions as in Definition 5.1.3.3. However, since we need Σord to be smooth, we shall make the following: Assumption 5.2.3.1 (for this subsection). The smoothness conditions defined by H on Σord and by H0 on Σ0 are compatible with each other. Lemma 5.2.3.2. Assumption 5.2.3.1 is satisfied, for example, when the group Hp bal r bal is of the form Up,0 (pr ), Up,1 (pr ), Up,1 (p ), Up,1,0 (pr1 , pr0 ), or Up,1,0 (pr1 , pr0 ), for some integers 0 ≤ r and 0 ≤ r1 ≤ r0 as in Definition 3.2.2.8. Proof. In all such cases, we can canonically identify SΦH with

1 N SΦH0

in SΦH0 ⊗ Q Z

for some integer N ≥ 1, where ΦH and ΦH0 are torus arguments for admissible filtrations compatible with the filtration D as in Definition 3.2.3.1. Then it is indeed possible that Σord is induced by a smooth Σ0 for MH0 . By construction, we also obtain an extension of the compatible choice Σord (resp. polord , ~ ord to a compatible choice Σ (resp. pol) for MH , although we cannot if defined) for M H assert that Σ is smooth in this case. (Thus, we will need the constructions in Propositions 1.3.1.11 and 2.2.2.3.) We also have an induced smooth Σord,0 , together with polord,0 if pol0 is also defined. Suppose that there is a smooth Σp (resp. polp ) for MHp such that Σ0 (resp. pol0 , if defined) is induced by Σp (resp. polp ) in the natural sense (similar to that in Definition 2.1.2.25). (These assumptions on Σ0 and ~ ord .) We shall Σord,0 can be met by refining any given cone decomposition Σord for M H assume that such choices have been made in the remainder of this subsection. Proposition 5.2.3.3. (Compare with Proposition 3.4.5.7.) With assumptions as min ~ above, let MHp , Mtor Hp ,Σp , and MHp be constructed over S0 = Spec(OF0 ,(p) ) as in [62].

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(This is possible because p is a good prime as in Definition 1.1.1.6, in which case rD = 0 and rH = rν ; see Definition 3.4.2.1.) Then we obtain a canonical open immersion ~ ord,tor M ,→ Mtor Hp ,Σp H0 ,Σord,0

(5.2.3.4)

~ ord0 ,→ MHp (as in (3.4.5.2)), a canonical extending the canonical open immersion M H quasi-finite ´etale surjective morphism ~ ord,tor ~ ord,tor M M × ~S0,rH H,Σord H0 ,Σord,0

(5.2.3.5)

~ S0

~ ord  M ~ ord0 × ~S0,r extending the canonical quasi-finite ´etale surjective morphism M H H H ~ S0

(as in (3.4.5.6)), and by composing (5.2.3.5) with (5.2.3.4) a canonical quasi-finite ´etale morphism ~ ~ ord,tor M → Mtor Hp ,Σp × S0,rH H,Σord

(5.2.3.6)

~ S0

~ ord → MHp × ~S0,r (as in extending the canonical quasi-finite ´etale morphism M H H ~ S0

(3.4.5.8)). Alternatively, we can construct the morphisms (5.2.3.4), (5.2.3.5), and (5.2.3.6) as a canonical open and closed subalgebraic stack, given by taking the schematic closure of Mord H , in a relatively representable functor of ordinary level-Hp structures of type (L ⊗ Zp , h · , · i, D) and their (unique if existent) extensions over degenerations. Z

Under the open immersion (5.2.3.4), the pullback of the degenerating family (G, λ, i, αHp ) of type MHp over Mtor Hp ,Σp admits (up to isomorphism) a unique exord ~ ord,tor ~ ord . tension to the degenerating family (G, λ, i, αH0,p , αH 0 ) of type MH0 over M 0 H ,Σord,0 p (Note that H0,p = Hp .) Under the quasi-finite morphism (5.2.3.5), the pullback of ord ~ ord over M ~ ord,tor the degenerating family (G, λ, i, αH0,p , αH is cano0 ) of type MH0 H0 ,Σord,0 p ord ~ nically isomorphic to the degenerating family (G, λ, i, αHp , α ) of type Mord over Hp

H

~ ord,tor M . Consequently, the pullback of H,Σord ωMtor := ∧top Lie∨ G/Mtorp Hp ,Σp

H ,Σp

∼ = ∧top e∗G Ω1G/Mtorp

H ,Σp

under (5.2.3.4) is canonically isomorphism to ωM ~ ord,tor

H0 ,Σord,0

:= ∧top Lie∨ ~ ord,tor G/M

and the pullback of ωM ~ ord,tor

H0 ,Σord,0

∼ = ∧top e∗G Ω1G/M ~ ord,tor

,

H0 ,Σord,0

under (5.2.3.5) is canonically isomorphic to

H0 ,Σord,0

top ∼ top e∗ Ω1 ord,tor . ωM Lie∨ ~ ord,tor := ∧ ~ ord,tor = ∧ G G/M ~ G/M H,Σord

H,Σord

H,Σord

Proof. Consider the open immersion tor Mord,tor Hp ,Σp ,→ MHp ,Σp

(5.2.3.7)

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ord representing the extensions of the ordinary level structure αG(Z over Mord Hp p) (extending (3.4.5.3)) as in condition (5) of Definition 3.4.2.10 (which is unique up to isomorphism if it exists). By comparing the universal property of Mord,tor Hp ,Σp given by (6) of Theorem 5.2.1.1 with the universal property of Mtor given by [62, Thm. Hp ,Σp 6.4.1.1(6)], we obtain the desired open and closed immersion (5.2.3.4) extending the open and closed immersion (3.4.5.5). The canonical morphism (5.2.3.5) exists by comparing the universal properties ~ ord,tor ~ ord,tor of M and M given by (6) of Theorem 5.2.1.1, which is quasi-finite and H,Σord H0 ,Σord,0 ´etale by comparing the sheaves of log 1-differentials using (4) of Theorem 5.2.1.1 (which is in turn based on Proposition 4.2.3.5) and by the assumption that both Σord and Σord,0 are induced by a smooth Σ0 for MH0 . Then we obtain (5.2.3.6) by composing (5.2.3.5) with (5.2.3.4) as in the proposition. Alternatively, we can construct (5.2.3.5) as a relatively representable functor parameterizing liftings of ordinary level structures and their extensions over degenerations (cf. condition (5) of Definition 3.4.2.10). If we construct the similar relative representable functor over the whole of Mtor Hp ,Σp , then we can also construct (5.2.3.6) as an open and closed subalgebraic stack. The remaining statements about the pullbacks of (G, λ, i, αHp ) are clear because the morphisms (5.2.3.7) and (5.2.3.5) are defined by comparing the degenerating families over the various partial toroidal compactifications, or by their interpretations as open and closed subalgebraic stacks in relative representable functors over Mtor Hp ,Σp .

Lemma 5.2.3.8. With assumptions asH in Proposition 5.2.3.3, the composition of min (5.2.3.4) with the canonical morphism Hp : Mtor Hp ,Σp → MHp induces a morphism ord,tor ord,min min ~ 0 ord,0 → M p with open image. Let M ~ 0 M denote this image (with its H ,Σ

H

H

~ ord,min canonical open subscheme structure). Then M is quasi-projective over ~S0,rH0 H0 H −1 ord,min ~ and Hp (MH0 ) is (set-theoretically) the image of (5.2.3.4) in Mtor Hp ,Σp , and hence H~ ord ord,tor ord,min ~ ~ 0 the induced morphism :M →M over ~S is proper and surH0

H0 ,Σord,0

jective. Consequently, since ωM ~ ord,tor

H0 ,Σord,0

0,rH0

H

is isomorphic to the pullback of ωMtor (see Hp ,Σp

Proposition 5.2.3.3), we have a canonical isomorphism  k ∼ ~ ord,min ~ ord,tor M , ω⊗ = Proj ⊕ Γ(M H0 ~ ord,tor H0 ,Σord,0 k≥0

M

 ) ,

H0 ,Σord,0

H ord ~ ord,tor ~ ord,min and the proper morphism ~ H0 : M → M is the Stein factorization H0 H0 ,Σord,0 (see [35, III-1, 4.3.3]) of itself (and hence has nonempty connected geometric fibers, by [35, III-1, 4.3.1, 4.3.3, 4.3.4] and its natural generalization to the context of algebraic stacks). Proof. Suppose x ¯ is a geometric point of the [(ΦHp , δHp , σ)]-stratum Z[(ΦHp ,δHp ,σ)] tor of MHp ,Σp . By comparing (6) of Theorem 5.2.1.1 with [62, Thm. 6.4.1.1(6)], or

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rather by the theory of degeneration on which the constructions are based (see Theorem 4.1.6.2), the condition of being in the image of (5.2.3.4) is a condition for the induced cusp label [(ΦHp , δHp )] and on the image of x ¯ under H the canonical morphism Z[(ΦHp ,δHp ,σ)] → [MZHHpp ] → Mmin , which is nothing but (¯ x). Therep H Hp H H −1 H ord,min ~ fore, once Hp (¯ x) ∈ MH0 , all other points in Hp ( Hp (¯ x)) is in the image of H −1 ord,min ~ 0 (5.2.3.4). This shows that Hp (M ) is (set-theoretically) the image of (5.2.3.4) H tor in MHp ,Σp . Since the complement of the (open) image of (5.2.3.4) in Mtor Hp ,Σp is H ord,min min ~ closed, and since Hp : Mtor  M is proper, we see that M is open in Hp ,Σp Hp H0 ord H ~ ~ ord,tor ~ ord,min is also proper. Mmin Hp , and that the induced morphism H0 : MH0 ,Σord,0 → MH0   H ⊗k min ∼ tor Since Hp : Mtor is defined (as Hp ,Σp → MHp = Proj ⊕ Γ(MHp ,Σp , ωMtorp p ) k≥0

H ,Σ

in [62, Sec. 7.2.3]) as the Stein factorization of the morphism from Mtor Hp ,Σp to a projective space defined by global sections of a sufficiently divisible power of ωMtor , HpH ,Σp its restriction to an open subscheme (of the source) containing all fibers of Hp with which it overlaps (or, equivalently, its pullback to an open subscheme of the target) is a proper morphism over its image, which can also be defined by a Stein H ord factorization. Hence, we can identify ~ H0 with the canonical morphism   ⊗k ∼ ~ ord,tor ~ ord,min ~ ord,tor M → M Proj ⊕ Γ( M , ω ) , = 0 0 ord,0 0 ord,0 ord,tor H ~ H ,Σ H ,Σ k≥0

M

H0 ,Σord,0

which is the Stein factorization of itself, as desired. Lemma 5.2.3.9. With assumptions as in Proposition 5.2.3.3, suppose moreover that Hp is neat; that Σp is projective, smooth, and equipped with a polarization ~ tor0 function polp , inducing a polarization function pol0 for Σ0 ; and that M H ,d0 pol0 is defined as in Proposition 2.2.2.1 (for some integer d0 ≥ 1). Then there is a canonical open and closed immersion tor ~ tor0 M H ,d0 pol0 ,→ MHp ,Σp

(5.2.3.10)

~ H0 ,→ MHp (as in (2.2.4.4), extending the canonical open and closed immersion M p now that H is neat), which is compatible with (2.2.4.3) and with the canonical H H tor min ~ tor0 ~ min morphisms ~ H0 : M H ,d0 pol0 → MH0 and Hp : MHp ,Σp → MHp . Moreover, the canonical finite morphism ~ tor ~ tor M H,d0 pol → MH0 ,d0 pol0

(5.2.3.11)

~ tor (cf. (2.2.2.4), with M H,d0 pol constructed as in Proposition 2.2.2.3) induces by composition with (5.2.3.10) a finite morphism tor ~ tor M H,d0 pol → MHp ,Σp .

(5.2.3.12)

0 ~ tor0 ~ tor Consequently, M H ,d0 pol0 is independent of the choices of pol and d0 , and MH,d0 pol is also independent of the choices of pol and d0 , because it is canonically isomorphic to the normalization of Mtor Hp ,Σp in MH under the composition of canonical morphisms tor MH → MH0 ,→ Mtor H0 ,Σ0 → MHp ,Σp (cf. Proposition 2.2.2.3).

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Proof. By comparing (6) of Theorem 1.3.1.3 with [62, Thm. 6.4.1.1(6)], there is a canonical open and closed immersion tor Mtor H0 ,Σ0 ,→ MHp ,Σp ⊗ Q

(5.2.3.13)

Z

(over S0 ) compatible with the canonical open and closed immersion min Mmin H0 ,→ MHp ⊗ Q,

(5.2.3.14)

Z

which can be pullback to S0 , and with the canonical H identified with the H of (2.2.4.3) min tor min morphisms H0 : Mtor → M and ⊗ Q : M H0 ,Σ0 H0 Hp ,Σp ⊗ Q → MHp ⊗ Q over S0 . Hp Z

Z

Z

For each integer d ≥ 0, we can compatibly construct H0 ,dpol0 , Hp ,dpolp , JH0 ,dpol0 , and JHp ,dpolp as in Definition 1.3.1.7 and [62, Def. 7.3.3.1], so that H0 ,dpol0 (resp. JH0 ,dpol0 ) is isomorphic to the pullback of Hp ,dpolp (resp. JHp ,dpolp ) under (5.2.3.13) (resp. (5.2.3.14)). By [62, Thm. 7.3.3.4], there is a canonical isomorphism H ∼ min NBlJHp ,d0 polp ( Hp ) : Mtor (5.2.3.15) Hp ,Σp → NBlJHp ,d0 polp (MHp ). By its very construction, this canonical isomorphism extends the canonical isomorphism H ∼ min NBlJH0 ,d0 pol0 ( H0 ) : Mtor (5.2.3.16) H0 ,Σ0 → NBlJH0 ,d0 pol0 (MH0 ) (as in Theorem 1.3.1.10, for the same d0 ). Since H0 ,d0 pol0 and Hp ,d0 polp are defined by the vanishing orders on boundary divisors in a compatible way, we see that J~H0 ,d0 pol0 (see Proposition 2.2.2.1) is isomorphic to the pullback of JHp ,d0 polp under the canonical isomorphism (2.2.4.3). Thus, we have the desired open and closed immersion (5.2.3.10) pulling the isomorphism (5.2.3.15) back to H ∼ ~ tor0 ~ min NBlJ~ 0 (~ H0 ) : M (M (5.2.3.17) H0 ) H ,d0 pol0 → NBlJ~ 0 0 0 H ,d0 pol

H ,d0 pol

H H in a way compatible with (2.2.4.3) and with the canonical morphisms ~ H0 and Hp . ~ tor The statements concerning M H,d0 pol are self-explanatory. Proposition 5.2.3.18. With the assumptions as in Lemma 5.2.3.9, there exists a canonical open immersion ~ ord,tor ~ tor M ,→ M (5.2.3.19) H,d0 pol,rH H,Σord (see Definition 2.2.3.5) compatible with (5.2.3.12) and (5.2.3.6) (and with the ca~ tor ~ tor nonical morphism M H,d0 pol,rH → MH,d0 pol ), whose composition with the canonical H~ ~ tor ~ min induces a morphism M ~ ord,tor ~ min morphism : M → M ord → M H,rH

H,d0 pol,rH

H,rH

H,Σ

H,rH

~ ord,min denote this image (with its canonical open subwith open image. Let M H H −1 ~ ord,min is quasi-projective and ~ ~ ord,min ) is (setscheme structure). Then M (M H

H,rH

H

~ tor theoretically) the image of (5.2.3.19) in M H,d0 pol,rH , and hence the induced morphism ord H~ ord,tor ord,min ~ ~ :M →M over ~S is proper and surjective. H

H,Σord

H

0,rH

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Consequently, since ωM ~ ord,tor

is canonically isomorphic to the pullback of

H,Σord

ωMtor under (5.2.3.6) (see Proposition 5.2.3.3), we have a canonical isomorphism Hp ,Σp   ⊗k ~ ord,min ∼ ~ ord,tor M Proj ⊕ Γ( M , ω ) , = H ~ ord,tor H,Σord M

k≥0

H,Σord

H ord ~ ord,tor ~ ord,min and the proper morphism ~ H0 : M → M is the Stein factorization H0 H0 ,Σord,0 of itself (and hence has nonempty connected geometric fibers, by [35, III-1, 4.3.1, 4.3.3, 4.3.4]). Proof. The existence of the canonical open immersion (5.2.3.19) follows from Pro~ tor position 5.2.3.3, from the fact that M H,d0 pol,rH is a normalization (see Lemma 5.2.3.9), and from Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11]). To prove the remaining statements, as in the proof of Lemma 5.2.3.8, the H −1 ~ ord,min ) is set-theoretically the image most crucial step is to show that ~ (M H,rH

H

~ tor of (5.2.3.19) in M H,d0 pol,rH . It suffices to verify this statement after pulled back to ∼ ~ ~ tor S0,rH and S0,rH ⊗ Fp (in two cases). In the former case, we have M H,d0 pol,rH ⊗ Q = Zp

Z

Mtor H,Σ,rH (see Definition 2.2.3.4), and the proof is similar to the proof of Lemma 5.2.3.8 using the theory of degeneration, by comparing (6) of Theorem 5.2.1.1 with (6) of Theorem 1.3.1.3. In the latter case, the proof follows from Lemma 5.2.3.8; ~ ord,tor ~ ord ⊗ Fp in M ⊗ Fp (see Theorem 5.2.1.1); and from the from the density of M H H,Σord Zp

Zp

~ ord,tor ~ ord,tor ⊗ Fp in⊗ Fp → M fact that the canonical quasi-finite morphism M H0 ,Σord,0 H,Σord Zp

Zp

duced by (5.2.3.5) is actually finite, because it is proper by Lemma 3.3.6.8 (with gp = 1 there; cf. the proof of Corollary 5.2.2.3, with g = 1 there). H −1 ~ ord,min ) is set-theoretically the (open) image of Once we know that ~ H,rH (M H H ord tor ~ ~ ord,tor (5.2.3.19) in MH,d0 pol,rH , we know that the induced morphism ~ H : M → H,Σord ord,min ~ ~ M over S0,r is proper and surjective. By the same Stein factorization arguH

H

ment as in the proof of Lemma 5.2.3.8, the last statement of the proposition follows from the assertions on ωM , and ωMtor in Proposition 5.2.3.3, and ~ ord,tor , ωM ~ ord,tor Hp ,Σp H,Σord

H0 ,Σord,0

from the finiteness of (5.2.3.12) (or rather (5.2.3.11)) in Lemma 5.2.3.9. 5.2.4

Boundary of Ordinary Loci

As in Section 1.3.2, let us describe the building blocks of Mord,tor in more detail. In H,Σord ~ ord,ZH , particular, we would like to describe and characterize the algebraic stacks M H ∼ ~ ord,ZH , C ~ ord , Ξ ~ ord , Ξ ~ ord (σ), Ξ ~ ord ~ ord /ΓΦ ,σ and M , ~Zord =Ξ H

ΦH ,δH

ΦH ,δH

ΦH ,δH

ΦH ,δH ,σ

[(ΦH ,δH ,σ)]

ΦH ,δH ,σ

H

~ ord ~ ord the formal algebraic stacks X ΦH ,δH ,σ and XΦH ,δH ,σ /ΓΦH ,σ in (2) of Theorem 5.2.1.1, and to describe canonical Hecke actions on collections of these geometric objects (compatible with those in Proposition 5.2.2.2). Throughout this subsection, let us fix the choice of a fully symplectic admissible ˆ as in Definitions 1.2.1.2 and 1.2.1.3, which we assume to be filtration Z of L ⊗ Z Z

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compatible with D as in Definition 3.2.3.1. Let us also fix a (noncanonical) choice of (LZ , h · , · iZ , hZ0 ) so that GZ can be defined as in Definition 1.2.1.9. Then we also have a boundary filtration DZ ∼ = D−1 for LZ ⊗ Zp ∼ = GrZ−1 ⊗ Zp determined by ˆ Z

Z

Z DZ,0 ∼ = D0−1 = D0 /(Z−2 ⊗ Zp ), which defines subgroups Pord DZ etc of G ⊗ Zp as in ˆ Z

Z

ord Definition 3.2.2.7, and defines quotients of subgroups of Pord Z,D (R) = PZ (R) ∩ PD (R) ord ord ∼ for each Z-algebra R as in Definition 3.2.3.9, so that PDZ (R) = Ph,Z,D (R). ˆ of standard form with respect to D For each open compact subgroup H of G(Z) ord,ΦH ~ ord,ZH as in the paragraphs con~ and M as in Definition 3.2.2.9, we can define MH H ~ ord taining and preceding (4.2.1.28), which are finite ´etale over the base change M Hh ,rH ... ~ ord (defined by MH and Mord as in Theorem 3.4.2.5) to ~S0,r . By Proposition of M Hh H h Hh ~ ord,ΦH → M ~ ord,ZH is finite ´etale and induces a 4.2.1.29, the canonical morphism M H H ∼ ~ ord,ZH ~ ord,ΦH /ΓΦ → . If H = U1bal (n) := U p (n0 )U1bal (pr ) MH canonical isomorphism M H H ∼M ~ ord,ΦH = ~ ord,ZH for some integer n0 ≥ 1 prime to p and some integer r ≥ 0, then M H

H

ord because there is a unique (ϕord −2,H , ϕ0,H ) inducing the prescribed (ϕ−2,H , ϕ0,H ). We n ord,Z ord,Φ ~ ord,Z ~ ~ n n ˆ (of := M . Moreover, for general H ⊂ G(Z) := M shall set M n n U1bal (n) standard form with respect to D as in Definition 3.2.2.9), by construction and by Proposition 3.4.4.1, we have the following two lemmas:

Lemma 5.2.4.1. (Compare with Lemma 1.3.2.1.) Let (Z, Φ, δ) and HGh,Z ,Φ be as in Lemma 1.3.2.1. Then HGh,Z ,Φ is of standard form with respect to DZ , which is a normal subgroup of HGh,Z (as in Definition 1.2.1.12) of equal depth (by definition— note that U1bal (n)Gh,Z = U1bal (n)G0h,Z for all integers n ≥ 1), and there is a canonical isomorphism ~ ord,ZH ∼ ~ ord M =M HGh,Z ,Φ ,rH , H

(5.2.4.2)

~ ord where M HG

is defined by (LZ , h · , · iZ , hZ0 ) as in Theorem 3.4.2.5, and where the subscript “rH ” means base change to ~S0,rH . If H0 is an open compact subgroup of H of standard form (with respect to D), then the corresponding morphism h,Z ,Φ

H0 ~ ord,Z ~ ord,ZH M →M H0 H

(5.2.4.3)

can be canonically identified with the quasi-finite flat morphism ~ ord0 M H

Gh,Z ,Φ ,rH0

~ ord →M HG

h,Z ,Φ

,rH .

(5.2.4.4)

~ ord The collection {M HG

}HGh,Z ,Φ (with H of standard form as above) naturally ∼ carries a Hecke action by elements gh = (gh,0 , gh,p ) ∈ GZ (A∞,p ) × Pord DZ (Qp ) = ∞,p ord ∞ Gh,Z (A ) × Ph,Z,D (Qp ) ⊂ Gh,Z (A ) with gh,p satisfying the conditions defined by DZ ∼ = D−1 on LZ ⊗ Zp ∼ = GrZ−1 ⊗ Zp as in Section 3.3.4, realized by quasi-finite flat h,Z ,Φ

Z

ˆ Z

surjections pulling tautological objects back to ordinary Hecke twists. Such a Hecke action enjoys the properties (under various conditions) concerning ´etaleness, finiteness, being isomorphisms between formal completions along fibers over Spec(Fp ),

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and inducing absolute Frobenius morphisms on fibers over Spec(Fp ) for elements of Up type as in Proposition 5.2.2.2 and Corollaries 5.2.2.3, 5.2.2.4, and 5.2.2.5. (We omit the details for simplicity.) If moreover H0 is a normal subgroup of H of standard form and equal depth as in Definition 3.2.2.9, then the canonical mor~ ord ~ ord0 etale and) is an phism M ,rH0 → MHG ,Φ ,rH0 induced by (5.2.4.4) (is finite ´ H h,Z

Gh,Z ,Φ

0 HGh,Z ,Φ /HG -torsor. h,Z ,Φ

Lemma 5.2.4.5. (Compare with Lemma 1.3.2.5.) Let HG0h,Z be as in Definition 1.2.1.12, which is a normal subgroup of HGh,Z (and hence of HGh,Z ,Φ ) of equal depth (by definition—again, note that U1bal (n)Gh,Z = U1bal (n)G0h,Z for all integers n ≥ 1). Then there is a canonical isomorphism ~ ord ~ ord,ΦH ∼ M =M HG0 H

h,Z

,rH ,

(5.2.4.6)

which is compatible with (5.2.4.2) and with Hecke actions as in Lemma 5.2.4.1. The ~ ord,ΦH → M ~ ord,ZH → M ~ ord canonical morphisms M Hh ,rH can be identified with the canoH H ord ord ~ ~ ~ ord nical finite ´etale morphisms MHG0 ,rH → MHG ,Φ ,rH → M HG ,rH , on which ΓΦH h,Z

h,Z

h,Z

acts equivariantly (and trivially on the latter two objects) via the canonical homomorphism ΓΦH → HG0l,Z /HGl,Z ∼ = HGh,Z /HG0h,Z with image HGh,Z ,Φ /HG0h,Z . In parti~ ord,ΦH → M ~ ord,ZH is an HG ,Φ /HG0 -torsor. cular, the finite ´etale morphism M H

H

h,Z

h,Z

By Proposition 4.2.1.30 and its proof, we have the following: ~ ord Lemma 5.2.4.7. (Compare with Lemma 1.3.2.7.) The morphism C ΦH ,δH → ord,Φ H ~ MH depends only on HG1,Z , is an abelian scheme when the splitting of (1.2.1.14) defined by any splitting δ also splits (1.2.1.13) (and induces an isomorphism HG1,Z ∼ = HG0h,Z n HU1,Z ; cf. the condition in (2) of Lemma 4.2.1.19), and is a torsor under 0 ~ ord,grp := C ~ ord the abelian scheme C ΦH0 ,δH0 defined by any H of standard form with ΦH ,δH 0 × ∼ HG0 n HU , which is Q -isogenous to Hom (X, B)◦ . (This clarifies the H = G1,Z

h,Z

O

1,Z

~ ord,ΦH .) If p - [L# : L], so that the ~ ord abelian scheme torsor structure of C ΦH ,δH → MH index of φ : Y ,→ X and the degree of λB : B → B ∨ are both prime to p, then there is a canonical Z× (p) -isogeny ◦ ord HomO (X, Bpord r ) /H r p ,Uess,ord

1,Zpr ,Dpr

~ ord,grp →C ΦH ,δH

(cf. (4.1.4.31)), where r := depthD (H) (see Definition 3.2.2.9) and Hpord r ,Uess,ord

1,Zpr ,Dpr

(see Definition 4.1.5.18) can be canonically embedded in the kernel of the canonical ◦ ◦ (induced by (4.1.4.34)). separable isogeny HomO (X, Bpord r ) → HomO (X, B) If r = depthD (H) and if n0 ≥ 1 is any integer prime to p such that U p (n0 ) ⊂ Hp , bal r so that U1bal (n) = U p (n0 )Up,1 (p ) ⊂ H are both of standard form and equally deep, and if we fix the choice of (Zn and) Φn , then the canonical morphism ~ ord := C ~ ord C Φn ,δn Φ bal U1

(n)

,δU bal (n) 1

~ Φord,δ ,r = C ~ Φord,δ × ~S0,r , →C n H H n H H ~ S0,rH

(5.2.4.8)

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where rn := rU1bal (n) = max(rD , r), (is finite ´etale and) is an HG1,Z /U1bal (n)G1,Z ∼ = ord Hn,G ess nUess -torsor (see Definition 4.1.5.18 and Section 4.2.1 for the definition 1,Z h,Zn

n

bal ord (n)G1,Z are as in Definition 1.2.1.12, and of Hn,G ess nUess ), where HG1,Z and U1 1,Zn h,Zn induces an isomorphism ∼ ~ ord ~ Φord,δ /(HG /U1bal (n)G ) → C CΦH ,δH ,rn . 1,Z 1,Z n n

(5.2.4.9)

Proof. The statements are self-explanatory. Lemma 5.2.4.10. (Compare with Lemma 1.3.2.11.) Suppose n = n0 pr for some integer n0 ≥ 1 prime to p and some integer r ≥ 0. Suppose (B, λB , iB , ϕord −1,n ) and (Zn , Φn = (X, Y, φ : Y ,→ X, ϕ−2,n , ϕ0,n ), δn ) are the tautological objects over ~ ord,Φn = M ~ ord,Zn . The abelian scheme torsor S := C ~ ord,Φn is univer~ ord → M M n n n Φn ,δn ord ∨,ord sal for the additional structures (cn , cn ) over noetherian normal schemes over ~ ord,Φn (inducing dominant morphisms over irreducible components) satisfying cerM n tain symplectic and liftability conditions, which can be re-interpreted as follows: ~ ord,Φn = M ~ ord,Zn parameterizes tuples ~ ord → M S=C n n Φn ,δn (G\ , λ\ : G\ → G∨,\ , i\ , βn\ 0 , βp\,ord ) r over noetherian normal schemes flat over ~S0,rH , where: (a) G\ (resp. G∨,\ ) is an extension of B (resp. B ∨ ) by the split torus T (resp. T ∨ ) with character group X (resp. Y ), and λ\ : G\ → G∨,\ induces λT = φ∗ : T → T ∨ and λB : B → B ∨ . (b) i\ = (i\,0 , i\,#,0 ) is a pair of homomorphisms i\,0 : O → EndS (G\ ) and i\,#,0 : O → EndS (G∨,\ ) compatible with each other under λ\ : G\ → G∨,\ , inducing compatible O-structures on B, B ∨ , T , and T ∨ . (c) βn\ 0 = (βn\,0 , βn\,#,0 , νn\ 0 ) is a principal level-n0 structure of (G\ , λ\ , i\ ) of 0 0 ∼ p ˆ ˆ p ), where β \,0 : (Z−1,n )S → : G\ [n0 ] and βn\,#,0 type (L ⊗ Z , h · , · i, Z ⊗ Z n0 0 0 ˆ Z

Z

∼

∨,\ (Z# [n0 ] are O-equivariant isomorphisms preserving filtrations on −1,n0 )S → G both sides and inducing on the graded pieces the ϕ−2,n0 , ϕ−1,n0 , and ϕ0,n0 (by duality), respectively, induced by the given ϕ−2,n , ϕord −1,n , and ϕ0,n ; and where ∼ \ νn0 : ((Z/n0 Z)(1))S → µn0 ,S is an isomorphism, which are compatible with λ\ # and the canonical morphism Z−1,n0 → Z# −1,n0 induced by h · , · i. (Here Z is the # ˆ canonically dual to the filtration on L ⊗ Z, ˆ equipped with a filtration on L ⊗ Z Z

Z

canonical morphism Z → Z# , respecting the filtration degrees, induced by h · , · i. Then the splitting δn0 induced by δn corresponds under βn\ 0 to splittings of 0 → T [n0 ] → G\ [n0 ] → B[n0 ] → 0 and 0 → T ∨ [n0 ] → G∨,\ [n0 ] → B ∨ [n0 ] → 0.) Moreover, βn\ 0 satisfies the liftability condition that, for each integer m0 ≥ 1 such that n0 |m0 , there exists a finite ´etale covering of S over which there exists \ an analogous triple βm lifting the pullback of βn\ 0 . 0

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) is a principle ordinary level-pr structure , νp\,ord , βp\,ord,#,0 = (βp\,ord,0 (d) βp\,ord r r r r → of (G\ , λ\ , i\ ) of type (L ⊗ Zp , h · , · i, Z ⊗ Zp , D), where βp\,ord,0 : (Gr0D,pr )mult r S Z

βp\,ord,#,0 r

(Gr0D# ,pr )mult S

ˆ Z

: → G∨,\ [pr ]) are O-equivariant homoG [p ]) and morphisms inducing closed immersions, preserving filtrations on both sides, and inducing on the graded pieces the ϕ−2,pr , ϕord −1,pr , and ϕ0,pr (by duality), respectively, induced by the given ϕ−2,n , ϕord −1,n , and ϕ0,n ; and where × \,ord r νpr ∈ (Z/p Z)S is a section, which are compatible with λ\ and the canonical morphism φ0D : Gr0D → Gr0D# induced by h · , · i. (Then the splitting δpord induced r \,ord,0 \,ord )→ by δn corresponds under βpr to splittings of 0 → T [n] → image(βpr \,ord,#,0 ord,#,0 image(ϕord,0 ) → 0 and 0 → image(β ) → image(ϕ ) → 0.) Morepr −1,pr −1,pr satisfies the liftability condition that, for each integer r0 ≥ r, there over, βp\,ord r exists a quasi-finite ´etale covering of S over which there exists an analogous triple βp\,ord lifting the pullback of βp\,ord . r r0 \

r

Proof. The statements are self-explanatory. Proposition 5.2.4.11. (Compare with Proposition 1.3.2.12.) The abelian scheme ~ ord,ΦH is universal for the additional structures ~ ord torsor S := C ΦH ,δH → MH ∨,ord ~ ord,ΦH (inducing dominant (cord ) over noetherian normal schemes over M H , cH H morphisms over irreducible components) satisfying certain symplectic and liftability conditions, which can be interpreted as parameterizing tuples \ \,ord (G\ , λ\ : G\ → G∨,\ , i\ , βH ), p , βH p

(5.2.4.12)

where: (1) G\ , G∨,\ , λ\ , and i\ are as in Lemma 5.2.4.10. \ p ˆ p , h · , · i, Z ⊗ Z ˆ p ), structure of (G\ , λ\ , i\ ) of type (L ⊗ Z (2) βH p is a level-H ˆ Z

Z

which is a collection

\ {βH } , n0 n0 p

where n0 ≥ 1 runs over integers prime to p

\ such that U (n0 ) ⊂ H , such that each βH (where Hn0 := Hp /U p (n0 )) is n0 a subscheme of a ∨,\ [n0 ]) IsomS (Z−1,n0 ,S , G\ [n0 ]) × IsomS (Z# −1,n0 ,S , G S
 × IsomS ((Z/n0 Z)(1))S , µn0 ,S p

S

over S, where the disjoint union is over representatives (Zn0 , Φn0 , δn0 ) (with the same (X, Y, φ)) in (ZHp , ΦHp , δHp ) induced by (ZH , ΦH , δH ), that becomes the disjoint union of all elements in the Hn0 -orbit of some princiˆ p , h · , · i, Z ⊗ Z ˆ p ), as pal level-n0 structure βn\ 0 of (G\ , λ\ , i\ ) of type (L ⊗ Z ˆ Z

Z

\ in Lemma 5.2.4.10, for any Z lifting Zn ; and where βH is mapped to m 0

\ βH (under the canonical morphism, which we omit for simplicity) when n0 p - m0 and n0 |m0 . The Hn0 -orbit of the ϕ−1,n0 determined by such βn\ 0

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as in Lemma 5.2.4.10 then defines the level structure ϕ−1,Hp of (B, λB , iB ) (where (B, λB , iB , ϕ−1,Hp , ϕord −1,Hp ) is the pullback of the tautological tuple ord,ZH ~ over M ). H

\,ord (3) βH is an ordinary level-Hp structure of (G\ , λ\ , i\ ) of type p (L ⊗ Zp , h · , · i, Z ⊗ Zp , D), which is a subscheme of ˆ Z

Z

a



, G\ [pr ] × HomS (Gr0D# ,pr )mult HomS (Gr0D,pr )mult , G∨,\ [pr ] S S S  × × (Z/pr Z)S S

over S, where the disjoint union is over representatives (Zpr , Φpr , δpr ) (with the same (X, Y, φ)) in the (ZHp , ΦHp , δHp ) induced by (ZH , ΦH , δH ), that becomes the disjoint union of all elements in the Hpord r -orbit (where ord bal r r of Hpr := Hp /Up,1 (p )) of some principle ordinary level-p structure βp\,ord r (G\ , λ\ , i\ ) of type (L ⊗ Zp , h · , · i, D), as in Lemma 5.2.4.10. The Hpord r -orbit Z

\,ord of the ϕord as in Lemma 5.2.4.10 then defines −1,pr determined by such βpr of (B, λB , iB ). the ordinary level structure ϕord −1,Hp

~ ord Proof. This follows from Lemmas 5.2.4.7 and 5.2.4.10, by realizing C ΦH ,δH ,rn as a ord ~ quotient of some CΦn ,δn , and by finite flat descent. Proposition 5.2.4.13. (Compare with Propositions 1.3.2.14 and 5.2.4.11.) Fix any lifting (Z, Φ = (X, Y, φ, ϕ−2 , ϕ0 ), δ) of a representative (ZH , ΦH , δH ) of ~ ord,ΦH is universal for ~ ord [(ZH , ΦH , δH )]. The abelian scheme torsor C ΦH ,δH → MH × Z(p) -isogeny classes of tuples \,ord (G\ , λ\ : G\ → G∨,\ , i\ , j \ , j ∨,\ , [βˆp,\ ]HpG , βH ) p

(5.2.4.14)

1,Z

~ ord,ΦH (inducing dominant morover noetherian normal base schemes S over M H phisms over irreducible components), where: (1) G\ (resp. G∨,\ ) is a semi-abelian scheme which is the extension of an abelian scheme B (resp. B ∨ ) by a split torus T (resp. T ∨ ) over S, which is equivalent to a homomorphism c : X(T ) → B ∨ (resp. c∨ : X(T ∨ ) → B). (2) λ\ : G\ → G∨,\ is Q× -isogeny which is up to Z× (p) -isogenies an isogeny of semi-abelian schemes over S, inducing a Q× -isogeny λT : T → T ∨ between the torus parts, which is dual to a Q-isomorphism λ∗T : X(T ∨ ) ⊗ Q → Z

X(T ) ⊗ Q, and a Q× -polarization λB : B → B ∨ between the abelian parts Z

(cf. [62, Def. 1.3.2.19 and the errata]) which is up to Z× (p) -isogenies a polarization, so that c(N λ∗T ) = (N λB )c∨ when N is any locally constant function over S valued in positive integers such that (N λ∗T )(X(T ∨ )) ⊂ X(T ) and such that N λ\ : G\ → G∨,\ is an isogeny.

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(3) i\

:

O ⊗ Z(p)

EndS (G\ ) ⊗ Z(p) S is a homomorphism inducing

→

Z

Z

O ⊗ Z(p) -actions on G∨,\ , T , T ∨ , B, and B ∨ up to Z(p) × -isogeny, compatiZ

ble with each other under the homomorphisms between these objects introduced thus far. In particular, the induced homomorphism iB : O ⊗ Z(p) → Z

EndS (B) ⊗ Z(p) S satisfies the Rosati condition defined by λB (cf. [62, Def. Z

1.3.3.1]). ∼ ∼ (4) j \ : X ⊗ Z(p) S → X(T ) ⊗ Z(p) S and j ∨,\ : Y ⊗ Z(p) S → X(T ∨ ) ⊗ Z(p) S Z

Z

Z

Z

are isomorphisms of O ⊗ Z(p) -modules, such that there exists a section Z

\ \ ∨,\ ∗ r(j \ , j ∨,\ ) of (Z× )λT ◦ j ∨,\ . (p),>0 )S such that j ◦ φ = r(j , j \,ord (5) βH is an ordinary level-Hp structure of (G\ , λ\ , i\ ) of type p (L ⊗ Zp , h · , · i, Z ⊗ Zp , D) as in Proposition 5.2.4.11. (Note that the defiˆ Z

Z

\,ord nition of βH is insensitive to Z× (p) -isogenies.) p p,\ p p ˆ (6) [β ]HG is a rational level-H structure of (G\ , λ\ , i\ , j \ , j ∨,\ ) of type 1,Z

(L ⊗ A∞,p , h · , · i, Z ⊗ A∞,p , Φ), which is an assignment to each geometric ˆ Z

Z

point s¯ of S a rational level-Hp structure of (G\ , λ\ , i\ , j \ , j ∨,\ ) of type (L ⊗ A∞,p , h · , · i, Z ⊗ A∞,p , Φ) based at s¯ (cf. [62, Def. 1.3.8.7]), which ˆ Z

Z

p is a π1 (S, s¯)-invariant HG -orbit [βˆs¯p,\ ]HpG 1,Z

of triples

1,Z

βˆs¯\ = (βˆs¯p,\,0 , βˆs¯p,\,#,0 , νˆs¯p,\ ),

such that the assignments at any two geometric points s¯ and s¯0 of the same connected component of S determine each other (cf. [62, Lem. 1.3.8.6]), where: ∼ ∞,p ∼ (a) βˆs¯p,\,0 : Z−1 ⊗ A∞,p → Vp G\s¯ and βˆs¯p,\,#,0 : Z# → Vp G∨,\ s¯ −1 ⊗ A ˆ Z

ˆ Z

are O ⊗ A∞,p -equivariant isomorphisms preserving filtrations on both Z

sides, which are compatible with λ\ and the canonical morphism ∞ induced by h · , · i. Z−1 ⊗ A∞ → Z# −1 ⊗ A ˆ Z

∼

ˆ Z

(b) νˆs¯p,\ : A∞,p (1) → Vp Gm,¯s is an isomorphism of A∞,p -modules such ˆ p (1) to Tp Gm,¯s , where r(j \ , j ∨,\ )s¯ is the that r(j \ , j ∨,\ )s¯ νˆs¯\ maps Z \ value at s¯ of the above section r(j \ , j ∨,\ ) of (Z× (p),>0 )S such that j ◦φ = \ ∨,\ ∗ ∨,\ r(j , j )λT ◦ j . ∼ (c) The induced morphisms Gr−2 (βˆs¯p,\,0 ) : GrZ−2 ⊗ A∞,p → Vp Ts¯ and ˆ Z

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∼

#

Gr−2 (βˆs¯p,\,#,0 ) : GrZ−2 ⊗ A∞,p → Vp Ts¯∨ coincide with the compositions ˆ Z

ϕ−2 ⊗ A∞,p ˆ Z

∼

GrZ−2 ⊗ A∞,p

HomA∞,p (X ⊗ A∞,p , A∞,p (1))

→

ˆ Z

\ −1

Z

⊗ A

((j )

∞,p ∗

)

Z(p)

∼

HomA∞,p (X(T ) ⊗ A∞,p , A∞,p (1))

→

Z

ν ˆs¯p,\ ∼

∼

→ HomA∞,p (X(T ) ⊗ A∞,p , Vp Gm,¯s ) → Vp Ts¯ Z

and ∞,p ϕ# −2 ⊗ A ˆ Z

#

GrZ−2 ⊗ A∞,p

∼

HomA∞,p (Y ⊗ A∞,p , A∞,p (1))

→

ˆ Z

Z

((j ∨,\ )−1 ⊗ A∞,p )∗ Z(p)

∼

→

HomA∞ (X(T ∨ ) ⊗ A∞,p , A∞,p (1)) Z

ν ˆs¯p,\ ∼

∼

→ HomA∞ (X(T ∨ ) ⊗ A∞,p , Vp Gm,¯s ) → Vp Ts¯∨ , Z

#

∼

Z ˆ ˆ respectively, where ϕ# ˆ (Y ⊗ Z, Z(1)) is induced by ϕ0 −2 : Gr−2 → HomZ Z

by duality. p (d) Together with νˆ−1,¯ ˆs¯p,\ , the induced morphisms s := ν ∼ ϕˆp−1,¯s := Gr−1 (βˆs¯p,\,0 ) : GrZ−1 ⊗ A∞,p → Vp Bs¯ ˆ Z

and ∼

#

ˆp,\,#,0 ) : GrZ ⊗ A∞,p → Vp B ∨ ϕˆp,# −1 s¯ −1,¯ s := Gr−1 (βs¯ ˆ Z

determine each other by duality. By varying s¯ over geometric points p,0 of S, the (π1 (S, s¯)-invariant) HG -orbits of h,Z p ord (ϕˆp−1,¯s , νˆ−1,¯ s , ϕ−1,Hp , ϕ−2 , ϕ0 ) \,ord (where ϕord as in Proposition 5.2.4.11) de−1,Hp is determined by βHp termine a tuple ord ord ((B, λB , iB , ϕ−1,Hp , ϕord −1,Hp ), (ϕ−2,H , ϕ0,H ))

~ ord,ΦH (cf. [62, Prop. whose Z× (p) -isogeny class is parameterized by MH 1.4.3.4]).

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The Z× (p) -isogenies \,ord ) (G\ , λ\ : G\ → G∨,\ , i\ , j \ , j ∨,\ , [βˆp,\ ]HpG , βH p 1,Z

∼Z×

(p)

\,0

-isog.

\,0

(G , λ

\,0

:G

∨,\,0

→G

\,ord,0 , i , j \,0 , j ∨,\,0 , [βˆp,\,0 ]HpG , βH ) p \,0

1,Z

between tuples as in (5.2.4.14) are given by pairs of Z× (p) -isogenies (f \ : G\ → G\,0 , f ∨,\ : G∨,\,0 → G∨,\ ) such that we have the following: \ \ ∨,\ ∨,\ (i) There exists a section r(f \ , f ∨,\ ) of (Z× )f ◦ (p),>0 )S such that λ = r(f , f \,0 \ λ ◦f . (ii) f \ and f ∨,\ respect the compatible O ⊗ Z(p) -actions on G\ , G\,0 , G∨,\ , and Z

G∨,\,0 (defined by i\ and i\,0 ). (iii) j \ = (f \ )∗ ◦ j \,0 and j ∨,\,0 = (f ∨,\ )∗ ◦ j ∨,\ . \,ord \,ord,0 \ (iv) βH and βH are canonically identified under the Z× (p) -isogenies f and p p f ∨,\ . ∼ (v) For each geometric point s¯, the morphisms Vp (f \ ) : Vp G\s¯ → Vp G\,0 s¯ and ∼ Vp (f ∨,\ ) : Vp Gs∨,\,0 → Vp G∨,\ satisfy the condition that, for any represen¯ s¯ tatives βˆs¯p,\ = (βˆs¯p,\,0 , βˆs¯p,\,#,0 , νˆs¯p,\ ) and βˆs¯p,\,0 = (βˆs¯p,\,0,0 , βˆs¯p,\,#,0,0 , νˆs¯p,\,0 ) of p [βˆp,\ ]HpG and [βˆp,\,0 ]HpG , respectively, the HG -orbits of 1,Z 1,Z

1,Z

(Vp (f \ ) ◦ βˆs¯p,\,0 , Vp (f ∨,\ )−1 ◦ βˆs¯p,\,#,0 , r(f \ , f ∨,\ )s−1 ˆs¯p,\ ) ¯ ν and (βˆs¯p,\,0,0 , βˆs¯p,\,#,0,0 , νˆs¯p,\,0 ) coincide, where r(f \ , f ∨,\ )s¯ is the value at s¯ of the above section r(f \ , f ∨,\ ) of \ \ ∨,\ ∨,\ (Z× )f ◦ λ\,0 ◦ f \ . (p),>0 )S such that λ = r(f , f Proof. As in [62, Sec. 1.4.3] and in the proof of Proposition 1.3.2.14, this can be proved by replacing any tuple as in (5.2.4.14) up to Z× (p) -isogeny, as in the statement ∼

of this proposition, with a tuple such that j \ : X ⊗ Z(p) → X(T ) ⊗ Z(p) (resp. j ∨,\ : Z

∼

Z

Y ⊗ Z(p) → X(T ∨ ) ⊗ Z(p) ) maps X (resp. Y ) to X(T ) (resp. X(T ∨ )), and such that, Z

Z

at each geometric point s¯ of S, the assigned βˆs¯p,\ = (βˆs¯p,\,0 , βˆs¯p,\,#,0 , νˆs¯p,\ ) satisfies the ˆ p (resp. Z# ⊗ Z ˆ p , resp. condition that βˆs¯p,\,0 (resp. βˆs¯p,\,#,0 , resp. νˆs¯p,\ ) maps Z−1 ⊗ Z −1 ˆ Z

ˆ Z

p ˆ p (1)) to Tp G\s¯ (resp. Tp G∨,\ Z s ). Then the tuple determines and is s¯ , resp. T Gm,¯ determined by a tuple as in (5.2.4.12), as desired. (These can be simultaneously achieved because of the existence of the section r(j \ , j ∨,\ ) of (Z× (p),>0 )S . The proof is similar to that of [62, Prop. 1.4.3.4], and hence omitted.)

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Construction 5.2.4.15. (Compare with Construction 1.3.2.16.) Suppose that H = Hp Hp is of standard form as in Definition 3.2.2.9, and that Hp is neat. Consider the degenerating family ord ~ ord,tor (G, λ, i, αHp , αH )→M p H,Σord

(5.2.4.16)

~ ord as in Theorem 5.2.1.1. Let ~Zord = ~Zord of type M H [(ΦH ,δH ,σ)] be any stratum of ord,tor + ~ M such that σ ⊂ PΦH is a top-dimensional cone in ΣΦH (in Σord ). Let H,Σord ord

(G\ ord , λ\ ord , i\ ord ) → ~Z ~ Z

~ Z

(5.2.4.17)

~ Z

ord

denote the pullback of the (G, λ, i) in (5.2.4.16) to ~Z , the closure of ~Zord in ~ ord,tor ~ ord M . Since σ is top-dimensional, the canonical morphism ~Zord → C ΦH ,δH is an H,Σord ord

~ ord , its pullback to ~Z is undefined. isomorphism. Since αHp is defined only over M H ord does extend (necesOn the other hand, by condition (5) of Definition 3.4.2.10, αH p ord,tor ~ sarily uniquely) to the whole M ord . The goal of this construction is to define a H,Σ

partial pullback of αHp to ~Zord , which still retains some information of αHp , and to ord give a more precise description of the pullback of (the unique extension of) αH to p ord

~Z . (The argument will be very similar to that in Construction 1.3.2.16, but we will spell out the details for the sake of certainty.) Let n ≥ 1 be any integer such that U(n) ⊂ H, such that n = n0 pr , where n0 ≥ 1 is an integer such that U(n0 ) ⊂ Hp , and where r = depthD (Hp ) ≥ 0 (cf. bal r Definition 3.2.2.9) is the integer such that Up,1 (p ) ⊂ Hp ⊂ Up,0 (pr ), and let us fix any choice of (Zn , Φn , δn ). Consider any top-dimensional cone σ 0 contained in σ that is smooth for the integral structure defined by SΦn , we have a canonical morphism ~ ord ~ ord X etale), inducing a morphism from Φn ,δn ,σ 0 → XΦH ,δH ,σ (which might not be finite ´ 0 ~Zord = of the source to the σ-stratum ~Zord = ~Zord the σ -stratum ~Zord n [(Φn ,δn ,σ 0 )] [(ΦH ,δH ,σ)] of the target (although the scheme-theoretic preimage of latter might not be the former), which can be identified with the canonical morphism (5.2.4.8). Let us denote the pullback of (5.2.4.17) to ~Zord n by (G\Zn , λ\Zn , i\Zn ) → ~Zord n .

(5.2.4.18)

~ ord Over each affine open formal subscheme Spf(R, I) of X Φn ,δn ,τ , such that S0 = Spec(R/I) is the τ -stratum of S = Spec(R), where both R and R/I are regular domains, we have a degenerating family (GS , λS , iS , αn0 ,η , αpord r ,η ) → S of type ~ ord = M ~ ord M . A priori, the level structure α is defined only over the generic n0 ,η n U1bal (n) point η of S (and it only extends to the largest open subscheme of S over which the pullback of GS is an abelian scheme). But since n0 is prime to the residue characteristics of S, by the same argument as in Construction 1.3.2.16, it induces a triple βn\ 0 ,S := (βn\,0 , βn\,#,0 , νn\ 0 ,S ) over S, with pullback βn\ 0 ,S0 := (βn\,0 , βn\,#,0 , νn\ 0 ,S0 ) 0 ,S 0 ,S 0 ,S0 0 ,S0

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to S0 . On the other hand, since (5.2.4.16) satisfies condition (5) of Definition 3.4.2.10, by Lemmas 4.1.4.19 and 4.1.4.20, ord,0 0 mult → Gη [pr ], αp\,ord r ,η = (αpr ,η : (GrD r )η p ×

r ord r → G∨ αpord,#,0 : (Gr0D# )mult r ,η η η [p ], νpr ,η ∈ (Z/p Z)η ) pr

satisfies Condition 4.1.4.1 and extends to a triple \,ord,0 βpord : (Gr0Dpr )mult → G\S [pr ], r ,S = (βpr ,S S ×

r → GS∨,\ [pr ], νp\,ord βp\,ord,#,0 : (Gr0D# )mult r ,S ∈ (Z/p Z)S ), r ,S S pr

with pullback to S0 denoted by βpord = (βp\,ord,0 , βp\,ord,#,0 , νp\,ord r ,S r ,S r ,S r ,S ). By the con0 0 0 0 ord ord ~ struction of XΦn ,δn ,τ , the triple βpr ,S induces, as in Proposition 4.1.4.21, the prescribed pair (ϕ−2,pr , ϕ0,pr ) in Φpr (induced by Φn ). As in Construction 1.3.2.16, by analyzing βn0 ,S as in the case of αn0 ,η as in [62, Sec. 5.2.2–5.2.3], we see that βn0 ,S retains almost all information of αn0 ,η , including the pairing e10,n0 to be compared with d10,n0 , as in [62, Lem. 5.2.3.12 and Thm. 5.2.3.14], except that it loses information about the pairing e00,n0 to be compared with d00,n0 . On the other hand, by construction, βpord r ,S retains all information of αpord Hence, if we denote the pullback of (5.2.4.17) to S0 by r ,η . (G\S0 , λ\S0 , i\S0 ) → S0 , then (G\S0 , λ\S0 , i\S0 , αn\ 0 ,S0 , αp\,ord r ,S ) → S0 determines and is de0 termined by (the prescribed (Zn , Φn , δn ) and) the pullback to S0 of the tautological ord ∨,ord ~ ord (up to isomorphisms inducing )) over C object ((B, λB , iB , ϕord −1,n ), (cn , cn Φn ,δn automorphisms of Φn ; i.e., elements of ΓΦn ; see Lemma 1.3.2.11). By patching over varying S, we obtain (with (G~\ ord , λ~\ ord , i~\ ord ) already defined as in (5.2.4.18)) a Zn Zn Zn tuple (G~\ ord , λ~\ ord , i~\ ord , β \ Zn

Zn

, β \,ord ) n0 ,~ Zord pr ,~ Zord n n

Zn

∼ ~ ord → ~Zord n = CΦn ,δn

(5.2.4.19)

such that the previous sentence is true with S0 replaced with ~Zord Since n . ord ord ∨,ord HG1,Z /U1bal (n)G1,Z acts compatibly on (β \ ~ ord , β \,ord ) and (ϕ , c , c ), n −1,n n r ~ ord n0 ,Zn

p ,Zn

the latter action being compatible with the HG1,Z /U1bal (n)G1,Z -torsor structure of (5.2.4.8), by forming the HG1,Z /U1bal (n)G1,Z -orbit (β \ p ~ ord , β \,ord~ ord ) of H ,Zn

(β \

Hp ,Zn

\,ord ~ ord , β r ~ ord ), we can descend (5.2.4.19) to a tuple

n0 ,Zn

p ,Zn

(G~\ ord , λ~\ ord , i~\ ord , β \ Z

Z

Z

Hp ,~ Zord

~ Φord,δ , , β \,ord~ ord ) → ~Zord ∼ =C H H H ,Z

(5.2.4.20)

p

where the first three entries form the pullback of (5.2.4.17) to ~Zord , which determines and is determined by (the prescribed (ZH , ΦH , δH ) and) the tautological object
ord ord ord ∨,ord ~ ord (B, λB , iB , ϕord ) →C (5.2.4.21) −1,H ), (ϕ−2,H , ϕ0,H ), (cH , cH ΦH ,δH (up to isomorphisms inducing automorphisms of ΦH ; i.e., elements of ΓΦH ). Since the tautological object (5.2.4.21) is independent of the choice of n0 , so is the tuple (5.2.4.20).

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321

By abuse of language, we say that (G\ ord , λ\ ord , i\ ord , β \ ~ Z

~ Z

ord

, β \,ord~ ord ) → ~Z p ~ ord

(5.2.4.22)

Hp ,Z

H ,Z

~ Z

ord

is the pullback of the degenerating family (5.2.4.16) to ~Z , with the convention ord that (as in the case of (G, λ, i, αHp , αH ) itself) β \ p ~ ord is defined only over ~Zord , p H ,Z

\

\

\

while (G , λ , i ) (resp.

β \,ord~ ord ) Hp ,Z

ord

is defined (resp. extends) over all of ~Z

as in

(5.2.4.17). As in the proof of (7) of Theorem 5.2.1.1, in every step of our construction of ord,tor ~ MH,Σord , the characteristic zero fiber of the degenerating families over the boundary charts we have used are the pullback from S0 = Spec(F0 ) to S0,rH = Spec(F0 [ζprH ]) of the corresponding ones over toroidal boundary charts for MH , the only deference being that we have only considered ordinary cusp labels in the construction for ~ ord,tor M . Therefore, there is a degenerating family H,Σord ~ ord,tor (G, λ, i, αH ) → M H,Σord

(5.2.4.23)

of type MH , with the same (G, λ, i) as in (5.2.4.16), where αH is defined only over ~ ord ⊗ Q, such that the pair (αHp , αord ) ⊗ Q is induced by αH as in Proposition M Hp H Z

Z

3.3.5.1. Therefore, by repeating the argument as in Construction 1.3.2.16 for αH , we obtain a pullback (G\ ord , λ\ ord , i\ ord , β \ ~ Z

~ Z

as in (5.2.4.22), where β \

H,~ Zord ⊗ Q

~ Z

) H,~ Zord ⊗ Q

ord

(5.2.4.24)

Z

is defined only over ~Zord ⊗ Q, such that the pair Z

Z

(β \ p ~ ord , β \,ord~ ord ) ⊗ Q H ,Z Hp ,Z Z

→ ~Z

is induced by

β \ ~ ord H,Z ⊗Q

by an obvious analogue of Propo-

Z

sition 3.3.5.1. (This finishes Construction 5.2.4.15.) Proposition 5.2.4.25. (Compare with Proposition 1.3.2.24.) By considering compatible Q× -isogenies (f : G\ → G\,0 , f ∨ : G∨,\,0 → G∨,\ ) inducing isomorphisms on the torus parts, we can define ordinary Hecke twists of the tau\ \,ord ~ ord tological object (G\ , λ\ , i\ , βH ) → C p , βH ΦH ,δH by elements g = (g0 , gp ) ∈ p ∞,p ord ∞ G1,Z (A ) × P1,Z,D (Qp ) ⊂ G1,Z (A ) such that the image of gp under the canonical ord homomorphism Pord 1,Z,D (Qp ) → Ph,D (Qp ) satisfies the condition defined by the filtration D−1 on GrZ−1 ⊗ Zp as in Section 3.3.4, and define the Hecke action of (such eleˆ Z ∞,p

~ ord ments of) G1,Z (A ) × Pord 1,Z,D (Qp ) on the collection {CΦH ,δH }HG1,Z with H of standard form, realized by quasi-finite flat surjections pulling tautological objects back to ordinary Hecke twists, which is compatible with the Hecke action of (suitable ele~ ord,ΦH }H 0 (with H of stanments of) Gh,Z (A∞,p ) × Pord h,Z,D (Qp ) on the collection {MH G h,Z

~ ord,ΦH (with varying H) and ~ ord dard form) under the canonical morphisms C ΦH ,δH → MH 0 ∞,p the canonical homomorphism G1,Z (A∞,p ) × Pord ) × Pord,0 1,Z,D (Qp ) → Gh,Z (A h,Z,D (Qp ).

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Such a Hecke action enjoys the properties (under various conditions) concerning ´etaleness, finiteness, being isomorphisms between formal completions along fibers over Spec(Fp ), and inducing absolute Frobenius morphisms on fibers over Spec(Fp ) for elements of Up type as in Proposition 5.2.2.2 and Corollaries 5.2.2.3, 5.2.2.4, and 5.2.2.5. (We omit the details for simplicity.) Over the subcollection indexed by HG1,Z with neat Hp (for H = Hp Hp of standard form), the Hecke action ~ ord of (suitable elements of) G1,Z (A∞,p ) × Pord 1,Z,D (Qp ) on {CΦH ,δH }HG1,Z is compatible with the Hecke action of (suitable elements of) P0Z (A∞,p ) × Pord,0 Z,D (Qp ) on the colord ord ~ ~ lection of strata {Z[(ΦH ,δH ,σ)] } above {CΦH ,δH }HG1,Z (cf. Proposition 5.2.2.2) under the canonical homomorphism P0 (A∞,p ) × Pord,0 (Qp ) → G1,Z (A∞,p ) × Pord (Qp ) ∼ = Z

1,Z,D

Z,D

∞ (P0Z (A∞,p ) × Pord,0 Z,D (Qp ))/U2,Z (A ). × By also considering Q -isogenies (f : G\ → G\,0 , f ∨ : G∨,\,0 → G∨,\ ) inducing Q× -isogenies on the torus parts, we can also define ordinary Hecke \ \,ord ~ ord twists of the tautological object (G\ , λ\ , i\ , βH ) → C p , βH ΦH ,δH by elements p ∞ (Q ))/U (A ) such that the image of gp ung = (g0 , gp ) ∈ (PZ (A∞,p ) × Pord p 2,Z Z,D ord ord der the canonical homomorphism PZ,D (Qp )/U2,Z (Qp ) → Ph,D (Qp ) satisfies the condition defined by the filtration D−1 on GrZ−1 ⊗ Zp as in Section 3.3.4, and ˆ Z

∞ define the Hecke action of (such elements of) (PZ (A∞,p ) × Pord Z,D (Qp ))/U2,Z (A ) ` ~ ord on the collection { CΦH ,δH }HPZ /HU2,Z (with H of standard form), where the disjoint unions are over classes [(ZH , ΦH , δH )] sharing the same ZH compatible with D, realized by quasi-finite flat surjections pulling tautological objects back to ordinary Hecke twists, which induces an action of G0l,Z (A∞ ) = ∞,p 0 ) × Pord,0 PZ (A∞ )/P0Z (A∞ ) ∼ = (PZ (A∞,p ) × Pord Z,D (Qp )) on the index Z,D (Qp ))/(PZ (A sets {[(ZH , ΦH , δH )]}, which is compatible with the Hecke action of (suitable ele∞ 0 ∞,p ∞ ∼ 0 ) × Pord,0 ments of) (PZ (A∞,p ) × Pord Z,D (Qp ))/UZ (A ) = Gl,Z (A ) ×(Gh,Z (A h,Z,D (Qp )) ` ~ ord,ΦH on the collection { M }H 0 (with H of standard form, with the same inH

G h,Z

dex sets and the same induced action of G0l,Z (A∞ )) under the canonical mor~ ord,ΦH (with varying H) and the canonical homomorphism ~ ord phisms C ΦH ,δH → MH ∞ 0 ∞ 0 ∞,p (PZ (A∞,p ) × Pord ) × Pord,0 Z,D (Qp ))/U2,Z (A ) → Gl,Z (A ) ×(Gh,Z (A h,Z,D (Qp )). Over p the subcollection indexed by HPZ /HU2,Z with neat H (for H = Hp Hp of standard ∞ form), the Hecke action of (suitable elements of) (PZ (A∞,p ) × Pord Z,D (Qp ))/U2,Z (A ) ` ~ ord on { CΦH ,δH }HPZ /HU2,Z is compatible with the Hecke action of (suitable ele~ ord ments of) PZ (A∞,p ) × Pord Z,D (Qp ) on the collection of strata {Z[(ΦH ,δH ,σ)] } above ` ~ ord { CΦH ,δH }HPZ /HU2,Z (cf. Proposition 5.2.2.2) under the canonical homomorphism ∞,p ∞ PZ (A∞,p ) × Pord ) × Pord Z,D (Qp ) → (PZ (A Z,D (Qp ))/U2,Z (A ). × In the Z(p) -isogeny class language as in Proposition 5.2.4.13, the morphism ~ ord ~ ord [g] : C ΦH0 ,δH0 → CΦH ,δH , ∞ 0 0 for g = (g0 , gp ) ∈ (PZ (A∞,p ) × Pord Z,D (Qp ))/U2,Z (A ) as above such that HPZ /HU2,Z ⊂ −1 g(HPZ /HU2,Z )g and such that [(ΦH , δH )] is g-assigned to [(ΦH0 , δH0 )] with a pair

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isomorphisms ∼

∼

(fX : X ⊗ Q → X 0 ⊗ Q, fY : Y 0 ⊗ Q → Y ⊗ Q) Z

Z

Z

Z

as in [62, Prop. 5.4.3.8], is characterized by \,ord [g]∗ (G\ , λ\ : G\ → G∨,\ , i\ , j \ , j ∨,\ , [βˆp,\ ]HpG , βH ) p 1,Z

∼

\,00

(G

Z× -isog. (p)

\,00

,λ

\,00

:G

∨,\,00

→G

\,00

,i

,

\,ord,00 fX ◦ j \,00 , fY−1 ◦ j ∨,\,00 , [βˆp,\,00 ◦ g0 ]HpG , βH ) p 1,Z

over CΦH0 ,δH0 , where \,ord (G\ , λ\ : G\ → G∨,\ , i\ , j \ , j ∨,\ , [βˆp,\ ]HpG , βH ) p 1,Z

and \,ord (G\,0 , λ\,0 : G\,0 → G∨,\,0 , i\,0 , j \,0 , j ∨,\,0 , [βˆp,\,0 ]H0,p , βH ) 0 p G1,Z

are representatives of the tautological ~ ord C , respectively, where

Z× (p) -isogeny

~ ord classes over C ΦH ,δH and

ΦH0 ,δH0

\,ord,00 (G\,00 , λ\,00 : G\,00 → G∨,\,00 , i\,00 , j \,00 , j ∨,\,00 , [βˆp,\,00 ]H0,p , βH ) p G1,Z

is the ordinary Hecke twists of the latter by gp (defined as in Proposition 3.3.4.9, with details omitted for simplicity) realized by some pair (G\,0 → G\,00 , G∨,\,00 → G∨,\,0 ) of isogenies of p-power degrees (with canonically induced additional structures), and where the rational level-Hp structure [βˆp,\,00 ◦ g0 ]HpG

1,Z

of (G\,00 , λ\,00 , i\,00 , fX ◦ j \,00 , fY−1 ◦ j ∨,\,00 ) of type (L ⊗ A∞,p , h · , · i, Z ⊗ A∞,p , Φ) is ˆ Z

Z

p ˆp,\,00 ◦ g0 , ~ ord determined at each geometric point s¯ of C ΦH0 ,δH0 by the HG1,Z -orbit of βs¯ where βˆs¯p,\,00 is any representative of the rational level-H0,p structure [βˆs¯p,\,00 ] 0,p of HG

1,Z

\,00

(G

\,00

,λ

\,00

,i

,j

\,00

,j

∨,\,00

) of type (L ⊗ A Z

ned to s¯ by [βˆ\,00 ]H0,p ).

∞,p

, h · , · i, Z ⊗ A

∞,p

0

, Φ ) based at s¯ (assig-

ˆ Z

G1,Z

Proof. As in the proof of Proposition 1.3.2.24, the first assertions of both of the first two paragraphs, and the whole third paragraph, can be justified as in the ~ ord . (We omit the details for simplicity.) As for the second assertions case of M H in both of the first two paragraphs, it suffices to note that the pullback of the ordinary Hecke twist of (5.2.4.16) is the ordinary Hecke twist of (5.2.4.20), the ~ ord latter of which can be identified with the tautological object over C ΦH ,δH under the ∼ ord ord ~ ~ canonical isomorphism Z[(ΦH ,δH ,σ)] → CΦH ,δH (for any top-dimensional σ, when Hp and hence H are neat).

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By Proposition 4.2.1.37 and its proof, we have the following: ~ ord Lemma 5.2.4.26. (Compare with Lemma 1.3.2.25.) The quotient Ξ ΦH ,δH depends only on HP0Z , and is a torsor under the torus EΦH with character group SΦH . If r = depthD (H) and if n0 ≥ 1 is any integer prime to p such that U p (n0 ) ⊂ Hp , bal r so that U1bal (n) = U p (n0 )Up,1 (p ) ⊂ H are both of standard form and equally deep, and if we fix the choice of (Zn and) Φn , then the canonical morphism ~ ord → Ξ ~ ord ~ ord × ~S0,r , (5.2.4.27) Ξ =Ξ Φn ,δn

ΦH ,δH ,rn

ΦH ,δH

~ S0,rH

n

where rn = rU1bal (n) = max(rD , r), (is finite ´etale and) is an HP0Z /U1bal (n)P0Z ∼ = ord ord ), where HP0Z -torsor (see page 254 for the definition of Hn,Gess nUess Hn,Gess nUess Z Z h,Zn

h,Zn

n

n

and U1bal (n)P0Z are as in Definition 1.2.1.12, and induces an isomorphism ∼ ~ ord ~ ord /(HP0 /U bal (n)P0 ) → Ξ ΞΦH ,δH ,rn . (5.2.4.28) Φn ,δn 1 Z Z Proof. The statements are self-explanatory. Lemma 5.2.4.29. (Compare with Lemmas 1.3.2.28 and 5.2.4.7.) Suppose n = n0 pr for some integer n0 ≥ 1 prime to p and some integer r ≥ 0. Suppose ord ∨,ord ) (B, λB , iB , ϕord −1,n ), (Zn , Φn = (X, Y, φ : Y ,→ X, ϕ−2,n , ϕ0,n ), δn ), and (cn , cn ord ord ord ~ ~ ~ are the tautological objects over CΦn ,δn . The torus torsor S := ΞΦn ,δn → CΦn ,δn is universal for the additional structure τnord over noetherian normal schemes over ~ ord (inducing dominant morphisms over irreducible components) satisfying cerC Φn ,δn tain symplectic and liftability conditions, which can be re-interpreted as follows: ~ ord → C ~ ord parameterizes tuples S=Ξ Φn ,δn Φn ,δn (G\ , λ\ : G\ → G∨,\ , i\ , τ, βn0 , βpord r ), where: (a) G\ , G∨,\ , λ\ , and i\ are as in Lemma 5.2.4.10. ∼ ⊗ −1 (b) τ : 1Y × X → (c∨ × c)∗ PB is a trivialization of biextensions such that ∗ (IdY × φ) τ is symmetric, and such that (iY (b) × IdX )∗ τ = (IdY × iX (b? ))∗ τ for all b ∈ O. Then τ induces homomorphisms ι : Y → G\ and ι∨ : X → G∨,\ compatible with the homomorphisms φ : Y ,→ X and λ\ : G\ → G∨,\ , and induces an O-equivariant homomorphism λ : G[n] → G∨ [n]. (c) βn0 = (βn0 0 , βn#,0 , νn0 ) is a principal level-n0 structure of (G\ , λ\ , i\ , τ ) 0 ∼ p ˆ ˆ p ), where β 0 : (L/n0 L)S → of type (L ⊗ Z , h · , · i, Z ⊗ Z G[n0 ] and βn#,0 : n0 0 Z

∼

ˆ Z

(L# /n0 L# )S → G∨ [n0 ] are O-equivariant isomorphism respecting the canonical ∼ filtrations on both sides, and νn0 : ((Z/n0 Z)(1))S → µn0 ,S is an isomorphism, inducing on the graded pieces the ϕ−2,n0 , ϕ−1,n0 , and ϕ0,n0 , respectively, induced by the given ϕ−2,n , ϕ−1,n , and ϕ0,n , which are compatible with the canonical morphisms L ,→ L# and λ : G[n0 ] → G∨ [n0 ]. Moreover, βn0 satisfies the liftability condition that, for each integer m0 ≥ 1 such that n0 |m0 , there exists a finite ´etale covering of S over which there exists an analogous triple βm0 lifting the pullback of βn0 .

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325

is, as in Lemma 5.2.4.10, an ordinary level-Hp structure of type (d) βpord = βp\,ord r r (L ⊗ Zp , h · , · i, Z ⊗ Zp , D), whose definition does not require τ (and the group ˆ Z

Z

schemes G[pr ] and G∨ [pr ]). Proof. The statements are self-explanatory. Proposition 5.2.4.30. (Compare with Propositions 1.3.2.31 and 5.2.4.11.) The to~ ord ~ ord rus torsor S := Ξ ΦH ,δH → CΦH ,δH is universal for the additional structure τH over ~ ord noetherian normal schemes over C ΦH ,δH (inducing dominant morphisms over irreducible components) satisfying certain symplectic and liftability conditions, which can be interpreted as parameterizing tuples ord (G\ , λ\ : G\ → G∨,\ , i\ , τ, βHp , βH ), p

where G\ , G∨,\ , λ\ , and i\ are as in Lemma 5.2.4.10, where τ is as in Lemma \,ord ord 5.2.4.29, where βH = βH is an ordinary level-Hp structure of (G\ , λ\ , i\ ) p p of type (L ⊗ Zp , h · , · i, Z ⊗ Zp , D) as in Lemma 5.2.4.11, for any Z ⊗ Zp lifting Zpr Z

ˆ Z

ˆ Z

and compatible with D; and where βHp is a level-Hp structure of (G\ , λ\ , i\ , τ ) ˆ p , h · , · i, Z ⊗ Z ˆ p ), which is a collection {βH }n , where n0 ≥ 1 runs of type (L ⊗ Z n0 0 Z

ˆ Z

over integers prime to p such that U p (n0 ) ⊂ Hp , such that each βHn0 (where Hn0 := Hp /U p (n0 )) is a subscheme of a

IsomS (L/n0 L)S , G[n0 ] × IsomS (L# /n0 L# )S , G∨ [n0 ] S
 × IsomS ((Z/n0 Z)(1))S , µn0 ,S S

over S, where the disjoint union is over representatives (Zn0 , Φn0 , δn0 ) (with the same (X, Y, φ)) in the (ZHp , ΦHp , δHp ) induced by (ZH , ΦH , δH ), that becomes the disjoint union of all elements in the Hn0 -orbit of some principal level-n0 structure ˆ p ), as in Lemma 5.2.4.29, for any ˆ p , h · , · i, Z ⊗ Z βn0 of (G\ , λ\ , i\ , τ ) of type (L ⊗ Z ˆ Z

Z

ˆ p lifting Zn ; and where βH Z⊗Z is mapped to βHn0 (under the canonical mor0 m0 ˆ Z

phism, which we omit for simplicity) when p - m0 and n0 |m0 . ∨ ∼ ˆ Let SΦH be the unique lattice in SΦ1 ⊗ Q such that S∨ Φ1 /SΦH = U2,Z (Z)/HU2,Z . Z

~ ord ~ ord Then S = Ξ ΦH ,δH → CΦH ,δH is torsor under the split torus EΦH with character group SΦH , equipped a homomorphism ~ Φord,δ ) : ` 7→ Ψ ~ ord SΦH → Pic(C ΦH ,δH (`) H H (by the torus torsor structure; see Proposition 4.2.1.46 and (4.2.1.49)), assigning ~ ord (`) over C ~ ord to each ` ∈ SΦH an invertible sheaf Ψ ΦH ,δH ΦH ,δH (up to isomorphism), together with isomorphisms ~ ord,∗ ~ ord ∆ ΦH ,δH ,`,`0 : ΨΦH ,δH (`)

⊗

OC ~ ord

ΦH ,δH

∼ ~ ord ~ ord (`0 ) → Ψ ΨΦH ,δH (` + `0 ) ΦH ,δH

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for all `, `0 ∈ SΦH , satisfying the necessary compatibilities with each other making ~ ord (`) an O ~ ord -algebra, such that ⊕ Ψ ΦH ,δH C ΦH ,δH

`∈SΦH



∼ ~ ord Ξ ΦH ,δH = SpecO

⊕

~ ord C ΦH ,δH

`∈SΦH

 ord ~ ΨΦH ,δH (`) .

When ` = [y ⊗ χ] for some y ∈ Y and χ ∈ X, we have a canonical isomorphism ~ ord (`) ∼ Ψ = (c∨ (y), c(χ))∗ PB . ΦH ,δH ~ ord Proof. This follows from Lemmas 5.2.4.26 and 5.2.4.29, by realizing Ξ ΦH ,δH ,rn as ord ~ a quotient of some ΞΦn ,δn , and by finite flat descent. For each rational polyhedral cone σ ⊂ (SΦH )∨ R as in Definition 1.2.2.2, we have an affine toroidal embedding   ord ord ord ∼ ~ ~ ~ ΞΦH ,δH ,→ ΞΦH ,δH (σ) = SpecO ⊕ ΨΦH ,δH (`) (5.2.4.31) ~ ord C ΦH ,δH

`∈σ ∨

~ ord , where (cf. (1.3.2.32) and (4.2.2.1)), both sides being relative affine over C ΦH ,δH ~ ord (σ) → C ~ ord Ξ smooth when the cone σ is smooth, with its σ-stratum ΦH ,δH ΦH ,δH is   ~ ord (`) as in (4.2.2.2) (cf. (1.3.2.33)), which ~ ord ⊕ Ψ Ξ = Spec ΦH ,δH ,σ

OC ~ ord

ΦH ,δH

`∈σ ⊥

ΦH ,δH

is by itself a torsor under the torus EΦH ,σ with character group σ ⊥ . For each ΓΦH -admissible rational polyhedral cone decomposition ΣΦH as in Definition 1.2.2.4, we have the toroidal embedding ord

ord

~ ~ ~ ord Ξ ΦH ,δH ,→ ΞΦH ,δH = ΞΦH ,δH ,ΣΦ

H

as in (4.2.2.4) (cf. (1.3.2.34)), the right-hand side being only locally of finite type ~ ord , with an open covering over C ΦH ,δH ord

~ Ξ ΦH ,δH =

∪

σ∈ΣΦH

~ ord (σ), Ξ ΦH ,δH

(5.2.4.32)

~ ord Ξ ΦH ,δH ,σ

(5.2.4.33)

(cf. (1.3.2.35)) inducing a stratification ord

a

~ Φ ,δ = Ξ H H

σ∈ΣΦH

` (cf. (1.3.2.36)). (The notation “ ” only means a set-theoretic disjoint union. The ord

~ Φ ,δ .) Concretely, if σ algebro-geometric structure is still the one inherited from Ξ H H ~ ord (σ) ⊂ Ξ ~ ord (ρ), but Ξ ~ ord is a face of ρ, then ρ∨ ⊂ σ ∨ and Ξ ΦH ,δH ΦH ,δH ΦH ,δH ,ρ is contained ord ord ord ~ ~ ~ in the closure of Ξ . The closure of Ξ in Ξ (ρ) is ΦH ,δH ,σ ΦH ,δH ,σ ΦH ,δH   ~ ord ~ ord (`) Ξ ⊕ Ψ (5.2.4.34) ΦH ,δH ,σ (ρ) := SpecO ΦH ,δH ~ ord C ΦH ,δH

`∈σ ⊥ ∩ ρ∨

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(cf. (1.3.2.37)). In this case, the open embedding ~ ord ~ ord Ξ Φ ,δ ,σ ,→ ΞΦ ,δ ,σ (ρ) H

H

H

page 327

(5.2.4.35)

H

(cf. (1.3.2.38)) is an affine toroidal embedding (as in [62, Def. 6.1.2.3]) for the torus ~ ord ~ ord torsor Ξ ΦH ,δH ,σ → CΦH ,δH . In Section 4.2.2, we have also defined ∧ ~ ord ~ ord , (5.2.4.36) X ΦH ,δH ,σ = (ΞΦH ,δH (σ))Ξ ~ ord ΦH ,δH ,σ

~ ord (σ) along its σ-stratum Ξ ~ ord (cf. (1.3.2.39)) the formal completion of Ξ ΦH ,δH ΦH ,δH ,σ . + ord ord ~ When σ ⊂ PΦH appears in ΣΦH ∈ Σ , the quotient XΦH ,δH ,σ /ΓΦH ,σ is isomorphic ∼ ~ ord,tor to the formal completion of M along its [(ΦH , δH , σ)]-stratum ~Zord [(ΦH ,δH ,σ)] = H,Σord ord 0 0 0 ~ Ξ /ΓΦ ,σ , as in Theorem 5.2.1.1. If there is a surjection (Z , Φ , δ )  ΦH ,δH ,σ

H

H

H

H

(ZH , ΦH , δH ) such that σ is mapped to a face of a cone ρ ⊂ P+ Φ0H under the canonical + mapping PΦH → PΦ0H , and if ρ ∈ ΣΦ0H ∈ Σord , then ~Zord [(Φ0 ,δ 0 ,ρ)] is contained in the H

H

ord

ord

~ ord,tor ~ closure ~Z[(ΦH ,δH ,σ)] of ~Zord [(ΦH ,δH ,σ)] in MH,Σord , and the completion of Z[(ΦH ,δH ,σ)] along ~Zord0 0 is canonically isomorphic to [(ΦH ,δH ,ρ)]

∧ ~ ord ~ ord X ΦH ,δH ,σ,ρ := (ΞΦH ,δH ,σ (ρ))Ξ ~ ord

(5.2.4.37)

ΦH ,δH ,ρ

~ ord ~ ord (cf. (1.3.2.40)), the formal completion of Ξ ΦH ,δH ,σ (ρ) along its ρ-stratum ΞΦH ,δH ,ρ . ~ ord Consider X ΦH ,δH

Lemma 5.2.4.38. (Compare with Lemma 1.3.2.41.)

=

ord

~ ord ~ X ΦH ,δH ,ΣΦH , the formal completion of ΞΦH ,δH along the union of the σ-strata + ~ ord Ξ ΦH ,δH ,σ for σ ∈ ΣΦH and σ ⊂ PΦH . Then we have a canonical morphism ~ ord,tor ~ ord X ΦH ,δH → MH,Σord

(5.2.4.39)

(cf. (1.3.2.42)) inducing a canonical isomorphism ∼ ~ ord,tor ∧ ~ ord /ΓΦ → X (M ord ) ΦH ,δH

H

H,Σ

(5.2.4.40)

∪~ Zord [(Φ

H ,δH ,σ)]

~ ord (cf. (1.3.2.43)), where ∪ ~Zord [(ΦH ,δH ,σ)] is the union of all strata Z[(ΦH ,δH ,σ)] with σ ∈ ∨ ∨ ΣΦH (and σ ⊂ P+ ~ ord,tor (resp. LieG∨ /M ~ ord,tor , ΦH ), under which the pullback of LieG/M H,Σord

H,Σord

∨ resp. λ∗ : Lie∨ ~ ord,tor → LieG/M ~ ord,tor ) can be canonically identified with the G ∨ /M H,Σord

pullback of Lie∨ ~ ord G \ /C

ΦH ,δH

Lie∨ ~ ord G\ /C Φ ,δ H

H

H,Σord

(resp. Lie∨ ~ ord G∨,\ /C

ΦH ,δH

, resp. (λ\ )∗ : Lie∨ ~ ord G∨,\ /C

→

ΦH ,δH

). For each stratum ~Zord [(ΦH ,δH ,σ)] , the isomorphism (5.2.4.40) is compa-

∼ ~ ord,tor ∧ ~ ord tible with the isomorphism X ΦH ,δH ,σ /ΓΦH ,σ → (MH,Σ )~ Zord

in (5) of Theo-

[(ΦH ,δH ,σ)]

~ ord ~ ord rem 5.2.1.1 (under the canonical morphisms X ΦH ,δH ,σ /ΓΦH ,σ → XΦH ,δH /ΓΦH and ~ ord,tor ~ ord,tor (M )∧ → (M )∧ Zord ). (Such isomorphisms are induced by H,Σord ~ H,Σord ∪ ~ Zord [(ΦH ,δH ,σ)]

[(ΦH ,δH ,σ)]

~ ord strata-preserving isomorphisms from ´etale neighborhoods of points of Ξ ΦH ,δH ,σ in ord,tor ord ord ~ ~ ~ ΞΦH ,δH (σ) to ´etale neighborhoods of points of Z[(ΦH ,δH ,σ)] in MH,Σord .)

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Proof. By using the various universal properties, the same argument in the proof of Lemma 1.3.2.41 also works here. Proposition 5.2.4.41. (Compare with Propositions 1.3.1.15, 1.3.2.24, 1.3.2.45, 5.2.2.2, 5.2.4.25.) By considering compatible Q× -isogenies (f : G\ → G\,0 , f ∨ : G∨,\,0 → G∨,\ ) compatible with the homomorphisms (ι : Y → G\ , ι∨ : X → G∨,\ ) inducing isomorphisms on the torus parts T and T ∨ and on the domains of ι and ι∨ , we can define ordinary Hecke twists of the tautoord ~ ord logical object (G\ , λ\ , i\ , τ, βHp , βH ) → Ξ ΦH ,δH by elements g = (g0 , gp ) ∈ p P0Z (A∞,p ) × Pord,0 Z,D (Qp ) such that the image of gp under the canonical homomorord,0 phism PZ,D (Qp ) → Pord h,D (Qp ) satisfies the condition defined by the filtration D−1 on GrZ−1 ⊗ Zp as in Section 3.3.4, and define the Hecke action of (such elements ˆ Z

~ ord of) P0Z (A∞,p ) × Pord,0 Z,D (Qp ) on the collection {ΞΦH ,δH }HP0Z with H of standard form, realized by quasi-finite flat surjections pulling tautological objects back to ordinary Hecke twists, which is compatible with the Hecke action of (suitable elements ~ ord of) P0Z (A∞,p ) × Pord,0 Z,D (Qp ) on the collection {CΦH ,δH }HG1,Z (with H of standard ~ ord ~ ord form) under the canonical morphisms Ξ ΦH ,δH → CΦH ,δH (with varying H) and ord,0 ∞,p 0 ∼ the canonical homomorphism P (A ) × P (Qp ) → G1,Z (A∞,p ) × Pord (Qp ) = Z

Z,D

1,Z,D

∞ (P0Z (A∞,p ) × Pord,0 Z,D (Qp ))/U2,Z (A ). Such a Hecke action enjoys the properties (under various conditions) concerning ´etaleness, finiteness, being isomorphisms between formal completions along fibers over Spec(Fp ), and inducing absolute Frobenius morphisms on fibers over Spec(Fp ) for elements of Up type as in Proposition 5.2.2.2 and Corollaries 5.2.2.3, 5.2.2.4, and 5.2.2.5. (We omit the details for simplicity.) By also considering Q× -isogenies (f : G\ → G\,0 , f ∨ : G∨,\,0 → G∨,\ ) compatible with the homomorphisms (ι : Y → G\ , ι∨ : X → G∨,\ ) inducing Q× on the torus parts T and T ∨ and on the domains of ι and ι∨ (possibly varying the isomorphism classes of the O-lattices X and Y ), we can also define ordinary ord ~ ord Hecke twists of the tautological object (G\ , λ\ , i\ , τ, βHp , βH ) → C ΦH ,δH by elep ∞,p ord ments g = (g0 , gp ) ∈ PZ (A ) × PZ,D (Qp ) such that the image of gp under the ord canonical homomorphism Pord Z,D (Qp ) → Ph,D (Qp ) satisfies the condition defined by Z the filtration D−1 on Gr−1 ⊗ Zp as in Section 3.3.4, and define the Hecke action of ˆ Z ` ~ ord (such elements of) PZ (A∞,p ) × Pord ΞΦH ,δH }HPZ (with H Z,D (Qp ) on the collection { of standard form), where the disjoint unions are over classes [(ZH , ΦH , δH )] sharing the same ZH compatible with D, realized by quasi-finite flat surjections pulling tautological objects back to ordinary Hecke twists, which induces an action of G0l,Z (A∞ ) = 0 ∞,p PZ (A∞ )/P0Z (A∞ ) ∼ ) × Pord,0 = (PZ (A∞,p ) × Pord Z,D (Qp )) on the index Z,D (Qp ))/(PZ (A sets {[(ZH , ΦH , δH )]}, which is compatible with the Hecke action of (suitable ele` ~ ord ∞ ments of) (PZ (A∞,p ) × Pord CΦH ,δH }HPZ /HU2,Z Z,D (Qp ))/U2,Z (A ) on the collection { (with H of standard form, with the same index sets and the same indu~ ord ~ ord ced action of G0l,Z (A∞ )) under the canonical morphisms Ξ ΦH ,δH → CΦH ,δH ∞,p ord (with varying H) and the canonical homomorphism PZ (A ) × PZ,D (Qp ) →

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∞ (PZ (A∞,p ) × Pord Z,D (Qp ))/U2,Z (A ). Any such Hecke action

~ ord : Ξ ~ ord ~ ord [g] Φ0 0 ,δ 0 0 → ΞΦH ,δH H

~ covering [g]

ord

H

~ ord ~ ord :C Φ0 0 ,δ 0 0 → CΦH ,δH induces a (finite flat) morphism H

H

~ ord ~ ord Ξ Φ0 0 ,δ 0 0 → ΞΦH ,δH H

H

~ ord C Φ0 0 ,δ 0

× ~ ord C Φ ,δ H

H

H0

H

~ ord between torus torsors over C 0 , which is equivariant with the morphism Φ0H0 ,δH 0 EΦ0H0 → EΦH dual to the homomorphism SΦH → SΦ0H0 induced by the pair of mor∼

∼

phisms (fX : X ⊗ Q → X 0 ⊗ Q, fY : Y 0 ⊗ Q → Y ⊗ Q) defining the g-assignment Z

Z

Z

Z

0 (Z0H0 , Φ0H0 , δH 0 ) →g (ZH , ΦH , δH ) of (ordinary) cusp labels (cf. [62, Def. 5.4.3.9]). 0 0 If g ∈ PZ (A∞,p ) × Pord Z,D (Qp ) is as above and if (ΦH0 , δH0 , ρ) is a g-refinement of (ΦH , δH , σ) as in [62, Def. 6.4.3.1], then there is a canonical morphism

~ ord : Ξ ~ ord ~ ord [g] Φ0 0 ,δ 0 0 (ρ) → ΞΦH ,δH (σ) H

(5.2.4.42)

H

~ ord : C ~ ord : Ξ ~ ord ~ ord , extending [g] ~ ord (cf. (1.3.2.46)) covering [g] →C → 0 0 ΦH ,δH Φ0H0 ,δH Φ0H0 ,δH 0 0 ord ord ord ~ ~ ~ to Ξ , and inducing a canonical morphism Ξ , mapping Ξ 0 0 ΦH ,δH

ΦH ,δH ,σ

ΦH0 ,δH0 ,ρ

~ ord : X ~ ord [g] Φ0 0 ,δ 0 H

∞,p

H0

,ρ

~ ord →X ΦH ,δH ,σ

(5.2.4.43) 0 0 (Φ0H0 , δH 0 , ΣΦ0

) × Pord Z,D (Qp )

) is as above and if (cf. (1.3.2.47)). If g ∈ PZ (A H0 is a g-refinement of (ΦH , δH , ΣΦH ) as in [62, Def. 6.4.3.2], then morphisms like (5.2.4.42) patch together and define a canonical morphism ~ ord : ΞΦ0 ,δ0 ,Σ0 [g] 0 H0 H0

Φ 0 H

→ ΞΦH ,δH ,ΣΦH

(5.2.4.44)

ord

~ ~ ord ~ ord , extending [g] : Ξ ~ ord (cf. (1.3.2.48)) covering [g] : C → C → 0 0 ΦH ,δH Φ0H0 ,δH Φ0H0 ,δH 0 0 ord ~ Ξ , and inducing a canonical morphism ΦH ,δH

~ ord : X ~ ord [g] Φ0 0 ,δ 0 H

,Σ0Φ0 H0 H0

~ ord →X ΦH ,δH ,ΣΦ

H

(5.2.4.45)

(cf. (1.3.2.49)) compatible with each (1.3.2.47) as above (under canonical morphisms). ~ ord0 If g ∈ PZ (A∞,p ) × PZ (Qp ) is as above and if we have a collection Σord,0 for M H ~ ord as in Definition 5.2.2.1, then that is a g-refinement of a collection Σord for M H the canonical morphism ~ ord,tor : M ~ ord,tor ~ ord,tor [g] →M H0 ,Σord,0 H,Σord 0 as in Proposition 5.2.2.2 is compatible with (5.2.4.43) when (Φ0H0 , δH 0 , ρ) is a g-refinement of (ΦH , δH , σ), under the canonical isomorphisms as in (5) of Theorem 0 0 5.2.1.1; and is compatible with (5.2.4.45) when (Φ0H0 , δH 0 , ΣΦ0 ) is a g-refinement H0 of (ΦH , δH , ΣΦH ), under the canonical isomorphisms as in Lemma 5.2.4.38.

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Proof. The assertions in the first two paragraphs can be justified as in the case ~ ord . (We omit the details for simplicity.) The third paragraph follows by of M H comparing the torus torsor actions of sufficiently divisible multiples of elements, for which we have explicit descriptions in Lemma 5.2.4.29 and Proposition 5.2.4.30. As for the last paragraph, since the canonical morphisms are defined by universal properties given in terms of degeneration data, their compatibility follows from the fact that (by the theory of degeneration as in Theorem 4.1.6.2, based on [62, Thm. 5.2.3.14] and (4.1.4.50), in particular) the ordinary Hecke twist of the tautological ord ~ ord,tor tuple over M by g defined using the ordinary level structure (αH0,p , αH 0 ) H0 ,Σord,0 p ord ~ 0 is compatible with the ordinary Hecke twist of the tautological tuple over M H ord ~ ord over Ξ 0 0 (ρ) by g defined using the ordinary level structure (βH0,p , β 0 ) over ΦH0 ,δH0

Hp

~ ord Ξ Φ0 0 ,δ 0 0 . H

H

We will continue the generalization of Section 1.3.2 in Section 7.1.2 below. Remark 5.2.4.46. Since all objects and morphisms in this subsection are defined by normalizations and by the various universal properties extending their analogues in characteristic zero, they are canonically compatible with the corresponding objects and morphisms in Section 1.3.2.

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Chapter 6

Partial Minimal Compactifications

The first goal of this chapter is to construct the partial minimal compactifications for the ordinary loci defined in Chapter 3, based on the partial toroidal compactifications constructed in Chapter 5 and on the total minimal compactifications constructed in Chapter 2 (which is in turn based on the projective minimal compactifications constructed in [62] in the good reduction case, for the auxiliary models). The second goal is to show that the partial toroidal compactifications are quasi-projective when the levels are neat away from p and when the compatible choices of smooth admissible cone decompositions are projective. The third goal is to show that the reductions of the partial minimal compactifications modulo powers of p are affine. These are all indispensable for the application of our work to the construction of p-adic modular forms as in, for example, [39]. 6.1

Homogeneous Spectra and Their Properties

In this section, we continue to assume the same settings as in Section 5.2. (We do not assume as in Section 5.2.3 that p is a good prime for the integral PEL datum (O, ?, L, h · , · i, h0 ) as in Definition 1.1.1.6.) 6.1.1

Construction of Quasi-Projective Models

~ ord,tor Let M be as in Theorem 5.2.1.1, and let H,Σord top ∼ top e∗ Ω1 ord,tor ωM Lie∨ ~ ord,tor := ∧ ~ ord,tor = ∧ G G/M ~ G/M H,Σord

H,Σord

(6.1.1.1)

H,Σord

be the Hodge invertible sheaf as usual. By [80, IX, 2.1] (cf. [28, Ch. V, Prop. N0 2.1] and [62, Prop. 7.2.1.1]), there exists an integer N0 ≥ 1 such that ω ⊗ ~ ord,tor is M

H,Σord

generated by its global sections. Then the global sections of

N0 k ω⊗ ~ ord,tor M

, for k ≥ 0,

H,Σord

define a morphism   N0 k ~ ord,tor ~ ord,tor M → Proj ⊕ Γ(M , ω⊗ ~ ord,tor ) H,Σord H,Σord k≥0

331

M

H,Σord

(6.1.1.2)

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over ~S0,rH (see [35, II, 3.7.4]). Let   k ~ ord,min := Proj ⊕ Γ(M ~ ord,tor M , ω⊗ H ~ ord,tor ) H,Σord M

k≥0

(6.1.1.3)

H,Σord

(as in [35, II, Sec. 2]), which is a scheme over ~S0,rH . By [35, II, 2.4.7], we have a canonical isomorphism   ⊗ Nk ~ ord,min ∼ ~ ord,tor M = Proj ⊕ Γ(M ord , ω ord,tor ) H

~ M

H,Σ

k≥0

H,Σord

for each integer N ≥ 1. Hence, the right-hand side of (6.1.1.2) is independent of the integer N0 ≥ 1 above, and (6.1.1.2) induces a canonical morphism H~ ord ~ ord,tor ~ ord,min :M (6.1.1.4) ord → M H

H

H,Σ

over ~S0,rH . (We have seen special cases of this in Propositions 5.2.3.8 and 5.2.3.18.) Since we do not know the finite generation of the graded algebra k ~ ord,min is pro~ ord,tor r ⊕ Γ(M , ω⊗ ~ ord,tor ) over OF0 ,(p) [ζp H ], we cannot assert that MH H,Σord M

k≥0

H,Σord

~ ord,min over jective over ~S0,rH . (We cannot even assert the quasi-projectivity of M H ~S0,r at this moment. As we will see soon, M ~ ord,min is indeed quasi-projective over H H ~S0,r , but almost never projective over ~S0,r .) H H ~ ord,min : To justify the absence of Σord in the notation of M H ~ ord,min in (6.1.1.3) is independent of the choice Lemma 6.1.1.5. The definition of M H of Σord . ord,tor

~ ~ ord,tor Proof. Suppose Σord,0 is a refinement of Σord , and suppose [1] :M → H,Σord,0 ord,tor ~ MH,Σord is the proper log ´etale surjection as in Proposition 5.2.2.2 such that the ord ~ ord,tor family (G, λ, i, αHp , αord ) → M ) → ord,0 is the pullback of (G, λ, i, αHp , α Hp

Hp

H,Σ

~ ord,tor M . Then we have ωM ~ ord,tor H,Σord

H,Σord,0

ord,tor

~ ∼ = ([1]

∗

) ωM ~ ord,tor , by definition. MoreoH,Σord

ver, as in [28, Ch. V, Rem. 1.2(b)] and in the proof of [62, Lem. 7.1.1.4], we have ~ ord,tor O tor = 0 for all i > 0 and [1] ~ ord,tor O tor = O tor by [50, Ch. I, Sec. Ri [1] MH,Σ0 MH,Σ0 MH,Σ ∗ ∗ 3], which implies by the projection formula [35, 0I , 5.4.10.1] that the canonical mor⊗k k ~ ord,tor ~ ord,tor phism Γ(M , ω⊗ ~ ord,tor ) → Γ(MH,Σord,0 , ω ~ ord,tor ) is an isomorphism for each H,Σord M

M

H,Σord

H,Σord,0

k ≥ 0. Hence, the canonical morphism     ⊗k ⊗k ~ ord,tor ~ ord,tor , ω , ω ) Proj ⊕ Γ(M ) → Proj ⊕ Γ( M ord,0 ord ord,tor ord,tor ~ ~ H,Σ H,Σ k≥0

M

H,Σord,0

k≥0

M

H,Σord

is also an isomorphism, as desired. To construct minimal compactifications as in [62, Sec. 7.2.3] using the technique of Stein factorizations, it is desirable to start with a proper morphism with target a ~ ord,tor , scheme quasi-projective over ~S0,rH . However, this is not straightforward for M H because it is not proper over ~S0,rH in general. We need the help of the auxiliary

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moduli problems and their compactifications as in Section 2.2 (see also Section 3.4.6). ˆ p ) and Hp = Proposition 6.1.1.6. Suppose H = Hp Hp such that Hp ⊂ G(Z bal r Up,1 (p ) for some integer r ≥ 0. Up to replacing Σord with a refinement, we may assume that it is smooth and projective, and that there exists a cone decom~ ord,tor position Σord over aux and an analogous partial toroidal compactification MHaux ,Σord aux ~S0,aux,r := Spec(OF ,(p) [ζpr ]) such that there is a (necessarily unique) proper 0,aux

morphism ~ ord,tor ~ ord,tor ord M →M H,Σord Haux ,Σ

(6.1.1.7)

aux

(over ~S0,aux,r ) extending (3.4.6.2), mapping the [(ΦH , δH , σ)]-strata ~Zord [(ΦH ,δH ,σ)] of ord,tor ord,tor ord ~ ~ ~ of M M ord to the [(ΦH , δH , σaux )]-stratum Z ord when H,Σ

aux

[(ΦHaux ,δHaux ,σaux )]

aux

Haux ,Σaux

(ΦHaux , δHaux , σaux ) is assigned to (ΦH , δH , σ) (see Definition 2.1.2.25). Under a0 a (6.1.1.7), the pullback of ω ⊗ is canonically isomorphic to ω ⊗ ~ ord,tor ~ ord,tor for the M

M

Haux ,Σord aux

H,Σord

integers a0 ≥ 1 and a ≥ 1 as in Lemma 2.1.2.35. ord Proof. Starting with the degenerating family (G, λ, i, αHp , αH ) of type p ord,tor ord ~ ~ , we would like to construct a degenerating family MH over MH ord,O O O O ~ ord over M ~ ord,tor , α , αH ) of type M , and show that it is the (GO , λ , i p aux aux aux Haux Haux H,Σord aux,p ord ) of pullback of the tautological degenerating family (Gaux , λaux , iaux , αHpaux , αH aux,p ord,tor ~ ~ ord over M under some (necessarily unique) morphism (6.1.1.7) extype M ord Haux

H,Σ

O O tending (3.4.6.2). For this purpose, we first construct GO aux , λaux , and iaux as in the ord,O O proof of Proposition 2.1.2.29. Then we can construct αHpaux and αHaux,p as in the proof of Lemma 3.4.1.4, verify as in the proof of Proposition 2.1.2.29 the universal ~ ord,tor ord stated as in (6) of Theorem 5.2.1.1, and obtain the desired property of M Haux ,Σaux ~ ord,tor ord when morphism (6.1.1.7) mapping ~Zord to ~Zord of M [(ΦH ,δH ,σ)]

[(ΦHaux ,δHaux ,σaux )]

Haux ,Σaux

(ΦHaux , δHaux , σaux ) is assigned to (ΦH , δH , σ) as in Definition 2.1.2.25. To show that the morphism (6.1.1.7) is proper, we apply the valuative criterion over the spectrum of a complete discrete valuation ring with algebraically closed ~ ord of residue field, with generic point mapped to the open dense subscheme M H ord,tor ord,tor ~ ~ MH,Σord , and verify the universal property of MH,Σord stated as in (6) of Theorem 5.2.1.1. As in the proof of [62, Prop. 6.3.3.17], since the base ring is a complete discrete valuation ring, the cone decomposition Σord does not impose any condition in the verification of this universal property. Thus, the only condition to verify is the extensibility condition (5) in the definition of degenerating families, which is satisfied in this case because the construction of the degenerating family ord,O O O O (GO , αH ) above is compatible with (and implicitly used) this aux , λaux , iaux , αHp aux,p aux extensibility condition. Proposition 6.1.1.8. In Proposition 6.1.1.6, we can choose Σord , Σord aux , and

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~ ord,tor ord such that the canonical morphism M Haux ,Σaux  H~ ord k ~ ord,tor ord → M ~ ord,min := Proj ⊕ Γ(M ~ ord,tor ord , ω ⊗ord,tor :M Haux ~ Haux Haux ,Σ Haux ,Σ aux

aux

k≥0

M

 )

Haux ,Σord aux

is proper (and surjective), and is the Stein factorization (see [35, III-1, 4.3.3]) of itself (and hence has nonempty connected geometric fibers, by [35, III-1, 4.3.1, 4.3.3, 4.3.4] and its natural generalization to the context of algebraic stacks). Concretely, up to replacing Hp with an open compact subgroup, we may assume ord p moreover that Haux is neat, that Σord aux is projective with a collection polaux of polariord zation functions, and that Haux , Σord aux , and polaux fit into the setup of the beginning of Section 5.2.3 (with Assumption 5.2.3.1 automatically satisfied by Lemma 5.2.3.2), ord so that Σord aux (resp. polaux ) extends to some projective (but possibly nonsmooth) Σaux ~ tor (resp. polaux ) such that M Haux ,d0,aux polaux is defined (for some integer d0,aux ≥ 1; see Proposition 2.2.2.3), and so that there is a canonical open immersion ~ ord,tor ord ,→ M ~ tor M Haux ,d0,aux polaux ,r Haux ,Σ aux

(cf. (5.2.3.19)—here rHaux = rν(Haux ) = r by definition, as explained in Remark ~ tor 3.4.2.2) inducing by composition with the structural morphism M Haux ,d0,aux polaux ,r → min ~ M a canonical morphism Haux ,r

~ ord,tor ord → M ~ min , M Haux ,r Haux ,Σ aux

inducing a commutative diagram ~ ord,tor M H,Σord

~ ord,tor ord /M

~ tor /M Haux ,d0,aux polaux ,r

 ord,min ~ MH

 ~ ord,min /M Haux

 ~ min /M Haux ,r

Haux ,Σaux

in which the vertical and the left top horizontal arrows are all proper, in which the left bottom horizontal arrow is finite, and in which the right top and bottom arrows are open immersions making the right-hand square Cartesian, such that the canonical morphisms H~ ord OM ~ ord,min → H,∗ OM ~ ord,tor H

H,Σord

and H~ ord OM ~ ord,min → Haux ,∗ OM ~ ord,tor Haux

Haux ,Σord aux

~ ord,min is quasi-projective over ~S0,r . are isomorphisms. Consequently, M H H Proof. Suppose H0,p is an open compact subgroup of Hp , which defines an open compact subgroup H0 = H0,p Hp of H = Hp Hp . Suppose Σord,0 is a 1-refinement of ~ ord,tor : M ~ ord,tor Σord as in Definition 5.2.2.1. Then the canonical surjection [1] 0 ord → H ,Σ

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~ ord,tor M is proper by Proposition 5.2.2.2. Thus, to prove the first paragraph in H,Σord p the proposition, we may compatibly replace Hp and Haux with sufficiently small p subgroups as above, and assume that Haux is also neat. We are also allowed to replace Σord with suitable refinements. By Proposition 5.2.3.18, by suitably choosing Σord aux (possibly at the expense of ~ ord,tor ord → M ~ ord,min such replacing Σord with a refinement), we can construct M Haux Haux ,Σaux ~ tor ~ min that it is the (proper and surjective) pullback of M → M Haux ,r

Haux ,d0,aux polaux ,r

~ ord,min → M ~ min under M Haux ,r (on the target), and such that it satisfies the other Haux ~ ord,tor ord , M ~ ord,min , M ~ tor ~ min statements (concerning M Haux ,d0,aux polaux ,r , and MHaux ,r ) in Haux Haux ,Σaux ~ ord,tor ~ ord,min ) this proposition. Then the remaining statements (concerning M ord and M H

H,Σ

follow from Proposition 6.1.1.6 as formal consequences.

Lemma 6.1.1.9. (Compare with Lemma 3.4.6.1.) With the setting as at the beginbal r ning of this section (but no longer assuming that Hp = Up,1 (p )), suppose H and Haux are as in Lemma 3.4.1.6. Then there exist compatible choices of smooth and projective Σord and Σord aux , which can be achieved by compatibly replacing any given ~ ord,tor ~ ord,tor ord are defined as in Thechoices with refinements, such that M ord and M H,Σ

Haux ,Σaux

orem 5.2.1.1, and such that there is a (necessarily unique, but possibly nonproper) morphism ~ ord,tor ~ ord,tor ord M →M H,Σord Haux ,Σ

(6.1.1.10)

aux

~ ord,tor extending (3.4.6.2), mapping the [(ΦH , δH , σ)]-strata ~Zord [(ΦH ,δH ,σ)] of MH,Σord ~ ord,tor to the [(ΦHaux , δHaux , σaux )]-stratum ~Zord when [(ΦHaux ,δHaux ,σaux )] of MHaux ,Σord aux (ΦHaux , δHaux , σaux ) is assigned to (ΦH , δH , σ) (see Definition 2.1.2.25). Under a0 a is canonically isomorphic to ω ⊗ (6.1.1.10), the pullback of ω ⊗ ~ ord,tor for the ~ ord,tor M

Haux ,Σord aux

M

H,Σord

integers a0 ≥ 1 and a ≥ 1 as in Lemma 2.1.2.35. ˆ we may assume that Σord is induced by some smooth Σp If Haux = Gaux (Z), aux aux for MGaux (Zˆ p ) , so that Proposition 5.2.3.3 and Lemma 5.2.3.8 apply (with (5.2.3.4) ~ ord,tor there being M ,→ Mtor in our notation here). p ord ˆ Gaux (Z),Σ aux

ˆ p ),Σaux Gaux (Z

Proof. As in the proof of Lemma 3.4.1.6, this is because the morphism (6.1.1.7) at sufficiently higher levels and with sufficiently refined cone decompositions induces the morphism (6.1.1.10).

Corollary 6.1.1.11.

With the assumptions as in Lemma 6.1.1.9, there exists a

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commutative diagram ~ ord /M Haux _

~ ord M H _ #

% ~H /M aux

~H M

 ~ ord,tor M H,Σord

 ~ ord,tor ord /M Haux ,Σ

aux

#

 ~ min M H

$  ~ min /M Haux

extending the morphisms (3.4.6.2) and (6.1.1.10), which is compatible with other canonical morphisms. For N1 ≥ 1 as in Proposition 2.2.1.2 and for integers a0 ≥ 1 and a ≥ 1 as aN1 ~ ord,tor in Lemma 2.1.2.35, the pullback of ω ⊗ ~ min to MH,Σord is canonically isomorphic to MH

aN1 ω⊗ ~ ord,tor . M

H,Σord

Proof. The upper-left square and its commutativity follow from Lemmas 3.4.6.1 ~H →M ~H →M ~ min and M ~ min are as in and 6.1.1.9. The canonical morphisms M aux H Haux ~ H and M ~ ord → M ~H ~ ord → M Proposition 2.2.1.2. The canonical morphisms M aux Haux H are as in Proposition 3.4.6.3. The remaining morphisms are compatibly induced by ~ min as normalizations. ~ min , and M ~ min , M ~ ord , M the universal properties of M Haux H H Haux ˆ and that For the second paragraph, we may assume that Haux = Gaux (Z), p ord Σaux is induced by some smooth Σaux for MGaux (Zˆ p ) , as in the last paragraph of

aN1 Lemma 6.1.1.9. By Proposition 2.2.1.2, ω ⊗ ~ min is canonically isomorphic to the

⊗ a0 N1 pullback of ωM min ⊗ a0 N1 ωM min

Gaux (ˆ Zp )

MH

. On the other hand, by [62, Thm. 7.2.4.1], the pullback of

Gaux (ˆ Zp )

to Mtor G

p ˆp aux (Z ),Σaux

⊗ a0 N1 is canonically isomorphic to ωM tor

p Gaux (ˆ Zp ),Σaux

⊗ a0 N1 5.2.3.8, the pullback of ωM tor

isomorphic to

a0 N1 ω⊗ ~ ord,tor M

Haux ,Σord aux

p Gaux (ˆ Zp ),Σaux

; by Lemma

~ ord,tor ord = M ~ ord,tor to M is canonically ord ˆ Haux ,Σ G (Z),Σ aux

aux

aux

a0 N1 ; and, by Lemma 6.1.1.9, the pullback of ω ⊗ ~ ord,tor M

to

Haux ,Σord aux

aN1 ~ ord,tor ~ min and M ~ min are defined M is canonically isomorphic to ω ⊗ Haux ~ ord,tor . Since MH H,Σord M

as normalizations of Mmin ˆ p ) , we Gaux (Z ⊗ aN1 canonically isomorphic to ω ~ ord,tor , M ord

H,Σord

aN1 ~ ord,tor see that the pullback of ω ⊗ ~ min to MH,Σord is MH

as desired.

H,Σ

Theorem 6.1.1.12. With the setting as at the beginning of this section (but no bal r longer assuming that Hp = Up,1 (p )), the canonical morphism (6.1.1.4)   H~ ord ⊗k ~ ord,tor ~ ord,min = Proj ⊕ Γ(M ~ ord,tor :M ord → M ord , ω ord,tor ) H

H,Σ

H

k≥0

H,Σ

~ M

H,Σord

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H ord is proper and is the Stein factorization of itself. Consequently, ~ H (is surjective and) has nonempty connected geometric fibers (by [35, III-1, 4.3.1, 4.3.3, 4.3.4] and its natural generalization to the context of algebraic stacks), and the canonically induced morphism H~ ord OM (6.1.1.13) ~ ord,min → H,∗ OM ~ ord,tor H

H,Σord

N0 ~ ord,tor is an isomorphism. Moreover, the invertible sheaf ω ⊗ ~ ord,tor over MH,Σord descends M

H,Σord

~ ord,min . By abuse of notation, we shall to an ample invertible sheaf O(1) over M H ⊗ N0 denote O(1) by ω ~ ord,min , even when ωM ~ ord,min itself is not defined. MH

H

~ ord,tor ~ min in Corollary 6.1.1.11 induces a canoThe canonical morphism M →M H H,Σord nical morphism ~ ord,tor ~ min , →M M H,rH H,Σord

(6.1.1.14)

which maps the [(ΦH , δH , σ)]-stratum ~Zord [(ΦH ,δH ,σ)] (see (2) of Theorem 5.2.1.1) to the [(ΦH , δH )]-stratum ~Z[(ΦH ,δH )],rH (see Definition 2.2.3.5), and which factors canonically as a composition ~ min ~ ord,tor ~ ord,min → M M →M H,rH H H,Σord

(6.1.1.15)

inducing a canonical open immersion ~ ord,min ,→ M ~ min , M H,rH H k under which the pullback of ω ⊗ ~ min to MH

~ ord,min M H

(6.1.1.16)

k is canonically isomorphic to ω ⊗ ~ ord,min , MH

when both are defined for some integer k (divisible by both aN1 and N0 ). In parti~ ord,min is quasi-projective over ~S0,r . cular, M H H bal r bal r Proof. Suppose Up,1 (p ) ⊂ Hp ⊂ Up,0 (pr ). Let H0 = Hp Hp = Up,1 (p ), and let ord,0 ord ord,0 ~ Σ be a compatible collection for MH0 such that Σ is a 1-refinement of Σord as ord,tor ~ ~ ord,tor ~ ord,tor in Definition 5.2.2.1. Then the canonical surjection [1] :M 0 ord → M ord H ,Σ

H,Σ

is proper by Proposition 5.2.2.2. Thus, to show that (6.1.1.4) is proper, it suffices to show that it is so with H (resp. Σord ) replaced with H0 (resp. Σord,0 ), which follows from the first paragraph of Proposition 6.1.1.8. Once the properness of (6.1.1.4) is known, since the canonical morphism (6.1.1.4) N0 k (for the original H and Σord ) is defined by global sections of ω ⊗ ~ ord,tor , for k ≥ 0, it M

H,Σord

follows that (6.1.1.4) is the Stein factorization of itself, and that, for each k ≥ 0, the N0 k ~ ord,tor invertible sheaf ω ⊗ ~ ord,tor over MH,Σord descends to an ample invertible sheaf O(1) M

H,Σord

~ ord,min (by definition of M ~ ord,min ; see (6.1.1.3)). over M H H ~ The induced morphism (6.1.1.14) maps ~Zord [(ΦH ,δH ,σ)] to Z[(ΦH ,δH )],rH by the definition of the latter by taking closures and exclusions. The canonical factorization (6.1.1.15) exists by the last paragraph of Corollary 6.1.1.11. The induced morphism (6.1.1.16) is an open immersion by Zariski’s main theorem (see [35, III-1,

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4.4.3, 4.4.11]), by the last paragraph of Corollary 6.1.1.11, and by the fact that ~ ord ] (cf. (3.4.6.5), Proposition 2.2.1.2, and Corolit is an open immersion over [M H k lary 6.1.1.11). When k is divisible by both aN1 and N0 , the pullback of ω ⊗ ~ min to MH

k ~ ord,min under (6.1.1.16) is canonically isomorphic to ω ⊗ord,min M , because their pulH ~ MH

~ ord,tor lbacks to M are isomorphic, and because (6.1.1.13) is an isomorphism. (See H,Σord the argument at the end of the proof of [62, Thm. 7.2.4.1], which is based on the projection formula (see [35, 0I , 5.4.10.1]), used in [62, Lem. 7.2.2.1].) 6.1.2

Local Structures and Stratifications

~ ord,min is normal. Proposition 6.1.2.1. (Compare with [62, Prop. 7.2.3.3].) M H ~ ord,tor Proof. Since M is normal because it is smooth over the normal base scheme H,Σord ~S0,r = Spec(OF ,(p) [ζprH ]), and since the canonical morphism (6.1.1.13) is an H

0

isomorphism by Theorem 6.1.1.12, the proposition follows from [62, Lem. 7.2.3.1]; or, alternatively, from the second half of the proof of [10, Sec. 6.7, Lem. 2] (ignoring the statement about finite generation). Corollary 6.1.2.2. (Compare with [62, the paragraph following Prop. 7.2.3.3].) ~ ord,min is flat over ~S0,r = Spec(OF ,(p) [ζprH ]). M H 0 H Proof. Since OF0 ,(p) [ζprH ] is a localization of the ring of integers of a number field, ~ ord,min → ~S0,r is flat because M ~ ord,min is normal and all its maximal points M H H H ~ ord,tor (see [36, 0, 2.1.2]) are of characteristic zero (as those of M ord are). H,Σ

By Theorem 6.1.1.12, the canonical morphism (6.1.1.4) has nonempty connected N0 geometric fibers, and the pullback of ω ⊗ ~ ord,tor to each such connected geometric M

H,Σord

fiber is trivial. By [28, Ch. V, Prop. 2.2] or [62, Prop. 7.2.1.2], this shows that the isomorphism class of the abelian part of G is constant on each of such fibers. In H ord ~ ord , then it has only one closed point. particular, if a geometric fiber of ~ H meets M H ord,tor ~ ~ ord is open in M Since M ord , and since the formation of coarse moduli spaces H

H,Σ

~ ord ] is an open subalgebraic space commutes with flat base change, we see that [M H ord H ord,tor ord,tor ord,min ~ ~ ~ ~ of [MH,Σord ]. The morphism H : MH,Σord → MH factors as H~ ord [ ] ord,tor ord,tor H ~ ~ ord,min , ~ M MH,Σord → [MH,Σord ] → H ~ ord is the factorization whose restriction to M H H~ ord ~ ord → [M ~ ord ] M H H

[

|~ ord ]

M ~ ord,min . →H M H

H

Applying Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11], and the formulation H ord in [62, Prop. 7.2.3.4] for algebraic spaces) to [~ ], and taking into account the H

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H ord ~ ord,min is normal (see Proposition 6.1.2.1), we see that [~ ] is an isofact that M H H ~ ord,min containing the image of [M ~ ord ]. (We morphism over an open subscheme of M H H ord ~ will see below that the image of [MH ] is actually open, with complements given by H ord closed subschemes, and hence [~ | ord ] is an open immersion.) ~ M H

H

H ord More generally, suppose that a fiber of ~ H meets the [(ΦH , δH , σ)]-stratum ~Zord [(ΦH ,δH ,σ)] . Let (ΦH , δH , σ) be any representative of the class [(ΦH , δH , σ)]. By ~ ord,tor )∧ ~ ord,tor along (5) of Theorem 5.2.1.1, the formal completion (M of M H

~Zord [(ΦH ,δH ,σ)]

H

~ Zord [(Φ

H ,δH ,σ)]

~ ord X ΦH ,δH ,σ /ΓΦH ,σ .

♥

is canonically isomorphic to Let ω denote the pulord top ~ lback of ωM Lie∨ ~ ord,tor to XΦH ,δH ,σ /ΓΦH ,σ . Let ωB ~ := ωB/ ~ ord,ZH := ∧ ~ M ~ ord,ZH , ~ M B/ H,Σord

H

H

~ ord,ZH . By abuse of notation, we ~ is the tautological abelian scheme over M where B H shall also denote the pullback of ωB~ by the same notation. Lemma 6.1.2.3. (Compare with [62, Lem. 7.1.2.1].) There is a canonical isomortop ~ ord phism ♥ ω ∼ = (∧Z X) ⊗ ωB~ over the formal algebraic stack X ΦH ,δH ,σ /ΓΦH ,σ . Z

Proof. The proof of [62, Lem. 7.1.2.1] works verbatim here. ~ ord,ZH by ~pord , which is ~ ord Let us denote the structural morphism C ΦH ,δH ΦH ,δH → MH proper and smooth because it is an abelian scheme torsor over the finite ´etale cover ~ ord,ΦH of M ~ ord,ZH . For simplicity of notation, as in [62, Def. 7.1.2.2], for each M H H ` ∈ SΦH , let ord,(`) ~ ord FJΦH ,δH := (~pord ΦH ,δH )∗ (ΨΦH ,δH (`)).

(6.1.2.4)

Consider the following composition of canonical morphisms (cf. [62, (7.1.2.3)]): k ~ ord,tor , ω ⊗ord,tor Γ(M ) H ~ M

H,Σord

~ ord,tor )∧ord → Γ((M ~ H Z

[(ΦH ,δH ,σ)]

→

hY

k ∼ ~ ord , ω⊗ ~ ord,tor ) = Γ(XΦH ,δH ,σ /ΓΦH ,σ ,

∼ =

ord,(`)

`∈σ ∨

ω⊗ k )

iΓΦH ,σ ⊗k ((∧top X) ⊗ ω ) ) ~ Z B

⊗

OC ~ ord

⊗

O~ ord,ZH

(6.1.2.5)

Z

ΦH ,δH

~ ord,ZH , FJ Γ(M H ΦH ,δH

♥

H,Σord

~ Φord,δ , Ψ ~ ord Γ(C ΦH ,δH (`) H H

`∈σ ∨

hY

M

iΓΦH ,σ ⊗k ((∧top X) ⊗ ω ) ) . ~ Z B Z

M H

Definition 6.1.2.6. (Compare with [62, Def. 7.1.2.4].) The above composition (6.1.2.5) is called the Fourier–Jacobi morphism along (ΦH , δH , σ), which we k ~ ord,tor , ω ⊗ord,tor denote by FJord ) has a ΦH ,δH ,σ . The image of an element f ∈ Γ(MH ~ M

H,Σord

natural expansion FJord ΦH ,δH ,σ (f ) =

X `∈σ ∨

ord,(`)

FJΦH ,δH ,σ (f )

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ord,(`)

where the sum can be infinite and where each FJΦH ,δH ,σ (f ) lies in ord,(`)

ord,(`)

~ ord,ZH , FJ FJCΦH ,δH (k) := Γ(M H ΦH ,δH

⊗

O~ ord,ZH

⊗k ((∧top ). ~) Z X) ⊗ ωB

M H

Z

The expansion FJord ΦH ,δH ,σ (f ) is called the Fourier–Jacobi expansion of f along ord,(`)

(ΦH , δH , σ), with Fourier–Jacobi coefficients FJΦH ,δH ,σ (f ) of each degree ` ∈ σ∨ . By the same argument as in [62, Sec. 7.1.2], we do not really need the Fourier–Jacobi coefficients of degrees outside P∨ ∩ σ ∨ , and the Fourier–Jacobi expansions ΦH = σ∈ΣΦH

are naturally invariant under the action of ΓΦH . We have an induced morphism h Y iΓΦH ord,(`) k ~ ord,tor , ω ⊗ord,tor FJord )→ FJCΦH ,δH (k) (6.1.2.7) ΦH ,δH : Γ(MH ~ M

H,Σord

`∈P∨ Φ

H

(cf. [62, (7.1.2.6)]). Definition 6.1.2.8. (Compare with [62, Def. 7.1.2.7].) The above morphism (6.1.2.7) is called the Fourier–Jacobi morphism along (ΦH , δH ), which we dek ~ ord,tor , ω ⊗ord,tor note by FJord ) has ΦH ,δH as above. The image of an element f ∈ Γ(MH ~ M

H,Σord

a natural expansion FJord ΦH ,δH (f ) =

ord,(`)

X

FJΦH ,δH (f ),

`∈P∨ ΦH ord,(`)

ord,(`)

where each FJΦH ,δH (f ) lies in FJCΦH ,δH (k). The expansion FJord ΦH ,δH (f ) is called the Fourier–Jacobi expansion of f along (ΦH , δH ), with Fourier–Jacobi ord,(`) coefficients FJΦH ,δH (f ) of each degree ` ∈ P∨ ΦH . Definition 6.1.2.9. (Compare with [62, Def. 7.1.2.10].) The constant term of k ~ ord,tor , ω ⊗ord,tor a Fourier–Jacobi expansion FJord ) is ΦH ,δH (f ) of an element f ∈ Γ(MH ~ M

H,Σord

the Fourier–Jacobi coefficient

ord,(0) FJΦH ,δH (f )

∈

ord,(0) FJCΦH ,δH (k)

in degree zero.

The same arguments as in [62, Sec. 7.1.2] gives the following: Proposition 6.1.2.10. (Compare with [62, Prop. 7.1.2.8, 7.1.2.9, and 7.1.2.13; see also the errata].) The Fourier–Jacobi morphism FJΦH ,δH satisfies the following properties: (1) FJΦH ,δH can be computed by any FJΦH ,δH ,σ as in Definition 6.1.2.6. The definition is independent of the σ we use. (2) FJΦH ,δH is independent of the ΓΦH -admissible smooth rational polyhedral cone decomposition ΣΦH of PΦH we use.

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k ~ ord,tor , ω ⊗ord,tor Γ(M ) along the H ~

∈

(3) The value of each element f

page 341

M

H,Σord

~ ord,tor [(ΦH , δH , σ)]-stratum ~Zord [(ΦH ,δH ,σ)] of MH,Σord is determined by its conord,(0)

ord,(0)

stant term FJΦH ,δH (f ), which is a ΓΦH -invariant element in FJCΦH ,δH (k). ord,(0) ∼ ~ ord,ΦH → M ~ ord,ZH )∗ O ~ ord,ΦH and In particular, since FJ = (M H

ΦH ,δH

H

MH

~ ord,ZH (see Proposition 4.2.1.29), the value of f is ~ ord,ΦH /ΓΦ ∼ M = M H H H constant along the fibers of the structural morphism ~Zord [(ΦH ,δH ,σ)] → ord,ZH ~ . We say in this case that it depends only on the abelian part of MH (G, λ, i, αHp , αord ) over ~Zord . Hp

[(ΦH ,δH ,σ)]

Proof. The same arguments as in the proofs of [62, Prop. 7.1.2.8, 7.1.2.9, and 7.1.2.13] work verbatim here. (The error in the statement of [62, Prop. 7.1.2.13] is due to changes necessitated by errors in other parts of the book, which does not invalidate the argument of the proof there.) By (3) of Proposition 6.1.2.10, applied to those k ≥ 0 divisible by N0 , we see that H~ ord ~ ord,min factors through ~Zord ~ ord,ZH . | ord : ~Zord → M → M H

~ Z[(Φ

H ,δH ,σ)]

[(ΦH ,δH ,σ)]

H

[(ΦH ,δH ,σ)]

H

This induces a morphism ~ ord,ZH → M ~ ord,min M H H

(6.1.2.11)

from an algebraic stack to a scheme, each of whose geometric fibers has only one single point. The argument used in proving (1) of Proposition 6.1.2.10 (or rather [62, Prop. 7.1.2.8]) shows the following: Lemma H~ ord | ord H

~ Z[(Φ

6.1.2.12. (Compare with [62, Lem. 7.2.3.6].) H ord ~ ord,min and ~ |~ ord : ~Zord [(ΦH ,δH ,σ)] → MH H Z

[(Φ0 ,δ 0 ,σ 0 )] H H

H ,δH ,σ)]

Two restrictions : ~Zord0 0 0 → [(ΦH ,δH ,σ )]

~ ord,min M H

have the same image and induce the same morphism as in (6.1.2.11) (up to the canonical identification between the sources) when there exist representati0 0 , σ 0 )], respectively, ves (ΦH , δH , σ) and (Φ0H , δH , σ 0 ) of [(ΦH , δH , σ)] and [(Φ0H , δH 0 0 such that (ΦH , δH ) and (ΦH , δH ) are equivalent and represent the same cusp label 0 [(ΦH , δH )] = [(Φ0H , δH )]. ~ ord 0 . By Theorem 6.1.1.12, Let us denote this common image by ~Zord [(ΦH ,δH )] = Z[(Φ0H ,δH )] ~Zord ~Z[(Φ ,δ )],r (see Definition 2.2.3.5). is an open subscheme of H H H [(ΦH ,δH )] We claim that the converse is also true: Proposition 6.1.2.13. (Compare with [62, Prop. 7.2.3.7].) If the intersection of H~ ord H~ ord ~Zord ) and ~Zord ) is [(ΦH ,δH )] := image( H |~ [(Φ0 ,δ 0 )] := image( H |~ Zord Zord H

[(ΦH ,δH ,σ)]

H

[(Φ0 ,δ 0 ,σ 0 )] H H

0 nonempty, then the two cusp labels [(ΦH , δH )] and [(Φ0H , δH )] are the same. (In this ~Zord0 0 .) case, we saw above that ~Zord = [(ΦH ,δH )] [(Φ ,δ )] H

H

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Proof. The proof of [62, Prop. 7.2.3.7] works almost verbatim here. But let us spell out the details for the sake of certainty. Suppose there exists a geometric point x ¯ in the intersection of ~Zord [(ΦH ,δH )] and ord H ~Zord0 0 . Let C be any proper irreducible curve in the fiber of ~ ~ ord,tor :M → [(ΦH ,δH )] H H,Σord ord,min ~ MH over x ¯. By [28, Ch. V, Prop. 2.2] or [62, Prop. 7.2.1.2] as before, the ~ ord,tor pullback of G → M ord to C is globally an extension of an isotrivial abelian H,Σ

scheme by a torus. If we take any geometric point z¯ of C, and take the pullback of ord ~ ord,tor ~ ord,tor (G, λ, i, αHp , αH )→M to the strict local ring of M at z¯ completed along H,Σord H,Σord p ~ ord over a base ring Rz¯ the curve C, then we obtain a degenerating family of type M H

that fits into the setting of Section 4.1.6. Then, by Theorem 4.1.6.2, this pullback (Rz¯) → DEGPEL,... (Rz¯) defines an object in the essential image of DEGPEL,Mord M ord H H and hence, in particular, a cusp label (ΦH , δH ). (The key point here is that the ~ ord,tor pullback of G → M to C is globally an extension of an abelian scheme by a split H,Σord torus.) Thus, there is a locally constant association of a cusp label [(ΦH , δH )] over H ord each such proper irreducible curve C. Since the fiber of ~ over x ¯ is connected, H

we see that the associated cusp label [(ΦH , δH )] must be globally constant over the 0 )], as desired. whole fiber. This forces [(ΦH , δH )] = [(Φ0H , δH Corollary 6.1.2.14. (Compare with [62, Cor. 7.2.3.8].) The subschemes ~Zord [(ΦH ,δH )] form a stratification a ~ ord,min = ~Zord M (6.1.2.15) [(ΦH ,δH )] H [(ΦH ,δH )]

~ ord,min by locally closed subscheme, with [(ΦH , δH )] running through a complete of M H 0 )]-stratum set of ordinary cusp labels (see Definition 3.2.3.8), such that the [(Φ0H , δH ~Zord0 0 lies in the closure of the [(ΦH , δH )]-stratum ~Zord if and only if there [(ΦH ,δH )] [(ΦH ,δH )] 0 0 is a surjection from the cusp label [(ΦH , δH )] to the cusp label [(ΦH , δH )] as in ` Definition 1.2.1.18. (The notation “ ” only means a set-theoretic disjoint union. ~ ord,min .) The algebro-geometric structure is still that of M H Proof. According to (2) of Theorem 5.2.1.1, the closure of the [(ΦH , δH , σ)]-stratum 0 0 0 ~Zord ~ ord,tor ~ ord 0 0 such ,σ )] [(ΦH ,δH ,σ)] in MH,Σord is the union of the [(ΦH , δH , σ )]-strata Z[(Φ0H ,δH 0 0 0 that [(ΦH , δH , σ )] is a face of [(ΦH , δH , σ)] as in Definition 1.2.2.19. Since the morH ord ~ ord,tor ~ ord,min is proper, we see that the closure of ~Zord phism ~ :M ord → M H

H,Σ

H

[(ΦH ,δH ,σ)]

~ ord,tor ~ ord,min , which is by definiin M is mapped to the closure of ~Zord [(ΦH ,δH )] in MH H,Σord tion the union of those ~Zord 0 )] such that there is a surjection from [(ΦH , δH )] [(Φ0H ,δH 0 0 to [(ΦH , δH )]. By Proposition 6.1.2.13, this union is disjoint. Hence, we may con~ ord,tor ) that clude (by induction on the incidence relations in the stratification of M H,Σord ord,min ~ (6.1.2.15) is indeed a stratification of M . H

As a byproduct:

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Corollary 6.1.2.16. (Compare with [62, Cor. 7.2.3.11].) If σ is top-dimensional H~ ord ∨ ~ ord in P+ |Zord : ~Zord ΦH ⊂ (SΦH )R , then the morphism [(ΦH ,δH ,σ)] → Z[(ΦH ,δH )] H ~ [(ΦH ,δH ,σ)]

(induced by restriction) is proper. Proof. Since σ is a top-dimensional cone, [(ΦH , δH , σ)] can be a face of another 0 0 [(Φ0H , δH , σ 0 )] (see Definition 1.2.2.19) only when [(ΦH , δH )] 6= [(Φ0H , δH )]. Then ord ~ (2) of Theorem 5.2.1.1 and Proposition 6.1.2.13 imply that Z[(ΦH ,δH ,σ)] is a closed H ord ~ ord,tor subalgebraic stack of the preimage (~ H )−1 (~Zord [(ΦH ,δH )] ). Since MH,Σord is proper H ord ~ ord over ~S0,rH , the induced morphism ~ H |~Zord : ~Zord [(ΦH ,δH ,σ)] → Z[(ΦH ,δH )] is also [(ΦH ,δH ,σ)]

proper, as desired. Combining Corollary 6.1.2.16 with Lemma 6.1.2.12 and with Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11], and the formulation in [62, Prop. 7.2.3.4] for algebraic spaces), we obtain the following: Corollary 6.1.2.17. (Compare with [62, Cor. 7.2.3.12; see also the errata].) The ~ ord,ZH ] → ~Zord morphism [M H [(ΦH ,δH )] induced by (6.1.2.11) is finite and induces a bijection on geometric points. ~ ord,1 be the Proposition 6.1.2.18. (Compare with [62, Prop. 7.2.3.13].) Let M H ~ ord,min formed by the union of the strata in (6.1.2.15) of coopen subscheme of M H ~ ord,1 of the canonical surjection dimension at most one. Then the pullback to M H ord H~ ord H ord,min ~ ~ ord,tor ~ [ ] : [M ]  M induced by is an isomorphism (regardless of the ord H

H

H,Σ

H

~ ord,min ). choice of Σord in the construction of M H Proof. The proof of [62, Prop. 7.2.3.13] also works here. Proposition 6.1.2.19. (Compare with [62, Prop. 7.2.3.16; see also the errata].) Let [(ΦH , δH )] be an ordinary cusp label, and let (ΦH , δH ) be a representative of ~ ord,min over the [(ΦH , δH )]-stratum [(ΦH , δH )]. Let x ¯ be a geometric point of M H ~Zord , which by abuse of notation we also identify as a geometric point of [(ΦH ,δH )] ord,ZH ~ [MH ] by Corollary 6.1.2.17. Let Aut(¯ x) be the group of automorphisms of ~ ord,ZH (cf. [62, Sec. A.7.5]). Let (M ~ ord,min )∧ denote the completion of x ¯ → M x ¯ H H ord,min ~ ~ ord,ZH ])∧ denote the completion the strict localization of M at x ¯. Let ([M x ¯ H H ~ ord,ZH ] at x ~ ord,ZH ), and of the strict localization of [M ¯ (as a geometric point of M H

H

ord,(`)

ord,(`)

let (FJΦH ,δH )∧ x ¯ denote the pullback of FJΦH ,δH under the canonical morphism ord,ZH ∧ ~ ~ ord,ZH . For convenience, let us also use the notation of the va(MH )x¯ → M H rious sheaves supported on x ¯ to denote their underlying rings or modules. Then we have a canonical isomorphism iAut(¯x) × ΓΦH h Y ord,(`) ∼ (6.1.2.20) O(M (FJΦH ,δH )∧ ~ ord,min )∧ = x ¯ x ¯ H

`∈P∨ Φ

H

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of rings, which is adic if we interpret the product on the right-hand side as the completion of the elements that are finite sums with respect to the ideal generated by ord,(0) the elements without constant terms (i.e., with trivial projection to (FJΦH ,δH )∧ x ¯ ). Let ord ∧ ~ us denote by (Z ) the completion of the strict localization of Z[(Φ ,δ )] at x ¯. ¯ [(ΦH ,δH )] x

H

H

~ ord,min )∧ to ([M ~ ord,ZH ])∧ , Then (6.1.2.20) induces a structural morphism from (M x ¯ x ¯ H H ∧ ~ ord,min )∧ whose pre-composition with the canonical morphism (~Zord ) → ( M x ¯ x ¯ H [(ΦH ,δH )] ord,ZH ∧ ord ∧ ∼ ~ ~ defines a canonical isomorphism (Z ) → ([M ]) . ¯ [(ΦH ,δH )] x

x ¯

H

Proof. The proof of [62, Prop. 7.2.3.16] works almost verbatim here. However, given the importance of this proposition, we shall spell out the details. By [35, III-1, 4.1.5 and 4.3.3], with natural generalizations to the context of algebraic stacks, the ring O(M x)-invariants in the ~ ord,min )∧ is isomorphic to the Aut(¯ x ¯ H H ord ~ ord,tor ring of regular functions over the completion of M along the fiber of ~ H : H,Σord ˜ ~ ord,min at x ~ ord,tor → M ¯. By Proposition 6.1.2.13, the preimage ~Zord M H [(ΦH ,δH )] := H,Σord ord H~ ord −1 ord H ~ ( H ) (~Z[(ΦH ,δH )] ) of ~Zord [(ΦH ,δH )] under H is the union ˜ ord ~Z [(ΦH ,δH )] =

∪ [(ΦH ,δH ,σ)]

~Zord [(ΦH ,δH ,σ)]

~ ord of those strata ~Zord [(ΦH ,δH ,σ)] over Z[(ΦH ,δH )] . According to (5) of Theorem 5.2.1.1 ∼ ~ ord,tor and [62, Lem. 6.2.5.27], there is a canonical isomorphism (M )∧ = H,Σord ~ Zord [(ΦH ,δH ,σ)]

~ ord X ΦH ,δH ,σ /ΓΦH ,σ for each representative (ΦH , δH , σ) of [(ΦH , δH , σ)]. Therefore, the H ord ~ ord,tor along the fiber of ~ ring of regular functions over the completion of M H,Σord

H

at x ¯ is isomorphic to the common intersection of the rings of regular functions ~ ord over the various completions of X ΦH ,δH ,σ /ΓΦH ,σ along the fibers of the structural H ord ~ ~ morphisms XΦH ,δH ,σ /ΓΦH ,σ → Mord,Z . In other words, it is isomorphic to the H ˆ FJord,(`) common intersection of the ΓΦH ,σ -invariants in the completions of ⊕ Φ ,δ `∈σ ∨

H

H

∼ ~ ord 0 0 for equivalent triples ~ ord along x ¯. Note that the identifications X ΦH ,δH ,σ = XΦ0H ,δH ,σ 0 0 0 (ΦH , δH , σ) and (ΦH , δH , σ ) involve the canonical actions of ΓΦH on the structural sheaves. Hence, the process of taking a common intersection also involves the process of taking ΓΦH -invariants. This shows the existence of (6.1.2.20). ∧ The claim that (6.1.2.20) is adic and that the composition (~Zord ¯ → [(ΦH ,δH )] )x ord,min ord,Z H ∧ ∧ ~ ~ (MH )x¯ → ([MH ])x¯ is an isomorphism follows from the fact that the sup∧ ∼ ~ ord,tor ~ ord port ~Zord of each formal completion (M /ΓΦ ,σ =X ord ) [(ΦH ,δH ,σ)]

H,Σ

is defined by the vanishing of the ideal that P∨ ΦH − {0} =

∩

σ∈ΣΦH ,σ⊂P+ Φ

H ,δH ,σ)]

~ ord (`) of ˆ Ψ ⊕ ΦH ,δH

`∈σ0∨

ΦH ,δH ,σ

~ Zord [(Φ

σ0∨ (because P∨ ΦH − {0}

H

~ ord (`), and ˆ Ψ ⊕ ΦH ,δH

`∈σ ∨ ⊂ σ0∨

for every σ ⊂

H

∨ P+ ΦH and because PΦH =

∩

σ∈ΣΦH

σ ∨ as explained in [62, Sec. 7.1.2]). Then we

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can conclude the proof by taking Aut(¯ x) × ΓΦH -invariants and by noting that ord,(0) ∧ Aut(¯ x) × ΓΦH ∼ Aut(¯ x) ∼ ∼ ((FJΦH ,δH )x¯ ) = (O(M = O[(M ~ ord,ZH ])∧ . ~ ord,ZH )∧ ) ~ ord,ZH )∧ ] = O([M H

x ¯

x ¯

H

x ¯

H

Corollary 6.1.2.21. (Compare with [62, Cor. 7.2.3.18].) The canonical finite surH ord ~ ord,ZH ]  ~Zord jection [M defined by ~ is an isomorphism. H

[(ΦH ,δH )]

H

Proof. The proof of Proposition 6.1.2.19 shows that the composition of the com∧ ~ ord,ZH ])∧ → (~Zord ~ ord,ZH ]  ~Zord pletion ([M x ¯ ¯ of the finite surjection [MH H [(ΦH ,δH )] )x [(ΦH ,δH )] H~ ord defined by H (described in Corollary 6.1.2.17) with the canonical structural iso∧ ∼ ~ ord,ZH ])∧ is the identity isomorphism. This forces morphism (~Zord ¯ → ([MH x ¯ [(ΦH ,δH )] )x ~ ord,ZH ]  ~Zord to be an isomorphism as the property of being an isomor[M H

[(ΦH ,δH )]

phism can be verified over the formal completions of the target. 6.2

Partial Minimal Compactifications of Ordinary Loci

In this section, we continue to assume the same settings as in Section 5.2. 6.2.1

Main Statements

~ ord can be described as follows: The partial minimal compactifications of M H Theorem 6.2.1.1. (Compare with [62, Thm. 7.2.4.1] and Theorem 1.3.1.5.) ~ ord,min quasi-projective and flat over ~S0,r = There exists a normal scheme M H H Spec(OF0 ,(p) [ζprH ]) (see Definition 2.2.3.3), such that we have the following: ~ ord as an open ~ ord,min contains the coarse moduli space [M ~ ord ] of M (1) M H H H fiberwise dense subscheme. ord ~ ord (2) Let (GM ~ ord , λM ~ ord , iM ~ ord , αHp , αHp ) be the tautological tuple over MH . Let H H H us define the invertible sheaf top ωM Lie∨ ~ ord := ∧ G H

~ ord ~ ord /MH M H

= ∧top e∗G~ ord Ω1G M H

~ ord ~ ord /MH M H

~ ord . Then there is a smallest integer N0 ≥ 1 such that ω ⊗ N0 is the over M H ~ ord MH

~ ord,min . pullback of an ample invertible sheaf O(1) over M H ~ ord → [M ~ ord ] is an isomorphism, and induces an If Hp is neat, then M H H ~ ord as an open fiberwise dense subscheme of M ~ ord,min . Moembedding of M H H reover, we have N0 = 1 with a canonical choice of O(1), and the restriction ~ ord is isomorphic to ω ~ ord . We shall denote O(1) by ω ~ ord,min , of O(1) to M H MH MH ~ ord,min . and interpret it as an extension of ω ~ ord to M MH

H

By abuse of notation, for each integer k divisible by N0 , we shall denote k O(1)⊗ k/N0 by ω ⊗ ~ ord,min itself is not defined. ~ ord,min even when ωM MH

H

~ ord,tor of M ~ ord as (3) For each (smooth) partial toroidal compactification M H H ord in Theorem 5.2.1.1, with a degenerating family (G, λ, i, αHp , αHp ) over

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~ ord,tor extending the tautological tuple (G ~ ord , λ ~ ord , i ~ ord , αHp , αord ) over M Hp H MH MH MH ~ ord , let M H

top top ∗ 1 ωM Lie∨ eG ΩG/M ~ ord,tor := ∧ ~ ord,tor = ∧ ~ ord,tor G/M H

be the graded

H

H

~ ord,tor extending ω ~ ord naturally. invertible sheaf over M H MH ord,tor ⊗k ~ algebra ⊕ Γ(MH , ω ~ ord,tor ), with its natural algebra MH k≥0

Then the structure

induced by tensor products, is independent of the choice (of the Σord used ~ ord,tor . in the definition) of M H ~ ~ The normal scheme Mord,min H  (quasi-projective and flat  over S0,rH ) is canok ~ ord,tor , ω ⊗ord,tor nically isomorphic to Proj ⊕ Γ(M ) , and there is a canok≥0

H

~ M H

H ord ~ ord,tor → M ~ ord,min determined by ω ~ ord,tor nical proper morphism ~ H : M H H MH H~ ord,∗ ⊗ N0 ∼ and the universal property of Proj, such that H O(1) = ω ~ ord,tor over MH H~ ord ord,tor ~ M , and such that the canonical morphism OM ~ ord,min → H,∗ OM ~ ord,tor H H H ord,tor ~ is an isomorphism. Moreover, when we vary the choices of M ’s, the H H ord morphisms ~ H ’s are compatible with the canonical morphisms among the ~ ord,tor ’s as in Proposition 5.2.2.2. M H H ord,∗ ⊗ k H ord When Hp is neat, we have ~ ω ord,min ∼ ω ~ ord,tor ∼ = ω ~ ord,tor and ~ = H

(4)

k ω⊗ ~ ord,min . M H ~ ord,min has M H

~ M H

MH

H,∗ MH

a natural stratification by locally closed subschemes a ~ ord,min = ~Zord M [(ΦH ,δH )] , H [(ΦH ,δH )]

with [(ΦH , δH )] running through a complete set of ordinary cusp labels (see Definition 1.2.1.7, [62, Def. 5.4.2.4], and Definition 3.2.3.8), such that the 0 [(Φ0H , δH )]-stratum ~Zord 0 )] lies in the closure of the [(ΦH , δH )]-stratum [(Φ0H ,δH ord 0 0 ~Z H )] [(ΦH ,δH )] if and only if there is a surjection from the cusp label [(ΦH , δ` to the cusp label [(ΦH , δH )] as in Definition 1.2.1.18. (The notation “ ” only means a set-theoretic disjoint union. The algebro-geometric structure ~ ord,min .) The analogous assertion holds after pulled back to is still that of M H fibers over ~S0,rH . Each [(ΦH , δH )]-stratum ~Zord [(ΦH ,δH )] is canonically isomorphic to the coarse ord,Z H ~ moduli space [M ] (which is a scheme) of the corresponding algebraic H ord,ZH ~ stack M (separated, smooth, and of finite type over ~S0,r ) associated H

H

with the cusp label [(ΦH , δH )] as in (4.2.1.28). Let us define the O-multi-rank of a stratum ~Zord [(ΦH ,δH )] to be the O-multirank of the cusp label represented by (ΦH , δH ) (see [62, Def. 5.4.2.7]). The ∼ ~ ord only stratum with O-multi-rank zero is the open stratum ~Zord [(0,0)] = [MH ],

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and those strata ~Zord [(ΦH ,δH )] with nonzero O-multi-ranks are called ordinary cusps. H ord ~ ord,tor is a surjection (5) The restriction of ~ H to the stratum ~Zord [(ΦH ,δH ,σ)] of MH to the stratum ~Zord of Mord,min . This surjection is smooth when Hp H

[(ΦH ,δH )]

∨ is neat, and is proper if σ is top-dimensional in P+ ΦH ⊂ (SΦH )R . ∼ ord,Z H ~ Under the above-mentioned identification [M ] → ~Zord H [(ΦH ,δH )] on the target, this surjection can be viewed as the quotient by ΓΦH ,σ (see [62, Def. 6.2.5.23]) of a torsor under a torus EΦH ,σ over an abelian scheme torsor ~ ord,ΦH of the algebraic stack M ~ ord,ZH ~ ord C etale cover M ΦH ,δH over the finite ´ H H ~ ord,ZH ] (which is a scheme), where the torus over the coarse moduli space [M H

EΦH ,σ is as in (5) of Theorem 1.3.1.5. (6) There is a canonical open immersion ~ ord,min ⊗ Q ,→ Mmin M H,rH H

(6.2.1.2)

Z

(see Definition 2.2.3.4) over S0,rH extending the canonical isomorphism ord ∼ ∼ [Mord H ] = [MH,rH ] = [MH ] × S0,rH over S0,rH (see the definition of MH S0

in Theorem 3.4.2.5), which is compatible with any canonical open immersion (5.2.1.2) in (7) of Theorem 5.2.1.1 and with the canonical morphisms H~ ord H ~ ord,tor ~ ord,min and :M : Mtor → Mmin . Under (6.2.1.2), the ord → M H

H

H,Σ

⊗k pullback of ωM min

H,rH

H

H,Σ

H

k is canonically isomorphic to the pullback of ω ⊗ ~ ord,min , MH

when both are defined for some integer k. The open immersion (6.2.1.2) induces isomorphisms ∼ ~Zord [(ΦH ,δH )] ⊗ Q → Z[(ΦH ,δH )],rH

(6.2.1.3)

Z

(see Definition 2.2.3.4), compatible with (5.2.1.3), when the cusp label [(ΦH , δH )] is ordinary; otherwise, the pullback of Z[(ΦH ,δH )],rH under (6.2.1.2) is empty. The canonical open immersion (6.2.1.2) extends to a canonical open immersion ~ ord,min ,→ M ~ min M H,rH H

(6.2.1.4)

k (see Definition 2.2.3.5) over ~S0,rH . Under (6.2.1.4), the pullback of ω ⊗ ~ min

MH,r

H

k is canonically isomorphic to ω ⊗ ~ ord,min , when both are defined for some inMH

~ ord,min is the pullback of teger k; and the [(ΦH , δH )]-stratum ~Zord [(ΦH ,δH )] of MH ~ min . the [(ΦH , δH )]-stratum ~Z[(ΦH ,δH )],rH of M H,rH Proof. With the ingredients we have provided, the proof is almost identical to that ~ ord,min is rather indirect, of [62, Thm. 7.2.4.1]. However, since the construction of M H we shall spell out the details for the sake of certainty.

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~ ord,min to be the normal scheme (quasi-projective and flat over Let us take M H ~S0,r ) constructed in Section 6.1.1. The first concern is whether its properties as H described in the theorem depend on the choice of Σord for the toroidal compactifi~ ord,tor = M ~ ord,tor cation M involved in the construction. It is clear that statements H H,Σord (1), (4), and (5) are satisfied regardless of the choice of Σord . Let us verify that this is also the case for statements (2) and (3). Suppose Σord,0 is a refinement of Σord as in Definition 1.2.2.16, suppose the ~ ord,tor : M ~ ord,tor ~ ord,tor morphism [1] etale surjection as in ord,0 → M ord is the proper log ´ H,Σ

H,Σ

Proposition 5.2.2.2, and suppose the invertible sheaves ωM are ~ ord,tor and ωM ~ ord,tor H,Σord H,Σord,0 ord ord H H ~ ord,tor ~ ord,min and ~ ~ ord,tor defined as in (6.1.1.1). Let ~ H,Σ : M → M : M → H H,Σ0 H,Σord H,Σord,0 H~ ord H~ ord ord,tor ord,min ~ ~ M be the two canonical morphisms. Then H,Σord,0 = H,Σord ◦ [1] and H ord H ord,tor ~ ∗ ~ ∼ ∼ ⊗ N0 [1] OM = OM ~ ord,tor ~ ord,tor implies that ( H,Σ ) O(1) = ω ~ ord,tor if and only if ∗ H,Σord,0

H ord (~

M

H,Σord

⊗ N0 ) O(1) ∼ = ωM ~ ord,tor H,Σord,0 ∗

H,Σord

(for the same O(1) and N0 ). In other words, we can

H,Σord,0

move freely between different choices of Σ by taking pullbacks or push-forwards, and there is a choice of O(1) with the smallest value of N0 ≥ 1 that works for all Σ. From now on, let us fix a choice of Σord and suppress it from the notation. We ~ ord,min would like to show that ω ~ ord extends to an ample invertible sheaf over M H

MH

when Hp is neat. H ord ~ ord,tor → M ~ ord,min to M ~ ord,1 By Proposition 6.1.2.18, the pullback of ~ H : M H H H ord,tor ord,tor ~ ~ is an isomorphism because the canonical morphism M → [M ] is an isoH H ~ ord,1 as an open subspace of morphism when Hp is neat. Therefore, we can view M H ord,tor ~ M and consider the restriction ωM ~ ord,tor |M1H , where ωM ~ ord,tor is as in statement H H H ord,1 ord,min ~ ~ (3). Since the complement of M in M has codimension at least two (by H H ~ ord,1 ) and since M ~ ord,min is noetherian and normal, it suffices to show definition of M H

H

that the coherent sheaf (see [32, VIII, Prop. 3.2]) ~ ord,min )∗ (ω ~ ord,tor | ~ ord,1 ) ~ ord,1 ,→ M ωM ~ ord,min := (MH H M M H

H

H

is an invertible sheaf. By fpqc descent (see [33, VIII, 1.11]), it suffices to verify this ~ ord,min . statement over the completions of strict localizations of M H ~ ord,min , Let x ¯ be a geometric point over some [(ΦH , δH )]-stratum ~Zord [(ΦH ,δH )] in MH ~ ord,tor that maps surjectiand consider any [(ΦH , δH , σ)]-stratum ~Zord in M [(ΦH ,δH ,σ)]

H

vely to ~Zord [(ΦH ,δH )] . Let (ΦH , δH , σ) be any representative of [(ΦH , δH , σ)]. Since p H is neat, H = Hp Hp is also neat, and our choice of Σord (see Definitions ~ ord 1.2.2.13 and 5.1.3.1) forces ΓΦH ,σ to act trivially on X ΦH ,δH ,σ (by [62, Lem. ord,tor ∧ ord ∼ ~ ~ 6.2.5.27]). Therefore, we have (MH )~Zord = XΦH ,δH ,σ (by (5) of The[(ΦH ,δH ,σ)]

~ ord,min )∧ orem 5.2.1.1). Let (M H ~ Zord

[(ΦH ,δH )]

~ ord,min denote the formal completion of M H

along the [(ΦH , δH )]-stratum ~Zord [(ΦH ,δH )] . Then we have a composition of canonical

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∼ ~ ord,tor )∧ ~ ord morphisms X ΦH ,δH ,σ = (MH ~ Zord

[(ΦH ,δH ,σ)]

page 349

349

~ ord,min )∧ → (M H ~ Zord

. By abuse of

[(ΦH ,δH )]

~ ord,min )∧ notation, let us denote the pullback of this composition from (M H ~ Zord

[(ΦH ,δH )]

~ ord,min )∧ of the strict localization of M ~ ord,min at x to the completion (M ¯ by x ¯ H H ord,tor ord,min ord ∧ ∧ ∧ ∼ ~ ~ ~ (X ) ( M ) → ( M ) . According to Proposition 6.1.2.19, = ¯ x ¯ x ¯ ΦH ,δH ,σ x H H ~ ord,min )∧ → (M ~ ord,ZH )∧ such that the further there is a structural morphism (M x ¯ x ¯ H H ord,tor ord,min ord ∧ ∧ ∼ ~ ~ ~ ord,ZH )∧ agrees with ~ composition (X )x¯ → (M )∧ ¯ = (MH x ¯ → (MH x ¯ ΦH ,δH ,σ )x H ∧ ~ ord,ZH )∧ induced by the structural morphism ~ ord the morphism (X ) → ( M x ¯ ¯ ΦH ,δH ,σ x H ~ ord,ZH of X ~ ord ~ ord ~ ord X →M . Over (X )∧ , the pullback ♥ ω of ω ~ ord,tor ΦH ,δH ,σ ~ ord,tor from M H

¯ ΦH ,δH ,σ x

H

is

ΦH ,δH ,σ isomorphic to (∧top Z

~ ord,min )∧ , (M x ¯ H

MH

X) ⊗ ωB~ by Lemma 6.1.2.3, which does descend Z

~ ord,ZH also makes sense there. to because the pullback of ωB~ from M H ~ ord,1 in the normal scheme M ~ ord,min has codimension Since the complement of M H H ord,min ~ ~ ord,min )∧ ) has to agree to (M at least two, the pullback of ωM ~ ord,min (from MH x ¯ H H ~ ord,min )∧ . In particular, it is invertible, as with the pullback of (∧top X) ⊗ ω ~ to (M Z

Z

B

H

x ¯

desired. H ord ∼ H ~ ord O ord,tor by The~ ord,tor → M ~ ord,min satisfies O ~ ord,min → Since ~ H : M ~ H H MH H,∗ M H orem 6.1.1.12, we see that two locally free sheaves E and F of finite rank over H ord H ord ~ ord,min are isomorphic if and only if (~ )∗ E ∼ M = (~ H )∗ F. Indeed, for the nontriH H H ord H ord H ord H ord vial implication we just need E ∼ = ~ H,∗ (~ H )∗ E ∼ = ~ H,∗ (~ H )∗ F ∼ = F (by the proH~ ord ∗ ∼ ~ ord,tor , we have jection formula (see [35, 0I , 5.4.10.1])). Since ( H ) ωM ~ ord,min = ωM H H ord ord H~ H~ N0 ∗ ∼ ∼ ⊗ord,tor ω ~ ord,tor = ωM has ~ ord,min , and the O(1) above such that ( H ) O(1) = ωM ~ H,∗ M H H H ⊗ N0 ∼ to satisfy O(1) = ω ord,min . This shows that ω ~ ord,min is ample and finishes the ~ M H

MH

verification of statements (2) and (3). Finally, statement (6) is a consequence of Proposition 2.2.1.2, (7) of Theorem 5.2.1.1, Theorem 6.1.1.12, Lemma 6.1.2.12, Proposition 6.1.2.13, and Corollary 6.1.2.14.

~ ord,min can be constructed as in Remark 6.2.1.5. When p is a good prime, M H Section 5.2.3, without any logical dependence on the construction by normalization in Section 2.2. Proposition 6.2.1.6 (base change properties). (Compare with [62, Prop. 7.2.4.3].) We can repeat the construction of Mord,min with ~S0,rH = H r Spec(OF0 ,(p) [ζp H ]) replaced with each (quasi-separated) locally noetherian normal ~ ord,min quasi-projective and flat scheme S over ~S0,rH , and obtain a normal scheme M H,S over S, with analogous characterizing properties described as in Theorem 6.2.1.1 (with Proj( · ) replaced with ProjS ( · ), and with Γ( · ) replaced with direct images

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over S), together with a canonical finite morphism ~ ord,min → M ~ ord,min × S. M H,S H

(6.2.1.7)

~ S0,rH

If S 0 → S is a morphism between locally noetherian normal schemes, then we also have a canonical finite morphism ~ ord,min ~ ord,min × S 0 . M →M (6.2.1.8) H,S 0 H,S S

Moreover, these finite morphisms satisfy the following properties: (1) If S → ~S0,rH (resp. S 0 → S) is flat, then (6.2.1.7) (resp. (6.2.1.8)) is an isomorphism. ~ ord,min × S (resp. M ~ ord,min × S 0 ) is noetherian and normal, then (2) If M H H ~ S0,rH

~ S0,rH

(6.2.1.7) (resp. (6.2.1.8)) is an isomorphism (by Zariski’s main theorem; see [35, III-1, 4.4.3, 4.4.11]). (3) Suppose s¯ is a geometric point of S. Then (6.2.1.8) (with S 0 replaced with s¯) is an isomorphism if the following condition is satisfied: ~ ord,min × s¯, char(¯ s) - # Aut(¯ x). (6.2.1.9) ∀ geometric points x ¯ of M H

~ S0,rH

(As in Proposition 6.1.2.19, Aut(¯ x) is the group of automorphisms of x ¯→ ~ ord,ZH , ~ ord,ZH × S or x ~ ord,ZH × s¯, or equivalently that of x ¯ → M ¯ → M M H H H ~ S0,rH

~ S0,rH

~ ord,min .) In this case, the if x ¯ is over the [(ΦH , δH )]-stratum ~Zord [(ΦH ,δH )] of MH ~ ord,min × s¯ is normal because M ~ ord,min is. geometric fiber M H,S H,¯ s S

(4) Suppose (6.2.1.9) is satisfied by all geometric points s¯ of S. (This is the case, for example, if Hp is neat. In general, there is a nonzero constant c depen~ ord such that # Aut(¯ x)|c ding only on the linear algebraic data defining M H ord,min ord,min ~ ~ for all geometric points x ¯ of MH ). Then the scheme MH × S ~ S0,rH

is normal, (6.2.1.7) is an ~ ord,min → S morphism M H,S mal fibers; see [35, IV-2, noetherian normal scheme

isomorphism (by property (2) above), and the is normal (i.e., flat with geometrically nor6.8.1 and 6.7.8]). Moreover, for every locally ~ ord,min × S 0 is normal, S 0 over S, the scheme M H,S S

(6.2.1.8) is an isomorphism (again, by property (2) above), and the mor~ ord,min phism M → S 0 is normal. H,S 0 Proof. We may assume that S and S 0 are affine, noetherian normal, and connected, because property (1) (and the convention that all schemes are quasi-separated) ~ ord,min along intersections of affine open allows us to patch the construction of M H,S subschemes of S. ~ ord,tor = M ~ ord,tor ~ ord,min , so that Let us take any M as in the construction of M H H H,Σord we have the canonical surjection   H~ ord k ~ ord,tor  M ~ ord,min = Proj ⊕ Γ(M ~ ord,tor , ω ⊗ord,tor :M ) H H H ~ H k≥0

MH

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which is proper and is the Stein factorization of itself, by Theorem 6.1.1.12. If we ~ ord,min over S, then we obtain a canonical morphism repeat the construction of M H H~ ord ~ ord,tor × S :M H,S

H

~ S0,rH

 ~ ord,min := Proj ⊕ Γ(M ~ ord,tor × S, (ω ~ ord,tor M H,S H M k≥0

H

~ S0,rH

⊗

O~S

 OS )⊗ k ) .

0,rH

H ord ~ ord,tor × S → M ~ ord,min × S is Since the base change morphism ~ H × S : M H H ~ S0,rH

proper, and since

N0 ω⊗ ⊗ ~ ord,tor M H O~S

~ S0,rH

~ S0,rH

OS is the pullback of the ample invertible sheaf

0,rH

N0 ω⊗ ~ ord,min MH

⊗

O~S

OS for the same integer N0 ≥ 1 as in (2) of Theorem 6.2.1.1, the

0,rH

H ord H ord Stein factorization of ~ H × S can be identified with the composition of ~ H,S ~ S0,rH

~ ord,min × S, which is the desired ~ ord,min → M with a canonical finite morphism M H H,S ~ S0,rH

~ ord,min M H,S

is a normal scheme quasi-projective morphism (6.2.1.7). Consequently, H~ ord over S, and H,S is also proper and is the Stein factorization of itself. The morphism S → ~S0,rH either is flat or factors through a closed point s of ~S0,r . In the former case, the morphism M ~ ord,min → S is the pullback of M ~ ord,min → H,S

H

H

~S0,r , which is flat by Theorem 6.2.1.1. In the latter case, the morphism M ~ ord,min → H H,S ~ ord,min → s, which is automatically flat. Thus, M ~ ord,min → S S is the pullback of M H,s

H,S

is always flat. The case of (6.2.1.8) is similar, with ~S0,rH (resp. S) replaced with S (resp. S 0 ). Now, property (1) has already been explained. Property (2) is self-explanatory, ~ ord,min and M ~ ord,min because M are noetherian normal by construction. H,S H,S 0 Let us prove property (3). Suppose that the condition (6.2.1.9) is satisfied. Since (6.2.1.7) is an isomorphism if it is so over the completions of strict local rings at geometric points of the target, and since the formation of Aut(¯ x)-invariants commutes with the base change from S to s¯ because char(¯ s) - # Aut(¯ x), by Proposition 6.1.2.19 (see, in particular, (6.1.2.20)), it suffices to show that, for each `0 ∈ P∨ ΦH ord,(`0 ) ∧ ∼ ) = with stabilizer ΓΦ ,` in ΓΦ , the formation of ΓΦ ,` -invariants in (FJ H

0

H

H

0

ΦH ,δH

x ¯

∧ ~ ord )∧ ~ ord Γ((C ¯. By ¯ , (ΨΦH ,δH (`0 ))x ¯ ) also commutes with the base change from S to s ΦH ,δH x Proposition 4.2.1.29, there exists a finite index normal subgroup Γ0ΦH of ΓΦH such ~ ord,ΦH , and such that the induced action of ΓΦ /Γ0 that Γ0ΦH acts trivially on M H ΦH H ~ ord,ΦH makes M ~ ord,ΦH → M ~ ord,ZH an ´etale (ΓΦ /Γ0 )-torsor. Hence, it suffices on M H ΦH H H H ~ ord,ΦH lifting x ~ ord,ZH , the formato show that, for each geometric point y¯ → M ¯→M H H ~ ord )∧ ~ ord (`0 ))∧ tion of invariants of Γ0ΦH ,`0 = ΓΦH ,`0 ∩ Γ0ΦH in Γ((C , ( Ψ y ¯ y¯ ), where ΦH ,δH ΦH ,δH ∧ ~ ( · ) denote the pullback to the completion of the strict localization of Mord,ΦH at y¯

y¯, commutes with the base change from S to s¯, for each `0 ∈ P∨ ΦH .

H

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Let X 0 and Y 0 be admissible sub-O-lattices of X and Y , respectively, such that φ(Y 0 ) ⊂ X 0 , such that `0 lies in the subgroup S0ΦH of SΦH defined by the same construction of SΦH using the embedding φ0 : Y 0 → X 0 induced by φ, and such that `0 is positive in Φ0H in the sense that, up to choosing a Z-basis y1 , . . . , yr of Y 0 , and by completion of squares for quadratic forms, there exists some integer N ≥ 1 such that N ·`0 can be represented as a positive definite matrix of the form ue t u, where e and u are matrices with integer coefficients, and where e = diag(e1 , . . . , er ) is diagonal with 0 positive entries. In this case, Γ0ΦH ,`0 acts on Φ0H via a discrete subgroup ΓΦH ,`0 of 0 the compact orthogonal subgroup of GLR (Y ⊗ R) preserving the above-mentioned Z

positive definite matrix by conjugation, which is necessarily finite. Consider the abe~ ord,ΦH defined by the same construction of C ~ ord,0 → M ~ ord , lian scheme torsor C ΦH ,δH H ΦH ,δH 0 0 0 using the embedding φ : Y → X instead of φ : Y → X, with a canonical morphism ~ ord,ΦH which is also an abelian scheme torsor, under which ~ ord ~ ord,0 C ΦH ,δH → CΦH ,δH over MH ~ ord (`0 ) descends to an invertible sheaf Ψ ~ ord,0 (`0 ), which is relatively amthe Ψ ΦH ,δH

ΦH ,δH

~ ord,ΦH because some positive tensor power of Ψ ~ ord,0 (`0 ) is isomorphic to ple over M H ΦH ,δH the pullback of the line bundle ⊗ (pr∗i (IdB , λB )∗ PB )⊗ ei over B under the finite 1≤i≤r

u∗

can.

~ ord,0 → Hom (Y, B) → Hom (Y, B) over morphism given by the composition C Z Z ΦH ,δH ~ ord,ΦH , because λB is a polarization (cf. [62, Def. 1.3.2.16]), and because all the M H 0 ei ’s are positive. Then Γ0ΦH ,`0 acts via the finite quotient ΓΦH ,`0 introduced above ∼ ~ ord )∧ ~ ord,0 )∧ ~ ord,0 (`0 ))∧ ~ ord (`0 ))∧ on Γ((C y¯ , (Ψ y¯ ). y¯ , (Ψ y¯ ) = Γ((C ΦH ,δH

ΦH ,δH

0

ΦH ,δH

If H is neat, then ΓΦH ,`0 is also neat and ~ ord,0 (`0 ) is relatively ample over rally, since Ψ ΦH ,δH ~ ord,0 )∧ , (Ψ ~ ord,0 (N · `0 ))∧ ) induces on ⊕ Γ((C N ≥0

ΦH ,δH y¯

ΦH ,δH

y¯

ΦH ,δH

must be trivial. More gene0 H ~ Mord,Φ , the action of ΓΦH ,`0 H 0

a faithful action of ΓΦH ,`0 on

~ ord,0 )∧ ~ ord,0 ∧ (C ΦH ,δH y¯ (cf. [81, Sec. 21, Thm. 5]). By construction, (CΦH ,δH )y¯ appears (up 00 ~ ord,ZH , to some identification) in the partial toroidal boundary construction of M H

00

~ ord,ZH is isomorphic to the stratum of M ~ ord,min labeled by the cusp label where M H H 00 00 00 [(ZH , ΦH , δH )] induced by [(ZH , ΦH , δH )] by the admissible surjections X → X 00 := X/X 0 and Y → Y 00 := Y /Y 0 (see [62, Lem. 5.3.1.14 and 5.4.2.11]). Therefore, there exists a degeneration (over a complete discrete valuation ring with fraction field 00 ~ ord,ZH such that k(z)) of an object parameterized by some functorial point z → M H

00

0 ~ ord,ZH above z. ΓΦH ,`0 is a subquotient of Aut(¯ z ) for any geometric point z¯ → M H Since char(¯ s) - # Aut(¯ z ) by the assumption that the condition (6.2.1.9) is satisfied, 0 0 it follows that char(¯ s) - #ΓΦH ,`0 . Therefore, the formation of ΓΦH ,`0 -invariants in ∧ ~ ord,0 )∧ ~ ord,0 Γ((C ¯, for each ΦH ,δH y¯ , (ΨΦH ,δH (`0 ))y¯ ) commutes with the base change from S to s `0 ∈ P∨ , and property (3) follows. ΦH It remains to prove property (4). Note that the assertions involving S 0 (in the last sentence) follow from the assertions involving only S, by [35, IV-2, 6.8.2 and 6.14.1] and property (2). To prove the assertions involving only S, we may replace S with its localizations, and assume that it is local. Let S1 be the localization of ~S0,rH at the

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image under S → ~S0,rH of the closed point of S. Since S1 is a localization of ~S0,rH , ~ ord,min → M ~ ord,min × S1 we know by property (1) that the canonical morphism M H H,S1 ~ S0,rH

~ ord,min M H,S1

~ ord,min M H

is an isomorphism, so that → S1 is the pullback of → ~S0,rH . Since the geometric points of S1 are either of characteristic zero or dominated by those ~ ord,min → S1 follows from of S, by [35, IV-2, 6.7.7], the normality of fibers of M H,S1 ~ ord,min × S ∼ ~ ord,min × S → S, the the normality of the geometric fibers of M =M H,S1

H

S1

~ S0,rH

latter of which follows from property (3) (and from the assumption that (6.2.1.9) is ~ ord,min → S1 satisfied by all geometric points s¯ of S). Thus, the (flat) morphism M H,S1 ~ ord,min × S → S is also a normal is normal. By [35, IV-2, 6.8.2], the pullback M H,S1

S1

~ ord,min × S is ~ ord,min × S ∼ morphism. By [35, IV-2, 6.14.1], the scheme M = M H H,S1 S1

~ S0,rH

normal. By property (2), this implies that (6.2.1.7) is an isomorphism, and hence ~ ord,min → S is normal, as desired. that the morphism M H,S 6.2.2

Hecke Actions

Let us state the following analogue of Proposition 5.2.2.2 for partial minimal compactifications. Proposition 6.2.2.1. (Compare with [62, Prop. 7.2.5.1] and Propositions 1.3.1.14, 2.2.3.1, 3.4.4.1 and 5.2.2.2.) Suppose we have an element g = (g0 , gp ) ∈ ∞ G(A∞,p ) × Pord D (Qp ) ⊂ G(A ) (see Definition 3.2.2.7), and suppose we have two ˆ such that H0 ⊂ gHg −1 , and such that open compact subgroups H and H0 of G(Z) 0 H and H are of standard form as in Definition 3.2.2.9. Let rH (resp. rH0 ) be defined by H (resp. H0 ) as in Definition 3.4.2.1. Suppose moreover that gp satisfies ~ ord : M ~ ord is defined (see ~ ord0 → M the conditions given in Section 3.3.4, so that [g] H

H

Proposition 3.4.4.1). Then there is a canonical quasi-finite surjection ~ ord,min : M ~ ord,min ~ ord,min [g] →M H0 H ~ ord ] : [M ~ ord0 ] → [M ~ ord ] induced by extending the canonical quasi-finite surjection [[g] H H ord ord ord ~ ~ ~ defined in Proposition the canonical quasi-finite surjection [g] : M 0 → M H

H

3.4.4.1. If the levels Hp and Hp0 at p are equally deep as in Definition 3.2.2.9, or if gp is of twisted Up type as in Definition 3.3.6.1 and depthD (Hp0 ) − depthD (gp ) = ~ ord,min is finite. depth (Hp ) > 0, then the surjection [g] D

If L ⊗ Zp ⊂ gp (L ⊗ Zp ), then there is a canonical morphism Z

Z ord,min

~ ([g]

~ )∗ : ([g]

ord,min

k ⊗k )∗ ω ⊗ ~ ord,min → ω ~ ord,min MH

MH0

(6.2.2.2)

k whenever ω ⊗ ~ ord,min is defined (which is compatible with the canonical isomorphism M H between the pullback of ω ⊗ k and ω ⊗ k over Mord0 ∼ = MH0 ,r 0 ). If gp ∈ Pord (Zp ), MH

MH0

H

H

D

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then the following diagram  ~ ord,min M H0 ~ ord,min [g]



~ ord,min  M H

~ min /M H0 ,rH0 



~ min [g] r 0 ,r H

(6.2.2.3) H

~ min /M H,rH

(see Definition 2.2.3.5) is (commutative and) Cartesian, and (6.2.2.2) is an isomorphism compatible with the corresponding one in Proposition 2.2.3.1. ~ ord,min maps the [(Φ0 0 , δ 0 0 )]-stratum ~Zord0 0 Moreover, the surjection [g] H H [(ΦH0 ,δH0 )] of ord,min ord,min ord ~ 0 ~ M to the [(ΦH , δH )]-stratum ~Z of M if and only if there are H

[(ΦH ,δH )]

H

0 0 0 representatives (ΦH , δH ) and (Φ0H0 , δH 0 ) of [(ΦH , δH )] and [(ΦH0 , δH0 )], respectively, 0 0 such that (ΦH , δH ) is g-assigned to (ΦH0 , δH0 ) as in [62, Def. 5.4.3.9]. If Σord = {ΣΦH }[(ΦH ,δH )] and Σord,0 = {Σ0Φ0 0 }[(Φ0 0 ,δ0 0 )] are two compatible H H H ~ ord choices of admissible smooth rational polyhedral cone decomposition data for M H ~ ord0 , respectively, such that Σord,0 is a g-refinement of Σord as in Definition and M H

~ ord,min : M ~ ord,min ~ ord,min is compatible 5.2.2.1, then the canonical surjection [g] M H0 H ord,tor ord,tor ord,tor ~ ~ ~ with the surjection [g] : MH0 ,Σord,0  MH,Σord given by Proposition 5.2.2.2. If g = g1 g2 , where g1 = (g1,0 , g1,p ) and g2 = (g2,0 , g2,p ) are elements of G(A∞,p ) × Pord D (Qp ), each having a setup similar to that of g, then we have ord,min ord,min ord,min ~ ~ ord = [g~2 ]ord ◦ [g~1 ]ord [g] = [g~2 ] ◦ [g~1 ] , extending the identity [g] in Proposition 3.4.4.1. min min in Proposition 1.3.1.14 caFinally, the finite surjection [g] : Mmin H0 → MH min min ~ min ⊗ Q nonically induces a finite surjection [g] :M 0 → Mmin . Then [g] rH0 ,rH

can be identified with the pullback of

min [g]rH0 ,rH

H ,rH0

H,rH

Z

~ ord,min ⊗ Q (on the target) under to M H Z

~ min ⊗ Q is finite. (6.2.1.2) in (6) of Theorem 6.2.1.1. In particular, [g] Z

Proof. Let Σord = {ΣΦH }[(ΦH ,δH )] and Σord,0 = {Σ0Φ0 0 }[(Φ0 0 ,δ0 0 )] be any two comH H H patible choices of admissible smooth rational polyhedral cone decomposition data ~ ord and M ~ ord0 , respectively, such that Σord,0 is a g-refinement of Σord as in for M H

H

Definition 5.2.2.1. (Such compatible choices of cone decompositions always exH ord ~ ord,tor ~ ord,min ist after refinements, by Proposition 5.1.3.2.) Let ~ H : M  M H H,Σord H~ ord ord,tor ord,min ~ ~ and H0 : MH0 ,Σ0,ord  MH0 be the surjections given by (3) of Theorem ~ ord,tor : M ~ ord,tor ~ ord,tor 6.2.1.1. Let [g]  M be the canonical surjection given H0 ,Σord,0 H,Σord ~ ord : M ~ ord0 → M ~ ord by Proposition 5.2.2.2 extending the canonical morphism [g] H H defined by the Hecke action of g, under which the ordinary Hecke twist of the ord ~ ord,tor tautological family (G, λ, i, αH0,p , αH by g (defined by Proposition 0 ) → M 0 H ,Σord,0 p ~ ord,tor 3.3.4.21 and Lemma 3.1.3.2) is the pullback (G0 , λ0 , i0 , α0 p , αord,0 ) → M 0 ord,0 of H

Hp

H ,Σ

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ord ~ ord,tor the tautological family (G, λ, i, αHp , αH )→M , equipped with a Q× -isogeny H,Σord p × [g −1 ]ord : G → G0 , which is up to Z× (p) -isogeny an isogeny (not just a Q -isogeny) when L ⊗ Zp ⊂ gp (L ⊗ Zp ). Z

Z

Consider the invertible sheaf 0 ωM ~ ord,tor

H0 ,Σord,0

:= ∧top Lie∨ ~ ord,tor G 0 /M

H0 ,Σord,0

= ∧top e∗G Ω1G0 /M ~ ord,tor

H0 ,Σord,0

~ ord,tor . We claim that ~ ord,tor over M , which is the pullback of ωM ~ ord,tor under [g] H0 ,Σord,0 H,Σord

there exists an integer N00 > 0 such that, for each k divisible by N00 , the k-th 0 0 descends to a (necessarily unique) intensor power (ωM )⊗ k of ωM ~ ord,tor ~ ord,tor H0 ,Σord,0

H0 ,Σord,0

~ ord,min vertible sheaf over M (see [62, Lem. 7.2.2.1]), which we abusively denote H0 0 ⊗k by (ωM . By the same argument as in the proof of Theorem 6.2.1.1, it ~ ord,min ) H0

suffices to show that there exists an integer N00 > 0 such that, for each stratum ~Zord ~ ord,min , for each stratum ~Zord ~ ord,tor [(ΦH0 ,δH0 )] in MH0 [(ΦH0 ,δH0 ,σ 0 )] in MH0 ,Σord,0 that maps surjectively to ~Zord , and for each representative (ΦH0 , δH0 , σ 0 ) of [(ΦH0 , δH0 , σ 0 )], [(ΦH0 ,δH0 )]

0

the N00 -th tensor power ( ♥ ω 0 )⊗ N0 of the pullback

♥

0 ω 0 of ωM ~ ord,tor

under the ca-

H0 ,Σord,0

~ ord,tor ~ ord nonical morphism X ΦH0 ,δH0 ,σ 0 → MH0 ,Σord,0 (see (5) of Theorem 5.2.1.1) descends to ~ ord,ZH0 ]. By ~ ord an invertible sheaf under the canonical morphism X 0 → [M H

ΦH0 ,δH0 ,σ

~ ord considering the pullback of the Q× -isogeny [g −1 ]ord : G → G0 to X ΦH0 ,δH0 ,σ 0 , ord,Z 0 0 0 0 H ~ there exists an O -lattice X and an abelian scheme B over MH such that ♥ 0 ∼ ω = (∧top X 0 ) ⊗ ωB 0 , where ωB 0 := ∧top Lie∨ ord,Z 0 (see Lemma 6.1.2.3, or Z

~ B 0 /M H

Z

H

rather the proof of [62, Lem. 7.1.2.1]). Hence, it suffices to take any integer N00 > 0 0 ⊗ N0 such that ΓH0 acts trivially on (∧top X 0 )⊗ N0 , and such that ωB 0 0 descends to Z ~ ord,ZH0 ], for any X 0 and B 0 as above, which exists by applying the analogue of [M H ~ ord,min (2) of Theorem 6.2.1.1 to the finitely many strata of M . H0 On the other hand, by (2) of Theorem 6.2.1.1, there exists an integer N0 > 0 k such that ω ⊗ ~ ord,min is defined for each k divisible by N0 , in which case the k-th MH

0 tensor power (ωM ~ ord,tor

H0 ,Σord,0

0 )⊗ k of ωM ~ ord,tor

H0 ,Σord,0

k is the pullback of ω ⊗ ~ ord,min under the MH

H ord ~ ord,tor H ord ~ ord,tor ~ ord,tor ~ ord,min , and hence ~ composition ~ H ◦ [g] :M →M ◦ [g] can H H H0 ,Σord,0 be identified with the composition   ord,tor 0 ⊗k ~ ord,tor ~ M → Proj ⊕ Γ( M , (ω ) ) ord,tor 0 ord,0 0 ord,0 H ,Σ

k≥0, N0 N00 |k

~ ord,min ∼ →M = Proj H of canonical morphisms.



⊕ k≥0, N0 |k

H ,Σ

~ M

H0 ,Σord,0

k ~ ord,min , ω ⊗ord,min Γ(M ) H ~



(6.2.2.4)

MH

0 Since (ωM ~ ord,tor

)⊗ k descends to an invertible sheaf

H0 ,Σord,0

0 ⊗k ~ ord,min (ωM over M when N00 |k, the above morphism (6.2.2.4) factors H0 ~ ord,min ) H0

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H ord ~ ord,tor ~ ord,min through ~ H0 : M → M and induces a (necessarily surjective) morH0 H0 ,Σord,0 ~ ord,min : M ~ ord,min ~ ord,min , under which (ω 0 ord,min )⊗ k is the pullback of phism [g] →M H0 H ~ M H0

k 0 ω⊗ ~ ord,min when N0 N0 |k. MH

0 But then (ωM ~ ord,tor

)⊗ k descends to a (necessarily unique) invertible sheaf

H0 ,Σord,0

k 0 ⊗k ~ ord,min , ~ ord,min (ωM over M , which is just the pullback of ω ⊗ H0 ~ ord,min ) ~ ord,min under [g] MH0

H0

whenever N0 |k.

0 Since ωM ~ ord,tor

H0 ,Σord,0

ord,tor

~ ∼ = ([g]

)∗ ωM ~ ord,tor , any sufficiently large H,Σord

p-power multiple of [g −1 ]ord , which can be [g −1 ]ord itself when L ⊗ Zp ⊂ gp (L ⊗ Zp ), induces a morphism ~ ord,tor )∗ : ([g] ~ ord,tor )∗ ω ~ ord,tor → ω ~ ord,tor ([g] M M H,Σord

Z

Z

,

(6.2.2.5)

H0 ,Σord,0

which in turn induces the desired morphism (6.2.2.2) whenever N0 |k. If the levels Hp and Hp0 at p are equally deep, or if gp is of twisted Up type as in Definition 3.3.6.1 and depthD (Hp0 ) − depthD (gp ) = depthD (Hp ) > 0, then the ~ ord,tor is proper by Proposition 5.2.2.2, and hence the induced surjection surjection [g] ~ ord,min is finite. [g] × −1 ord ] : Suppose gp ∈ Pord D (Zp ) (in this paragraph). Then the Q -isogeny [g × 0 G → G is a Z(p) -isogeny, and hence the morphisms (6.2.2.5) and (6.2.2.2) induced by [g −1 ]ord itself (not by nontrivial p-power multiples) are isomorphisms. ~ ord,min is finite by the previous paragraph. By the conMoreover, we know that [g] ~ min struction of [g] rH0 ,rH by various universal properties (see Definition 2.2.3.4 and the proof of Proposition 2.2.3.1), we obtain the commutative diagram (6.2.2.3), and the canonical isomorphism (6.2.2.2) is compatible with the corresponding one in H ord ~ ord0 is the canoProposition 2.2.3.1. By the fact that the restriction of ~ to M H0

H

~ ord,min to [M ~ ord0 ] ~ ord0 → [M ~ ord0 ], we see that the restriction of [g] nical morphism M H H H ord ~ ~ ord0 ] → [M ~ ord ] induced by the canonical suris the canonical surjection [[g] ] : [M H H ~ ord : M ~ ord0  M ~ ord defined by the Hecke action of g. Consequently, by jection [g] H H ~ ord,min ~ ord,min and M (and by Zariski’s main theorem; noetherian normality of M H H0 ord,min ~ ~ ord,min see [35, III-1, 4.4.3, 4.4.11]), M 0 coincides with the normalization of M H

in or

~ ord ~ ord0 under the composition of canonical morphisms M ~ ord0 [g]→ M ~ ord M H H H ord ord ~ ~ equivalently (by the construction of MH0 and MH in Theorem ord ~ ord

H

~ ord,min , M H

,→ 3.4.2.5, and

by the construction of [g] and [g] in Proposition 3.4.4.1) the normalization ~ ord,min in Mord0 ∼ 0 of M M under the composition of canonical morphisms = H ,rH0 H H ord

[g] ~ ord,min . Hence, the commutative diagram (6.2.2.3) is CarMord → Mord H0 H → MH min ord ~ ~ min tesian, because [g] rH0 ,rH is the normalization of MH,rH in MH0 under the canonical ord

[g] ord ~ ord,min ,→ M ~ min (by the construction in Definition morphism Mord H0 → MH → MH H,rH

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2.2.3.5 and the proof of Proposition 2.2.3.1, and by Proposition 3.4.4.1 again for the comparison between [g]ord and [g]). ~ ord,min follow ~ ord,min The statements about the images of the strata of M under [g] H0 ~ ord,tor from the corresponding statements about the images of the strata of M 0 ord unH ,Σ

ord,tor

~ der [g] . The last two paragraphs of this proposition follows from the last two paragraphs of Proposition 5.2.2.2 (by choosing compatible choices of cone decompositions, which is always possible after refinements, by Proposition 5.1.3.2, such ~ ord,tor , [g~1 ]ord,tor , and [g~2 ]ord,tor are all defined). that [g] Corollary 6.2.2.6. (Compare with [62, Cor. 7.2.5.2] and Corollary 2.2.3.2.) Supˆ such that H and H0 are pose we have two open compact subgroups H and H0 of G(Z) of standard form as in Definition 3.2.2.9, such that H0 is a normal subgroup of H, and such that depthD (H) = depthD (H0 ). Then the canonical morphisms defined in ~ ord,min Proposition 6.2.2.1 induces a canonical action of the finite group H/H0 on M , H0 ord,min ord,min ord,min ord,min ~ ~ ~ ~ 0 M , where M is the and the canonical surjection [1] :M H,rH0

H

H,rH0

~ ord,min × ~S0,r 0 , can be identified with the quotient of M ~ ord,min normalization of M H H0 H ~ S0,rH

by this action. ~ ord,min is quasi-projective Proof. The existence of such an action is clear. Since M H0 ord,min 0 ~ ~ over S0,rH0 and normal, the quotient MH0 /(H/H ) exists as a scheme over ~S0,rH0 (cf. [25, V, 4.1]). Then it follows from Zariski’s main theorem (see [35, III-1, 4.4.3, ~ ord,min ~ ord,min over ~S0,r 0 (with 4.4.11]) that the induced morphism M /(H/H0 ) → M H0 H,rH0 H noetherian normal target) is an isomorphism, because it is finite by Proposition 6.2.2.1, and because it is generically an isomorphism (over Mord H , by the moduli ord as in Theorem 3.4.2.5, and by the characterizaand M interpretations of Mord 0 H H tion of coarse moduli spaces as geometric and uniform categorical quotients in the category of algebraic spaces; see [62, Sec. A.7.5]). Corollary 6.2.2.7. (Compare with Corollaries 3.4.4.3 and 5.2.2.3.) With the setting as in Proposition 6.2.2.1, the morphism ~ ord,min : M ~ ord,min ~ ord,min →M [g] H0 H ~ ord,min : M ~ ord,min ~ ord,min is finite. (cf. Definition 3.4.4.2) induced by [g] →M H0 H ~ ord,min : Proof. As in the proof of Proposition 6.2.2.1, we may assume that [g] ~ ord,tor : M ~ ord,min ~ ord,min is induced by some [g] ~ ord,tor ~ ord,tor M → M → M . Since H0 H H0 ,Σord,0 H,Σord ord H~ ord H ~ ord,tor ~ ord,min and ~ ~ ord,tor the canonical morphisms : M : M 0 0 ord,0 → M 0 ord → H

H ,Σ

H

H

H,Σ

~ ord,tor : ~ ord,min are proper and surjective, and since the canonical morphism [g] M H

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~ ord,tor ~ ord,tor M ⊗ Fp → M ⊗ Fp is proper by Lemma 3.3.6.8 (see the proof of CorolH0 ,Σord,0 H,Σord Z

Z

lary 5.2.2.3), it follows that the quasi-finite morphism ~ ord,min : M ~ ord,min ~ ord,min ⊗ Fp [g] ⊗ Fp → M H0 H Z

Z

is also proper, which is then finite (cf. [35, IV-3, 8.11.1]). Hence, the morphism ~ ord,min : M ~ ord,min ~ ord,min is also finite, as desired. [g] →M 0 H

H

Corollary 6.2.2.8. (Compare with Corollaries 3.4.4.4 and 5.2.2.4, and Example 3.4.4.5.) With the setting as in Proposition 6.2.2.1, if g = (g0 , gp ) ∈ ˆ p ) × Pord (Zp ), if H0,p = g0 Hp g −1 in G(Z ˆ p ), if H0 ord = (gp Hp g −1 )ord in G(Z D p p 0 ord p MD (Zp ) (see (3.3.3.5)), and if H and hence H0,p are neat, then (rH0 = rH and) ~ ord,min : M ~ ord,min ~ ord,min is an isomorphism. (See the the induced morphism [g] →M 0 H

H

~ ord,min : remark at the end of Corollary 3.4.4.4.) Consequently, the morphism [g] ~ ord,min ~ ord,min is ´etale in a (Zariski) neighborhood of M ~ ord,min M →M ⊗ Fp . (Despite H0 H H0 Z

min

min the last statement of Proposition 6.2.2.1, the morphism [g] : Mmin in H0 → MH Proposition 1.3.1.14 might not be ´etale along cusps Z[(ΦH0 ,δH0 )] labeled by cusp labels ~ ord,min [(ΦH0 , δH0 )] not compatible with D, which do not meet M ⊗ Q.) H0 Z

~ ord,min : M ~ ord,min Proof. By Corollary 6.2.2.7, the induced morphism [g] → H0 ord,min ord,min ord ~ ~ ~ 0 is an isoM is finite. By Corollary 3.4.4.4, the restriction of [g] to M H H ~ ord,min morphism. Since Hp and H0,p are neat, by (4) of Proposition 6.2.1.6, M ⊗ Fp 0 H

and

~ ord,min M H

Z

⊗ Fp are normal. Hence, by Zariski’s main theorem (see [35, III-1, Z

~ ord,min : M ~ ord,tor ~ ord,min ⊗ Fp ⊗ Fp → M 4.4.3, 4.4.11]), the induced finite morphism [g] H H0 Z

Z

~ ord,min : M ~ ord,min ~ ord,min (being a is necessarily an isomorphism. Hence, [g] → M H0 H finite morphism between formal schemes flat over Zp , which is an isomorphism between the fibers over Spec(Fp )) is an isomorphism. As for the last statement, ~ ord,min : M ~ ord,min ~ ord,min is ´etale at the points of M ~ ord,tor [g] → M ⊗ Fp because it 0 0 H

H

H

Z

induces isomorphisms between the formal completions (see [35, IV-4, 17.6.3]), and ~ ord,min hence it is ´etale at a neighborhood of M ⊗ Fp because ´etaleness is an open H0 Z

condition on the source of a morphism (see [33, I, 4.5]). Corollary 6.2.2.9 (elements of Up type). (Compare with Corollaries 3.4.4.6 and 5.2.2.5.) Suppose in Proposition 6.2.2.1 that g0 = 1 and gp is of Up type as in Definition 3.3.6.1 (so that it is of twisted Up type and depthD (gp ) = 1). Then the induced morphism ~ ord,min : M ~ ord,min ~ ord,min ⊗ Fp [g] ⊗ Fp → M H0 H Z

Z

(6.2.2.10)

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is finite and coincides with the composition of the absolute Frobenius morphism ~ ord,min ⊗ Fp → M ~ ord,min FM ⊗ Fp ~ ord,min ⊗ Fp : MH0 H0 H0

Z

Z

Z

with the canonical finite morphism ~ ord,min : M ~ ord,min ~ ord,min ⊗ Fp . [1] ⊗ Fp → M H0 H Z

(6.2.2.11)

Z

ord

If Hp0 = Hpord as open compact subgroups of Mord D (Zp ) (see (3.3.3.5)), then 0 (rH = rH and) the canonical morphism (6.2.2.11) is an isomorphism by Corollary 6.2.2.8, and the composition ~ ([1]

ord,min −1

∼

~ ord,min ⊗ Fp M H

→

Z

)

~ ord,min M ⊗ Fp H0

~ ord,min [g]

→

Z

~ ord,min ⊗ Fp M H Z

coincides with the absolute Frobenius morphism ~ ord,min ⊗ Fp → M ~ ord,min ⊗ Fp . FM ~ ord,min ⊗ Fp : MH H H

Z

Z

Z

Proof. The first paragraph of the corollary follows from the corresponding first ~ ord,min : M ~ ord,min ~ ord,min is induced paragraph of Corollary 5.2.2.5, because [g] →M 0 H

H

~ ord,tor : M ~ ord,tor ~ ord,tor by some [g] → M as in the proof of Proposition 6.2.2.1. H0 ,Σord,0 H,Σord The second paragraph of the corollary follows from the first paragraph and from Corollary 6.2.2.8. Remark 6.2.2.12. (Compare with Remarks 3.4.4.9 and 5.2.2.8.) By Kunz’s theorem [54] (cf. [76, Sec. 42, Thm. 107]), the absolute Frobenius morphisms FM ~ ord,min ⊗ Fp and FM ~ ord,min ⊗ Fp in Corollary 6.2.2.9 are not flat in general, because H0

Z

H

Z

~ ord,min ~ ord,min ⊗ Fp are not regular in general (except in very special M ⊗ Fp and M H0 H Z

Z

cases). Therefore, while familiar facts in the modular curve case might remain true in general, some proofs might have to be modified due to the failure of flatness of such absolute Frobenius morphisms. 6.2.3

Quasi-Projectivity of Partial Toroidal Compactifications

Theorem 6.2.3.1. (Compare with [62, Thm. 7.3.3.4] and Theorem 1.3.1.10.) Suppose Hp is neat, and suppose Σord is projective with a compatible collection polord of polarization functions as in Definition 5.1.3.3. (Such Σord and polord exist by Proposition 5.1.3.4.) For each integer d ≥ 1, suppose ~H,dpolord is defined over ~ ord,tor = M ~ ord,tor M as in Definition 1.3.1.7, and suppose J~H,dpolord is defined over H H,Σord ord,min ~ M as in Definition 1.3.1.8. Then there exists an integer d0 ≥ 1 such that the H

following are true:

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H ord (1) The canonical morphism (~ H )−1 J~H,d0 polord ·OM H,d0 polord of cohe~ ord,tor → ~ H rent OM ~ ord,tor -ideals is an isomorphism, which induces a canonical morphism H

NBlJ~

H,d0 pol

H~ ord ~ ord,tor → NBl ~ ( ):M H J H ord

H,d0 polord

~ ord,min ) (M H

by the universal property of the normalization of blow-up (see [62, Def. 7.3.2.2]). H ord (2) The canonical morphism NBlJ~ (~ H ) above is an isomorphism. ord H,d0 pol

H ord ~ ord,tor → M ~ ord,min is projective, and hence In particular, the morphism ~ H : M H H ord,tor ~ M is a scheme quasi-projective (and smooth) over ~S0,rH . If Condition 1.3.1.9 H is satisfied, then the above two statements (1) and (2) are true for all d0 ≥ 3. Proof. Let us begin with the reduction to the case Condition 1.3.1.9 (cf. [4, Ch. IV, Sec. 2, p. 329], [28, Ch. V, Sec. 5, p. 178], and [62, Cond. 7.3.3.3]) is satisfied. By [62, Lem. 7.3.1.7], or rather by the original [28, Ch. V, Lem. 5.3], there exists a normal open compact subgroup H0,p of Hp such that, for H0 = H0,p Hp (while 0 H = Hp Hp , with the same Hp ), Condition 1.3.1.9 is satisfied by the Σord,(H ) = 0 {ΣΦH0 }[(ΦH0 ,δH0 )] and polord,(H ) = {polΦH0 }[(ΦH0 ,δH0 )] induced by Σord and polord 0 as in [62, Constr. 7.3.1.6], and such that Σord,(H ) is smooth. ~ ord,tor ~ ord,tor 0 , ~ord 0 (H0 ) , Suppose that Theorem 6.2.3.1 is true for M =M H0 H0 ,Σord,(H ) H0 ,d0 pol ord,tor ord 0 ~ ~ and J is quasi-projective 0 (H0 ) for some integer d ≥ 1. In particular, M 0 0

H0 ,d0 pol

H

and smooth over ~S0,rH . ~ ord ~ ord By construction, the surjections Ξ ΦH0 ,δH0 (σ)  ΞΦH ,δH (σ) are finite flat (with possible ramification along the boundary strata) whenever (ΦH , δH ) is induced by (ΦH0 , δH0 ) and σ is a cone in the cone decomposition ΣΦH0 = ΣΦH of PΦH0 ∼ = ord,tor ord,tor ord,tor ~ ~ ~ PΦ . Therefore, the canonical surjection [1] :M 0  M ord (given H0 ,Σord,(H

H

)

H,Σ

by Proposition 5.2.2.2) is finite flat. It is the unique finite flat extension of the ~ ord0  M ~ ord (since H0 = H0,p Hp and H = Hp Hp canonical (finite ´etale) surjection M H H ord,tor ~ only differ away from p). Since MH0 ,Σord,(H0 ) is quasi-projective and smooth over ~ ord,tor ~S0,r , the quotient by H/H0 is also quasi-projective and isomorphic to M ord H,Σ

H

over ~S0,rH (by [25, V, 4.1] and by Zariski’s main theorem [35, III-1, 4.4.3, 4.4.11] as in the proof of [62, Cor. 7.2.5.2] and Corollary 6.2.2.6). Moreover, we know that ord,tor ∼ ~ ~ord by construction. Hence, we have verified all the )∗~ord 0 = ([1] H,polord H0 ,polord,(H ) assumptions of [62, Prop. 7.3.2.3], whose application completes the reduction. H ord H ~ ord,tor → M ~ ord,min parallels that of Since the description of ~ :M : Mtor → H

H

H

H

H

Mmin H (see Proposition 6.1.2.19 and [62, Prop. 7.2.3.16]), the remainder of the proof, namely the verification of the theorem under Condition 1.3.1.9, is analogous to that of [62, Thm. 7.3.3.4; see also the errata].

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Corollary 6.2.3.2. Under the assumptions in Theorem 6.1.1.12, suppose moreover that Hp is neat, that Σord is projective smooth and extends to a projective (but possibly nonsmooth) Σ for MH , with a collection pol of polarization functions, and ~ tor that M H,d0 pol is still defined as in Proposition 2.2.2.3, for some integer d0 ≥ 1. (If we extend Σord to a projective smooth Σ for MH , which is possible by Proposition 5.1.3.4, then we only need Proposition 2.2.2.1.) Then we have a commutative diagram  ~ tor ~ ord,tor /M (6.2.3.3) M H,d0 pol,rH H,Σord 

~ ord,min  M H

 /



~ min M H,rH

of canonical morphisms over ~S0,rH , in which the top horizontal arrow is also an ~ ord,tor open immersion (over ~S0,rH ) extending the open immersion M × S0,rH ,→ H,Σord ~ S0,rH

Mtor H,Σ,rH

(over S0,rH ) in (7) of Theorem 5.2.1.1, and a fortiori the induced canonical morphism ~ ord,min ~ ord,tor →M M H H,Σord

~ tor M H,d0 pol,rH

× ~ min M H,r

(6.2.3.4)

H

is an isomorphism. (That is, the diagram (6.2.3.3) is Cartesian.) Proof. The existence of the commutative diagram (6.2.3.3) in which the top horizontal arrow is an open immersion is implied by the existence of the isomorphism (6.2.3.4). By assumption, polord is the restriction of pol, so that ~ ord,min → M ~ min )∗ J~H,d pol J~H,d0 polord ∼ = (M H 0 H because they both define the schematic closure of the closed subscheme of ~ ord,min ⊗ Q defined by (M ~ ord,min ⊗ Q → Mmin )∗ JH,d pol . Hence, we obtain a canoM 0 H H H Z

Z

nical isomorphism NBlJ~

H,d0 polord

~ ord,min ) → NBl ~ ord,min ~ min ∗ ~ (M H (M →M ) JH,d H

H

0 pol

~ ord,min ), (M H

which can be identified with the desired isomorphism (6.2.3.4) by Theorem 6.2.3.1 and by the construction in Proposition 2.2.2.3 (or Proposition 2.2.2.1, if Σ is projective smooth). 6.3 6.3.1

Full Ordinary Loci in p-Adic Completions Hasse Invariants

~ Consider any toroidal compactification Mtor ˆ p ) (over S0,aux = Spec(OF0,aux ,(p) )) Gaux (Z carrying a tautological semi-abelian scheme Gaux as in [62, Thm. 6.4.1.1]. (In this subsection, we shall suppress the notation for cone decompositions for simplicity.)

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Let Fabs : Mtor G

ˆp )

aux (Z

phism, and let

⊗ Fp → Mtor G

⊗ Fp denote the absolute Frobenius mor-

ˆp )

aux (Z

Z

Z

Faux : Gaux ⊗ Fp → (Gaux ⊗ Fp )(p) := F∗abs (Gaux ⊗ Fp ) Z

Z

Z

Mtor ˆ p ) ⊗ Fp , Gaux (Z Z

be the relative Frobenius morphism of Gaux ⊗ Fp → an isogeny of semi-abelian schemes over Mtor G

ˆ p ⊗ Fp aux (Z ) Z

Z Mtor ˆ p ) ⊗ Fp . Gaux (Z Z

which is

Since Gaux ⊗ Fp → Z

is flat, by [25, VIIA , 4.3], there is a canonical morphism Vaux : (Gaux ⊗ Fp )(p) → Gaux ⊗ Fp , Z

Z

called the (relative) Verschiebung morphism of Gaux ⊗ Fp → Mtor G

ˆp aux (Z )

Z

⊗ Fp , such Z

that Vaux ◦ Faux = [p], the multiplication by p on Gaux ⊗ Fp . Then Vaux is also an Z

isogeny, and induces a morphism Lie∨ (Vaux ) : Lie∨ (Gaux ⊗ Fp )/(Mtor

⊗ Fp )

Gaux (ˆ Zp ) Z

Z

→ Lie∨ (Gaux ⊗ Fp )(p) /(Mtor

⊗ Fp ) Gaux (ˆ Zp ) Z

Z

∼ = F∗abs (Lie∨ (Gaux ⊗ Fp )/(Mtor

⊗ Fp ) Gaux (ˆ Zp ) Z

Z

).

By taking top exterior powers (of the same degree), we obtain a morphism ∧top Lie∨ (Vaux ) : ωMtor

Gaux (ˆ Zp )

→

F∗abs (ωMtor G

Zp ) aux (ˆ

⊗ Fp Z

⊗p ⊗ Fp ) ∼ = ωMtor

Gaux (ˆ Zp )

Z

⊗ Fp , Z

or equivalently a section Hasseaux ∈ Γ(Mtor G

⊗(p−1)

ˆp )

aux (Z

⊗ Fp , ωMtor Z

Gaux (ˆ Zp )

⊗ Fp ).

(6.3.1.1)

Z

The value of Hasseaux at each geometric point s¯ of Mtor G

ˆp aux (Z )

⊗ Fp is the so-called Z

Hasse invariant of the pullback (Gaux )s¯ of the semi-abelian scheme Gaux to s¯, which is nonzero exactly when (Gaux )s¯ is an ordinary semi-abelian variety, or equivalently when the abelian part of (Gaux )s¯ is an ordinary abelian variety (because the nonvanishing of the Hasse invariant is equivalent to the separability of the Verschiebung morphism, which is in turn equivalent to the triviality of the local-local part, as in [81, p. 147], of the p-torsion subgroup scheme of the abelian part of (Gaux )s¯). Remark 6.3.1.2. By definition, the formation of Hasse invariants is compatible with separable isogenies, because they induce isomorphisms between sheaves of differentials. ~ ord,tor over Let us also consider any smooth partial toroidal compactification M H ~S0,r carrying a tautological semi-abelian scheme G as in Theorem 5.2.1.1. Then, H similar to the case of Hasseaux above, we can define a section ~ ord,tor ⊗ Fp , ω ⊗(p−1) Hasseord H ∈ Γ(MH ~ ord,tor ⊗ Fp ). Z

MH

Z

(6.3.1.3)

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~ ord,tor ⊗ Fp is the Hasse inThe value of Hasseord ¯ of M H at each geometric point s H Z

variant of the pullback Gs¯ of the semi-abelian scheme G to s¯, which is always nonzero because the abelian part of Gs¯ is always an ordinary abelian variety. This is consistent with the following: Lemma 6.3.1.4. With the setting as in Lemma 6.1.1.9, suppose (with suitable choices of cone decompositions, up to refinement if necessary) there is a morphism ~ ord ~ ord,tor → Mtor M ˆ p ) ] given by the ˆ p ) extending the morphism MH → [MGaux (Z H Gaux (Z can. can. ord ~ ~ composition M → MH → M → [M ˆp ˆ p ] (see Propositions 2.2.1.1 H

Gaux (Z )

Gaux (Z )

~ ord,tor ⊗ Fp under the canonical and 3.4.6.3). Then the pullback of Hasseaux to M H Z

~ ord,tor ⊗ Fp → (M ~ tor ~ morphism M H,d0 pol × S0,rH ) ⊗ Fp is nowhere zero. H ~ S0

Z

Z

If a0 ≥ 1 and a ≥ 1 are integers as in Lemma 2.1.2.35, then the pullback of 0 ~ ord,tor ⊗ Fp is the multiple of (Hasseord )a by a global unit. to M Hasseaaux H H Z

~ ord,tor → Mtor Proof. The pullback of Gaux (resp. G∨ aux ) under MH G × a1

morphic to G

∨ × a2

×

(G )

× a2

×

(resp. G

~ ord,tor M H

∨ × a1

(G )

ˆp aux (Z )

must be iso-

), because it is already

~ ord,tor M H

so over the open dense subscheme Mord H by Proposition 2.1.1.15 (and by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5]). Then the lemma follows from the fact that an abelian variety is ordinary if and only if its dual is (cf. Lemma 3.1.1.5). Remark 6.3.1.5. The definition of Hasseord and Lemma 6.3.1.4 may seem redundant, but we will indeed need them in Proposition 6.3.2.2 below. p ˆ p ) is a neat open compact sub⊂ Gaux (Z Proposition 6.3.1.6. Suppose that Haux min group. Then we can replace MGaux (Zˆ p ) (resp. MG (Zˆ p ) , resp. Mtor ˆ p ) ) with the G (Z aux

aux

scheme MHpaux (resp. Mmin , resp. Mtor with any choice of cone decompositions) Hp Hp aux aux ~ over S0,aux in the constructions above. The invertible sheaf ωMtorp descends to Haux

an ample invertible sheaf ωMmin p

Haux

on Mmin (by [62, Thm. 7.2.4.1]), and for each Hp aux

integer k ≥ 0 the canonical morphism ⊗k ⊗k Γ(Mmin ⊗ Fp , ωM ⊗ Fp ) → Γ(Mtor ⊗ Fp , ωM tor min Hp Hp aux aux p p Haux

Z

Z

Z

Haux

⊗ Fp ) Z

is an isomorphism, under which we can pullback Hasseaux and its powers (which are min a priori defined on Mtor ⊗ Fp . ˆ p ) ⊗ Fp ) to MHp aux G (Z aux

Z

Z

Proof. This follows from [62, Prop. 7.2.4.3 and Cor. 7.2.4.8]. Corollary 6.3.1.7. Let a0 ≥ 1 and a ≥ 1 be integers as in Lemma 2.1.2.35, p and let N1 be as in Proposition 2.2.1.2 (for some choice of Haux ). Then we can

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Compactifications with Ordinary Loci ⊗ aN (p−1) ⊗ Fp ), MH Z

1 0 N1 ~ min ⊗ Fp , ω pullback Hasseaaux to a section of Γ(M H ~ min

Z

which we denote

1 by HasseaN by abuse of notation. H Suppose moreover that the image of H under the canonical homomorphism ˆ → G(Z ˆ p ) is neat (which means, a fortiori, that H is also neat), so that (by G(Z) p ˆ p) Lemma 2.1.1.18) there exists some neat open compact subgroup Haux ⊂ Gaux (Z p such that H is mapped into Haux = Haux Gaux (Zp ) under the homomorphism ˆ → Gaux (Z) ˆ given by (2.1.1.10). (This condition is satisfied when H = Hp Hp G(Z) is of standard form as in Definition 3.2.2.9 and Hp is neat.) Then (with this choice p of Haux ) we can take N1 = 1 in the above paragraph.

Proof. This follows from Proposition 6.3.1.6, because the canonical morphism min ~ min → Mmin ~ min M in this case. (The second paraH ˆ p ) factors through MH → MHp aux Gaux (Z graph is self-explanatory.) Corollary 6.3.1.8. Suppose we have an element g = (g0 , gp ) ∈ G(A∞,p ) × G(Zp ) ⊂ ˆ such G(A∞ ), and suppose we have two open compact subgroups H and H0 of G(Z) min ~ ~ min ~ min that H0 ⊂ gHg −1 , so that we have a canonical finite surjection [g] :M H0 → MH (over ~S0 ) and a canonical isomorphism ~ ([g]

min

⊗ k(p−1) ∼ → MH

)∗ ω ~ min

⊗ k(p−1) MH0

ω ~ min

(6.3.1.9)

~ min over M H0 whenever the right-hand side is defined for some k ≥ 1, as in Proposition 2.2.3.1. Up to replacing k with a more divisible integer, suppose that both HassekH and HassekH0 are defined as (some powers of the ones defined) in Corollary 6.3.1.7. Then the canonical morphism min

~ ([g]

⊗ k(p−1) ⊗ Fp ) MH Z ⊗ k(p−1) min ~ 0 ⊗ Fp , ω Γ(M ⊗ Fp ) H ~ min M Z Z H0

~ min ⊗ Fp , ω )∗ : Γ(M H ~ min Z

→

(6.3.1.10)

induced by (6.3.1.9) sends HassekH to HassekH0 . ~ min (see Proposition 2.2.3.1), it is induced by Proof. By the construction of [g] min tor some [g0 ] : Mmin → Mmin : ˆ p ) , which is in turn induced by some [g0 ] H0,p G (Z aux

Mtor → Mtor H0,p G aux

ˆp aux (Z ) HassekH0

aux

(with some suitable cone decompositions). By definition of

HassekH and in Corollary 6.3.1.8, by Proposition 6.3.1.6, and by the den⊗ F (resp. MGaux (Zˆ p ) ⊗ Fp ) in Mtor ⊗ Fp (resp. Mtor sity of MH0,p p ˆ p ) ⊗ Fp ), the aux H0,p G (Z Z

aux

Z

aux

Z

Z

morphism (6.3.1.10) is induced by the corresponding morphism ⊗ kaux (p−1)

[g0 ]∗ : Γ(MGaux (Zˆ p ) ⊗ Fp , ωM Z

Gaux (ˆ Zp )

⊗ kaux (p−1)

→ Γ(MH0,p ⊗ Fp , ωM aux Z

0,p Haux

⊗ Fp ) Z

⊗ Fp ) Z

(for some integer kaux ≥ 1 related to k), and it suffices to show that this morphism sends (the restriction of) Hasseaux to its pullback under the canonical morphism

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MH0,p ⊗ Fp → MGaux (Zˆ p ) ⊗ Fp . Since the morphism [g0 ] aux Z

page 365

is defined by a Hecke

Z

twist of the universal object on MH0,p (see [62, Sec. 6.4.3]) using its level structure aux away from p (see Proposition 3.3.2.1 for how it is realized as an Z× (p) -isogeny), this follows from the fact that the formation of Hasse invariants is compatible with separable isogenies (see Remark 6.3.1.2). 6.3.2

Nonordinary and Full Ordinary Loci

Definition 6.3.2.1. Let Hasseaux be defined as in (6.3.1.1). (1) The locus where Hasseaux vanishes on Mtor G

ˆp aux (Z )

⊗ Fp defines a cloZ

sed subalgebraic stack with reduced structures, which we denote by non-ord (Mtor . ˆ p ) ⊗ Fp ) G (Z aux

Z

(2) The set-theoretic image of (Mtor G

ˆp aux (Z )

cal morphism Mtor G

ˆ p ⊗ Fp aux (Z ) Z

⊗ Fp )non-ord under the proper canoni-

→ Mmin G

Z ˆp )

aux (Z

⊗ Fp defines a closed subscheme Z

with reduced structures, which we denote by (Mmin G

ˆp aux (Z )

(3) The set-theoretic preimage of (Mmin G

aux

⊗ Fp )non-ord . Z

non-ord ~ min ⊗ Fp unin M H ˆ p ) ⊗ Fp ) (Z Z

Z

der the canonical morphism in Proposition 2.2.1.2 defines a closed sub~ min ⊗ Fp )non-ord . scheme with reduced structures, which we denote by (M H Z

~ H ⊗ Fp )non-ord , ([M ~ H ⊗ Fp ])non-ord , and We define in the same way (M Z

Z

non-ord ~ tor (M using other canonical morphisms in Propositions H,d0 pol ⊗ Fp ) Z

2.2.1.1, 2.2.1.2, and 2.2.2.1. We use similar notation for their base changes to other rings. (4) Consider the notation of formal completions in Fraktur as in Definition ~ min,non-ord , 3.4.4.2, we shall also denote the schemes defined in (3) by M H ~ non-ord , [M ~ H ]non-ord , and M ~ tor,non-ord , which we now view as closed subM H H,d0 pol schemes with reduced structures in their obvious ambient formal schemes or formal algebraic stacks, respectively. We call these closed subschemes the nonordinary loci of the various schemes. We call the open complements of these closed subschemes the full ordinary loci of the various schemes of formal schemes, and denote them with the superscripts “full-ord” (replacing “non-ord”). Proposition 6.3.2.2. The open immersion (6.1.1.16) induces an open immersion ~ min , ~ ord,min ,→ M M H,rH H whose image factors through an open immersion ~ ord,min ,→ M ~ min,full-ord M H H,rH that is also closed.

(6.3.2.3)

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Suppose that Hp is neat, that Σord extends to a (projective smooth) cone decomposition Σ for MH , with a collection pol of polarization functions (which is possible ~ tor by Proposition 5.1.3.4), so that M H,d0 pol is defined, for some integer d0 ≥ 1, as in Proposition 2.2.2.1. Then the commutative diagram (6.2.3.3) induces a commutative diagram  ~ ord,tor ~ tor,full-ord   ~ tor /M /M M H,d0 pol,rH H,d0 pol,rH H,Σord  ord,min   ~ MH

 ~ min,full-ord   /M H,rH

 ~ min /M H,rH

of formal schemes, in which the vertical arrows are all proper and surjective, in which all horizontal arrows are open immersions, and in which the two horizontal arrows at the left-hand side are also closed immersions. Proof. There are two kinds of statements to be proved. The first kind is to show ~ ord,min ,→ M ~ min does not meet that the open image of the canonical morphism M H,rH H ~ min,non-ord , and that in the setup of the second paragraph the open image of M H,rH ~ ord,tor ~ tor ~ tor,non-ord . (The the canonical morphism M does not meet M ord ,→ M H,Σ

H,d0 pol,rH

H,d0 pol,rH

latter statement implies the former statement.) The second kind is to show that the ~ min,full-ord is closed, and that in the setup of ~ ord,min ,→ M induced open immersion M H,rH H ~ ord,tor ~ tor,full-ord is also the second paragraph the induced open immersion M ord ,→ M H,d0 pol,rH

H,Σ

closed. (Again, the latter statement implies the former statement.) For statements of both kinds, we are allowed to replace H = Hp Hp with some H0 = H0,p Hp0 , where H0,p is a neat open compact subgroup of Hp , and where bal r bal r Hp0 = Up,1 (p ) if Up,1 (p ) ⊂ Hp ⊂ Up,0 (pr ) for some integer r ≥ 0 (and replace ord Σ and Σ with suitable cone decompositions), because then the morphisms from the new setup to the current setup will consist of proper morphisms compatible ~ ord,tor with each other (cf. Propositions 5.2.2.2 and 6.2.2.1 for the case of M and H,Σord ord,min tor min ~ ~ ~ M —the case of M and M are obvious because they are proper H

H,d0 pol,rH

H,rH

by themselves), such that the nonordinary loci in the new setup is the precise pullback from the current setup. Then we may assume that we are in the setup of the second paragraph, so that both Lemma 6.3.1.4 and Corollary 6.2.3.2 are applicable. Hence, the statements of the first kind follow immediately from Lemma 6.3.1.4, ~ ord,tor because the pullback of every positive power of Hasseaux is nowhere zero on M . H,Σord As for statements of the second kind, since the vertical arrows are proper, it ~ ord,tor ~ tor,full-ord ⊗ Fp suffices to show that the canonical open immersion M ⊗ Fp ,→ M H,d0 pol,rH H,Σord Z

Z

is closed. Note that this statement can be verified by replacing d0 with a sufficiently large multiple d0,aux . By Proposition 6.1.1.6, it suffices to show that the canonical ~ ord,tor ord ⊗ Fp ,→ M ~ tor,full-ord open immersion M Haux ,d0,aux polaux ,r ⊗ Fp is closed (for sufficiently Haux ,Σ aux

Z

Z

~ tor,full-ord large d0,aux ≥ 1 such that M Haux ,d0,aux polaux ,r can be compatibly defined). Then we

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may assume that there is a semi-abelian scheme Aaux with additional structures λaux ~ tor,full-ord ⊗ Fp . After pulled back to any strict local base scheme S, and iaux over M H,d0 pol,rH Z

r the quasi-finite flat group scheme Gaux,S [pr ] (resp. G∨ aux,S [p ]) admits a canonical r mult ∨ r mult subgroup scheme Gaux,S [p ] (resp. Gaux,S [p ] ) of multiplicative type, which ), such that (resp. (Gr0D# ,pr )mult is finite flat and of the same rank as (Gr0Daux ,pr )mult S S aux r r mult ∨ r ∨ r mult the quotient Gaux,S [p ]/Gaux,S [p ] (resp. Gaux,S [p ]/Gaux,S [p ] ) is a quasifinite ´etale group scheme. (This is possible because S is in characteristic p > 0, and because the abelian part of every fiber of Gaux is ordinary. Then we can construct such a subgroup scheme by putting together the torus part and the multiplicative part of the abelian part of the torsion points; see [62, Sec. 3.4.1] for a review of the definition of the torus part and the abelian parts.) Thus, the existence of principal ordinary level-pr structures is an open and closed condition, as desired.

Proposition 6.3.2.4. For any given integers i ≥ 0 and j ≥ 1, the scheme ~ min ⊗ (Z/pj Z))full-ord is affine. ~ min,full-ord ⊗ (Z/pj Z) ∼ M = (M H,i H,i Z

Z

Proof. By Corollary 6.3.1.7 and Definition 6.3.2.1, with N1 as in Proposition 2.2.1.2 p ~ min ⊗ Fp , ω ⊗ aN1 (p−1) ⊗ Fp ) (for some choice of Haux ), the section HasseaN1 ∈ Γ(M H ~ min Z

MH

Z

~ min ⊗ Fp (as its vanishing locus), and ~ min ⊗ Fp )non-ord of M defines the subscheme (M H H Z

Z

~ min ⊗ Fp . ~ min ⊗ Fp )non-ord of M ~ min ⊗ Fp defines the subscheme (M its pullback to M H,i H,i H,i Z

Z

Z

aN1 ~ min Since ω ⊗ ~ min is ample over MH (see Proposition 2.2.1.2), and since the canonical MH

~ min ⊗ Fp )full-ord ~ min is finite (for each i ≥ 0), this shows that (M ~ min → M morphism M H,i H H,i j−1

Z

j−1

is affine. Since x ≡ y (mod p) implies xp ≡ yp (mod pj ) (in any ring), min j ~ by patching over affine open subschemes of MH ⊗ (Z/p Z), we can uniquely lift Hassep

j−1

aN1

Z

gument as above shows that desired.

Z ⊗ pj−1 aN1 (p−1) ⊗ (Z/pj Z)). Then the MH Z ~ min ⊗ (Z/pj Z))full-ord is affine (for each i (M H,i Z

~ min ⊗ (Z/pj Z), ω to Γ(M H ~ min

same ar≥ 0), as

Corollary 6.3.2.5. Suppose that Hp is neat (so that H = Hp Hp satisfies the condi~ ord,min ⊗ (Z/pj Z) ∼ tion in Corollary 6.3.1.7). Then, for each integer j ≥ 1, both M = H Z

~ ord,min ⊗ (Z/pj Z) and M ~ min ⊗ (Z/pj Z))full-ord are af~ min,full-ord ⊗ (Z/pj Z) ∼ M = (M H,rH H H,rH fine.

Z

Z

Z

Proof. This follows from Propositions 6.3.2.2 and 6.3.2.4. For the sake of completeness, let us include a condition which ensures that the open and closed immersion (6.3.2.3) is actually an isomorphism when Hp = G(Zp ). Condition 6.3.2.6. The group G(Zp ) acts transitively on the set of maximal totally isotropic O ⊗ Zp -modules D0 of L ⊗ Zp satisfying the same conditions as D does as Z

Z

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in Lemma 3.2.2.1 and Assumption 3.2.2.10. Lemma 6.3.2.7. Suppose that Condition 6.3.2.6 holds, and that H = Hp Hp with Hp = G(Zp ). Then the canonical morphism ~ ord ⊗ Fp → (M ~ H,r ⊗ Fp )full-ord M H H Z

(6.3.2.8)

Z

(induced by (3.4.6.4)) is a bijection on geometric points, and hence (by Zariski density of the image) the open and closed immersion (6.3.2.3) is an isomorphism. When Hp is neat, by Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11]), the canonical ~ ord → M ~ H,r −(M ~ H,r ⊗ Fp )non-ord between noetherian normal schemes morphism M H H H Z

over Spec(Z(p) ), lifting the morphism (6.3.2.8) over Spec(Fp ), is also an isomorphism. ~ H,r ⊗ Fp )full-ord , by the conProof. Given a geometric point s = Spec(k) → (M H Z

~ H,r as a normalization, we may assume that there exists an abelian struction of M H scheme A over S = Spec(R), where R is a complete discrete valuation ring with fraction field K of characteristic zero and algebraically closed residue field k of ~ H,r characteristic p > 0, and assume that there exist a morphism ξ : S → M H non-ord ~ (see Proposition 2.2.1.1) lifting Spec(k) → (MH,rH ⊗ Fp ) , as in Section 3.2.1. Z

Then, by extending the pullback of the tautological object over MH to the noetherian normal S (by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5], and by extending isomorphisms between finite ´etale group schemes), A also carries a polarization λ and an O-endomorphism structure i, and a level Hp -structure αHp (see [62, Def. 1.3.7.6] and Definition 3.3.1.4). Moreover, since Hp = G(Zp ), by ord,0 Proposition 3.2.1.1, (A, λ, i) also carries an ordinary level-Hp structure αH of p 0 (A, λ, i) of type (L ⊗ Zp , h · , · i, D ) (as in Definition 3.3.3.4), for some filtration D0 of Z

L ⊗ Zp satisfying the same conditions as D does as in Lemma 3.2.2.1 and Assumption Z

3.2.2.10. In general, D0 can be different from D. Since Hp = G(Zp ) acts transitively on the set of such D0 , there exists gp ∈ G(Zp ) which defines an isomorphism matching (L ⊗ Zp , h · , · i, D, D# , φD ) with Z

(L ⊗ Zp , h · , · i, D0 , D0,# , φ0D ), and hence (without modifying (A, λ, i, αHp )) the tuple Z

ord,0 ord,0 (A, λ, i, αHp , αH ) → S (with αH defined by D0 ) can be canonically identified p p ord ~ ord (with αord defined by with a tuple (A, λ, i, αHp , αH ) → S parameterized by M H Hp p ~H D), with (αHp , αord ) ⊗ Q induced by αH as in Proposition 3.3.5.1. Thus, ξ : S → M Hp

Z

~ ord → M ~ H,r (see (3.4.6.4)). Since the geometric point must factor through S → M H H non-ord ~ s = Spec(k) → (MH,rH ⊗ Fp ) is arbitrary, the lemma follows, as desired. Z

Lemma 6.3.2.9. Condition 6.3.2.6 is true when either p is a good prime as in Definition 1.1.1.6, or when the Iwasawa decomposition G(Qp ) = Pord D (Qp )G(Zp ) holds. By [12, Prop. 4.4.3] (see also [14, (18) on p. 392] for a more explicit statement),

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the latter is true when, for example, G ⊗ Qp is connected (which is the case when Z

O ⊗ Q involves no simple factor of type D as in [62, Def. 1.2.1.15]; cf. [53, Sec. 7, Z

p. 393]) and when G(Zp ) is maximal open compact in G(Qp ). Proof. When p is a good prime, this follows from the Gram–Schmidt process as in [62, Prop. 1.2.4.5] (with R = Zp and k = Fp ). Otherwise, by the same [62, Prop. 1.2.4.5] (but with R = k = Qp there), we know that there exists some element gp ∈ G(Qp ) such that D0,0 ⊗ Qp = gp (D0 ⊗ Qp ). By the Iwasawa decomposition Zp

Zp

ord G(Qp ) = Pord D (Qp )G(Zp ), since PD (Qp ) stabilizes D ⊗ Qp , we may assume that Zp

gp ∈ G(Zp ), as desired. (We note that the underlying O ⊗ Zp -modules GrD and Z

GrD0 are isomorphic for more basic reasons: As in the proof of Lemma 3.2.2.6, we may assume that they are O0 ⊗ Zp -modules, for some maximal O0 as in Condition Z

1.2.1.1. Then it follows from [93, Thm. 18.10] that they are isomorphic because their Qp -spans are.) Remark 6.3.2.10. Although Lemma 6.3.2.9 can be improved, we omit the further discussions for the sake of simplicity. 6.3.3

Nonemptiness of Ordinary Loci

~ ord ⊗ Fp is even noSo far we have not touched upon the question of whether M H Z

nempty. When p is good (as in Definition 1.1.1.6), one can show that it follows ~ ord ⊗ Fp is nonempty if and from [102] (and the surjectivity of (3.4.5.6)) that M H Z

only if Assumption 3.2.2.10 holds. (It suffices to verify this when H = Hp G(Zp ). By [102], the full ordinary locus of MHp ⊗ Fp is open and dense when the first half of Z

Assumption 3.2.2.10 holds. Such a full ordinary locus contains a nonempty ordinary locus when the second half of Assumption 3.2.2.10 also holds—note that, by Lemmas 6.3.2.7 and 6.3.2.9, this is automatic except when O ⊗ Q involves some factor of Z

type D, or when G(Zp ) fails to be a maximal open compact in G(Qp )—by [12, Cor. 3.3.2], up to modifying the choice of the integral PEL datum, the latter can always be avoided.) This simple criterion, however, does not necessarily apply when p is not good. We shall record in this subsection some simple-minded (but rather restrictive) implication of the construction of partial toroidal compactifications. ~ ord,tor Proposition 6.3.3.1. In the construction of M in Theorem 5.2.1.1, if H,Σord ord,Z H ~ M ⊗ Fp is nonempty for some ZH (forming part of the representative of some H Z

~ ord ⊗ Fp and M ~ ord,tor cusp label [(ZH , ΦH , δH )]), then M ⊗ Fp are nonempty. H H,Σord Z

Z

~ ord,tor )∧ Proof. By (5) of Theorem 5.2.1.1, the formal completion (M H ~ Zord

of

[(ΦH ,δH ,σ)]

~ ord,tor M H

along the [(ΦH , δH , σ)]-stratum ~Zord [(ΦH ,δH ,σ)] is canonically isomorphic to the

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~ ord ~ ord ~ ord formal algebraic stack X ΦH ,δH ,σ /ΓΦH ,σ . Since ΞΦH ,δH and ΞΦH ,δH ,σ are both torus ~ ord torsors over the abelian scheme torsor C etale cover ΦH ,δH over the nontrivial finite ´ ord,Φ ord,Z H H ~ ~ ~ ord (σ) is an affine toroidal embedding ~ ord M of M , and since Ξ ,→ Ξ ΦH ,δH ΦH ,δH H H ~ ord,ZH ⊗ Fp is nonempty, then Ξ ~ ord ~ ord , if M over C ΦH ,δH ,σ ⊗ Fp is nonempty, and ΦH ,δH H Z

Z

~ ord ~ ord ~ ord Ξ ΦH ,δH ⊗ Fp is nonempty and open dense in ΞΦH ,δH (σ) ⊗ Fp . Since XΦH ,δH ,σ is the Z

Z

~ ord (σ) (which contains Ξ ~ ord formal completion of Ξ ΦH ,δH ΦH ,δH as an open subalgebraic ord ord ~ ~ stack) along Ξ , this shows that X ⊗ Fp is nonempty, which implies ΦH ,δH ,σ ΦH ,δH ,σ Z

~ ord,tor that M ⊗ Fp is nonempty. Moreover, the nonemptiness and open density of H,Σord Z

~ ord ~ ord ~ ord Ξ ΦH ,δH ⊗ Fp in ΞΦH ,δH (σ) ⊗ Fp implies that MH ⊗ Fp is also nonempty. Z

Z

Z

ˆ Corollary 6.3.3.2. If there exists a fully symplectic admissible filtration Z on L ⊗ Z Z

with respect to (L, h · , · i) as in Definition 1.2.1.3 that is compatible with D as in De~ ord ⊗ Fp and M ~ ord,tor ⊗ Fp finition 3.2.3.1 such that GrZ−1 = Z−1 /Z−2 = {0}, then M H H,Σord Z

are nonempty.

Z

Proof. This follows from Proposition 6.3.3.1 because, in this case, we can extend the H-orbit ZH of Z to a representative (ZH , ΦH , δH ) of an ordinary cusp label (as in ~ ord,ZH ⊗ Fp is trivially nonempty. Definition 3.2.3.8), and the zero-dimensional M H Z

Remark 6.3.3.3. In practice, in the setup of Corollary 6.3.3.2, we may choose D after finding a Z such that GrZ−1 is trivial. Although this seems very restrictive, it is applicable whenever G admits a rational parabolic subgroup with abelian unipotent radical. This is thus applicable, for example, to the construction of Galois representations for cohomological automorphic representations of general linear groups over CM or totally real fields (without any polarizability condition) in [39].

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Chapter 7

Ordinary Kuga Families

In this chapter, we continue to assume the same settings as in Section 5.2. Our main goal is to generalize the results in Section 1.3.3 to the context of ordinary loci. We will first introduce some analogues of the definitions and results in Section 1.3.3, and explain how the proofs in [61] can be translated into this context. 7.1

Partial Toroidal Compactifications

7.1.1

Parameters for Ordinary Kuga Families

e etc be chosen as in Section 1.2.4. (To e h · , · ie, e e δ), Let Q, O0 , Q−2 , Q0 , (L, h0 ), (e Z, Φ, make sure Theorems 1.3.3.15 and 7.1.4.1 below are compatible, we need to make e ⊗ Zp by identical choices.) The filtration D of L ⊗ Zp defines a filtration e D of L Z

setting

Z

e ⊗ Zp . e D1 = 0 ⊂ e D0 = (e Z−2 ⊗ Zp ) ⊕ D0 ⊂ e D−1 = L ˆ Z

(7.1.1.1)

Z

e is compatible with e e δ) Then (e Z, Φ, D (as in Definition 3.2.3.1). ˆ of standard form with respect e ⊂ G( e Z) Consider any open compact subgroup H bal r p e ep ⊂ Uep,0 (pr ) e e e (p ) ⊂ H to D as in Definition 3.2.2.9, so that H = H Hp and Uep,1 × rνe e and νe(H ep ) = ker(Z× for r = deptheD (H); e ≤ r). (Here p → (Z/p Z) ) (where rν bal r r e p ) as in Definition 3.2.2.8.) Let r e be Uep,1 (p ) and Uep,0 (p ) are subgroups of G(Z H as in Definition 3.4.2.1. b=H e b (see Definition 1.2.4.4) is of standard form of depth r in the Note that H G following sense: Definition 7.1.1.2. (Compare with Definition 3.2.2.9.) For any integer r ≥ 0, let bal r bal r r r b e us write Ubp,1 (p ) := Uep,1 (p )G b and Up,0 (p ) := Up,0 (p )G b (cf. Definition 1.2.4.4). b b We say that an open compact subgroup Hp ⊂ G(Zp ) is of standard form with respect to D if there exists an integer r ≥ 0 such that bal r bp ⊂ Ubp,0 (pr ). Ubp,1 (p ) ⊂ H bp , and write r = depthD (H bp ). (The In this case, we say that r is the depth of H e notation makes sense because D is uniquely determined by D.) 371

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ˆ is of standard form with b ⊂ G( b Z) We say that an open compact subgroup H p ˆ p ) and H b=H b H bp , where H b p ⊂ G( b Z bp ⊂ G(Z b p ), respect to D if it is of the form H bp is of standard form with respect to D. In this case, we set depthD (H) b := such that H b depthD (Hp ). bp and H bp0 of G(Z b p ) (resp. H b and H b0 We say that two open compact subgroups H ˆ of standard form with respect to D are equally deep if depth (H b Z)) bp ) = of G( D 0 0 b b b depthD (Hp ) (resp. depthD (H) = depthD (H )). We shall suppress the term “with respect to D” when the choice of D is clear from the context. bp (resp. H) b is of standard form, then so is Remark 7.1.1.3. By definition, if H b b bp ) = depthD ((H bp )G ) (resp. (Hp )G (resp. HG ; see Definition 1.2.4.4), and depthD (H b = depthD (H bG )). depthD (H) ˆ be of standard form (with respect to D) as in Definition 3.2.2.9, Let H ⊂ G(Z) and let rH be as in Definition 3.4.2.1 (so that Theorem 5.2.1.1 and its consequences e H e p , and r e be as in the previous paragraph, such that H e p is neat hold). Let H, H e b e b (as and such that H satisfies Condition 1.2.4.7 (which involves H). Then H = H G we have seen above) satisfies the following: b is of standard form with respect to D as in Definition 7.1.1.2, Condition 7.1.1.4. H p b b H is neat, and HG (see Remark 7.1.1.3) is also of standard form with respect to D as in Definition 3.2.2.9, so that rHb G is defined as in Definition 3.4.2.1. b alone, without referring to H.) e We shall (This is a condition when we consider H e b assume that H (or H) satisfies moreover the following: Condition 7.1.1.5. rHe = rHb G (≥ rH ). Remark 7.1.1.6. (Compare with Remark 1.2.4.10.) For each H as above, there e satisfying these conditions, which we may also require to satisfy Conditions exists H 1.2.4.8 and 1.2.4.9, because the pairing h · , · ie is the direct sum of the pairings on Q−2 ⊕ Q0 and on L. e is compatible with e e δ) Since (e Z, Φ, D (as in Definition 3.2.3.1), it induces a eϕ e e = (X, e Ye , φ, representative (e ZHe , Φ e e0,He ), δeHe ) of an ordinary cusp label e, ϕ H −2,H e e , δe e )] at level H e (as in Definition 3.2.3.8). Let Σ e ord be any compatible [(e Z e, Φ H

H

H

~e ord choice of admissible smooth rational polyhedral cone decomposition data for M e H that is projective (see Definitions 5.1.3.1 and 5.1.3.3). Let σ e ⊂ P+ be any tope Φ f H

ee dimensional nondegenerate rational polyhedral cone in the cone decomposition Σ ΦH f e ord . in Σ Definition 7.1.1.7. (Compare with Definition 1.2.4.11.)

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e ord,++ is the set of all triples κ e Σ e ord , σ e (1) K e = (H, e) as above (such that H Q,H satisfies Conditions 1.2.4.7 and 7.1.1.5). e ord,+ is the subset of K e ord,++ consisting of elements κ e Σ e ord , σ (2) K e = (H, e) such Q,H Q,H e satisfies Condition 1.2.4.8. that H e ord is the subset of K e ord,+ consisting of elements κ e Σ e ord , σ (3) K e = (H, e) such Q,H

Q,H

e also satisfies Condition 1.2.4.9. that H ˘ e , δ˘ e , τ˘)] having [(Φ e e , δe e , σ The equivalence classes [(Φ H H H H e )] as a face has been described in Section 1.2.4 (following Definition 1.2.4.11), and the ordinary cusp e e , δe e )] as a face have ˘ e , δ˘ e )] (see Definition 3.2.3.8) having [(e ˘ e, Φ ZHe , Φ labels [(Z H H H H H ˘ such that Z ˘ e , δ˘ e ) which are H-orbits e ˘ δ) ˘ e, Φ ˘, Φ, ˘ (as representatives (Z of triples (Z H H H in (1.2.4.12)) satisfies the additional property that it is compatible with e D (as in Definition 3.2.3.1). In this case, the filtration Z induced by e Z as in Section 1.2.4 (following Definition 1.2.4.11) is compatible with D (cf. (1.2.4.13)). Hence, we have: e Lemma 7.1.1.8. Under the canonical surjective assignment (given the splitting δ; ˘ e , δ˘ e )] at le˘ e, Φ see (3) following Definition 1.2.4.11) from the set of cusp labels [(Z H H H e e e e vel H admitting a surjection to [(ZHe , ΦHe , δHe )], to the set of cusp labels [(ZH , ΦH , δH )] at level H, the ordinary ones are mapped to ordinary ones, and the preimage of the set of ordinary ones is the set of ordinary ones. The induced assignment from ordinary ones to ordinary ones is bijective if we assume Condition 1.2.4.8, and is e e e ) ⊂ H. still surjective if we only assume GrZ−1 (H Pe Z

Then we have the diagram (1.2.4.14) and the morphisms (1.2.4.18), (1.2.4.19), and (1.2.4.20); we define σ ˘ to be the image of σ e ⊂ P+ under the first more Φ f H

e˘ phism in (1.2.4.20); we consider as in Definition 1.2.4.21 the subsets Σ ΦH σ and f,˘ + e e ΣΦ˘ ,˘σ of ΣΦ˘ f and the subgroups ΓΦ˘ f,ΦH , ΓΦ˘ f,ΦH ,˘σ , ΓΦ˘ c , and ΓΦ˘ c,ΦH of ΓΦ˘ f ; H

f H

H

H

H

b ˘ , (S b ˘ )∨ , pr b and, most importantly, we define S Φc Φc R (S ˘ H

b ˘ , and P b+ P ˘ Φc Φ H

H

∨ Φ c )R H

H

H

b ˘ )∨ , : (SΦ˘ f )∨ R → (SΦ c R H

H

as in Definition 1.2.4.29, (1.2.4.32), (1.2.4.33), and (1.2.4.34),

c H

b˘ = and define the ΓΦ˘ c -admissible rational polyhedral cone decomposition Σ ΦH c H b {pr b τ )} e of P ˘ as in Corollary 1.2.4.40. (All of these are verbatim ∨ (˘ (SΦ )R ˘ c H

τ˘∈ΣΦ ˘

σ f,˘ H

ΦH c

as in Section 1.2.4.) Then we have: b ord = Lemma 7.1.1.9. (Compare with Lemma 1.2.4.42.) The collection Σ b ˘ } ˘ ˘ , where [(Φ ˘ b , δ˘ b )] runs through equivalence classes of H-orbits b {Σ of ΦH H H c [(ΦH c,δH c)] ˘ compatible with (Φ, e as in Definition 1.2.4.17 (with Z ˘ δ) e δ) ˘ and representatives (Φ, e ˘ e Z suppressed in the notation), such that Z is compatible with D (as in Definition ˘ e , δ˘ e )] is ordinary), defines a compatible ˘ e, Φ 3.2.3.1) (so that the cusp label [(Z H H H choice of admissible smooth rational polyhedral cone decomposition data analogous to the notion for MH in Definition 1.2.2.13. There is an obvious notion of refinements for such collections, analogous to that in [62, Def. 6.4.2.8].

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b ord extends to some Σ b as in Lemma 1.2.4.42. Conversely, Moreover, each such Σ b as in Lemma 1.2.4.42 induces (by restriction to ordinary cusp labels) a valid each Σ b ord for this lemma. Σ e ord = {Σ e ˘ } ˘ ˘ (with Proof. This follows from the corresponding facts for Σ ΦH f [(ΦH f)] f,δH indices running through all ordinary cusp labels). (The statements concerning extensibility and restrictions follow from the corresponding ones in Propositions 5.1.3.2 and 5.1.3.4.) Remark 7.1.1.10. (Compare with Remark 1.2.4.43.) Here we omit the precise definition of a compatible choice of admissible smooth rational polyhedral cone decomposition data because we can only construct partial toroidal compactifications b ord defined by some Σ e ord and σ of Kuga families for those Σ e. Definition 7.1.1.11. (Compare with Definition 1.2.4.44.) We say that two κ e1 and ord,++ e κ e2 in KQ,H (see Definition 7.1.1.7) are equivalent if they determine the same b Σ b ord ). In this case, we shall abusively write κ = [e κ = (H, κ1 ] = [e κ2 ]. Then we ord,++ ord b b to be the set of all such κ = (H, Σ ), with a partial order κ0 = take K Q,H

b0 , Σ b ord,0 )  κ = (H, b Σ b ord ) when H b0 ⊂ H b 0 and when Σ b ord,0 is a refinement of Σ b ord (H ord,+ ord,++ (see Definition 7.1.1.9). We also take the subset KQ,H (resp. Kord Q,H ) of KQ,H e ord,++ under the canonical e ord,+ (resp. K e ord ) of K to be the image of the subset K Q,H

Q,H

Q,H

e ord,++  Kord,++ , with an induced partial order denoted by the same surjection K Q,H Q,H symbol . bG ⊂ Lemma 7.1.1.12. (Compare with Lemma 1.2.4.45.) In Lemma 1.2.4.45, if H p e ˆ e e G(Z) satisfies Condition 7.1.1.4, then we may assume that H = H Hp is of standard e p , and satisfies both Conditions 1.2.4.7 and 7.1.1.5. form with neat H Proof. This follows from the proof of Lemma 1.2.4.45, because the restriction of ν e → Gm to P e 0 factors as P e0 → G → νe : G Gm . e e Z Z Lemma 7.1.1.13. (Compare with Lemma 1.2.4.46.) For each neat open compact ˆ satisfying Condition 7.1.1.4, there exists some element κ = b of G(Z) subgroup H ord,++ ord ord b b b (H, Σ ) ∈ KQ,H , which lies in Kord,+ Q,H (resp. KQ,H ) if H also satisfies Condition 1.2.4.8 (resp. both Conditions 1.2.4.8 and 1.2.4.9). b is induced by some H e=H ep H ep with Proof. By Lemmas 1.2.4.45 and 7.1.1.12, H p e neat H as in Definition 1.2.4.4, which we assume to also satisfy Conditions 1.2.4.7 ~e ord e ord for M and 7.1.1.5. By Propositions 5.1.3.2 and 5.1.3.4, there exists some Σ e . H b ord to be induced by Σ e ord as in Lemma 7.1.1.9, and take κ = (H, b Σ b ord ). Let us take Σ Then, by definition, we have κ ∈ Kord,++ . The remaining statements of the lemma Q,H also follow by definition. Lemma 7.1.1.14. (Compare with Lemma 1.2.4.47.) The partial order  among ord elements in Kord,++ (resp. Kord,+ Q,H Q,H , resp. KQ,H ) is directed; that is, if we are given

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b Σ b ord ) and κ0 = (H b0 , Σ b ord,0 ), then there exists some κ00 = (H b 00 , Σ b ord,00 ) two κ = (H, 00 00 0 00 b to be any open compact such that κ  κ and κ  κ . Moreover, we can take H b∩H b 0 (which can be H b∩H b 0 itself) satisfying Condition 7.1.1.4 (with subgroup of H b there replaced with H b 00 here). H Proof. The same argument of the proof of Lemma 1.2.4.47 works here. Now we consider some compatibility conditions between a collection Σord for ~ ord and elements of K e ord,++ or Kord,++ . M H Q,H Q,H e ord,++ : e Σ e ord , σ First consider the following condition on an element κ e = (H, e) in K Q,H

Condition 7.1.1.15. (Compare with [61, Cond. 3.8] and Condition 1.2.4.48.) For ˘ e , δ˘ e , τ˘) (with Z ˘ e , δ˘ e ) is a ˘ e suppressed in the notation) such that (Z ˘ e, Φ each (Φ H H H H H H ˘  X, e s˘ : representative of an ordinary cusp label which admits a surjection (sX˘ : X Y + e ˘ e e Y  Y ) to (e ZHe , ΦHe , δHe ), and such that τ˘ ⊂ PΦ˘ is a cone in the cone decomposition f H

e ˘ (in Σ e ord ) having a face σ Σ ˘ that is a ΓΦ˘ f -translation (see Definition 1.2.2.3) of Φf H

H

the image of σ e ⊂ P+ under the first morphism in (1.2.4.20) (induced by (sX˘ , sY˘ )), e Φ f H

the image of τ˘ in PΦH under the (canonical) second morphism in (1.2.4.20) is ord contained in some cone τ ⊂ P+ ). ΦH in the cone decomposition ΣΦH (in Σ Remark 7.1.1.16. The only difference between Conditions 1.2.4.48 and 7.1.1.15 is that we only consider ordinary cusp labels in the latter. By Lemma 1.2.4.35 (and Remark 7.1.1.16), if κ = [e κ] ∈ Kord,++ is the eleQ,H ment determined by κ e, then Condition 7.1.1.15 for κ e is equivalent to the following condition for κ: Condition 7.1.1.17. (Compare with [28, Ch. VI, Def. 1.3] and Condition b ˘ (where τb = pr b ˘ e , δ˘ e , τ˘) is in 1.2.4.49.) For each τb ∈ Σ τ ) for some (Φ Φc H H (S ˘ )∨ (˘ H

Φc R H

b ˘ in Σ b ord ), the image of τb in P+ under (1.2.4.37) is the cone decomposition Σ ΦH Φc H

ord contained in some cone τ ⊂ P+ ). ΦH in the cone decomposition ΣΦH (in Σ

Remark 7.1.1.18. The only difference between Conditions 1.2.4.49 and 7.1.1.17 is that we only consider ordinary cusp labels in the latter. Definition 7.1.1.19. (Compare with Definition 1.2.4.50.) For ? = ++, +, or ∅, to be the subset of Kord,? let us take Kord,? Q,H consisting of elements κ satisfying Q,H,Σord Condition 7.1.1.17. b ord with a refineSince Condition 7.1.1.17 can be achieved by replacing any given Σ ord,? ment (in the same set), we see that each KQ,H,Σ is nonempty and has an induced directed partial order. e Q,H ⊂ K e+ ⊂ K e ++ , KQ,H ⊂ Thus, we have defined analogues of all the sets K Q,H Q,H ++ + ++ K+ Q,H ⊂ KQ,H , and KQ,H,Σ ⊂ KQ,H,Σ ⊂ KQ,H,Σ in Definitions 1.2.4.11, 1.2.4.44,

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and 1.2.4.50. We would like to analyze the relation between the sets we defined there and here. Definition 7.1.1.20. (Compare with Definitions 1.2.4.11, 1.2.4.44, and 1.2.4.50.) For each ?1 = ++, +, or ∅, and for each ?2 = Σ or ∅: e std,?1 is the (nonempty) set of all κ e ?1 such that H e Σ, e σ e is (1) K e = (H, e) ∈ K Q,H Q,H of standard from with respect to e D as in Definition 3.2.2.9. std,?1 b Σ) b ∈ K?1 b (2) KQ,H,? is the (nonempty) set of all κ = (H, Q,H,?2 such that H is 2 of standard from with respect to D as in Definition 7.1.1.2. Proposition 7.1.1.21. (Compare with Proposition 1.2.4.52.) Suppose H is of standard form with respect to D as above. For each ?1 = ++, +, or ∅, and for each e ord,?1 and Kord,?1 are nonempty, and the natural canonical ?2 = Σ or ∅, the sets K Q,H Q,H,?2 morphisms e ord,?1 e std,?1 → K ord : K Q,H

Q,H

and ord,?1 1 ord : Kstd,? Q,H,?2 → KQ,H,?2

are surjective and compatible with each other under the various canonical maps. 1 Common refinements for finite subsets exist in any sets of the form Kstd,? Q,H,?2 or 1 Kord,? Q,H,?2 . When doing so, we may allow varying levels or twists by Hecke actions, and we may vary ?1 and ?2 as well (in any order). For any such refinement κ = b Σ) b or (H, b Σ b ord ), we may prescribe H b to be any allowed open compact subgroup of (H, ˆ b b b 0, G(Z) in the context, we may require Σ to be finer than any cone decomposition Σ ord b b and we may require Σ or Σ to be invariant under any choice of an open compact b ∞ ) normalizing H. b If κ = (H, b Σ b ord ) ∈ Kord,?1 , if H b0 ⊂ H b is an subgroup of G(A

Q,H,?2

b ˘ )∨ defined by S b˘ open compact subgroup such that the integral structures on (S ΦH ΦH c0 R c0 b ˘ are identical for each ordinary cusp label [(Φ ˘ b 0 , δ˘ b 0 )] at level H b 0 inducing and S H

ΦH c

H

˘ b , δ˘ b )] at level H, b and if Σ b ord,0 is the collection induced by an ordinary cusp label [(Φ H H b ord at level H b 0 , then κ0 := (H b0 , Σ b ord,0 ) belongs to the same set Kord,?1 (without the Σ Q,H,?2 b ord,0 ). If κ = (H, b Σ b ord ) ∈ Kord,?1 and κ0 = (H b0 , Σ b 0 ) ∈ Kstd,?1 such need to refine Σ Q,H,?2

Q,H,?2

b 00 , Σ b 00 ) ∈ Kstd,?1 such that that κ  ord(κ0 ), then there exists an element κ00 = (H Q,H,?2 1 ord(κ00 ) = κ and κ00  κ0 , and we may assume that κ00  κ000 for any κ000 ∈ Kstd,? Q,H,?2 000 such that ord(κ ) = κ.

Proof. These follow from the corresponding existence, refinement, and extensibility f for e and pol statements in Propositions 1.2.2.17, 5.1.3.2, and 5.1.3.4 for collections Σ ord ord ~ e e and for collections Σ f e e . The second last statement is e ord and pol M for M H H obvious. As for the last statement, note that ordinary cusp labels (by their very definition, see Definition 3.2.3.8) can only admit surjections to ordinary cusp labels, and hence refinements of cone decompositions over non-ordinary cusp labels do not necessitate further refinements over the ordinary cusp labels.

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For later references, let us also define the following: Definition 7.1.1.22. For each Zp -algebra R, we also define the following quoee(R) and P e ord (R) (see Definitions 1.2.1.10, 1.2.1.11, 1.2.4.3, tients of subgroups of P Z e D 3.2.2.7, and 3.2.3.9): e ord (R) := P ee(R) ∩ P e ord (R). (1) P Z e e Z,e D D e ord,0 (R) := P e 0 (R) ∩ P e ord (R). (2) P e Z,e D

e Z

e D

b ord e ord (R) = P e ord,0 (R)/U e e(R). (3) P D (R) := P1,e 2,Z e Z,e D Z,e D ord b ord e e e (4) U (R) := U (R)/( U (R) ∩ U (R)), which can be canonically identified e e D Z Z e D e e ord (R) → with the image of Ue(R) under the canonical homomorphism P Z

e D

e ord (R). M e D b ord e ord,0 (R)/U e ord (R), which can be canonically identified with the (5) M D (R) := Pe e D Z,e D e ord,0 (R) under the canonical homomorphism P e ord (R) → M e ord (R), image of P e Z,e D

e D

e D

b ord )(R) := which is (under the splitting δe above) isomorphic to (Mord nU D D ord ord b MD (R) n UD (R). ep of G(Z e p ) of standard Definition 7.1.1.23. For each open compact subgroup H ep ), we define H b p = (H ep ) b as in form as in Definition 3.2.2.9, with r = deptheD (H G b e Definition 1.2.4.4, so that (by definition) r = depthD (Hp ) = deptheD (Hp ). Then b pr := H bp /Ubbal (pr ) H p,1 is a subgroup of bal r ∼ b ord Ubp,0 (pr )/Ubp,1 (p ) = MD (Z/pr Z),

and we set can. 
−1 can. b ord ∼ r r bpord := M b ord b bbal r b pr ) (7.1.1.24) H (H D (Zp ) → MD (Z/p Z) → Up,0 (p )/Up,1 (p ) (cf. (3.3.3.5)), b ord ord := H bpord ∩ U b ord H D (Zp ), b p,U D

and b ord ord := H bpord /H b ord ord . H b p,M p,U D

D

Then (by definition) b ord ord = ((H bp )G )ord H p,M D

(where the right-hand side is defined using Definition 1.2.4.4 and (3.3.3.5)) as subgroups of Mord D (Zp ), and we have an exact sequence b ord ord → H bpord → H b ord ord → 1 1→H b p,M p,U D

(7.1.1.25)

D

compatible with the canonical exact sequence ord b ord b ord 1→U D (Zp ) → MD (Zp ) → MD (Zp ) → 1.

(7.1.1.26)

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Definition 7.1.1.27. (Compare with Definitions 1.2.4.53, 1.2.4.54, and 3.2.3.9.) ˘ is compatible with e Suppose Z D as in (3.2.3.2). For each Zp -algebra R, we define the b ˘ (R) (see Definition 1.2.4.53): following quotients of subgroups of P Z b ord (R) := (P e ord (R) ∩ P e 0 (R))/U e e(R). (Because of the compatibility be(1) P ˘ 2,Z ˘ e Z,D Z,e D Z b ˘ (R), b ˘ (R), U b˘ (R), U ˘ and e tween Z D, we do not define new groups for Z Z 2,Z Z b ˘ (R) , G b ˘ (R), and G b 0 (R) here.) U 1,Z

l,Z

l,˘ Z

b ord (R) := P e ord (R)/Z e˘ (R) ∼ b ord (R)/Z b˘ (R) is the subgroup of ele(2) P = P Z Z ˘ ˘ h,˘ Z,D Z,e D Z,e D ∼ b ments of G ˘ (R) = Gh,Z (R) preserving the filtration D−1 induced by e D on h,Z

˘ GrZ−1

⊗ Zp ∼ = GrZ−1 ⊗ Zp as in Definition 3.2.3.1. ˆ Z

ˆ Z

b ord,0 (R) := P e ord,0 (R)/U e e(R) is the kernel of the canonical homomorphism (3) P 2,Z ˘ ˘ Z,D Z,e D ˘ ˘ b ord (R) → G b 0 (R). (e ν −1 GrZ−2 , GrZ0 ) : P ˘ Z,D

l,˘ Z

b ord,0 (R)/U b ˘ (R). b ord (R) := P e ord,0 (R) ∼ (4) P =P 2,Z ˘ ˘ 1,˘ Z,D Z,D Z,e D ord,0 ord ord b b b e e ˘ (R) ∼ P (R)/U (5) P (R) := P (R)/U = h,˘ Z,D 1,˘ Z,e D ord,0 ∼ b b P1,˘Z,D (R)/U˘Z (R) =

∼ e ord,0 (R)/U e ˘ (R) ∼ = =P Z ˘ Z,e D ord ord ord ord ∼ ∼ b b b e e P˘Z,eD (R)/Z˘Z,eD (R) = P˘Z,D (R)/Z˘Z (R) = Ph,˘Z,D (R). 1,Z

1,˘ Z,D

1,˘ Z (R)

b Then the canonical homomorphism G(R) → G(R) induces the following canonical homomorphisms: b ord (R) → Pord (R). (1) P Z,D ˘ Z,D ord b (2) P (R) → Pord (R). h,˘ Z,D

h,Z,D

b ord,0 (R) → Pord,0 (3) P Z,D (R). ˘ Z,D ord ord b (4) P1,˘Z,D (R) → P1,Z,D (R). b ord,0 (R) → Pord,0 (R). (5) P h,˘ Z,D

h,Z,D

We also consider (see Definition 1.2.4.54) e ord (R) := P e ˘ (R) ∩ P ee(R) ∩ P e ord (R). P ˘ e Z Z Z,e Z,e D D Then we have the canonical isomorphism ∼ e e ord (R)/P e ord,0 (R) → e ˘0 (R) = G e ˘ e(R), P P˘Z,eZ (R)/P ˘ l,Z,Z ˘ Z Z,e Z,e D Z,e D

and the canonical homomorphism e ord (R)/U e ˘ (R) → Pord P Z,D (R)/U2,Z (R). ˘ 2,Z Z,e Z,e D 7.1.2

Boundary of Ordinary Loci, Continued

Let us continue the study in Section 5.2.4, which generalized part of Section 1.3.2. Let us continue with the setup of Section 7.1.1, with the same choices of e etc as in Section 1.2.4. Let κ e e δ), e Σ e ord , σ (L, h · , · ie, e h0 ), (e Z, Φ, e = (H, e) be any element ord,++ e e h · , · ie, e in the set KQ,H as in Definition 7.1.1.7. The data of O, (L, h0 ), e D, and

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~e ord ˆ (of standard form with respect to e e ⊂ G( e Z) H D) then define M e as in Theorem H p ord e e 3.4.2.5. Since H is neat and Σ is projective (and smooth), by Theorems 5.2.1.1 ~e ord ~e ord,tor ~e ord,tor and 6.2.3.1, we have a partial toroidal compactification M = MH, e e eΣ e ord of M H H ~ which is quasi-projective and smooth over S0,rH . We are mainly interested in comf ~e ord,tor ~ ord,tor paring the boundary structures of MH, under suitable conditions. eΣ e ord and M H,Σord ˘ satisfying In the remainder of this subsection, let us fix the choice of a Z (1.2.4.12), so that we have the groups and homomorphisms defined in Definitions 1.2.4.53, 1.2.4.54, and 7.1.1.27. Suppose that an ordinary cusp label [(ZH , ΦH , δH )] at level H is canonically ˘ e , δ˘ e )] at level H e (necessarily ˘ e, Φ assigned (as in Lemma 1.2.4.15) to a cusp label [(Z H H H e e also ordinary) admitting a surjection to [(e ZHe , ΦHe , δHe )], so that we have (1.2.4.18), (1.2.4.19), and (1.2.4.20), and the definitions following them. Lemma 7.1.2.1. (Compare with Lemma 1.3.2.50.) By comparing the universal properties, we obtain a canonical morphism ~e ord ~ Φord,δ , C ˘ f,δ˘f → C Φ H H H

(7.1.2.2)

H

˘∨,ord ), which is an orbit of ´etale-locally-defined pairs by sending (˘ cord e ,c e H H (˘ cord n :

1 ˘ nX

→ Bp∨,ord , c˘n∨,ord : r

1 ˘ nY

→ Bpord r )

for some integer n = n0 pr where n0 ≥ 1 is an integer prime to p such that Uep (n0 ) ⊂ e p and where r = depthe(H ep ), to the orbit (cord , c∨,ord ) of ´etale-locally-defined pairs H H D H (cord n :

1 nX

, cn∨,ord : → Bp∨,ord r

1 nY

→ Bpord r ),

∨,ord with (cord ) induced by (˘ cord ˘∨,ord ) by restrictions to n1 X and n1 Y , where X n , cn n ,c n ˘ X e and s ˘ : Y˘  Ye , and Y are the kernels of the admissible surjections sX˘ : X Y respectively. (This definition canonically extends to a compatible definition in the Z× (p) -isogeny class language in Proposition 5.2.4.13, which we omit for simplicity.) ˘

f ~e ord,ΦH ∼ ~ ord,ΦH and there is a canonical homomorphism bG = H, then M If H =M e H H

~e ord,grp ~ ord,grp C ˘ f,δ˘f → C Φ ΦH ,δH H

(7.1.2.3)

H

~ ord,ΦH , which can be identified with the (cf. (1.3.2.52)) of abelian schemes over M H canonical homomorphism ˘ B)◦ → Hom (X, B)◦ HomO (X, O

(7.1.2.4) ord,grp

~e ~ ord,ΦH , and the C (cf. (1.3.2.53)) up to canonical Q× -isogenies over M ˘ f,δ˘f - and Φ H H

H

~e ord ~ ord,ΦH and C ~ ord,ΦH , re~ ord ~ ord,grp -torsor structures of C C ˘ f,δ˘f → M Φ ΦH ,δH → MH H ΦH ,δH H H spectively, are compatible with each other under (7.1.2.2) and (7.1.2.3). Moreover,

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~ ord,ΦH , which is canonically the kernel of (7.1.2.3) is an abelian scheme over M H × ◦ e B) of (7.1.2.4), and (7.1.2.2) is a torsor Q -isogenous to the kernel HomO (X, ~ ord under the pullback to C ΦH ,δH of this abelian scheme. ˘

f ~e ord,ΦH ~e ord The abelian scheme torsor C → M and the finite ´etale covering ˘ e ˘ H Φ f,δ f H

˘

H

˘

f f ~e ord,ΦH ~e ord,ZH b = H e b and M → M depend (up to canonical isomorphism) only on H e e H H G

˘

c ~b ord,ΦH ~b ord ˘ b , δ˘ b ) (see Definition 1.2.4.17). We shall denote them as C ˘ b, Φ (Z → M ˘ b ˘ H Φ c,δ c H H H H

˘

H

˘

c c ~b ord,ΦH ~b ord,ZH and M →M when we want to emphasize this (in)dependence. b b H H

Proof. The first paragraph is self-explanatory. As for the second paragraph, by Lemma 5.2.4.7, it suffices to verify the statements in the case H = U1bal (n) and e = Uebal (n) for some integer n = n0 pr . (Then the third paragraph also follows by H 1 ~e ord ~e ord ~ ord ~ ord Lemma 5.2.4.7.) In this case, C ˘ n ,δ˘n and C ˘ f,δ˘f = C Φ Φ ΦH ,δH = CΦn ,δn are abelian H

˘

H

˘

f ~e ord,ΦH ~e ord,Zn ∼ ~ ord,ΦH ~ ord,Zn . For simplicity, let us denote schemes over M = M =M = MH e n H n ~ ord,Zn . the kernel of (7.1.2.2) by C, viewed as a scheme over M n By Proposition 4.2.1.30 and its proof, C is necessarily the extension of a finite flat ~ ord,Zn ) of ´etale-multiplicative type by an abelian scheme over group scheme π 0 (C/M n ord,Zn ~ Mn . By Lemma 7.1.2.1, we know that C ⊗ Q is an abelian scheme Q× -isogenous

Z

~ ord,ΦH . Hence, e B)◦ → M to the pullback of the ordinary abelian scheme HomO (X, H ~ ord,Zn ) must be trivial, and C is necessarily an abelian scheme, which is π 0 (C/M n ~ ord,ΦH over all of M ~ ord,Zn (by noetherian nore B)◦ → M Q× -isogenous to HomO (X, n H ord,Z ~ n and by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5]). mality of M n ~ ord , as explained at the end of Hence, h is a torsor under the pullback of C to C Φn ,δn the proof of Lemma 1.3.2.50. Proposition 7.1.2.5. (Compare with Proposition 1.3.2.55.) Under the canonical morphisms as in (7.1.2.2), and under the canonical homomorphisms b ˘ (A∞,p ) × P b ord G (Qp ) → G1,Z (A∞,p ) × Pord 1,Z,D (Qp ) 1,Z 1,˘ Z,D and ∞ e ˘ e(A∞,p ) × P e ord (Qp ))/U e ˘ (A∞ ) → (PZ (A∞,p ) × Pord (P Z,D (Qp ))/U2,Z (A ), ˘ 2,Z Z,Z Z,e Z,e D

b ˘ (A∞,p ) × P b ord (Qp ) on the collection the Hecke action of (suitable elements of) G 1,Z 1,˘ Z,D ~b ord {C ˘ c,δ˘c }H bb Φ H

H

G

b of standard form) is compatible with the Hecke action of (with H 1,˘ Z

~ ord (suitable elements of) G1,Z (A∞,p ) × Pord 1,Z,D (Qp ) on the collection {CΦH ,δH }HG1,Z (with H of standard form; see Proposition 5.2.4.25); the Hecke action of (suitable elements ~e ord e ord (Qp ))/U e ˘ (A∞ ) on the collection {` C e ˘ e(A∞,p ) × P of) (P ˘ f,δ˘f }H e e /H ee Φ 2,Z Z,Z ˘ Z,e Z,e D H

H

P˘ e Z,Z

U ˘ 2,Z

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e of standard form) is compatible with the Hecke action of (suitable ele(with H ` ~ ord ∞ ments of) (PZ (A∞,p ) × Pord CΦH ,δH }HPZ /HU2,Z Z,D (Qp ))/U2,Z (A ) on the collection { ∞ e ˘ e(A ) on the index sets (with H of standard form); and the induced action of G l,Z,Z e e {[(e Z e , Φ e , δ e )]} is compatible with the induced action of G0 (A∞ ) on the index H

H

l,Z

H

sets {[(ZH , ΦH , δH )]} (again see Proposition 5.2.4.25) under the canonical homoe ˘ e(A∞ ) → G0 (A∞ ). morphism G l,Z l,Z,Z These Hecke actions induce a Hecke action of (suitable elements of) the subb ˘ (A∞,p ) × P b ord (Qp ))/U b ˘ (A∞ ) of (P e ˘ e(A∞,p ) × P e ord (Qp ))/U e ˘ (A∞ ) on group (P Z 2,Z 2,Z ˘ Z,Z ˘ Z,D Z,e Z,e D ` ~b ord b of standard form), which is compatible the collection { C Φ˘ c,δ˘c }Hb b /Hb b (with H H

H

P˘ Z

U ˘ 2,Z

∞ with the Hecke action of (suitable elements of) (PZ (A∞,p ) × Pord Z,D (Qp ))/U2,Z (A ) on ` ~ ord the collection { CΦH ,δH }HPZ /HU2,Z (with H of standard form) under the canonical

~b ord ~ ord b morphisms C ˘ c,δ˘c → C Φ ΦH ,δH (with varying H and H) and the canonical homoH H ∞ b ˘ (A∞ ) → (PZ (A∞,p ) × Pord b ˘ (A∞,p ) × P b ord (Qp ))/U morphism (P Z,D (Qp ))/U2,Z (A ); Z 2,Z ˘ Z,D b 0 (A∞ ) of G e ˘ e(A∞ ) on the index sets and the induced action of the subgroup G l,˘ Z

l,Z,Z

˘ b , δ˘ b )]} is compatible with the induced action of G0 (A∞ ) on the index sets ˘ b, Φ {[(Z l,Z H H H b 0 (A∞ ) → G0 (A∞ ). {[(ZH , ΦH , δH )]} under the canonical homomorphism G ˘ l,Z l,Z Proof. As in the case of (1.3.2.51), the canonical morphisms as in (7.1.2.2) correspond to pushouts of extensions of B (resp. B ∨ ) by T˘ (resp. T˘∨ ) under the canonical ˘ (resp. homomorphism T˘ → T (resp. T˘∨ → T ∨ ) induced by the restriction from X Y˘ ) to X (resp. Y ). Hence, the realizations of the Hecke twists are compatible in the desired ways. (We omit the details for simplicity.) is a top-dimensional nondegenerate rational polyhedral cone Suppose σ e ⊂ P+ e Φ f H

e e in Σ e ord , and suppose σ in the cone decomposition Σ ˘ is the image of σ e ⊂ P+ e Φf Φ H

f H

b ˘ (see Definition under the first morphism in (1.2.4.20). Then we have σ ˘⊥ = S ΦH c b=H e b , and we have the following: 1.2.4.29) for any such σ ˘ ∨ , where H G Proposition 7.1.2.6. (Compare with Proposition 1.3.2.56, Lemma 5.2.4.29 and Proposition 5.2.4.30.) The scheme   ~e ord ~e ord ∼ ˘ Ξ Spec ⊕ ( `) Ψ = ˘ f,δ˘f,˘ ˘ f,δ˘f Φ σ Φ O H

H

ord ~ e˘ C ˘ Φ f,δ f H H

˘ σ⊥ `∈˘

H

H

~e ord e˘ ˘ ⊥ , which over C ˘ f,δ˘f is a torsor under the split torus E Φ ΦH σ with character group σ f,˘ H H b ˘ , which b ˘ with character group S is canonically isomorphic to the split torus E ΦH ΦH c c b ∨ /S b∨ ∼ bb = H e e 0 /H e e (see Definition 1.2.4.53). We have S depends only on H = P˘Z0

P˘Z

U2,eZ

ˆ H b ˘ := S b˘ b ˘ (Z)/ b b , where S U 2,Z Φ1 Φb U ˘ 2,Z

G(ˆ Z)

SΦ˘ 1  SΦ e 1 (see Definition 1.2.4.29).

˘1 Φ

˘c Φ H

is the kernel of the canonical homomorphism

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~e ord ~e ord → C The torus torsor S := Ξ ˘ f,δ˘f is universal for the additio˘ f,δ˘f,˘ σ Φ Φ H H H H bal ord ∨,ord b b /Ub (n) b -orbits of ´etale-locally-defined ), which are H nal structures (b τ , τb b H

1

P˘Z

b H

P˘Z

pairs (b τn , τbn∨ ) (for some integer n = n0 pr where n0 ≥ 1 is an integer prime b p and where r = depthD (H bp ), in which case we set to p such that Ubp (n0 ) ⊂ H bal p bal r b b b U1 (n) := U (n0 )Up,1 (p )), where: ∼

⊗ −1 1 (1) τbnord = τbn0 : 1 1 Y × X,S ˘)∗ PB is a trivialization of biex→ (˘ c∨ ˘ YS × c n0 | n n0 tensions. ∼ ⊗ −1 (2) τbn∨,ord = τbn∨0 : 1 1 Y˘ × X,S → (˘ c∨ ˘|XS )∗ PB is a trivialization of biexn0 × c n0 tensions. (3) τbn0 and τbn∨0 satisfy the analogues of the usual O-compatibility condition. (4) τbn0 and τbn∨0 satisfy the symmetry condition that τbn0 |1Y × Y˘ ,S and τbn∨0 |1Y˘ × Y,S coincide under the canonical isomorphism induced by the swapping isomor∼ phism 1Y × Y˘ ,S → 1Y˘ × Y,S and the symmetry automorphism of PB . (5) τbn0 |1 1 Y × X,S = τbn∨0 |1 1 Y × X,S . n0

n0

~b ord ~e ord We shall denote Ξ ˘ c,δ˘c when we want to emphasize that (by Lemma ˘ f,δ˘f,˘ Φ σ by ΞΦ H H H H b = H e b and (Z ˘ b , δ˘ b ) (see Definition 1.2.4.17), ˘ b, Φ 5.2.4.26) it depends only on H G H H H but does not depend on the choice of σ ˘. ~b ord ~b ord b ˘ -torsor structure of Ξ The E → C ˘ ˘ ˘ c,δ˘c defines a homomorphism Φ , δ Φ Φc c c H

H

H

H

H

~b ord ~b ord ˘ b ˘ → Pic(C S ˘ c,δ˘c ) : `˘ 7→ Ψ ˘ c,δ˘c (`), Φ Φ Φc H

H

H

H

H

~b ord ~b ord ˘ over C b ˘ an invertible sheaf Ψ assigning to each `˘ ∈ S ˘ c,δ˘c (`) ˘ c,δ˘c (up to isoΦ Φ ΦH c H H H H morphism), together with isomorphisms ~b ord,∗ ~b ord ˘ ∆ ˘ `˘0 : Ψ ˘ c,δ˘c (`) ˘ c,δ˘c,`, Φ Φ H

H

H

H

⊗

O ~ ord b˘ C Φ

ord ~b ord 0 ∼ ~ b Φ˘ ,δ˘ (`˘ + `˘0 ) Ψ ˘ c,δ˘c (`˘ ) → Ψ Φ c c H

H

H

H

˘ c,δH c H

˘ `˘0 ∈ S b ˘ , satisfying the necessary compatibilities with each other making for all `, Φc H

~b ord ˘ an O ord Ψ ˘ c,δ˘c (`) Φ ~b

⊕ ˘ S b˘ `∈ Φ

H

c H

H

CΦ ˘

-algebra, such that

˘ c,δH c H

~b ord Ξ = SpecO ˘ c,δ˘c ∼ Φ H H

 ~b ord ˘ Ψ ( `) . ˘ ˘ c,δ c Φ

 ⊕ ord ~ b˘ C ˘ Φ c,δ c H H

˘ S b˘ `∈ Φ

H

H

c H

˘ such that either y ∈ Y or χ ∈ X, we When `˘ = [y ⊗ χ] for some y ∈ Y˘ and χ ∈ X have a canonical isomorphism ~b ord ˘ ∼ Ψ c∨ (y), c˘(χ))∗ PB . = (˘ ˘ c,δ˘c (`) Φ H

H

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˘ n , then the canonical morphism ˘n and) Φ If we fix the choice of (Z ~b ord ~b ord ~b ord Ξ S0,rn ˘ n ,δ˘n → Ξ ˘ c,δ˘c,rn = Ξ ˘ c,δ˘c × ~ Φ Φ Φ H

H

H

H

(7.1.2.7)

~ S0,r f H

(cf. (1.3.2.57)), where rn = rUbbal (n) = rU1bal (n) 1 b b 0 /Ubbal (n) b 0 -torsor, and induces an isomorphism H 1

P˘Z

=

max(rD , r), is an

P˘Z

ord ∼ ~ ~b ord b b 0 /Ub1bal (n) b 0 ) → b Φ˘ ,δ˘ ,r Ξ Ξ ˘ n ,δ˘n /(H Φ P P c c n ˘ Z

H

˘ Z

(7.1.2.8)

H

(cf. Lemma 5.2.4.26; cf. also (1.3.2.58)). ord

~e Proof. These follow from the corresponding properties of Ξ ˘ f,δ˘f as in Lemmas Φ H H 5.2.4.26 and 5.2.4.29, and Proposition 5.2.4.30, because the restriction from SΦ˘ f to H b ˘ (see Definition 1.2.4.29) corresponds to taking orbits of restrictithe subgroup S ΦH c

ons of τ˘n0 : 1 the pairs

∼

⊗ −1 → (˘ c∨ × c˘)∗ PB to 1

˘ Y˘ × X,S (b τnord , τbn∨,ord ) 1 n0

=

(b τn0 , τbn∨0 )

1 n0

˘ Y × X,S

and 1

1 n0

Y˘ × X,S ,

which form

as above.

For each rational polyhedral cone ρ˘ ⊂ (SΦ˘ f )∨ ˘ as a face, we have an R having σ H affine toroidal embedding   ~e ord ~e ord ~e ord ˘ Ξ ,→ Ξ (˘ ρ ) := Spec Ψ ⊕ ( `) (7.1.2.9) ˘ f,δ˘f,˘ ˘ f,δ˘f,˘ ˘ f,δ˘f Φ σ Φ σ Φ O H

H

H

H

e C ˘ Φ

˘ f,δH f H

H

`∈˘ σ ⊥ ∩ ρ˘∨

H

(cf. (1.3.2.59)) as in (5.2.4.31). b ˘ )∨ , we have an affine In general, for each rational polyhedral cone ρb ⊂ (S ΦH c R toroidal embedding   ~b ord ~b ord ~b ord ˘ Ψ ( `) (7.1.2.10) ⊕ Ξ ,→ Ξ (b ρ ) := Spec ˘ ˘ ˘ ˘ ˘ ˘ Φ c,δ c Φ c,δ c Φ c,δ c O H

H

H

H

ord ~ b˘ C ˘ Φ c,δ c H H

H

˘ ρ∨ `∈b

H

(cf. (1.3.2.60)). By Proposition 7.1.2.6, (7.1.2.9) and (7.1.2.10) can be canonically identified b = H e b , when (Z ˘ b , δ˘ b ) is determined by (Z ˘ e , δ˘ e ) as in Defini˘ b, Φ ˘ e, Φ when H G H H H H H H tion 1.2.4.17, and when ρb = pr(Sb ˘ )∨ (˘ ρ). (Hence, (7.1.2.9) depends only on these Φc R H

induced parameters.)

~b ord ~b ord Both sides of (7.1.2.10) are relative affine over C ρ) → ˘ c,δ˘c (b ˘ c,δ˘c , where Ξ Φ Φ H

H

H

H

~b ord ~b ord C is smooth when the cone ρ b is smooth. The ρ b -stratum of Ξ ρ) is ˘ ˘ ˘ c,δ˘c (b ΦH Φ c,δH c H H   ~b ord ~b ord ˘ Ξ ⊕ Ψ (7.1.2.11) ˘ c,δ˘c,b ˘ c,δ˘c (`) Φ ρ := SpecO Φ H

H

ord ~ b˘ C ˘ Φ c,δ c H H

˘ ρ˘⊥ `∈

H

H

~e ord (cf. (1.3.2.61)), which is canonically isomorphic to Ξ ˘ f,δ˘f,ρ˘. The affine morphism Φ H

~b ord Ξ ˘ ,δ˘ Φ c H

ρ c,b H

H

~b ord ∼ ˘ b˘ → C ˘ c,δ˘c is a torsor under the torus E Φ Φ c,b ρ = EΦ ˘ with character f,ρ H H H

H

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group ρb⊥ ∼ = ρ˘⊥ . (Note that these two instance of ⊥ are taken in different ambient b ˘ of spaces.) For each ΓΦ˘ c -admissible rational polyhedral cone decomposition Σ ΦH c H b ˘ as in Definition 1.2.4.40, we have (as in (4.2.2.4)) a toroidal embedding P ΦH c

ord

ord

~b ord ~b ~b Ξ ˘ c,δ˘c = Ξ ˘ c,δ˘c,Σ b˘ ˘ c,δ˘c ,→ Ξ Φ Φ Φ Φ H

H

H

H

H

H

(7.1.2.12) c H

~b ord (cf. (1.3.2.62)), the right-hand side being only locally of finite type over C ˘ c,δ˘c , Φ H H with an open covering ord

~b Ξ ˘ c,δ˘c = Φ H

H

~b ord Ξ ρ) ˘ c,δ˘c (b Φ

(7.1.2.13)

a ~ ord b Φ˘ ,δ˘ ,bρ Ξ c c

(7.1.2.14)

∪ b˘ ρ b∈Σ Φ

H

H

c H

(cf. (1.3.2.63)) inducing a stratification ord

~b Ξ ˘ c,δ˘c = Φ H

H

H

b˘ ρ b∈Σ Φ

H

c H

` (cf. (1.3.2.64)). (The notation “ ” only means a set-theoretic disjoint union. The ord

~b algebro-geometric structure is still the one inherited from Ξ ˘ c,δ˘c .) Let Φ H

ord

H

ord

~b ~b ρ))∧ X ˘ c,δ˘c (b ˘ c,δ˘c,b Φ ρ := (ΞΦ ~ ord H

H

H

H

b˘ Ξ Φ

(7.1.2.15)

˘ ,ρ c,δH c b H

~b ord ~b ord (cf. (1.3.2.65)), the formal completion of Ξ ρ) along its ρb-stratum Ξ ˘ c,δ˘c (b ˘ c,δ˘c,b Φ Φ ρ, H

~e ord which is canonically isomorphic to X ˘ ord

H

H

˘f,˘ ΦH σ ,ρ˘ f,δH

ord

H

~e ord , the formal completion of Ξ ˘

ρ), ˘f,˘ ΦH σ (˘ f,δH

ord

~e ~e ~e ρ), along its ρ˘-stratum Ξ the closure of Ξ ˘ f,δ˘f (˘ ˘ f,δ˘f,ρ˘ (cf. (1.3.2.40)). ˘ f,δ˘f,˘ Φ Φ σ in ΞΦ H H H H H H Also, let us define ~b ord ~b ord X ˘ c,δ˘c = X ˘ c,δ˘c,Σ b˘ Φ Φ Φ H

H

H

H

(7.1.2.16)

c H

ord

~b (cf. (1.3.2.66) and Lemma 5.2.4.38) to be the formal completion of Ξ ˘ c,δ˘c,Σ b˘ Φ Φ H

~b ord b+ . b ˘ and ρb ⊂ P along the union of the σ b-strata Ξ b∈ Σ ˘ c,δ˘c,b Φ ρ for ρ ˘ Φc Φ H

H

H

H

c H

c H

Proposition 7.1.2.17. (Compare with Propositions 1.3.2.67, 5.2.2.2, 5.2.4.25, 5.2.4.41, and 7.1.2.5.) There is a Hecke action of (suitable elements of) ~b ord ord,0 0 ∞,p b (A ) × P b b P (Qp ) on the collection {Ξ ˘ c,δ˘c }H b b 0 (with H of standard form), Φ ˘ ˘ Z Z,D H

H

P ˘ Z

realized by quasi-finite flat surjections pulling tautological objects back to ordinary Hecke twists, which is compatible with the Hecke action of (suitable elements ~b ord b of standard b ˘ (A∞,p ) × P b ord (Qp ) on the collection {C of) G (with H ˘ c,δ˘c }H bb Φ 1,Z 1,˘ Z,D H

H

G

1,˘ Z

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~b ord ~b ord b and form) under the canonical morphisms Ξ ˘ c,δ˘c (with varying H) ˘ c,δ˘c → C Φ Φ H H H H ord,0 0 ∞,p ∞,p ord b (A ) × P b b ˘ (A ) × G b (Qp ) = the canonical homomorphism P (Qp ) → G 1,Z ˘ ˘ Z 1,˘ Z Z,D ord,0 0 ∞ ∞ b (A ) × P b b ˘ (A ). Such a Hecke action enjoys the properties (under va(P )/U ˘ Z

2,Z

˘ Z,D

rious conditions) concerning ´etaleness, finiteness, being isomorphisms between formal completions along fibers over Spec(Fp ), and inducing absolute Frobenius morphisms on fibers over Spec(Fp ) for elements of Up type as in Proposition 5.2.2.2 and Corollaries 5.2.2.3, 5.2.2.4, and 5.2.2.5. (We omit the details for simplicity.) b ord (Qp ) on b ˘ (A∞,p ) × P There is also a Hecke action of (suitable elements of) P Z ˘ Z,D ` ~b ord b the collection { ΞΦ˘ c,δ˘c }Hb b (with H of standard form), where the disjoint uniH

H

P˘ Z

˘ b , δ˘ b )] sharing the same Z ˘ b compatible with D, realized ˘ b, Φ ons are over classes [(Z H H H H by quasi-finite flat surjections pulling tautological objects back to ordinary Hecke b 0 (A∞ ) = P b ˘ (A∞ )/P b 0 (A∞ ) on the index sets twists, which induces an action of G l,˘ Z

˘ Z

Z

˘ b , δ˘ b )]}, which is compatible with the Hecke action of (suitable elements of) ˘ b, Φ {[(Z H H H ~b ord ∞,p b ˘ (A ) × P b ord (Qp ))/U b ˘ (A∞ ) on the collection {` C b of (P (with H ˘ c,δ˘c }H b b /H bb Φ Z 2,Z ˘ Z,D H

H

P˘ Z

U

2,˘ Z

b 0 (A∞ )) standard form, with the same index sets and the same induced action of G l,˘ Z b˘ ˘ → C b ˘ ˘ (with varying H) b and the canounder the canonical morphisms Ξ ΦH ΦH c,δH c c,δH c b ˘ (A∞,p ) × P b ˘ (Qp ) → (P b ˘ (A∞,p ) × P b ord (Qp ))/U b ˘ (A∞ ). nical homomorphism P Z

Z

˘ Z,D

Z

2,Z

Any such Hecke action ~b ord ~b ord ~g ]ord : Ξ [b ˘ 0 ,δ˘0 → Ξ ˘ c,δ˘c Φ Φ 0 0 c H

~g ] covering [b

ord

~b :C ˘0 Φ

H

c H

H

ord

ord ,δ˘0c0 c0 H H

~b →C ˘ c,δ˘c induces a morphism Φ H

ord

~b Ξ ˘0 Φ

,δ˘0c0 c0 H H

H

~b ord →Ξ ˘ c,δ˘c Φ H

H

~b ord C ˘ 0 ,δ˘0 Φ 0

× ~bord C ˘ ˘ ,δ Φ c H

c H

c0 H

c H

~b ord between torus torsors over C ˘ 0 ,δ˘0 , which is equivariant with the morphism Φ c0 H c0 H b˘ → S b ˘ 0 induced by the pair b˘ 0 → E b ˘ dual to the homomorphism S E Φ Φc Φc Φ c0 H

H

of morphisms (fX˘

∼ ˘ ⊗Q → ˘ 0 ⊗ Q, f ˘ : X X Y Z

c0 H

H

∼ : Y˘ 0 ⊗ Q → Y˘ ⊗ Q) defining

Z

Z

Z

˘ 0 , δ˘0 ) →gb (Z ˘ b , δ˘ b ) of cusp labels (which is the ˘0b0 , Φ ˘ b, Φ the gb-assignment (Z b0 H b0 H H H H H b ˘ (A∞ ) = e ˘ (A∞ ) ∩ P e 0 (A∞ ) lifting gb ∈ P ge-assignment for any element ge ∈ P Z

Z

e Z

e ˘ (A∞ ) ∩ P e 0 (A∞ ))/U e ˘ (A∞ ), which is nevertheless independent of the choice of (P Z 2,Z e Z ge; cf. Lemma 1.2.4.42 and [62, Def. 5.4.3.9]). b ˘ (A∞,p ) × P b ord (Qp ) is as above and if (Φ ˘ 0 , δ˘0 , ρb0 ) is a gb-refinement of If gb ∈ P Z ˘ b0 H b0 Z,D H ˘ b , δ˘ b , ρb) (cf. Lemma 1.2.4.42 and [62, Def. 6.4.3.1]), then there is a canonical (Φ H

H

morphism ~b ord ~b ord ~g ]ord : Ξ [b ρ0 ) → Ξ ρ) ˘ 0 ,δ˘0 (b ˘ c,δ˘c (b Φ Φ 0 0 c H

c H

H

H

(7.1.2.18)

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~b ord ~b ord ~g ]ord : C ~g ]ord : (cf. (1.3.2.68) and (5.2.4.42)) covering [b ˘ 0 ,δ˘0 → C ˘ c,δ˘c , extending [b Φ Φ 0 0 c H

H

c H

H

~b ord ~b ord ~b ord ~b ord Ξ → Ξ 0 to ΞΦ ˘ 0 ,δ˘0 ˘ c,δ˘c , mapping Ξ ˘ 0 ,δ˘0 ,b ˘ c,δ˘c,b Φ Φ Φ ρ ρ , and inducing a canonical H H H H c0 H c0 c0 H c0 H H morphism ~b ord ~g ]ord : X [b ˘ 0 ,δ˘0 Φ 0

,b ρ0 c0

c H

H

~b ord →X ˘ c,δ˘c,b Φ ρ H

(7.1.2.19)

H

b ˘ (A∞,p ) × P b ord (Qp ) is as above and if (cf. (1.3.2.69) and (5.2.4.43)). If gb ∈ P Z ˘ Z ˘ 0 , δ˘0 , Σ ˘ b , δ˘ b , Σ b ˘ ) (cf. Lemma 1.2.4.42 and [62, b 0 0 ) is a gb-refinement of (Φ (Φ ˘ b0 H b0 Φc H H Φ H H

c0 H

Def. 6.4.3.2]), then morphisms like (7.1.2.18) patch together and define a canonical morphism ord

ord

~b ~g ]ord : Ξ [b ˘0 Φ

b0 ,δ˘0c0 ,Σ ˘0 c0

H

Φ c0 H

H

~b →Ξ ˘ c,δ˘c,Σ b˘ Φ Φ H

H

(7.1.2.20) c H

~b ord ~b ord ~g ]ord : ~g ]ord : C (cf. (1.3.2.70) and (5.2.4.44)) covering [b ˘ c,δ˘c , extending [b ˘ 0 ,δ˘0 → C Φ Φ 0 0 c H

H

c H

H

~b ord ~b ord Ξ ˘ c,δ˘c , and inducing a canonical morphism ˘ 0 ,δ˘0 → Ξ Φ Φ 0 0 c H

H

c H

H

~b ord ~g ]ord : X [b ˘ 0 ,δ˘0 Φ 0

b0 ,Σ ˘0 c0

c H

Φ c0 H

H

~b ord →X ˘ c,δ˘c,Σ b˘ Φ Φ H

H

(7.1.2.21) c H

(cf. (1.3.2.71) and (5.2.4.45)) compatible with each (7.1.2.19) as above (under canonical morphisms). Proof. By Proposition 7.1.2.6 (see in particular (7.1.2.8)), and by finite flat descent, the assertions in the first three paragraphs are reduced to the ones for the principal levels, which then follow from the cor` ~e ord responding assertions for the collection { Ξ ˘ f,δ˘f }H e e 0 (by restricting the Φ H

H

P ˘ Z

e ˘ (A∞,p ) × P e ord (Qp ) to suitable elements of action of suitable elements of P Z ˘ Z,e D ∞,p 0 ∞,p 0 ord e e e e e (P˘Z (A ) ∩ PeZ (A )) ×(P˘Z (Qp ) ∩ PeZ (Qp ) ∩ PeD (Qp ))), because the tautological ~b ord ~b ord objects over Ξ are canonically induced by those over ˘ b bal ˘ n ,δ˘n = Ξ Φ Φ ,δ˘ b bal U1

~e ord ~e ord Ξ ˘ n ,δ˘n = Ξ ˘ e bal Φ Φ U1

(n)

,δ˘Ue bal (n) .

(n)

U1

(n)

The assertions in the last paragraph then follow from

1

the universal properties of toroidal embeddings (cf. [62, Prop. 6.2.5.11]). Lemma 7.1.2.22. (Compare with Lemmas 1.3.2.72 and 7.1.2.1.) By comparing the universal properties, we obtain a canonical morphism ord

~e ~ ord Ξ ˘ f,δ˘f → Ξ Φ ΦH ,δH H

H

(7.1.2.23)

ord covering (7.1.2.2), by sending τ˘H etale-locally-defined trivialie , which is an orbit of ´

zations τ˘nord = τ˘n0 : 1

∼

1 n0

˘ Y˘ × X,S

⊗ −1 → (˘ c∨ ˘)∗ PB (for some integer n = n0 pr where n0 × c

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e p and where r = depthe(H ep )), n0 ≥ 1 is an integer prime to p such that Uep (n0 ) ⊂ H D ord ord to the orbit τH of ´etale-locally-defined trivializations τn = τn0 = τ˘n0 |1 1 Y × X,S . n0

The morphisms (7.1.2.23) and (7.1.2.2) induce a canonical morphism ~e ord ~ ord Ξ ˘ f,δ˘f → Ξ ΦH ,δH Φ H

H

× ~ ord C Φ ,δ H

~e ord C ˘ f,δ˘f Φ H

H

(7.1.2.24)

H

~e ord between torus torsors over C ˘ f,δ˘f , equivariant with the homomorphism EΦ ˘f → Φ H H H EΦH dual to the canonical homomorphism SΦH → SΦ˘ f (see (1.2.4.18)). H Suppose the image of a rational polyhedral cone ρ˘ ⊂ (SΦ˘ f )∨ R under the (canoH nical) second morphism in (1.2.4.20) is contained in some rational polyhedral cone ρ ⊂ (SΦH )∨ R . Then there is a canonical morphism ~e ord ~ ord Ξ ρ) → Ξ ˘ f,δ˘f (˘ Φ ΦH ,δH (ρ) H

H

(7.1.2.25)

~e ord (cf. (1.3.2.75)) covering (7.1.2.2) and extending (7.1.2.23), mapping Ξ ˘ f,δ˘f,ρ˘ to Φ H H ord ~ ΞΦH ,δH ,ρ and inducing a canonical morphism ~e ord ~ ord X ˘ f,δ˘f,ρ˘ → X Φ ΦH ,δH ,ρ H

(7.1.2.26)

H

e ˘ and ΣΦ are cone decompositions of P ˘ and PΦ , re(cf. (1.3.2.76)). If Σ H H ΦH ΦH f f e ˘ under the (canonical) second morspectively, such that the image of each ρ˘ in Σ ΦH f

phism in (1.2.4.20) is contained in some ρ ∈ ΣΦH , then morphisms like (7.1.2.25) patch together and define a canonical morphism ord

ord

~e Ξ ˘ f,δ˘f,Σ e˘ Φ Φ H

H

f H

~ →Ξ ΦH ,δH ,ΣΦ

H

(7.1.2.27)

(cf. (1.3.2.77)) covering (7.1.2.2), extending (7.1.2.23), and inducing a canonical morphism ~e ord X ˘ f,δ˘f,Σ e˘ Φ Φ H

H

f H

~ ord →X ΦH ,δH ,ΣΦ

H

(7.1.2.28)

(cf. (1.3.2.78)) compatible with each (7.1.2.26) as above (under canonical morphisms). Proof. The statements are self-explanatory. Lemma 7.1.2.29. (Compare with Lemmas 1.3.2.79, 7.1.2.1, and 7.1.2.22.) By comparing the universal properties (cf. Proposition 7.1.2.6), we obtain a canonical morphism ~b ord ~ ord Ξ ˘ c,δ˘c → Ξ Φ ΦH ,δH H

H

(7.1.2.30)

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ord ∨,ord covering (7.1.2.2), by sending the pair (b τH bHb ), which is an orbit of ´etaleb ,τ ord ∨,ord ∨ locally-defined pairs (b τn , τbn ) = (b τn0 , τbn0 ) (for some integer n = n0 pr where b p and where r = depthD (H bp )), n0 ≥ 1 is an integer prime to p such that Ubp (n0 ) ⊂ H ord ord ∨ to the orbit τH of ´etale-locally-defined τn = τn0 = τbn0 |1 1 Y × X,S = τbn0 |1 1 Y × X,S , n0

as in Proposition 7.1.2.6. The morphisms (7.1.2.30) and (7.1.2.2) induce a canonical morphism ~b ord ~ ord Ξ ˘ c,δ˘c → Ξ ΦH ,δH Φ H

H

× ~ ord C Φ ,δ H

~b ord C ˘ c,δ˘c Φ H

n0

(7.1.2.31)

H

H

ord

~b (cf. (1.3.2.81)) between torus torsors over C ˘ c,δ˘c , equivariant with the surjective Φ H H b homomorphism E ˘ → EΦ (see Proposition 7.1.2.6) dual to the canonical injective ΦH c

homomorphism SΦH

H

b ˘ (see Definition 1.2.4.29). ,→ S Φc H

b ˘ )∨ under (1.2.4.37) is Suppose the image of a rational polyhedral cone ρb ⊂ (S ΦH f R contained in some rational polyhedral cone ρ ⊂ (SΦH )∨ R . Then there is a canonical morphism ~b ord ~ ord (ρ) Ξ ρ) → Ξ ˘ c,δ˘c (b Φ ΦH ,δH H

(7.1.2.32)

H

(cf. (1.3.2.82) and (7.1.2.25)) covering (7.1.2.2) and extending (7.1.2.30), mapping ~b ord ~ ord Ξ ˘ c,δ˘c,b Φ ρ to ΞΦH ,δH ,ρ and inducing a canonical morphism H

H

~b ord ~ ord X ˘ c,δ˘c,b Φ ρ → XΦH ,δH ,ρ H

(7.1.2.33)

H

b˘ b ˘ and ΣΦ are cone decompositions of P (cf. (1.3.2.83) and (7.1.2.26)). If Σ H ΦH ΦH c c b and PΦ , respectively, such that the image of each ρb in Σ ˘ under (1.2.4.37) is ΦH c

H

contained in some ρ ∈ ΣΦH , then morphisms like (7.1.2.32) patch together and define a canonical morphism ord

~b Ξ ˘ c,δ˘c,Σ b˘ Φ Φ H

H

c H

ord

~ Φ ,δ ,Σ →Ξ Φ H H

H

(7.1.2.34)

(cf. (1.3.2.84) and (7.1.2.27)) covering (7.1.2.2), extending (7.1.2.30), and inducing a canonical morphism ~b ord X ˘ c,δ˘c,Σ b˘ Φ Φ H

H

c H

~ ord →X ΦH ,δH ,ΣΦ

H

(7.1.2.35)

(cf. (1.3.2.85) and (7.1.2.28)) compatible with each (7.1.2.33) as above (under canonical morphisms). (See Lemma 7.2.5.34 below for statements concerning the relative log 1-differentials of (7.1.2.34).) Proof. The statements are self-explanatory.

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Proposition 7.1.2.36. (Compare with Propositions 1.3.2.90 and 7.1.2.5.) Une and H), and under the canonical morphisms as in (7.1.2.23) (with varying H ord,0 0 ∞,p 0 e e der the canonical homomorphisms P˘Z (A ) × P˘Z,eD (Qp ) → PZ (A∞,p ) × Pord,0 Z,D (Qp ) ∞,p ord ∞,p ord e e and P˘ e(A ) × P (Qp ) → PZ (A ) × P (Qp ), the Hecke action of (suitZ,Z

˘ Z,e Z,e D

Z,D

e 0 (A∞,p ) × P e ord,0 (Qp ) P ˘ ˘ Z Z,e D

able elements of)

~e ord e on the collection {Ξ ˘ f,δ˘f }H e e 0 (with H Φ H

H

P ˘ Z

of standard form) is compatible with the Hecke action of (suitable elements of) ~ ord P0Z (A∞,p ) × Pord,0 Z,D (Qp ) on the collection {ΞΦH ,δH }HP0Z (with H of standard form); e ˘ e(A∞,p ) × P e ord (Qp ) on the and the Hecke action of (suitable elements of) P Z,Z

` ~e ord collection { Ξ ˘ f,δ˘f }H ee Φ H

H

P˘ e Z,Z

˘ Z,e Z,e D

e of standard form) is compatible with the (with H

Hecke action of (suitable elements of) PZ (A∞,p ) × Pord Z,D (Qp ) on the collection ` ~ ord { ΞΦH ,δH }HPZ (with H of standard form), where the index sets are as in Proposition 5.2.4.41. These Hecke actions are all compatible with those in Proposition 7.1.2.5. They are also compatible with extensions to toroidal embeddings and their formal completions. Proof. As in the case of Proposition 7.1.2.5, the canonical morphisms as in (7.1.2.30) correspond to pushouts of extensions of B (resp. B ∨ ) by T˘ (resp. T˘∨ ) under the canonical homomorphism T˘ → T (resp. T˘∨ → T ∨ ) induced by the re˘ (resp. Y˘ ) to X (resp. Y ). Hence, the realizations of the Hecke striction from X twists are compatible in the desired ways. (We omit the details for simplicity.) Proposition 7.1.2.37. (Compare with Propositions 1.3.2.91, 7.1.2.5 and 7.1.2.36.) b and H), and unUnder the canonical morphisms as in (7.1.2.30) (with varying H ord,0 0 ∞,p 0 b b der the canonical homomorphisms P˘Z (A ) × P˘Z,D (Qp ) → PZ (A∞,p ) × Pord,0 Z,D (Qp ) ∞,p ord ∞,p ord b b and P˘ (A ) × P (Qp ) → PZ (A ) × PZ,D (Qp ), the Hecke action of (suitaZ

˘ Z,D

ord

~b b 0 (A∞,p ) × P b ord,0 (Qp ) on the collection {Ξ b ble elements of) P ˘ c,δ˘c }H b b 0 (with H Φ ˘ ˘ Z Z,D H

H

P ˘ Z

of standard form) is compatible with the Hecke action of (suitable elements of) ~ ord P0Z (A∞,p ) × Pord,0 Z,D (Qp ) on the collection {ΞΦH ,δH }HP0Z (with H of standard form); b ˘ (A∞,p ) × P b ord (Qp ) on the collection and the Hecke action of (suitable elements of) P Z

˘ Z,D

` ~b ord b { Ξ ˘ c,δ˘c }H b b (with H of standard form) is compatible with the Hecke action of Φ H H P˘ Z ` ~ ord (suitable elements of) PZ (A∞,p ) × Pord ΞΦH ,δH }HPZ (with Z,D (Qp ) on the collection { H of standard form), where the index sets are as in Proposition 5.2.4.41. These Hecke actions are all compatible with those in Proposition 7.1.2.5. They are also compatible with extensions to toroidal embeddings and their formal completions. Proof. As in the proofs of Propositions 1.3.2.91 and 7.1.2.17, the Hecke action ~b ord b ˘ (A∞,p ) × P b ord (Qp ) on the collection {` Ξ of (suitable elements of) P ˘ c,δ˘c }H bb Φ Z ˘ Z,D H

H

P˘ Z

b of standard form) is induced by the Hecke action of (suitable elements of) (with H

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~e ord e 0 (A∞,p ) × P e ord,0 (Qp ) on the collection {` Ξ e P ˘ f,δ˘f }H e e 0 (with H of standard form). Φ ˘ ˘ Z Z,e D H

H

P ˘ Z

Hence, these statements follow from the corresponding statements of Proposition 7.1.2.36. Remark 7.1.2.38. (Compare with Remark 5.2.4.46.) As in Remark 5.2.4.46, since all objects and morphisms in this subsection are defined by normalizations and by the various universal properties extending their analogues in characteristic zero, they are canonically compatible with the corresponding objects and morphisms in Section 1.3.2. 7.1.3

Ordinary Kuga Families and Their Generalizations

~ ord Consider the abelian scheme GM ~ ord over MH and its semi-abelian extension G over H ord,tor ord,tor ~ ~ M = M as in Theorem 5.2.1.1. Let Q be any O-lattice. By (4) of H H,Σord ~ ord is defined and is Proposition 3.1.2.4, the abelian scheme Hom (Q, G ~ ord )◦ → M O

H

MH

ordinary.

Definition 7.1.3.1. (Compare with Definition 1.3.3.3.) An ordinary Kuga fa~ ord is an (ordinary) abelian scheme Ngrp → M ~ ord that is Q× -isogenous mily over M H H ◦ to HomO (Q, GM ~ ord ) for some O-lattice Q. H

Definition 7.1.3.2. (Compare with Definition 1.3.3.4.) An generalized ordi~ ord is a torsor N → M ~ ord under some ordinary Kuga nary Kuga family over M H H ~ ord as in Definition 7.1.3.1. family Ngrp → M H Then the following four lemmas and proposition can be proved by the same arguments as before: Lemma 7.1.3.3. (Compare with [61, Lem. 2.6] and Lemma 1.3.3.5.) The abelian scheme HomZ (Q∨ , G∨ ~ ord ) is isomorphic to the dual abelian scheme of M H

HomZ (Q, GM ~ ord ). H

Lemma 7.1.3.4. (Compare with [61, Lem. 2.9] and Lemma 1.3.3.6.) Let jQ : Q∨ ,→ Q be as in Lemma 1.2.4.1. Then the isogeny ∨ ∨ λM ~ ord ,jQ ,Z : HomZ (Q, GM ~ ord ) → HomZ (Q , GM ~ ord ) H

H

H

∨ induced canonically by jQ and λM ~ ord : GM ~ ord → GM ~ ord is a polarization. H

H

H

Proposition 7.1.3.5. (Compare with [61, Prop. 2.10 and Cor. 2.12] and Proposi◦ × tion 1.3.3.7.) The abelian scheme HomO (Q∨ , G∨ ~ ord ) is Q -isogenous to the dual M H

◦ ∨ ,→ Q as in abelian scheme of HomO (Q, GM ~ ord ) . Moreover, given any jQ : Q H Lemma 1.2.4.1, the composition ◦ λM ~ ord ,jQ : HomO (Q, GM ~ ord ) ,→ HomZ (Q, GM ~ ord ) H H H (7.1.3.6) λM ~ ord ,j ,Z Q H ◦ ∨ → HomZ (Q∨ , G∨ )  (Hom (Q, G ) ) ord ~ O ~ ord M M H

H

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∨ induced canonically by jQ and the polarization λM ~ ord : GM ~ ord → GM ~ ord is a polariH H H zation.

~ ord be as in Definition 7.1.3.7. (Compare with Definition 1.3.3.9.) Let N → M H ~ ord to be the dual abelian Definition 7.1.3.2. Then we define the dual N∨ → M H ~ ord of Ngrp → M ~ ord . scheme Ngrp,∨ → M H H Remark 7.1.3.8. (Compare with Remark 1.3.3.10.) By [92, XIII, Prop. 1.1], N∨ = ~ ord ) → M ~ ord (which can be ~ ord is canonically isomorphic to Pic0 (N/M Ngrp,∨ → M H H H defined as in the case of abelian schemes; cf. [62, Def. 1.3.2.1]). Note that this is ~ ord is a nontrivial always a group scheme, with its identity section, even when N → M H grp ord ~ torsor of N → MH . Definition 7.1.3.9. (Compare with Definition 1.3.3.11.) By abuse of notation, we ∨ ∨ denote by LieN/M ~ ord (resp. LieN/M ~ ord , resp. LieN∨ /M ~ ord , resp. LieN∨ /M ~ ord ) the locally H

H

H

H

∨ ∨ free sheaf LieNgrp /M ~ ord (resp. LieNgrp /M ~ ord , resp. LieNgrp,∨ /M ~ ord , resp. LieNgrp,∨ /M ~ ord ) H H H H ~ ord , although N → M ~ ord might have no section. over M H

H

Lemma 7.1.3.10. (Compare with Lemma 1.3.3.12.) We have: ∼ Lie∨ (LieN/M ~ ord , OM ~ ord ), ~ ord = HomOM ~ ord N/M H

Lie∨ ~ ord N∨ /M H Ω1N/M ~ ord H

H

H

H

∼ = HomOM~ ord (LieN∨ /M ~ ord , OM ~ ord ), H

H

H

∼ ~ ord )∗ Lie∨ ~ ord , = (N → M H N/M H

∼ ∨ ~ ord ∗ ∨ ~ ord , Ω1N∨ /M ~ ord = (N → MH ) LieN∨ /M H

H

~ ord )∗ Ω1 ord ∼ (N → M = Lie∨ ~ ord , H ~ N/M N/M H

H

~ ord )∗ Ω1 ∨ ord ∼ (N∨ → M = Lie∨ ~ ord , H ~ N ∨ /M N /M H

H

~ ord )∗ ON ∼ R (N → M = LieN∨ /M ~ ord , H 1

H

~ ord )∗ ON∨ ∼ R1 (N∨ → M = LieN/M ~ ord . H H

The relative de Rham cohomology ~ ord ) := Ri (N → M ~ ord )∗ (Ω• ord ) H idR (N/M H H ~ N/M H

and its Hodge filtration and Gauss–Manin connection ∇ are canonically isomorphic ~ ord ). to those of H idR (Ngrp /M H In Theorem 1.3.3.15, we used the isomorphisms in Corollary 1.3.3.13 and denoted, for example, the extension of HomO (Q, LieGM /MH ) to Mtor as H H × HomO (Q, LieG/Mtor ). However, such isomorphisms involve Q -isogenies which H might not induce isomorphisms between Lie algebras (or their duals) in mixed characteristics, and this is indeed a concern because we allow the residue characteristics to ramify in the integral PEL datum. Therefore, we need to be more precise

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in the sheaves of modules we use. Indeed, it is now better to have not only the ~ ord,tor of G ~ ord → M ~ ord , but also the semi-abelian semi-abelian extension G → M H H MH grp ord ~ extensions of the Kuga families N → MH as in Definition 7.1.3.1, their torsors ~ ord as in Definition 7.1.3.2, and the duals of all these, to M ~ ord,tor . N→M H

H

◦ By Proposition 3.1.3.4, the abelian subscheme HomO (Q, GM ~ ord ) of the abelian H × rk (Q) ~ ord extends to the semi-abelian subscheme Hom (Q, G ~ ord ) ∼ over M = G Z Z

MH ◦

~ ord M H

H

scheme HomO (Q, G) of the semi-abelian scheme HomZ (Q, G) ∼ = G× rkZ (Q) over ord,tor ~ MH . ~ ord be a generalized ordinary Kuga family as in Definition 7.1.3.2, Let N → M H ~ ord as in Definition which is a torsor under some ordinary Kuga family Ngrp → M H ◦ grp × ~ ord . over M 7.1.3.1, together with a Q -isogeny h : Z := HomO (Q, GM ~ ord ) → N H H tor ◦ Let Z := HomO (Q, G) be as in Proposition 3.1.3.4. By definition, there exist ~ ord is noetherian normal, an integer N ≥ 1 such that N h is an isogeny. Since M H tor ~ ord,tor , and we by Lemma 3.1.3.2, N h extends to an isogeny Z → Next over M H × tor −1 formally define the Q -isogeny h to be N times this extended isogeny. (Since ~ ord is noetherian normal, this is well defined by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], M H or [62, Prop. 3.3.1.5].) ~ ord,tor Definition 7.1.3.11. We can say that the semi-abelian scheme Next → M H ord,tor ~ is the extended ordinary Kuga family over M . It is determined (up to H ~ ord . (It does not depend on the ~ ord to M isomorphism) by its restriction Ngrp → M H H ord grp ord ~ ~ structure of N → MH as a torsor of N → MH .) By [80, IV, 7.1] (see also [62, Thm. 3.4.3.2]), there is also a dual semi-abelian ~ ord,tor extending the dual abelian scheme Ngrp,∨ → M ~ ord . scheme Next,∨ → M H H 7.1.4

Main Statements

The partial toroidal compactifications of Kuga families and their generalizations can be described as follows: Theorem 7.1.4.1. (Compare with [61, Thm. 2.15] and Theorem 1.3.3.15.) Let Q be as in Theorem 1.3.3.15. Let H = Hp Hp be as at the beginning of Section 3.3.5, and let rH be as in Definition 3.4.2.1. Suppose that Hp is neat, and that Σord is as ~ ord,tor = M ~ ord,tor in Definition 5.1.3.1, so that (by Theorem 5.2.1.1) M is an algebraic H H,Σord space separated, smooth, and of finite type over ~S0,rH . (By Theorem 6.2.3.1, if Σord ~ ord,tor ~ is projective as in Definition 5.1.3.3, then M ord is quasi-projective over S0,r .) H,Σ

Consider the abelian scheme GM ~ ord H

H

~ ord in (1) of Theorem 5.2.1.1. Consider over M H

ord,+ ord,++ ord,+ ord,++ the sets Kord and Kord Q,H ⊂ KQ,H ⊂ KQ,H Q,H,Σord ⊂ KQ,H,Σord ⊂ KQ,H,Σord as in Definitions 7.1.1.11 and 7.1.1.19, with compatible directed partial orders. These sets parameterize the following data:

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b Σ b ord ) ∈ Kord,++ , let Hκ := H bG (which satisfies Condition (1) For each κ = (H, Q,H 7.1.1.4 and is contained in H; see Definition 1.2.4.4) and rκ := rHκ , so that ~ ord is a quasi-finite ´etale cover of M ~ ord := M ~ ord × ~S0,r , inducing a M κ Hκ H,rκ H ~ S0,rH

~ ord M Hκ

~ ord , M H

quasi-finite flat morphism → which is finite when Hκ and H are equally deep as in Definition 3.2.2.9. (This is the case, for example, b Σ b ord ) ∈ Kord,+ and hence Hκ = H.) when κ = (H, Q,H ~ ord → M ~ ord (see Then there is an generalized ordinary Kuga family N Hκ

κ

Definition 7.1.3.2), which is a torsor under an ordinary Kuga family ~ ord,grp → M ~ ord (see Definition 7.1.3.1) with a Q× -isogeny N κ Hκ ◦ ~ ord,grp κisog : HomO (Q, GM ~ ord ) → Nκ Hκ

of abelian schemes over sion

~ ord , M Hκ

together with an open fiberwise dense immer-

~ ord ,→ N ~ ord,tor κtor : N κ κ ~ ord,tor is quasi-projective of schemes over ~S0,rκ , such that the scheme N κ ~ ord in N ~ ord,tor and smooth over ~S0,rκ , and such that the complement of N κ κ ord ~ (with its reduced structure) is a relative Cartier divisor E∞,κ with simple normal crossings. ~ ord,tor has a stratification by locally closed subschemes The scheme N κ a ~bord ~ ord,tor = N Z[(Φ˘ c,δ˘c,bτ )] , κ H

H

˘ c,δ˘c,b [(Φ τ )] H H

˘ b , δ˘ b , τb)] running through a complete set of equivalence classes of with [(Φ H H ˘ b , δ˘ b , τb) (as in Lemma 1.2.4.42) with the underlying Z ˘ b (suppressed in (Φ H H H b + and the notation by our convention) compatible with D, and with τb ⊂ P ˘c Φ H b ˘ ∈ Σ. b (The notation “`” only means a set-theoretic disjoint τb ∈ Σ ΦH c ~ ord,tor .) In this union. The algebro-geometric structure is still that of N ord b ˘0 ˘ 0 , δ˘0 , τb0 )]-stratum ~Z stratification, the [(Φ [(Φ b H b H

κ

,δ˘0c,b τ 0 )]

c H

ord

lies in the closure of

H

b ˘ ˘ ˘ b , δ˘ b , τb)] is a face of ˘ b , δ˘ b , τb)]-stratum ~Z the [(Φ [(ΦH τ )] if and only if [(ΦH H H H c,δH c,b ˘ 0 , δ˘0 , τb0 )] as in Lemma 1.2.4.42. The analogous assertion holds after [(Φ b b H H ord ~ ord = ~Z b pulled back to fibers over ~S0,rκ . In particular, N κ [(0,0,{0})] is an open fiberwise dense stratum in this stratification. ord b ˘ ˘ ~ ˘ b , δ˘ b , τb)]-stratum ~Z The [(Φ [(Φ c,δ c,b τ )] is smooth over S0,rκ and isomorphic H H H

H

~b ord to the support of the formal scheme X ˘ c,δ˘c,b Φ τ (see (7.1.2.15)) for every H H ˘ ˘ ˘ ˘ representative (ΦHb , δHb , τb) of [(ΦHb , δHb , τb)], which is the completion of an ~b ord ~b ord affine toroidal embedding Ξ τ ) (along its τb-stratum Ξ ˘ c,δ˘c,b ˘ c,δ˘c (b Φ Φ τ ) of a H

H

H

H

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~b ord ~b ord torus torsor Ξ ˘ c,δ˘c over a finite ˘ c,δ˘c over an abelian scheme torsor C Φ Φ H

H

H

˘

H

˘

c c ~b ord,ΦH ~b ord,ZH ´etale cover M of the scheme M (quasi-projective over ~S0,rκ ) in b b H H Lemma 7.1.2.1 and Proposition 7.1.2.6. ord ~ ord,tor )∧ord ~ ord,tor along ~Z b ˘ ˘ The formal completion (N of N κ κ [(Φ c,δ c,b τ )] ~

b ˘ Z [(Φ

H

˘ ,τ c,δH c b )] H

H

ord

~b is canonically isomorphic to X ˘ c,δ˘c,b Φ τ; H

~ ord,tor )∧ ord (N κ ~ b ˘ ∪Z [(Φ

ord

H

and the formal completion

ord

˘ ,τ c,δH c b )] H

~b , where ∪ Z ˘ c,δ˘c,b [(Φ τ )] is the union of all strata H

H

~b ~b ord b ˘ , is canonically isomorphic to X Z[(Φ˘ c,δ˘c,bτ )] with τb ∈ Σ ˘ c,δ˘c /ΓΦ ˘ c (cf. Φ Φ c H H H H H H (5) of Theorem 5.2.1.1 and Lemma 5.2.4.38). (Such isomorphisms can be induced by strata-preserving isomorphisms between ´etale neighborhoods ~bord ~ ord,tor and ´etale neighborhoods of points of of points of Z ˘ c,δ˘c,b [(Φ τ )] in Nκ H

H

~b ord ~b ord τ ).) Ξ ˘ c,δ˘c (b ˘ c,δ˘c,b Φ τ in ΞΦ H H H H ord,tor ~ ord,min extending ~ ord,tor → M ~ admits a canonical surjection N Each Nκ κ H ord ord ~ , and the latter is the pullback of ~ → M the canonical surjection N κ H ~ ord ,→ M ~ ord,min on the tarthe former under the canonical morphism M H H ~ ord ~ ord,min and N ~ ord → M ~ ord,tor → M get (see Theorem 6.2.1.1). Both N κ

κ

H

H

are proper when Hκ and Hκ0 are equally deep as in Definition 3.2.2.9. ord b ˘ ˘ ~ ord,tor to ˘ b , δ˘ b , τb)]-stratum ~Z Such a morphism maps the [(Φ [(ΦH τ )] of Nκ H H c,δH c,b ord,min ~ the [(ΦH , δH )]-stratum ~Zord of M if and only if the cusp label [(ΦH ,δH )]

H

˘ b , δ˘ b )] as in Lemma 1.2.4.15. [(ΦH , δH )] is assigned to the cusp label [(Φ H H ord,+ ~ ord . If ~ ord = M If κ ∈ KQ,H , then Hκ = H, and hence rκ = rH and M H Hκ ~ ord is an ordinary Kuga ~ ord = M ~ ord,grp → M ~ ord = N κ ∈ Kord , then N Q,H

κ

κ

Hκ

H

family. b0 , Σ b ord,0 )  κ = (H, b Σ) b in Kord,++ , we have For each relation κ0 = (H Q,H b 0 ⊂ Hκ = H bG and hence rκ0 ≥ rκ ; and there is a surjection Hκ0 = H G ~ ord,tor → N ~ ord,tor fκtor 0 ,κ : Nκ0 κ extending a canonical quasi-finite flat surjection ~ ord ~ ord fκ0 ,κ : N κ0 → Nκ , ~ ord,tor ~ ord,tor × ~S0,r 0 exinducing a canonical log ´etale surjection N →N κ κ0 κ ~ S0,rκ

~ ord ~ ord × M ~ ord equivaritending a canonical finite ´etale surjection N κ 0 → Nκ Hκ0 ~ ord M Hκ

×

ant with the canonical Q -isogeny isog ~ ord,grp ~ ord,grp × M ~ ord , fκgrp ◦ ((κ0 )isog )−1 : N →N 0 ,κ := κ κ Hκ0 κ0 ~ ord M Hκ

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such that Ri (fκtor 0 ,κ )∗ ONtor = 0 for i > 0. These surjections are compatible κ0 ~ ord,min . The morphism f tor with the canonical morphisms to M κ0 ,κ is proper H ~ ord ~ ord is finite ´etale, if Hκ and log ´etale, and the morphism fκ0 ,κ : N →N 0 κ

κ

Hκ0 are equally deep as in Definition 3.2.2.9. (2) For each κ ∈ Kord,++ , the structural morphism Q,H,Σord ~ ord → M ~ ord = M ~ ord × ~S0,r fκ : N κ H,rκ H κ ~ S0,rH

~ ord → M ~ ord in (1) with the (the composition of the structural morphism N κ Hκ ord ord ~ ~ canonical morphism MHκ → MH,rκ ) extends (necessarily uniquely) to a surjection ~ ord,tor ~ ord,tor → M ~ ord,tor = M ~ ord,tor =M × ~S0,r , f tor : N ord ord κ

κ

H,rκ

H,Σ

H,Σ

,rκ

~ S0,rH

κ

~ ord,tor which is log smooth (as in [45, 3.3] and [43, 1.6]) if we equip N κ and Mtor H,rκ with the canonical (fine) log structures given respectively by the relative Cartier divisors with (simple) normal crossings ~Eord ∞,κ and ord ord ~ ~ ~ D∞,H,rκ := D∞,H × S0,rκ (see (1) above and (3) of Theorem 5.2.1.1). ~ S0,rH

Then we have the following commutative diagram: +NCD ~ ord,tor ~ ord   /N N κ κ smooth surjective

log smooth surjective

fκ

 ~ ord   M H,rκ

+NCD

quasi-projective smooth

fκtor

 ~ ord,tor /M H,rκ

' separated smooth of finite type

/ ~S0,rκ

ord b ˘ ˘ ~ ord,tor to ˘ b , δ˘ b , τb)]-stratum ~Z The morphism fκtor maps the [(Φ [(ΦH τ )] of Nκ H H c,δH c,b ord,tor ~ the [(ΦH , δH , τ )]-stratum ~Zord of M if and only if (the cusp label [(ΦH ,δH ,τ )]

H

˘ b , δ˘ b )] as in Lemma 1.2.4.15 and) [(ΦH , δH )] is assigned to the cusp label [(Φ H H b the image of τb ∈ ΣΦ˘ c under (1.2.4.37) is contained in τ ∈ ΣΦH as in CondiH

~b ord ~ ord tion 1.2.4.49. In this case, the compatible morphisms X ˘ c,δ˘c,b Φ τ → XΦH ,δH ,τ H

~b ord and X ˘ c,δ˘c,Σ b˘ Φ Φ H

H

c H

~ ord →X ΦH ,δH ,ΣΦ

H

induced by

fκtor

H

(and the canonical isomor-

phisms in (1) above and in (5) of Theorem 5.2.1.1) coincide with the canonical morphisms as in (7.1.2.33) and (7.1.2.35). (These morphisms can be induced by compatible morphisms between ´etale neighborhoods of points of the supports of formal schemes in relevant ambient schemes as in (1) above, compatible with all stratifications.) ~ ord → M ~ ord and f tor : N ~ ord,tor → M ~ ord,tor are proper The morphisms fκ : N κ κ κ Hκ H,rκ when Hκ and H are equally deep as in Definition 3.2.2.9 (e.g., when κ ∈ Kord,+ ; cf. (1) above). Q,H,Σord

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Compactifications with Ordinary Loci tor tor If κ0  κ, then we have the compatibility fκtor = fκ,r ◦fκtor 0 ,κ , where fκ,r 0 := 0 κ0 κ tor f × ~S0,r 0 . κ

κ

~ S0,rκ

(3) Suppose κ ∈ Kord,+ (not just in Kord,++ , so that Hκ = H and rκ = Q,H,Σord Q,H,Σord ~ ~ rH , and so that the base change from S0,rH to S0,rκ in (2) is unnecessary). In this case, we also consider as in Definition 7.1.3.11 the extended or~ ord,ext → M ~ ord,tor , a semi-abelian scheme extendinary Kuga family N κ H ~ ord,grp → M ~ ord , together with the semiding the ordinary Kuga family N κ H ord,tor ord,ext,∨ ~ ~ abelian scheme Nκ → MH extending the dual abelian scheme ~ ord,grp,∨ → M ~ ord , so that the locally free sheaves Lie~ ord,ext ~ ord,tor , N κ H Nκ /MH ∨ ~ ord,tor Lie∨ , and Lie , Lie over M ord,ext,∨ ord,tor ord,ext ord,tor ord,ext,∨ ord,tor ~ ~ ~ ~ ~ ~ Nκ

Nκ

/MH

/MH

Nκ

H

/M H

∨ extend the locally free sheaves LieN ~ ord /M ~ ord , LieN ~ ord,∨ ~ ord , and ~ ord /M ~ ord , LieN /M κ κ H H κ H ∨ ~ ord , respectively. (See Definition 7.1.3.9.) LieN over M ord ~ ord,∨ ~ H /MH κ For simplicity, let us suppress the subscripts “κ” from the notation. (All canonical isomorphisms will be required to be compatible with the canonical 0 isomorphisms defined by pulling back under fκtor 0 ,κ for each relation κ  κ ord,+ in KQ,H,Σ .) Then the following are true:

(a) Let Ω1N ~ ord,tor /~ S

0,rH

[d log ∞] and Ω1M ~ ord,tor /~ S H

[d log ∞] denote the shea0,rH

ves of modules of log 1-differentials over ~S0,rH given by the (respective) canonical log structures defined in (2). Let 1

ΩN ~ ord,tor /M ~ ord,tor H := (Ω1N ~ ord,tor /~ S

0,rH

[d log ∞])/((f tor )∗ (Ω1M ~ ord,tor /~ S H

Then there is a canonical isomorphism ∼ 1 (f tor )∗ Lie∨ ~ ord,ext ~ ord,tor = Ω~ ord,tor N

N

/MH

0,rH

[d log ∞])). (7.1.4.2)

~ ord,tor /M H

~ ord,tor , extending the canonical isobetween locally free sheaves over N morphism ∼ 1 f ∗ Lie∨ (7.1.4.3) ~ ord /M ~ ord = ΩN ~ ord /M ~ ord N H

H

~ ord (see Lemma 7.1.3.10). over N (b) For each integer b ≥ 0, there exist canonical isomorphisms a

Rb f∗tor (ΩN ~ ord,tor /M ~ ord,tor ) H ∼ = (∧b LieN ~ ord,ext,∨ /M ~ ord,tor ) H

⊗

O~ ord,tor M

(∧a Lie∨ ~ ord,ext /M ~ ord,tor ). N

(7.1.4.4)

H

H

and a

Rb f∗tor (ΩN ~ ord,tor /M ~ ord,tor H

⊗

O~Nord,tor

a ∼ = Rb f∗tor (ΩN ~ ord,tor /M ~ ord,tor ) H

I~Eord )

⊗

∞

O~ ord,tor M H

ID ~ ord

∞,H

(7.1.4.5)

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397

~ ord,tor , where I~ ord (resp. I~ ord ) is the of locally free sheaves over M H D E ∞

∞,H

ON ~ ord,tor -ideal (resp. OM ~ ord,tor -ideal) defining the relative Cartier diH ord ord ~ ~ ~ ord ) (with its reduced structure), compavisor E =E (resp. D ∞

∞,κ

∞,H

tible with cup products and exterior products, extending the canonical ~ ord induced by the canonical isomorphism isomorphism over M H (7.1.4.6) Rb f∗ (O~ ord ) ∼ = ∧b Lie~ ord,∨ ~ ord .

(c)

N /M H N • 1 • Let ΩN ~ ord,tor /M ~ ord,tor := ∧ ΩN ~ ord,tor /M ~ ord,tor be the log de Rham complex H H tor ord,tor ~ ~ associated with f :N → Mord,tor (with differentials inherited H from Ω•N ). Let the (relative) log de Rham cohomology be ~ ord /M ~ ord H

defined by ~ ord,tor /M ~ ord,tor ) := Ri f tor (Ω•~ ord,tor ~ ord,tor ). H ilog-dR (N ∗ N /M H H Then the (relative) Hodge spectral sequence a

a+b ~ ord,tor /M ~ ord,tor ) E1a,b := Rb f∗tor (ΩN ~ ord,tor /M ~ ord,tor ) ⇒ H log-dR (N H H (7.1.4.7) degenerates at E1 terms, and defines a Hodge filtration on ~ ord,tor /M ~ ord,tor ) with locally free graded pieces given by H ilog-dR (N H a Rb f∗tor (ΩN ~ ord,tor /M ~ ord,tor ) for integers a + b = i, extending the canonical H ~ ord /M ~ ord ). Hodge filtration on H i (N H

dR

As a result, for each integer i ≥ 0, there is a canonical isomorphism ∼ ~ ord,tor /M ~ ord,tor ) → ~ ord,tor /M ~ ord,tor ), ∧i H 1log-dR (N H ilog-dR (N H H compatible with the Hodge filtrations defined by (7.1.4.7), extending the canonical isomorphism ∼ ~ ord /M ~ ord ) → ~ ord /M ~ ord ) ∧i H 1 (N H i (N H

dR

H

dR

~ ord (defined by cup product). over M H (d) For each jQ : Q∨ ,→ Q as in Lemma 1.2.4.1, the Q× -polarization λMH ,jQ in Proposition 1.3.3.7 extends canonically to a Q× -polarization ∨

◦ ◦ λM ~ ord ,jQ : HomO (Q, GM ~ ord ) → (HomO (Q, GM ~ ord ) ) , H

H

H

which induces a Q× -polarization ~ ord,grp → N ~ ord,grp,∨ λN ~ ord ,jQ : N and defines canonically (as in [23, 1.5]) a perfect pairing ~ ord /M ~ ord ) ⊗ Q) ⊗ Q : (H 1 (N h · , · iλ ~ ord ,j M Q H

H

dR

Z

Z

~ ord /M ~ ord ) ⊗ Q) ×(H 1dR (N H Z

→ OM ~ ord (1) ⊗ Q. H

(This is an abuse of notation because h · , · iλM~ ord ,j H

Z

is not yet defined.) Q

~ ord,tor /M ~ ord,tor ) is (under the restriction morphism) Then H 1log-dR (N H canonically isomorphic to the unique subsheaf of ~ ord ,→ M ~ ord,tor )∗ (H 1 (N ~ ord /M ~ ord )) (M H

H

satisfying the following conditions:

dR

H

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~ ord,tor /M ~ ord,tor ) is locally free of finite rank over i. H 1log-dR (N H OM ~ ord,tor . H

1

ii. The sheaf f∗tor (ΩN ~ ord,tor /M ~ ord,tor ) can be identified with the subsheaf H ord,tor ~ ord ,→ M ~ of (M )∗ (f∗ (Ω1N H H ~ ord /M ~ ord )) formed (locally) by sectiH

~ ord,tor /M ~ ord,tor ). (Here ons that are also sections of H 1log-dR (N H ~ ord ,→ we view all sheaves canonically as subsheaves of (M H ord,tor 1 ord ~ ord ~ ~ MH )∗ (H dR (N /MH )).) ~ ord,tor /M ~ ord,tor ) ⊗ Q is self-dual under the push-forward iii. H 1log-dR (N H Z

~ ord ,→ M ~ ord,tor )∗ h · , · iλ (M ⊗ Q. H H ~ ord ,j M H

Q

Z

(e) The Gauss–Manin connection ~ ord /M ~ ord /M ~ ord ) → H • (N ~ ord ) ⊗ Ω1 ord ∇ : H •dR (N H H dR ~ M /~ S OM ~ ord

H

0,rH

(7.1.4.8)

H

extends to an integrable connection ~ ord,tor /M ~ ord,tor ) ∇ : H •log-dR (N H ~ ord,tor /M ~ ord,tor ) → H •log-dR (N H

⊗

O~ ord,tor

(7.1.4.9)

1

ΩM ~ ord,tor /~ S0,r H

H

M H

~ ord , called the extended Gauss–Manin conwith log poles along D ∞,H nection, satisfying the usual Griffiths transversality with the Hodge filtration defined by (7.1.4.7). (4) (Hecke actions; cf. Propositions 3.4.4.1 and 5.2.2.2) Suppose we have b ∞,p ) = × P b ord an element gb = (b g0 , gbp ) ∈ G(A D (Qp ) with image gh = ∞,p ord (gh,0 , gh,p ) ∈ G(A ) × PD (Qp ) (see Definition 3.2.2.7) under the ca∞,p b ∞,p ) × P b ord ) × Pord nonical homomorphism G(A D (Qp ), and D (Qp ) → G(A 0 ˆ such that suppose we have two open compact subgroups H and H of G(Z) −1 0 0 H ⊂ gh Hgh , and such that H and H are of standard form as in Definition 3.2.2.9. Suppose moreover that gh,p satisfies the conditions given in Section ord ~ ord0 → M ~ ord 3.3.4 (with gp there replaced with gh,p here), so that [g~h ] :M H

H

is defined (see Proposition 3.4.4.1). Suppose Σord,0 = {Σ0Φ0 0 }[(Φ0 0 ,δ0 0 )] is a H H H compatible choice of admissible smooth rational polyhedral cone decomposi~ ord0 , which is a gh -refinement of Σord = {ΣΦ }[(Φ ,δ )] as tion data for M H H H H ord,+ ord,++ in Definition 5.2.2.1. Consider the sets Kord ⊂ K ⊂ K and 0 0 Q,H Q,H Q,H0 ord,+ ord,++ ord KQ,H0 ,Σord,0 ⊂ KQ,H0 ,Σord,0 ⊂ KQ,H0 ,Σord,0 as in Definitions 7.1.1.11 and 7.1.1.19 (for H0 and Σord,0 ), with compatible directed partial orders, parameterizing generalized ordinary Kuga families and their compactifications with properties as in (1), (2), and (3) above. The sets Kord,++ etc and Q,H ord,++ KQ,H0 etc (and the objects they parameterize) satisfy the compatibility with gb (and gh ) in the sense that the following are true:

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b Σ b ord ) ∈ Kord,++ (resp. Kord,+ , resp. Kord ), and (a) For each κ = (H, Q,H Q,H Q,H ˆ such that H b 0 ⊂ G( b Z) b 0 ⊂ gbHb b g −1 (so for each open compact subgroup H −1 0 b b that Hκ = HG and Hκ0 = HG satisfy Hκ0 ⊂ gh Hκ gh ), there exists an b0 , Σ b ord,0 ) ∈ Kord,++ element κ0 = (H (resp. Kord,+0 , resp. Kord 0 ) such 0 Q,H

Q,H

Q,H

that there exists a (necessarily unique) quasi-finite flat surjection ~g ]ord : N ~ ord ~ ord [b κ0 → Nκ ord

covering the compatible surjections [g~h ]

(7.1.4.10)

ord ~ ord0 → M ~ ord , [g~h ] :M H H rκ0 ,rκ :

~ ord : M ~ ord0 ~ ord ~ ord → M ~ ord given by ProposiM H ,rκ0 → MH,rκ , and [gh ] Hκ0 Hκ ~ ord ~ ord × M ~ ord tion 3.4.4.1, inducing a finite flat surjection N κ 0 → Nκ Hκ0 ~ ord M Hκ

of abelian scheme torsors equivariant with the isogeny (not just a Q× -isogeny) ~ ord,grp ~ ord,grp × M ~ ord N →N κ Hκ0 κ0 ~ ord M Hκ

induced

by

(κ0 )isog ,

κisog ,

and

the

Q× -isogeny

→

GM ~ ord

H 0 κ

~ ord ~ ord GM ~ ord × MHκ0 realizing GM ~ ord × MHκ0 as an ordinary Hecke twist Hκ

Hκ

~ ord M Hκ

~ ord M Hκ

× of GM ~ ord by gh (which is the pullback of the Q -isogeny GM ~ ord → H0

H 0 κ

GM ~ ord H

~ ord0 as an ordinary Hecke twist ~ ord0 realizing G ~ ord × M × M H H M H

~ ord M H

~ ord M H

~ ord ~ ord of GM ~ ord by gh ). (Here all the base changes from MH to MH0 and H0

ord ~ ord to M ~ ord use the surjections denoted by [g~h ] .) The chafrom M Hκ Hκ0

~g ]ord ⊗ Q is ´etale. racteristic zero pullback [b Z

b Σ b ord ) and H b 0 as in (4a) such that κ ∈ Kord,++ (resp. (b) For each κ = (H, Q,H ord,+ ord 0 0 b ord,0 b K , resp. K ), there is an element κ = (H , Σ ) ∈ Kord,++ 0 Q,H

Q,H

Q,H

~g ]ord is defined as in (4a) (see such that [b (resp. resp. b ord,0 is a gb-refinement of Σ b ord (cf. Lemma (7.1.4.10)), and such that Σ 1.2.4.42 and Definition 5.2.2.1), which extends to a (necessarily unique) surjection Kord,+ Q,H0 ,

Kord Q,H0 )

~g ]ord,tor : N ~ ord,tor ~ ord,tor [b →N κ κ0

(7.1.4.11)

such that ord,tor

~g ] Ri [b ∗

ON ~ ord,tor = 0 κ0

(7.1.4.12)

b ord,0 is gb-induced by Σ b ord (cf. Lemma 1.2.4.42 and for all i > 0. If Σ ord,tor ~g ] Definition 5.2.2.1), then [b is quasi-finite. The characteristic ~g ]ord,tor ⊗ Q is log ´etale. zero pullback [b Z

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Compactifications with Ordinary Loci ord b ˘0 ˘ 0 , δ˘0 , τb0 )]-stratum ~Z Under (7.1.4.11), the [(Φ [(Φ 0 0 b b H H

~ ord,tor of N κ0

,δ˘0c0 ,b τ 0 )] c0 H H

ord b ˘ ˘ ~ ord,tor if and ˘ b , δ˘ b , τb)]-stratum ~Z is mapped to the [(Φ [(ΦH τ )] of Nκ H H c,δH c,b ˘ b , δ˘ b , τb) and (Φ ˘ 0 , δ˘0 , τb0 ) of only if there are representatives (Φ H

H

b0 H

b0 H

˘ b , δ˘ b , τb)] and [(Φ ˘ 0 , δ˘0 , τb0 )], respectively, such that (Φ ˘ 0 , δ˘0 , τb0 ) [(Φ b0 H b0 b0 H b0 H H H H ˘ b , δ˘ b , τb) (cf. Lemma 1.2.4.42 and [62, Def. is a gb-refinement of (Φ H H ~b ord 6.4.3.1]). In this case, the compatible morphisms X → 0 ˘ 0 ,δ˘0 ,b Φ 0 0 τ ~b ord ~b ord X and X ˘ ˘ ˘ 0 ,δ˘0 Φ c,δ c,b τ Φ 0 H

H

b0 ,Σ ˘0 c0 Φ H c0 H

c H

~b ord →X ˘ c,δ˘c,Σ b˘ Φ Φ H

H

c H

c H

induced by (7.1.4.11) (and

c H

the canonical isomorphisms in (1) above) coincide with the canonical morphisms as in (7.1.2.19) and (7.1.2.21). If κ ∈ Kord,++ (resp. Kord,+ , resp. Kord Q,H,Σord ), we may assume in Q,H,Σord Q,H,Σord the above that κ0 ∈ Kord,++ (resp. Kord,+ , resp. Kord Q,H0 ,Σord,0 ), Q,H0 ,Σord,0 Q,H0 ,Σord,0 so that (7.1.4.11) covers the surjection ord,tor ~ ord,tor [g~h ]rκ0 ,rκ : M H0 ,Σord,0 ,r

~ ord,tor →M H,Σord ,rκ

κ0

ord,tor ~ ord,tor ~ ord,tor induced by the surjection [g~h ] : M → M given by H0 ,Σord,0 H,Σord Proposition 5.2.2.2. ~g ]ord,tor is defined as in (4b) for some κ ∈ Kord,+ ord and (c) Suppose [b Q,H,Σ

κ0 ∈ Kord,+ (not just in Kord,++ and Kord,++ ). Then there Q,H0 ,Σord,0 Q,H,Σord Q,H0 ,Σord,0 is a canonical morphism ord,tor

~g ] ([b

~g ]ord,tor )∗ H a+b (N ~ ord,tor /M ~ ord,tor ) )∗ : ([b κ log-dR H,Σord ~ ord,tor ~ ord,tor /M ) → H a+b log-dR (Nκ0 H0 ,Σord,0

respecting the Hodge filtrations and compatible with the canonical morphisms ord,tor

~g ] ([b

~g ]ord,tor )∗ Ω1~ ord,tor ~ ord,tor → Ω1~ ord,tor ~ ord,tor , )∗ : ([b Nκ N 0 /M 0 ord,0 /M ord κ

H,Σ

ord,tor

ord,tor

ord,tor

ord,tor

ord,tor

ord,tor

([g~h ]rκ0 ,rκ )∗ : ([g~h ]

([g~h ]rκ0 ,rκ )∗ : ([g~h ] ([g~h ]rκ0 ,rκ )∗ : ([g~h ]

H ,Σ

∗

) LieN ~ ord,ext,∨ ~ ord,tor → LieN ~ ord,ext,∨ /M ~ ord,tor /M κ κ0

H,Σord

H0 ,Σord,0

∨ )∗ Lie∨ ~ ord,ext /M ~ ord,tor → LieN ~ ord,ext /M ~ ord,tor N κ

H,Σord

κ0

H0 ,Σord,0

)∗ LieG∨ /M ~ ord,tor → LieG∨ /M ~ ord,tor H,Σord

ord,tor ord,tor ∨ ([g~h ]rκ0 ,rκ )∗ : ([g~h ] )∗ Lie∨ ~ ord,tor → LieG/M ~ ord,tor G/M H,Σord

,

H0 ,Σord,0

,

H0 ,Σord,0

~ ord,tor and N ~ ord,tor and the canonical isomorphisms in (3) for N . The κ κ0 characteristic zero pullbacks of these canonical morphisms are isomorphisms.

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(d) If the levels Hκ,p and Hκ0 ,p at p are equally deep as in Definition b 3.2.2.9 (by Remark 7.1.1.3, this is equivalent to the condition that H 0 b and H are equally deep as in Definition 7.1.1.2), or if gh,p is of twisted Up type as in Definition 3.3.6.1 and depthD (Hκ0 ,p ) − depthD (gh,p ) = depthD (Hκ,p ) > 0, then the surjection (7.1.4.10) is finite, and the surjection (7.1.4.11) is proper. (e) If gh,p ∈ Pord D (Zp ), then the morphism ~g ]ord : N ~ ord × ~S0,r 0 ~ ord [b κ0 → Nκ rκ0 κ ~ S0,rκ

~g ]ord : N ~ ord ~ ord is ´etale, and the surjection canonically induced by [b κ0 → Nκ ~g ]ord,tor : N ~ ord,tor ~ ord,tor × ~S0,r 0 [b →N κ κ0 rκ0 κ ~ S0,rκ

ord,tor

~g ] ~ ord,tor ~ ord,tor is log ´etale, canonically induced by [b : N → N κ κ0 b 0 is and the canonical morphisms in (4c) are all isomorphisms. If Σ ord,tor ~g ] b then [b gb-induced by Σ, is quasi-finite ´etale (not just log ´etale). rκ 0 0 b ∞,p ) × P b ord (f ) If we have an element gb ∈ G(A D (Qp ) with image 0 ∞,p ord gh ∈ G(A ) × PD (Qp ) under the canonical homomorphism ∞,p b ∞,p ) × P b ord ) × Pord G(A D (Qp ), with a similar setup such D (Qp ) → G(A ord ord,tor ord ord ~ ~ 0 0 ~ 00 → N ~ 0 and [b ~ ord,tor ~ ord,tor that [b g] : N g] : N → N are 00 0 κ

κ

κ

κ

ord ~ ord compatibly defined for some κ00 ∈ Kord,++ g~0 gb] : N κ00 → Q,H00 , then [b ord,tor ~ ord,tor ~ ord,tor are also compatibly defined ~ ord and [b g~0 gb] : N → N N κ κ κ00 ord ord ord,tor ~ ~g ]ord ◦ [b 0 and satisfy the identities [b g gb] = [b g~0 ] and [b g~0 gb] =

ord,tor ord,+ ord,+ 0 00 ~g ]ord,tor ◦ [b [b g~0 ] . If κ ∈ Kord,+ Q,H , κ ∈ KQ,H0 , and κ ∈ KQ,H00 , ord ord ord,tor ~g ]ord )∗ and ([b we also have ([b g~0 gb] )∗ = ([b g~0 ] )∗ ◦ ([b g~0 gb] )∗ = ord,tor

~g ] ([b g~0 ] )∗ ◦ ([b (g) The morphism

ord,tor

)∗ (in both applicable senses above). ~g ]ord : N ~ ord ~ ord [b κ0 → Nκ

(7.1.4.13)

(cf. Definition 3.4.4.2) induced by (7.1.4.10) is finite. The morphism ~g ]ord,tor : N ~ ord,tor ~ ord,tor [b →N κ κ0

(7.1.4.14)

b 0 is gb-induced by induced by (7.1.4.11) is proper, and is finite flat if Σ ord b b Σ. If gbp ∈ PD (Zp ), then the morphism ~g ]ord,tor : N ~ ord,tor ~ ord,tor × S ~ 0,r 0 [b →N κ κ0 rκ 0 κ

(7.1.4.15)

~ 0,r S κ

induced by (7.1.4.14) is proper log ´etale (because it is log ´etale by (4e)). b 0 is gb-induced by Σ, b then (7.1.4.15) is finite ´etale (because If moreover Σ it is quasi-finite ´etale by (4e); cf. [35, IV-3, 8.11.1]).

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ˆ p) × P b Z b ord b 0,p = gb0 H b p gb−1 , if (H bp0 )ord = (h) If gb = (b g0 , gbp ) ∈ G( D (Zp ), if H 0 bp gbp−1 )ord (cf. (7.1.1.24)), and if Σ b 0 is gb-induced by Σ, b then (b gp H (rκ0 = rκ and) the induced morphisms (7.1.4.13) and (7.1.4.14) are isomorphisms. (These conditions are true, in particular, when gb = 1 b = H bp H bp and H b0 = H b 0,p H bp0 satisfy H b 0,p = H b p and and when H bp0 )ord = H bpord ; cf. see the remark at the end of Corollary 3.4.4.4.) (H (i) (elements of Up type.) Suppose gb0 = 1 and gbp is the image of an e ord,0 (Qp ) under the canonical morphism element gep of Up type in P e Z,e D ord,0 ord e b PeZ,eD (Qp ) → PD (Qp ) (cf. Definition 7.1.1.22). Then gh,0 = 1, gh,p b ord,0 is also a 1-refinement of Σ b ord . is an element of Up type, and Σ The morphism ~g ]ord : N ~ ord ~ ord [b κ0 ⊗ Fp → Nκ ⊗ Fp Z

(7.1.4.16)

Z

induced by (7.1.4.10) is finite flat and coincides with the composition of the (finite flat) absolute Frobenius morphism ~ ord ~ ord F~ ord :N 0 ⊗ Fp → N 0 ⊗ Fp κ

Nκ0 ⊗ Fp

κ

Z

Z

Z

with the canonical finite flat morphism ~ ord : N ~ ord ~ ord [1] κ0 ⊗ Fp → Nκ ⊗ Fp . Z

(7.1.4.17)

Z

On the other hand, the morphism ~g ]ord,tor : N ~ ord,tor ⊗ Fp ~ ord,tor ⊗ Fp → N [b κ κ0 Z

(7.1.4.18)

Z

induced by (7.1.4.11) is proper and coincides with the composition of the (finite flat) absolute Frobenius morphism ~ ord,tor ~ ord,tor ⊗ Fp F~ ord,tor :N ⊗ Fp → N 0 0 Nκ0

κ

⊗ Fp Z

κ

Z

Z

with the canonical proper morphism ~ ord,tor : N ~ ord,tor ~ ord,tor ⊗ Fp . [1] ⊗ Fp → N κ κ0 Z

(7.1.4.19)

Z

b ord,0 is gb-induced by Σ b ord . Then Σ b ord,0 is also Suppose moreover that Σ ord b 1-induced by Σ , and the above morphisms (7.1.4.18) and (7.1.4.19) are finite flat. b 0 )ord = H b ord as open compact subgroups of M b ord (Zp ) (see If (H p p D ord,0 b b ord , so that (7.1.1.24)), then we can take Σ to be g-induced by Σ (rκ0 = rκ and) the canonical morphism (7.1.4.19) is an isomorphism by (4h), and so that the composition ord,tor −1

~ ([1]

~ ord,tor ⊗ Fp N κ

∼

)

~ ord,tor N ⊗ Fp κ0

→

Z

ord,tor ~ [b g]

→

~ ord,tor ⊗ Fp N κ

Z

Z

coincides with the (finite flat) absolute Frobenius morphism ~ ord,tor ⊗ Fp → N ~ ord,tor ⊗ Fp . F~ ord,tor :N Nκ

⊗ Fp Z

κ

κ

Z

Z

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(j) Suppose gb0 = 1 and gbp is the image of an element gep of Up type in e ord,0 (Qp ) under the canonical morphism P e ord,0 (Qp ) → P b ord P D (Qp ) (cf. e e Z,e D Z,e D Definition 7.1.1.22), so that gh,0 = 1 and gh,p is an element of PD (Qp ) of Up type (cf. Definition 3.3.6.1). b Σ b ord ) ∈ Kord,+ ord and κ0 = (H b0 , Σ b ord,0 ) ∈ Suppose that κ = (H, Q,H,Σ ord,+ b0 b K 0 ord,0 , that depthD (H ) − 1 = depthD (H) > 0 (see Definition Q,H ,Σ

bp0 )ord = H bpord as open compact subgroups of M b ord 7.1.1.2), that (H D (Zp ) ord,0 b b that depthD (H0 ) − 1 = (see (7.1.1.24)), that Σ is gb-induced by Σ, depthD (H) > 0 (see Definition 3.2.2.9), that (Hκ0 ,p )ord = (Hκ,p )ord as ord,0 open compact subgroups of Mord is D (Zp ) (see (3.3.3.5)), and that Σ ord gh -induced by Σ as in Definition 5.2.2.1. Then (rκ0 = rκ = rH0 = rH and) the canonical morphism ord,tor ~ ord,tor ~ ord,tor →M [g~h ] :M H0 ,Σord,0 H,Σord

and (7.1.4.11) are finite flat surjections, which induce (as in Corollary 5.2.2.5 and statement (4i) above, by composition with inverses of canonical forgetful isomorphisms) the absolute Frobenius morphisms FM ~ ord,tor ⊗ Fp and FN ~ ord,tor ⊗ Fp , respectively; and (7.1.4.11) induces a κ H,Σord Z

Z

finite surjection ord,tor ~ ord,tor := N ~ ord,tor ~ ord,tor → ([g~h ] )∗ N N κ κ κ0

× ~ ord,tor M ord

~ ord,tor , M H0 ,Σord,0

H,Σ

(7.1.4.20) which induces the relative Frobenius morphism F(N ~ ord,tor ~ ord,tor ⊗ Fp )/(M κ

⊗ Fp )

:

H,Σord Z

Z

~ ord,tor ⊗ Fp → F∗~ ord,tor N κ M

⊗ Fp

H,Σord Z

Z


~ ord,tor ⊗ Fp . N κ Z

(5) (Q× -isogenies.) Let gl = (gl,0 , gl,p ) be an element of GLO ⊗ A∞,p (Q ⊗ A∞,p ) × GLO ⊗ Zp (Q ⊗ Qp ) = GLO ⊗ A∞ (Q ⊗ A∞ ). Z

Z

Z

Z

Z

Z

ˆ in Q ⊗ A∞ determines a unique O-lattice Q0 Then the submodule gl (Q ⊗ Z) Z

Z

(up to isomorphism), together with a unique choice of an isomorphism ∼

[gl ]Q : Q ⊗ Q → Q0 ⊗ Q, Z

Z ∼

ˆ with inducing an isomorphism Q ⊗ A∞ → Q0 ⊗ A∞ matching gl (Q ⊗ Z) Z

Z

Z

ˆ (and in particular gl (Q ⊗ Zp ) = Q ⊗ Zp with Q0 ⊗ Zp if gl,p ∈ Q0 ⊗ Z Z

Z

Z

GLO ⊗ Zp (Q ⊗ Zp )), and inducing a canonical Q× -isogeny Z

Z

Z

◦ ◦ [gl ]∗Q : HomO (Q0 , GM ~ ord ) → HomO (Q, GM ~ ord ) H

H

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Compactifications with Ordinary Loci ord,+ ord,++ defined by [gl ]Q . Consider the sets Kord and Q0 ,H ⊂ KQ0 ,H ⊂ KQ0 ,H ord,+ ord,++ ord KQ0 ,H,Σord ⊂ KQ0 ,H,Σord ⊂ KQ0 ,H,Σord as in Definitions 7.1.1.11 and 7.1.1.19 (with Q replaced with Q0 ), with compatible directed partial orders, parameterizing generalized ordinary Kuga families and their compactifications with properties as in (1), (2), and (3) above. The sets Kord,++ etc and Q,H ord,++ KQ0 ,H etc (and the objects they parameterize) satisfy the compatibility with gl in the sense that the following are true: b Σ b ord ) ∈ Kord,++ (resp. Kord,+ , resp. Kord ), (a) For each κ = (H, Q,H

Q,H

Q,H

b0 , Σ b ord,0 ) ∈ Kord,++ there is an element κ0 = (H (resp. Kord,+ , Q0 ,H,Σord Q0 ,H,Σord ord 0 b ⊂ Hκ = H bG , such that the resp. KQ0 ,H,Σord ) such that Hκ0 = H G × Q -isogeny ∗,ord,grp ~ ord,grp ~ ord,grp × M ~ ord [g~l ]κ0 ,κ := κisog ◦ [gl ]∗Q ◦ ((κ0 )isog )−1 : N →N κ Hκ0 κ0 ~ ord M Hκ

is an isogeny (not just a quasi-isogeny), and such that there is a (necessarily unique) quasi-finite flat surjection ∗,ord ~ ord ~ ord [g~l ]κ0 ,κ : N κ0 → Nκ

(7.1.4.21)

~ ord ~ ord × M ~ ord of abelian inducing a finite flat surjection N → N κ κ0 Hκ0 ~ ord M Hκ ∗,ord,grp

scheme torsors equivariant with the isogeny [g~l ]κ0 ,κ ∗,ord teristic zero pullback [g~l ] 0 ⊗ Q is finite ´etale. κ ,κ

. The charac-

Z

b Σ b ord ) as in (5a), there is an element κ0 = (b) For each κ = (H, ord ~ ∗,ord b0 , Σ b ord,0 ) ∈ Kord,++ (H (resp. Kord,+ Q0 ,H Q0 ,H , resp. KQ0 ,H ) such that [gl ]κ0 ,κ is defined as in (5a) (see (7.1.4.21)) and extends to a (necessarily unique) surjection ∗,ord,tor ~ ord,tor ~ ord,tor , [g~l ]κ0 ,κ :N →N κ κ0

(7.1.4.22)

∗,ord,tor Ri ([g~l ]κ0 ,κ )∗ ON ~ ord,tor = 0

(7.1.4.23)

such that κ0

∗,ord,tor for all i > 0. The characteristic zero pullback [g~l ]κ0 ,κ ⊗ Q is proper Z

log ´etale. If κ ∈ Kord,++ (resp. Kord,+ , resp. Kord Q,H,Σord ), then we may Q,H,Σord Q,H,Σord assume in the above that κ0 ∈ Kord,++ (resp. Kord,+ , resp. Q0 ,H,Σord Q0 ,H,Σord ord KQ0 ,H,Σord ). Then (7.1.4.22) is compatible with the canonical mor~ ord,tor → M ~ ord,tor , f tor ~ ord,tor ~ ord,tor , and phisms f tor : N : N → M 0 0 κ

κ

~ ord,tor . ~ ord,tor → M M H,rκ0 H,rκ

H,rκ

κ

κ

H,rκ0

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∗,ord,tor (c) Suppose [g~l ]κ0 ,κ is defined as in (5b) for some κ ∈ Kord,+ and Q,H,Σord

κ0 ∈ Kord,+ (not just in Kord,++ and Kord,++ ). Then, for Q0 ,H,Σord Q,H,Σord Q0 ,H,Σord each integer i ≥ 0, there is a canonical morphism ∗,ord,tor ~ ord,tor /M ~ ord,tor ) ([g~l ]κ0 ,κ )∗ : H ilog-dR (N κ H,Σord

~ ord,tor ~ ord,tor /M ) → H ilog-dR (N κ0 H,Σord extending the canonical morphism ~ ord ~ ord /M ~ ord ) → H i (N ~ ord ([gl ]∗κ0 ,κ )∗ : H idR (N κ H κ0 /MH ) dR induced by [gl ]Q , respecting the Hodge filtrations and inducing canonical morphisms ∗,ord,tor a ([g~l ]κ0 ,κ )∗ : Rb f~∗ord,tor (ΩN ~ ord,tor ~ ord,tor ) /M κ H a

→ Rb f~∗ord,tor (ΩN ~ ord,tor /M ~ ord,tor ) 0 H κ

(for integers a + b = i) compatible (under the canonical isomorphisms ~ ord,tor ~ ord,tor and N ) with the canonical morphisms in (3) for N κ κ0 ∗,ord,ext ∨ ([g~l ]κ0 ,κ )∗ : Lie∨ ~ ord,ext /M ~ ord,ext /M ~ ord,tor ~ ord,tor → LieN N κ

H,Σord

κ0

H,Σord

and ∗,ord,ext,∨ d[g~l ]κ0 ,κ : LieN ~ ord,ext,∨ ~ ord,tor → LieN ~ ord,ext,∨ /M ~ ord,tor /M κ H,Σord

κ0

H,Σord

induced by the morphisms ∗,ord,ext ~ ord,ext ~ ord,ext [g~l ]κ0 ,κ :N →N κ κ0

and ∗,ord,ext,∨ ~ ord,ext,∨ → N ~ ord,ext,∨ [g~l ]κ0 ,κ :N κ κ0 ∗,ord,grp ~ ord,grp ~ ord,grp and its dual respectively, induced by [g~l ]κ0 ,κ :N →N κ κ0 ∗,ord,grp,∨ ord,grp,∨ ord,tor ord,grp,∨ ~ ~ 0 ~ ~ [gl ]κ0 ,κ :N →N over M . (In fact, all these κ κ H,Σord morphisms are induced by [gl ]∗Q .) The characteristic zero pullbacks of these canonical morphisms are isomorphisms. (d) If the levels Hκ,p and Hκ0 ,p at p are equally deep as in Definition 3.2.2.9, then the surjection (7.1.4.21) is finite, and the surjection (7.1.4.22) is proper. ∗,ord,grp (e) If gl,p ∈ GLO ⊗ Zp (Q ⊗ Zp ), then [g~l ]κ0 ,κ is a Z× (p) -isogeny, the surZ

Z

jection (7.1.4.21) is quasi-finite ´etale, the surjection (7.1.4.22) is log ´etale, and the canonical morphisms in (5c) are all isomorphisms.

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(f ) If we have an element gl0 of GLO ⊗ A∞ (Q ⊗ A∞ ) with a simiZ

Z

∗,ord ∗,ord,tor lar setup such that [g~l0 ]κ00 ,κ0 and [g~l0 ]κ00 ,κ0 are compatibly defi∗,ord ∗,ord,tor ned for some κ00 ∈ Kord,++ , then [g~l gl0 ]κ00 ,κ and [g~l gl0 ]κ00 ,κ Q00 ,H,Σord ∗,ord are also compatibly defined and satisfy the identities [g~l gl0 ]κ00 ,κ = ∗,ord ∗,ord ∗,ord,tor ∗,ord,tor ∗,ord,tor [g~l ]κ0 ,κ ◦ [g~l0 ]κ00 ,κ0 and [g~l gl0 ]κ00 ,κ = [g~l ]κ0 ,κ ◦ [g~l0 ]κ00 ,κ0 .

If

κ ∈ Kord,+ , κ0 ∈ Kord,+ , and κ00 ∈ Kord,+ , we also Q,H,Σord Q0 ,H,Σord Q00 ,H,Σord ∗,ord ∗,ord ∗,ord ∗,ord,tor ∗ ∗ ∗ have ([g~l gl0 ]κ00 ,κ ) = ([g~l0 ]κ00 ,κ0 ) ◦ ([g~l ]κ0 ,κ ) and ([g~l gl0 ]κ00 ,κ )∗ = ∗,ord,tor ∗,ord,tor ([g~l0 ]κ00 ,κ0 )∗ ◦ ([g~l ]κ0 ,κ )∗ . (g) The morphism ∗,ord ~ ord ~ ord [g~l ]κ0 ,κ : N κ0 → Nκ

(7.1.4.24)

(cf. Definition 3.4.4.2) induced by (7.1.4.21) is finite. The morphism ∗,ord,tor ~ ord,tor ~ ord,tor , [g~l ]κ0 ,κ :N →N κ κ0

(7.1.4.25)

induced by (7.1.4.22) is proper, and there exist choices of κ0 , which can be assumed to satisfy κ0  κ00 for any given κ00 , such that (7.1.4.25) is finite flat. If gl,p ∈ GLO ⊗ Zp (Q ⊗ Zp ), then the morphism Z

Z

∗,ord,tor

~ 0,r 0 ~ ord,tor ~ ord,tor × S [g~l ]κ0 ,κ,rκ0 : N →N κ κ0 κ

(7.1.4.26)

~ 0,r S κ

induced by (7.1.4.25) is proper log ´etale (because it is log ´etale by (5e)), and there exists choices of κ0 , which can be assumed to satisfy κ0  κ00 for any given κ00 , such that (7.1.4.15) is finite ´etale (by (5e); cf. [35, IV3, 8.11.1]). ˆ if H b 0,p = H b p , and if (H bp0 )ord = H bpord (cf. (h) If gl ∈ GLO ⊗ Zˆ (Q ⊗ Z), Z

Z

(7.1.1.24)), then there exist choices of κ0 , which can be assumed to satisfy κ0  κ00 for any given κ00 , such that (rκ0 = rκ and) the induced morphisms (7.1.4.24) and (7.1.4.25) are isomorphisms. (These b=H bp H bp conditions are true, in particular, when gl = 1 and when H 0 0,p b 0 0,p p 0 ord ord b b b b b b and H = H Hp satisfy H = H and (Hp ) = Hp ; cf. see the remark at the end of Corollary 3.4.4.4.) (6) After pulled back to the characteristic zero fibers, the objects and morphisms in this theorem are canonically compatible with those in Theorem ~g ]ord,tor ⊗ Q, 1.3.3.15 (cf. Proposition 7.1.1.21). (In particular, f tor ⊗ Q, [b 0 κ ,κ

∗,ord,tor

and [g~l ]κ0 ,κ

∗,ord,tor

⊗ Q, where Z

fκtor 0 ,κ

ord,tor

~g ] is as in (1), [b

Z

Z

is as in (7.1.4.11),

and [g~l ]κ0 ,κ is as in (7.1.4.22), are always proper, without the additional conditions on depths of levels at p.)

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With the of [61, Thm. we will spell that, when κ

ingredients we have already provided, the proof is similar to that 2.15; see also the errata]. Nevertheless, for the sake of certainly, out the details in the next few sections. The proof will clarify ∨ ∈ Kord,+ ~ ord /M ~ ord , LieN ~ ord /M ~ ord , Q,H , the locally free sheaves such as LieN κ H κ H ∨ ord ~ , and their extensions Lie~ ord,ext ~ ord,tor , ~ ord , and Lie~ ord,∨ ~ ord over M

LieN ~ ord,∨ /M κ

H

Nκ

H

/MH

Lie∨ ~ ord,ext ~ ord,tor , N /M κ H

Nκ

Lie∨ ~ ord,ext,∨ ~ ord,tor N /M κ H

LieN ~ ord,ext,∨ ~ ord,tor , and /M κ H vely, are the correct ones to use in the statements.

/MH

~ ord,tor , respectiover M H

Remark 7.1.4.27. (Compare with Remarks 1.1.2.1, 1.3.1.4, 1.3.3.33, 3.4.2.8, and 5.2.1.5.) By varying the choices of L and Q inducing the same L ⊗ Z(p) and Q ⊗ Z(p) , Z

Z

e inducing the same L e ⊗ Z(p) , we can respectively, and hence varying the choices of L Z

b in the parameter κ = (H, b Σ) b to be any open compact (in practice) allow the H ∞ p b b b b b b ∞,p ) and H bp is of subgroup of G(A ) of the form H = H Hp , where Hp ⊂ G(A standard form as in Definition 7.1.1.2. Nevertheless, this can be achieved by varying the lattice Q alone, and hence is already incorporated in (5) of Theorem 7.1.4.1. 7.2

Main Constructions of Compactifications and Morphisms

7.2.1

Partial Toroidal Boundary Strata

e etc be chosen as in Section 1.2.4, and let e e h · , · ie, e e δ), Let (L, h0 ), (e Z, Φ, D be defined ~ ord and by D as in Section 7.1.1. (The choice of D is implicit in the construction of M H e e ord , σ hence a prerequisite of Theorem 5.2.1.1 and its consequences.) Let κ e = (H, Σ e) ord,++ e ord,++ as in Definition 1.2.4.11, and let κ = [e be any element in the set K κ ] ∈ K Q,H Q,H be as in Definition 1.2.4.44. e h · , · ie, e By Theorem 3.4.2.5 and Proposition 3.4.6.3, the data of O, (L, h0 ), e D, and ord ~e ep (where H e p is neat by assumption) define a scheme M e=H ep H H e quasi-projective H ~S0,r (see Definition 2.2.3.3 and Condition 7.1.1.5). Since H e p is neat over ~S0,rH = κ f e ord is projective (and smooth), by Theorems 5.2.1.1 and 6.2.3.1, we have a and Σ ~e ord,tor ~e ord,tor partial toroidal compactification M =M eΣ e ord which is quasi-projective and e H H, e satisfies Conditions 1.2.4.7 and 7.1.1.5, by construction smooth over S0,rH . Since H f (see Propositions 4.2.1.29 and 4.2.1.30), we have f f ~e ord,ΦH ~e ord,ZH ∼ ∼ ~ ord , M =M =M e e H H Hκ

e

e

(7.2.1.1)

f ~e ord,ΦH ~e ord bG = GreZ−1 (H e e 0 ) = GreZ−1 (H e e ), and C where Hκ = H → M is an abee e e H Φ , δ P Pe f f

e

e Z

Z

H

H

~e ord,grp × lian scheme torsor under an abelian scheme C ef canonically Q -isogenous to e f,δ Φ H H b saHom (Q, G ~ ord )◦ (which is the C in the proof of Proposition 4.2.1.30). If H O

M Hκ

~ ord = M ~ ord . If H b also tisfies Condition 1.2.4.8, then we have Hκ = H and hence M Hκ H

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~e ord,grp ~e ord ~ ord = M ~ ord is an abelian satisfies Condition 1.2.4.9, then C ef → M ef = C e f,δ e f,δ Φ Φ Hκ H H H H H scheme, not just a torsor. Remark 7.2.1.2. (Compare with Remark 1.3.4.1 and Proposition 4.1.6.1.) The isomorphism (7.2.1.1) means we do not need to consider nontrivial twisted objects (ϕ e∼ e∼ eord eord e−2,He , ϕ e0,He ) and ϕ e−1,He = αHκ (resp. e, ϕ e ) (resp. (ϕ e, ϕ e )) above (ϕ −2,H 0,H −2,H 0,H ord p ϕ eord e = (αHκ , αHκ,p )). −1,H

Since σ e is a top-dimensional nondegenerate rational polyhedral cone in the cone e e in Σ e ord , by (2) of Theorem 5.2.1.1, the locally closed stratum decomposition Σ Φf H

~eord Z[(Φ ef,e e f,δ σ )] (not its closure) is a zero-dimensional torus bundle over the abelian H

H

ord ~e ord ~ ord . In other words, ~Z e e e scheme torsor C over M e e [(Φ f,δ f,e σ )] is canonically isoΦ f,δ f Hκ H

H

H

H

ord ~e ord ~ ord to be this stratum ~Z e e e morphic to C ef . Let us define N e f,δ Φ [(ΦH σ )] , and denote κ e f,δH f,e H H ord ord ord ~ ~ ~ ~ the canonical morphism Nκe → MH,rκ = MH × S0,rκ by fκe (which is the com~ S0,rH

~ ord ). Let us ~ ord → M ~ ord → M ~ ord and M position of the canonical morphisms N H,rκ Hκ Hκ κ e ~e ord,grp ord,grp × ◦ ~ denote the canonical Q -isogeny HomO (Q, GM := C Φ eisog . ef by κ e f,δ ~ ord ) → Nκ e H

Hκ

H

ord ~e ord ~ ord = ~Z e e e e ord Note that N is canonically isomorphic to C ef for every Σ e f,δ Φ [( Φ , δ ,e σ )] κ e f H f H H H ee . and every top-dimensional cone σ e in Σ ΦH f

Lemma 7.2.1.3. (Compare with Lemma 1.3.4.2.) The abelian scheme torsor ~e ord C ef (see (5) of Theorem 6.2.1.1 and Definition 1.2.1.15) and the canonical isoe f,δ Φ H

H

f ~e ord,ΦH ~e ord,grp ∼ ~ ord depend geny HomO (Q, GMH )◦ → C =M ef of abelian schemes over M e f,δ e Φ H Hκ H H b e b ˆ (up to canonical isomorphism) only on the open compact subgroup H = HG b of G(Z) 0 e Moreover, if H e is any open (see Definitions 1.2.4.3 and 1.2.4.4) determined by H. ˆ still satisfying Conditions 1.2.4.7 and 7.1.1.4 such that e Z) compact subgroup of G( ˆ ∼ ˆ U( ˆ induced by the splitting e0 = H bG n H b b under the isomorphism G( b Z) b Z) H = G(Z)n

e

b G

U

~e ord ~e ord,grp ∼ ~e ord,grp δe (cf. Condition 1.2.4.9), then we have C ef0 = C ef0 = C Φ ef . e f0 ,δ e f,δ e f0 ,δ Φ Φ H H H H H H

~e ord Proof. This follows from the very construction of C ef (see the proof of Propoe f,δ Φ H H e sition 4.2.1.30), which is (up to canonical isomorphism) insensitive to replacing H with an open compact subgroup still satisfying Conditions 1.2.4.7 and 7.1.1.4 that b=H eb. defines the same H G ~ ord and κ Consequently, N eisog depend (up to canonical isomorphism) only on the κ e ˆ determined by H b of G( b Z) e (see Definitions 1.2.4.3 and open compact subgroup H 1.2.4.4).

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409

ord ~e ord,tor ~ ord,tor to be the closure of ~Z e e e Let us take N eΣ e ord . Then we obtain [(ΦH σ )] in MH, κ e f,δH f,e tor ord ~ ~ ord,tor . Certainly, the canonical open fiberwise dense immersion κ e : N ,→ N κ e κ e ord,tor ~ b but also on the choices of Σ e e and σ N depends not only on H e. κ e

ΦH f

ep Lemma 7.2.1.4. (Compare with Lemma 1.3.4.3.) Under the assumption that H ord,tor ~e e=H ep H ep is neat), the closure of every stratum in M is neat (and hence H eΣ e ord has H, no self-intersection. Proof. The same argument of the proof of Lemma 1.3.4.3 (or rather of [61, Lem. 3.1]) works here. e Σ e ord , σ Corollary 7.2.1.5. (Compare with Corollary 1.3.4.4.) For each κ e = (H, e) ∈ ord ord,tor ~ ~ ord,tor ++ ord ~ ~ e e e e e e ord is quasi-projective and e K of N σ )] in MH, Σ Q,H , the closure Nκ κ e = Z[(Φ e f,δH f,e H ord,tor ord ~ ~ in N (with its reduced structure) smooth over ~S0,r , and the complement of N κ e

κ

κ e

is a relative Cartier divisor with simple normal crossings. Proof. Combine Lemma 7.2.1.4, (3) of Theorem 5.2.1.1, and Theorem 6.2.3.1. ord,tor

~e ~ ord,tor . By (2) of The stratification of M eΣ e ord induces a stratification of N H, κ ~ ord,tor are parameterized by equivalence classes Theorem 5.2.1.1, the strata of N κ ˘ e , δ˘ e , τ˘)] having [(Φ e e , δe e , σ [(Φ H H H H e )] as a face (as in Definition 1.2.2.19), which have been described in Sections 1.2.4 (following Definition 1.2.4.11) and 7.1.1 (following ˘ e , δ˘ e )] is an ordinary cusp label (as in Defini˘ e, Φ Definition 7.1.1.7), such that [(Z H H H ˘ ˘ ˘ tion 3.2.3.8). With (Z, Φ, δ) etc chosen as in Section 1.2.4, and with σ ˘ being the image of σ e ⊂ P+ under the first morphism in (1.2.4.20), by Corollary 1.2.4.26, we e Φ f H

e+ may take τ˘ ∈ Σ ˘ Φ

σ c,˘ H

(see Definition 1.2.4.21) having σ ˘ as a face, whose ΓΦ˘ c -orbit is H

˘ e , δ˘ e , τ˘) is a representative of some [(Φ ˘ e , δ˘ e , τ˘)] having well defined, such that (Φ H H H H e e , δe e , σ [(Φ H H e )] as a face. By construction (see Propositions 4.2.1.37 and 4.2.1.46, (4.2.2.1), (4.2.2.2), and (4.2.2.3)), the scheme   ~e ord ~e ord ∼ ˘ Ψ ( `) (7.2.1.6) Ξ Spec ⊕ = ˘ ˘ ˘ ˘ Φ f,δ f Φ f,δ f O H

H

ord ~ e˘ C ˘ Φ f,δ f H H

˘ ˘ `∈S Φ

H

H

f H

~e ord is a torsor over C ˘ f,δ˘f under (the pullback of) the split torus EΦ ˘f Φ H

H

=

H

ord

~e ˘ is the subsheaf of O ord HomZ (SΦ˘ f , Gm ), where Ψ ˘ f,δ˘f (`) Φ ~ e H

O ~eord CΦ ˘

˘ f,δH f H

H

H

ΞΦ ˘

(considered as an

˘ f,δH f H

˘ -algebra, by abuse of language) on which EΦ˘ f acts by the character `, H

with its τ˘-stratum ~e ord ∼ Ξ ˘ f,δ˘f,˘ Φ τ = SpecO H H

 ord ~ e˘ C ˘ Φ f,δ f H H

 ~e ord ˘ ⊕ Ψ ( `) ˘ ˘ Φ f,δ f

˘ τ⊥ `∈˘

H

H

(7.2.1.7)

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~e ord ˘ together with the affine ⊕ Ψ ˘ f,δ˘f (`), Φ

ord defined by the sheaf of ideals If = τ˘

H

˘ τ∨ `∈˘ 0

H

toroidal embedding ~e ord ~e ord Ξ τ) ∼ = SpecO ˘ f,δ˘f ,→ Ξ ˘ f,δ˘f (˘ Φ Φ H

H

H

H

 ~e ord ˘ ⊕ Ψ ˘ f,δ˘f (`) Φ

 ord ~ e˘ C ˘ Φ f,δ f H H

H

˘ τ∨ `∈˘

(7.2.1.8)

H

~e ord along τ˘. (All the schemes are relatively affine over C ˘ f,δ˘f , and all the morphisms Φ H H are the canonical ones dual to the obvious morphisms between O ~eord -algebras.) CΦ ˘

ord

˘ f,δH f H

ord

~e ~e The closure Ξ τ ) of the σ ˘ -stratum on Ξ τ ) is defined by the sheaf of ˘ f,δ˘f,˘ ˘ f,δ˘f (˘ Φ σ (˘ Φ H

ideals

H

H

H

~e ord ˘ and hence we have a canonical isomorphism Ψ ˘ f,δ˘f (`), Φ

⊕

H

˘ σ ∨ ∩ τ˘∨ `∈˘ 0

H

~e ord τ) ∼ Ξ = SpecO ˘ f,δ˘f,˘ Φ σ (˘ H

H



ord ~ e˘ C ˘ Φ f,δ f H H

⊕ ˘ σ ⊥ ∩ τ˘∨ `∈˘

 ~e ord ˘ , Ψ ˘ f,δ˘f (`) Φ H

(7.2.1.9)

H

ord

~e whose τ˘-stratum is canonically isomorphic to the scheme Ξ ˘ f,δ˘f,˘ Φ τ above, which (as H

H

~e ord τ )) is defined by the sheaf of ideals a closed subscheme of Ξ ˘ f,δ˘f,˘ Φ σ (˘ H

H

ord If σ ˘ ,˘ τ :=

⊕ ˘ σ ⊥ ∩ τ˘∨ `∈˘ 0

~e ord ˘ Ψ ˘ f,δ˘f (`). Φ H

(7.2.1.10)

H

~e ord ~e ord ~e ord τ ) along Ξ Let X ˘ f,δ˘f,˘ ˘ f,δ˘f,˘ ˘ f,δ˘f,˘ Φ τ, σ (˘ Φ σ ,˘ τ denote the formal completion of ΞΦ H

H

H

H

H

H

~e ord which can be canonically identified as a closed formal subscheme of X ˘ f,δ˘f,˘ Φ τ , the H

H

~e ord ~e ord formal completion of Ξ τ ) along its τ˘-stratum Ξ ˘ f,δ˘f (˘ ˘ f,δ˘f,˘ Φ Φ τ , inducing the closures H H H H e e ˘ e , δ˘ e , τ˘)-model. (See of the [(Φ e , δ e , σ e)]-strata on every ordinary good formal (Φ H

H

H

H

Section 5.1.1.7 for the definition of good formal models, and see Definition 5.1.2.9 for e e , δe e , σ the labeling of the strata by equivalence classes of triples of the form [(Φ H H e )].) By (5) of Theorem 5.2.1.1, the strata-preserving canonical isomorphism ~e ord,tor ∧ (M eΣ e ord )~ ord H, e ˘ Z [(Φ

˘ ,˘ f,δH f τ )] H

~e ord ~e ord ∼ =X ˘ f,δ˘f,˘ ˘ f,δ˘f,˘ ˘ f,˘ Φ τ = XΦ τ /ΓΦ τ H H H H H

(where ΓΦ˘ f,˘τ is trivial by [62, Lem. 6.2.5.27]) induces a canonical isomorphism H

~ ord,tor )∧ord (N κ e ~ e ˘ Z [(Φ

˘ ,˘ f,δH f τ )] H

~e ord ∼ =X ˘ f,δ˘f,˘ Φ σ ,˘ τ. H H

~e ord,tor (Alternatively, one may refer directly to the gluing construction of M in Section e H 5.1.3, based on the crucial Proposition 5.1.2.7; cf. [62, Sec. 6.3.3].)

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411

Justification for the Parameters

e ++ . The goal of So far we have parameterized the objects we constructed by κ e∈K Q,H this subsection is to show that the equivalence classes κ = [e κ] ∈ Kord,++ , with the Q,H natural directed partial  order among them (see Definitions 7.1.1.11 and 7.2.2.20), form a more natural parameter set for the objects we have constructed. (See Proposition 7.2.2.19 below.) Construction 7.2.2.1. (Compare with Construction 1.3.4.6.) e ord,++ , consider the degenerating family e Σ e ord , σ (H, e) in K Q,H

For each κ e =

ord e ei, α ~ ord,tor e λ, (G, eHe p , α eH e ) → MH, eΣ e ord

(7.2.2.2)

p

~e ord of type M e as in Theorem 5.2.1.1. As in Construction 5.2.4.15, let H b bi) → N ~ ord,tor b λ, (G, κ e

(7.2.2.3)

ord ~ ord,tor , the closure of N ~ ord = ~Z e e e denote the pullback of (7.2.2.2) to N [(Φ f,δ f,e σ )] in κ e κ e H

H

~e ord ~ ord,tor . Note that N ~ ord is canonically isomorphic to C M e is topef because σ e f,δ Φ ord κ e e e H,Σ H

H

~e ord dimensional. Although α eHe p is defined only over M e , by proceeding as in ConH struction 5.2.4.15, we can define a (partial) pullback ord ~ ord,tor b bi, α b λ, e \ e\ e\ e\ , βe\,ord ) → N (G, bHb p , α bH b ) := (G , λ , i , βH κ e ep e H p

(7.2.2.4)

p

~ ord,tor , with the convention that (as in the of the degenerating family (7.2.2.2) to N κ e ord e ei, α e ei) (resp. e λ, e λ, case of (G, eHe p , α eH ) itself) α b is defined only over Nκe , while (G, p b ep H ~ ord,tor as in (7.2.2.3). By construction, α eord ) is defined (resp. extends) over all of N κ e

ep H

the pullback ~e ord ord b~ ord , bi~ ord , α ~ ord ∼ b ~ ord , λ C ) → N (G b , α b = ef , e f,δ p b Φ κ e bp H N N N H H H κ e

κ e

(7.2.2.5)

κ e

~ ord determines and is determined by (the prescribed (e e e , δe e ) of (7.2.2.4) to N ZHe , Φ κ e H H and) the tautological object ~e ord ∨,ord
ord (A, λ, i, αHpκ , αH ), (e cord , e c ) → C ef e f,δ Φ e e κ,p H H H

H

(7.2.2.6)

e e ; i.e., elements of Γ e ). Here (up to isomorphisms inducing automorphisms of Φ H Φf H

f ~e ord,ZH ∼ ~ ord . As explaiis the tautological object over M = M e H Hκ ned in the proof of Proposition 4.2.1.29, and also in Remark 1.3.4.1, we do not need to consider nontrivial twisted objects (ϕ eord eord e−2,He , ϕ e0,He ) and e, ϕ e ) above (ϕ −2,H 0,H e the tautological object e δ), ϕ eord = (αHp , αord ). With the fixed choice of (e Z, Φ,

ord ) (A, λ, i, αHpκ , αH κ,p

e −1,H

κ

e

Hκ,p

b and hence so is the tuple (7.2.2.5). Thus, the notation (7.2.2.6) depends only on H,

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ord (b αHb p , α bH b p ) is justified. As at the end of Construction 5.2.4.15, by considering the degenerating family

~e ord,tor e ei, α e λ, (G, eHe ) → M eΣ e ord H,

(7.2.2.7)

e e , with the same (G, e ei) as in (7.2.2.2), where α e λ, of type M eHe is defined only over H ord ~e ord M αHe p , α eH eHe as in Proposition e ⊗ Q, such that the pair (e H e ) ⊗ Q is induced by α p

Z

Z

3.3.5.1, we obtain a (partial) pullback b bi, α ~ ord,tor b λ, (G, b b) → N H

(7.2.2.8)

κ e

~ ord ⊗ Q, such that the pair as in (1.3.4.9), where α bHb is defined only over N κ e Z

ord (b αHb p , α bH bHb in an obvious analogue of Proposition 3.3.5.1. b ) ⊗ Q is induced by α p

Z

b bi, α b λ, Construction 7.2.2.9. (Compare with Construction 1.3.4.12.) Let (G, bHb p , ord,tor ord,tor ord ~ b ~ b be as in (7.2.2.4), and let (G, λ, bi, α bHb ) → Nκe be as in (7.2.2.8) α bHb ) → Nκe p ~ ord,tor centered in Construction 7.2.2.1. Consider any morphism ξ : Spec(V ) → N κ e ~ ord,tor such that V is a complete discrete valuation ring at a geometric point s¯ of N κ e

with fraction field K, and such that η := Spec(K) is mapped to the generic point of the irreducible component containing the image of s¯. Suppose the image of s¯ lies on ord ~e ord,tor e ˘ ˘ ˘ e , δ˘ e , ρ˘)] is represented ˘ e , δ˘ e , ρ˘)]-stratum ~Z of M the [(Φ eΣ e ord , where [(Φ H, [( Φ , δ , ρ)] ˘ H H H H f H f H ˘ ϕ˘ e , ϕ˘ e ), δ˘ e ) representing some ˘ e , δ˘ e , ρ˘) with (Z ˘ e = (X, ˘ Y˘ , φ, ˘ e, Φ by some (Φ H

H

H

H

−2,H

0,H

H

ordinary cusp label as in Section 7.1.1. (As in Construction 1.3.4.12, we avoid using ˘ e , δ˘ e , τ˘) because the symbol τ will be used for another the more familiar notation (Φ H H purpose below.) For simplicity, let us fix compatible choices of representatives eϕ e and (Z ˘ ϕ˘−2 , ϕ˘0 ), δ), ˘ as in Sections 1.2.4 e = (X, e Ye , φ, ˘ = (X, ˘ Y˘ , φ, ˘, Φ (e Z, Φ e−2 , ϕ e0 ), δ) e and 7.1.1, in their H-orbits. ~e ord Since X S0,rκ , there exists a complete regular ˘ f,δ˘f,ρ˘ is formally smooth over ~ Φ H H ∼ V and such that the morphism local ring Ve and an ideal Ie ⊂ Ve such that Ve /Ie = ord

~e ~ ord,tor extends to a morphism ξe : Spf(Ve , I) e →X Spec(V ) → N ˘ f,δ˘f,ρ˘, which induΦ κ e H H e e e ces a dominant morphism from Spec(V ) to Spec(R), where R is the local ring of ~e ord X ¯. Let ˘ f,δ˘f,ρ˘ at the image of s Φ H

H

‡ ord,‡ e‡ , ei‡ , α e‡ , λ (G eH eH ) → Spec(Ve ) ep , α e

(7.2.2.10)

p

denote the pullback of (7.2.2.2) under the composition of ξe with the canonical ~e ord,tor ~e ord morphism X ˘ f,δ˘f,ρ˘ → M eΣ e , and let Φ H, H

H

‡ ord,‡ b‡ , bi‡ , α b‡ , λ (G bH bH ) → Spec(V ) bp , α b

(7.2.2.11)

p

denote the pullback of (7.2.2.4) under ξ. Similarly, let e‡ , ei‡ , α e‡ , λ (G e‡ ) → Spec(Ve ) e H

(7.2.2.12)

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413

denote the pullback of (7.2.2.7) under the composition of ξe with the canonical ~e ord,tor ~e ord morphism X eΣ e , and let ˘ f,δ˘f,ρ˘ → M Φ H, H

H

‡ b‡ , bi‡ , α b‡ , λ (G bH b ) → Spec(V )

(7.2.2.13)

‡ ‡ ord,‡ denote the pullback of (7.2.2.4) under ξ. Then α eH αH eH ) over η as in e induces (e ep , α e p

ord,‡ ‡ ‡ bH ) over η in an analogous way. (We Proposition 3.3.5.1, and α bH αH b b induces (b bp , α p

omit the details for simplicity.) As in (6) of Theorem 5.2.1.1, (7.2.2.10) defines an object in the essential image e ...ord (Ve ), which corresponds to a tuple of DEGPEL,M e ord (V ) → DEG e f H

PEL, M H f

\,ord,‡ e ‡ , λ e‡ , i e‡ , X e ‡ , Ye ‡ , φe‡ , e (B c‡ , e c∨,‡ , τe‡ , [e αH ]) e B B

e in the essential image of DDPEL,M e ord (V ) → DD f H

...ord (Ve ) under (4.1.6.4) in The-

ef PEL, M H

\,ord,‡ orem 4.1.6.2, where [e αH ] is represented by some e

\,ord,‡ α eH = (e Z‡He , ϕ eord,‡ eord,‡ e‡−1,He p , ϕ eord,‡ e e, ϕ e =(ϕ e ), −2,H −1,H −1,H p

ord,‡ ∨,ord,‡ ord,‡ eord,‡ , e ϕ eord,‡ cH ,e cHe , τeHe ) e , δH e e 0,H

induced by some \,‡ ‡,∼ ‡ ‡ e‡ , e e‡ , ϕ α eH e‡−1,He , ϕ e‡,∼ c∨,‡ eH e = (ZH e e−2,H e, ϕ e , δH e cH e, e e ,τ e) 0,H H \,‡ e ‡ , Ye ‡ , φe‡ , [e over η as in Section 4.1.6. Note that (X αH e ]) determines some cusp la‡ ‡ ‡ e , δe )] equivalent to the cusp label [(Z ˘ e , δ˘ e )] represented by the ˘ e, Φ bel [(e ZHe , Φ e H e H H H H ˘ introduced above (where the (ϕ e ˘ δ) ˘ e is induced ˘, Φ, H-orbit of the (Z e‡ ,ϕ e‡ ) in Φ e −2,H

‡,∼ (ϕ e‡,∼ e0, e, ϕ e) −2,H H

e 0,H

H

by as in the corrected [62, Def. 5.4.2.8] in the errata). For simplicity, we shall use entries in this last representative to replace their isomorphic (or equivalent) objects, and say in this case that (ϕ e‡,∼ e‡,∼ eord,‡ eord,‡ e, ϕ e ) and (ϕ e, ϕ e ) induce −2,H 0,H −2,H 0,H (ϕ˘−2,He , ϕ˘0,He ). ˘ e e ‡ , λ e‡ , i e‡ , X, ˘ Y˘ , φ, By definition, the pullback of (B c‡ , e c∨,‡ ) to the subscheme B B b‡ , bi‡ ) → Spec(V ). Let us denote it by b‡ , λ Spec(V ) of Spec(Ve ) depends only on (G ˘ b b ‡ , λ b‡ , i b‡ , X, ˘ Y˘ , φ, (B c‡ , b c∨,‡ ). B B eϕ e e e = (X, e Ye , φ, Note that the H-orbit (e ZHe , Φ e−2,He , ϕ e0,He ), δeHe ) is part of the data of κ e. H ˘ ˘ By Lemma 1.2.4.16, it makes sense to consider Z b , (ϕ˘ b , ϕ˘ b ), and δ b , which are H

−2,H

0,H

H

∼ ˘ ∼ ˘ ˆ Z(1)), ˆ ˆ and δ, b ˘ ⊗ Z, ˘, (ϕ˘−2 : Gr˘Z−2 → the H-orbits of Z HomZˆ (X ϕ˘0 : GrZ0 → Y˘ ⊗ Z), Z

Z

ord ˘ which induces the H-orbit b b respectively. Then we have the orbit H-orbit δ˘Hb of δ, δbH b e L e (with e Z−1,n reof δ˘ord . Moreover, by extending restrictions to subgroups of L/n ˘−1,n ) as in Constructions 1.3.4.6 and 7.2.2.1, (e placed with its subgroup Z α‡ p , α eord,‡ ) e H

induces an ordinary level-Hκ structure

ϕ e‡−1,He

=

(ϕ e‡−1,He p , ϕ eord,‡ ep ) −1,H

ep H

b ‡ , λ b‡ , i b‡ ) of (B B B

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‡ ord,‡ depending only on (b αH bH ), which we denote by ϕ bord,‡ b‡−1,Hb p , ϕ bord,‡ bp , α b b = (ϕ b ), −1,H −1,H p

p

b ‡ , λ b‡ , i b‡ ) ⊗ Q induwhich is compatible with the level-Hκ structure ϕ b‡−1,Hb of (B B B Z

‡ ced by α eH e as in Proposition 3.3.5.1. Then it also makes sense to consider the ord,‡ b H-orbit (ϕ eord,‡ eord,‡ b−2, bord,‡ b, ϕ b ), which we denote by (ϕ b, ϕ b ), which is a subscheme −2,H 0,H H 0,H ord,‡ b U(n)-orbits, b b which can be identified with a system of H/ of (ϕ˘ b , ϕ˘ b ) × ϕ −2,H

0,H

˘ ZH c

b −1,H

b e b ⊂ H, b which surjects under the where n ≥ 1 are integers such that U(n) := U(n) G ord,‡ two projections to the orbits defining (ϕ˘−2,Hb , ϕ˘0,Hb ) and ϕ b−1, b and is compatible H ‡,∼ b with the H-orbit (ϕ b‡,∼ b0, ˘−2,Hb , ϕ˘0,Hb ) × ϕ b‡−1,Hb . b, ϕ b ) defined as a subscheme of (ϕ −2,H H ˘ ZH c

‡,∼ b In this case, we say that (ϕ b−2, b‡,∼ bord,‡ bord,‡ b, ϕ b ) and (ϕ b, ϕ b ) induce the H-orbit H 0,H −2,H 0,H (ϕ˘−2,Hb , ϕ˘0,Hb ). By proceeding as in Construction 1.3.4.12, we can define b \,‡ b ∨,\,‡ (b ι‡ : Y → G ι∨,‡ : X → G ) η ,b η

and (b τ ‡ : 1Y

∼

˘ × X,η

⊗ −1 → (b c∨,‡ |Y × b c‡ )∗ PB b ‡ ,η , ∼

⊗ −1 c∨,‡ × b c‡ |X )∗ PB τb∨,‡ : 1Y˘ × X,η → (b b ‡ ,η )

(satisfying certain familiar compatibility conditions, which we omit for simplicity), ∨,‡ ∨,‡ ‡ ∨,‡ ‡ b p -orbits (b and define the H ι‡Hb p , b ιH c‡Hb p , b cH bHb p , τbH bH b p ) and (b bp , τ b p ) determined by α bp , b which is induced by the H-orbits (b ι‡ , b ι∨,‡ ) and (b c‡ , b c∨,‡ , τb‡ , τb∨,‡ ) determined by b H

b H

b H

b H

b H

b H

∨,ord,‡ ‡ ) (see ,e cH carries no more information than (e cord,‡ α bH ιord,‡ e e b . On the other hand, e e H H p

p

p

Proposition 4.1.5.20 and Definitions 4.1.5.22 and 4.1.5.23), and (e cord,‡ ,e c∨,ord,‡ ) is e e H H

~e ord ∨,ord , e c ) over C the pullback of the tautological object (e cord ˘ f,δ˘f , which depends Φ e e H H H H b only on H. Hence, it makes sense to define ∨,ord,‡ (b ιord,‡ ,b ιH ) := (b ι‡Hb p , b ι∨,‡ b b b p ), H H ∨,‡ ord,‡ ∨,ord,‡ ‡ (b τH , τbHb ) := (b τH bH b bp , τ b p ),

(b cord,‡ ,b c∨,ord,‡ ) := (e cord,‡ ,e c∨,ord,‡ ). b b e e H H H H In summary, given the family (1.3.4.9) in Construction 1.3.4.6, each morphism ξ : Spec(V ) → Ntor κ e as above determines a tuple \,ord,‡ ‡ ˘ b b , λ b‡ , i b‡ , X, ˘ Y˘ , φ, (B c‡ , b c∨,‡ , τb‡ , τb∨,‡ , [b αH ]), (7.2.2.14) b B B \,ord,‡ where [b αH ] is an equivalence class of b \,ord,‡ ord,‡ ord,‡ ∨,ord,‡ ord,‡ ∨,ord,‡ bord , b ˘ b, ϕ α bH = (Z bord,‡ bord,‡ ,b cHb , τbHb , τbHb ) (7.2.2.15) b cH b b, ϕ b, ϕ b , δH b H b−2,H −1,H 0,H

(whose precise definitions we omit for simplicity), induced by some ˘ b, ϕ α b\,‡ = (Z b‡,∼ , ϕ b‡ ,ϕ b‡,∼ , δ˘ b , b c‡ , b c∨,‡ , τb‡ , τb∨,‡ ) b H

H

b −2,H

b −1,H

b 0,H

H

b H

b H

b H

b H

(7.2.2.16)

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(cf. (1.3.4.16)) over η, in a way analogous to that in Section 4.1.6. Given a tuple as in (7.2.2.14), if we set ord b ‡ b‡ , i b‡ , ϕ (B ‡ , λB ‡ , iB ‡ , ϕord −1,Hκ ) := (B , λB b) B b−1,H

and (c‡ , c∨,‡ , τ ‡ ) := (b c‡ |X , b c∨,‡ |Y , τb‡ |1Y ×X,η ), \,ord,‡ and define [αH ] using similar restrictions, then the tuple κ \,ord,‡ ]) (B ‡ , λB ‡ , iB ‡ , X, Y, φ, c‡ , c∨,‡ , τ ‡ , [αH κ

(7.2.2.17)

(V ) → DDPEL,... (V ) defines an object in the essential image of DDPEL,Mord M ord H H κ

κ

\,‡ (because we can define [αH ] as in (1.3.4.17) in Construction 1.3.4.12). On κ ord,‡ ‡ b‡ , bi‡ , α b‡ , λ bH ) → Spec(V ) is deterthe other hand, the pullback (G bH b bp , α p

‡ ord,‡ b‡ , bi‡ , α b ‡η , λ bH mined up to isomorphism by its generic fiber (G η η bH b p ,η , α b ,η )

→

p

Spec(K), which (up to isomorphism) determines and is determined by a tuple ~e ord ‡ ord,‡ ord,‡ ∨,ord,‡ , α ), (e c , e c )) → Spec(K) parameterized by C ((G‡η , λ‡η , i‡η , αH p ef . e f,δ Φ H ,η e e ,η κ,p κ H,η H,η H

H

‡ ord,‡ The abelian part (G‡η , λ‡η , i‡η , αH , αH ) extends to a degenerating family p κ,p ,η κ ,η ‡ ord,‡ (G‡ , λ‡ , i‡ , αH p,α Hκ,p ) κ

(7.2.2.18)

~ ord over Spec(V ) (with α‡ p still defined only over η = Spec(K)) which of type M Hκ Hκ defines an object in the essential image of DEGPEL,Mord (V ) → DEGPEL,... (V ) M ord H H κ

κ

‡ (because we can defined αH as in (1.3.4.18) in Construction 1.3.4.12). By the theκ ory of two-step degenerations (see [28, Ch. III, Thm. 10.2] and [62, Sec. 4.5.6]), and by analyzing endomorphism structures and level structures as in [62, Sec. 5.1–5.3] and Section 4.1, under (4.1.6.4) in Theorem 4.1.6.2, this last object (7.2.2.18) corresponds to the above object (7.2.2.17) in the essential image of DDPEL,Mord (V ) → DDPEL,... (V ). M ord Hκ Hκ ¯ As in Construction 1.3.4.12, by checking the values of their entries on K-points, ¯ is any fixed algebraic closure of K, the tuple over Spec(V ) as in (7.2.2.14) where K

determines and is determined by the tuple (7.2.2.11) (up to isomorphism, over Spec(K)). ˘→ As in (4.2.2.7), the pair (b τ ‡ , τb∨,‡ ) defines compatible morphisms υτb‡ : Y × X Z and υτb∨,‡ : Y˘ × X → Z (using the discrete valuation υ : Inv(V ) → Z of V ), which define the same element b ˘ )∨ υτb‡ = υτb∨,‡ ∈ (S Φc R H

(see (1.2.4.29)). On the other hand, as in (6) of Theorem 5.2.1.1, τe‡ defines a ˘ → Z, which defines an element morphism υτe‡ : Y˘ × X υτe‡ ∈ ρ˘ ⊂ P+ ˘ , Φ f H

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where ρ˘ is as above. Since (b τ ‡ , τb∨,‡ ) is defined by extending restrictions of υτe‡ , we see that υτb‡ = υτb∨,‡ ∈ ρb = pr(Sb ˘

∨ Φ c )R H

b˘ (˘ ρ) ⊂ P Φc H

(see (1.2.4.41)). If ρ˘ is replaced with another representative, then ρb is replaced with a translation under the action of ΓΦ˘ c . (This finishes Construction 7.2.2.9.) H

Proposition 7.2.2.19. (Compare with Proposition 1.3.4.19.) Suppose κ e = 0 e ord,++ such that κ0 = [e e Σ, e σ e0 , Σ e 0, σ (H, e) and κ e0 = (H e0 ) are elements in K κ ]  κ = [e κ] Q,H ord,++ ord,tor ord b ~ b λ, bi, α in K (see Definition 7.1.1.11). Let (G, b bp , α b ) → N (resp. Q,H

H

κ e

bp H

ord,0 0 b0 , bi0 , α ~ ord,tor b0 , λ (G bH bH ) denote the pullback of the degenerating family b 0,p , α b 0 ) → Nκ e0 p

~e ord,tor ~e ord,tor ord,0 0 ord e0 , ei0 , α e ei, α e0 , λ e λ, eH (G, eHe p , α eH eH eΣ e ord (resp. (G e 0 ,Σ e ord,0 ), and let e 0,p , α e ) → MH, e ) → MH p

p

0 ~ ord,tor b bi, α ~ ord,tor (resp. (G b0 , bi0 , α b λ, b0 , λ ) denote the pullback of the (G, bHb ) → N bH b 0 ) → Nκ e0 κ e

~e ord,tor ~e ord,tor 0 e0 , ei0 , α e ei, α e0 , λ e λ, eH degenerating family (G, eHe ) → M e 0 ,Σ e ord,0 ), as eΣ e ord (resp. (G H, e 0 ) → MH tor tor ~ ord,tor in Construction 7.2.2.1. Then there is a canonical surjection f 0 : N 0 → N κ e ,e κ

κ e

κ e

ord,0 0 ~ ord,tor b0 , bi0 , α b0 , λ bH is canonically isomorphic to the pulunder which (G bH b 0,p , α b 0 ) → Nκ e0 p

0 ord b bi, α ~ ord,tor , and under which (G b0 , bi0 , α ~ ord,tor b0 , λ b λ, bH lback of (G, bHb p , α bH b p ) → Nκ b 0 ) → Nκ e e0 tor b b b is canonically isomorphic to the pullback of (G, λ, i, α bHb ) → Nκe . ~ ord,tor of N ~ ord = e ord,++ , the closure N e Σ, e σ In particular, for each κ e = (H, e) ∈ K Q,H

κ e

κ e

~eord ~e ord,tor ~ ord ,→ etor : N Z[(Φ in M ef,e eΣ e ord and the open (fiberwise dense) embedding κ e f,δ H, σ )] κ e H H ord,tor ~ b Σ b ord ) N depend (up to canonical isomorphism) only on the pair κ = [e κ] = (H, κ e

in Kord,++ . It satisfies the descriptions of stratifications and completions in the Q,H third, fourth, and fifth paragraphs of (1) of Theorem 1.3.3.15. ~ ord,tor ~ ord,tor induces a morphism N ~ ord,tor The morphism fκetor → N → 0 ,e κ : Nκ e0 κ e κ e0 ord,tor ~ ~ N × S0,r 0 which is ´etale locally given by equivariant morphisms between κ e

~ S0,rκ

κ

toric schemes mapping strata to strata, and hence (this induced morphism) is log ´etale essentially by definition (see [45, Thm. 3.5]). Moreover, as in [28, Ch. V, Rem. 1.2(b)] and in the proof of [62, Lem. 7.1.1.4], we have Ri (fκetor 0 ,e ~ ord,tor = 0 κ )∗ ON κ e0

for i > 0 by [50, Ch. I, Sec. 3].

b 0 ) is determined by some H e (resp. H e 0 ) saH and 7.1.1.5. By Lemma 7.2.1.3, we may reb0 ∩ H b (resp. H e 0 ∩ H), e in which case we have a H ~e ord ~e ord ∼ ~ ord,tor canonical (forgetful) morphism fκe0 ,eκ : N C → C = = e ef ∼ e f,δ e 0 Φ ΦH κ e f0 ,δH f0 H H ord,0 ord,0 ∨,ord,0 ord,tor 0 0 0 0 ~ (by construction). Suppose ((G , λ , i , α p , α ), (e c ,e c )) (resp. N b (resp. Proof. Suppose H tisfying Conditions 1.2.4.7 b 0 (resp. H e 0 ) with place H

κ e

ord ), (e cord c∨,ord ))) ((G, λ, i, αHpκ , αH e ,e e κ,p H H

Hκ0

Hκ0 ,p

e0 H

e0 H

~e ord is the tautological object over C ef0 (resp. e f0 ,δ Φ H

H

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~e ord C ef ), as in Construction 7.2.2.1, which determines and is determined by e f,δ Φ H H ord,0 ord,p 0 0 b0 , bi0 , α ~ ord bN , biN , α ~ ord ), b ord , λ b ,λ (G bH bHb p , α bH )→N e κ e κ e ~ ~ ord N ~ ord bH b 0,p , α e0 (resp. (GNκ κ e b 0 ) → Nκ b N N κ e0

κ e0

κ e0

p

p

ord,0 ord 0 b bi, α b0 , bi0 , α ~ ord,tor b λ, b0 , λ (resp. (G, bHb p , α bH the pullback of (G bH bH b ) → b 0,p , α b 0 ) → Nκ e0 p

p

~ ord,tor ) to N ~ ord ~ ord ). N (resp. N κ e0 κ e κ e

Then fκe0 ,eκ is also the canonical morphism ~e ord determined by the universal property of C ef , under which the pullback e f,δ Φ H H ∨,ord ord ord b cHe , e cHe )) is canonically isomorphic to the H-orbit of ((G, λ, i, αHpκ , αHκ,p ), (e

ord,0 ord,0 0 0 )); c∨,ord,0 ((G0 , λ0 , i0 , αH cord,0 c∨,ord,0 )) of ((G0 , λ0 , i0 , αH cord,0 p,α p ,α Hκ,p ), (e Hκ0 ,p ), (e e0 e ,e e e0 , e κ H H H H κ0 b~ ord , bi~ ord , α ~ ord under b ~ ord , λ or, rather, such that the pullback of (G b bp , α bord ) → N Nκ e

Nκ e

H

Nκ e

κ e

bp H

ord,0 0 b0 , bi0 , α ~ ord b b 0 ord , λ fκe0 ,eκ is canonically isomorphic to the H-orbit (G bH ~ ~ ord N ~ ord bH bp , α e0 b ) → Nκ N N κ e0

ord,0 0 b0 , bi0 , α ~ ord b 0 ord , λ of (G bH ~ ~ ord N ~ ord bH b 0,p , α e0 . b 0 ) → Nκ N N κ e0

κ e0

κ e0

κ e0

p

κ e0

p

~ ord,tor Since N is noetherian normal, by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], κ e0 ord,0 0 ~ ord b0 , bi0 , α b 0 ord , λ bH or [62, Prop. 3.3.1.5], since the family (G ~ ~ ord N ~ ord bH e0 is b 0,p , α b 0 ) → Nκ N N κ e0

κ e0

κ e0

p

ord ~ ord unb~ ord , bi~ ord , α b ~ ord , λ canonically isomorphic to the pullback of (G bHb p , α bH b p ) → Nκ e Nκ Nκ Nκ e e e ord,tor ord ord tor ~ ~ ~ der fκe0 ,eκ , as soon as fκe0 ,eκ : Nκe0 → Nκe extends to a morphism fκe0 ,eκ : Nκe0 → ord,0 ord,tor ord,tor 0 b0 b0 0 ~ ~ b ,α b )→N 0 is canonically isomorphic N , we know that (G , λ , i , α b b 0,p b0 κ e H H p tor ord ~ ord,tor under f tor b bi, α b λ, ) → N to the pullback of (G, bHb p , α bH κ e0 ,e κ . Such an extension fκ e0 ,e κ bp κ e ord,tor ord,tor ord ord ~ ~ ~ ~ (resp. Nκe0 ). is necessarily unique, because Nκe (resp. Nκe0 ) is dense in Nκe ~ ord extends locally. ~ ord → N Hence, it suffices to show that fκe0 ,eκ : N 0 κ e κ e ~eord ord,tor ~ Let s¯ be any geometric point of Nκe0 on the Z ˘ f0 ,δ˘f0 ,ρ˘0 )] -stratum [(Φ H H ord,tor ~e 0 ˘ ˘ ˘ ˘ )] is represented by some (ΦHe 0 , δ˘He 0 , ρ˘0 ) with of MHe 0 ,Σ e ord,0 , where [(ΦH e 0 , δH e0 , ρ κ e

˘ ϕ˘ e 0 , ϕ˘ e 0 ), δ˘ e 0 ) representing some cusp label as in Section ˘ e0 = (X, ˘ Y˘ , φ, ˘ e0 , Φ (Z H H −2,H 0,H H e = 1.2.4. For simplicity, let us fix compatible choices of representatives (e Z, Φ e e ˘ ˘ e e ˘ ˘ ˘ ˘ (X, Y , φ, ϕ e−2 , ϕ e0 ), δ) and (Z, Φ = (X, Y , φ, ϕ˘−2 , ϕ˘0 ), δ), as in Sections 1.2.4 and e 0 -orbits. As in Construction 7.2.2.9, each morphism ξ 0 : Spec(V ) → 7.1.1, in their H ord,tor ~ 0 ~ ord,tor N centered at a geometric point s¯ of N , where V is a complete discrete 0 κ e

κ e

valuation ring with fraction field K, and where η := Spec(K) is mapped to the generic point of the irreducible component containing the image of s¯, determines a tuple ˘ b b ‡ , λ b‡ , i b‡ , X, ˘ Y˘ , φ, (B c‡ , b c∨,‡ , τb‡ , τb∨,‡ , [b α\,ord,‡ ]) B

as in (7.2.2.14), where

B

\,ord,‡ [b αH ] b0

b0 H

is an equivalence class of

\,ord,‡ ord,‡ ord,‡ ord,‡ ∨,ord,‡ bord ˘ b0 , ϕ α bH = (Z b−1, bord,‡ cord,‡ c∨,ord,‡ , τbH bHb 0 ) b0 , b b0 b0 , ϕ b0 , ϕ b 0 , δH b0 , b b0 b0 , τ H b−2,H H 0,H H H

as in (7.2.2.15), induced by some \,‡ ‡ ∨,‡ ˘ b0 , b ˘ b0 , ϕ α bH b‡,∼ b‡−1,Hb 0 , ϕ b‡,∼ c‡Hb 0 , b c∨,‡ bH bH b 0 = (ZH b0 , ϕ b 0 , δH b0 , τ b0 , τ b0 ) −2,H 0,H H

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as in (7.2.2.16) over η, in a way analogous to that in Section 4.1.6, and the pair b + in Σ b ˘ . (We (b τ ‡ , τb∨,‡ ) defines an element υτb‡ = υτb∨,‡ in ρb0 for some ρb0 ⊂ P ˘ Φ c0 Φ c0 H

H

should have denoted all these entries with some extra 0 in their superscripts, because ord,0 0 b0 , bi0 , α ~ ord,tor b0 , λ they are determined by the pullback of (G bH bH . But we b 0,p , α b 0 ) → Nκ e0 p

b omit them for the sake of simplicity.) By forming H-orbits, we obtain a tuple \,ord,‡ ‡ ‡ ∨,‡ ‡ ∨,‡ ˘ b b , λ b‡ , i b‡ , X, ˘ Y˘ , φ, (B c ,b c , τb , τb , [b α ]), where

b B B H \,ord,‡ [b αHb ] is an equivalence class of \,ord,‡ ord,‡ ord,‡ ord,‡ ∨,ord,‡ ord,‡ ∨,ord,‡ bord , b ˘ b, ϕ α bH = (Z b−1, bord,‡ ,b cHb , τbHb , τbHb ) b cH b b, ϕ b, ϕ b , δH b H b−2,H H 0,H

induced by some \,‡ ‡,∼ ∨,‡ ‡ ∨,‡ ˘b, b ˘ b, ϕ α bH b−2, b‡−1,Hb , ϕ b‡,∼ c‡Hb , b cH bHb , τbH b = (ZH b, ϕ b , δH b ,τ b ) H 0,H

over η, and the pair (b τ ‡ , τb∨,‡ ) defines the same element υτb‡ = υτb∨,‡ in ρb0 . By 0 b b Hence, under the canonical isomorphism assumption, Σ is a refinement of Σ. 0 ∼ b b b + in Σ b ˘ , so that υτb‡ = υτb∨,‡ PΦ˘ c0 = PΦ˘ c , we have ρb ⊂ ρb for some cone ρb ⊂ P ˘ Φc Φ H

H

H

c H

lies in ρb.

˘f ...ord,Φ˘ H f ~e ord,ΦH ee in By the universal property of M (as the normalization of M e H H ˘f ord,Φ H e b M , which depends only on H; see Definition 1.2.1.15, (4.2.1.27), the dee H

˘b = ˘ b, Φ finition preceding (4.2.1.28), and Proposition 4.2.1.29), the data (Z H H ord,‡ ord,‡ ord,‡ ‡ ˘ ϕ˘ b , ϕ˘ b ), δ˘ b ), (ϕ ˘ Y˘ , φ, b , λ b‡ , i b‡ , ϕ (X, b−2,Hb , ϕ b0,Hb ) and (B b ) on the torus −2,H 0,H H B B b−1,H and abelian parts, which are induced by the corresponding data (ϕ b‡,∼ b‡,∼ b, ϕ b ) and −2,H 0,H ˘

f ~e ord,ΦH ‡ b ‡ , λ b‡ , i b‡ , ϕ (B . e b ) over η, define a canonical morphism ξ1 : Spec(V ) → MH B B b−1,H ˘f ... Φ ord ord ~e H ~e e Φ˘ ,n in By the universal property of C ˘ f,δ˘f → M e (as the normalization of C Φ ord,H f H H H ord CΦH ,δH for some n; see (4.2.1.26), the definition preceding (4.2.1.28), and Propo∨,ord,‡ sition 4.2.1.30), the additional data (b cord,‡ ,b cH ) lifting (b cord,‡ , b c∨,ord,‡ ), induced b b H

~e ord by (b c‡Hb , b c∨,‡ ˘ f,δ˘f lifting ξ1 . b ) over η, define a canonical morphism ξ0 : Spec(V ) → C Φ H H H By the construction of   ~e ord ~e ord ∼ ˘ ⊕ ΨΦ˘ f,δ˘f (`) ΞΦ˘ f,δ˘f,˘σ = SpecO H

H

ord ~ e˘ C ˘ Φ f,δ f H H

˘ σ⊥ `∈˘

H

H

~e ord over C ˘ f,δ˘f , which we can canonically identify as Φ H H   ~b ord ~b ord ∼ ˘ ( `) Ξ Spec ⊕ Ψ ˘ c,δ˘c ˘ c,δ˘c = Φ Φ O H H H H ord ~ b˘ C ˘ Φ c,δ c H H

˘ S b˘ `∈ Φ

c H

ord

~b over C ˘ c,δ˘c (see Proposition 7.1.2.6), it enjoys the universal property (similar Φ H

H

~e ord ~e ord to that of Ξ ˘ f,δ˘f ; see (4.2.1.25), the definition preceding (4.2.1.28), ˘ f,δ˘f → C Φ Φ H

H

H

H

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and Propositions 4.2.1.30 and 4.2.1.46) such that the final part of the data ord,‡ ∨,ord,‡ ‡ ∨,‡ (b τH , τbHb ) lifting (b τ ‡ , τb∨,‡ ), induced by (b τH bH b b, τ b ) over η, determines a cano~e ord nical morphism ξ˜K : Spec(K) → Ξ ˘ f,δ˘f,˘ Φ σ lifting ξ0 under the canonical morphism H

H

~e ord ~e ord Ξ τ ‡ , τb∨,‡ ) lies in ˘ f,δ˘f,˘ ˘ f,δ˘f . Since the element υτb‡ = υτb∨,‡ defined by (b Φ σ → CΦ H H H H 0 ρb ⊂ ρb, by the construction of   ~e ord ~e ord ∼ ˘ ΞΦ˘ f,δ˘f,˘σ (˘ ρ) = SpecO ⊕ ΨΦ˘ f,δ˘f (`) H

H

ord ~ e˘ C ˘ Φ f,δ f H H

H

˘ σ ⊥ ∩ ρ˘∨ `∈˘

H

(see (7.2.1.9)), which we can canonically identify as   ~b ord ~b ord ˘ ΞΦ˘ c,δ˘c (b ρ) = SpecO ⊕ ΨΦ˘ c,δ˘c (`) H

H

ord ~ b˘ C ˘ Φ c,δ c H H

H

˘ ρ∨ `∈b

H

b and on ρb∨ ∼ (see (7.1.2.10)), which depends only on H ˘ ⊥ ∩ ρ˘∨ , and by the same =σ argument as in the proof of Proposition 4.2.2.8, the morphism ξ˜K extends to a ~e ord ρ) lifting ξ0 under the canonical morphism morphism ξ˜ : Spec(V ) → Ξ ˘ f,δ˘f,˘ Φ σ (˘ H

H

~e ord ~e ord ρ) → C ˘-stratum Ξ ˘ f,δ˘f , which maps the special point of Spec(V ) to the ρ ˘ f,δ˘f,˘ Φ Φ σ (˘ H

H

H

H

~e ord ~e ord ρ). (Alternatively, we can noncanonically lift υτb‡ = υτb∨,‡ to Ξ ˘ f,δ˘f,˘ ˘ f,δ˘f,ρ˘ of Ξ Φ σ (˘ Φ H

H

H

H

~e ord ~e ord ρ) directly, and invoke the elements of ρ˘ ⊂ P+ ˘ f,δ˘f and Ξ ˘ f,δ˘f (˘ Φ ˘ , work with ΞΦ Φ H

f H

H

H

H

original Proposition 4.2.2.8.) Since V is complete, ξ˜ induces a morphism ξˆ from ~e ord ~e ord ρ) along its ρ˘-stratum Spf(V ) to X ˘ f,δ˘f,˘ ˘ f,δ˘f,˘ σ (˘ Φ σ ,ρ˘, the formal completion of ΞΦ H

~e ord Ξ ˘ ,δ˘ Φ f H

H

H

H

~e ord . Then the composition of ξˆ with the canonical morphism X ˘ f,δ˘f,˘ Φ σ ,ρ˘ → H H ord,tor ~ gives a canonical morphism ξ : Spec(V ) → N .

˘ f,ρ H

~ ord,tor N κ e κ e As explained in Construction 7.2.2.9, ξη := ξ|η : η = Spec(K) → ord,0 0 b0 , bi0 , α ~ ord is determined by the pullback (G b 0η , λ bH N η η bH b 0,p ,η , α κ e b ,η ) → Spec(K) of p

ord,0 0 b0 , bi0 , α ~ ord,tor ~ ord,tor b0 , λ , whose bH under ξη0 := ξ 0 |η : Spec(K) → N (G bH b 0,p , α b 0 ) → Nκ e0 κ e0 p

ord,0 0 b0 , bi0 , α b b0 , λ H-orbit (G bH η η η bH b p ,η , α b ,η ) → Spec(K) is (as explained in the first paragraph p

ord b bi, α ~ ord,tor under b λ, of this proof) isomorphic to the pullback of (G, bHb p , α bH b p ) → Nκ e ord,tor ord,tor 0 ord ~ ~ ~ . Hence, the composition of ξη : Spec(K) → Nκe0 with fκe0 ,eκ : Nκe0 → Nκe ~ ord , and ξ : Spec(V ) → N ~ ord,tor can be ξη = fκe0 ,eκ ◦ ξη0 by the universal property of N κ e κ e ~ ord . interpreted as a (necessarily unique) extension of fκe0 ,eκ ◦ ξη0 : Spec(K) → N κ e 0 ~ ord,tor Since ξ 0 : Spec(V ) → N and s ¯ (the prescribed center of ξ ) are arbi0 κ e ~ ord,tor trary, and since N is noetherian normal, this shows that fκe0 ,eκ extends to 0 κ e

tor fκetor 0 ,e κ , as desired. The argument also shows that the restriction of fκ e0 ,e κ to the ord,tor ord ord ~e ~e e ˘ ˘ 0 ∼ ˘ e 0 , δ˘ e 0 , ρ˘0 )]-stratum ~Z [(Φ ˘ f0 ,δ˘f0 )ρ˘0 of M e 0 ,Σ e ord,0 coincides with the [(Φ f0 ,δ f0 ,ρ˘ )] = (ΞΦ H H H H

H

H

H

~e ord ~e ord canonical morphism (Ξ ˘ f0 ,δ˘f0 )ρ˘0 → Ξ ˘ f,δ˘f,ρ˘ on geometric points. Since the images Φ Φ H

H

H

H

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~e ord b = pr(Sb ˘ of such morphisms cover Ξ ˘ f,δ˘f,ρ˘, because ρ Φ H

ρb0 = pr(Sb ˘

∨ Φ c )R H

H

(˘ ρ0 ), the morphism fκetor 0 ,e κ is surjective.

∨ Φ c )R H

(˘ ρ) is covered by the cones

By considering ordinary good algebraic models in Section 5.1.2 and by arguing as ~ ord,tor ~ ord,tor in the paragraph preceding [61, Lem. 5.10], the morphism fκetor →N 0 ,e κ : Nκ e0 κ e ~e ord ~e ord 0 is ´etale locally given by the canonical morphism ΞΦ˘ f0 ,δ˘f0 ,˘σ0 (˘ ρ ) → ΞΦ˘ f,δ˘f,˘σ (˘ ρ), H

H

H

H

~e ord because the tautological data (as in (7.2.2.14)) over Ξ ρ0 ) is the pul˘ f0 ,δ˘f0 ,˘ Φ σ 0 (˘ H

~e ord lback of the one over Ξ ρ). ˘ f,δ˘f,˘ Φ σ (˘ H

ord

H

By construction, the induced morphism

H

ord

~e ~e Ξ ρ0 ) → Ξ ρ) × ~S0,rκ0 is log ´etale and equivariant with respect ˘ f0 ,δ˘f0 ,˘ ˘ f,δ˘f,˘ Φ σ 0 (˘ Φ σ (˘ H

H

H

H

~ S0,rκ

e˘ e˘ to the canonical homomorphism E ΦH σ 0 → EΦ σ between tori, which (by Prof0 ,˘ f,˘ H position 1.3.2.56 again) can be canonically identified with the canonical log ´etale ~b ord ~b ord 0 morphism Ξ (b ρ ) → Ξ ρ), equivariant with respect to the canonical ho˘ ˘ ˘ c,δ˘c (b ΦH Φ c0 ,δH c0 H H b b momorphism E ˘ → E ˘ between tori (dual to the canonical homomorphism ΦH c0

ΦH c

b˘ → S b ˘ up to canonical identifications; see Definition 1.2.4.29). S ΦH ΦH c c0 ~ ord,tor satisfies the descriptions We note that the above argument shows that N κ e of stratification and formal completions as in the third, fourth, and fifth paragraphs of (1) of Theorem 7.1.4.1, because they follow from (2) and (5) of Theorem 5.2.1.1 e tor (for M eΣ e ord ), from Lemma 5.2.4.38, and from the justifications provided in Section H, 5.2.4. The remainder of the proposition then follows. Thanks to Lemma 7.2.1.3 and Proposition 7.2.2.19, we can make the following: e ord,++ Definition 7.2.2.20. (Compare with Definition 1.3.4.20.) For κ e ∈ K Q,H ord,++ which defines κ = [e κ] ∈ KQ,H (see Definition 7.1.1.11), we shall denote ~ ord,grp and κ ~ ord ,→ N ~ ord,tor by κisog : κ eisog : Hom (Q, G ~ ord )◦ → N etor : N O

κ e

κ e

MHκ

κ e

~ ord,grp and κtor : N ~ ord ,→ N ~ ord,tor , respectively. For κ HomO (Q, GM e and ~ ord ) → Nκ κ κ ◦

Hκ

e ord,++ such that κ0 = [e κ e0 in K κ0 ]  κ = [e κ] in Kord,++ , we shall denote the caQ,H Q,H grp isog ord ord ~ ord,grp → ~ 0 → N ~ , f 0 := κ e ◦ ((e κ0 )isog )−1 : N nonical morphisms fκe0 ,eκ : N κ e

κ e

κ e

κ e ,e κ

~ ord,grp × M ~ ord , and f tor ~ ord,tor ~ ord,tor by fκ0 ,κ : N ~ ord ~ ord grp N →N κ0 → Nκ , fκ0 ,κ := Hκ0 κ e0 ,e κ : Nκ κ e e0 κ e ~ ord M Hκ

isog

κ

~ ord,grp ~ ord,grp × M ~ ord , and f tor ~ ord,tor → N ~ ord,tor , ◦ ((κ0 )isog )−1 : N → N κ κ κ0 ,κ : Nκ0 Hκ0 κ0 ~ ord M Hκ

respectively. (That is, we drop the tildes in all such notations.) We shall denote by ~bord ~ ord,tor , which is the [(Φ ˘ b , δ˘ b , τb)]-stratum of N ˘ e , δ˘ e , τ˘)]-stratum Z ˘ ˘ the [(Φ [(ΦH τ )] c,δH c,b

H

H

κ

H

H

~eord ~e ord ~b ord ∼ ~ ord,tor under the canonical identification Z[(Φ˘ f,δ˘f,˘τ )] ∼ Ξ Ξ = = ˘ f,δ˘f,˘ ˘ c,δ˘c,b Φ τ Φ τ of Nκ e H H H H H H

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~ ord,tor and N ~ ord,tor (up to canonical isomorphism) (when (Φ ˘ b , δ˘ b , τb) is between N κ κ e H H ˘ e , δ˘ e , τ˘) as in Section 1.2.4). determined by (Φ H H b Σ) b and κ0 = (H b0 , Σ b 0 ) are elements of Corollary 7.2.2.21. Suppose that κ = (H, ord,++ 0 KQ,H satisfying κ  κ, and that Hκ and Hκ0 are equally deep as in Definition ~ ord,tor ~ ord,tor (see Proposition 3.2.2.9. Then the canonical morphism f tor : N → N 0 0 κ ,κ

κ

κ

7.2.2.19) is proper. e ord,++ . Take a third element e Σ, e σ Proof. Suppose κ = [e κ] for some κ e = (H, e) ∈ K Q,H b 00 , Σ b 00 ) ∈ Kord,++ such that H b 00 = H b 0 (and so that Hκ00 = Hκ0 ), such κ00 = (H Q,H e ord,++ e 00 , Σ e 00 , σ that κ00  κ0 , and such that κ00 = [e κ00 ] for some κ e00 = (H e00 ) ∈ K Q,H

e 00 ⊂ H e are equally deep as in Definition 3.2.2.9, such that Σ e 00 is a such that H 00 e refinement of Σ, and such that σ e is contained in σ e. Then we have canonical tor ~ ord,tor → N ~ ord,tor ~ ord,tor ~ ord,tor such that surjections fκtor and f : N → N 00 ,κ0 : Nκ00 00 0 κ κ ,κ κ κ00 tor tor tor tor fκ00 ,κ = fκ0 ,κ ◦ fκ00 ,κ0 (see Proposition 7.2.2.19), and fκ00 ,κ is proper because it is the ~ ord,tor → M ~ ord,tor , restriction (to a closed subscheme) of the canonical surjection M e 00 ,Σ e 00 eΣ e H H, e and H e 00 are equally deep (see Proposition 5.2.2.2). the latter being proper because H tor Consequently, fκtor 0 ,κ is also proper, because fκ00 ,κ0 is surjective.

7.2.3

Extensibility of fκ

The goal of this subsection is to show that, when κ ∈ Kord,++ (see Definitions Q,H,Σord ord ord ~ ~ 7.1.1.11 and 7.1.1.19), the structural morphism fκ : Nκ → MH,rκ (see Section ~ ord,tor = ~ ord,tor → M 7.2.1) extends (necessarily uniquely) to a morphism f tor : N κ

κ

H,rκ

~ ord,tor between the compactifications. Recall that fκ is the composition of the M H,Σord ,rκ ~ ord . ~ ord → M ~ ord with the canonical morphism M ~ ord → M structural morphism N Hκ

Hκ

κ

H,rκ

Let us begin with an arbitrary element κ ∈ Kord,++ (not necessarily in Q,H ord,++ e ord,++ e e ord , σ K κ] for some κ e = (H, Σ e) ∈ K ord ), which is of the form κ = [e Q,H

Q,H,Σ

e ord,++ord ). (not necessarily in K Q,H,Σ By Construction 7.2.2.1, we have the pullback (7.2.2.4) ord b:G ~ ord,tor b λ b→G b ∨ , bi, α (G, bHb p , α bH b ) → Nκ p

of (7.2.2.2) to b ~ ord scheme G N κ

(resp. G∨ ~ ord ), N κ

~ ord,tor (see Definition 7.2.2.20), such that the semi-abelian =N κ e e ~ ord (resp. G b∨ = G e ∨ ) (cf. (7.2.2.5)) is the extension of G~ ord =G ~ ord ~ ord Nκ Nκ N N κ κ ord ord ~ ~ the pullback under fκ : Nκ → MH,rκ of the abelian scheme GM ~ ord H,r ~ ord,tor N κ

κ

~ ord ~ ord with e~ ord (resp. Te∨ ) over N (resp. G∨ κ ~ ord ) over MH,rκ , by the split torus TN ~ ord M N κ

H,rκ

κ

e (resp. Ye ), parameterized by the tautological object e e → character group X c : X ord ~e ~ ord ∼ G∨ c∨ : Ye → GN = C ef . By taking the abelian parts e f,δ ~ ord ) over Nκ Φ ~ ord (resp. e N κ

κ

H

H

ord b bi, and (b of λ, αHb p , α bH b ), we obtain a polarization, an O-endomorphism structure, p

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b ~ ord , which and the ordinary level structure (αHp , αord Hp ) on the abelian part of GN κ ~ ord to N ~ ord by agree with the pullbacks of the data λ, i, and (αHp , αord ) over M Hp

H,rκ

κ

~ ord → M ~ ord . By noetherian normality of (the closure) N ~ ord,tor (of N ~ ord in fκ : N κ κ κ H,rκ ord,tor ~e M ), and by [92, IX, 2.4], [28, Ch. I, Prop. 2.9], or [62, Prop. 3.3.1.7], the e H embedding b ~ ord = G e ~ ord Te~ ord ,→ G Nκ

Nκ

Nκ

~ ord extends (uniquely) to an embedding of group schemes over N κ b=G e ~ ord,tor Te~ ord,tor ,→ G Nκ

Nκ

~ ord,tor , and the quotient of group schemes over N κ b Te~ ord,tor = G e ~ ord,tor /Te~ ord,tor G := G/ Nκ

Nκ

(7.2.3.1)

Nκ

~ ord can be iden~ ord,tor whose restriction G~ ord to N is a semi-abelian scheme over N κ κ Nκ b tified with G~ ord , the abelian part of G~ ord . Similarly, we obtain Nκ

Nκ

∨

b ∨ /Te∨ord,tor = G e ∨ord,tor /Te∨ord,tor , G := G ~ ~ ~ N N N κ

κ

(7.2.3.2)

κ

∨ GN ~ ord κ

~ ord can be identified ~ ord,tor whose restriction to N a semi-abelian scheme over N κ κ b ∨ . By [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, with G∨ , the abelian part of G ord ord ~ ~ N N κ

κ

∨

Prop. 3.3.1.5], the semi-abelian G carries (unique) additional structures λ : G → G ~ ord is the pullback of the tautological and i, such that the restriction of (G, λ, i) to N κ ord ~ ord . Together with ~ ord → M ~ tuple (GM ~ ord , λM ~ ord , iM ~ ord ) over MH,rκ by fκ : Nκ H,rκ H,rκ

H,rκ

H,rκ

the ordinary level structure (αHp , αord ~ ord Hp ) already defined on the abelian part GN κ b of G~ ord , we obtain a degenerating family N

~ ord,tor (G, λ, i, αHp , αord Hp ) → Nκ

(7.2.3.3)

~ ord . By applying the same construction to (7.2.2.8), we obtain a degeneof type M H rating family ~ ord,tor (G, λ, i, αH ) → N (7.2.3.4) κ ord ~ of type MH , with the same (G, λ, i), where αH is defined only on N ⊗ Q, such κ

Z

that the pair (αHp , αord Hp ) ⊗ Q is determined by αH as in Proposition 3.3.5.1. Z

Proposition 7.2.3.5. Suppose κ ∈ Kord,++ ; i.e., κ is an element of Kord,++ Q,H Q,H,Σord ~ ord,tor = satisfying Condition 7.1.1.17 (for the same Σord in the definition of M H,rκ

~ ord,tor e ord,++ord ; i.e., κ M ). (In this case, κ = [e κ] for some κ e∈K e is an element of H,Σord ,rκ Q,H,Σ ord,++ ~ ord → e K satisfying Condition 7.1.1.15.) Then the structural morphism fκ : N Q,H

κ

~ ord extends (necessarily uniquely) to a morphism f tor : N ~ ord,tor → M ~ ord,tor M κ κ H,rκ H,Σord ,rκ between the compactifications, which satisfies the descriptions concerning stratifications and formal completions in the second paragraph of (2) of Theorem 7.1.4.1.

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~ ord,tor = N ~ ord,tor be as in Proof. Let ξ : Spec(V ) → N κ κ e ‡ ord,‡ 7.2.2.9. As explained there, the pullback (G‡ , λ‡ , i‡ , αH p,α Hκ,p ) κ of (7.2.3.3) under ξ is a degenerating family as in (7.2.2.18) in (V ) → DEGPEL,... (V ), which corresponds image of DEGPEL,Mord M ord H H κ

10374-main

Construction → Spec(V ) the essential to an object

κ

\,ord,‡ (B ‡ , λB ‡ , iB ‡ , X, Y, φ, c‡ , c∨,‡ , τ ‡ , [αH ]) as in (7.2.2.17) in the essential image of κ ... (V ) → DD DDPEL,Mord ord (V ), and the morphism υτ ‡ : Y × X → Z defined PEL, M H H κ

κ

by τ ‡ as in (4.2.2.7) (using the discrete valuation υ : Inv(V ) → Z of V ) defines an element υτ ‡ ∈ P+ b = pr(Sb ˘ )∨ (˘ ρ) under (1.2.4.37). If κ ΦH in the image ρ of ρ Φc R H

satisfies Condition 7.1.1.17, which means any such ρ is contained in some cone in the cone decomposition ΣΦH (in Σord ), then the condition in (6) of Theorem 5.2.1.1 is satisfied for all such ξ, and hence it follows that fκ extends to a (necessarily unique) fκtor , as desired. It satisfies the descriptions concerning stratifications and formal completions in the second paragraph of (2) of Theorem 7.1.4.1 because the universal property of Mtor H,Σ given by (6) of Theorem 1.3.1.3 is given in terms of the degeneration data, which determines the (approximations of) the invertible sheaves ~ ord (`), and the same is true for the constructions of the canonical morsuch as Ψ ΦH ,δH phisms in Lemmas 5.2.4.38 and 7.1.2.29, and Proposition 7.1.2.17, using the various universal properties (all given in terms of degeneration data). Now let us resume the notation system at the end of Section 7.2.1 (using τ˘ instead of ρ˘). By construction (and the proof of Proposition 7.2.3.5), for κ = [e κ] = e Σ, e σ [(H, e)] ∈ Kord,++ , we have a commutative diagram Q,H,Σord ~ ord,tor o N κ

~e ord X ˘ f,δ˘f,˘ Φ σ ,˘ τ

~e ord /C ˘ ,δ˘ Φ

 ~ ord X ΦH ,δH ,τ,rκ

 ~ ord /C ΦH ,δH ,rκ

H

f H

H

f H

(7.2.3.6)

f tor

 ~ ord,tor o M H,rκ

of canonical morphisms (where all subscripts “rκ ” mean base changes from ~S0,rH to ~S0,r ) whenever the image of τ˘ under the (canonical) second morphism in (1.2.4.20) κ is contained in τ . It is worth recording the following observations: b Σ b ord ) ∈ Kord,++ extends to some κ0 = (H, b Σ) b ∈ Lemma 7.2.3.7. Suppose κ = (H, Q,H std,++ b ord extends to K (see Definition 7.1.1.20), which means in particular that Σ Q,H

b as in Lemma 1.2.4.42. That is, κ = [e e Σ e ord , σ some Σ κ] for some κ e = (H, e) ∈ ord,++ ord,++ 0 e e e e KQ,H , and κ e extends to some κ e = (H, Σ, σ e) ∈ KQ,H , which means in partiord ~ e e extends to some projective smooth e ord for M cular that the projective smooth Σ H e e Σ for MHe , which can always be achieved by Proposition 5.1.3.4. Hence, we set κ0 = [e κ0 ] ∈ Kord,++ . Then there are canonical open immersions Q,H ~ ord,tor ⊗ Q ,→ Ntor N = Ntor (7.2.3.8) 0 0 × S0,r κ

Z

κ ,rκ

κ

S0

κ

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inducing an isomorphism ∼ b ~bord b ˘ ˘ Z[(Φ˘ c,δ˘c,bτ )] ⊗ Q → Z ˘ c,δ˘c,b τ )] × S0,rκ [(Φ τ )],rκ := Z[(Φ c,δ c,b H

H

H

Z

H

H

H

(7.2.3.9)

S0

˘ b of (see Definition 7.2.2.20; cf. Definition 2.2.3.4) when the underlying Z H ˘ ˘ [(ΦHb , δHb , τb)] (suppressed in the notation by our convention; cf. Lemma 1.2.4.42) b ˘ ˘ is compatible with D; otherwise, the pullback of Z [(ΦH τ )],rκ under (7.2.3.8) is c,δH c,b empty. Proof. This follows from (2) and (7) of Theorem 5.2.1.1, because by construction ord ~e ord,tor ~ ord,tor is the closure of Nκ = ~Z e e e N eΣ e ord . κ [(Φ f,δ f,e σ )] in MH, H

H

Lemma 7.2.3.10. With the same setting as in Lemma 7.2.3.7, suppose there exist Σord and Σ such that Σord extends to Σ as in Proposition 5.1.3.2, and such that b Σ b ord ) ∈ Kord,++ord and κ0 = (H, b Σ) b ∈ Kstd,++ . (This is always possible up κ = (H, Q,H,Σ Q,H,Σ b with a refinement satisfying Condition 7.1.1.17 for this Σ.) Then the to replacing Σ diagram ~ ord,tor ⊗ Q N κ

(7.2.3.8)

Z

/ Ntor κ0 ,rκ

fκtor ⊗ Q

 ~ ord,tor M ⊗Q H,Σord ,rκ

fκtor 0

Z

Z

(7.2.3.11)

(5.2.1.2)

 / Mtor H,Σ,rκ

of canonical morphisms is commutative and Cartesian. are induced Proof. The diagram (7.2.3.11) is commutative because fκtor and fκtor 0 ord,tor tor ~ by compatible universal properties of MH,Σord and MH,Σ (see (6) of Theorem 1.3.1.3 and (6) and (7) of Theorem 5.2.1.1). By (2) and (7) of Theorem 5.2.1.1, the open image of (5.2.1.2) is the union of the strata Z[(ΦH ,δH ,τ )],rH of Mtor H,Σ,rκ (as in (2) of Theorem 1.3.1.3) whose underlying cusp labels [(ΦH , δH )] are ordinary. Similarly, by Lemma 7.2.3.7 (which is based on the same argument), the open image of (7.2.3.8) is the union of the strata e ˘ ˘ e tor e e e 0 Z e Σ,r e κ having Nκ = Z[(Φ [(ΦH τ )],rκ of MH, σ )],rκ as a face, whose underlying f,δH f,˘ f,δH f,e H ˘ ˘ e , δ e )] are ordinary. Hence, it follows from Lemma 7.1.1.8 that the cusp labels [(Φ H

H

diagram (7.2.3.11) is Cartesian, as desired. tor Corollary 7.2.3.12. For every κ ∈ Kord,++ Q,H,Σ , the characteristic zero fiber fκ ⊗ Q Z

~ ord,tor → M ~ ord,tor of the morphism fκtor : N is proper (although fκtor might not be). κ H,Σord ,rκ b with some refinement, we may assume that the assumptiProof. Up to replacing Σ ons of Lemma 7.2.3.10 are satisfied. Then, since the diagram (7.2.3.11) is Cartesian, tor the properness of fκtor ⊗ Q follows from that of fκtor : Ntor κ0 → MH,Σ (see (2) of TheZ

orem 1.3.3.15).

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7.2.4

Proposition 7.2.4.1. With the setting as in Proposition 7.2.3.5, suppose moreover that Hκ and H are equally deep as in Definition 3.2.2.9. Then the morphism ~ ord,tor → M ~ ord,tor fκtor : N is proper. κ H,Σord ,rκ b Σ) b ∈ Kord,++ord , under the assumption Proof. Starting with H, Σord , and κ = (H, Q,H,Σ e Σ e ord , σ that Hκ and H are equally deep, by making compatible choices of κ e = (H, e) ∈ ord,++ ord,+ 0 ord,0 0 0 e ord,0 0 e e e K κ]), H , Σ , κ e = (H , Σ ,σ e) ∈ K ord (such that κ = [e 0 ord,0 , Q,H,Σ

Q,H ,Σ

b0 , Σ b ord,0 ) := [e eaux , Σ e ord , σ κ0 = (H κ] ∈ Kord,+ , Haux , Σord eaux = (H aux , κ aux eaux ) ∈ Q,H0 ,Σord,0 ord,+ ord,+ ord e b b KQaux ,Haux ,Σord , and κaux = (Haux , Σaux ) := [e κaux ] ∈ KQaux ,Haux ,Σord , which are aux aux all equally deep of some depth r at p (in the obvious sense), which can always be achieved by replacing the collections of cone decompositions with refinements, we obtain by Proposition 5.2.2.2 (with g = 1), by (7.1.4.11) (with gb = 1) and (4d) of Theorem 7.1.4.1, and by Proposition 6.1.1.6 a diagram ~e ord,tor o o M eΣ e ord H, O

proper can.

~e ord,tor M e 0 ,Σ e ord,0 H O

can.

can.

?

?

~ ord,tor o o N κ fκtor



proper can.

~e ord,tor /M e aux ,Σ e ord H O aux can.

proper

~ ord,tor N κ0 fκtor 0

proper



~ ord,tor ~ ord,tor o proper o M M H,Σord ,rκ can. H0 ,Σord,0 ,r

κ0

proper

?

~ ord,tor /N κaux 

fκtor aux

~ ord,tor ord /M Haux ,Σ ,r aux

in which the upward arrows are canonical closed immersions, the leftward arrows are canonical proper surjections, the rightward arrows are proper morphisms, (where the middle ones are induced by the top ones,) and the downward arrows are surjections. Since all the morphism are induced by universal properties (using (6) of Theorem 1.3.1.3), the diagram is commutative. Therefore, to show that fκtor is proper, it suffices to show that fκtor is proper (and hence so is fκtor Therefore, for the 0 ). aux purpose of proving this proposition, by replacing objects and morphism in the first column of the above diagram with those auxiliary ones in the third column, we may assume the following: • p is a good prime for (O, ?, L, h · , · i, h0 ), and hence also for e h · , · ie, e (O, ?, L, h0 ) (see Remark 1.2.4.2); p e ep = Uebal (pr ); e e e p is neat and where H • H = H Hp , where H p,1 p p bal r • Hκ = H = H Hp , where H is neat and where Hp = Up,1 (p ); e with subgroups that are equally deep at p, and and we are free to replace H and H ord ord e to replace Σ and Σ with refinements. (Then rκ = rH and hence we may drop the subscripts “rκ ” from the notation for base changes from ~S0,rH to ~S0,rκ .)

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Since Assumption 5.2.3.1 is satisfied by Lemma 5.2.3.2, by Lemma 5.2.3.9 and e ord with a refinement, we have compatible Proposition 5.2.3.18, up to replacing Σ canonical morphisms ~e tor ~e ord,tor M f κ e de0 pol,r eΣ e ord ,→ M H, H,

(7.2.4.2)

~e tor e torp p M f κ →M e de0 pol,r H, e ,Σ e H

(7.2.4.3)

(as in (5.2.3.19)) and

ord

f extending (Σ f ) as in Proposition 5.1.3.4, e pol) e ord , pol (as in (5.2.3.12)) for some (Σ, p e e e e Consider the some integer d0 ≥ 1, and some collection Σ for MHe p inducing Σ. tautological degenerating family ord,tor

~e e ei, α e λ, (G, eHe p , α epord r ) → MH, eΣ e ord

(7.2.4.4)

~e ord ord ep = Uebal (pr ). Then the because H epord eH of type M r e , where we denote α H p,1 e by α p

ord,tor

~e e ei, α e λ, subtuple (G, eHe p ) → M eΣ e ord extends to a degenerating family H, ~e tor e ei, α e λ, (G, eHe p ) → M f κ e de0 pol,r H,

(7.2.4.5)

e e p , by taking the pullback under (7.2.4.3) of the tautological degenerating of type M H e tor . Hence, by (6) of Theorem 1.3.1.3, the image family of the same type over M e p ,Σ ep H

~e tor of (7.2.4.2) can be characterized as the maximal open subscheme of M f κ such e de0 pol,r H, ord that the pullback of (7.2.4.5) carries an ordinary level structure α e pr . ~ ord,tor is quasi-projective over ~S0,r and M ~ ord,tor Since N is separated over ~S0,rκ , κ κ H,Σord tor the morphism fκ is separated and finite type. Moreover, the restriction of fκtor ~ ord is the proper morphism fκ : N ~ ord → M ~ ord (underlying an abelian scheme to N κ κ H ord ord,tor ~ ~ torsor), and Nκ ⊗ Q is open and dense in Nκ by construction. Also, fκtor ⊗ Q : Z

Z

~ ord,tor ~ ord,tor ⊗ Q → M N ⊗ Q is proper by Corollary 7.2.3.12. Therefore, in order κ H,Σord ,rκ Z

Z

to show that fκtor is proper (by the valuative criterion), it suffices to show that, for each complete discrete valuation ring V with fractional field K of characteristic zero and algebraically closed residue field k of characteristic p, each morphism ~ ord ⊗ Q such that ξ0 := fκ ◦ ξ˘0 : Spec(K) → M ~ ord ⊗ Q extends ξ˘0 : Spec(K) → N κ H,rκ Z

Z

~ ord,tor to a morphism ξ : Spec(V ) → M also extends to a morphism ξ˘ : Spec(V ) → H,Σord ~e ord,tor ~ ord,tor . Note that ξ˘0 induces a morphism ξ : Spec(K) → M N eΣ e ord ⊗ Q. Since 0 H, κ Z

~e tor M S0,rκ , the morphism ξ 0 extends to f κ is projective (and hence proper) over ~ e de0 pol,r H, ~e tor a morphism ξ : Spec(V ) → MH, f κ . We would like to verify the following: e de0 pol,r Claim 7.2.4.6. The image of ξ is contained in the image of (7.2.4.2).

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Since all objects involved are locally of finite presentation, we may assume that V is the completion of a localization of an algebra of finite type over Z. (This might not be absolutely necessary, but will simplify the various constructions.) Then, by ~e tor lifting (the finitely many) generators to local rings of M f κ (together with e de0 pol,r H, the generators of the ideals to localize, and by taking suitable completions), there e and prime exists a complete noetherian normal domain Ve , with fractional field K e e e e e ideals I ⊂ J ⊂ V , such that V is complete with respect to J, such that Ve /Ie ∼ = V, e Ie defines the maximal ideal of the complete discrete valuation ring and such that J/ ~e tor V , together with a morphism ξe : Spec(Ve ) → M f κ inducing the morphism e de0 pol,r H, tor ~ ~e ord,tor eeef ˘ e →M ξ : Spec(V ) = Spec(Ve /I) eΣ e ord , and H,d0 pol,rκ extending ξ0 : Spec(K) → MH, e e . Now Claim 7.2.4.6 follows from the e →M inducing a morphism ξe0 : Spec(K) H,rκ following: Claim 7.2.4.7. The image of ξe is contained in the image of (7.2.4.2). Let us denote by e e , ei e , α ee, λ e (G e p ,V e ) → Spec(V ) V V V eH the pullback of (7.2.4.5), and by e e , ei e , α e e,λ (G ee K

K

K

e) H,K

(7.2.4.8)

e → Spec(K)

(7.2.4.9)

e e , ei e , α e e . Let e of the tautological tuple (G ee ,λ the pullback under K e ) over MH Mf M f M f eH H

H

H

ord αpord,0 epord,#,0 , νepord eH, αHe p ,Ke , α epord (e αHe p ,Ke , α eH eK e r ,K r ,K r ,K r ,K e = (e e )) be determined by α e p ,K e ) = (e e,α e e as in Section 4.1.6. Then the pullback of (7.2.4.8) to Spec(K) is canonically isomore e , ei e , α e e,λ e with the first three entries as in (7.2.4.9). phic to (G e e p e ) → Spec(K), K

K

K

H ,K

We shall denote with subscripts “V ” (resp. K) the pullbacks of these objects under ξ (resp. ξ 0 ). e e and e h · , · ie, e Since p is a good prime for (O, ?, L, h0 ) (see Remark 1.2.4.2), λ K e ee : hence λVe are both prime-to-p polarizations, which induce an isomorphism λ V ∼ × e e [pr ] → G e ∨ [pr ]. Since νeord ∈ (Z/pr Z) G always uniquely extends to some e r e e V Spec(K) V p ,K ×

r νepord epord e ) , we see that α r ,V r ,K e ∈ (Z/p Z)Spec(V e extends to some ordinary level structure ord,0 ord,#,0 ord ord e = (e α α e ,α e , νe ) over V if and only if α eord,0 : (Gre0 r )mult ,→ e pr ,V

e pr ,V

e pr ,V

e pr ,V

e pr , K

D,p

e Spec(K)

0 e e [pr ] extends to some α e e [pr ]. Thus, to verify Claim G epord,0 )mult e ) ,→ G r ,V e : (Gre K V D,pr Spec(V

7.2.4.7 (which is equivalent to the extensibility of α epord epord r ,K r ,V e to α e ), it suffices to verify the following: e denote the finite flat subgroup scheme of multiplicative type Claim 7.2.4.10. Let H r e e 0 of multiplicaof GVe [p ] uniquely lifting the maximal finite flat subgroup scheme H e 0 [pr ] (by [26, IX, 3.6 bis] and [35, III-1, 5.1.4]; see also [34, IX, 6.1]), tive type of G e e over Spec(k). Then the generic fiber H e e of H e where G0 is the special fiber of G K ord,0 coincides with image(e αpr ,Ke ) (see Definition 3.3.3.1).

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Let Ve 1 denote the localization of Ve at the kernel of Ve  V , which has residue field K. Since the image of Spec(K) under ξe is contained in the open image of e1 (7.2.4.2), so is the image of Spec(Ve 1 ). Since α epord r ,K e already extends to Spec(V ), ord,0 e K [pr ], and (by [26, IX, 3.6 bis] ,→ G it makes sense to write α e r : (Gre0 r )mult p ,K

D,p

Spec(K)

and [35, III-1, 5.1.4] again, applied to Ve 1 ), Claim 7.2.4.10 follows from the following: Claim 7.2.4.11. Let H denote the finite flat subgroup scheme of multiplicative type e V [pr ] uniquely lifting the maximal finite flat subgroup scheme H 0 of multiplicaof G e 0 [pr ], then the generic fiber H K of H coincides with image(e tive type of G αpord,0 r ,K ). Let us denote by ord (GV , λV , iV , αHp ,K , αH ) → Spec(V ) p ,V

(7.2.4.12)

~ ord,tor the pullback under ξ : Spec(V ) → M of the tautological degenerating family H,Σord ord,tor ord ~ (G, λ, i, αHp , αpr ) → MH,Σord , and denote its further pullback to Spec(K) by replacing the subscript “V ” with “K”. (Here we understand that αHp ,K is defined only over Spec(K).) ~e ord,tor ~ ord ⊂ M e K is Since the image of Spec(K) under ξe is on N eΣ e ord , we know that G H, κ e and an extension of GK by the pullback TeK of a torus Te with character group X, ord,0 ord,0 r e image(e αpr ,K ) is an extension of image(αpr ,K ) by TK [p ]. As in Section 7.2.3, since V is normal (as a discrete valuation ring), by [92, IX, 2.4], [28, Ch. I, Prop. 2.9], e K extends to an embedding TeV ,→ G eV . or [62, Prop. 3.3.1.7], the embedding TeK ,→ G r e Therefore, by [26, IX, 3.6 bis] and [35, III-1, 5.1.4], TV [p ] is automatically a closed subgroup scheme of any H in Claim 7.2.4.11. By [92, IX, 1.4], [28, Ch. I, Prop. e V /TeV is canonically 2.7], or [62, Prop. 3.3.1.5], the quotient semi-abelian scheme G isomorphic to GV . Hence, Claim 7.2.4.11 follows from the following: Claim 7.2.4.13. Let H denote the finite flat subgroup scheme of multiplicative type of GV [pr ] uniquely lifting the maximal finite flat subgroup scheme H0 of multiplicative type of G0 [pr ], where G0 is the special fiber of G over Spec(k), then the generic fiber HK of HV coincides with image(αpord,0 r ,K ). But this is trivially verified because αpord r ,K extends to the ordinary level structure ord αpr ,V of (GV , λV , iV ), in which case H = image(αpord,0 r ,V ), by comparison over the characteristic p special fiber over Spec(k) (and by [26, IX, 3.6 bis] and [35, III-1, 5.1.4] once again). Proposition 7.2.4.14. Suppose no longer that Hκ and H are equally deep as in ~ ord,tor → Definition 3.2.2.9. For each κ ∈ Kord,++ , there is a canonical surjection N κ Q,H ord,min ord ord ~ ~ ~ MH extending the canonical surjection Nκ → MH (which is the composition ~ ord → M ~ ord with the canonical morphism M ~ ord → M ~ ord ), and the latof fκ : N κ H,rκ H,rκ H ~ ord ,→ M ~ ord,min ter is the pullback of the former under the canonical morphism M H

H

on the target (see Theorem 6.2.1.1). More generally, such a morphism maps the

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ord b ˘ ˘ ~ ord,tor to the [(ΦH , δH )]-stratum ~Zord ˘ b , δ˘ b , τb)]-stratum ~Z [(Φ [(ΦH τ )] of Nκ [(ΦH ,δH )] H H c,δH c,b ~ ord,min if and only if the cusp label [(ΦH , δH )] is assigned to the cusp label of M H ˘ b , δ˘ b )] as in Lemma 1.2.4.15. Such surjections are compatible with the cano[(Φ H H ~ ord,tor → N ~ ord,tor (defined by Proposition 7.2.4.1) when nical morphisms fκtor 0 ,κ : Nκ0 κ ~ ord → M ~ ord and κ0  κ in Kord,++ . When Hκ and H are equally deep, both N κ

Q,H

H

~ ord,tor → M ~ ord,min are proper. N κ H b Σ) b ∈ Kord,+ (i.e., Hκ = H) because, by ProProof. We may assume that κ = (H, Q,H ~ ord,min → M ~ ord,min , position 6.2.2.1, there is a canonical quasi-finite surjection M Hκ H ~ ord (as the unique open stratum) is the pullback of M ~ ord . Let Σord be and M Hκ H ~ ord as in Definition 1.2.2.13. Take any element any compatible choice for M H b 00 , Σ b 00 ) ∈ Kord,+ ord such that H b = H b 00 , and such that Σ b 00 is a refineκ00 = (H Q,H,Σ

b so that κ00  κ in Kord,+ . Then Hκ00 = Hκ = H, and hence we have ment of Σ, Q,H ~ ord,tor (by Proposition 7.2.2.19 and ~ ord,tor → N canonical proper surjections fκtor 00 ,κ : Nκ00 κ ord,tor tor ~ ord,tor ~ Corollary 7.2.2.21) and f 00 : N 00 →M ord (by Proposition 7.2.4.1). κ

κ

H,Σ

These proper surjections are their own Stein factorizations (see [35, III-1, 4.3.3]), by the normality of the target schemes, by [62, Lem. 7.2.3.1], and by Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11]). That is, the canonical morphisms ON → (fκtor 00 ,κ )∗ O~ ord,tor ~ ord,tor N κ

(7.2.4.15)

tor OM ~ ord,tor → (fκ00 )∗ ON ~ ord,tor

(7.2.4.16)

κ00

and κ00

H,Σord

are isomorphisms. (These are special cases of (7.1.4.4) with a = b = 0.) top ∗ 1 top ~ ord,tor is the b eGb ΩG/ Let ωN Lie∨ ord,tor := ∧ ~κ ~ ord,tor = ∧ b N ~ ord,tor , where G → Nκ b N G/ κ

κ

tautological semi-abelian scheme as in (7.2.2.4), and ωN ~ ord,tor be similarly defined, κ00 so that ∼ tor ∗ ~ ord,tor . ωN ~ ord,tor = (fκ00 ,κ ) ωN κ 00

(7.2.4.17)

κ

On the other hand, by (7.2.3.1), we have tor ∗ ∼ top e ωN ~ ord,tor = (∧Z X) ⊗(fκ00 ) ωM ~ ord,tor 00 κ

H,Σord

Z

∼ e → (cf. [62, Lem. 7.1.2.1]). Hence, by choosing any isomorphism (∧top X) Z, we Z have

∼ tor ∗ ~ ord,tor . ωN ~ ord,tor = (fκ00 ) ωM 00 κ

(7.2.4.18)

H,Σord

H ord ~ ord,tor By composing fκtor with the canonical proper surjection ~ H : M → 00 H,Σord ~ ord,min , we obtain a canonical proper surjection M H ~ ord,tor ~ ord,min , N →M κ00 H

(7.2.4.19)

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which is canonically determined by the canonical isomorphisms   k ~ ord,min ∼ ~ ord,tor , ω ⊗ord,tor M ) = Proj ⊕ Γ(M H H ~ MH k≥0   ord,tor ⊗ k ∼ ~ 00 , ω~ ord,tor ) = Proj ⊕ Γ(N κ Nκ00

k≥0

(by (3) of Theorem 6.2.1.1, and (7.2.4.15), (7.2.4.16), and (7.2.4.18)). By (7.2.4.17), we also have a canonical isomorphism   k ~ ord,min ∼ ~ ord,tor , ω ⊗ord,tor M ) , = Proj ⊕ Γ(N κ H ~ Nκ

k≥0

which induces the desired canonical morphism ~ ord,tor → M ~ ord,min , N κ H

(7.2.4.20)

~ ord,tor → N ~ ord,tor is (7.2.4.19). whose composition with fκtor 00 ,κ : Nκ00 κ Note that (7.2.4.20) is proper because (7.2.4.19) is proper and fκtor 00 ,κ is surjective. (This uses the simplifying assumption that Hκ = H.) ~ ord under f tor ~ ord under (7.2.4.19) is the preimage of M The preimage of M κ00 : H H ord,tor ord,tor ord ~ ~ ~ Nκ00 → MH,Σord , which is nothing but Nκ00 (which can be shown in many ways— e.g., by comparing the ranks of torus parts, or by comparing the dimensions of abe~ ord ~ ord b=H b 0 , by Lemma 7.2.1.3, N lian parts). Since H κ00 is mapped isomorphically to Nκ ~ ord under (7.2.4.19) is N ~ ord . This shows that under fκtor 00 ,κ . Hence, the preimage of MH κ ord ord ~ ~ is the pullback of (7.2.4.20) under the cano→ M the canonical surjection N κ H ~ ord ,→ M ~ ord,min on the target. More generally, by (4) of Theorem nical morphism M H H ~ ord,min under the ca6.2.1.1, the preimage of the [(ΦH , δH )]-stratum ~Zord [(ΦH ,δH )] of MH H ord ~ ord,tor ~ ord,min is the union of the strata ~Zord nonical morphism ~ H : M →M H [(ΦH ,δH ,σ)] H,Σord with the same underlying cusp label [(ΦH , δH )]. Hence, its further preimage in ~ ord,tor ~ ord,tor , are described as in the stateN , and the image of this preimage in N κ κ00 ment of the proposition, because of the corresponding strata-mapping properties of ~ ord,tor → M ~ ord,tor ~ ord,tor ~ ord,tor and f tor the canonical morphisms f tor :N →N 00 : N 00 00 00 ord κ ,κ

κ

κ

κ

κ

H,Σ

(see Propositions 7.2.2.19 and 7.2.3.5). e Σ, e σ From now on, let us fix a choice of some κ e = (H, e) ∈ Kord,++ , inducing Q,H,Σord ord,++ e some κ = [e κ] ∈ KQ,H,Σord . For simplicity, we shall suppress κ, Σ, and Σ from the notation when their choices are clear from the context. 7.2.5

Log Smoothness of f tor

We would like to show that the morphism f tor is log smooth (as in [45, 3.3] and ~ ord,tor and M ~ ord,tor with the canonical fine log structures [43, 1.6]) if we equip N H,rκ given respectively by the relative Cartier divisors with simple normal crossings ~ ord and M ~ ord,tor − M ~ ord with their reduced ~ ord,tor − N given by the complements N H,rκ

H,rκ

~ ord,tor is the pullback of the one of structures. (Note that the log structure of M H,rκ ~ ord,tor defined by M ~ ord,tor − M ~ ord . In what follows, we will freely state related M H

H

H

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~ ord,tor that are already know for M ~ ord,tor .) Moreover, for each of the facts about M H,rκ H sheaves to be introduced below, we will denote with the subscript “free” its free quotients defined by the image under ( · ) → ( · ) ⊗ Q (as in Definition 3.4.3.1). Z

According to [45, 3.12] (cf. [61, Lem. 3.11]), we have the following: Lemma 7.2.5.1. To show that the morphism f tor is log smooth, it suffices to show that the first morphism in the canonical exact sequence (f tor )∗ (Ω1M ~ ord,tor /~ S H,rκ

0,rκ

[d log ∞]) → Ω1N ~ ord,tor /~ S

0,rκ

[d log ∞] (7.2.5.2)

1

→ ΩN ~ ord,tor /M ~ ord,tor → 0 H,r κ

is injective, and that

1 ΩN ~ ord,tor /M ~ ord,tor H,rκ

is locally free of finite rank.

By (4) of Theorem 5.2.1.1, the extended Kodaira–Spencer morphism [62, Def. ~ ord,tor induces an isomorphism 4.6.3.44] for G → M H,rκ ∼

1 KSG/M ~ ord,tor ,free → ΩM ~ ord,tor /~ ~ ord,tor /~ S0,r : KS(G,λ,i)/M S κ

H,rκ

H,rκ

H,rκ

0,rκ

[d log ∞]

~e ord,tor ~ ord,tor , while the extended Kodaira–Spencer morphism for G e → M over M e H H,rκ induces an isomorphism KS e

ord,tor ~ ef /~ S0,rκ G/M H

∼

: KS

ord,tor

~ ef ee e λ, (G, i)/M H

,free

→ Ω1~ ord,tor ef M H

/~ S0,rκ

[d log ∞]

~e ord,tor . By (7.2.3.1) and (7.2.3.2), we have a commutative diagram over M e H 0

/ Lie∨ ~ ord,tor G/N λ

/ Lie∨e

~ ord,tor G~Nord,tor /N

∗

 0 / Lie∨ ∨ ~ ord,tor G /N

e∗ λ

/ Lie∨e∨

~ ord,tor /N ~ Nord,tor

e λ∗ e =φ∗ T



~ ord,tor G~ ord,tor /N N

/ Lie∨e T

/ Lie∨e∨ T



~ Nord,tor

~ ord,tor /N

/0

(7.2.5.3)

/0

in which the horizontal rows are exact. Using the sheaves and vertical arrows in this diagram, we can define KS(G,λ,i)/N ~ ord,tor , KS(G, ee ~ ord,tor , and e λ, i)~Nord,tor /N as in [62, Def. 6.3.1] and Definitions 1.3.1.2 and 3.4.3.1. KS(Te,λ e ,i e )~ ord,tor /N ~ ord,tor T T N ord ~ ord,tor is canonically isomorphic to the pullback of Since (G, λ, i, αHp , α ) → N Hp

ord ~ ord,tor under f tor : N ~ ord,tor → M ~ ord,tor , we have a canonical (G, λ, i, αHp , αH )→M H,rκ H,rκ p isomorphism

∼ (f tor )∗ KS(G,λ,i)/M ~ ord,tor ,free = KS(G,λ,i)/N ~ ord,tor ,free . H,rκ

On the other hand, KS(G, ee ~ ord,tor is canonically isomorphic to the pullback e λ, i)~Nord,tor /N ord,tor ~ of KS , and KS e ord,tor to N ~ ord,tor is canonically isomor~ ee ef e λ, (G, i)/M H

(T ,λTe ,iTe )~Nord,tor /N

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phic to the pullback of the sheaf KS(Te,λ e ,i e )~ T

T S0,r κ

/~ S0,rκ

:= (Lie∨ Te

~ S0,r

κ

⊗ /~ S0,rκ O ~ S

Lie∨ Te∨

0,rκ

λ∗Te (y) ⊗ z − λ∗Te (z) ⊗ y / (b? x) ⊗ y − x ⊗(by)

~ S0,r

κ

) /~ S0,rκ

! x∈Lie∨ e T

/~ S0,rκ ~ S0,r κ

y,z∈Lie∨ e∨ T

~ S0,rκ

/~ S0,r

κ

b∈O

similarly defined by the split tori Te and Te∨ over ~S0,rκ with respective character e and Ye . By the commutativity in the above diagram, there is a canonical groups X surjection KS(G, ~ ord,tor , ee ~ ord,tor  KS(Te,λ e ,i e )~ ord,tor /N e λ, i)~ ord,tor /N T

N

(7.2.5.4)

T N

whose kernel K := ker(KS(G, ~ ord,tor ) ee ~ ord,tor  KS(Te,λ e ,i e )~ ord,tor /N e λ, i)~ ord,tor /N T

N

(7.2.5.5)

T N

contains KS(G,λ,i)/N ~ ord,tor as a natural subsheaf. ~e ord,tor ˘ e , δ˘ e , τ˘)]-stratum is along the [(Φ Because the ´etale local structure of M e H H H ord ~e the same as that of ΞΦ˘ f,δ˘f (˘ τ ), the calculation in the proof of Proposition 4.2.3.5 H H induces by restriction (to the closure shows that the isomorphism KS e ~e ord,tor ~ G/MH f

/S0,rκ

~ ord,tor of the [(Φ e e , δe e , σ N H H e )]-stratum) an isomorphism ∼

Kfree → Ω1N ~ ord,tor /~ S

0,rκ

[d log ∞]

(7.2.5.6)

making the diagram  (f tor )∗ KS(G,λ,i)/M ~ ord,tor ,free



/ Kfree

H,rκ

KS

~ G/M

ord,tor ~ /S0,r κ H,rκ

o

(f tor )∗ (Ω1M ~ ord,tor /~ S H,rκ

o (7.2.5.6)



/ Ω1 ~ ord,tor /~ N S

[d log ∞]) 0,rκ

 [d log ∞] 0,rκ

commutative. In particular, the bottom arrow (which is the first morphism in (7.2.5.2)) is injective, and the isomorphism (7.2.5.6) induces a canonical isomorphism ∼

1

Kfree /KS(G,λ,i)/N ~ ord,tor /M ~ ord,tor ~ ord,tor ,free → ΩN H,r

(7.2.5.7)

κ

~ ord,tor . We would like to verify the following: of coherent sheaves over N Claim 7.2.5.8. Kfree /KS(G,λ,i)/N ~ ord,tor ,free is locally free of finite rank. Then it will follow from Lemma 7.2.5.1 and the above that f tor is log smooth, and the proof of (2) of Theorem 7.1.4.1 will be complete.

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Let us define K0 := K/KS(G,λ,i)/N ~ ord,tor

(7.2.5.9)

~ ord,tor . over N Lemma 7.2.5.10. The canonical morphism 0 Kfree /KS(G,λ,i)/N ~ ord,tor ,free → Kfree

(7.2.5.11)

is an isomorphism. Proof. It suffices to verify this over the completions of the strict local rings of ~ ord,tor at points of characteristic p, which are complete noetherian normal domains N whose spectra we denote by S. Over each such S, we have compatible (noncanonical) ordinary level structures → G\S [p∞ ], α eord = (e αord,0 : (Gre0D )mult S ∞ α eord,#,0 : (Gre0D# )mult → G∨,\ eord ) S S [p ], ν

and → GS [p∞ ], αord = (αord,0 : (Gr0D )mult S ∨

αord,#,0 : (Gr0D# )mult → GS [p∞ ], ν ord S ) (where the subscripts “S” mean pullbacks to S) defining a commutative diagram 0

/ TeS [p∞ ] e∗ λTe =φ

0

 / Te∨ [p∞ ] S

/ image(e αord,0 ) eS λ

 / image(e αord,#,0 )

/ image(αord,0 ) S

/0

(7.2.5.12)

λS

 / image(αord,#,0 )

/0

canonically dual to the pullback of (7.2.5.3) to S, under canonical isomorphisms as in (5) and (6) of Proposition 3.2.1.1. The diagram (7.2.5.12) admits (noncanonieS , because cal) splittings, namely splittings of the two exact rows compatible with λ h · , · ie is the direct sum of the pairings on Q−2 ⊕ Q0 and on L (and because the ordinary level structures match the diagram (7.2.5.12) with the corresponding diagram of constant objects). Such splittings induce (by duality) splittings of the pullback of eS . Hence, (7.2.5.3) to S, namely splittings of the two exact rows compatible with λ 0 we have a noncanonical isomorphism between KS and KS(G,λ,i)/N ~ ord,tor ,S ⊕ KS over S, and also a corresponding one between their free quotients. Hence, (7.2.5.11) is an isomorphism, as desired. e G) (resp. Hom (X, e G∨ ), resp. Hom (Ye , G), By Proposition 3.1.3.4, HomO (X, O O ∨ e resp. HomO (Y , G )) is relatively representable by an extension of a quasifinite flat group scheme of ´etale-multiplicative type by a semi-abelian scheme e G)◦ (resp. Hom (X, e G∨ )◦ , resp. Hom (Ye , G)◦ , resp. Hom (Ye , G∨ )◦ ) HomO (X, O O O ord,tor ~ e and λ : G → G∨ over MH,rκ . Then the homomorphisms φe : Ye ,→ X

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~ ord,tor induce homomorphisms Hom (X, e G∨ )◦ → Hom (Ye , G∨ )◦ and over M O O H,rκ HomO (Ye , G)◦ → HomO (Ye , G∨ )◦ with kernels that are quasi-finite group schemes ~ ord,tor , and hence the fiber product of ´etale-multiplicative type over M H,rκ ...ord,ext e e G∨ ) C := HomO (X, × HomO (Ye , G) (7.2.5.13) e ,G∨ ) HomO (Y

...ord,ext ~ ord,tor ) of ´etalee is also an extension of a quasi-finite group scheme π 0 (C /M H,rκ ...ord,ext,◦ ord,tor ~ e multiplicative type by a semi-abelian scheme C over MH,rκ . Lemma 7.2.5.14. We have compatible canonical isomorphisms ...ord,ext K0 ∼ = (f tor )∗ Lie∨ e C

(7.2.5.15)

~ ord,tor /M H,rκ

and ...ord,ext K0free ∼ = (f tor )∗ Lie∨ e C

∼ ...ord,ext = (f tor )∗ Lie∨ e C


~ ord,tor free /M H,rκ

∼ ...ord,ext,◦ = (f tor )∗ Lie∨ e C

(7.2.5.16)

~ ord,tor ,free /M H,rκ ~ ord,tor /M H,rκ

.

...ord,ext,◦ ~ ord,tor , it follows that e is a semi-abelian scheme over M Since C H,rκ ~ ord,tor , and that K0 is locally ...ord,ext,◦ ord,tor is locally free of finite rank over M Lie∨ free H,r e C

~ /M H,rκ

κ

~ ord,tor

free of finite rank over N

.

Proof. By definition, we have canonical isomorphisms Lie∨ Hom

e ∨ ~ ord,tor Z (X,G )/MH,rκ

∼ e ∨ , Lie∨ ∨ ~ ord,tor ) = HomZ (X G /M H,rκ

∼ e O ~ ord,tor ), Lie∨ ∨ ~ ord,tor ) = HomO~ ord,tor (HomZ (X, M G /M M

∼ =

H,rκ

H,rκ

H,rκ

∨ HomO~ ord,tor (LieTe ord,tor /M ~ ord,tor , LieG∨ /M ~ ord,tor ) H,rκ ~ M H,rκ M H,rκ

H,rκ

∼ = Lie∨ Te

~ ord,tor ~ ord,tor /MH,rκ M H,rκ

⊗

O~ ord,tor

Lie∨ ~ ord,tor G ∨ /M H,rκ

M H,rκ

and Lie∨ Hom

~ ord,tor e ,G)/M

Z (Y

H,rκ

∼ = HomZ (Ye ∨ , Lie∨ ~ ord,tor ) G/M H,rκ

∨ ∼ = HomO~ ord,tor (HomZ (Ye , OM ~ ord,tor ), LieG/M ~ ord,tor ) H,rκ

M H,rκ

∼ =

H,rκ

∨ HomO~ ord,tor (LieTe∨ ~ ord,tor , LieG/M ~ ord,tor ) /M H,rκ M H,rκ ~ ord,tor M H,rκ

H,rκ

∼ = Lie∨ ~ ord,tor G/M H,rκ

⊗

O~ ord,tor M H,rκ

Lie∨ Te∨

~ ord,tor M H,rκ

~ ord,tor /M H,rκ

.

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...ord,ext e Since C is (by definition) the subgroup scheme of the ordinary semie G∨ ) over which the tautological obabelian scheme HomZ (Ye , G) × HomZ (X, ~ ord,tor M H,rκ

∨

ject (e c, e c ) is O-equivariant and satisfies the compatibility e cφe = λe c∨ , and ~ ord,tor is canonically isomorphic to the pullback of since (G, λ, i, αHp , αord Hp ) → N ord,tor ord ~ ~ ord,tor → M ~ ord,tor , the pullback (G, λ, i, αHp , αHp ) → M under f tor : N H,rκ H,rκ ...ord,ext ord,tor is canonically isomorphic to the quotient of (f tor )∗ Lie∨ ~ /M H,rκ

e C



Lie∨ Te

⊗ ~ ord,tor /N ~ Nord,tor O~Nord,tor

 + Lie∨ ~ ord,tor G/N

⊗

O~Nord,tor

Lie∨ ∨ ~ ord,tor G /N

 

Lie∨ Te∨

~ Nord,tor

~ ord,tor /N

,

as a subsheaf of 

Lie∨ e G

~ ord,tor /N ~ Nord,tor

 / Lie∨ ~ ord,tor G/N

⊗

O~Nord,tor

⊗

O~Nord,tor



Lie∨ e∨ G

~ Nord,tor

~ ord,tor /N



Lie∨ ∨ ~ ord,tor , G /N

by relations as in Definition 4.2.3.4, which is by definition K0 . Hence, we have the canonical isomorphism (7.2.5.15). ...ord,ext,◦ ...ord,ext ~ ord,tor is a semi-abelian scheme and π (C ~ ord,tor ) e e Since C /M /M 0 H,rκ H,rκ ∨ is quasi-finite flat of ´etale-multiplicative type, Lie...ord,ext,◦ ord,tor is locally of e C

~ /M H,rκ

~ ord,tor , and the canonical morphism Lie∨ ...ord,ext finite rank over M H,rκ e C

...ord,ext,◦ Lie∨ e C

~ ord,tor /M H,rκ

~ ord,tor /M H,rκ

→

induces a canonical isomorphism ...ord,ext Lie∨ e C

∼

~ ord,tor ,free /M H,rκ

...ord,ext,◦ → Lie∨ e C

~ ord,tor /M H,rκ

.

(7.2.5.17)

Since the formation of free quotients is compatible with pulling back under f tor , the canonical isomorphisms (7.2.5.15) and (7.2.5.17) induce the canonical isomorphisms in (7.2.5.16). The remaining statements of the lemma are self-explanatory. Thus, we have verified Claim 7.2.5.8 and proved (2) of Theorem 7.1.4.1. Now suppose κ ∈ Kord,+ (not just in Kord,++ , so that Hκ = H and rκ = Q,H,Σord Q,H,Σord rH ). This is the setting in (3) of Theorem 7.1.4.1, where the semi-abelian scheme ~ ord,ext = N ~ ord,ext → M ~ ord,tor extending N ~ ord,grp = N ~ ord,grp → M ~ ord is also defined N κ κ H H (as in Definition 7.1.3.11). bal r Consider any n = n0 pr such that p - n0 , U p (n0 ) ⊂ Hp , and Up,1 (p ) ⊂ Hp ⊂ r ord Up,0 (p ). By condition (5) of Definition 3.4.2.10, (by abuse of language) αH exp ord,tor ord,0 ~ tends to an ordinary level structure of (G, λ, i) over M . Although αpr (resp. H

ord αpord,#,0 ) is only ´etale locally defined by αH , its schematic image image(αpord,0 )= r r p ord,#,0 αpord,0 ((Gr0Dpr )mult ) = αpord,#,0 ((Gr0D# )mult r r ~ ord,tor ) (resp. image(αpr ~ ord,tor )) depends M M H

pr

H

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ord only on αH and descends to a finite flat subgroup scheme of multiplicative type p ~ ord,tor , which we (by abuse of notation) still denote by of G (resp. G∨ ) over M H image(αpord,0 ) (resp. image(αpord,#,0 )). As in Section 4.2.1, let us define r r ord,0 ) G  Gord pr := G/image(αpr

(7.2.5.18)

G∨  G∨,ord := G∨ /image(αpord,#,0 ), r pr

(7.2.5.19)

and respectively, together with morphisms Gord pr  G

(7.2.5.20)

 G∨ , Gp∨,ord r

(7.2.5.21)

and respectively, such that the compositions (7.2.5.20) ◦ (7.2.5.18) and (7.2.5.21) ◦ (7.2.5.19) are multiplications by pr . (See (4.1.4.31), (4.1.4.32), (4.1.4.34), (4.1.4.33), (4.2.1.1), (4.2.1.2), (4.2.1.3), and (4.2.1.4). The restrictions of the morphisms ~ ord exist as duals of the restrictions of (7.2.5.19) and (7.2.5.20) and (7.2.5.21) to M H ~ ord,tor by [92, IX, 1.4], [28, Ch. I, Prop. (7.2.5.18), respectively, and extends to M H 2.7], or [62, Prop. 3.3.1.5].) Note that the kernels of (7.2.5.20) and (7.2.5.21) are quasi-finite ´etale, because they are ´etale locally subgroups of the constant group −1 schemes (Gr−1 ~ ord,tor and (GrD# )M ~ ord,tor , respectively. Dpr )M H

pr

H

e G∨,ord ) (resp. HomO ( n1 Ye , Gord By Proposition 3.1.3.4 (as above), HomO ( n1 X, pr pr )) is representable by an extension of a quasi-finite flat group scheme of ´etale~ ord,tor , and the fiber product multiplicative type by a semi-abelian scheme over M H ...ord,ext en e Gp∨,ord := HomO ( n1 X, C ) × HomO ( n1 Ye , Gord (7.2.5.22) r pr ) e ,G∨ ) HomO (Y

...ord,ext ~ ord,tor ) of ´etalee /M is also an extension of a quasi-finite group scheme π 0 (C H ...ord,ext,◦ n ~ ord,tor . Note that en multiplicative type by a semi-abelian scheme C over M H ...ord,ext,◦ ...ord,ext e e are closed subgroup schemes of the semi-abelian scheme Cn and C n ~ ord,tor . The group H e ord ess acts e G∨,ord HomZ ( n1 X, ) × HomZ ( n1 Ye , Gord pr pr ) over MH e n,U ~ ord,tor M H

e Zn

~ ord , naturally on the (abelian scheme) pullback of this semi-abelian scheme to M H ord,tor ~ and this action extends canonically over MH by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5]. Let ...ord,ext ...ord,ext e H,n en e ord ess C := C /H (7.2.5.23) e e n,U e Zn

(cf. (4.2.1.14)—we do not form the quotient by

e ord ess H e n,G

and other groups here,

h,e Zn

~ ord,tor ), so that because we are already working over M H ...ord,ext,◦ ...ord,ext,◦ ∼ ee e e ord ess . C /H =C H,n n e n,U e Zn

(7.2.5.24)

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By construction, we have the following commutative diagram ...ord,ext,◦   e H,n C e

 e G∨,ord / Hom ( 1 X, ) pr Z n

 ...ord,ext,◦   e C

× ~ ord,tor M H

e G∨ ) / HomZ (X,

 e ord ess HomZ ( n1 Ye , Gord ) /H r p e n,U e Zn

 × ~ ord,tor M H

HomZ (Ye , G)

(7.2.5.25) ~ ord,tor , in which all horizontal of canonical morphisms of semi-abelian schemes over M H arrows are closed immersions. Lemma 7.2.5.26. In (7.2.5.25), the vertical arrows are (quasi-finite and) unramified. Also, the first vertical arrow is an ´etale isogeny between semi-abelian schemes. Proof. The first statement is true because the vertical arrows are homomorphisms with quasi-finite ´etale kernels (because (7.2.5.20) and (7.2.5.21) are). Since the first vertical arrow is a homomorphism between semi-abelian schemes, it is surjective and automatically flat by [35, IV-3, 11.3.10 a)⇒b) and 15.4.2 e0 )⇒b)] (cf. the proof of [62, Lem. 1.3.1.11]), and hence is an ´etale isogeny. By construction, under the canonical isomorphism (7.2.1.1), we have a canonical isomorphism ...ord,ext,◦ ...ord,◦ ~ ord ∼ ~ ord e H,n eΦ M C × M C × (7.2.5.27) = e ef H H ... H,n

~ ord,tor M H

ord,e ZH

e M H

...ord,eZH ...ord,◦ e ~ ord , where C eΦ as in (4.2.1.15), is defined over M of abelian schemes over M e H H f H,n

~e ord in the toroidal boundary construction of M e . H ord ~ ~e ord ord ∼ ~ e e e C Recall (see Section 7.2.1) that N = ef , which (by Propoe f,δ Φ , δ ,e σ )] κ = Z[(Φ f f H H H H

~e ord,grp ~ ord,grp = C sition 4.2.1.30) is a torsor under an abelian scheme N ef , which (as in e f,δ κ Φ H ...H ord,◦ ~ ord e the proof of Proposition 4.2.1.30) is canonically isomorphic to C Φ e H ,n ... × MH . ord,e ZH

e M H

Combining this with (7.2.5.27), we obtain a canonical isomorphism ... ∼ e ord,ext,◦ ~ ord,grp → ~ ord N C H,n × M e κ H

(7.2.5.28)

~ ord,tor M H

~ ord . Since M ~ ord,tor is noetherian normal, by [92, IX, of abelian schemes over M H H 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5], we obtain a canonical isomorphism ... ∼ e ord,ext,◦ ~ ord,ext → N C H,n . (7.2.5.29) e κ

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~ ord,tor . Combining this with the first vertical arrow of semi-abelian schemes over M H in (7.2.5.25), we obtain (see Lemma 7.2.5.26) a canonical quasi-finite ´etale isogeny ...ord,ext,◦ ~ ord,ext → C e N (7.2.5.30) κ

of semi-abelian schemes over

~ ord,tor . M H

Hence, we obtain the following:

Corollary 7.2.5.31. The canonical isogeny (7.2.5.30) induces a canonical isomorphism ...ord,ext,◦ Lie∨ e C

∼

~ ord,tor /M H

→ Lie∨ ~ ord,ext /M ~ ord,tor . N κ

(7.2.5.32)

H

Consequently, we obtain a canonical isomorphism ∼ 1~ ord,tor ~ ord,tor (f tor )∗ Lie∨ ~ ord,ext /M ~ ord,tor = ΩN /MH N

(7.2.5.33)

H

by combining (7.2.5.7), Lemmas 7.2.5.10 and 7.2.5.14, and (7.2.5.32). This isomorphism (7.2.5.33) gives the desired isomorphism (7.1.4.2). (The re~ ord is compatible with the composition of isomorphisms striction of (7.2.5.33) to N (7.1.4.3) because of the same calculation in the proof of Proposition 4.2.3.5.) Thus, we have also proved (3a) of Theorem 7.1.4.1. The same argument above (based on Proposition 4.2.3.5) also shows the following: Lemma 7.2.5.34. (Compare with Lemma 1.3.2.79. Lemma 7.1.2.29.) Consider the morphisms

This is a continuation of

~b ord ~e ord ~ Φord,δ ,r := C ~ Φord,δ × ~S0,r C = C ˘ ˘ ˘ f,δ˘f → C Φ c,δ c Φ κ H H κ H H H

H

H

H

(7.2.5.35)

~ S0,rH

and ord

~b Ξ ˘ c,δ˘c,Σ b˘ Φ Φ H

H

c H

ord

~ Φ ,δ ,Σ →Ξ Φ H H

ord

H

,rκ

~ Φ ,δ ,Σ := Ξ Φ H H

H

× ~S0,rκ

(7.2.5.36)

~ S0,rH

\ ~ ord induced by (7.1.2.2) and (7.1.2.34), respectively. Over C ΦH ,δH (where G etc are tautologically defined), we have an extension ...ord,\,◦ ...ord,◦ ...ord,\,◦ e H,n 0 → E H,n → C → C H,n → 0 (7.2.5.37) b b

...ord,\,◦ ...ord,\,◦ e H,n of a semi-abelian scheme by a torus, where the definition of C (resp. C H,n , b ...ord,ext,◦ ...ord,◦ ee resp. E H,n ) is similar to that of C , but with G etc replaced with G\ etc H,n b

(resp. B etc, resp. T etc), which can be identified up to compatible Q× -isogenies with an extension e T )◦ → Hom (X, e G\ )◦ → Hom (X, e B)◦ → 0. 0 → HomO (X, O O

(7.2.5.38)

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Then (7.2.5.35) is smooth, (7.2.5.36) is log smooth, and we have canonical isomorphisms ~b ord ∼ ~ Φord,δ )∗ Lie∨ ...ord,\,◦ ~ ord ( C (7.2.5.39) Ω1~ ord = ˘ c,δ˘c → C Φ H H /C C ord b˘ C Φ

~

H

˘ /CΦ ,δ ,r c,δH c H H κ H

H

H,n

ΦH ,δH

(cf. (1.3.2.54)) and ord

1

Ω~ ord b˘ Ξ Φ

ord

˘ ,Σ b ˘ c,δH c Φ H

~ /Ξ Φ

H ,δH ,ΣΦH ,rκ

c H

~b ∼ = (Ξ ˘ c,δ˘c,Σ b˘ Φ Φ H H

~ Φord,δ )∗ Lie∨ ...ord,\,◦ →C H H e C H,n c

c H

~ ord /C Φ ,δ H

H

(7.2.5.40)

(cf. (1.3.2.86)), where 1

Ω~ ord b˘ Ξ Φ

~

ord

/ΞΦ ,δ ,Σ ˘ ,Σ b ΦH ,rκ H H ˘ c,δH c Φ H c H

ord

H

~ b˘ Ξ Φ

ord

~b ((Ξ ˘ c,δ˘c,Σ b˘ Φ Φ H

:= (Ω1 ord

c H

~ Φ ,δ ,Σ →Ξ Φ H H

H

[d log ∞])/

~ ord /C ˘ ,Σ b ΦH ,δH ˘ c,δH c Φ H c H

∗ 1 ,rκ ) Ω~ ord ΞΦ ,δ H

H ,ΣΦH ,rκ

~ ord /C Φ ,δ H

[d log ∞]) H

is the sheaf of modules of relative log 1-differentials. Moreover, the canonical morphism ord

~b Ξ ˘ c,δ˘c,Σ b˘ Φ Φ H

H

~b ord C ˘ c,δ˘c Φ

ord

~ Φ ,δ ,Σ →Ξ Φ H H

c H

H

×

H

~ ord C Φ ,δ H

ord

∼ ~ Φ ,δ ,Σ ,r =Ξ ΦH κ H H

H

H

× ~ ord C Φ ,δ ,rκ H

(7.2.5.41)

~b ord C ˘ c,δ˘c Φ H

H

H

(induced by (7.1.2.2) and (7.1.2.34), or by (7.2.5.35) and (7.2.5.36); cf. (1.3.2.87)) induces a canonical short exact sequence ord

~b 0 → (Ξ ˘ c,δ˘c,Σ b˘ Φ Φ H

H

c H

ord

~b ∗ 1 →C ˘ c,δ˘c ) Ω ~ ord Φ H

H

b˘ C Φ

~ ord ˘ /C ΦH ,δH ,rκ c,δH c H

1

→ Ω~ ord b˘ Ξ Φ

~

ord

(7.2.5.42)

/ΞΦ ,δ ,Σ ˘ ,Σ b ΦH ,rκ H H ˘ c,δH c Φ H c H

1

→ Ω~ ord b˘ Ξ Φ

~

→0

ord

ord

/ΞΦ ,δ ,Σ × ˘ ,Σ b ΦH ~ ord H H ˘ c,δH c Φ H c C H ΦH ,δH

~b C ˘ Φ

˘ c,δH c H

(cf. (1.3.2.88)), where 1

Ω~ ord b˘ Ξ Φ

~

ord

× /ΞΦ ,δ ,Σ ˘ ,Σ b ΦH ~ ord H H ˘ c,δH c Φ H c C H ΦH ,δH

ord

~b ((Ξ ˘ c,δ˘c,Σ b˘ Φ Φ H

H

c H

:= (Ω1 ord

~bord C ˘ ˘ ,δ Φ c H

~bord

~ b˘ Ξ Φ

[d log ∞])/

/C Φ ˘ ,Σ ˘ b ˘ ,δ ˘ c,δH c Φ c H c H H c H

c H

ord

~ Φ ,δ ,Σ )∗ Ω1 ord →Ξ Φ H H H

~ Ξ Φ

H ,δH ,ΣΦH

~bord /C ˘ ˘ ,δ Φ c H

[d log ∞])

c H

is the sheaf of modules of relative log 1-differentials, which is exact and has locally free terms, which can be canonically identified with the pullback under ord ~b ~ ord ΞΦ˘ c,δ˘c,Σ →C b˘ ΦH ,δH of the canonical short exact sequence Φ H

H

c H

...ord,\,◦ ~ ord 0 → Lie∨ /C C H,n

ΦH ,δH

...ord,\,◦ → Lie∨ e C H,n c

~ ord /C Φ ,δ H

...ord,◦ ~ ord → Lie∨ /C E H

c H,n

ΦH ,δH

→0

(7.2.5.43)

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of locally free sheaves. Hence, (7.2.5.41) is also log smooth (by [45, 3.12]). If p - [L# : L] as in Definition 1.1.1.6 and hence λ is prime-to-p, and if O is maximal at p, then we may assume in the above that (7.2.5.37) and (7.2.5.38) can be identified up to Q× -isogenies which are separable up to Z× (p) -isogenies, and hence ord

~b that (7.2.5.42) can be identified up with the pullback under Ξ ˘ c,δ˘c,Σ b˘ Φ Φ H

H

of the canonical short exact sequence e Lie∨ ~ ord 0 → HomO (X, B/C

ΦH ,δH

e Lie∨ \ ~ ord ) → HomO (X, G /C

c H

~ ord →C ΦH ,δH

)

ΦH ,δH

e Lie∨ ~ ord → HomO (X, T /C

)→0

(7.2.5.44)

ΦH ,δH

(cf. (1.3.2.89)). Proof. The statements are self-explanatory. (For the last paragraph, note that, e Ye )], [X ˘ Y˘ )], and [X : φ(Y )] are also prime-to-p, e : φ( ˘ : φ( under the assumption, [X ∨,ord  G∨ , as in (7.2.5.20) and the canonical homomorphisms Gord pr → G and Gpr and (7.2.5.21), are ´etale.) 7.2.6

Equidimensionality of f tor

Let us resume the context of the diagram (7.2.3.6) and take a closer look at it. (Then we no longer suppose that κ ∈ Kord,+ .) By the construction of f tor , Q,H,Σord ~ ord,tor , the preimage given any stratum ~Zord of M H,rκ

[(ΦH ,δH ,τ )],rκ

~eord Z[(ΦH ,δH ,τ )] := (f tor )−1 (~Zord [(ΦH ,δH ,τ )],rκ ) ~eord e˘ has a stratification formed by Z ˘ runs through cones in Σ ˘ f,δ˘f,˘ [(Φ τ )] , where τ ΦH f H H satisfying the following conditions: (1) τ˘ ⊂ P+ ˘ . Φ f H

(2) τ˘ has a face σ ˘ that is a ΓΦ˘ f -translation of the image of σ e ⊂ P+ e Φ H

f H

under

the first morphism in (1.2.4.20). (3) The image of τ˘ under the (canonical) second morphism in (1.2.4.20) is contained in τ ⊂ P+ ΦH . ~ ord,tor )∧ord The formal completion (N ~

admits a canonical morphism

e Z [(Φ

H ,δH ,τ )]

~ ord,tor )∧ord (N ~

~ Φord,δ ,r = C ~ Φord,δ × ~S0,r , →C κ H H κ H H

e Z [(Φ

~ S0,rH

H ,δH ,τ )]

whose pre-composition with the canonical morphism ~ ord,tor )∧ord (N ~ e ˘ Z [(Φ

˘ ,˘ f τ )] f,δH H

~ ord,tor )∧ord → (N ~ e Z [(Φ

H ,δH ,τ )]

,

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~eord ~eord for every stratum Z ˘ f,δ˘f,˘ [(Φ τ )] of Z[(ΦH ,δH ,τ )] , coincides with the composition of H

H

~e ord ~e ord ~ ord canonical morphisms X →C ˘ f,δ˘f,˘ ˘ f,δ˘f → C Φ σ ,˘ τ Φ ΦH ,δH ,rκ by its very construction. H H H H tor Since f is ´etale locally given by morphisms between toric schemes equivariant under (surjective) morphisms between tori, to determine whether f tor is equidimensional (cf. [28, Ch. VI, Def. 1.3 and Rem. 1.4] and [61, Sec. 3D]), it suffices to determine whether the relative dimension of each of the induced (smooth) morphism ~eord ~ ord ), the relative diZ[(Φ˘ f,δ˘f,˘τ )] → ~Zord ~ ord (N [(ΦH ,δH ,τ )] between strata is at most dimM H H H ~ ord → M ~ ord . mension of f : N H,rκ

By abuse of language, we define the R-dimension of a cone to be the R-dimension ord ~e ord,tor ~ ord = ~Z e e e of its R-span. Then the codimension of N is dimR (e σ) = e [(ΦH σ )] in MH f,δH f,e ∨ dimR ((SΦ ) ) because σ e is top-dimensional. The codimension of ef R H

~eord Z ˘

ord

˘f,˘ [(ΦH τ )] f,δH

~e ∼ =Ξ ˘ f,δ˘f,˘ Φ τ H H

~eord ~e ord,tor ~ ord,tor is equal to dimR (˘ τ ). Therefore, the codimension of Z in M ˘ f,δ˘f,˘ e [(Φ τ )] in N H H H ∨ is equal to dimR (˘ τ ) − dimR (e σ ) = dimR (˘ τ ) − dimR ((SΦ e f )R ). On the other hand, the H ord,tor ∼ ~ ~ ord in M is dimR (τ ), and so is the codicodimension of ~Zord =Ξ H

ΦH ,δH ,τ

[(ΦH ,δH ,τ )]

∼ ~ ord ~ ~ ord,tor ~ ord mension of ~Zord [(ΦH ,δH ,τ )],rκ = ΞΦH ,δH ,τ,rκ := ΞΦH ,δH ,τ × S0,rκ in MH,rκ . Hence, ~ S0,rH

we have (as in [61, (3.16)]) ord

dim~Zord

[(ΦH ,δH ,τ )],rκ

~e (Z ˘ f,δ˘f,˘ [(Φ τ )] ) H

H

(7.2.6.1)

∨ = dimM τ ) − dimR ((SΦ ~ ord (N) − (dimR (˘ e f )R )) + dimR (τ ). H

H

Let τ 0 denote the image of τ˘ in (SΦH )∨ ˘, we have τ 0 ⊂ τ . R . By assumption on τ If τ 0 = τ , then ∨ dimR (τ ) = dimR (τ 0 ) ≤ dimR (˘ τ ) − dimR ((SΦ e f )R ), H

and hence (7.2.6.1) implies dim~Zord

[(ΦH ,δH ,τ )],rκ

~eord ~ ord ). (Z ˘ f,δ˘f,˘ ~ ord (N [(Φ τ )] ) ≤ dimM H

H

H,rκ

ord

~e tor (If this is true for all Z is equidimensional.) On the other ˘ f,δ˘f,˘ [(Φ τ )] , then f H H 0 hand, suppose τ ( τ . Then there exists a face of τ 00 of τ 0 such that τ 00 ⊂ τ and dimR (τ 00 ) < dimR (τ ). Note that τ 00 is the image of at least one face of τ˘ satisfying the three conditions in the first paragraph of this subsection. By dropping redundant basis vectors, we may assume moreover that this face τ˘00 of τ˘ satisfies ∨ dimR (τ 00 ) = dimR (˘ τ 00 ) − dimR ((SΦ e f )R ). Then we have H

∨ dimR (τ ) > dimR (τ 00 ) = dimR (˘ τ 00 ) − dimR ((SΦ e f )R ), H

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and hence (7.2.6.1) implies ord

dim~Zord

[(ΦH ,δH ,τ )],rκ

~e ~ ord ), (Z ˘ f,δ˘f,˘ ~ ord (N [(Φ τ 00 )] ) > dimM H

H

H,rκ

which means f tor cannot be equidimensional. This motivates the following strengthening of Condition 7.1.1.15 on an element e ord,++ : e Σ e ord , σ κ e = (H, e) in K Q,H ˘ e , δ˘ e , τ˘) such Condition 7.2.6.2. (Compare with [61, Cond. 3.17].) For each (Φ H H + ord ord,tor ~ ~ that Z[(Φ˘ ,δ˘ ,˘τ )] is a stratum in N , the image of τ˘ ⊂ PΦH under the (canof H

f H

nical) second morphism in (1.2.4.20) is exactly some cone τ ⊂ P+ ΦH in the cone ord decomposition ΣΦH (in Σ ). As in the case of Condition 7.1.1.17, if κ = [e κ] ∈ Kord,++ is the element determined Q,H by κ e, then Condition 7.2.6.2 for κ e is equivalent to the following condition for κ: b˘ Condition 7.2.6.3. (Compare with [28, Ch. VI, Def. 1.3].) For each τb ∈ Σ ΦH c ˘ e , δ˘ e , τ˘) is in the cone decomposition Σ b ˘ in (where τb = pr b τ ) for some (Φ ∨ (˘ H

(SΦ )R ˘

H

ΦH c

c H

b ord ), the image of τb in P+ under (1.2.4.37) is exactly some cone τ ⊂ P+ in Σ ΦH ΦH the cone decomposition ΣΦH (in Σord ). Proposition 7.2.6.4. (Compare with [61, Prop. 3.18].) The following are equivalent: (1) Condition 7.2.6.3 is satisfied. ~ ord,tor → M ~ ord,tor is equidimensional (with relative (2) The morphism f tor : N H,rκ ~ ord → M ~ ord ). dimension equal to the one of f : N H,rκ

(3) The morphism f tor is flat. (4) The morphism f tor is log integral (see [45, Def. 4.3]). Proof. The equivalence between Condition 7.2.6.3 and equidimensionality has been ~ ord,tor and M ~ ord,tor are regular (because they are explained above. Since both N H,rκ smooth over ~S0,r = Spec(OF ,(p) [ζprκ ])), the equidimensionality and flatness of f tor κ

0

are equivalent by [35, IV-3, 15.4.2 b)⇔e’)]. By [45, Prop. 4.1(2)], the log integrality of f tor is equivalent to the flatness of each of the canonical homomorphisms Z[τ ∨ ] ,→ Z[˘ τ ∨ ] (defined when ~Zord is mapped to ~Zord ˘ ,δ˘ ,˘ [(ΦH ,δH ,τ )],rκ ), which is equivalent [(Φ τ )] f H

f H

to the equidimensionality of each such homomorphism (by the smoothness of Z[τ ∨ ] and Z[˘ τ ∨ ] over Z, and by [35, IV-3, 15.4.2 b)⇔e’)] again), which is equivalent to Condition 7.2.6.3 by the same (dimension comparison) argument. Proposition 7.2.6.5. (Compare with [28, Ch. VI, Rem. 1.4] and [61, Prop. 3.18].) e ord Condition 7.2.6.3 can be achieved by replacing both the cone decompositions Σ ord and Σ with some refinements.

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Proof. Since this is a question only about cone decompositions, the same argument of the proof of [61, Prop. 3.18] works here. Remark 7.2.6.6. (Compare with [61, Rem. 3.20].) We will not need Propositions 7.2.6.4 and 7.2.6.5 in what follows. We supply them here because knowing equidimensionality, flatness, or log integrality of f tor is useful in many applications. 7.2.7

Hecke Actions

The aim of this subsection is to explain the proof of statements (4) and (5) of Theorem 7.1.4.1, with (4c) and (5c) conditional on (3b) and (3c) of Theorem 7.1.4.1. These statements might seem elaborate, but they are self-explanatory and based on the following simple idea: Since N and Ntor are constructed using the toroidal e e , we can use the Hecke actions on M e e and their (compatible) compactifications of M H H extensions to toroidal compactifications provided by Proposition 5.2.2.2. b 0 , Σord,0 , gl , and Q0 be as in (4) and (5) of Theorem 7.1.4.1. (For proving Let gb, H (4) and (5) of Theorem 7.1.4.1, we may assume in what follows that either gb = 1 or gl = 1, although the theory works in a more general context.) Let ge = (e g0 , gep ) be ee(A∞,p ) × P e ord (Qp ) such that, under the canonical isomorphism any element in P Z e Z,e D ee(A∞,p ) × P e e(A∞ ) n (P e 0 (A∞,p ) × P e ord,0 (Qp )) e ord (Qp ) ∼ P =G e e l,Z Z e Z Z,e D Z,e D induced by the splitting δe (as in Definition 1.2.4.3), ge is mapped to (gl−1 , ge0 ) for e 0 (A∞,p ) × P e ord,0 (Qp ) that is mapped to gb under the canonical some element ge0 ∈ P e e Z Z,e D morphism e 0 (A∞,p ) × P e ord,0 (Qp ) → G(A b ∞,p ) × P b ord P D (Qp ). e Z

e Z,e D

e Σ, e σ e 0 be a (necessarily neat) subgroup Suppose κ = [e κ] for some κ e = (H, e). Let H ˆ such that we have the following: e Z) of G( ep0 is of standard form (where H e 0,p ⊂ H e p is necessarily neat) such e0 = H e 0,p H • H that rHe 0 ≥ rHe , and such that gep satisfies the conditions analogous to those given in Section 3.3.4. e 0 ⊂ geHe e g −1 . • H e 0 also satisfies Conditions 1.2.4.7 and 7.1.1.5. • H e0 = H b 0 when H b 0 is prescribed (as in (4) of Theorem 7.1.4.1). • H b G e 0 satisfies Condition 1.2.4.8 or 1.2.4.9 when H e does. • H (These are possible by Lemma 1.2.4.45.) By Proposition 5.2.2.2, there exists e ord,0 such that the canonical morphism [e some choice of projective smooth Σ g] : ord ord ord,tor ord,tor ~ ord,tor ~e ~e ~ ~ e e 0 e ord,0 → M e e e ord . By replacing MHe 0 → MHe extends canonically to [e g] :M H ,Σ H,Σ ord,0 e Σ with a refinement such that it satisfies Condition 7.1.1.15 (with Σord,0 and) ~g ]ord,tor sends the stratum with some choice of σ e0 , and such that the morphism [e

~eord ~eord Z[(Φ˘ f0 ,δ˘f0 ,eσ0 )] to Z ˘ f,δ˘f,e [(Φ σ )] , we see that the induced morphism from the closure of H

H

H

H

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~eord Z ˘

˘f0 ,e [(ΦH σ 0 )] f0 ,δH tor

~eord to the closure of Z g] ˘ f,δ˘f,e [(Φ σ )] gives the existences of the morphisms [b H

H

tor

and [b g ] when gl = 1 (resp. [gl ]∗κ0 ,κ and ([gl ]∗κ0 ,κ ) when gb = 1) as in (4a) and (4b) e0 , Σ e 0, σ (resp. (5a) and (5b)) of Theorem 7.1.4.1, where κ0 = (H e0 ) lies in KQ,H0 ,Σord,0 (resp. KQ0 ,H,Σ ), which satisfy (4d), (4e), and (4f) (resp. (5e) and (5f)) thanks to the ~g ]ord,tor , except that (7.1.4.12) corresponding statements of Proposition 5.2.2.2 for [e and (7.1.4.23) still have to be explained. (As explained at the end of the proof of Proposition 7.2.3.5, the description concerning stratifications and formal com~g ]ord,tor is constructed using pletions are true because such a canonical morphism [e (6) of Theorem 5.2.1.1, which is consistent with the constructions of the canonical morphisms in Lemmas 5.2.4.38 and 7.1.2.29, and Proposition 7.1.2.17, using the various universal properties given in terms of degeneration data.) Also, (4g) and (5g) follow from Corollaries 3.4.4.3 and 5.2.2.3; (4h) and (5h) follow from Corollaries 3.4.4.4 and 5.2.2.4; (4i) follows from Corollaries 3.4.4.6 and 5.2.2.5; and (4j) follows from Corollary 5.2.2.5 and (4i) (where the finite flatness of (7.1.4.20), which is a finite morphism between regular schemes, is automatic by [35, IV-3, 15.4.2 e0 )⇒b)]; cf. [62, Lem. 6.3.1.11]). As in the case of showing Ri (fκtor 0 ,κ )∗ O~ ord,tor = 0 for i > 0 in Proposition N tor

κ0

tor

7.2.2.19, since the morphisms [b g] and ([gl ]∗κ0 ,κ ) are ´etale locally given by tor equivariant morphisms between toric schemes, we have Ri [b g ]∗ ON ~ ord,tor = 0 and κ0

tor

Ri ([gl ]∗κ0 ,κ )∗ ON ~ ord,tor = 0 for i > 0 (by [50, Ch. I, Sec. 3]), which are (7.1.4.12) and κ0

(7.1.4.23) of Theorem 7.1.4.1. The remaining statements in (4c) and (5c) of Theorem 7.1.4.1 now follow if we assume statements (3b) and (3c) of Theorem 7.1.4.1. (See Section 7.3.6 below.)

7.3

Calculation of Formal Cohomology

7.3.1

Setting

b Σ b ord ) = Throughout this section, unless otherwise specified, we assume that κ = (H, ord,+ ord,+ ord e e e [e κ] ∈ KQ,H,Σord for some κ e = (H, Σ , σ e) ∈ KQ,H,Σord , so that Hκ = H and ord ord ~ ord,grp → ~ ~ rκ = rH , and so that f : N → MH is a torsor under an abelian scheme N ord,tor ord,tor ord ord,ext ~ ~ ~ ~ M , which extends to a semi-abelian scheme N →M = M ord (where H

H

H,Σ

the subscripts “κ” are suppressed for the sake of simplicity); and we fix the choice of ~ ord,tor . The aim of this section an arbitrary (locally closed) stratum ~Zord [(ΦH ,δH ,τ )] of MH is to calculate the relative cohomology of the pullback of the structural morphism ~ ord,tor → M ~ ord,tor to the formal completion (M ~ ord,tor )∧ f tor : N . (See (5) of H H ~ Zord [(ΦH ,δH ,τ )]

Theorem 5.2.1.1 for a description of this formal completion. See also the first paragraph of Section 7.2.6 for a description of the formal completion (Ntor )∧ e Z [(ΦH ,δH ,τ )]

of Ntor

~eord tor −1 ~ ord along Z ) (Z[(ΦH ,δH ,τ )] ).) [(ΦH ,δH ,τ )] = (f

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~ ord Let ID ~ ord be the OM ~ ord,tor -ideal defining the relative Cartier divisor D∞,H (with ∞,H

H

its reduced structure) in (3) of Theorem 5.2.1.1, and let I~Eord = I~Eord be the ∞ ∞,κ ~ ord ~ ord ON ~ ord,tor -ideal defining the relative Cartier divisor E∞ = E∞,κ (with its reduced structure) in (1) of Theorem 7.1.4.1. (The subscripts “κ” are suppressed for the sake of simplicity.) Note that we have a canonical inclusion (f tor )∗ ID ~ ord ,→ I~ Eord

(7.3.1.1)

∞

∞,H

of ON as the radical of (f tor )∗ ID ~ ord -ideals, realizing I~ ~ ord . Eord ∞ ∞,H For M being one of the following quasi-coherent ONtor -modules ONtor , I~Eord , ∞ b tor and (f tor )∗ ID ~ ord , we will show that the relative cohomology R f∗ (M ) is locally ∞,H

free and canonically isomorphic to the (putative) answers given in the statements of Theorem 7.1.4.1. As a byproduct of the method, we will also investigate the . Since ) or O~eord := ker(ONtor  O~eord cases of M being I~eord Z[(Φ

Z[(Φ

H ,δH ,τ )]

Z[(Φ

H ,δH ,τ )]

H ,δH ,τ )]

f is proper (and since the choice of ~Zord [(ΦH ,δH ,τ )] is arbitrary), by Grothendieck’s fundamental theorem [35, III-1, 4.1.5] (and by fpqc descent for the property of local freeness [33, VIII, 1.11]), it suffices to show these over the pullback f tor : ~ ord,tor )∧ ~ ord,tor )∧ (Ntor )∧ord → (M of f tor to (M . These will tor

H

~ e Z

[(ΦH ,δH ,τ )]

H

~ Zord [(Φ

H ,δH ,τ )]

~ Zord [(Φ

H ,δH ,τ )]

be carried out in the remainder of this section. + ++ 0+ For simplicity of notation, we will denote by OX (resp. OX , resp. OX ) the tor ∗ ) under any morphism X → pullback of I~Eord (resp. (f ) ID ord ~ ord , resp. I~ e ∞

∞,H

Z[(Φ

H ,δH ,τ )]

Ntor from a formal scheme. For example, the pullback of I~Eord under the canonical + → Ntor will be denoted O(N tor )∧

morphism (Ntor )~∧ord e Z [(Φ

H ,δH ,τ )]

∞

.

ord ~ e Z [(ΦH ,δH ,τ )]

The definitions and arguments in this section will follow those in [61, Sec. 4] very closely, but we will take this opportunity to clarify or correct some flaws in the exposition there. 7.3.2

Formal Fibers of f tor

The definitions and arguments in this subsection will follow those in [61, Sec. 4A] very closely. Definition 7.3.2.1. ΓΦ˘ f,τ is the subgroup of elements in ΓΦ˘ f stabilizing (both) X H H and Y and inducing an element in ΓΦH ,τ (the subgroup of ΓΦH formed by elements mapping τ to itself). Since we have tacitly assumed that ΓΦH ,τ is trivial by Conditions 1.2.2.9 and [62, Lem. 6.2.5.27], ΓΦ˘ f,τ is also the subgroup of elements in ΓΦ˘ f fixing (both) X and H H Y. Let ΓΦ˘ c,ΦH be as in Definition 1.2.4.21. By Lemma 1.2.4.25, ΓΦ˘ c,ΦH maps σ ˘, H H the image of σ e in PΦ˘ f , to itself. On the other hand, by Condition 1.2.2.9 (and H

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Lemma 1.3.4.3), if a cone τ˘ ⊂ P+ ˘ Φ

f H

e ˘ has a face that is a Γ ˘ -translation in Σ Φ f,τ Φf H

H

˘. of σ ˘ , then it cannot have a different face that is also a ΓΦ˘ f,τ -translation of σ H

e˘ e ˘ consisting of cones τ˘ satisfying Definition 7.3.2.2. Σ ΦH σ ,τ is the subset of ΣΦ f,˘ f H the following conditions (cf. similar conditions in the first paragraph of Section 7.2.6): (1) τ˘ ⊂ P+ ˘ . Φ f H

(2) τ˘ has σ ˘ as a face. (3) The image of τ˘ under the (canonical) second morphism in (1.2.4.20) is contained in τ ⊂ P+ ΦH . e˘ e˘ e˘ In other words, Σ ΦH σ ,τ is the subset of ΣΦ σ = ΣΦ σ (see (1) of Definition f,˘ c,˘ f,˘ H H 1.2.4.21) consisting of cones τ˘ whose image under the (canonical) second morphism in (1.2.4.20) is contained in τ ⊂ P+ ΦH . Thus, to obtain a complete list of representatives of the equivalence classes ord e ˘ e , δ˘ e , τ˘)] parameterizing the strata of ~Z [(Φ [(ΦH ,δH ,τ )] , it suffices to take representaH H e tives of ΣΦ˘ f,˘σ,τ modulo the action of ΓΦ˘ c,ΦH . (That is, we do not have to consider H H ΓΦ˘ c,ΦH -translations of σ ˘ .) H

~e ord ~e ord Let Ξ (τ ) denote the toroidal embedding of Ξ ˘ ˘ ˘ f,δ˘f formed by gluing the Φ f,δ f Φ H

H

H

H

~e ord ~e ord affine toroidal embeddings Ξ τ ) over C ˘ runs through cones ˘ f,δ˘f (˘ ˘ f,δ˘f , where τ Φ Φ H

H

H

H

~e ord in ΣΦ˘ f,˘σ,τ . To minimize confusion, we shall distinguish between Ξ τ1 ) and ˘ f,δ˘f (˘ Φ H

H

H

~e ord ˘ e , δ˘ e , τ˘1 )] = [(Φ ˘ e , δ˘ e , τ˘2 )]. For each τ˘ as above (having Ξ τ2 ) even when [(Φ ˘ f,δ˘f (˘ Φ H H H H H

H

~e ord σ ˘ as a face), recall that we have denoted the closure of the σ ˘ -stratum of Ξ τ) ˘ f,δ˘f (˘ Φ H

H

~e ord ~e ord ~e ord τ ), let by Ξ τ ). Let Ξ ˘ f,δ˘f,˘ ˘ f,δ˘f,˘ ˘ f,δ˘f,˘ Φ σ (˘ Φ σ (τ ) denote the union of all such ΞΦ σ (˘ H

H

H

H

H

H

~e ord ~e ord ~e ord Ξ ˘ f,δ˘f,˘ ˘ f,δ˘f,˘ ˘ f,δ˘f,τ denote the union of all such Ξ σ ,τ denote the Φ Φ τ , and let XΦ H

H

H

H

H

H

~e ord ~e ord formal completion of Ξ ˘ f,δ˘f,˘ ˘ f,δ˘f,τ . Φ σ (τ ) along ΞΦ H

H

H

H

~e ord e˘ For each τ˘ ∈ Σ ˘ f,δ˘f,τ formed by the Φ f,˘ σ ,τ , consider the open subscheme Uτ˘ of ΞΦ H

H

H

~e ord ~e ord union of all (locally closed) strata of Ξ ˘ f,δ˘f,τ that contains the stratum Ξ ˘ f,δ˘f,˘ Φ Φ τ H

H

H

H

~e ord in its closure, and consider the open formal subscheme Uτ˘ of X ˘ f,δ˘f,˘ Φ σ ,τ supported H

H

~e ord ~e ord on Uτ˘ . The subscheme Uτ˘ of Ξ τ ) given ˘ f,δ˘f,τ is the closed subscheme of Ξ ˘ f,δ˘f (˘ Φ Φ H

H

H

H

~e ord ~e ord ~e ord τ ) and Ξ by the intersection of Ξ ˘ f,δ˘f (˘ ˘ f,δ˘f,τ (as an open subscheme of Ξ ˘ f,δ˘f,τ ), Φ Φ Φ H

H

H

H

H

H

~e ord ~e ord and the formal subscheme Uτ˘ of X τ) ˘ f,δ˘f,˘ ˘ f,δ˘f,˘ Φ σ ,τ is the formal completion of ΞΦ σ (˘ H

H

H

H

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along Uτ˘ . The collection {Uτ˘ }τ˘∈Σ e˘

Φ f,˘ σ ,τ H

447

~e ord forms an open covering of Ξ ˘ f,δ˘f,τ . We Φ H

H

~e ord can interpret X ˘ f,δ˘f,˘ e˘ Φ σ ,τ as constructed by gluing the collection {Uτ˘ }τ˘∈Σ H

page 447

H

of

Φ f,˘ σ ,τ H

formal schemes along their intersections (of supports). Definition 7.3.2.3.

(1) τ˘σ˘∨ is the intersection of (˘ τ 0 )∨ ˘0 running through faces of τ˘ ˘ f ) for τ 0 (in SΦ H e˘ in Σ ˘ itself). ΦH σ ,τ (including τ f,˘ ∨ (2) τ˘σ˘ ,+ is the intersection of (˘ τ 0 )∨ ˘0 running through faces of τ˘ ˘ f ) for τ 0 (in SΦ H in ΣΦ˘ f (including τ˘ itself) that also has σ ˘ as a face. H ∨ (3) τ+ is the intersection of (τ 0 )∨ (in S ) for τ 0 running through faces of τ Φ 0 H in ΣΦH (including τ itself). Then we have the canonical isomorphism 

 ~e ord ~e ord ˘ / ⊕ Ψ ˘ ⊕ Ψ ( `) ( `) Uτ˘ ∼ = SpecO ˘ ˘ ˘ ˘ Φ f,δ f Φ f,δ f ord ~ e˘ C ˘ Φ f,δ f H H

H

˘ τ∨ `∈˘

~e ord of schemes affine over C ˘ f,δ˘f . As O ~ ord Φ e H

H

CΦ ˘

phism

H

H

H

-modules, we have a canonical isomor-

H

˘ τ∨ `∈˘ σ ˘

H

˘ f,δH f H



~e ord ~e ord ˘ / ⊕ Ψ ˘ ∼ ⊕ Ψ = ˘ f,δ˘f (`) ˘ f,δ˘f (`) Φ Φ

˘ τ∨ `∈˘

H

˘ τ∨ `∈˘ σ ˘

H

⊕ ∨ ˘ τ ∨ −˘ `∈˘ τσ ˘

~e ord ˘ Ψ ˘ f,δ˘f (`). Φ H

H

If we equip τ˘∨ − τ˘σ˘∨ with the semigroup structure induced by the canonical bi~e ord ˘ as an jection (˘ τ ∨ − τ˘σ˘∨ ) → τ˘∨ /˘ τσ˘∨ , then we may interpret ⊕ Ψ ˘ f,δ˘f (`) Φ O ~eord CΦ ˘

∨ ˘ τ ∨ −˘ `∈˘ τσ ˘

H

H

-algebra, with algebra structure given by the isomorphisms

˘ f,δH f H

~e ord,∗ ~e ord ˘ ∆ ˘ `˘0 : Ψ ˘ f,δ˘f (`) ˘ f,δ˘f,`, Φ Φ H

H

H

H

O ~ ord e˘ C Φ

∼ =

inherited from those of O~e ord ΞΦ ˘

~e ord ˘ Ψ ˘ f,δ˘f (`) Φ H

H

H

H

H

H

˘ f,δH f H

ord

~e ˘ if `˘ + `˘0 ∈ τ˘∨ − τ˘∨ and by Ψ ˘ f,δ˘f (`) Φ σ ˘

⊕ ˘ ˘ `∈S Φ

˘ f,δH f H

ord ~e ord 0 ∼ ~ e Φ˘ ,δ˘ (`˘ + `˘0 ) Ψ ˘ f,δ˘f (`˘ ) → Ψ Φ f f

⊗

H

~e ord 0 Ψ ˘ f,δ˘f (`˘ ) → 0 Φ

⊗

O ~ ord e˘ C Φ

H

f H

H

H

˘ f,δH f H

otherwise. Then we have a canonical isomorphism   ~e ord ˘ . Uτ˘ ∼ ⊕ Ψ ( `) = SpecO ˘ ˘ Φ f,δ f ord ~ e˘ C ˘ Φ f,δ f H H

∨ ˘ τ ∨ −˘ `∈˘ τσ ˘

H

H

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By definition, we have τ˘∨ − τ˘σ˘∨ =



∪

τ˘0 face of τ˘ e˘ in Σ Φ ,˘ σ ,τ

(˘ τ 0 )⊥ ∩ τ˘∨




⊂σ ˘ ⊥ ∩ τ˘∨ .

f H

The formal scheme Uτ˘ , being the formal completion of   ~e ord ~e ord ˘ Ξ ⊕ Ψ τ) ∼ = SpecO ˘ f,δ˘f,˘ ˘ f,δ˘f (`) Φ σ (˘ Φ H

H

ord ~ e˘ C ˘ Φ f,δ f H H

H

˘ σ ⊥ ∩ τ˘∨ `∈˘

H

along Uτ˘ , can be canonically identified with the relative formal spectrum of the ~e ord ~e ord ˘ over C ˆ denotes the compleˆ Ψ OCe ˘ ˘ -algebra ⊕ ˘ f,δ˘f (`) ˘ f,δ˘f , where ⊕ Φ Φ Φ f,δ f H H

H

˘ σ ⊥ ∩ τ˘∨ `∈˘

H

H

ord

tion of the sum with respect to the O~e ord ΞΦ ˘

H

(˘ τ) ˘ ,˘ f,δH f σ H

⊕

-ideal

˘ σ ⊥ ∩ τ˘∨ `∈˘ σ ˘

~e ˘ Note Ψ ˘ f,δ˘f (`). Φ H

H

that all the above canonical isomorphisms correspond to canonical isomorphisms ~e ord ˘ (with of O ~eord -algebras formed by sums of sheaves of the form Ψ ˘ f,δ˘f (`) Φ CΦ ˘

O ~eord CΦ ˘

H

˘ f,δH f H

˘ f,δH f H

-algebra structures inherited from that of O~e ord ΞΦ ˘

let us write OUτ˘ ∼ = ∼ OU0+ = τ ˘ OUτ˘ ∼ =

ˆ ⊕ ˘ σ ⊥ ∩ τ˘∨ `∈˘

H

). By abuse of language,

˘ f,δH f H

~e ord ˘ Ψ ˘ f,δ˘f (`), Φ H

H

ord

ˆ ⊕ ˘ σ ⊥ ∩ τ˘∨ `∈˘ σ ˘

⊕ ∨ ˘ τ ∨ −˘ `∈˘ τσ ˘

~e ˘ Ψ ˘ f,δ˘f (`), Φ H

H

~e ord ˘ Ψ ˘ f,δ˘f (`). Φ H

H

By Condition 1.2.2.9 (and Lemma 1.3.4.3), the action of ΓΦ˘ c,ΦH induces only H the trivial action on each stratum it stabilizes. Therefore, the quotient morphism ~e ord ~e ord X ˘ f,δ˘f,˘ ˘ f,δ˘f,˘ ˘ c,ΦH σ ,τ /ΓΦ Φ σ ,τ → XΦ H

H

H

H

(7.3.2.4)

H

of formal schemes over ~S0,rH is a local isomorphism. The morphism (7.3.2.4) is not ~e ord ~e ord defined over C ˘ f,δ˘f when the action of ΓΦ ˘ f,δ˘f is nontrivial. Nevert˘ c,ΦH on C Φ Φ H H H H H ~ ord , and hence heless, since Γ ˘ acts trivially on ΦH , it acts trivially on C ΦH ,δH

ΦH c,ΦH

~ ord . (7.3.2.4) is defined over C ΦH ,δH Proposition 7.3.2.5. (Compare with [61, Prop. 4.3].) There is a canonical isomorphism ~ ord,tor )∧ord (N ~ e Z [(Φ

H ,δH ,τ )]

~e ord ∼ =X ˘ f,δ˘f,˘ ˘ c,ΦH Φ σ ,τ /ΓΦ H H H

~ ord , characterized by the identifications of formal schemes over C ΦH ,δH ~ ord,tor )∧ord (N ~ e ˘ Z [(Φ

˘ ,˘ f,δH f τ )] H

~e ord ∼ =X ˘ f,δ˘f,˘ Φ σ ,˘ τ H H

(7.3.2.6)

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~e ord of formal schemes over C ˘ f,δ˘f (compatible with the canonical morphisms Φ H

H

~ ord,tor )∧ord (N ~ e ˘ Z [(Φ

˘ ,˘ f,δH f τ )] H

~ ord,tor )∧ord → (N ~ e Z [(Φ

H ,δH ,τ )]

ord

~e ~ ord ). (The formation of the formal completion here is similar and C ˘ f,δ˘f → C Φ ΦH ,δH H H to the one in (5) of Theorem 5.2.1.1.) ~e ord e τ˘ denote the completion of Ξ e˘ Proof. Let τ˘ ∈ Σ . Let U τ ) along Uτ˘ , ˘ f,δ˘f (˘ Φ ΦH ,˘ σ ,τ f H H which contains Uτ˘ as a closed formal subscheme (with the same support Uτ˘ ). ~e ord Since Uτ˘ is the union of (Ξ ˘0 running through faces of τ˘ in ˘ f,δ˘f )τ˘0 with τ Φ H H e τ˘ e˘ Σ , which are cones in P+ , the tautological degeneration data over U ˘f Φ H

ΦH σ ,τ f,˘

satisfies the positivity condition (with respect to the ideal defining Uτ˘ ), and we obtain by Mumford’s construction as in Section 4.2.2 a degenerating family ~e ord ord e ♥ei, ♥ α e e ♥ λ, eHe p , ♥ α eH ( ♥ G, e (cf. Definition 3.4.2.10), which we e ) → Uτ˘ of type MH p

call a Mumford family (cf. Definition 4.2.2.21). Note that a Mumford family is defined in the sense of relative schemes, namely as a functorial assignment to e τ˘ a degenerating family of type each affine open formal subscheme Spf(R) of U ord ~e M e over Spec(R). Therefore, (6) of Theorem 5.2.1.1 applies, and implies the H ~e ord,tor e τ˘ → M under which existence of a canonical (strata-preserving) morphism U e H ~e ord,tor ord ord e ♥ei, ♥ α e e ♥ λ, e e e e ep , α ( ♥ G, . eHe p , ♥ α eH e e ) → Uτ˘ is the pullback of (G, λ, i, α e ) → MH H eH p

p

ord,tor

~e e e 0 e˘ Moreover, if τ˘ ∈ Σ e ΦH σ ,τ , then the morphisms from Uτ˘ and from Uτ˘ to MH f,˘ e τ˘ ∩ U e τ˘0 . agree over the intersection U e e , δe e , σ By taking the closures of the [(Φ H H e )]-strata (not as closed subschemes of the supports, but as closed formal subschemes), and by arguing as in the proof ~ ord,tor for all τ˘ in of Proposition 7.2.2.19, we obtain canonical morphisms Uτ˘ → N e˘ Σ ΦH σ ,τ , which patch together, cover all strata above [(ΦH , δH , τ )], and induce the f,˘ desired isomorphism (7.3.2.6). 0

Let us also consider OU+τ˘ and OU++ . By definition, the OUτ˘ -ideal OU+τ˘ is isomorτ ˘ ~e ord -ideal defining the complement of Ξ phic to the pullback of the O~e ord ˘ f,δ˘f,˘ Φ σ ΞΦ ˘

ord

H

(˘ τ) ˘ ,˘ f,δH f σ H

~e in Ξ τ ). (In general, this is different from the O~e ord ˘ f,δ˘f,˘ Φ σ (˘ H

H

(˘ τ) ˘ ,˘ f,δH f σ H

ΞΦ ˘

H

-ideal defining

~e ord ~e ord the closed subscheme Ξ of Ξ τ ).) The above descriptions imply the ˘ f,δ˘f,˘ ˘ f,δ˘f,˘ Φ τ Φ σ (˘ H H H H following simple but important facts: Lemma 7.3.2.7. (Compare with [61, Lem. 4.1].) Suppose τ˘ and τ˘0 are two cones e˘ in Σ ˘0 is a face of τ˘. Then: Φ f,˘ σ ,τ such that τ H

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(1) We have a canonical open immersion Uτ˘0 ,→ Uτ˘ (resp. Uτ˘0 ,→ Uτ˘ ) of formal ~e ord subschemes of X ˘ f,δ˘f,˘ Φ σ ,τ . H H (2) The compatible canonical restriction morphisms OUτ˘ → OUτ˘0 , OU+τ˘ → OU+τ˘0 , OU++ → OU++ , τ ˘ τ ˘0 OU0+ → OU0+ τ ˘ τ ˘0 correspond to the compatible canonical morphisms ord

ord

ˆ ⊕ ˘ σ ⊥ ∩ τ˘∨ `∈˘

ˆ ⊕ ˘ σ ⊥ ∩ τ˘∨ `∈˘ σ ˘ ,+

ˆ ⊕ ∨) ˘ σ ⊥ ∩(˘ `∈˘ τ ∨ +τ+

~e ˘ → Ψ ˘ f,δ˘f (`) Φ H

H

~e ord ˘ → Ψ ˘ f,δ˘f (`) Φ H

H

~e ord ˘ → Ψ ˘ f,δ˘f (`) Φ H

H

ˆ ⊕ ˘ σ ⊥ ∩(˘ `∈˘ τ 0 )∨

˘ σ ⊥ ∩ τ˘∨ `∈˘ σ ˘

of O ~eord CΦ ˘

~e ˘ → Ψ ˘ f,δ˘f (`) Φ H

H

H

ˆ ⊕ ˘ σ ⊥ ∩(˘ `∈˘ τ 0 )∨ σ ˘ ,+

H

~e ord ˘ Ψ ˘ f,δ˘f (`), Φ H

ˆ ⊕ ∨) ˘ σ ⊥ ∩((˘ `∈˘ τ 0 )∨ +τ+

ord

ˆ ⊕

~e ˘ Ψ ˘ f,δ˘f (`), Φ

H

~e ord ˘ Ψ ˘ f,δ˘f (`), Φ H

H

ord

ˆ ⊕ ˘ σ ⊥ ∩(˘ `∈˘ τ 0 )∨ σ ˘

~e ˘ Ψ ˘ f,δ˘f (`) Φ H

H

ˆ in each -algebras, respectively, where the two instances of ⊕

˘ f,δH f H

expression denote the completions of the sums with respect to the sheaves ~e ord ~e ord ˘ and ˘ respectively, and of ideals ⊕ Ψ ⊕ Ψ ˘ f,δ˘f (`) ˘ f,δ˘f (`), Φ Φ ∨

˘ σ ⊥ ∩ τ˘∨ `∈˘ σ ˘ ∨ and +τ+

H

H

0 ∨

˘ σ ⊥ ∩(˘ `∈˘ τ 0 )∨ σ ˘

∨ +τ+

H

H

are defined by viewing SΦH as a subsemigroup (˘ τ) where τ˘ of SΦ˘ f using the (canonical) first morphism SΦH ,→ SΦ˘ f in (1.2.4.18). H H (3) The canonical restriction morphism OUτ˘ → OUτ˘0 corresponds to the canonical morphism ⊕ ∨ ˘ τ ∨ −˘ `∈˘ τσ ˘

of O ~eord CΦ ˘

˘ f f,δH H

~e ord ˘ → Ψ ˘ f,δ˘f (`) Φ H

H

⊕ ˘ τ 0 )∨ −(˘ `∈(˘ τ 0 )∨ σ ˘

~e ord ˘ Ψ ˘ f,δ˘f (`) Φ H

H

~e ord ~e ord ˘ to Ψ ˘ when -algebras, which maps Ψ ˘ f,δ˘f (`) ˘ f,δ˘f (`) Φ Φ H

H

H

H

`˘ ∈ (˘ τ ∨ − (˘ τ 0 )∨ τ ∨ − τ˘σ˘∨ ) ∩((˘ τ 0 )∨ − (˘ τ 0 )∨ σ ˘ ) = (˘ σ ˘ ), and to zero otherwise. (4) The correspondences in (2) and (3) above are canonically compatible with each other.

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By (5) of Theorem 5.2.1.1, we have a canonical isomorphism ~ ord,tor )∧ord (M ~ H Z

[(ΦH ,δH ,τ )]

∼ ~ ord =X ΦH ,δH ,τ .

(7.3.2.8)

By Proposition 7.3.2.5 and by the very constructions, we may identify the pullback ~e ord ~ ord,tor )∧ of f tor to (M with the canonical morphism X ˘ f,δ˘f,˘ ˘ c,ΦH → ord Φ σ ,τ /ΓΦ H ~ Z H

[(ΦH ,δH ,τ )]

H

H

~ ord X ΦH ,δH ,τ . By abuse of notation, we shall also denote this pullback by ~e ord ~ ord f tor : X ˘ f,δ˘f,˘ ˘ c,ΦH → XΦH ,δH ,τ . Φ σ ,τ /ΓΦ H

H

H

e˘ For each τ˘ ∈ Σ τ ] denote the image of Uτ˘ under (7.3.2.4), which is ΦH σ ,τ , let U[˘ f,˘ ~ ord . By admissibility of Σ ˘ , we know isomorphic to Uτ˘ as a formal scheme over C ΦH ,δH ord

ΦH f

~e e˘ that the set Σ ˘ f,δ˘f,˘ ˘ c,ΦH is finite. Then XΦ ˘ c,ΦH can be constructed σ ,τ /ΓΦ ΦH σ ,τ /ΓΦ f,˘ H H H H of formal schemes over their by gluing the finite collection {U[˘τ ] }[˘τ ]∈Σ e˘ /Γ ˘ Φ f,˘ σ ,τ H

intersections. Let us denote by

Φ c,ΦH H

tor ~ ord f[˘ τ ] → XΦH ,δH ,τ τ ] : U[˘

the restriction of f tor to U[˘τ ] . If we choose a representative τ˘ of [˘ τ ], then we can tor ord tor ~ ~ ord identify f[˘ : U → X with the canonical morphism f : Uτ˘ → X [˘ τ] ΦH ,δH ,τ τ˘ ΦH ,δH ,τ τ] ~e ord ~ ord induced by the canonical morphism X ˘ f,δ˘f,˘ Φ σ ,τ → XΦH ,δH ,τ . Let us denote by H

H

~ ord gτ˘ : Uτ˘ → X ΦH ,δH ,τ

~e ord C ˘ f,δ˘f , Φ

× ~ ord C Φ ,δ H

H

H

H

~e ord ~ Φord,δ , h:C ˘ f,δ˘f → C Φ H H H

H

and ~ ord hτ : X ΦH ,δH ,τ

~e ord ~ ord C ˘ f,δ˘f → X Φ ΦH ,δH ,τ

× ~ ord C Φ ,δ H

H

H

H

the canonical morphisms. Then we have a canonical identification fτ˘tor = hτ ◦ gτ˘ . ~e ord (Note that gτ˘ is a morphism between affine formal schemes over C ˘ f,δ˘f , and that Φ H H ord ~ ~ hτ is the pullback of h to the affine formal scheme XΦH ,δH ,τ over CΦord .) H ,δH ~ ord , For simplicity, let us view O ~ ord and O~ ord as sheaves over C XΦ

H ,δH ,τ

Z[(Φ

H ,δH ,τ )]

ΦH ,δH

~ ord ~ ord ~ ord ~ ord and suppress (X ΦH ,δH ,τ → CΦH ,δH )∗ and (Z[(ΦH ,δH ,τ )] → CΦH ,δH )∗ from the noord ord ~ ~ tation. For push-forwards (to C ΦH ,δH ) of sheaves over XΦH ,δH ,τ , we shall use the ˆ notation ⊕ to denote the completion with respect to (the push-forward of) the ideal of definition of OX . ~ ord ΦH ,δH ,τ

Based on Lemma 7.3.2.7, we have the following important facts: Lemma 7.3.2.9. (Compare with [61, Lem. 4.6].)

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e˘ (1) For each τ˘ ∈ Σ σ ,τ , and each integer d ≥ 0, we have the canonical ΦH f,˘ isomorphisms Rd (fτ˘tor )∗ OUτ˘ ∼ = Rd (fτ˘tor )∗ OU+τ˘ ∼ = ∼ Rd (fτ˘tor )∗ OU++ = τ ˘ ∼ Rd (fτ˘tor )∗ OU0+ = τ ˘ Rd (fτ˘tor )∗ OUτ˘ ∼ = of OX ~ ord

ˆ ⊕ ˘ σ ⊥ ∩ τ˘∨ `∈˘

~e ord ˘ Rd (hτ )∗ (Ψ ˘ f,δ˘f (`)), Φ H

H

ord

ˆ ⊕ ˘ σ ⊥ ∩ τ˘∨ `∈˘ σ ˘ ,+

~e ˘ Rd (hτ )∗ (Ψ ˘ f,δ˘f (`)), Φ H

ˆ ⊕ ∨) ˘ σ ⊥ ∩(˘ `∈˘ τ ∨ +τ+

H

~e ord ˘ Rd (hτ )∗ (Ψ ˘ f,δ˘f (`)), Φ H

H

ord

ˆ ⊕ ˘ σ ⊥ ∩ τ˘∨ `∈˘ σ ˘

⊕ ∨ ˘ τ ∨ −˘ `∈˘ τσ ˘

~e ˘ Rd (hτ )∗ (Ψ ˘ f,δ˘f (`)), Φ H

H

~e ord ˘ Rd (hτ )∗ (Ψ ˘ f,δ˘f (`)) Φ H

H

-modules.

ΦH ,δH ,τ

(2) For each γ ∈ ΓΦ˘ c,ΦH , we have a commutative diagram H

gτ˘

~ ord X ΦH ,δH ,τ



~e ord C ˘ f,δ˘f Φ

× ~ ord C Φ ,δ H

/ Uγ τ˘

γ

Uτ˘

H

 γ

H

~ ord /X ΦH ,δH ,τ

H

gγ τ˘

× ~ ord C Φ ,δ H

~e ord C ˘ f,δ˘f Φ H

H

H

hτ

hτ





~ ord X ΦH ,δH ,τ

~ ord X ΦH ,δH ,τ

of formal schemes, (naturally) compatible with the commutative diagram

gτ˘

~ ord Ξ ΦH ,δH ,τ

 H



ord

~e C ˘ f,δ˘f Φ

× ~ ord C Φ ,δ

/ Uγ τ˘

γ

Uτ˘

H

H

hτ

 ~ ord Ξ ΦH ,δH ,τ

H

γ

~ ord /Ξ ΦH ,δH ,τ

gγ τ˘ ord

~e C ˘ f,δ˘f Φ

× ~ ord C Φ ,δ H

H

H

H

hτ

 ~ ord Ξ ΦH ,δH ,τ

of their supports. Then the canonical morphisms in (1) are compatible with the canonical isomorphisms γ ∗ OUγ τ˘ → OUτ˘ , γ ∗ OU+γ τ˘ → OU+τ˘ , γ ∗ OU++ → OU++ , γ ∗ OU0+ → OU0+ , and γ ∗ OUγ τ˘ → OUτ˘ induced by the γτ ˘ τ ˘ γτ ˘ τ ˘ ord ∼ ~ ~e ord ~e ord ˘ → ˘ over C e Φ˘ ,δ˘ (`) canonical isomorphisms γ ∗ : γ ∗ Ψ Ψ ˘ f,δ˘f , re˘ f,δ˘f (γ `) Φ Φ f H f H H H H H spectively.

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e˘ (3) For each integer d ≥ 0, if τ˘0 is a face of τ˘ in Σ σ ,τ , then the canonical ΦH f,˘ morphisms Rd (fτ˘tor )∗ OUτ˘ → Rd (fτ˘tor 0 )∗ OU 0 , τ ˘ + Rd (fτ˘tor )∗ OU+τ˘ → Rd (fτ˘tor 0 )∗ O Uτ˘0 , ++ Rd (fτ˘tor )∗ OU++ → Rd (fτ˘tor 0 )∗ O Uτ˘0 , τ ˘ 0+ Rd (fτ˘tor )∗ OU0+ → Rd (fτ˘tor 0 )∗ O Uτ˘0 , τ ˘

Rd (fτ˘tor )∗ OUτ˘ → Rd (fτ˘tor 0 )∗ OU 0 τ ˘ induced by restriction from Uτ˘ to Uτ˘0 correspond to the morphisms ˆ ⊕ ˘ σ ⊥ ∩ τ˘∨ `∈˘

ˆ ⊕ ˘ σ ⊥ ∩ τ˘∨ `∈˘ σ ˘ ,+

ˆ ⊕ ∨) ˘ σ ⊥ ∩(˘ `∈˘ τ ∨ +τ+

ˆ ⊕ ˘ σ ⊥ ∩ τ˘∨ `∈˘ σ ˘

⊕ ∨ ˘ τ ∨ −˘ `∈˘ τσ ˘

of OX ~ ord

~e ord ˘ → Rd h∗ (Ψ ˘ f,δ˘f (`)) Φ H

H

~e ord ˘ → Rd h∗ (Ψ ˘ f,δ˘f (`)) Φ H

H

~e ord ˘ → Rd h∗ (Ψ ˘ f,δ˘f (`)) Φ H

H

~e ord ˘ → Rd h∗ (Ψ ˘ f,δ˘f (`)) Φ H

H

~e ord ˘ → Rd h∗ (Ψ ˘ f,δ˘f (`)) Φ H

H

-modules, respectively.

ˆ ⊕ ˘ σ ⊥ ∩(˘ `∈˘ τ 0 )∨

~e ord ˘ Rd h∗ (Ψ ˘ f,δ˘f (`)), Φ H

ˆ ⊕ ˘ σ ⊥ ∩(˘ `∈˘ τ 0 )∨ σ ˘ ,+

~e ord ˘ Rd h∗ (Ψ ˘ f,δ˘f (`)), Φ H

ˆ ⊕ ∨) ˘ σ ⊥ ∩((˘ `∈˘ τ 0 )∨ +τ+

ˆ ⊕ ˘ σ ⊥ ∩(˘ `∈˘ τ 0 )∨ σ ˘

H

H

~e ord ˘ Rd h∗ (Ψ ˘ f,δ˘f (`)), Φ H

H

~e ord ˘ Rd h∗ (Ψ ˘ f,δ˘f (`)), Φ

⊕ ˘ τ 0 )∨ −(˘ `∈(˘ τ 0 )∨ σ ˘

H

H

~e ord ˘ Rd h∗ (Ψ ˘ f,δ˘f (`)) Φ H

H

All of these morphisms send

ΦH ,δH ,τ

ord

~e ~e ord ˘ (identically) to Rd h∗ (Ψ ˘ when it is defined on Rd h∗ (Ψ ˘ f,δ˘f (`)) ˘ f,δ˘f (`)) Φ Φ H H H H both sides, and to zero otherwise. 7.3.3

Relative Cohomology and Local Freeness

The definitions and arguments in this subsection will follow those in [61, Sec. 4B] very closely. ~ ord,tor )∧ ~ ord By (7.3.2.8), we shall identify (M with X ΦH ,δH ,τ , and identify H ~ Zord [(ΦH ,δH ,τ )]

~Zord ~ ord ~ ord [(ΦH ,δH ,τ )] with ΞΦH ,δH ,τ . For simplicity of notation, we shall use XΦH ,δH ,τ and ~Zord [(ΦH ,δH ,τ )] more often than their counterparts. ~e ord ~ ord ~ ord,grp Recall that C ef ) is a torsor under an abelian scheme C e f,δ ΦH ,δH (resp. C Φ ΦH ,δH H

H

f ~e ord,ΦH ~e ord,grp ~ ord,ΦH (resp. M ~ ord,ZH (resp. (resp. C etale cover M ) of M ef ) over the finite ´ e f,δ e H Φ H H

e

H

H

f ~e ord,ZH M ) (see Section 4.2 and, in particular, Propositions 4.2.1.29 and 4.2.1.30). e H Since the pairing h · , · ie is the direct sum of the pairings on Q−2 ⊕ Q0 and on ef ef ~e ord,ΦH ~e ord,ZH ∼ ∼ ~ ord,ΦH and M ~ ord,ZH (cf. Lemmas 5.2.4.1 and L, we have M = M = M e e H H H H

e

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ord ord 5.2.4.5). Let (B, λB , iB , ϕ−1,Hp , ϕord eord eord −1,Hp ) and (ϕ−2,H , ϕ0,H ) (resp. (ϕ e, ϕ e )) −2,H 0,H f ~ ord,ΦH ~ ord,ΦH (resp. M ). Let T (resp. T ∨ ) be the be the tautological tuples over M H e H split torus with character group X (resp. Y ). For simplicity of notation, we shall denote the pullbacks of B, B ∨ , T , and T ∨ , respectively, by the same symbols. ~ ord The pullback of G (resp. G∨ ) to X ΦH ,ZH ,τ (as a formal scheme, rather than as a relative scheme as in the case of Mumford families) is an extension of B (resp. B ∨ ) by T (resp. T ∨ ), and this extension is a pullback of the tautological extension G\ ~ ord . For simplicity, we shall also denote the pullbacks of G\ (resp. G∨,\ ) over C ΦH ,δH ∨,\ and G , respectively, by the same symbols. By Lemma 7.1.2.1, the morphism ~e ord ~ ord h:C ˘ f,δ˘f → C Φ ΦH ,δH is proper and smooth, and is a torsor under the pullback to H H ord ~ ord,ΦH . ~ e B)◦ → M C of an abelian scheme Q× -isogenous to Hom (X,

e

O

ΦH ,δH

H

e σ˘ ,τ of the cones τ˘ in Σ e˘ Consider the union N ΦH σ ,τ , which has a closed covef,˘ e σ˘ ,τ ) of the cones τ˘ in Σ e˘ ring by the closures τ˘cl (in N ΦH σ ,τ (with natural incidence f,˘ relations inherited from those of the cones τ˘ as locally closed subsets of (SΦ˘ f )∨ R ). H

By definition, the nerve of the open covering {Uτ˘ }τ˘∈Σ e˘

Φ f,˘ σ ,τ H

valently the open covering {Uτ˘ }τ˘∈Σ e˘

Φ f,˘ σ ,τ H

formal schemes {Uτ˘ }τ˘∈Σ e˘

~e ord of X ˘ f,δ˘f,˘ Φ σ ,τ , or equiH

H

~e ord of Ξ ˘ f,δ˘f,˘ Φ σ (τ ) (by the supports of the H

H

), is naturally identified with the nerve of the closed

Φ f,˘ σ ,τ H

covering {˘ τ cl }τ˘∈Σ e˘

Φ f,˘ σ ,τ H

e σ˘ ,τ . of N

e σ˘ ,τ here differs from that in [61, Sec. 4B]. Remark 7.3.3.1. Our description of N (The description in [61, Sec. 4B] is misleading because it abusively identifies the homology of the nerve with the cohomology of the dual one realized by the unions of closures of cones. We take this opportunity to present the clarified and corrected exposition here.) Accordingly, if we set e σ˘ ,τ /Γ ˘ Nσ˘ ,τ := N Φ c,ΦH , H

cl e˘ and let [˘ τ ] denote the closure of [˘ τ ] in Nσ˘ ,τ , for each [˘ τ] ∈ Σ ˘ c,ΦH . Then ΦH σ ,τ /ΓΦ f,˘ H the nerve of the open covering {U[˘τ ] }[˘τ ]∈Σ (7.3.3.2) e˘ /Γ ˘ Φ f,˘ σ ,τ H

Φ c,ΦH H

ord

~ ord,tor )∧ord of (N ~ e Z [(Φ

H ,δH ,τ )]

~e ∼ =X ˘ f,δ˘f,˘ ˘ c,ΦH , or equivalently the open covering Φ σ ,τ /ΓΦ H H H

{Uτ˘ }[˘τ ]∈Σ e˘

˘ ,Φ σ ,τ /ΓΦ Φ f,˘ c H H H

ord

ord

~e ∼ ~e ˘ ˘ /Γ ˘ of Z of the supports of formal schemes, is naturally [(ΦH ,δH ,τ )] = ΞΦ ΦH f,δH f,τ c,ΦH H identified with the nerve of the closed covering cl {[˘ τ ] }[˘τ ]∈Σ (7.3.3.3) e˘ /Γ ˘ Φ f,˘ σ ,τ H

Φ c,ΦH H

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of Nσ˘ ,τ . e σ˘ ,τ in closer detail. By choosing Let us analyze the topological structure of N ˘ e ⊗ Q (over Q), we obtain the some (noncanonical) splitting of sX˘ ⊗ Q : X ⊗ Q  X Z

Z

Z

decomposition (1.2.4.31) of (SΦ˘ f )∨ b˘ R , inducing the projection pr(S

∨ Φ c )R H

H

in (1.2.4.32)

b ˘ (see Definition 1.2.4.29) because σ e is a (defined over Q). Note that σ ˘⊥ = S Φc H

top-dimensional cone in P+ e . Then Lemmas 1.2.4.38 and 1.2.4.39 also imply the Φ f H

following:

Corollary 7.3.3.4. (Compare with Corollary 1.2.4.40.) The set {pr(Sb ˘

∨ Φ c )R H

(˘ τ )}τ˘∈Σ e˘

Φ f,˘ σ ,τ H

of rational polyhedral cones defines a ΓΦ˘ c,ΦH -admissible rational polyhedral H cone decomposition (cf. Definition 1.2.2.4) of   e σ˘ ,τ ) = pr(Sb ˘ )∨ (N ∪ pr(Sb ˘ )∨ (˘ τ) (7.3.3.5) Φc R H

e˘ τ˘∈Σ Φ

Φc R H

σ ,τ f,˘ H

in the sense that we have the following: (1) Every pr(Sb ˘

∨ Φ c )R H

(˘ τ ) is a nondegenerate rational polyhedral cone.

(2) The union (7.3.3.5) e σ˘ ,τ ). pr(Sb ˘ )∨ (N

is

(3) {pr(Sb ˘

is invariant under the action of ΓΦ˘ c,ΦH in the

Φc R H

∨ Φ c )R H

(˘ τ )}τ˘∈Σ e˘

Φ f,˘ σ ,τ H

disjoint

and

defines

a

stratification

of

H

sense that ΓΦ˘ c,ΦH permutes the cones in it. Under this action, the set H of ΓΦ˘ c,ΦH -orbits is finite. H

Proof. The same argument of the proof of Corollary 1.2.4.40 also works here (see Definition 7.3.2.2). Corollary 7.3.3.6. The projection pr(Sb ˘

∨ Φ c )R H

e σ˘ ,τ to pr b valence from N (S ˘

∨ Φ c )R H

in (1.2.4.32) induces a homotopy equi-

e σ˘ ,τ ). (N

Proof. Any continuous section x ˜0 as in Lemma 1.2.4.39 defines a continuous map e σ˘ ,τ ) → N e σ˘ ,τ whose pre- and post-compositions with pr b pr(Sb ˘ )∨ (N (S ˘ )∨ are homoΦc R H

topic to the identity morphisms. Hence, the corollary follows. Lemma 7.3.3.7. The preimage of pr(Sb ˘

∨ Φ c )R H

Φc R H

e σ˘ ,τ ) under the identification (N

∨ ∼ b ∨ (ΓΦ˘ c,ΦH )∨ ˘ c )R (in (1.2.4.32), induced by (1.2.4.31)) is the subset R ⊕(SΦH )R = (SΦ H H (ΓΦ˘ c,ΦH )∨ R × τ. H

Proof. By Condition 7.1.1.17, we know that τ˘ ⊂ (ΓΦ˘ c,ΦH )∨ ˘ ∈ R × τ for every τ H ∨ e˘ Σ ˘0 ˘ c,ΦH )R × τ is contained in τ Φ f,˘ σ ,τ . The question is whether every point of (ΓΦ H

H

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e ˘ ,τ as the union of the cones τ˘ in Σ e˘ e˘ for some τ˘ ∈ Σ ΦH σ ,τ . By definition of Nσ ΦH σ ,τ f,˘ f,˘ (see Definition 7.3.2.2), the answer is in the affirmative, because of the following   0 y elementary fact: If t is a (partitioned) real symmetric matrix with entries in yz R such that z is positive definite, and if x0 is a particular positive definite matrix with entries in R, then there exists a sufficiently large real number t0 such that   tx0 y is positive definite for all t ≥ t0 . t y z Lemma 7.3.3.8. The projection pr(ΓΦ˘

)∨ c,ΦH R H

∨ ∨ : (ΓΦ˘ c,ΦH )∨ ˘ c,ΦH )R : (y, z) 7→ y R ⊕(SΦH )R → (ΓΦ H

H

∨ induces a homotopy equivalence from (ΓΦ˘ c,ΦH )∨ ˘ c,ΦH )R . R × τ to (ΓΦ H

H

Proof. By choosing any point z0 in τ , we obtain a continuous section ∨ (ΓΦ˘ c,ΦH )∨ ˘ c,ΦH )R × τ : y 7→ (y, z0 ) R → (ΓΦ H

H

whose pre- and post-compositions with pr(ΓΦ˘

|

)∨ c,ΦH R H

(ΓΦ ˘

)∨ c,ΦH R H

×τ


are homotopic

to the identity morphisms. Hence, the lemma follows, as desired. Lemma 7.3.3.9. (Compare with [61, Lem. 4.21].) The topological space Nσ˘ ,τ is homotopic to the real torus ∨ TΦ ˘ c,ΦH )R /ΓΦ ˘ c,ΦH , e f,ΦH := (ΓΦ H

H

H

whose cohomology groups (by contractibility of (ΓΦ˘ c,ΦH )∨ R ) are H

∼ j ˘ H j (TΦ , Z) ∼ = ∧j (HomZ (ΓΦ˘ c,ΦH , Z)) e f,ΦH , Z) = H (ΓΦ c,ΦH H

H

H

~ ord ⊗ Q, we have a canonical isomorphism for each integer j ≥ 0. Over C ΦH ,δH Z

H j (ΓΦ˘ c,ΦH , Z) ⊗ OC~ ord H

Z

ΦH ,δH

⊗Q ∼ = ∧j (HomO (Q∨ , LieT ∨ /C~ ord

ΦH ,δH

Z

)) ⊗ Q.

(7.3.3.10)

Z

Proof. By Corollary 7.3.3.6 and Lemma 7.3.3.8, the projection ∨ ∨ ∨ ∨ (SΦ ˘ c,ΦH )R ⊕(SΦH )R → (ΓΦ ˘ c,ΦH )R : (x, y, z) 7→ y e f )R ⊕(ΓΦ H

H

H

e σ˘ ,τ to (Γ ˘ defines a homotopy equivalence from N )∨ . This homotopy equivaΦH c,ΦH R lence (defined by projection) is compatible with the action of (ΓΦ˘ c,ΦH )∨ R , because H the action is defined by translations on the second factor (ΓΦ˘ c,ΦH )∨ . R Therefore, H e σ˘ ,τ /Γ ˘ Nσ˘ ,τ = N is homotopic to the real torus T e = (Γ ˘ )∨ /Γ ˘ . ΦH c,ΦH

ΦH f,ΦH

ΦH c,ΦH R

ΦH c,ΦH

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The canonical isomorphism (7.3.3.10) then follows from the composition of the following canonical isomorphisms H j (ΓΦ˘ c,ΦH , Z) ⊗ OC~ ord H

ΦH ,δH

Z

⊗Q Z

∼ = (∧j (HomZ (ΓΦ˘ c,ΦH , Z))) ⊗ OC~ ord H

⊗Q

ΦH ,δH

Z

Z

∼ e X), Z))) ⊗ O ~ ord = (∧j (HomZ (HomO (X, C

ΦH ,δH

Z

∼ = ∧j (HomO (Q∨ , HomZ (Y, OC~ ord

ΦH ,δH

∼ = ∧ (HomO (Q , LieT ∨ /C~ ord ∨

j

ΦH ,δH

⊗Q Z

))) ⊗ Q Z

))) ⊗ Q Z

induced by the canonical isomorphisms e X) ⊗ Q ∼ ΓΦ˘ c,ΦH ⊗ Q ∼ = HomO (X, = HomO (Q, Y ) ⊗ Q, H

Z

Z

Z

as desired. Lemma 7.3.3.11. Let τ˘ = R>0 v1 +· · ·+R>0 vn be a nonzero smooth nondegenerate rational polyhedral cone in (SΦ˘ f )∨ R , where v1 , . . . , vn are nonzero vectors, and let K H

× be a cone in (SΦ˘ f )∨ R (i.e., a subset stable under the multiplicative action of R>0 ) H

such that 0 6∈ K and τ˘K := τ˘ ∩ K is convex. (Here τ˘ is the closure of τ˘ in (SΦ˘ f )∨ R .) H Up to reordering v1 , . . . , vn if necessary, suppose moreover that, for some nonzero m ≤ n, we have (R≥0 v1 + · · · + R≥0 vm ) − {0} ⊂ K

(7.3.3.12)

(R≥0 vm+1 + · · · + R≥0 vn ) ∩ K = ∅.

(7.3.3.13)

but In this case, τ˘0 := R>0 v1 +· · ·+R>0 vm is the largest face of τ˘ such that its closure τ˘0 (in (SΦ˘ f )∨ ˘0 −{0} ⊂ K (so that τ˘0 −{0} ⊂ τ˘K ). Consider the continuous R ) satisfies τ H map F : [0, 1] × τ˘K → τ˘K defined by sending (t, x1 v1 + · · · + xm vm + xm+1 vm+1 + · · · + xn vn ) to x1 v1 + · · · + xm vm + (1 − t)xm+1 vm+1 + · · · + (1 − t)xn vn . Then F defines a deformation retract from τ˘K to its subset τ˘0 − {0}. The construction of F is compatible with restrictions to faces τ˘00 of τ˘ that still satisfy the condition of this lemma. Proof. The statements are self-explanatory. (The condition (7.3.3.12) is needed for the compatibility with restrictions to faces. The condition (7.3.3.13) is needed for the deformation retract F to be defined at t = 1.)

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e σ˘ ,τ : Definition 7.3.3.14. For each `˘ ∈ SΦ˘ f , define the following subsets of N H

˘ ˘ ⊥ ∩ τ˘∨ . e `˘ is the union of τ˘ ∈ Σ e˘ (1) N σ ˘ ,τ Φ f,˘ σ ,τ such that ` ∈ σ H

˘ ˘ ˘ ⊥ ∩ τ˘∨ . e `,+ e˘ (2) N ˘∈Σ σ ˘ ,+ σ ˘ ,τ is the union of τ Φ f,˘ σ ,τ such that ` ∈ σ H

˘ ˘ ˘ ⊥ ∩ τ˘∨ . e `,0+ e˘ (3) N ˘∈Σ σ ˘ σ ˘ ,τ is the union of τ Φ f,˘ σ ,τ such that ` ∈ σ H

Lemma 7.3.3.15. Suppose there exist nonzero rational vectors v1 , . . . , vn in (SΦ˘ f )∨ R such that H

σ ˘ = R>0 v1 + · · · + R>0 vm , and such that τ˘ = R>0 v1 + · · · + R>0 vm + R>0 vm+1 + · · · + R>0 vn e˘ is a cone in Σ Φ f,˘ σ ,τ . Then we have the following criteria: H

˘ vi i ≥ 0 for all m + 1 ≤ i ≤ n. (1) `˘ ∈ σ ˘ ⊥ ∩ τ˘∨ if and only if h`, ⊥ ∨ ˘ ˘ vi i > 0 for all m + 1 ≤ i ≤ n. (2) ` ∈ σ ˘ ∩ τ˘σ˘ ,+ if and only if h`, ˘ vi i ≥ 0 for all m+1 ≤ i ≤ n and h`, ˘ vi i > 0 for (3) `˘ ∈ σ ˘ ⊥ ∩ τ˘σ˘∨ if and only if h`, + all m + 1 ≤ i ≤ n such that vi ∈ PΦ and such that the image of vi under e f H

the (canonical) second morphism in (1.2.4.20) is contained in τ ⊂ P+ ΦH . Proof. These follow immediately from the definitions. (See Definitions 7.3.2.2, 7.3.2.3, and 7.3.3.14.) ˘ ˘ e `˘ , N e `,+ e `,0+ Proposition 7.3.3.16. For each `˘ ∈ SΦ˘ f , the subsets N σ ˘ ,τ ˘ ,τ of σ ˘ ,τ , and Nσ H e σ˘ ,τ (in Definition 7.3.3.14) all have contractible or empty complements in N e σ˘ ,τ . N

Proof. We may and we shall assume that `˘ ∈ σ ˘ ⊥ , because otherwise all subsets in question will be empty (and the lemma becomes trivial). Since `˘ ∈ σ ˘ ⊥ , the e conditions for each cone τ˘ ∈ ΣΦ˘ f,˘σ,τ to be in each of the four subsets (see DefiH nition 7.3.3.14) depend only on the image τb = pr(Sb ˘ )∨ (˘ τ ) under the projection Φc R H

pr(Sb ˘

in (1.2.4.32). By Lemma 1.2.4.39, as in Corollary 7.3.3.6 (and its proof),

∨ Φ c )R H

the projection pr(Sb ˘

∨ Φ c )R H

˘ e `,+ e σ˘ ,τ , N e `˘ , N induces homotopy equivalences from N σ ˘ ,τ σ ˘ ,τ , ˘

˘ ˘ ˘ b b` e `,0+ e ˘ ,τ − N e `˘ , N e σ˘ ,τ − N e `,+ e ˘ ,τ − N e `,0+ e ˘ ,τ , N e N σ ˘ ,τ , σ ˘ ,τ σ ˘ ,τ , Nσ σ ˘ ,τ , and Nσ σ ˘ ,τ to their images Nσ ˘

˘

˘

˘

˘

`,+ `,0+ ` `,+ `,0+ b b b b b b b b e ,N e e ˘ ,τ − N e ,N e σ˘ ,τ − N e , and N e σ˘ ,τ − N e N σ ˘ ,τ σ ˘ ,τ , Nσ σ ˘ ,τ σ ˘ ,τ σ ˘ ,τ , respectively. By Corollary 7.3.3.4, each of such images has an induced cone decomposition (in the obvious b e σ˘ ,τ sense) by subsets of {pr b τ )} e . Using Lemma 7.3.3.7 to identify N ∨ (˘ (SΦ )R ˘ c H

with (ΓΦ˘ c,ΦH )∨ R × τ , we have

τ˘∈ΣΦ ˘

σ ,τ f,˘ H

H

pr(Sb ˘

∨ Φ c )R H



(˘ τ ) − {0} ∩ (ΓΦ˘ c,ΦH )∨ R ×{0} = ∅ H

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 e˘ for all τ˘ ∈ Σ Φ f,˘ σ ,τ , because a real symmetric matrix H

x y t y0

 can be positive semi-

definite only when y = 0. (The proof is elementary.)
∨ b0 b ˘ − (Γ ˘ For simplicity, let us denote P Φ c,ΦH )R ×{0} by P . Let Φc H

H

˘ yi < 0} b 0˘ := {y ∈ P b 0 : h`, P ` 0 for all 1 ≤ i ≤ n. (2) ` ∈ σ+ Proof. These follow immediately from the definitions. (See Definitions 8.2.2.3 and 8.2.3.1.) Proposition 8.2.3.3. (Compare with Proposition 7.3.3.16.) For each ` ∈ SΦH , the e ` and N e `,+ of N e (in Definition 8.2.3.1) both have contractible or empty subsets N e complements in N. Proof. For simplicity, let us denote PΦH − {0} by P0 . Let P0` 0 or d > 0. Using the spectral sequence (8.2.3.7), by Lemmas 8.2.3.12 and 8.2.3.15, we have   ˆ FJord,d,(`) (N ) , H c (N, H d (M + )) ∼ = H c ΓΦ , ⊕ H

`∈P∨,+ Φ

ΦH ,δH

H

which admits a filtration with graded pieces given by subquotients of   Y ord,d,(`) H c ΓΦH , FJΦH ,δH (N )

(8.2.3.19)

`∈ΓΦH ·`0

for `0 running through representatives of ΓΦH -orbits in P∨,+ ΦH . By assumption, we ord,d,(`)

may assume that N is chosen such that FJΦH ,δH (N ) = 0 for all d > 0 and d c + ` ∈ P∨,+ ΦH . Hence, we have H (N, H (M )) = 0 when d > 0. On the other hand, suppose c > 0. By (3) of Definition 8.2.3.8, and by Lemma 8.2.3.17, we have Y
ΓΦ ord,d,(`) ord,d,(` ) FJΦH ,δH (N ) ∼ = CoindΓΦH ,` FJΦH ,δH 0 (N ) H

0

`∈ΓΦH ·`0


ΓΦH ord,d,(` ) = Coind{Id} FJΦH ,δH 0 (N ) (see [11, Ch. III, Sec. 5]), and hence (8.2.3.19) is equal to zero for all c > 0, by Shapiro’s lemma (see [11, Ch. III, (6.2)]). Thus, we also have H c (N, H d (M + )) = 0 when c > 0, as desired. 8.2.4

Formal Fibers of Canonical Extensions

~ ord , we have a commutative diagram of canonical morBy the construction of X ΦH ,δH phisms ~ ord,tor )∗ Lie ~ ord,tor ~ ord (X ΦH ,δH → MH G/M

∼

H

λ



~ ord,tor )∗ Lie ∨ ~ ord,tor ~ ord (X ΦH ,δH → MH G /M H

∗ ~ ord ~ ord / (X ~ ord ΦH ,δH → CΦH ,δH ) LieG\ /C

ΦH ,δH

∼/



λ\

∗ ~ ord ~ ord (X ~ ord ΦH ,δH → CΦH ,δH ) LieG∨,\ /C

ΦH ,δH

(8.2.4.1)

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(see Lemma 5.2.4.38). As in Remark 8.1.2.5, consider the pairing h · , · iλ\ : Lie∨ ~ ord G∨,\ /C

(1) × LieG\ /C~ ord

ΦH ,δH

ΦH ,δH

→ OC~ ord

(1)

ΦH ,δH

defined by the composition Lie∨ ~ ord G∨,\ /C

ΦH ,δH

Id ⊗ dλ\

→

(1) ⊗ LieG\ /C~ ord

ΦH ,δH

Lie∨ ~ ord G∨,\ /C

ΦH ,δH

can.

(1) ⊗ LieG∨,\ /C~ ord

ΦH ,δH

→ OC~ ord

(1).

ΦH ,δH

Let us define (cf. (8.1.3.11)) ord,ΦH ,δH E~M ord D,0

:= IsomO ⊗ O ~ ord Z

C

ΦH ,δH

(Gr0D,0 ⊗ OC~ ord R0

((Lie∨ ~ ord G∨,\ /C

ΦH ,δH

ΦH ,δH

, Gr−1 ~ ord D,0 ⊗ OC

(1), LieG\ /C~ ord

ΦH ,δH

R0

ΦH ,δH

, h · , · iλ\ , OC~ ord

(1)),

ΦH ,δH

, h · , · iφD,0 , OC~ ord

(1))),

(8.2.4.2)

ΦH ,δH

ord which is an Mord D,0 -torsor with the group MD,0 acting as automorphisms on −1 0 (GrD,0 ⊗ OC~ ord , GrD,0 ⊗ OC~ ord , h · , · iφD,0 , OC~ ord (1)). By construction, and R0

ΦH ,δH

R0

ΦH ,δH

ΦH ,δH

by (4) of Proposition 8.1.3.6, the commutative diagram (8.2.4.1) induces a canonical isomorphism ∗ ~ ord,ΦH ,δH ∼ ~ ord,tor )∗ E~ord,can ~ ord ~ ord ~ ord (X = (X ΦH ,δH → MH ΦH ,δH → CΦH ,δH ) EMord Mord D,0

(8.2.4.3)

D,0

of Mord D,0 -torsors. Definition 8.2.4.4. (Compare with Definition 8.1.3.13.) Let R be any R0 -algebra. For each W ∈ RepR (Mord D,0 ), we define Mord D,0 ⊗ R ord,ΦH ,δH E~M (W ) ord ,R D,0

:=

ord,ΦH ,δH (E~M ⊗ ord D,0 R0

R0

R)

×

W.

Lemma 8.2.4.5. Let R be any R0 -algebra. ord,ΦH ,δH (1) The assignment E~M ( · ) defines an exact functor from RepR (Mord ord ,R D,0 ) D,0

to the category of quasi-coherent OC~ ord

-modules.

ΦH ,δH

(2) For each W ∈ RepR (Mord D,0 ), we have a canonical isomorphism ∗ ~ ord,ΦH ,δH ~ ord,tor )∗ E~ord,can ~ ord ~ ord ~ ord (X (W ) ∼ (W ), = (X ΦH ,δH → MH ΦH ,δH → CΦH ,δH ) EMord ,R Mord ,R D,0

D,0

(8.2.4.6) functorial in W , of quasi-coherent OX ~ ord

-modules.

ΦH ,δH

Proof. The proof of the first statement is the same as that of Lemmas 1.4.1.10 and 8.1.2.10. The second statement follows from the very constructions of both sides of (8.2.4.6), and from the canonical isomorphism (8.2.4.3).

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Let (c : X → B ∨ , c∨ : Y → B) be the tautological pair of morphisms over ∼ ∼ ord ~ ord ~ CΦH ,δH . Then γ = (γX : X → X, γY : Y → Y ) ∈ ΓΦH acts on C ΦH ,δH by ∨ ∨ ∨ sending (c, c ) to the pair of pre-compositions (cγX , c γY ). Since λB c = cφ and φγY = γX φ, we still have the compatibility relation λB (c∨ γY ) = cφγY = (cγX )φ. By [62, Lem. 3.4.4.2], we have a commutative diagram / γ ∗ G\

/T

0

o

0

 /T

/B

γ ∗ λ\

λT ∗ γX

/B

0

/0

\

λB

λ



 / T∨ ∗ γY

(8.2.4.7)

o

 / G\ λT

0

/0

λB

 / B∨

/ γ ∗ G∨,\

o

o

  / G∨,\

 

/ T∨

/ B∨

/0 

/0

of canonically defined morphisms, in which the horizontal rows are exact, inducing the corresponding commutative diagrams between sheaves of relative Lie algebras and their duals. ∼

Remark 8.2.4.8. Although γ ∗ G∨ → G∨ is an isomorphism of semi-abelian schemes that are extensions of B by T , and it induces the identity morphism on B, it does not induce the identity morphism on T . Hence, we cannot view γ ∗ G∨ as the ∼ ~ ord . Similarly, although γ ∗ G∨,\ → tautological object over C G∨,\ as extensions ΦH ,δH ∨ ∨ ∗ ∨,\ ~ ord . of B by T , we cannot view γ G as the tautological object over C ΦH ,δH Lemma 8.2.4.9. For each γ ∈ ΓΦH , we have a canonically defined isomorphism ∼

ord,ΦH ,δH ord,ΦH ,δH γ ∗ : γ ∗ E~M → E~M ord ord D,0

D,0

~ ord of Mord D,0 -torsors over CΦH ,δH . ∗ ∗ (γ1 (γ2 )) when γ = γ2 ◦ γ1 in

Such morphisms satisfy the compatibility γ ∗ = γ1∗ ◦ ΓΦH . Consequently, for each W ∈ RepR (Mord D,0 ), we have a canonically defined isomorphism ∼

ord,ΦH ,δH ord,ΦH ,δH γ ∗ : γ ∗ E~M (W ) → E~M (W ), ord ,R ord ,R D,0

D,0

∗

also satisfying the compatibility γ =

γ1∗

◦

(γ1∗ (γ2∗ ))

when γ = γ2 ◦ γ1 in ΓΦH .

Proof. These follow from the very constructions of the objects and from the commutativity of (8.2.4.7). Proposition 8.2.4.10. Let R be any R0 -algebra, and let W be any object of ord,can RepR (Mord Then the pullback of the automorphic sheaf E~M ord ,R (W ) (resp. D,0 ). D,0

ord,sub ~ ord,tor to (M ~ ord,tor )∧ E~M ord ,R (W )) over MH ˜ ord H ~ D,0

Z[(Φ

∼ ~ ord /ΓΦ (see (8.2.2.7)) is = X H ΦH ,δH

H ,δH )]

formally canonical (resp. subcanonical) as in Definition 8.2.3.8.

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Proof. Let M

:=

ord,ΦH ,δH (W ). By E~M ord ,R D,0 ∼ iσ : M |Uσ → gσ∗ N

~ ord /ΓΦ ) ((X H ΦH ,δH

→

~ ord,tor )∗ E~ord,can (W ) and N M H Mord ,R

:=

D,0

construction, the isomorphism (8.2.4.6) induces an isomorphism

for each σ ∈ Σ+ ΦH . Hence, (1) of Definition 8.2.3.8 is verified. Since the isomorphism (8.2.4.6) is based on (8.2.4.1), in which the horizontal isomorphisms (which is dependent on the rigidification of ΦH over Uσ ) is twisted by γ (in the same way as the diagram of relative Lie algebras induced by (8.2.4.7)) ∼ under the isomorphism γ : Uσ → Uγσ in (2) of Lemma 8.2.2.10, the canonical iso∼ morphism γ ∗ M |Uγσ → M |Uσ induces by pre- and post-compositions with iσ and ∼ ∗ −1 ∗ γ (iγσ ) the same isomorphism γ ∗ : γ ∗ gγσ N → gσ∗ N defined by adjunction by the isomorphism given by Lemma 8.2.4.9. Hence, (2) and (3) of Definition 8.2.3.8 are also verified, as desired. ∨,+ Lemma 8.2.4.11. Let ` be any element of PΦ . Then the invertible sheaf H ord,Z H ord ord ~ ~ ~ Ψ is relatively ample. ΦH ,δH (`) over CΦH ,δH → MH

Proof. Since ` pairs positively with all elements in PΦH , up to choosing a Z-basis y1 , . . . , yr of Y , and by completion of squares for quadratic forms, there exists some integer N ≥ 1 such that N · ` can be represented as a positive definite matrix of the form ue t u, where e and u are matrices with integer coefficients, and where e = diag(e1 , . . . , er ) is diagonal with positive entries. Consider the finite u∗ can. ~ ord morphism defined by the composition C → Hom (Y, B) → Hom (Y, B) (see ΦH ,δH

Z

Z

Section 4.2.1 for the first morphism), where B is the tautological abelian scheme over ~ ord,δH , under which a positive tensor power of Ψ ~ ord (`) is isomorphic to a positive M ΦH ,δH H tensor power of the pullback of the ample line bundle ⊗ (pr∗i (IdB , λB )∗ PB )⊗ ei 1≤i≤r

over B. Since λB is a polarization (cf. [62, Def. 1.3.2.16]), and since all the ei ’s are ~ ord (`) is relatively ample, as desired. positive, we see that Ψ ΦH ,δH ~ ord,ZH , we consider the completion (M ~ ord,ZH )∧ of For each geometric point x ¯ of M x ¯ H H ord,Z H ~ the strict localization of M at x ¯, and denote (abusively) by the same notation H ~ ord,ZH . ~ ord,ZH )∧ of h : C ~ ord ~ ord )∧ the pullback h : (C x ¯ ¯ → (MH ΦH ,δH → MH ΦH ,δH x Corollary 8.2.4.12. Suppose N is a coherent OC~ ord

-module such that, at

ΦH ,δH

~ ord,ZH , M H

~ ord each geometric point x ¯ of the pullback (N )∧ x ¯ of N from CΦH ,δH to ord ∧ ~ (C ¯) by co¯ admits an exhaustive and separated filtration (depending on x ΦH ,δH )x a herent O(C~ ord )∧ -submodules {N }a∈Z such that each graded piece Gra (N ) := ¯ ΦH ,δH x

~ ord )∧ N a /N a+1 is isomorphic to the pullback (from Spec(R0 ) to (C ¯ ) of some ΦH ,δH x (a) (finitely generated) R0 -module N . Then ord,d,(`) ~ ord FJΦH ,δH (N ) = Rd h∗ (Ψ ΦH ,δH (`)

⊗

OC ~ ord

ΦH ,δH

for all d > 0 and ` ∈ P∨,+ ΦH .

N)=0

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505

~ ord,ZH is proper, by [35, III-1, 4.1.5], for our purpose, ~ ord Proof. Since h : C ΦH ,δH → MH ~ ord,ZH , and replace we may fix the choice of an arbitrary geometric point x ¯ of M H ord,ZH ∧ ord,ZH ~ ~ )x¯ . Since any exhaustive to (MH all objects with their pullbacks from MH and separated filtration defines a filtration spectral sequence, we are reduced to ~ ord )∧ the case that N is the pullback (from Spec(R0 ) to (C ¯ ) of some finitely ΦH ,δH x generated R0 -module N . Since R0 is a Dedekind domain, by the same reduction argument as in the proof of [62, Lem. 7.1.1.4], we may assume that N = R0 /n for some (possibly zero) ideal n of R0 , and work over N after making the base change from R0 . Hence, we are reduced to showing (after base change from R0 ~ ord (`))∧ to R0 /n) that Rd h∗ ((Ψ x ¯ ) = 0 for all d > 0, which then follows from ΦH ,δH Lemma 8.2.4.11, [81, Sec. 16], and [35, III-1, 4.1.5], because h is an abelian scheme torsor over a finite ´etale cover. (Alternatively, without reducing to the case that N = R0 /n, but still using the fact that R0 is a Dedekind domain, we may invoke the perfect complex construction as in [81, Sec. 5, Thm.] (see also [7, III, 3.7 and 3.7.1]) and the “universal coefficient theorem” as in the proof of [71, Thm. 8.2].) The question is how to verify the rather elaborate condition in Corollary 8.2.4.12. Let us define (cf. (8.2.4.2)) ord,ΦH ,δH E~M ord D,0,fil

:= IsomO ⊗ O ~ ord ((Lie∨ ~ ord ~ ord (1), LieG\ /C G∨,\ /C Z

(Gr0D,0

C

⊗ OC~ ord

R0

ΦH ,δH

ΦH ,δH

ΦH ,δH

ΦH ,δH

, Gr−1 D,0

⊗ OC~ ord

R0

, h · , · iλ\ , OC~ ord (1), filtrations), ΦH ,δH

, h · , · iφD,0 , OC~ ord

ΦH ,δH

(1), filtrations)), (8.2.4.13)

ΦH ,δH

where the filtrations on Lie∨ ~ ord G∨,\ /C

ΦH ,δH

(1) and LieG\ /C~ ord

are defined by the

ΦH ,δH

commutative diagram 0

/ Lie ~ ord T /C

ΦH ,δH

dλT

0

/ Lie \ ~ ord G /C dλ\



/ Lie ∨ ~ ord T /C

ΦH ,δH

/ Lie

ΦH ,δH



/ Lie ∨,\ ~ ord G /C

ΦH ,δH

~ ord B/C Φ ,δ H



/0

(8.2.4.14)

H

dλB

/ Lie ∨ ~ ord B /C

/0

ΦH ,δH

of canonically defined morphisms, in which the horizontal rows are exact; and where ~ord,ΦH ,δH the filtrations on Gr0D,0 (1) and Gr−1 D,0 can be any filtrations such that EMord D,0,fil

is indeed an ´etale torsor. (For example, they can be defined by any liftings of (4.1.4.14) and (4.1.4.15) to admissible filtrations over Zp and the isomorphisms ∼ ∼ ˜0 → ˜ 0 and Gr−1 ˜0 → ˜ Gr−1 ⊗ R Gr−1 ⊗ R ⊗R Gr−1 # # ⊗ R0 in Section 8.1.1. Then D,0

R0

D

Zp

D ,0

R0

D

Zp

we can deduce the existence of sections over geometric points in characteristic p by the deformation theory of ordinary abelian varieties, and in characteristic zero, or in any good characteristics, by classification of pairings. Afterwards, we can deduce the existence of ´etale locally defined sections as in the proof of Lemma 1.4.1.7.) We ord shall fix the choice of this filtrations on Gr0D,0 (1) and Gr−1 D,0 , and denote by MD,0,fil

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the subgroup scheme of Mord D,0 defined by elements stabilizing this chosen filtration. ord,ΦH ,δH ord ~ ord is an M Then E~M ord D,0,fil -torsor over CΦH ,δH as the subscript suggests. D,0,fil

Let us define the category RepR0 (Mord D,0,fil ) as in Definition 8.1.2.7. Then, as in Lemmas 1.4.1.10 and 8.1.2.10 (whose proofs are essentially the same), we have an ord,ΦH ,δH ord exact functor E~M ord ,R ( · ) from RepR0 (MD,0,fil ) to the category of quasi-coherent OC~ ord

ΦH ,δH

0

D,0,fil

-modules. Moreover, if we view an object W of RepR0 (Mord D,0 ) as an object

ord of RepR0 (Mord D,0,fil ) by restriction to MD,0,fil , then we have a canonical isomorphism ord,ΦH ,δH ∼ ~ord,ΦH ,δH (W ). E~M ord ,R (W ) = EMord ,R D,0,fil

0

D,0

0

(8.2.4.15)

ord Let Mord D,0,uni be the normal subgroup scheme of MD,0,fil defined by sections of inducing the trivial automorphisms on the graded pieces.

Mord D,0,fil

Lemma 8.2.4.16. Suppose W is a finitely generated object of RepR0 (Mord D,0,fil ). Then W admits an exhaustive and separated filtration {W a }a∈Z by (finitely generated) ord subobjects in RepR0 (Mord D,0,fil ), such that MD,0,uni acts trivially on each graded pieces Gra (W ) := W a /W a+1 . Proof. Let W 0 be the R0 -submodule of W spanned by (u−Id)W for all sections u of −1 ord −1)p, where pup−1 is Mord D,0,uni . For any section p of MD,0,fil , we have p(u−1) = (pup ord ord also a section of the normal subgroup scheme MD,0,uni of MD,0,fil . Therefore, W 0 and W/W 0 are subobject and quotient objects of W in RepR0 (Mord D,0,fil ), respectively, and ord 0 MD,0,uni acts trivially on the quotient object W/W . Since W is finitely generated over R0 , the submodule W 0 of W is a proper (and possibly zero) submodule. (To show this, we can reduce modulo the maximal ideal of R0 , base extend to the algebraically closure, and apply the Lie–Kolchin theorem.) By replacing W with W 0 , and by repeating this process (which terminates in finitely many steps because W is finitely generated), we obtain the desired exhaustive and separated filtration as in the statement of the lemma. Corollary 8.2.4.17. Suppose W is a finitely generated object of RepR0 (Mord D,0,fil ). ord,ΦH ,δH ~ Then N := EMord ,R (W ) satisfies the condition in Corollary 8.2.4.12 (and hence D,0,fil

ord,d,(`) FJΦH ,δH (N

0

) = 0 for all d > 0 and ` ∈ P∨,+ ΦH ).

Proof. By Lemma 8.2.4.16 (and by the obvious analogue of Lemmas 1.4.1.10 and ord,ΦH ,δH 8.1.2.10 for E~M ord ,R ( · )), we may assume that W is a finitely generated object of D,0,fil

0

ord RepR0 (Mord D,0,fil ) on which MD,0,uni acts trivially. ~ ord,ZH . By construction, over the Let x ¯ be an arbitrary geometric point of M H ord ∧ ~ ord ~ formal completion (C ΦH ,δH )y¯ of the strict localization of CΦH ,δH at any geometric ord,ΦH ,δH point y¯ above x ¯, we have a section of E~M giving an isomorphism between ord D,0,fil

~ ord )∧ the pullbacks to (C ΦH ,δH y¯ of (Lie∨ ~ ord G∨,\ /C

ΦH ,δH

(1), LieG\ /C~ ord

ΦH ,δH

, h · , · iλ\ , OC~ ord

ΦH ,δH

(1), filtrations)

(8.2.4.18)

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and (Gr0D,0 ⊗ OC~ ord

ΦH ,δH

R0

, Gr−1 ~ ord D,0 ⊗ OC R0

ΦH ,δH

, h · , · iφD,0 , OC~ ord

(1), filtrations), (8.2.4.19)

ΦH ,δH

inducing isomorphisms between the graded pieces. Since the graded pieces of H ,δH ~ ord,ZH , the section of E~ord,Φ (8.2.4.18) are pullbacks of objects defined over M H Mord D,0,fil

~ ord )∧ over any such (C ΦH ,δH y¯ defines a section of ord,ΦH ,δH E~M ord D,0,gr

:= IsomO ⊗ O ~ ord Z

C

ΦH ,δH

(Gr(Lie∨ ~ ord G∨,\ /C

Gr(Gr0D,0 ⊗ OC~ ord R0

ΦH ,δH

ΦH ,δH

, Gr−1 ~ ord D,0 ⊗ OC R0

(1), LieG\ /C~ ord

ΦH ,δH

ΦH ,δH

, h · , · iλ\ , OC~ ord

(1)),

ΦH ,δH

, h · , · iφD,0 , OC~ ord

(1)),

(8.2.4.20)

ΦH ,δH

ord,ΦH ,δH ord,ΦH ,δH , over all of → E~M in the image of the canonical morphism E~M ord ord D,0,gr

D,0,fil

∧ ~ ord )∧ ~ ord (C ¯ (not just over (CΦH ,δH )y¯ ). ΦH ,δH x ∧ ~ord,ΦH ,δH Consequently, the construction of (N )∧ x ¯ = (EMord ,R (W ))x ¯ depends only on D,0,fil

0

ord,ΦH ,δH ~ ord )∧ the local choices of liftings of the above global section E~M over (C ord ¯ ΦH ,δH x D,0,gr

ord,ΦH ,δH ord,ΦH ,δH to local sections of E~M (under the canonical morphism E~M → ord ord D,0,fil

D,0,fil

ord,ΦH ,δH ord E~M ), and such choices form a torsor under Mord ord D,0,uni . Since MD,0,uni acts D,0,gr

trivially on W , we see that (N )∧ x ¯ is constant in the sense that it is isomorphic to ~ ord )∧ the pullback (from Spec(R0 ) to (C ¯ ) of the R0 -module W , as desired. ΦH ,δH x Proposition 8.2.4.21. Let R be any R0 -algebra, let W be any object of ~ord,ΦH ,δH (W ). Then FJord,d,(`) (N ) = 0 for all d > 0 RepR (Mord D,0 ), and let N := EMord ,R ΦH ,δH D,0

∨,+ and ` ∈ PΦ . H

Proof. Since this is a statement independent of the R-module structure of W , we may view W as an R0 -module. By construction, we have a canonical isomorord,ΦH ,δH ord,ΦH ,δH phism E~M (W ) ∼ (W ) of quasi-coherent OC~ ord -modules. Since = E~Mord ,R ord ,R D,0

D,0

0

ΦH ,δH

taking cohomology and tensor products commutes with direct limits, and since ord,ΦH ,δH E~M ( · ) is an exact functor (by Lemma 8.2.4.5), by writing W as a direct ord ,R D,0

0

limit of finitely generated R0 -subobjects in RepR (Mord D,0 ) (which is possible because is of finite type over R ), we may assume that W is finitely generated object of Mord 0 D,0 RepR (Mord ). Then it follows from (8.2.4.15) and Corollary 8.2.4.17 that N satisfies D,0 the condition in Corollary 8.2.4.12, and hence the proposition follows. Remark 8.2.4.22. For general W , even when W is locally free of finite ord,can rank, it is not true that the pullback of the automorphic sheaf E~M ord ,R (W ) to D,0

~ ord,tor )∧ (M ˜ ord H ~ Z[(Φ

∼ ~ ord /ΓΦ (see (8.2.2.7)) admits an exhaustive and separa=X H ΦH ,δH

H ,δH )]

ted filtration such that each of the graded pieces is the pullback of some coherent ~ ord,ZH (let alone from Spec(R0 )). This is why we need a different sheaf from M H trivialization over each Uσ in (1) of Definition 8.2.3.8.

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8.2.5

End of the Proof

Proof of Theorem 8.2.1.3. As in the proof of Proposition 8.2.4.21, we may reduce the question to the case that R = R0 and W is a finitely generated ob~ord,sub ject of RepR (Mord ~ ord,tor -module. Since D,0 ), so that EMord ,R (W ) is a coherent OM H D,0 H~ ord ord,tor ord,min ~ ~ : MH → MH is proper, by [35, III-1, 4.1.5], in order to show (8.2.1.4) H for all b > 0, we may fix the choice of an arbitrary (locally closed) stratum ~Zord ~ ord,min , and replace all objects with their pullbacks from M ~ ord,min H [(ΦH ,δH )] of MH ~ ord,min )∧ to (M . Then the theorem follows as a combination of Propositions H

~ Zord [(Φ

H ,δH )]

8.2.3.18, 8.2.4.10, and 8.2.4.21. 8.3

Constructions over the Total Models

ord ~ ord to the In this section, our goal is to extend the construction of E~M ord ,R ( · ) over MH D,0

~ H (see Proposition 2.2.1.1), and to extend construction of certain E~Mord ( · ) over M D,0 ,R ord,sub ord,can ~ ord,tor the constructions of E~ ord ( · ) and E~ ord ( · ) over M ord to the constructions MD,0 ,R

MD,0 ,R

H,Σ

can ~ tor ~sub of certain E~M ord ,R ( · ) and EMord ,R ( · ) over MH,d0 pol , respectively, when H is neat D,0

D,0

~ tor and when Σord extends to some projective Σ (so that M H,d0 pol is defined as in Proposition 2.2.2.1 for some integer d0 ≥ 1). When Assumption 3.2.2.10 holds, ord we will also construct the analogues of these with Mord D,0 replaced with PD,0 , such that they specialize to the constructions in the previous subsection. (Since we can ord canonically view objects of RepR (Mord D,0 ) as objects of RepR (PD,0 ), we would like ~can E~Pcan ord ,R ( · ) to be compatible with EMord ,R ( · ) as in Lemma 1.4.1.10, and as in Lemma D,0

D,0

~ tor 8.1.2.10 when Assumption 3.2.2.10 holds. On the other hand, since M H,d0 pol is often ord,can ~ far from smooth, we do not expect EGord ,R ( · ) and the corresponding extended D,0

Gauss–Manin connections, induced by the one in Proposition 8.1.3.6, to further ~ tor extend to the whole M H,d0 pol .) Throughout this section, we shall retain Assumption 8.1.2.1; and we shall also ~ H etc with the normalizations of their base changes from Spec(OF ,(p) ) replace M 0 to Spec(R0 ), and replace S0 = Spec(F0 ) and ~S0 = Spec(OF0 ,(p) ) with Spec(R0 ⊗ Q) Z

and Spec(R0 ), respectively. 8.3.1

Principal Bundles

Let us begin with some preliminary constructions in characteristic zero. Construction 8.3.1.1. Over S0 , we can define the principal Gord D,0 -bundle (resp. ord -bundle, resp. M -bundle) E (resp. E , resp. E ) over MH as in DePord ord ord ord GD,0 PD,0 MD,0 D,0 D,0 finition 1.4.1.3 (resp. Definition 1.4.1.4, resp. Definition 1.4.1.5) by the tautological objects over MH , which satisfy the analogue of Lemma 1.4.1.7. These are possible

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because of Lemma 8.1.1.7 (in characteristic zero). Thus, for each W in RepR (Gord D,0 ) ord (resp. RepR (Pord ), resp. Rep (M )), if we set R := R ⊗ Q and W := W ⊗ Q, Q Q R D,0 D,0 Z

Z

then we can define EGord (WQ ) (resp. EPord (WQ ), resp. EMord (WQ )) over D,0 ,RQ D,0 ,RQ D,0 ,RQ MH ⊗ RQ as in Definition 1.4.1.9, which satisfy the analogue of Lemma 1.4.1.10. R0

(We only define these sheaves for WQ , but not for W itself.) For each Σ of admissible can smooth rational polyhedral cone decomposition data for MH , we can define EG ord , D,0

sub can sub can sub can sub can sub EG ord , EPord , EPord , EMord , EMord , EGord ,R ( · ), EGord ,R ( · ), EPord ,R ( · ), EPord ,R ( · ), Q Q Q Q D,0

D,0

D,0

D,0

D,0

D,0

D,0

D,0

D,0

can sub EM ord ,R ( · ), and EMord ,R ( · ) as in Definition 1.4.2.9, which satisfy the analogue of Q Q D,0 D,0 ˜ Q := R ⊗ K Lemma 1.4.2.10. For each R0 -algebra K as in Lemma 8.1.1.7, set R R0

˜ Q := W ⊗ K. Then E ord ˜ (W ˜ Q ) (resp. E ord ˜ (W ˜ Q ), resp. E ord ˜ (W ˜ Q )) and W G ,RQ P ,RQ M ,RQ D,0

R0

D,0

D,0

˜ Q ) (resp. E ˜ (W ˜ Q ), resp. E ˜ is isomorphic to the usual EG0 ,R˜ Q (W ˜ Q (WQ )) conP0 ,RQ M0 ,R ˜ structed as in [61, Sec. 6] and Section 1.4.1 over MH ⊗ RQ , and these isomorphisms R0

extend to isomorphisms between the canonical (or subcanonical) extensions of the sources and the targets. ~ H , over In mixed characteristics, we will only construct principal bundles over M ~ ~ which we have the degenerating family (A, λ,~i, α ~ H ) of type MH as in Proposition 2.2.1.1. Let us define the principal Mord -bundle D,0 E~Mord := IsomO ⊗ O~ ((Lie∨ ~ H , h · , · i~ ~ H (1)), ~ M ~ (1), LieA/ ~ ∨ /M λ , OM A D,0 Z

(Gr0D,0

⊗ R0

MH

−1 OM ~ H , GrD,0

H

(8.3.1.2)

⊗ OM ~ H , h · , · iφD,0 , OM ~ H (1)))

R0

~ H (cf. Definition 8.1.2.4), which extends the principal Mord -bundle E ord over over M MD,0 D,0 MH in Construction 8.3.1.1. When Condition 8.1.1.2 holds, let us also define the principal Pord D,0 -bundle ∨ ~ ~ E~Pord := IsomO ⊗ O~ ((H dR ~ H (1), LieA ~ (1)), ~ ∨ /M 1 (A/MH ), h · , · iλ , OM D,0 Z

H

MH

0 ((Gr0D,0 ⊕ Gr−1 ~ H , h · , · iφD,0 , OM ~ H (1), GrD,0 ⊗ OM ~ H )) D,0 ) ⊗ OM R0

(8.3.1.3)

R0

~ H (cf. Definition 8.1.2.3), which extends the principal Pord -bundle E ord over over M PD,0 D,0 MH in Construction 8.3.1.1. To justify these terminologies, we need the following: Lemma 8.3.1.4. (Compare with Lemmas 1.4.1.7 and 8.1.2.6.) The relative scheme ~ H is an ´etale torsor under (the pullback of) the group scheme Mord . E~Mord over M D,0 D,0 ~ H is an fpqc torsor When Condition 8.1.1.2 holds, the relative scheme E~Pord over M D,0

under (the pullback of) the group scheme Pord D,0 . Proof. Let O0 be any maximal order in O ⊗ Q containing O as in the beginning Z

ˆ of Section 8.1.1, which satisfies Condition 1.2.1.1, so that the O-action on L ⊗ Z Z

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extends to an O0 -action on the same module. By using the level structures in characteristic zero, and by [62, Cor. 1.3.5.4], the O-structure i : O → EndMH (A) uniquely extends to an O0 -structure i0 : O0 → EndMH (A), and LieA/MH with its O ⊗ Q = O0 ⊗ Q-module structure given naturally by i0 also satisfies the deterZ

Z

~ H is minantal condition in [62, Def. 1.3.4.1] given by (L ⊗ R, h · , · i, h0 ). Since M Z

noetherian and normal, by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, Prop. 3.3.1.5], ~ i0 uniquely extends to an O0 -structure ~i0 : O0 → EndM ~ H (A). Since the deter0 minantal condition is closed, LieA/ ~ H with its O ⊗ Z(p) -module structure given ~ M Z

naturally by ~i0 also satisfies the determinantal condition in [62, Def. 1.3.4.1] given by (L ⊗ R, h · , · i, h0 ). Since the filtrations D and D# are O0 ⊗ Zp -equivariant, Z

Z

as explained in the proof of Lemma 3.2.2.6, and by the assumptions made in the beginning of Section 8.1.1, O0 acts equivariantly on the source and target of φ0D,0 . ~ H at points of finite type over ~S0 , by Lemma Over the formal completions of M 8.1.1.7 in characteristic zero, and by [91, 3.23 c) and d), and the reductions in Ch. ~ ~λ,~i0 ) define 6] in characteristic p, the infinitesimal deformations of pullbacks of (A, sections of the pullback of the relative subscheme IsomO0 ⊗ O~ ((Lie∨ ~ H , h · , · i~ ~ H (1)), ~ M ~ (1), LieA/ ~ ∨ /M λ , OM A Z

(Gr0D,0

⊗ R0

H

MH

−1 OM ~ H , GrD,0

⊗ OM ~ H , h · , · iφD,0 , OM ~ H (1)))

R0

(note the difference between O and O0 ), which induce sections of the pullof E~Mord D,0 back of E~Mord . Hence, by Artin’s approximation theory (cf. [3, Thm. 1.10 and Cor. D,0 2.5]), as in the proof of Lemma 8.1.2.6, it follows that E~Mord is an ´etale Mord -torsor, D,0

D,0

as desired. When Condition 8.1.1.2 holds, the analogous statements for Pord D,0 can be similarly proved. 8.3.2

Automorphic Bundles

Following Lemma 8.3.1.4, by fpqc descent of quasi-coherent sheaves (see [33, VIII, 1.3]), we can make the following: Definition 8.3.2.1. (Compare with Definitions 1.4.1.9 and 8.1.2.8.) Let R be any R0 -algebra. For each W in RepR (Mord D,0 ), we define Mord D,0 ⊗ R

E~Mord (W ) := (E~Mord ⊗ R) D,0 ,R D,0

R0

×

W,

R0

~ H ⊗ R associated with W . It is called an called the automorphic sheaf over M R0

automorphic bundle if W is locally free of finite rank over R, in which case ~ H ⊗ R. When Condition 8.1.1.2 E~Mord (W ) is also locally free of finite rank over M D,0 ,R R0

ord holds, we define similarly E~Pord (W ) for W ∈ RepR (Pord D,0 ) by replacing MD,0 with D,0 ,R

Pord D,0 in the above expression.

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Lemma 8.3.2.2. (Compare with Lemmas 1.4.1.10 and 8.1.2.9.) Let R be any R0 -algebra. The assignment E~Mord ( · ) defines an exact functor from D,0 ,R ord ~ H. Rep (M ) to the category of quasi-coherent sheaves over M R

D,0

Proof. By ´etale descent, the proof is similar to that of Lemma 1.4.1.10. Lemma 8.3.2.3. (Compare with Lemmas 1.4.1.10 and 8.1.2.10.) Let R be any R0 -algebra. Suppose that Condition 8.1.1.2 holds. (1) The assignment E~Pord ( · ) defines an exact functor from RepR (Pord D,0 ) to D,0 ,R ~ the category of quasi-coherent sheaves over MH . ord (2) If we view an object of W ∈ RepR (Mord D,0 ) as an object of RepR (PD,0 ) via ord ord the canonical homomorphism PD,0 → MD,0 , then we have a canonical isomorphism E~Pord (W ) ∼ (W ). = E~Mord D,0 ,R D,0 ,R

(3) Suppose W ∈ RepR (Pord D,0 ) has a decreasing filtration by subobjects a Fa (W ) ⊂ W in RepR (Pord D,0 ) such that each graded piece GrF (W ) := a a+1 ord F (W )/F (W ) can be identified with an object of RepR (MD,0 ). Then E~Pord (W ) has a filtration Fa (E~Pord (W )) := E~Pord (Fa (W )) with graded D,0 ,R D,0 ,R D,0 ,R pieces E~Mord ,R (GraF (W )). D,0

Proof. By fpqc descent, the proof is still similar to that of Lemma 1.4.1.10. Lemma 8.3.2.4. With the setting as in Definition 8.3.2.1, we have a canonical isomorphism ∼ E~Mord (W ) ⊗ Q → EMord (WQ ) D,0 ,R D,0 ,RQ

(8.3.2.5)

Z

over MH ⊗ RQ . When Condition 8.1.1.2 holds, the analogous statement is true if R0

ord we replace Mord D,0 with PD,0 . ∼ Proof. We have (8.3.2.5) because we have E~Mord ⊗ Q → EMord by definition (see D,0 D,0 Z

Construction 8.3.1.1 and (8.3.1.2)). The analogous statement for Pord D,0 can be similarly proved. ~ and ω ~ Lemma 8.3.2.6. (Compare with Lemmas 1.4.1.11 and 8.1.2.11.) Let A MH be as in Proposition 2.2.1.1. For any R0 -algebra R, the pullback of LieA/ ~ H (resp. ~ M ~ H ⊗ R is canonically isomorphic to E~ ord (W ) for W = Lie∨~ ~ , resp. ω ~ ) to M A/MH

Gr−1 D,0 ⊗ R (resp. R0

MH

∨ (Gr−1 D,0 )

R0

⊗ R, resp. R0

MD,0 ,R

∨ ∧top (Gr−1 D,0 )

⊗ R). R0

Proof. This follows from the definition (see (8.3.1.2)), and from Lemma 8.3.2.2.

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8.3.3

Canonical Extensions

Definition 8.3.3.1. With the setting as in Definition 8.3.2.1, suppose moreover that H is neat and that Σ is projective, with a collection pol of polarization functions, ~ tor ~ so that M H,d0 pol over S0 = Spec(OF0 ,(p) ) is defined. Then, by Lemma 8.3.2.4, we ~ H ⊗ R) and E can can patch E~Mord (W ) (over M (WQ ) (over Mtor H,Σ ⊗ RQ ) into a Mord ,RQ D,0 ,R R0

R0

D,0

quasi-coherent sheaf can E~Mord (W ) ∪ EM ord ,R (WQ ) Q D,0 ,R D,0

over the open subscheme ~ H ⊗ R) ∪(Mtor ⊗ RQ ) (M H,Σ R0

of

~ tor M H,d0 pol

R0

⊗ R (see Proposition 2.2.2.1), and define the quasi-coherent sheaf  
can ~ H ⊗ R) ∪(Mtor ⊗ RQ ) ,→ M ~ tor E~M (M ⊗R ord ,R (W ) := H,Σ H,d pol 0 D,0 ∗ R0 R0 R0   can E~Mord (W ) ∪ E ord ,R (WQ ) ,R M Q D,0 R0

D,0

~ tor over M H,d0 pol ⊗ R. Similarly, we define the quasi-coherent sheaf R0

sub E~M ord ,R (W ) D,0

can sub ~ tor over M H,d0 pol ⊗ R by replacing EMord ,R (WQ ) with EMord ,R (WQ ) in the above definiR0

D,0

D,0

can tion of E~M ord ,R (W ). D,0 When Condition 8.1.1.2 holds, we also define similar assignments W 7→ ord ord ~sub E~Pcan ord ,R (W ) and W 7→ EPord ,R (W ) by replacing MD,0 with PD,0 in the above conD,0 D,0 structions.

Lemma 8.3.3.2. With the setting as in Lemma 8.3.3.1, we have compatible canonical isomorphisms ∼ E~can (W ) ⊗ Q → E can (WQ ) ord ord MD,0 ,R

Z

MD,0 ,RQ

and ∼ sub sub E~M ord ,R (W ) ⊗ Q → EMord ,R (WQ ) Q D,0

over

Mtor H,Σ

Mord D,0 with

⊗ RQ . R0 Pord D,0 .

Z

D,0

When Condition 8.1.1.2 holds, the same is true if we replace

Proof. This follows immediately from the definitions. Lemma 8.3.3.3. When Condition 8.1.1.2 holds, if we view an object W ∈ ord ord RepR (Mord D,0 ) as an object of RepR (PD,0 ) via the canonical homomorphism PD,0 → ∼ ~can ~can Mord D,0 , then we have canonical isomorphisms EPord ,R (W ) = EMord ,R (W ) and D,0

D,0

∼ ~sub E~Psub ord ,R (W ) = EMord ,R (W ), compatible with each other, and with the isomorphism D,0 D,0 in Lemma 8.3.2.3.

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Proof. This is because the analogues statements are true for EPcan ord ,R (WQ ), Q D,0

~sub EPsub ord ,R (WQ ), and EPord ,R (W ). Q D,0

D,0

Lemma 8.3.3.4. With the setting as in Definition 8.3.3.1, suppose moreover that R is an integral domain flat over R0 , and that W is locally free over R. Then can ~ tor ~sub E~M ord ,R (W ) and EMord ,R (W ) are torsion-free over MH,d0 pol ⊗ R, and their restrictiD,0

R0

D,0

~ H ⊗ R) ∪(Mtor ⊗ RQ ) of M ~ tor ons to the open subscheme (M H,Σ H,d0 pol ⊗ R (see DefiniR0

R0

R0

tion 8.3.3.1) are locally free. When Condition 8.1.1.2 holds, the analogues statements are true if we replace ord Mord D,0 with PD,0 . Proof. Since MH is smooth over S0 and since RQ is an integral domain, the local ~ tor rings of MH ⊗ RQ are integral domains. Since M H,d0 pol is flat over R0 , since MH is R0

~ tor open and dense in the noetherian normal scheme M H,d0 pol , and since R is flat over tor ~ R0 , it follows that the local rings of MH,d0 pol are also integral domains. By definition (see Construction 8.3.1.1 and Definition 8.3.2.1), when W is locally free over can sub ~ (W ) ∪ EM R, the sheaves E~Mord ord ,R (WQ ) and EMord ,R (W ) ∪ EMord ,R (WQ ) are loD,0 ,R D,0 Q Q D,0

D,0

~ H ⊗ R) ∪(Mtor ⊗ RQ ), and hence their direct images E~can cally free over (M (W ) H,Σ Mord ,R R0

R0

D,0

sub ~ tor and E~M ord ,R (W ) are also torsion-free over MH,d0 pol ⊗ R, as desired. The analogous R0

D,0

statement for Pord D,0 can be similarly proved. 8.3.4

Compatibility with the Constructions over the Ordinary Loci

Proposition 8.3.4.1. For each R flat over R0 , and for each W in RepR (Mord D,0 ) that is flat over R0 , there is a canonical isomorphism ord ∼ ~ ord ⊗ R → M ~ H ⊗ R)∗ E~ ord (W ), E~M ord ,R (W ) = (MH MD,0 ,R D,0

R0

(8.3.4.2)

R0

(cf. (3.4.6.4)) and there are compatible canonical isomorphisms ord,can ∗ ~ can ∼ ~ ord,tor ~ tor E~M ord ,R (W ) = (MH,Σord ⊗ R → MH,d0 pol ⊗ R) EMord ,R (W )

(8.3.4.3)

ord,sub ∗ ~ sub ∼ ~ ord,tor ~ tor E~M ord ,R (W ) = (MH,Σord ⊗ R → MH,d0 pol ⊗ R) EMord ,R (W )

(8.3.4.4)

D,0

R0

R0

D,0

and D,0

R0

R0

D,0

(cf. (5.2.3.19)) when Hp and hence H = Hp Hp are neat, and when Σord extends to ~ tor some projective Σ, with a collection pol of polarization functions (so that M H,d0 pol is defined as in Proposition 2.2.2.1). These canonical isomorphisms are compatible with each other. When Condition 8.1.1.2 holds, the analogous statements are true if we replace ord Mord D,0 with PD,0 .

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Proof. We have the canonical isomorphism (8.3.4.2) because, by Proposition ~ ~λ,~i) (see Proposition 2.2.1.1) under 3.4.6.3, the pullback of the tautological (A, ~ ord . (3.4.6.4) is canonically isomorphic to the tautological (A, λ, i) over M H As for (8.3.4.3) and (8.3.4.4), by construction, and thanks to the canonical isomorphism (8.3.4.2), we already have the isomorphisms (8.3.4.3) and (8.3.4.4) (compatible with (8.3.4.2) and with each other) in an open dense subscheme whose complement has codimension at least two. Now the question is whether the direct images of the open immersion from such an open dense subscheme to the whole scheme coincide with the canonical or subcanonical extensions; or, equivalently, whether the canonical or subcanonical extensions coincide with the direct images of their restrictions to such an open dense subscheme. (Note that we start with existing extensions here.) This is a local question, which we can verify after ´etale localization or rather just over the completions of strict local rings, without having to assume that the sheaves are automorphic. Since R and W are flat, by writing flat modules as direct limits of finitely generated free modules, and by using the fact that taking direct images (under quasi-compact and quasi-separated morphisms) commute with arbitrary direct limits, it suffices to treat the case that R = R0 and ~ ord,tor is regular. W = R is trivial, which is then known because the scheme M H,Σord ord The analogous statements for PD,0 can be similarly proved. 8.3.5

Pushforwards to the Total Minimal Compactifications

Definition 8.3.5.1. With the setting as in Definition 8.3.2.1, suppose moreover that H is neat. Let Σ be any collection for MH (which might not be projective), so that Mtor H,Σ is defined over S0 . Let the quasi-coherent sheaf can E~Mord (W ) ∪ EM ord ,R (WQ ) Q D,0 ,R D,0

over the open subscheme ~ H ⊗ R) ∪(Mtor ⊗ RQ ) (M H,Σ R0

R0

~ tor of M H,d0 pol ⊗ R be defined as in Definition 8.3.3.1 (by Lemma 8.3.2.4), and define R0

the quasi-coherent sheaf  
can,min ~ H ⊗ R) ∪(Mtor ⊗ RQ ) → M ~ min ⊗ R E~M (M ord ,R (W ) := H,Σ H D,0 ∗ R0 R0 R0   can E~Mord ,R (W ) ∪ E ord (WQ ) D,0

MD,0 ,RQ

~ min ⊗ R, using Proposition 2.2.1.2 and the canonical morphisms over M H R0

Mmin H in (3) of Theorem 1.3.1.5. Similarly, we define the quasi-coherent sheaf sub,min E~M ord ,R (W ) D,0

H H

: Mtor H →

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sub ~ min ⊗ R by replacing E can (WQ ) with EM over M ord ,R (WQ ) in the above definition H Mord ,R R0

D,0

D,0

of E~Pcan ord ,R (W ). These are well defined (i.e., independent of the choice of Σ) because D,0

of the projection formula [35, 0I , 5.4.10.1], because of the isomorphism (1.4.3.3) tor (with g = 1 and H0 = H there), and because, under any morphism Mtor H,Σ  MH,Σ0 as in [62, Prop. 6.4.2.3], the pushforward of the ideal sheaf defining the boundary of Mtor H,Σ0 with reduced subscheme structure is canonically isomorphic to the ideal sheaf defining the boundary of Mtor H,Σ0 with reduced subscheme structure (cf. (1.3.3.19), with Q = {0} there). When Condition 8.1.1.2 holds, we also define similar assignments W 7→ ord ord ~sub,min E~Pcan,min ord ,R (W ) and W 7→ EPord ,R (W ) by replacing MD,0 with PD,0 in the above conD,0 D,0 structions. Lemma 8.3.5.2. With the setting as in Lemma 8.3.5.1, we have compatible canonical isomorphisms ∼ H (WQ ) E~can,min (W ) ⊗ Q → E can ord ord MD,0 ,R

H,∗

Z

MD,0 ,RQ

and

over Mmin H

∼ H sub,min E~M E sub ,RQ (WQ ) ord ,R (W ) ⊗ Q → H,∗ Mord D,0 D,0 Z H min → M is as in (3) of Theorem 1.3.1.5. When ⊗ RQ , where H : Mtor H H

R0

ord Condition 8.1.1.2 holds, the same is true if we replace Mord D,0 with PD,0 .

Proof. This follows immediately from the definitions. Lemma 8.3.5.3. In Definition 8.3.5.1, if Σ is projective, with a collection pol of ~ tor ~ polarization functions, as in Definition 8.3.3.1, so that M H,d0 pol over S0 is defined, then we have compatible canonical isomorphisms H ∼~ E~can,min (W ) = E~can (W ) ord ord H,∗ MD,0 ,R

MD,0 ,R

and H sub,min ∼~ E~sub (W ) E~M ord ,R (W ) = H,∗ Mord ,R D,0

D,0

H ~ tor ~ min is as in Definition 2.2.2.1. Consequently, ~ min , where ~ : M over M H H,d0 pol → MH H ~ H ⊗ R) ∪(Mtor ⊗ RQ ) is the preimage of (M ~ H ⊗ R) ∪(Mmin ⊗ RQ ) under since (M R0

H,Σ

R0

R

H

R

0 0 H ~ tor ~ min ⊗ R, by Lemma 8.3.5.2, the canonical morphism ~ H,∗ : M ⊗ R → M H H,d0 pol

R0

R0

can,min ~sub,min E~M ord ,R (W ) and EMord ,R (W ) are canonically isomorphic to the pushforwards of D,0

D,0

the quasi-coherent sheaves H E~Mord ,R (W ) ∪(

H,∗

can EM ord ,R (WQ )) Q

H,∗

sub EM ord ,R (WQ )), Q

D,0

D,0

and E~Mord (W ) ∪( D,0 ,R

H

D,0

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respectively, over the open subscheme ~ H ⊗ R) ∪(Mmin ⊗ RQ ) (M H R0

R0

~ min ⊗ R (cf. Definition 8.3.3.1). of M H R0

When Condition 8.1.1.2 holds, the analogous statements are true if we replace ord MD,0 with Pord D,0 . can,min ~sub,min Proof. Since the sheaves E~M ord ,R (W ) and EMord ,R (W ) in Definition 8.3.5.1 are (up D,0

D,0

to canonical isomorphisms) independent of the choice of Σ, we may assume (up to a refinement) that Σ is projective, with a collection pol of polarization functions. Then it suffices to note that the pushforward under a composition of morphisms is the composition of the pushforwards of the individual morphisms. The analogous statement for Pord D,0 can be similarly proved. Corollary 8.3.5.4. For each R flat over R0 , and for each W in RepR (Mord D,0 ) that is flat over R0 , there are compatible canonical isomorphisms H~ ord ord,can ~ ord,min ⊗ R → M ~ min ⊗ R)∗ E~can,min E~ (W ) ∼ (W ) (8.3.5.5) = (M H H H,∗ Mord ,R Mord ,R D,0

R0

R0

D,0

and H~ ord ord,sub ~ ord,min ⊗ R → M ~ min ⊗ R)∗ E~sub,min E~ (W ) ∼ (W ) = (M H H H,∗ Mord ,R Mord ,R D,0

R0

R0

(8.3.5.6)

D,0

(cf. (3) and (6) of Theorem 6.2.1.1) when Hp and hence H = Hp Hp are neat, and when Σord extends to some projective Σ, with a collection pol of polarization ~ tor functions (so that M H,d0 pol is defined as in Proposition 2.2.2.1). When Condition 8.1.1.2 holds, the analogous statements are true if we replace ord Mord D,0 with PD,0 . ~ ord that extends to some projective Σ, Proof. By choosing a collection Σord for M H with a collection pol of polarization functions, these follow from Lemma 8.3.5.3, Corollary 6.2.3.2, and Proposition 8.3.4.1. Lemma 8.3.5.7. With the setting as in Definition 8.3.5.1, suppose R is noetherian and flat over R0 , and suppose W is flat over R0 and is finitely generated and (S1 ) can,min ~sub,min over R (see [35, IV-2, 5.7.2]). Then E~M ord ,R (W ) and EMord ,R (W ) are coherent D,0

D,0

~ min ⊗ R. over M H R0

can,min ~sub,min Proof. Let us explain the case of E~M ord ,R (W ), because the case of EMord ,R (W ) D,0

D,0

~ min is flat over R0 , and since M ~ H is fiberwise dense in M ~ min is similar. Since M H H min min ~ ~ by Proposition 2.2.1.7, the complement of (MH ⊗ R) ∪(MH ⊗ RQ ) in MH ⊗ R R0

R0

R0

is of codimension at least two. Hence, [35, IV-2, 5.11.4], it suffices to show H bycan can,min ~ ~ that the restrictions EMord (W ) and H,∗ EPord ,RQ (WQ ) of E~M ord ,R (W ) to MH ⊗ R D,0 ,R D,0

D,0

R0

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and Mmin H ⊗ RQ (see Lemma 8.3.5.2), respectively, are coherent and (S1 ) over some R0

~ H is noetherian and open subsets containing all points of codimension one. Since M ~ H is flat over the normal, it is (S2 ) by Serre’s criterion [35, IV-2, 5.8.6]. Since M ~ ~ Dedekind scheme S0 , by [35, IV-2, 5.7.6], the fibers of MH → ~S0 are (S1 ). By ~ H ⊗ R, being Lemma 8.3.1.4 and Definition 8.3.2.1, the sheaf E~Mord (W ) over M D,0 ,R R0

´etale locally isomorphic to the pullback of W , is coherent by ´etale descent, and is (S1 ) by [35, IV-2, 6.4.2]. Similarly, since Mtor H,Σ is smooth over S0 , the sheaf tor EPcan ord ,R (WQ ) over MH,Σ ⊗ RQ is also coherent and (S1 ). By [62, Prop. 7.2.3.13], Q R0 H D,0 tor min : M → M is an isomorphism over the open subscheme M1H of Mmin H,Σ H H formed H by the Hunion of all strata (see (4) of Theorem 1.3.1.5) of codimension at most one. 1 Thus, H,∗ EPcan ord ,R (WQ ) is coherent and (S1 ) over MH ⊗ RQ , which contains all Q R0

D,0

points of Mmin H ⊗ RQ of codimension one, as desired. R0

Corollary 8.3.5.8. (Compare with Lemma 8.3.3.4.) With the setting as in Definition 8.3.5.1, suppose moreover that R is an integral domain flat over R0 , and that can,min ~sub,min W is locally free over R. Then E~M ord ,R (W ) and EMord ,R (W ) are torsion-free over D,0

D,0

~ min ⊗ R. M H R0

When Condition 8.1.1.2 holds, the analogous statements are true if we replace ord Mord D,0 with PD,0 . Proof. Since R is flat over R0 , the canonical morphism OM ~ min H

H~ H,∗

OM ~ tor

H,d0 pol

⊗R

⊗R

→

R0

(cf. (2.2.2.2)) is an isomorphism, and it follows from Lemmas

R0

can,min ~ min ⊗ R ~sub,min 8.3.5.3 and 8.3.3.4 that E~M ord ,R (W ) and EMord ,R (W ) are torsion-free over MH D,0

R0

D,0

under the assumptions we made on R and W . Lemma 8.3.5.9. (Compare with Lemmas 1.4.2.11, 8.1.3.15, and 8.3.2.6.) Let a ≥ 1 be as in Lemma 2.1.2.35, and let N1 ≥ 1 be as in Proposition 2.2.1.2, so that aN1 ω⊗ ~ min is defined. For any flat R0 -algebra R, and for each integer k ≥ 1, the pullback MH

kaN1 ~ min ⊗ R is canonically isomorphic to E~can,min of ω ⊗ to M (W0⊗ kaN1 ⊗ R) for H ~ min Mord ,R MH

top

W0 := ∧

R0 ∨ (Gr−1 ) . D,0

D,0

R0

~ H ⊗ R and Proof. By Definition 8.3.5.1, it suffices to verify this separately over M R0

~ H ⊗ R, this follows from Lemma 8.3.2.6. Over Mmin ⊗ RQ , this ⊗ RQ . Over M H R0 R0 R0 H ⊗ kaN1 ∼ ⊗ kaN1 is a combination of Lemma 1.4.2.11 and the fact that ωMmin = H,∗ ωM (by tor H H (2) and (3) of Theorem 1.3.1.5, and by the projection formula [35, 0I , 5.4.10.1]). Mmin H

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aN1 Lemma 8.3.5.10. Let a ≥ 1 be as in Lemma 2.1.2.35, and let ω ⊗ ~ min and W0 be MH

as in Lemma 8.3.5.9. For each W in RepR (Mord D,0 ), and for each integer k ≥ 1, we have compatible canonical isomorphisms can,min ⊗ ka0 N1 ∼ ~ can,min kaN1 ) = EMord ,R (W ) ⊗ ω ⊗ E~M ord ,R (W ⊗ W0 ~ min R0

D,0

MH

OM ~ min

D,0

H

and sub,min kaN1 ⊗ k ∼ ~ sub,min E~M ) = EMord ,R (W ) ⊗ ω ⊗ ord ,R (W ⊗ W0 ~ min . R0

D,0

MH

OM ~ min

D,0

H

Proof. As in the proof of Lemma 8.3.5.9, by Definition 8.3.5.1, it suffices to verify ~ H ⊗ R, this follows from ~ H ⊗ R and Mmin ⊗ RQ . Over M these separately over M H R0

R0

R0

the construction in Definition 8.3.2.1 and from Lemma 8.3.2.6 that k EMord (W ⊗ W0⊗ k ) ∼ (W ) ⊗ ω ⊗ = EMord ~ D,0 ,R D,0 ,R

MH

OM ~

R0

H

for each integer k and each W in RepR (Gord D,0 ). On the other hand, by Lemma 8.3.5.9, by (2) and (3) of Theorem 1.3.1.5, and by the projection formula [35, 0I , 5.4.10.1], the verification of the lemma over Mmin H ⊗ RQ can be reduced to the verification of R0

the corresponding statements ⊗ kaN1 ∼ can can EM ) = EMord ,RQ (WQ ) ord ,R (WQ ⊗ W0 Q R0

D,0

D,0

⊗

OMtor

⊗ kaN1 ωM tor H

H,Σ

and ⊗ kaN1 ∼ sub sub EM ) = EMord ,RQ (WQ ) ord ,R (WQ ⊗ W0 Q R0

D,0

D,0

⊗

OMtor

⊗ kaN1 ωM tor H

H,Σ

sub can over Mtor H,Σ ⊗ RQ , which are true by the construction of EMord ,RQ ( · ) and EMord ,RQ ( · ) R0

D,0

D,0

⊗k ⊗k can (cf. Construction 8.3.1.1), and by the analogue EM ⊗ RQ ) ∼ = ωM tor ⊗ RQ ord ,R (W0 Q H

R0

D,0

R0

of Lemma 1.4.2.11, for each integer k. 8.3.6

Hecke Actions

Proposition 8.3.6.1. (Compare with Propositions 1.4.3.1, 1.3.1.14, and 2.2.3.1, (4) of Theorem 1.3.3.15, and Proposition 8.1.4.1.) Suppose that we have an element g = (g0 , gp ) ∈ G(A∞,p ) × G(Zp ) ⊂ G(A∞ ); that we have two open compact ˆ such that H0 ⊂ gHg −1 ; and that the image Hp of H subgroups H and H0 of G(Z) ˆ → G(Z ˆ p ) is neat (and so that H is neat, under the canonical homomorphism G(Z) ~ :M ~ min : M ~ H0 → M ~ H is induced by the [g] ~ min ~ min in Proposition and so that [g] →M 0 H

H

2.2.3.1). For each W ∈ RepR (Mord D,0 ), there is (by abuse of notation) a canonical isomorphism ∼ ~ ~ ∗ : [g] ~ ∗ E~ ord (W ) → EMord (W ) [g] MD,0 ,R D,0 ,R

(8.3.6.2)

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~ H0 , where the first E~ ord (W ) is defined over M ~ H, of quasi-coherent sheaves over M MD,0 ,R ~ H0 . and where the second is defined over M The canonical isomorphism (8.3.6.2) is compatible with the canonical isomorphism ∼

[g]∗ : [g]∗ EMord (WQ ) → EMord (WQ ) D,0 ,RQ D,0 ,RQ

(8.3.6.3)

p over MH (as in (1.4.3.2)). If gp ∈ Pord D (Zp ) (cf. Example 3.3.4.5), and if H = H Hp 0 0,p 0 p and H = H Hp are of standard form (with neat H ), then (8.3.6.2) is also compatible with the canonical isomorphism (8.1.4.2) (under the canonical isomorphism induced by (8.3.4.2)). If g = g1 g2 , where g1 and g2 are elements of G(A∞,p ) × G(Zp ), each having a ~ ∗ = [g~1 ]∗ ◦ [g~2 ]∗ whenever the involved setup similar to that of g, then we have [g] isomorphisms are defined. When Condition 8.1.1.2 holds, the analogous statements are true if we replace ord MD,0 with Pord D,0 . p

Proof. Since Hp is neat, H0 ⊂ g0 Hp g0−1 is also neat (and so is H0 = H0,p Hp0 ). By Proposition 2.2.3.1, the canonical surjection [g] : MH0 → MH (over S0 = Spec(F0 )) ~ : M ~ H0 → M ~ H . By construction, the extends to a canonical finite surjection [g] pullback of the tautological object (A, λ, i, αH ) over MH under [g] : MH0 → MH is 0 ) of the tautological object canonically isomorphic to the Hecke twist (A0 , λ0 , i0 , αH × (A, λ, i, αm ) over MH0 by g, realized by a Z(p) -isogeny [g] : A → A0 over MH0 , because gp ∈ G(Zp ). (Here, for simplicity, we use the same notation (A, λ, i) for ~ H0 is noetherian and both the tautological objects over MH and over MH0 .) Since M ~ 0 0 normal, and since MH is dense in MH , by [92, IX, 1.4], [28, Ch. I, Prop. 2.7], or [62, ~ H (see Proposition 2.2.1.1) under ~ ~λ,~i) over M Prop. 3.3.1.5], the pullback of the (A, 0 ~ ~ ~ [g] : MH0 → MH is the unique extension of (A , λ0 , i0 ) (up to canonical isomorphism) ~ H0 , which we denote by (A ~ 0 , ~λ0 , ~i0 ), and the Z× -isogeny [g] : A → A0 uniquely over M (p) extends to a Z× -isogeny (p) ~ ∗:A ~→A ~0, [g] ∼

(8.3.6.4) ∗

~ ~ ~ : Lie ~ ~ → Lie ~ 0 ~ ∼ which induces isomorphisms [g] ~ H and [g]∗ : ~ M ∗ A/MH0 A /MH0 = [g] LieA/ ∗ ∼ ~ Lie ~ ∨ ~ , which respect the pairings defined by ~λ Lie∨ ∨ ~ → Lie∨ 0,∨ ~ ∼ = [g] ~ /M 0 A H ~0

~ A

/MH0

A /MH

ˆp ˆp and λ up to the unique number r0 in Z× (p),>0 such that r0 ν(g0 )Z = Z . For each W in RepR (Pord D,0 ), these two isomorphisms induced by (8.3.6.4) define the desired ~ H0 , which extends the canonical isomorphism (8.3.6.3). isomorphism (8.3.6.2) over M The morphism (8.3.6.2) is compatible with (8.3.6.3) by construction. If gp ∈ Pord D (Zp ), then (8.3.6.2) is also compatible with the canonical morphism (8.1.4.2) ~ ord and [g] ~ are (under the canonical isomorphism induced by (8.3.4.2)), because [g] × ~ ord0 compatible (see (6.2.2.3)) and hence must induce the same Z -isogeny over M (p)

extending the ordinary Hecke twist of the tautological object over Mord H0 by [g].

H

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The second last paragraph of the proposition (for Mord D,0 ) is true because, by ~ = [g~2 ] ◦ [g~1 ] between morphisms of Proposition 2.2.3.1, we have the identity [g] ~ = [g~2 ]◦ [g~1 ] between Z× -isogenies, and hence schemes, which induces the identity [g] (p)

~ ∗ = [g~1 ]∗ ◦ [g~2 ]∗ between isomorphisms like (8.3.6.2). also the desired identity [g] When Condition 8.1.1.2 holds, the analogous statements for Pord D,0 can be similarly proved, by using the isomorphism ∗ ∼ dR ~ ~ ~ ∗ : H dR (A/ ∼ ~ ~ H0 ) → ~ M ~0 ~ [g] H dR 1 1 (A /MH0 ) = [g] H 1 (A/MH )

induced by (8.3.6.4). Corollary 8.3.6.5. With the same setting as in Proposition 8.3.6.1, there are (by abuse of notation) a canonical isomorphism ~ ([g]

min

∼ can,min ~ min )∗ E~can,min )∗ : ([g] (W ) → E~M ord ,R (W ) Mord ,R D,0

(8.3.6.6)

D,0

and a canonical morphism ~ ([g]

min

sub,min ~ min )∗ E~sub,min )∗ : ([g] (W ) → E~M ord ,R (W ) Mord ,R D,0

(8.3.6.7)

D,0

~ min of quasi-coherent sheaves over M H0 , where the sheaves at the left-hand sides are min ~ defined over MH , and where the sheaves at the right-hand sides are defined over ~ min M H0 , which are compatible with each other and with (8.3.6.2). The canonical morphisms (8.3.6.6) and (8.3.6.7) are compatible with the canonical morphisms induced by (1.4.3.3) and (1.4.3.4). If gp ∈ Pord D (Zp ) (cf. Example 3.3.4.5), then they are also compatible with the canonical morphisms induced by (8.1.4.3) and (8.1.4.4) under the canonical isomorphisms induced by (the pushforwards from the partial toroidal compactifications to the partial minimal compactifications of) (8.3.4.3) and (8.3.4.4). If g = g1 g2 , where g1 and g2 are elements of G(A∞,p ) × G(Zp ), each having a ~ min )∗ = ([g~1 ]min )∗ ◦ ([g~2 ]min )∗ whenever setup similar to that of g, then we have ([g] the involved morphisms are defined. When Condition 8.1.1.2 holds, the analogous statements are true if we replace ord Mord D,0 with PD,0 . can,min ~sub,min Proof. By the constructions of E~M ord ,R (W ) and EMord ,R (W ) as pushforwards (see D,0

D,0

Definition 8.3.5.1), these follow from the corresponding statements of Proposition ~ H and M ~ H0 , and of Proposition 1.4.3.1 over Mtor and Mtor0 0 (by 8.3.6.1 over M H,Σ H ,Σ Lemma 8.3.5.2; and from the compatibility stated in (6) of Theorem 7.1.4.1). Corollary 8.3.6.8. With the same setting as in Proposition 8.3.6.1, suppose H0 is a normal subgroup of H. Then the canonical morphisms (8.3.6.2), (8.3.6.6), and can (8.3.6.7) induce compatible actions of the finite group H/H0 on E~M ord ,R (W ) over D,0

sub,min ~ H0 and on E~can,min ~ min ~ M (W ) and E~M ord ,R (W ) over MH0 , covering those on MH0 and Mord ,R D,0

D,0

~ min M H0 (cf. [62, Cor. 7.2.5.2] and Corollary 2.2.3.2).

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521

ˆ in ProposiProof. The statements are self-explanatory, by taking g ∈ H ⊂ G(Z) tion 8.3.6.1 and Corollary 8.3.6.5. Proposition 8.3.6.9. With the same setting as in Proposition 8.3.6.1, suppose moreover that H0 is a normal subgroup of H, so that the actions H/H0 as in Corollary 8.3.6.8 are defined; that R is flat over R0 ; and that W is locally free over R. Then the canonical morphisms induced by (8.3.6.2), (8.3.6.6), and (8.3.6.7) by adjunction induce canonical isomorphisms 0 ∼ ~ E~ ord (W )
H/H , E~Mord (W ) → [1] (8.3.6.10) ∗ MD,0 ,R D,0 ,R
H/H0 ∼ can,min ~ min E~can,min E~M (W ) , ord ,R (W ) → [1]∗ Mord ,R

(8.3.6.11)


H/H0 ∼ sub,min ~ min E~sub,min , (W ) E~M ord ,R (W ) → [1]∗ Mord ,R

(8.3.6.12)

D,0

D,0

and D,0

D,0

~ :M ~ min : M ~ min ~ H0 → M ~ H and [1] ~ min where [1] H0 → MH are as in Proposition 2.2.3.1 and can,min ~sub,min Corollary 2.2.3.2, and where the E~Mord (W ), E~M ord ,R (W ), and EMord ,R (W ) at the D,0 ,R D,0

D,0

~ H and M ~ min (resp. M ~ H0 left-hand sides (resp. right-hand sides) are defined over M H min ~ and MH0 ). When Condition 8.1.1.2 holds, the analogous statements are true if we replace ord Mord D,0 with PD,0 . Proof. By Lemma 8.3.5.3, in order to justify the canonical isomorphisms (8.3.6.10), (8.3.6.11), and (8.3.6.12), it suffices to establish the canonical isomorphisms 0 ~ E~ ord (W )
H/H , (W ) → [1] (8.3.6.13) E~Mord ∗ MD,0 ,R D,0 ,R min

H H,∗

can EM ord ,R (WQ ) → [1]∗ Q

H,∗


H/H0 can EM , ord ,R (WQ ) Q

(8.3.6.14)

H,∗


H/H0 sub EM , ord ,R (WQ ) Q

(8.3.6.15)

H

D,0

D,0

and min

H H,∗

sub EM ord ,R (WQ ) → [1]∗ Q D,0

H

D,0

sub can where the EM ord ,R (WQ ) and EMord ,R (WQ ) at the left-hand sides (resp. right-hand Q Q D,0

D,0

tor sides) are defined over Mtor H,Σ (resp. MH0 ,Σ0 ) for some projective smooth collection Σ for MH0 (resp. Σ0 for MH , which is a 1-refinement of Σ invariant under the action min min of H/HH0 ), andHwhere [1] : Mmin H0 → MH is as in Proposition 1.3.1.14. Or, since min tor tor tor [1] ◦ H0 = H ◦[1] , where [1] : MH0 ,Σ0 → Mtor H,Σ is as in Proposition 1.3.1.15, we shall consider
H/H0 tor can can EM (8.3.6.16) ord ,R (WQ ) → [1]∗ EMord ,R (WQ ) Q Q D,0

D,0

and
H/H0 tor sub sub EM , ord ,R (WQ ) → [1]∗ EMord ,R (WQ ) Q Q D,0

D,0

(8.3.6.17)

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instead of (8.3.6.14) and (8.3.6.15). Let us start with (8.3.6.13). By Corollary 2.2.3.2 (the argument of whose proof also works for the normalizations of the base changes from Spec(OF0 ,(p) ) to Spec(R0 ) of the schemes involved), we have a canonical isomorphism
H/H0 ∼ ~ OM . (8.3.6.18) ~ H → [1]∗ OM ~ 0 H

Since R is flat over R0 , we have an induced isomorphism
H/H0 ∼ ~ OM . ~ H ⊗ R → [1]∗ OM ~ 0 ⊗R H

R0

(8.3.6.19)

R0

Since W is locally free over R, by Lemma 8.3.1.4 and Definition 8.3.2.1, E~Mord (W ) D,0 ,R ~ H ⊗ R. Therefore, by (8.3.6.2) (with g = 1) and by the prois locally free over M R0

jection formula [35, 0I , 5.4.10.1], and by (8.3.6.19), the restriction of the canonical morphism induced by adjunction by (8.3.6.2) (with g = 1) induces the desired isomorphism (8.3.6.13). tor Next let us consider (8.3.6.16) and (8.3.6.17). Since the morphism [1] : tor tor MH0 ,Σ0 → MH,Σ is proper, its admits a Stein factorization (see [35, III-1, 4.3.3])
tor [1]∗ OMtor0 0 → Mtor Mtor H,Σ , H0 ,Σ0 → Z := SpecO H ,Σ

Mtor H,Σ

where Z is a noetherian normal scheme finite over the projective smooth scheme 0 Mtor H,Σ . Hence, Z is also projective, and carries an action of H/H defined by the tor tor compatible isomorphisms [g] : Mtor H0 ,Σ0 → MH0 ,Σ0 defined by Proposition 1.3.1.15 for each g ∈ H. By Zariski’s main theorem (see [35, III-1, 4.4.3, 4.4.11]), the induced morphism Z/(H/H0 ) → Mtor H,Σ is an isomorphism, because it is generically the known isomorphism ∼

MH0 /(H/H0 ) → MH (see [62, Cor. 7.2.5.2]). (Alternatively, one may define the H/H0 -action on Z using tor the facts that the pullback of Z → Mtor H,Σ under MH ,→ MH,Σ on the target is the canonical finite ´etale cover MH0 → MH carrying a canonical action of H/H0 , and that Z is noetherian normal and hence, by Zariski’s main theorem, coincides with the normalization of Mtor H,Σ in MH0 under the canonical morphism MH0 → MH → Mtor . Therefore, it is not really necessary to assume that Σ0 is invariant under the H,Σ action of H/H0 .) Thus, we obtain the canonical isomorphism
H/H0 ∼ tor tor OMtor → [1] O , (8.3.6.20) M ∗ 0 0 H,Σ H ,Σ

which induces the canonical isomorphism ∼

tor

ID∞,H → [1]∗ ID∞,H0


H/H0

,

(8.3.6.21)

tor where ID∞,H (resp. ID∞,H0 ) is defined over Mtor H,Σ (resp. MH0 ,Σ0 ) as in Definition 1.4.2.9, because D∞,H and D∞,H0 are reduced subschemes defining normal crossings

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divisors, and D∞,H0 = ([1] induced isomorphisms OMtor H,Σ

tor −1

)

523


(D∞,H ) red . Since R is flat over R0 , we have

∼


H/H0

(8.3.6.22)


H/H0

(8.3.6.23)

tor

⊗ RQ

→ [1]∗ OMtor0

⊗ RQ

→ [1]∗ ID∞,H0

R0

page 523

⊗ H ,Σ0 R 0

RQ

and ID∞,H

∼

R0

tor

⊗ RQ

,

R0

tor Since EPcan ord ,R (WQ ) is locally free over MH,Σ ⊗ RQ , the desired isomorphism Q D,0

R0

(8.3.6.16) (resp. (8.3.6.17)) follows from (8.3.6.22) (resp. (8.3.6.23)), from (1.4.3.3) (with g = 1 and H0 = H there), and from the projection formula [35, 0I , 5.4.10.1], as desired. The analogous statements for Pord D,0 can be similarly proved.

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[54] E. Kunz, Characterizations of regular local rings of characteristic p, Amer. J. Math. 91, 3, pp. 773–784 (1969). [55] E. Kunz, K¨ ahler Differentials, Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig (1986). [56] W. Kuyk and J.-P. Serre (eds.), Modular Functions of One Variable III, Lecture Notes in Mathematics, Vol. 350. Springer-Verlag, Berlin, Heidelberg, New York (1973). [57] K.-W. Lan, Arithmetic compactification of PEL-type Shimura varieties, Ph. D. Thesis, Harvard University, Cambridge, Massachusetts (2008), errata and revision available online at the author’s website. [58] K.-W. Lan, Elevators for degenerations of PEL structures, Math. Res. Lett. 18, 5, pp. 889–907 (2011), errata available online at the author’s website. [59] K.-W. Lan, Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties, J. Reine Angew. Math. 664, pp. 163–228 (2012a), errata available online at the author’s website. [60] K.-W. Lan, Geometric modular forms and the cohomology of torsion automorphic sheaves, in [44], pp. 183–208 (2012b). [61] K.-W. Lan, Toroidal compactifications of PEL-type Kuga families, Algebra Number Theory 6, 5, pp. 885–966 (2012c), errata available online at the author’s website. [62] K.-W. Lan, Arithmetic compactification of PEL-type Shimura varieties, London Mathematical Society Monographs, Vol. 36. Princeton University Press, Princeton (2013), errata available online at the author’s website. [63] K.-W. Lan, Boundary strata of connected components in positive characteristics, Algebra Number Theory 9, 9, pp. 1955–2054 (2015a), an appendix to the article “Families of nearly ordinary Eisenstein series on unitary groups” by Xin Wan. [64] K.-W. Lan, Compactifications of splitting models of PEL-type Shimura varieties, preprint (2015b), to appear in Trans. Amer. Math. Soc. [65] K.-W. Lan, Compactifications of PEL-type Shimura varieties in ramified characteristics, Forum Math. Sigma 4, e1, pp. 1–98 (2016a). [66] K.-W. Lan, Higher Koecher’s principle, Math. Res. Lett. 23, 1, pp. 163–199 (2016b). [67] K.-W. Lan, Vanishing theorems for coherent automorphic cohomology, Res. Math. Sci. 3, 39, pp. 1–43 (2016c). [68] K.-W. Lan, Integral models of toroidal compactifications with projective cone decompositions, Int. Math. Res. Not. IMRN 2017, 11, pp. 3237–3280 (2017). [69] K.-W. Lan and B. Stroh, Relative cohomology of cuspidal forms on PEL-type Shimura varieties, Algebra Number Theory 8, 8, pp. 1787–1799 (2014). [70] K.-W. Lan and J. Suh, Vanishing theorems for torsion automorphic sheaves on compact PEL-type Shimura varieties, Duke Math. J. 161, 6, pp. 1113–1170 (2012). [71] K.-W. Lan and J. Suh, Vanishing theorems for torsion automorphic sheaves on general PEL-type Shimura varieties, Adv. Math. 242, pp. 228–286 (2013). [72] R. P. Langlands and D. Ramakrishnan (eds.), The Zeta Functions of Picard Modular Surfaces, based on lectures delivered at a CRM Workshop in the spring of 1988. Les Publications CRM, Montr´eal (1992). [73] J. Lubin, J.-P. Serre and J. Tate, Elliptic curves and formal groups, Summer Institute on Algebraic Geometry, Woods Hole (1964), available at http://www.ma.utexas.edu/users/voloch/lst.html.

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[74] K. Madapusi Pera, Toroidal compactifications of integral models of Shimura varieties of Hodge type, preprint (2015). [75] Y. I. Manin, The theory of commutative formal groups over fields of finite characteristic, Russian Math. Surveys 18, pp. 1–83 (1963). [76] H. Matsumura, Commutative Algebra, 2nd edn., Mathematics Lecture Note Series. The Benjamin/Cummings Publishing Company, Inc. (1980). [77] H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, Vol. 8. Cambridge University Press, Cambridge, New York (1986). [78] J. S. Milne, The points on a Shimura variety modulo a prime of good reduction, in [72], pp. 151–253 (1992). [79] B. Moonen, Models of Shimura varieties in mixed characteristic, in [94], pp. 267–350 (1998). [80] L. Moret-Bailly, Pinceaux de vari´et´es ab´eliennes, Ast´erisque, Vol. 129. Soci´et´e Math´ematique de France, Paris (1985). [81] D. Mumford, Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, Vol. 5. Oxford University Press, Oxford (1970), with appendices by C. P. Ramanujam and Yuri Manin. [82] D. Mumford, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24, 3, pp. 239–272 (1972). [83] D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 34, 3rd edn. Springer-Verlag, Berlin, Heidelberg, New York (1994). [84] A. Ogus, F-Crystals, Griffiths Transversality, and the Hodge Decomposition, no. 221 in Ast´erisque. Soci´et´e Math´ematique de France, Paris (1994). [85] G. Pappas and M. Rapoport, Local models in the ramified case, I. The EL-case, J. Algebraic Geom. 12, 1, pp. 107–145 (2003). [86] G. Pappas and M. Rapoport, Local models in the ramified case, II. Splitting models, Duke Math. J. 127, 2, pp. 193–250 (2005). [87] G. Pappas and M. Rapoport, Local models in the ramified case, III. Unitary groups, J. Inst. Math. Jussieu 8, 3, pp. 507–564 (2009). [88] G. Pappas and X. Zhu, Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math. 194, pp. 147–254 (2013). [89] R. Pink, Arithmetic compactification of mixed Shimura varieties, Ph.D. thesis, Rheinischen Friedrich-Wilhelms-Universit¨ at, Bonn (1989). [90] M. Rapoport, Compactifications de l’espace de modules de Hilbert–Blumenthal, Compositio Math. 36, 3, pp. 255–335 (1978). [91] M. Rapoport and T. Zink, Period Spaces for p-divisible Groups, no. 141 in Annals of Mathematics Studies. Princeton University Press, Princeton (1996). [92] M. Raynaud, Faisceaux amples sur les sch´emas en groupes et les espaces homog`enes, Lecture Notes in Mathematics, Vol. 119. Springer-Verlag, Berlin, Heidelberg, New York (1970). [93] I. Reiner, Maximal Orders, London Mathematical Society Monographs. New Series, Vol. 28. Clarendon Press, Oxford University Press, Oxford (1975). [94] A. J. Scholl and R. L. Taylor, Galois Representations in Arithmetic Algebraic Geometry, London Mathematical Society Lecture Note Series, Vol. 254. Cambridge University Press, Cambridge, New York (1998).

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[95] J.-P. Serre, Geom´etrie alg´ebrique et g´eom´etrie analytique, Ann. Inst. Fourier. Grenoble 6, pp. 1–42 (1955–1956). [96] D. C. Spencer and S. Iyanaga (eds.), Global Analysis. Papers in Honor of K. Kodaira. Princeton University Press, Princeton (1969). [97] B. Stroh, Compactification de vari´et´es de Siegel aux places de mauvaise r´eduction, Ph.D. thesis, Universit´e Henri Poincar´e, Nancy, France (2008). [98] B. Stroh, Compactification de vari´et´es de Siegel aux places de mauvaise r´eduction, Bull. Soc. Math. France 138, 2, pp. 259–315 (2010a). [99] B. Stroh, Compactification minimale et mauvaise r´eduction, Ann. Inst. Fourier. Grenoble 60, 3, pp. 1035–1055 (2010b). [100] A. Vasiu, Integral canonical models of Shimura varieties of preabelian type, Asian J. Mathematics 3, pp. 401–518 (1999). [101] A. Vasiu and T. Zink, Purity results for p-divisible groups and abelian schemes over regular bases of mixed characteristic, Documenta Math. 15, pp. 571–599 (2010). [102] T. Wedhorn, Ordinariness in good reductions of Shimura varieties of PEL-type, Ann. Sci. Ecole Norm. Sup. (4) 32, pp. 575–618 (1999). [103] C. A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, Vol. 38. Cambridge University Press, Cambridge, New York, Melbourne (1994). [104] Y. G. Zarhin, Endomorphisms of abelian varieties and points of finite order in characteristic p, Math. Notes 21, 6, pp. 415–419 (1977).

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Index

( · )mult , 156 S ( · )mult , 156 ( · , · )aux , 125 ( · , · )∗aux , 126, 133 (| · , · |), 34, 267 [ · ], 26 ◦ , 157 h · , · i, 22, 34 h · , · i00,n , 78 h · , · i00,n0 , 232 h · , · i02 , 219, 223 h · , · i10,n , 68 h · , · i10,n0 , 232 h · , · iord 10,pr , 223, 232 mult , 229, 232 (h · , · iord 10,pr )η h · , · i11 , 30, 219 h · , · i11,R , 30 h · , · i20 , 28 h · , · i(g) 20 , 40 (h · , · i20,n )mult , 232 η h · , · iM , 479 h · , · iQ , 43 h · , · iaux , 126 h · , · ican. , 118 h · , · i(g) , 39 h · , · iij,n , 231 h · , · iλ , 24, 482 h · , · iλMH ,jQ , 101 , 486 h · , · i λM ~ ord

h · , · iφD,0 , 478 h · , · ie, 44 ∼Q× -isog. , 72, 75 ∼Z× -isog. , 318, 323 √ (p) −1, 21 ?, 22 ?aux , 128 , 53, 54, 99, 108, 113, 374, 411, 416 [(0, 0, {0})]-stratum, 291 [0]-stratum, 291, 294 (1) (Tate twist), 21 ~ 521 [1], ~ min , 151, 521 [1] ~ ord , 211, 402 [1] ~ ord,min , 357, 359 [1] ~ ord,tor , 301, 304, 402 [1] [1]tor , 3, 522 A, 23, 197 a, 139 a0 , 139 a1 , 125 a2 , 125 ~ 141 A, ~ aux , 141 A AO aux , 126, 129, 199 AM aux , 126, 199 i A∨,(p ) , 180 Agaux ,daux , 126, 142 Agaux ,daux ,n , 127 Ag,d , 126, 142 Ag,d,n,n , 127 A(p) , 190

H

, 397, 486 h · , · i λM ~ ord ,j H

Q

⊗ Q, 397 h · , · i λM ~ ord ,j H

Q

Z

h · , · ican λ , 121, 485 h · , · iλ\ , 502 h · , · iφ , 28 531

page 531

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532

Compactifications with Ordinary Loci

10374-main

Compactifications with Ordinary Loci i

A(p ) , 180 ∞ A∨ s [p ], 164 ∞ ´ A∨ [p ]et , 164 s ∞ mult A∨ [p ] , 164 s As [p∞ ], 164 As [p∞ ]´et , 164 As [p∞ ]mult , 164 abelian scheme ordinary, see ordinary abelian scheme absolute Frobenius morphism, 181, 190, 211, 304, 359, 362, 402 flat, 212, 305 not flat, 359 admissible boundary component, 35 admissible filtration, 27 admissible radical, 34 admissible rational polyhedral cone decomposition, 35, 52, 455 projective, 35 smooth, 35 compatible choice, see compatible choice of admissible smooth rational polyhedral cone decomposition data admissible submodule, 32, 34 admissible surjection, 32 affine toroidal embedding, 81, 89, 267, 326, 327, 383, 410 algebraic construction, 4, 8 α ~ Gaux (Zˆp ) , 141 α ~ H , 141 αH , 422 αHp , 422 αord Hp , 422 ord αG(Z , 212, 307 p) αH , 23, 57, 58, 192 \ [αH ], 138 aux αHpaux , 333 ord αH , 333 aux,p ord,O αHaux,p , 200, 333 O p αH , 130, 200, 333 aux O αHaux , 129, 130 † αH , 59, 145 ‡ αH , 59 ♦ αH , 280 ♥ αH , 59, 270, 271 αHκ , 107, 408 ‡ αH , 112 κ \,‡ [αH ], 112 κ

\,ord,‡ [αH ], 415 κ αHpκ , 408 ‡ αH p , 415 κ

ord αH , 408 κ,p ord,‡ αH , 415 κ,p αHn , 192 αHn0 , 173, 192, 237 \,ord αH , 241 n pr 0

\,ord αH , 240 n \ [αH ], 138, 281 \ αH , 243, 281 \,‡ [αH ], 59 \,ord [αH ], 242, 281 \,ord αH , 241, 281 \,ord,‡ ], 292, 297 [αH αHp , 174, 192, 197, 206, 237, 241, 295 lifting, 174 † αH p , 296 ‡ αH p , 291, 296 ♦ αHp , 280 ♥ αHp , 270, 271, 296 ord , 180, 193, 198, 206, 240, 241, 295 αH p lifting, 182 ord,0 αH , 180 p ord,† αH , 296 p ord,‡ αH , 291, 296 p ♦

ord αH , 280 p ♥ ord αHp , 270, 271, 296 ord,#,0 αH , 180 p αHpr , 192 ord αH ord , 179, 193, 240 r p

ord,0 αH ord , 179 pr

ord,#,0 αH , 179 ord pr

α, ˆ 165 α bHb , 108, 412 ‡ α bH b , 109, 413 \,‡ [b αH b ], 112

\,‡ α bH b , 112, 414 \,ord,‡ [b αH ], 414 b

\,ord,‡ α bH , 414 b α bHb p , 411 ‡ α bH b p , 412 ord α bHb , 411 p

ord,‡ α bH , 412 b p

page 532

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Compactifications with Ordinary Loci

Index

α ˆ p , 165 α ˆ s¯, 130, 193 α ˆ sord ¯ , 182, 193, 199 α ˆ sord,0 , 182, 193, 199 ¯ α ˆ sord,#,0 , 182, 193, 199 ¯ α ˆ s¯,p , 193 α ˆ sp¯ , 174, 193, 199 [α ˆ sp¯ ]Hp , 174 α ˆ sO¯ , 130 α ˆ sO,p ¯ , 130, 200 , 200 [α ˆ sO,p ¯ ]Hp aux αn0 , 173, 198, 204, 234 (αn0 , νn0 ), 172 \,ord [αn ], 234 \,ord αn , 231 equivalence class, 234 αpr , 193 r , 178, 179, 193, 198, 200, 204, 217, αpord 229, 234 compatible with degeneration, 217 αpord,0 , 178, 179, 193, 206, 217, 229 r ord,0,\ αpr , 220, 230 , 200 αpord,0,O r , 178, 179, 193, 206, 217, 229, 230 αpord,#,0 r αpord,#,0,\ , 220, 230 r αpord,#,0,O , 200 r ord,O αpr , 200 αp#,mult , 193 r αpord,0 r ,S , 206, 220, 221 αpord,0,\ , 220 r ,S ord,#,0 αpr ,S , 220 αpord,#,0 , 206, 221 r αpord,#,0,\ , 220 r ,S α eHe , 108, 412 ‡ α eH e , 109, 412 \,‡ [e αH e ], 110

\,‡ α eH e , 110, 413 \,ord,‡ [e αH ], 413 e \,ord,‡ α eH , 413 e α eHe p , 411 ‡ α eH e p , 412 ord α eHe , 411 p

ord,‡ α eH , 412 e p

r , 426 α epord analytic construction, 4 comparison with, 124

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page 533

533

Artin’s approximation theory, 119, 122, 286, 482, 510 Artin’s criterion, 198, 202 Aut(¯ x), 343, 350 automorphic bundle, 4, 5, 8, 13, 120, 482, 510 canonical extension, see canonical extension subcanonical extension, see subcanonical extension automorphic sheaf, 120, 482, 510 auxiliary model, 9, 16, 17, 125–141, 198–202, 333–336, 425 B, 268 B (abelian scheme), 66, 70, 138, 216, 248, 281, 313, 315, 454 B (homomorphism), 269 B0 , 216 ~ 339 B, Baux (abelian scheme), 138 Baux (homomorphism), 138 B ‡ (abelian scheme), 59, 112, 292, 297, 415 B ‡ (homomorphism), 60, 292, 297 B ∨ , 70, 216, 315 ∨,ord Bη,p r , 224, 233, 245 ord r , 224, 233, 245 Bη,p B(G), 279, 282 B(G‡ ), 60, 292, 297 B( ♦ G), 281 b ‡ , 110, 413 B Bp∨,ord , 248, 379 r ord Bpr , 248, 379 (B, ?, V, h · , · i, h0 ), 22 e ‡ , 110, 413 B bad prime, 27 balanced Γ1 (pr ), 6, 13–15 βH , 80 βHn , 80 βHn0 , 325 \ βH , 314 n 0

\ βH , 69 n \ βH , 69 βHp , 325 \ βH p , 314 \,ord βH , 314–316 p ord βH , 325 p

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Compactifications with Ordinary Loci

\,ord βH ~ ord , 321 p ,Z \ βH,Z , 74 [βˆ\ ]HG1,Z , 70

[βˆp,\ ]Hp

G1,Z

, 315, 316

βˆs¯\ , 70, 316 βˆs¯\,0 , 71 [βˆs¯\ ]HG1,Z , 70 βˆs¯\,#,0 , 71 βˆs¯p,\,0 , 316 [βˆs¯p,\ ]Hp , 316 G1,Z

βˆs¯p,\,#,0 , 316 βn , 79 βn0 , 78, 79 βn0 , 324 βn0 0 , 324 βn\ 0 , 313 βn\,0 , 313 0 βn\,#,0 , 313 0 βn#,0 , 324 0 βn\ , 69 βn\,0 , 69 βn\,#,0 , 69 βn#,0 , 78, 79 βp\,ord , 313, 314 r βp\,ord,0 , 314 r \,ord,#,0 βpr , 314 r , 324, 325 βpord \ βeH e , 108 βe\ , 411 411

bottom level, 15, 17, 125 boundary moduli problem, 32 2, 23 c

page 534

Compactifications with Ordinary Loci

\ βH , 321 p ,~ Zord

ep H \,ord βeH ep ,

10374-main

(complex conjugation), 22 c, 68, 70, 138, 216, 249, 281, 315 ord ~Φ C , 262 1 ,δ1 ord ~ Φ ,δ , 260, 266, 295, 312, 314, 347, 379 C H H ~ ord,grp , 312, 379 C ΦH ,δH ord ~Φ C , 423, 438 H ,δH ,rκ ord ~ CΦH ,δH ,rn , 312, 380 ord ~Φ C , 312, 313 n ,δn caux , 138 c∨ aux , 138

c˘He , 85 c˘∨ e , 85 H c˘∨,ord , 379 e H

c˘ord e , 379 H c˘n , 85 c˘∨ n , 85 c˘∨,ord , 379 n c˘ord n , 379 c‡ , 59, 112, 292, 297, 415 ...ord,\,◦ C H,n , 438 ...ord C ΦH ,n , 255 ...ord,◦ C ΦH ,n , 256 ...ord,grp C ΦH ,n , 256 ...ord,grp,◦ C ΦH ,n , 261 ...ord C Φn , 249 ...ord,◦ C Φn , 250 ...ord,◦◦◦ , 250 C Φn c∨ , 68, 70, 138, 216, 249, 281, 315 c∨,‡ , 59, 112, 292, 297, 415 cH , 69, 243, 259, 266, 281 c∨ H , 69, 243, 259, 266, 281 c∨,ord , 241, 255, 266, 270, 281, 314, 379 H cHn , 243, 245 cHn0 , 236, 237 c∨ Hn0 , 236, 237 c∨ Hn , 243, 245 c∨,ord Hn , 240 cord Hn , 240 cord H , 241, 255, 266, 270, 281, 314, 379 c∨,ord , 239, 245 H ord pr

, 239, 245 cord H ord r p

~b ord C ˘ , 380, 394, 438 ˘ ,δ Φ c c H

H

b c‡ , 110, 413 b c∨,‡ , 110, 413 b c‡Hb , 111, 414 b c∨,‡ b , 111, 414 H b c∨,ord,‡ , 414 b H b cord,‡ , 414 b H b c‡Hb p , 414

b c∨,‡ b p , 414 H

e c∨,ord , 411 e H e cord e , 411 H b c‡n , 111 b c∨,‡ n , 111

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Compactifications with Ordinary Loci

Index

b ˘ ˘ , 86, 98 C ΦH c,δH c cn , 68, 245 cn0 , 233, 237, 249 c∨ n0 , 233, 237, 249 c‡n , 111 c∨ n , 68, 245 c∨,‡ n , 111 ∨,ord cn , 233, 249, 313, 379 cord n , 233, 249, 313, 379 c∨,ord , 224, 229, 233, 239, 245, 249 pr r cord , 224, 229, 233, 239, 245, 249 p CΦH ,δH , 58, 62, 67, 69, 259 grp CΦ , 67 H ,δH ord CΦ , 259 H ,δH CΦn ,δn , 67, 68 ~e ord C ˘ , 379, 409 ˘ ,δ Φ f f H

H

~e ord,grp C ˘ , 379 ˘ ,δ Φ f f H

H

~e ord C ˘n , 380 ˘ n ,δ Φ ~e ord CΦ e , 407 e ,δ f f H

H

e c‡ , 110, 413 ...ord,\,◦ eb C , 438 ...H,n ord,ext e Ce , 436 ...H,n ord,ext,◦ e Ce , 436 ...H,n ord,ext e C , 436 ...n ord,ext,◦ en C , 436 ...ord,ext e C , 434 ...ord,ext,◦ e C , 434 ...ord e C Φ˘ f,n , 418 H ...ord,◦ eΦ C , 437 e f H,n

e c∨,‡ , 110, 413 e cHe , 109 e c‡He , 110, 413 e c∨ e , 109 H e c∨,‡ e , 110, 413 H e c∨,ord , 414 e H

e c∨,ord,‡ , 413 e H e cord , 414 e H e cord,‡ , 413 e H ‡,1 e cn , 111 e c∨,‡,1 , 111 n e ˘ ˘ , 85, 86 C Φ f,δ f H

H

10374-main

page 535

535

e grp , 85 C ˘ ˘ ,δ Φ f H

f H

e ˘ ˘ , 86 C Φn ,δn e e e , 106 C ΦH f,δH f

e grp , 106 C e e ,δ Φ f H

f H

canonical extension, 4, 5, 8, 13, 122, 123, 486, 488, 509, 512 formal fiber, 501–507 canonical subgroup, 2, 13, 197 partial, 197 Cartier dual, 162, 188, 193 coarse moduli space, 26, 61 compatible choice of admissible smooth rational polyhedral cone decomposition data, 3, 7, 37, 53, 135, 288, 373 compatible, 135, 136 existence, 37, 288, 289, 374, 376 extension, 288, 289, 374, 376 gh -induced, 403 gh -refinement, 102, 398 gb-induced, 399 gb-refinement, 102, 399 g-induced, 300 g-refinement, 3, 65, 84, 123, 300 induced, 136, 374 projective, 37, 289 refinement, 37, 288, 289, 373, 376 compatible collection of polarization functions, 4, 5, 8 compatible with D cusp label, 172 Z, 172 cone, 34 face, 34 nondegenerate, 34 rational polyhedral, 34 smooth, 35 supporting hyperplane, 34 cone decomposition, 35, 52, 455 conn , 164 constant term, 340 convexity of polarization function, 35 cusp, 3, 32–33, 62 ordinary, 7, 347 cusp label, 27, 29, 58 assigment, 47, 85, 99 compatible with D, 171, 172 with cone, 36

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536

Compactifications with Ordinary Loci

Compactifications with Ordinary Loci

assignment, 136 face, 38, 98, 393 gb-refinement, 91, 103, 385, 400 g-refinement, 65, 84, 301, 329 induced, 136 with cone decomposition, 36 gb-refinement, 91, 386 g-refinement, 84, 300, 329 refinement, 37 surjection, 36 g-assignment, 65, 84, 300, 322, 329, 354 gb-assignment, 91, 385 ge-assignment, 91, 385 ordinary, 172, 373 surjection, 32, 33 D, 6, 10, 165, 168, 177, 197, 311 D (Cartier dual), 162 D−1 , 165, 168 D−1 , 171, 243 D−1 −1 , 171 D0−1 , 171 D1−1 , 171 D−1,pr , 171, 219–221, 231, 239, 243 D−1 −1,pr , 171, 219 D0−1,pr , 171, 219 D1−1,pr , 171, 219 D# −1,pr , 172, 219 D#,−1 −1,pr , 172, 219 D#,0 −1,pr , 172, 219 #,1 D−1,pr , 172, 219 D0 , 165, 168, 243 d0 , 4, 5, 8, 63, 147, 359, 360 d0,aux , 334, 366 d00,n , 74, 78 d00,n0 , 233, 320 D1 , 165, 168 d10,n , 68, 74, 78 d10,n0 , 233, 320 dord 10,pr , 226, 229, 233 ~ ord D ∞,H , 296 Daux , 10, 198 D−1 aux , 198 D0aux , 198 D1aux , 198 D# aux , 198 D0aux,Qp , 198 ∗,ord,ext,∨ [g~l ] 0 , 405 κ ,κ

D∞,H , 58

10374-main

dλL , 473 dλext L , 474 dλM ~ ord ,jQ , 475 H dλM ~ ord ,jQ ⊗ Q, 475 H

Z

dλ~Nord ,jQ , 475 dλ~Nord ,jQ ⊗ Q, 475 Z

dλ ~e ord ΨΦ e

ej ) e (` Q f,δH f H

⊗ Q, 474 Z

d log, 473 d log(L), 473 ~ ord d log(Ψ ΦH ,δH (`)), 277 d log q, 281 D0pr , 169, 218 Dipri , 197 D#,0 pr , 169, 218 dpol, 63 D# , 165, 168 D#,−1 , 165, 168 D#,0 , 165, 168 D#,1 , 165, 168 DQp , 168 D−1 Qp , 168 0 DQp , 168 D1Qp , 168 e D, 371 e D−1 , 371 e D0 , 371 de0 , 426 e D1 , 371 DZ , 311 DZ,0 , 311 DDPE,O (R, I), 216 DDPEL,... (V ), 415 M ord Hκ ... DDPEL, M ord (R, I), 242, 247, 281 H DDPEL,... (V ), 297 M ord H DDPEL,... ord (R, I), 234 Mn DDPEL,MHaux (V ), 138 DDfil.-spl. PEL,MH (R, I), 243 DDPEL,Mord (V ), 415 Hκ

DDPEL,MHκ (V ), 112 DDPEL,Mord (R, I), 247, 281 H DDPEL,Mord (V ), 297 H DDPEL,MH (R, I), 247, 281 DDPEL,MH (V ), 59, 138 ...ord (Ve ), 413 DD e PEL, M H f

e DDPEL,M e ord (V ), 413 f H

page 536

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Compactifications with Ordinary Loci

Index

e DDPEL,M e (V ), 110 f H de Rham cohomology, 24 de Rham homology, 24 DEGPE,O (R, I), 216 DEGPEL,... (V ), 415 M ord Hκ ... DEGPEL, M ord (R, I), 241, 247, 281 H DEGPEL,... (V ), 297 M ord H DEGPEL,... ord M n (R, I), 234 DDPEL,MHaux (V ), 138 DEGPEL,Mord (V ), 415 Hκ

DEGPEL,MHκ (V ), 112 DEGPEL,Mord (R, I), 247, 281 H DEGPEL,Mord (V ), 297 H DEGPEL,MH (R, I), 247 DEGPEL,MH (V ), 59, 138 ...ord (Ve ), 413 DEG e PEL, M H f

e DEGPEL,M e ord (V ), 413 f H

e DEGPEL,M e (V ), 110 f H degenerating family, 8 pullback to boundary stratum, 73–74, 319–321 ~ ord of type M Haux , 333 ~ ord of type M H , 205 of type MH , 57 of type MHaux , 137 of type MHpaux , 137 of type (PE, O), 145, 215 degeneration datum, 27 comparison with characteristic zero, 242–247 of (G, λ, i), 217 for general ordinary level structure, 235–242 for principal ordinary level structure, 217–235 degeneration pattern, 8, 60 δ, 29, 47 ~ ord,∗ ∆ ΦH ,δH ,`,`0 , 266, 325 δaux , 134 ˘ 47 δ, δ˘Hb , 48, 110, 380, 413 δ˘He , 47 δ (g) , 41 δH , 243, 248, 281 δHaux , 134 δHpaux , 135 ‡ δH , 60, 292, 297

10374-main

page 537

537

δHn , 243 δHn0 , 236 ord δH , 240 n ord δH , 241, 255, 266, 270, 281 ord δH ord , 239, 244 r p

ord,0 δH ord , 239, 244 pr

ord,#,0 δH , 239, 244 ord pr

ˆ 29 δ, ~b ord,∗ ∆ ˘ ,`, ˘ `˘0 , 382 ˘ ,δ Φ c H c H ord b δ b , 413 H

b ∗˘ ∆ Φ

˘ ,`, ˘ `˘0 c,δH c H

, 88

∆M ~ ord,tor , 299 H δn , 29, 248, 313 δn0 , 232, 313 ∆ord,∗ n,`,`0 , 251 δnord , 232 δpr , 244, 253 r , 232, 239, 244, 253, 314 δpord , 222, 232, 239, 244 δpord,0 r ord,0 mult (δpr )η , 230 δpord,#,0 , 222, 232, 239, 244 r )mult , 230 (δpord,#,0 r η ∗ ∆ΦH ,δH ,`,`0 , 80 e 45 δ, ~e ord,∗ ∆ ˘ ,`, ˘ `˘0 , 447 ˘ ,δ Φ f H f H e δHe , 46, 372 δe‡ , 110, 413 e H

ord,‡ δeH , 413 e depth of elements of twisted Up type, 195 ˆ 372 b Z), of subgroups of G( b of subgroups of G(Zp ), 371 ˆ 170 of subgroups of G(Z), of subgroups of G(Zp ), 170 depthD (gp ), 195 depthD (H), 170, 197 b 372 depthD (H), b p ), 371 depthD (H depthD (Hp ), 170 e 371 deptheD (H), Der~Nord,tor /M ~ ord,tor , 471 H Der~Nord /M ~ ord , 471 H determinantal condition on Lie algebra, 6, 23, 131, 141, 164, 203, 510

June 16, 2017 8:34

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Compactifications with Ordinary Loci

10374-main

Compactifications with Ordinary Loci

Dieudonn´e–Manin classification, 156 Diff −1 , 43 Diff −1 O/Z , 43 Disc, 95, 158 DiscO/Z , 95, 158 dual lattice, 22 dual semi-abelian scheme, 57, 205, 215, 392

sub,min E~M ord ,R ( · ), 514–517, 520, 521 D,0

can E~Mord ,R ( · ) ∪ EM ord ,R (( · )Q ), 512, 514 Q D,0 H D,0 can E~Mord ,R ( · ) ∪( H,∗ EMord ,R (( · )Q )), 515 Q D,0 D,0 H sub E~Mord ,R ( · ) ∪( H,∗ EM ord ,R (( · )Q )), 515 D,0

D,0

Q

E~Pord , 509 D,0

E~Pord ord , 481 D,0

e00,n , 74 e00,n0 , 320 e10,n , 74 e10,n0 , 320 eord 10,pr , 226 eA[pr ] , 178 eAs¯[p∞ ] , 182 eAs [p∞ ] , 164 ord E~G ord , 481 D,0

ord,can E~G , 487 ord D,0

ord E~G ord ,R ( · ), 482, 483 D,0

ord,can E~G ord ,R ( · ), 488, 508 D,0

ord,sub E~G ord ,R ( · ), D,0

488

~Eord ∞,κ , 393 ~Eord ∞ , 397 ord,ΦH ,δH E~M , 505 ord D,0,fil

ord,ΦH ,δH E~M ord ,R ( · ), 506 D,0,fil

0

ord,ΦH ,δH E~M , 507 ord D,0,gr

E~Mord , 509 D,0

ord E~M ord , 481 D,0

ord,can E~M , 487 ord D,0

ord,ΦH ,δH E~M , 502 ord D,0

E~Mord ,R ( · ), 508, 510–513, 518, 521 D,0

can E~M ord ,R ( · ), 508, 512, 513, 515 D,0

can,min E~M ord ,R ( · ), 514–517, 520, 521 D,0

ord E~M ord ,R ( · ), 482, 483, 508, 513 D,0

ord,can E~M ord ,R ( · ), 488, 508, 513, 516 D,0

ord,ΦH ,δH E~M ( · ), 502 ord ,R D,0

ord,sub E~M ord ,R ( · ), 488, 508, 513, 516 D,0

sub E~M ord ,R ( · ), 508, 512, 513, 515 D,0

E~Pord,can , 487 ord D,0

E~Pord ,R ( · ), 508, 510, 511, 513, 521 D,0

E~Pcan ord ,R ( · ), 508, 512, 513, 516 D,0

E~Pcan,min ord ,R ( · ), 515–517, 521 D,0

E~Pord ord ,R ( · ), 482, 483, 508, 513 D,0

E~Pord,can ord ,R ( · ), 488, 508, 513, 516 D,0

E~Pord,sub ord ,R ( · ), 488, 508, 513, 516 D,0

E~Psub ord ,R ( · ), 508, 512, 513, 516 D,0

sub,min E~M ord ,R ( · ), 515 D,0

E~Psub,min ord ,R ( · ), 515–517, 521 D,0

eB[n] , 68 eB[pr ]η , 228 ED,Z , 201 ED,Z,free , 204 ED,Z,tor , 204 ...ord,◦ , 438 E H,n ... b , 264 E 1 ...Φ ord E Φ1 , 264 ...ord E ΦH ,n,free , 258 ...ord E ΦH ,n , 258 ...ord , 258 E ...ΦH ,n,tor E Φn , 79 ... E Φn0 , 264 ...ord E Φn ,free , 251 ...ord E Φn , 251 ...ord E Φn ,tor , 251 EG0 , 118 can EG , 122 0 EG0 ,R ( · ), 120 can EG ( · ), 122 0 ,R sub EG0 ,R ( · ), 123 EG0 ,R˜ Q ( · ), 509 EGord , 508 D,0 can EG ord , 509 D,0

page 538

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Compactifications with Ordinary Loci

10374-main

Index

EGord ,RQ ( · ), 509 D,0

can EG ord ,R ( · ), Q D,0 sub EGord ,R ( · ), Q D,0

509 509

EGord ,R˜ Q ( · ), 509 D,0

sub EG ord , 509 D,0

b ˘ , 88, 381 E Φc H

λη

e

ord,0

image(αpr

)

, 228

eimage(ϕord,0 ) , 228 −1,pr

E∞ , 100 E∞,κ , 98 eλ , 173 eλη , 228 EM0 , 119 can EM , 122 0 EM0 ,R ( · ), 120 can ( · ), 123 EM 0 ,R sub ( · ), 123 EM 0 ,R EM0 ,R˜ Q ( · ), 509 EMord , 508 D,0 can EM ord , 509 D,0

EMord ,RQ ( · ), 509, 511, 519 D,0

can EM ord ,R ( · ), 509, 512, 515, 521 D,0

Q

sub EM ord ,R ( · ), 509, 512, 515, 521 D,0

Q

EMord ,R˜ Q ( · ), 509 D,0

sub EM ord , 509 D,0

eφm , 231 eφn , 232 EP0 , 119 EPcan , 122 0 EP0 ,R ( · ), 120 EPcan ( · ), 123 0 ,R ( · ), 123 EPsub 0 ,R EP0 ,R˜ Q ( · ), 509 EPord , 508 D,0 EPcan ord , 509 D,0

EPord ,RQ ( · ), 509, 511 D,0

EPcan ord ,R ( · ), 509, 512, 515 D,0

Q

EPsub ord ,R ( · ), 509, 515 D,0

Q

EPord ,R˜ Q ( · ), 509 D,0

EPsub ord , 509 D,0

EΦ˘ f , 92, 387, 409 H EΦH , 62, 76, 80, 258, 324, 325 EΦH ,σ , 62, 81, 326, 347

page 539

539

EΦn , 76 e˘ E ΦH σ , 88, 381 f,˘ easier unitary case, 18 elevator, 145 endomorphism structure, 23, 164, 172, 177, 191, 197 ε(g) , 39, 41 equally deep ˆ 372 b Z), subgroups of G( b subgroups of G(Zp ), 372 ˆ 170 subgroups of G(Z), subgroups of G(Zp ), 170 ´ et , 164 extended ordinary Kuga family, see ordinary Kuga family, extended extension of isogeny, 160–162 F , 22 f , 126, 199 f , 161, 162 (g) F−2 , 39 (g) F−2,h(C) , 42 (g)

(F−2,h(C) )⊥ , 42 (g) F−2,R , 41 (g) F−i , 39

F0 , 3, 23, 44 F0,aux , 9, 128 F00 , 117 (i) FA∨ /S , 181 a ~ ord F (EPord ,R (W )), 483 D,0

Fa (E~Pord ,R (W )), 511 D,0

Fa (EP0 ,R (W )), 120 ~ ord,tor /M ~ ord,tor )), 470 Fa (H ilog-dR (N H FA/M , 211 ~ ord ⊗ Fp H0 Z

•

Fa (Ω~Nord,tor /~S0,r ), 470 H FA/S , 190 (i) FA/S , 181 a F (W ), 120, 483, 511 Fabs , 362 Faux , 362 f,´ et , 162 F(g) , 39, 40 fκ , 99, 395 extensibility, 421–423 fκtor , 425 aux fκ0 ,κ , 99, 116, 394, 420 fκgrp 0 ,κ , 99, 116, 394, 420

June 16, 2017 8:34

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Compactifications with Ordinary Loci

10374-main

page 540

Compactifications with Ordinary Loci

fκtor 0 ,κ , 99, 116, 394, 420 properness, 421 fκe , 107, 408 fκe0 ,eκ , 114, 416 fκegrp 0 ,e κ , 116, 420 fκetor 0 ,e κ , 113, 416 fκtor , 99, 117, 395, 422 equidimensionality, 441–443 flatness, 442–443 formal fiber, 445–453 log integrality, 442–443 log smoothness, 430–435 properness, 424–428 FM ~ ord,min ⊗ Fp , 359 H

Z

FM ~ ord ⊗ Fp , 211 H

Z

FM ~ ord,min ⊗ Fp , 359 H0

Z

FM ~ ord ⊗ Fp , 211 H0 Z

FM ~ ord,tor

⊗ Fp

, 304

H0 ,Σord,0 Z

FM ~ ord,tor

⊗ Fp

, 304, 403

H,Σord Z

F~Nord,tor ⊗ Fp , 402 κ

Z

F(~Nord,tor ⊗ Fp )/(M ~ ord,tor κ

Z

⊗ Fp )

, 403

H,Σord Z

F~Nord,tor ⊗ Fp , 402 κ0

Z

F~Nord ⊗ Fp , 402 κ0

Z

FS , 181, 190 fσ , 495 f[σ] , 495 tor f[˘ τ ] , 451 fτ˘tor , 451 f tor , 451, see also fκtor face, 34 of cusp label with cone, 38, 98, 393 failure of Hasse’s principle, 26 fiber-wise geometric identity component, 157 group scheme of, 158 filtration admissible, 27 fully symplectic, 27 fully symplectic-liftable, 27 integrable, 27 split, 27 symplectic, 27 FJord ΦH ,δH , 340 ord,(0)

FJΦH ,δH (f ), 340

ord,(0)

(FJΦH ,δH )∧ x ¯ , 344 ord,d,(`) FJΦH ,δH (N ), 499 FJord ΦH ,δH (f ), 340 ord,(`) FJΦH ,δH , 339 ord,(`) FJΦH ,δH (f ), 340 ord,(`) (FJΦH ,δH )∧ x ¯ , 343 ord FJΦH ,δH ,σ , 339 FJord ΦH ,δH ,σ (f ), 339 ord,(`) FJΦH ,δH ,σ (f ), 339 ord,(0) FJCΦH ,δH (k), 340 ord,(`) FJCΦH ,δH (k), 340

formally canonical sheaf, 498 formally subcanonical sheaf, 499 Fourier–Jacobi coefficient along (ΦH , δH ), 340 along (ΦH , δH , σ), 340 expansion along (ΦH , δH ), 340 along (ΦH , δH , σ), 340 constant term, 340 morphism along (ΦH , δH ), 340 along (ΦH , δH , σ), 339 Fraktur, 210 Frobenius morphism absolute, 181, 190, 211, 304, 359, 362, 402 flat, 212, 305 not flat, 359 relative, 181, 190, 211, 304, 362, 403 full-ord , 365 full ordinary locus, 14, 365 fully symplectic filtration, 27 fully symplectic-liftable filtration, 27 G, 3, 22 G, 57, 58, 205, 215, 295 [g], 3, 65, 83, 84, 322 [[g]], 3, 65 [g −1 ], 194 [g −1 ]ord , 191, 194 G0 , 118 G[0] , 291 G0 , 216 [g0 ], 175 g0 , 149, 175, 191, 208, 518 [g0−1 ], 175, 191

June 16, 2017 8:34

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Compactifications with Ordinary Loci

10374-main

Index

G0 (R), 118 G1,Z (R), 31, 56 G(A), 22 G(A∞ ), 22 G(A∞,p ), 22 G(Ap ), 22 ~ 5, 149 [[g]], ~ 5, 149, 518–519 [g], min ~ [g] 0 , 152 i ,i

~ min , 5, 149, 518 [g] ~ min )∗ , 520 ([g] ord

~ [[g] ], 7, 353 ~ ord , 7, 209–211, 329, 519 [g] ~ ord,min , 7, 353, 357, 358 [g] ord,min

~ ([g] )∗ , 353 ord ∗ ~ ([g] ) , 489 ~ ord,tor , 7, 300, 303 [g] ord,tor

~ ([g] )∗ , 489 ord ~ [g] r 0 , 209, 210 H

~ ord,tor , 303 [g] rH0 ~ ∗ , 518, 519 [g] Gaux , 9, 128 Gaux , 333, 361 GO aux , 137, 333 GO,∨ aux , 137 GM aux , 137 GM,∨ aux , 137 G, 422 ∨ G , 422 G~Nord , 422 κ

∨

G~Nord , 422 κ ([g]tor )∗ , 123 Gord D,0 , 479 Gord D,0 (R), 479 G† , 59, 145, 296 G‡ , 59, 112, 291, 296, 415 ♦ G, 280 ... ord [g] , 208 G∨ , 57, 205, 215 G∨,\ , 68, 70, 313, 315, 454 Gess (R), 235 Gess (Z/nZ), 235 Gess (Z/pr Z), 237 [gh ], 102 gh , 67, 102, 398

page 541

541

gh,0 , 398 ord [g~h ] , 399 ord,tor [g~h ] , 403 ord [g~h ] , 399 rκ0 ,rκ

ord,tor [g~h ]r 0 ,rκ , 400 κ

ord,tor ([g~h ]r 0 ,rκ )∗ , 400 κ gh,p , 398 tor [gh ] , 103 Gh,Z , 30 Gess h,Zn , 235 ess,ord Gh,Z , 238 pr ,Dpr

G0h,Z (R), 31, 56 Gh,Z (R), 30, 56, 378 (Gh,Z n U1,Z )(R), 31 b 108, 411 G, [b g ], 91, 102 gb, 102, 398 gb0 , 398 b 1,˘Z (R), 56 G ~g ]ord , 385, 386, 399, 401, 402 [b ~g ]ord,tor , 399, 401, 402 [b ord,tor

~g ] ([b )∗ , 400 ord ~ [b g ]r 0 , 401 κ

~g ]ord,tor , 401 [b rκ0 b ‡ , 109, 412 G b 0 ˘(R), 56 G h,Z

b h,˘Z (R), 56, 378 G b 0 ˘(R), 56, 378 G l,Z b l,˘Z (R), 56 G b~ ord , 421 G Nκ

b~∨ord , 421 G Nκ b~ ord , 411 G Nκ e b N , 109 G κ e

gbp , 398 b G(R), 45, 55 [b g ]tor , 102 ([b g ]tor )∗ , 103 ♥ G, 59, 270, 271, 296 gl , 103, 403 gl,0 , 403 ∗,ord,tor [g~l ] 0 , 406 κ ,κ,rκ0

∗,ord [g~l ]κ0 ,κ , 404, 406

June 16, 2017 8:34

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542

Compactifications with Ordinary Loci

10374-main

Compactifications with Ordinary Loci

∗,ord,ext [g~l ]κ0 ,κ , 405 ∗,ord,ext,∨ [g~l ] 0 , 405 κ ,κ ∗,ord,ext

([g~l ]κ0 ,κ )∗ , 405 ∗,ord,grp [g~l ] 0 , 404, 405 κ ,κ

∗,ord,grp,∨ [g~l ]κ0 ,κ , 405 ∗,ord,tor [g~l ] 0 , 404, 406 κ ,κ ∗,ord,tor

([g~l ]κ0 ,κ )∗ , 405 ∗ [gl ]κ0 ,κ , 104 ([gl ]∗κ0 ,κ )∗ , 105, 405 [gl ]∗,tor κ0 ,κ , 104 ∗ ([gl ]∗,tor κ0 ,κ ) , 104, 105 gl,p , 403 [gl ]Q , 103, 403 [gl ]∗Q , 104, 403 ([gl ]∗Q )∗ , 105 Gess l,Zn , 235 Gess l,Zpr ,Dpr , 237 Gess,ord l,Zpr ,Dpr , 238 G(LZ ,h · , · iZ ) , 30 G0l,Z (R), 31, 56 Gl,Z (R), 31, 56, 57 GM ~ ord , 295 H GM ~ ord , 421 H,rκ

G∨ ~ ord , 421 M H,rκ

GMH , 58 [g]min , 3, 65, 358 G~Nord , 421 κ G~∨ , 421 Nord κ

G\ , 68, 70, 272, 313, 315, 454 [g]ord , 191, 209 gp , 149, 183, 191, 208, 518 of twisted Up type, 195, 209 of Up type, 194, 211 [gp−1 ], 211 gp,−1 , 183 [gp−1 ]ord , 184, 191 t −1 gp , 194 t gp,−1 , 183 gp,0 , 183 [gp ]ord , 184 gpord , 183 gp,# , 194 gp,#,0 , 183 G(Q), 22 G(Q)0 , 39

G(R), 22, 30, 56 G(R), 22 G(R)0 , 39 [g]ord rH0 , 209 [g]tor rH0 ,rH , 301 gσ , 495 [g]∗ , 123, 519 gτ˘ , 451 e 45 G, e 108, 411 G, e G1,eZ (R), 45, 55 e ‡ , 109, 412 G e G0h,eZ (R), 45 e ˘ e(R), 56, 57, 378 G l,Z,Z e 0 e(R), 45 G l,Z

e e(R), 45 G l,Z e~ ord , 421 G Nκ

e~∨ord , 421 G Nκ e~ ord,tor , 422 G Nκ

e ∨ord,tor , 422 G ~ Nκ e \ , 108, 411 G gep , 402, 403 b (G n U)(R), 45 tor [g] , 3, 65, 522 G(Z), 22 GZ , 30 G\ ord , 319 ~ Z

G\Z , 73 ˆ 22 G(Z), ˆ p ), 22 G(Z G(Z/nZ), 22 Γ0 (pr ), 6 Γ1 (pr ), 2, 6 balanced, 6, 13–15 ΓΦ˘ c , 48, 52, 98, 373, 394 H ΓΦ˘ c,ΦH , 48, 50, 373, 445 H

(ΓΦ˘ c,ΦH )∨ R , 50, 455 H ΓΦ˘ f , 48, 373 H ΓΦ˘ f,ΦH , 48, 373 H ΓΦ˘ f,ΦH ,˘σ , 48, 373 H ΓΦ˘ f,τ , 445 H ΓΦ˘ f,˘τ , 410 H ΓΦH , 34, 48, 66 ΓΦH ,`0 , 500 ΓΦH ,σ , 58, 62, 271, 290, 295, 347

page 542

June 16, 2017 8:34

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Compactifications with Ordinary Loci

10374-main

Index

ΓΦH ,τ , 445 ΓΦ e , 48 f H ΓΦ e ,ΦH , 50 f H

γ ∗ , 452, 496, 498, 503 Gauss–Manin connection, 25, 97, 101, 121, 391, 398, 470, 485 extended, 101, 121, 398, 469–471, 486, 508 generalized Kuga family, 4, see Kuga family, generalized ordinary, see ordinary Kuga family, generalized generalized ordinary Kuga family, see ordinary Kuga family, generalized generalized Up operator, 184 geometrically normal, 198, 350 geometrically unibranch, 198 good algebraic model ordinary, see ordinary good algebraic model good formal model ordinary, see ordinary good formal model good prime, 1, 9, 17, 23, 44, 128, 152–153, 198, 203, 212–213, 305–310, 349, 369, 425–428, 490–491 (GrZ−1,n0 , h · , · i11,n0 ), 231 GrZ−1,pr , 171 GrZ−1,R , 30 GrZ−1 , 30 GrZ0 (g), 31 Gr−2 (βˆs¯\,0 ), 71 Gr−2 (βˆs¯\,#,0 ), 71 GrZ−2,pr , 218, 221, 243 #

GrZ−2,pr , 218, 221, 243 #

(GrZ−2,pr )mult , 221 S (GrZ−2,pr )mult , 221 S GrZ−2 (g), 31 (g)

GrF−i , 39 GrV−i , 39 GrZ−i , 30 GrZ−i (g), 30 e GrZ−i , 45 Gra (N ), 504 GrD , 169 Gr−1 D , 165, 168, 477 (Gr−1 D−1,pr )η , 224 Gr0D−1,pr , 243

page 543

543

(Gr−1 #

)η , 224 D−1,pr Gr0D# , 243 −1,pr 0 GrD , 165, 168, 314 Gr−1 D,0 , 477 Gr0D,0 , 478 (Gr0D )mult , 165 s GriD , 170 Gr−1 D,pr , 169, 177 Gr0D,pr , 169, 218 (Gr0Dpr )mult , 217 η (Gr0D,pr )mult , 177, 314 S (Gr0D# )mult , 217 η pr

Gr−1 , 165, 168, 477 D# Gr0D# , 165, 169, 314 Gr−1 , 477 D# ,0 Gr0D# ,0 , 478 , 165 (Gr0D# )mult s Gr−1 , 169, 177 # r D ,p Gr0D# ,pr , 169, 218 , 177, 314 (Gr0D# ,pr )mult S GraF (W ), 120, 483, 511 • GraK (Ω~Nord,tor /~S0,r ), 470 H Z Gr (g), 31 Gram–Schmidt process, 369 Grauert–Riemenschneider vanishing theorem, 491 Griffiths transversality, 101, 398, 471 group scheme ´etale, 155 ´etale-multiplicative-type, 155 of fiberwise geometric connected components, 158 multiplicative-type, 155 H, 3, 23, 197 ord ~Φ h (morphism to C ), 451 H ,δH ord,ZH ∧ ~ h (morphism to (MH )x¯ ), 504 ~ ord,ZH ), 495, 504 h (morphism to M H h (polarization), 26 h−1 , 42 h0 , 22, 26 (h0 )−1 , 30, 42 h0,aux , 128 e h0 , 44 H dR 1 (A/MH ), 24 ~ ord H dR ~ ord /MH ), 486, 490 1 (GM H

June 16, 2017 8:34

ws-book961x669

544

Compactifications with Ordinary Loci

H

can H dR , 121 1 (GMH /MH ) log-dR ~ ord,tor ~ ord,tor (N /M H ), H1 tor tor H log-dR (N /M ), 121 H 1

ord ~ e Z [(ΦH ,δH ,τ )]

H d (O(0+ ~ Nord,tor )∧

), 461

H d (O(+~Nord,tor )∧

), 461

ord ~ e Z [(ΦH ,δH ,τ )]

XΦ ˘

), 461

˘ ,˘ f,δH f σ ,τ H

H d (O~0+ ord e˘ X Φ

H H

), 461

˘ ,˘ f,δH f σ ,τ H

f H

H d (O~e ord ΞΦ ˘

461 461

f H

), 461

˘ ,τ f,δH f H

H d (O~eord

h,˘ Z

, 56

b b 0 , 56 H G h,˘ Z

b b , 56 H G ˘ l,Z

b b 0 , 56 H G l,˘ Z

b b , 56 H P˘ Z b b 0 , 56 H P ˘ Z

b ord ord , 377 H p,MD b pord , 377 H b pr , 377 H b ordb ord , 377 H b Σ), b 53, 97 (H, b b ord ), 374, 393 (H, Σ b HU b , 45 b b , 56 H U ˘ 1,Z

bb H U

2,˘ Z

(O~+ord ), e˘ X ˘ ,˘ Φ f,δ f σ ,τ H H d (O~++ ), ord e˘ X ˘ ,˘ Φ ,δ σ ,τ d

), 461

~ e Z [(ΦH ,δH ,τ )]

H d (O~e ord

bb H G

p,UD

ord ~ e Z [(ΦH ,δH ,τ )]

(O(++ ~ Nord,tor )∧ord

b G , 45, 46, 56, 85, 97, 372, 393 H b b , 56 H G ˘ 1,Z

485

Haux , 129, 199, 333 p Haux , 9, 129, 141, 199, 334 Haux,p , 199 H d (M ) (over Nσ˘ ,τ ), 461 H d (M ) (over Nσ˘ ,τ ), 460 H d (M ) (over N), 498 ), 461 H d (O(~Nord,tor )∧

H

page 544

Compactifications with Ordinary Loci

~ ord can , 485, 490 H dR ~ ord /MH ) 1 (GM

d

10374-main

), 461

Z[(Φ ,δ ,τ )] H H H 1dR (A/MH ), 24 ~ ord can , 485 H 1dR (GM ~ ord /MH ) H H 1dR (GMH /MH )can , 121 ~ ord H idR (Ngrp /M H ), 391 i grp H dR (N /MH ), 97 ~ ord H idR (N/M H ), 391 i H dR (N/MH ), 97

HG1,Z , 31, 56, 67, 72, 312 HGh,Z , 31, 56, 66, 72, 311 p,0 HG , 317 h,Z HGh,Z ,Φ , 66, 311 HG0h,Z , 31, 56, 67, 70, 312 HGl,Z , 31, 56 HG0l,Z , 31, 56 Hh , 32 Hh0 , 32 Hh00 , 32 b 45, 47, 53, 86, 97, 107, 371, 374, 393, H, 408

, 56, 88, 381

b b , 56 H U˘ Z b b , 56 H Z˘ Z

Hκ , 97, 393 ~ ord,tor /M ~ ord,tor ), 397–398, 485 H 1log-dR (N H H 1log-dR (Ntor /Mtor H ), 101, 121 ~ ord,tor /M ~ ord,tor ), 397, 469 Hi (N log-dR

H

H ilog-dR (Ntor /Mtor H ), 100 Hn , 29, 66, 69, 80, 236 Hn0 , 172, 236, 314, 325 Hn0 ,Pess \Hn0 , 236 Zn 0 Hn,Gess , 62, 236 h,Z n

ord ess , 254 Hn,G h,Z n

ord,0 Hn,G ess , 254 h,Zn

0 ess , 252 Hn,G h,Zn Hn,Gess , 62, 67, 252 nUess 1,Zn h,Z n

ord ess nUess , 254 Hn,G 1,Zn h,Zn ess , 76, 252 Hn,Gess nU Zn h,Z n

ord ess nUess , 254 Hn,G Zn h,Zn Hn,Gess , 236 l,Z n

ord ess , 254 Hn,G l,Z n

ord,0 Hn,G ess , 254 l,Zn

0 ess , 252 Hn,G l,Z n

Hnord , 254

June 16, 2017 8:34

ws-book961x669

Compactifications with Ordinary Loci

10374-main

Index

Hn,Pess , 236 Zn 254 n Hn,> , 236 Hn,Uess , 62, 67, 236 1,Zn

h,e Z

l,˘ Z,e Z l,Z

e e 0 , 45 H G l,e Z

e p , 371 H e p , 371 H bal r Uep,1 (p )G b , 371 e e , 56 H P˘ Z,e Z

e e /H ee H P˘ e U Z ,Z

nUZ r ,D r p p

, 253

Hpord r ,Gess,ord

1,ess,ord nUZ r ,D r h,Zpr ,Dpr p p

Hpord r ,Gess,ord

, 238

H ord,0 ess,ord r

, 253

pr ,Dpr

Hp0 r ,Gess , 253 l,Z pr

r , 177, 197, 238, 315 Hpord Hpord r ,Pess,ord , 238 Zpr ,Dpr

ord r , 239 Hpord r ,Pess,ord \Hp Zpr ,Dpr

Hpord r 0 , 180 r ,> , 238 Hpord Hpord r ,Uess,ord

, 238

1,Zpr ,Dpr

Hpord r ,Uess,ord

, 238

2,Zpr ,Dpr

Hpord r ,Uess,ord Z r ,D r p

e Σ e ord , σ (H, e), 373 e Σ, e σ (H, e), 46, 85 e b , 45 H U e e , 45 H U e 1,Z

, 253

l,Zpr ,Dpr

p ,Gl,Z

, 57

e Z

, 253 ess,ord

h,Zpr ,Dpr

2,˘ Z

e e , 45 H PeZ e e 0 , 45 H P

h,Zpr ,Dpr

Hpord r ,Gess,ord

, 56

e e , 45 H G e

ord ess , 254 Hn,Z Zn H-orbit, 241 Hp , 6, 197, 490 Hp , 6, 177, 197 Hpord , 179, 197 Hpr , 236, 238 , 238 Hpord r ,Gess,ord ess,ord pr ,Dpr

, 45

ee H G

ord ess , 254 Hn,U Zn Hn,Zess , 236 Zn

pr ,Gh,Z

, 45

ee H G

h,e Z

ord ess , 254 Hn,U 2,Zn Hn,Uess , 236 Zn

H

ee H G

e e 0 , 45 H G

ord ess , 254 Hn,U 1,Zn Hn,Uess , 76, 236 2,Zn

ord,0

545

1,e Z

ord ess , Hn,P Z

, 238

p

Hpord r ,Zess,ord , 238 Zpr ,Dpr

HPZ , 31, 56, 66 HPZ /HU2,Z , 57 HPZ ,Φ , 66 HP0Z , 31, 56, 76, 324 hτ , 451 e 45, 85, 371 H, e b , 45, 86, 107, 371, 372, 408 H G

page 545

ee H U

2,e Z

, 45

e e , 45 H UeZ e e , 45 H ZeZ

HU1,Z , 31, 56, 67, 312 HU2,Z , 31, 56, 76, 80, 325 HUZ , 31, 56 HZZ , 31, 56, 66 Hasseaux , 362 0 Hasseaaux , 363 0 N1 Hasseaaux , 364 1 HasseaN , 364 H Hasseord H , 362 a (Hasseord H ) , 363 Hasse invariant, 361–365 healthy regular, 162 Hecke action, 9, 13, 16, 38, 55 of elements of Up type, 211, 304, 358–359, 402–403 of G1,Z (A∞ ), 75, 83, 87 of G1,Z (A∞,p ) × Pord 1,Z,D (Qp ), 321, 322, 380 of G(A∞ ), 3, 65–66, 123–124 of G(A∞,p ) × G(Zp ), 5, 149–150, 518–520 of G(A∞,p ) × Pord D (Qp ), 7, 208–210, 300–303, 353–358, 488–490

June 16, 2017 8:34

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546

Compactifications with Ordinary Loci

Gh,Z (A∞ ), 67, 75 Gh,Z (A∞,p ) × Pord h,Z,D (Qp ), 311, 321 b 1,˘Z (A∞ ), 87, 90 G b 1,˘Z (A∞,p ) × P b ord G 1,˘ Z,D (Qp ), 380, 384 ∞ b of G(A ), 102–103 b ∞,p ) × P b ord of G(A D (Qp ), 398–403, 443–444 b 0 ˘(A∞ ), 87, 90, 91, 381, 385 of G l,Z of G0l,Z (A∞ ), 75, 83, 87, 322, 328, 381 of G0l,Z (A∞ ) × Gh,Z (A∞ ), 75 of G0l,Z (A∞ ) ×(G0h,Z (A∞,p ) × Pord,0 h,Z,D (Qp )), 322 e ∞ ), 117 of G(A e of Gl,˘Z,eZ (A∞ ), 87, 381 of GZ (A∞ ), 67 of GZ (A∞,p ) × Pord DZ (Qp ), 311 ˆ p ) × Pord of G(Z (Z p ), 210–211, 303, 358 D of GLO ⊗ A∞ (Q ⊗ A∞ ), 103–105, Z

of of of of of of

page 546

Compactifications with Ordinary Loci

of of of of

of of

10374-main

Z

403–406, 443–444 b ˘Z (A∞ ), 90, 95 P b ˘Z (A∞,p ) × P b ˘ord (Qp ), 385, 389 P Z,D b ˘Z (A∞,p ) × P b ˘ord (Qp ))/U b 2,˘Z (A∞ ), (P Z,D 381, 385 b ˘Z (A∞ )/U b 2,˘Z (A∞ ), 87, 90 P b ˘0 (A∞ ), 90, 95 P Z b ˘0 (A∞,p ) × P b ord,0 (Qp ), 384, 389 P ˘ Z Z,D 0 ∞ e ˘(A ), 94 P Z e ˘0 (A∞,p ) × P e ord,0 (Qp ), 389 P Z

˘ Z,e D

e ˘ e(A∞ ), 94 of P Z,Z e ˘ e(A∞,p ) × P e ˘ord of P (Qp ), 389 Z,Z Z,e Z,e D ∞,p ord e e e 2,˘Z (A∞ ), of (P˘Z,eZ (A ) × P˘Z,eZ,eD (Qp ))/U 380 e ˘ e(A∞ )/U e 2,˘Z (A∞ ), 87 of P Z,Z of PZ (A∞ ), 75, 83, 94, 95 of PZ (A∞,p ) × Pord Z,D (Qp ), 322, 328, 389 ∞ of (PZ (A∞,p ) × Pord Z,D (Qp ))/U2,Z (A ), 322, 328, 381 of PZ (A∞ )/U2,Z (A∞ ), 75, 83, 87 of P0Z (A∞ ), 75, 83, 94, 95 of P0Z (A∞,p ) × Pord,0 Z,D (Qp ), 322, 328, 389 Hecke twist, 13 by G1,Z (A∞ ), 75 by G(A∞ ), 102 by G(A∞,p ), 175–177 by GZ (A∞ ) ∼ = Gh,Z (A∞ ), 67 ordinary, see ordinary Hecke twist b ˘Z (A∞ ), 90 by P

b ˘0 (A∞ ), 90 by P Z e ˘0 (A∞ ), 95 by P Z e ˘ e(A∞ ), 95 by P Z,Z by PZ (A∞ ), 83 by PZ (A∞ )/U2,Z (A∞ ), 75 by P0Z (A∞ ), 83 hereditary, 41, 169 higher Koecher’s principle, 492 Hodge filtration, 97, 101, 391, 397, 469, 470 invertible sheaf, 127, 331 spectral sequence, 100, 397, 469 Hodge invertible sheaf, 11 )◦ , 250 HomO ( n1 X, Bp∨,ord r ord ◦ 1 HomO ( n X, Bpr ) , 249 ◦ r ) , 250 HomO ( n1 Y, Bpord HomO (N , Z), 157 HomO (N, Z), 163 HomO (N, Z)◦ , 163 HomO (N, Z)◦ , 158 ◦ HomO (Q∨ , G∨ ~ ord ) , 390 M H

◦ HomO (Q∨ , G∨ MH ) , 96 ◦ ) HomO (Q, GM ~ ord , 390 H HomO (Q, GMH ), 95 HomO (Q, GMH )◦ , 95 HomO (X, B)◦ , 67, 312 ◦ r ) , 312 HomO (X, Bpord ∨ ◦ HomO (Y, B ) , 250 HomZ (Q∨ , G∨ MH ), 96 HomZ (Q∨ , G∨ ~ ord ), 390 M H

HomZ (Q, GM ~ ord ), 390 H HomZ (Q, GMH ), 96 i, 57, 58, 197, 205, 215, 295 iσ , 498 i[0] , 291 I0,0 , 217 ~i, 141 ~iaux , 141 iaux , 333 iO aux , 129, 130, 137, 199, 333 iM aux , 130, 137, 199 iB , 66, 70, 138, 216, 248, 281, 313, 316, 454 iBaux , 138 iB ‡ , 59, 112, 292, 297, 415 iBb ‡ , 110, 413 iBe ‡ , 110

June 16, 2017 8:34

ws-book961x669

Compactifications with Ordinary Loci

10374-main

Index

iBe ‡ , 413 i, 422 I~Dord , 397, 488 ∞,H

ID∞,H , 100, 123, 522 i† , 59, 145, 296 i‡ , 59, 112, 291, 296, 415 ♦ i, 280 I~Eord , 397 ∞ IE∞ , 100 bi, 108, 411 bi‡ , 109, 412 bi~ ord , 411 Nκ e biN , 109 κ e ♥ i, 59, 270, 271, 296 I` , 269 iM ~ ord , 295 H iMH , 58 \ i , 69, 70, 272, 313, 315, 316 i\,0 , 313 i\,#,0 , 313 i(p) , 190 , 269 IΦord H ,δH ,σ i+ σ , 499 ei, 108, 411 ei‡ , 109, 412 ei\ , 108, 411 Ieσ˘ord ,˘ τ , 410 Ieτ˘ord , 410 iX (b), 216 iY (b), 216 Iy,χ , 217, 269 Iy,φ(y) , 217 i\ ord , 319 ~ Z

i\Z , 73 I~eord

, 445

Z[(Φ

H ,δH ,τ )]

Igusa tower, v ord,0 image(αH ), 180 p ord,#,0 image(αH ), 180 p ord,0 image(αH ord ), 179 pr

ord,#,0 image(αH ), 179 ord pr

image(αpord,0 ), 178, 218 r image(αpord,#,0 ), 178, 218 r image(βp\,ord,0 ), 314 r image(βp\,ord,#,0 ), 314 r image(ϕord,0 −1,pr ), 224, 248, 314

page 547

547

image(ϕord,#,0 −1,pr ), 224, 248, 314 integrable filtration, 27 integral PEL datum, 2, 21 auxiliary choice, 9 Inv(R), 269, 292 Inv(S), 279, 282 Inv(V ), 60, 297 ord

~ Φ ,δ ), 268 Inv(Ξ H H inverse different, 43 ι, 77, 79, 227, 251, 272, 324 ıalg , 286 ι∨ , 324 ιord H , 241 b ι‡ , 111, 414 b ι∨,‡ , 111, 414 b ι‡Hb , 111, 414

b ι∨,‡ b , 111, 414 H b ι∨,ord,‡ , 414 b H b ιord,‡ , 414 b H

b ι‡Hb p , 414

b ι∨,‡ b p , 414 H b ι‡n , 111 b ι∨,‡ n , 111 ιn , 77 ιn0 , 251 ι∨ n , 78 ιord n , 251 ınat , 286 ιpr , 228 e ι‡ , 110 e ι‡,1 , 111 e ι∨,‡ , 110 e ι∨,‡,1 , 111 e ιord,‡ , 414 e H p

e ι‡n , 111 e ι‡,1 n , 111 e ι∨,‡ n , 111 ∨,‡,1 e ιn , 111 isotrivial torus, 216 Iwasawa decomposition, 368, 369 J~H,d0 pol , 5, 147, 361 J~H,d0 polord , 8, 360 ~H,d0 polord , 360 J~H,dpol , 147 J~H,dpolord , 359 ~H,dpolord , 359

June 16, 2017 8:34

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548

Compactifications with Ordinary Loci

10374-main

page 548

Compactifications with Ordinary Loci

J~H0 ,d0 pol0 , 148 j ∨,\ , 70, 315, 316 JH,d0 pol , 4, 5, 63 H,d0 pol , 63 JH,dpol , 63 H,dpol , 63 JHp ,dpolp , 309 Hp ,dpolp , 309 H,pol , 63 (d) JH,pol , 63 JH0 ,dpol0 , 64 j \ , 70, 315, 316 jQ , 43, 101, 390, 397, 486 •

Ka (Ω~Nord,tor /~S0,r ), 470 H KQ,H , 53, 97 ord KQ,H , 374, 392 Kord,+ Q,H , 374, 392 Kord,++ , 374, 392 Q,H K+ , 53, 97 Q,H ++ KQ,H , 53, 97 KQ,H,Σ , 55, 97 Kord Q,H,Σord , 375, 392 Kord,+ , 375, 392 Q,H,Σord Kord,++ , 375, 392 Q,H,Σord K+ Q,H,Σ , 55, 97 K++ Q,H,Σ , 55, 97 e KQ,H , 46 e ord K Q,H , 373 e ord,+ , 373 K Q,H e ord,++ , 373 K Q,H e + , 46 K Q,H e ++ , 46, 85 K Q,H Kstd Q,H,Σ , 376 Kstd,+ Q,H,Σ , 376 Kstd,++ Q,H,Σ , 376 Kstd Q,H , 376 e std K Q,H , 376 Kstd,+ Q,H , 376 e std,+ , 376 K Q,H Kstd,++ , 376 Q,H std,++ e KQ,H , 376

κ, 53, 97, 108, 116, 374, 393, 420 κaux , 425 κ eaux , 425 κisog , 97, 116, 393, 420

[e κ], 53, 108, 116, 374, 420 κ e, 46, 85, 373 [e κaux ], 425 κ eisog , 107, 408 κ etor , 107, 409 κtor , 98, 116, 393, 420 Kodaira–Spencer class extended, 471 isomorphism, 26, 207, 208 extended, 121, 273, 486 morphism, 25, 207, 208, 274–276, 487 extended, 59, 121, 272, 277, 281, 282, 284, 285, 287, 296, 431, 486, 487 Koecher’s principle, 492 Koszul filtration, 470 KS, 57, 121, 206, 272 KS, 207 KS(A,λ,i,αH )/MH , 26 KS(A,λ,i)/MH , 26 KSA/MH /S0 , 25, 26 KSA/S/~S0,r , 207 H KS(B,c)/C~ ord /~ S0,r , 274 H

ΦH ,δH

KS(B,c,c∨ ,λ\ ,i\ )/C~ ord / Φ1 ,δ1
KS(B,λB ,iB )/C~ ord free , 275 Φ1 ,δ1

KS(B,c,c∨ ,λ\ ,i\ )/C~ ord

, 274, 275

KS(B,c,c∨ ,λ\ ,i\ )/C~ ord

,free ,

ΦH ,δH ΦH ,δH

274, 275

/

KS(B,c,c∨ ,λ\ ,i\ )/C~ ord

ΦH ,δH

KS(B,λB ,iB )/C~ ord




ΦH ,δH

KS(B ∨ ,c∨ )/C~ ord

ΦH ,δH

free

/~ S0,rH ,

KS(B,λB ,iB )/C~ ord

, 275

KS(B,λ

, 275

, 275

274

ΦH ,δH

~ ord,ΦH B ,iB )/MH

KS(B,λ

~ ord,ΦH ,free B ,iB )/MH

KSB/M ~ ord,ΦH /~ S H

KS

ord

~ B/Ξ Φ

H ,δH

, 274

, 274 0,rH

/~ S0,rH

, 274

KSfree , 206, 273, 486 KS(G,λ,i)/~Nord,tor , 431 KS( ♦ G, ♦ λ, ♦ i)/S,free , 281, 282, 284 KS ♦ G/S/~S0,r , 281, 282, 284 H KS(G,λ,i,αHp ,αord )/S , 206 Hp

KS(G,λ,i,αHp ,αord )/S,free , 206 Hp

KS(G,λ,i,αH )/S , 57 KS(G,λ,i)/M ~ ord,tor ,free , 296 H

June 16, 2017 8:34

ws-book961x669

Compactifications with Ordinary Loci

10374-main

Index

KS(G,λ,i)/M ~ ord,tor ,free , 431 H,rκ

KS(G,λ,i)/Mtor , 59 H KS(G,λ,i)/S , 57, 206 KS(G,λ,i)/S,free , 206, 286, 287 , 296, 486 KSG/M ~ ord,tor /~ S0,rH H , 431 KSG/M ~ ord,tor /~ S 0,rκ

H,rκ

KSG/Mtor /S0 , 59, 121 H KS \ ~ ord , 272 ~ (G ,ι)/ΞΦ ,δ /S0,rH H H

KS(G\ ,ι)/Ξ ~ ord KS KS

ΦH ,δH

/~ S0,rH ,

ord

~ (G\ ,λ\ ,i\ )/Ξ Φ

H ,δH

ord

~ (G\ ,λ\ ,i\ )/Ξ Φ

,free

276 , 273

, 272

H ,δH

KSG/S/~S0,r , 286, 287 H KS e e e ~e ord,tor , 431 (G,λ,i)/MH f

,free

KS(G, ee e λ, i)~ /~ Nord,tor , 431 Nord,tor , 431 KS e ~e ord,tor ~ G/MH f

/S0,rκ

KS(T,λT ,iT )/~S0,r ,free , 276 H KS , 276 ord ~ (T,λT ,iT )/ΞΦ

H ,δH

,free

,free , Φ1 ,δ1

KS(T,λT ,iT )/Ξ ~ ord KS(T,λT ,iT )/Ξ ~ ord

ΦH ,δH

276

,free , 276

KS(Te,λ e ,i e )~ ord,tor /~Nord,tor , 431 T T N KS(Te,λ e ,i e )~ /~ S0,r , 432 T

T S0,r κ

κ

Kuga family, 4, 95 generalized, 4, 96 ordinary, see ordinary Kuga family toroidal compactification, 4, 97–106 Kunz’s theorem, 212, 305, 359 [L], 473 L, 22 L, 472 L0 , 117 Laux , 125 (Laux , h · , · iaux ), 128, 198 L# aux , 126, 198 ˘ 88 `, L(g) , 39 [Lgrp ], 473 Lgrp , 473 (L/n0 L, h · , · i), 172, 173 (L/pr L, h · , · i, Dpr ), 178, 179 (L, h · , · i), 22 (L, h · , · i, h0 ), 22 L# , 22 e 43 L, `ejQ , 474

page 549

549

e h · , · ie, e (L, h0 ), 43, 85, 371 e h · , · ie, e (L, h0 , e Z), 48 ∞,p (L ⊗ A , h · , · i, Z ⊗ A∞,p , Φ), 316 ˆ Z

Z

(L ⊗ A∞ , h · , · i, Z ⊗ A∞ , Φ), 70, 76 Z

ˆ Z

(LZ , h · , · iZ , hZ0 ), 30, 66, 311 (LZH , h · , · iZH , hZ0H ), 32 (LZn , h · , · iZn , hZ0n ), 30 ˆ p , h · , · i), 173, 174 (L ⊗ Z Z

ˆ p , h · , · i, Z ⊗ Z ˆ p ), 313, 314, 324, 325 (L ⊗ Z Z

ˆ Z

ˆ h · , · i, Z), 69, 79, 80 (L ⊗ Z, Z

(L ⊗ Zp , h · , · i, D), 179, 180 Z

(L ⊗ Zp , h · , · i, Z ⊗ Zp , D), 314, 315 Z

ˆ Z

Lagrange, 126 λ, 23, 57, 58, 79, 197, 205, 215, 295, 324 λ[0] , 291 ~λ, 141 ~λaux , 141 λaux , 333 λO aux , 126, 129, 137, 199, 333 λM aux , 126, 137, 199 λB , 66, 70, 138, 216, 248, 281, 313, 315, 454 λBaux , 138 λB ‡ , 59, 112, 292, 297, 415 λord B,η,pr , 226 λBb ‡ , 110, 413 λBe ‡ , 110, 413 λ, 422 λ† , 59, 145, 296 λ‡ , 59, 112, 291, 296, 415 ♦ λ, 280 λM ~ ord , 295 H b 108, 411 λ, b‡ , 109, 412 λ b~ ord , 411 λ Nκ e bN , 109 λ κ e b\,‡ , 111 λ ♥ λ, 59, 270, 271, 296 λL , 473 λext L , 474 λLgrp , 473 λM ~ ord ,jQ , 390, 397, 474 H λM ~ ord ,jQ ,Z , 390 H λMH , 58 λMH ,jQ , 96, 101

June 16, 2017 8:34

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Compactifications with Ordinary Loci

550

page 550

Compactifications with Ordinary Loci

λMH ,jQ ,Z , 96 λ~Nord ,jQ , 397, 474 λN,jQ , 101 λ\ , 68, 70, 272, 313, 315 λ(p) , 190 i λ(p ) , 181 , 474 λ ~e ord e

Lie∨ (Vaux ), 362 LieG∨ /S , 57 Lie∨ G∨ /S , 57 LieG/S , 57 Lie∨ G/S , 57 Lie~Nord,∨ /M ~ ord , 396

λ∗ , 57 λT , 68, 70, 313, 315 λ∗T , 70 e 108, 411 λ, e‡ , 109, 412 λ e\ , 108, 411 λ e\,‡ , 110 λ λ-Weil pairing, 173 λ\ ord , 319

Lie~Nord,ext,∨ /M ~ ord,tor , 396

ΨΦ e

10374-main

e f,δH f H

κ κ

H

κ

H

Lie~∨ ord,ext,∨ ~ ord,tor , 396 /M N κ

H

Lie~Nord,ext /M ~ ord,tor , 396 κ

H

Lie~∨ ord,ext ~ ord,tor , 396 N /M κ

H

Lie~Nord /M ~ ord , 396 κ

H

Lie~∨ ~ ord , 396 Nord κ /MH LieN∨ /M ~ ord , 391 H

~ Z λ\Z ,

73 level structure comparison with characteristic zero, 192–194 naive, see naive level structure ordinary, see ordinary level structure symplectic-liftability condition, 173 of type (L ⊗ A∞,p , h · , · i, Z ⊗ A∞,p , Φ), ˆ Z

Z

316 based at s¯, 316 of type (L ⊗ A∞ , h · , · i, Z ⊗ A∞ , Φ), 70 ˆ Z

Z

based at s¯, 70 ˆ p , h · , · i), 174 of type (L ⊗ Z Z

lifting, 174 principal, 173 ˆ p , h · , · i, Z ⊗ Z ˆ p ), 314, 325 of type (L ⊗ Z Z

H

Lie~∨ ord,∨ ~ ord , 396 N /M

(`jQ )

ˆ Z

principal, 313, 324 ˆ h · , · i), 23 of type (L ⊗ Z, Z

ˆ h · , · i, Z), 69, 80 of type (L ⊗ Z, Z

principal, 69, 79 level-H structure datum ˆ h · , · i), 243 of type (L ⊗ Z, Z

Lie–Kolchin theorem, 506 LieA∨ /MH , 24 Lie∨ A∨ /MH , 24 LieA/MH , 24 Lie∨ A/MH , 24 Lie algebra condition, 6, 23, 131, 141, 164, 203, 510

Lie∨ ~ ord , 391 N∨ /M H LieN∨ /MH , 96 Lie∨ N∨ /MH , 96 LieNgrp,∨ /M ~ ord , 391 H

Lie∨ ~ ord , 391 Ngrp,∨ /M H LieNgrp,∨ /MH , 96 Lie∨ Ngrp,∨ /MH , 96 LieNgrp /M ~ ord , 391 H

Lie∨ ~ ord , 391 Ngrp /M H LieNgrp /MH , 96 Lie∨ Ngrp /MH , 96 LieN/M ~ ord , 391 H

Lie∨ ~ ord , 391 N/M H LieN/MH , 96 Lie∨ N/MH , 96 liftability condition for βn , 79 for βn0 , 324 for βn\ 0 , 313 for βn\ , 69 , 314 for βp\,ord r for (cn , c∨ n ), 68 for τn , 78 local model, 15, 125 log de Rham cohomology, 100, 397, 469 log de Rham complex, 100, 397 log ´etale, 113, 416 log smooth, 99, 395 M , 445, 460, 498, 504 M0 , 118 M0 (R), 118

June 16, 2017 8:34

ws-book961x669

Compactifications with Ordinary Loci

10374-main

Index

~ ord,Φ1 , 262 M 1 ~ ord,Z1 , 262 M 1 ~ ord,tor M ord , 335 ˆ G (Z),Σ aux

aux

~ H ], 210 [M ~ [MH ], 5, 141 ~ H , 210 M ~ MH , 5, 9, 141, 336 ~ Haux , 336 M ~ tor M Haux ,d0,aux polaux ,r , 334 tor,full-ord ~ MHaux ,d0,aux polaux ,r , 366 ~ min M Haux , 336 ~ ord M Haux , 213 ~ Mord,min Haux , 334 ~ min M Haux ,r , 334 ~ ord,tor ord , 333 M Haux ,Σaux ~ tor M H,d0 pol,i , 5, 151 ~ tor M H,d0 pol,rH , 366 tor ~ MH,d0 pol,rH , 361 ~ tor,full-ord , 366 M H,d0 pol,rH ~ tor M H,d0 pol , 5, 9, 147 ~ tor,non-ord , 365 M H,d0 pol non-ord ~ tor , (M H,d0 pol ⊗ Fp ) Z ord ~H M ,rH0 , 311 G

551

normality, 338 ~ ord,min )∧ (M x ¯ , 343, 494 H ~ ord,ΦH , 260, 265, 295, 311, 347, 379 M H ~ ord,tor , 294, 295, see also M ~ ord,tor M ord H

H,Σ

~ ord,tor )∧ (M x ¯ , 349 H ~ ord,ZH ], 343, 346 [M H ~ ord,ZH , 259, 265, 295, 311, 346 M H ~ ord,ZH ])∧ ([M x ¯ , 343 H ~ ord,ZH )∧ (M , x ¯ 494, 504 H tor ~ MH0 ,d0 pol0 , 148 ~ H,r , 214 M H ~ min M H,rH , 365 ~ min M H,rH , 347 ~ Mmin,full-ord ⊗ (Z/pj Z), 367 H,rH

Z

~ min,full-ord , 365 M H,rH j full-ord ~ min (M , 367 H,rH ⊗ (Z/p Z)) Z

~ H,r ⊗ Fp )full-ord , 368 (M H Z

~ ord M H,rκ , 393 ~ ord,min , 349 M H,S

365

h,Z

~ H,i , 5, 151 M ~ min M H,i , 5, 151 ~ min,full-ord , 367 M H,i full-ord ~ min , 367 (M H,i ⊗ Fp ) Z non-ord ~ min , 367 (M H,i ⊗ Fp ) Z j full-ord ~ min (M , 367 H,i ⊗ (Z/p Z)) Z

~ ord M Hκ , 407 min ~ MH , 5, 143, 336 ~ min,non-ord , 365 M H non-ord ~ min , 365 (M H ⊗ Fp ) Z

~ H ]non-ord , 365 [M ~ Mnon-ord , 365 H ~ ord [M H ], 7, 10, 214, 338, 345 quasi-projectivity, 10 ~ ord M H , 7, 10, 203, 213 ~ ord,1 , 343, 348 M H ~ ord,min , 357, 365, 367 M H ~ ord,min , 5, 7, 11, 309, 332, 345 M H flatness, 338

page 551

~ ord,tor [M ], 338 H,Σord ~ ord,tor M , ord 303, 366 H,Σ

~ ord,tor M , 7, 11, 14, 289, 294–295, 309 H,Σord ~ ([MH ⊗ Fp ])non-ord , 365 Z

~ H ⊗ Fp )non-ord , 365 (M Z

~ min M H , 9 ~ ord M n0 pr , 204 ~ ord M n , 204 n ~ ord,Φ , 311, 313 M n ord,Zn ~ Mn , 311, 313, 380 Mord D,0,fil , 505 Mord D,0 , 479 Mord D,0 (R), 479 Mord D,0,uni , 506 Mord D , 170 Mord D (R), 169 b ord (Mord nU D D )(R), 377 ...ord MHaux , 199 ...ord MHh , 244, 255 ...ord MH , 6, 10, 197 ...ord,ΦH , 255 MH ...ord MH (S), 197 ...ord,ZH , 255 MH

June 16, 2017 8:34

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552

Compactifications with Ordinary Loci

Compactifications with Ordinary Loci

...ord Mn0 pr , 198 ...ord Mn , 198 ...ord,Zn , 248 Mn [MGaux (Zˆp ) ], 141 MGaux (Zˆp ) , 129, 141, 335 Mmin ˆp ) , 143 Gaux (Z non-ord (Mmin , 365 ˆ p ) ⊗ Fp ) Gaux (Z Z

Mtor ˆp ),Σp , 335 Gaux (Z aux

Mtor ˆp ) , 361 Gaux (Z non-ord (Mtor , 365 ˆ p ) ⊗ Fp ) Gaux (Z Z

[MH ], 3, 26, 61 MH , 3, 8, 23, 57 [MHaux ], 129 MHaux , 129 Mmin Haux , 139 [MHpaux ], 129 MHpaux , 9, 129 MHpaux Gaux (Zp ) , 129 p Mmin Haux , 9, 139 p p Mtor Haux ,Σaux , 9, 136 tor MHaux ,Σaux , 136 Mtor H,d0 pol , 64 MHGh,Z , 66 MHGh,Z ,Φ , 66 MHh , 32, 66, 244 ~ ord M Hh , 274, 293, 311 MH0h , 32 MH00h , 32 ~ ord M H ,r , 274, 293, 311 h

H

MH,i , 151 Mmin H,i , 151 MHκ , 97, 106 Mmin H , 3, 8, 61 Mord H , 6, 10, 203 H Mord,Φ , 259 H ord,ZH MH , 259 [MHp ], 152 MHp , 152, 212, 305 Mmin Hp , 152, 305 Mord Hp , 212 Mord,tor Hp ,Σp , 306 Mtor Hp ,Σp , 305 Z p [MHHp ], 308 ΦH MH , 32, 58, 62, 66, 259 Mtor H0 ,d0 pol0 , 64 MH,rH , 203

Mmin H,rH , 347 H MZH,r , 259 H MH,rν , 203 Mtor H,Σ,i , 151 Mtor H,Σ,rH , 297 [Mtor H,Σ ], 124 Mtor H,Σ , 3, 8, 58 Mtor H , 58 [MZHH ], 62 MZHH , 32, 58, 62, 66, 259 ˘ c ~b ord,ΦH M , 380, 394 b H ˘ c ~b ord,ZH , 380, 394 MHb b ord M D (R), 377 ˘ c b ΦH M , 86, 98 b H ˘

c b ZH M b , 86, 98 H Φn Mn , 68 MZnn , 30, 66, 68 MPE,O (R, I), 217 MPEL,... (R, I), 242, 247 M ord H (R, I), 235 MPEL,... M ord n MPEL,Mord (R,I) , 247 H MPEL,MH (R,I) , 247 + M , 499 ~e tor , 426 M f e de pol,r H, 0

κ

~e ord M e , 372, 379, 407 H ˘ f ~e ord,ΦH MHe , 379 e f ~e ord,ΦH MHe , 407 ~e ord,tor MHe , 379, 407 ord,˘ ZH f ~e M , 380 e H ef ~e ord,ZH MHe , 407 ~e ord,tor MH, e Σ e ord , 379, 407 ˘ ~e ord,Zn Mn , 380 e eord (R), 377 M D ...ord,Φ˘ H f e He M , 418 ˜ (m) , 25 M H e e , 46, 85 M H e e p , 426 M H e tor M e p ,Σ e p , 426 H ˘

f e ΦH M e , 85, 86 H e tor M e e , 85, 106

H,Σ

10374-main

page 552

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10374-main

Index

e tor M e , 85, 106 H e ˘Znn , 86 M

553 ˘

`,0+ b e σ˘ ,τ , 458 N ˘ `,+

Mess,ord Zpr ,Dpr , 238 maximal point, 266, 267, 466 maximal totally isotropic submodule, 168 minimal compactification of MH , 3, 61–62 mixed Shimura varieties, 14, 16 module of universal finite differentials, 286 µp∞ ,S , 156 µp∞ ,s , 164 mult , 156, 161, 162, 164 multi-rank, 28, 117 Mumford family, 59, 82, 272, 296, 449 Mumford’s construction, 217

b e σ˘ ,τ , 458 N e N` , 497 e `,+ , 497 N e σ˘ ,τ , 454 N e `σ˘˘ ,τ , 458 N ˘ e `,0+ N , 458

N, 96, 390 N , 498, 504, 507 N, 496 N0 , 331, 345 n0 , 172 N1 , 139 N a , 504 ~ ord,tor N κaux , 425 ~ ord N κ , 401 ~ ord N κ , 393, 420 ~ ord,ext N , 396 κ ~ Nord,grp,∨ , 396 κ ~ ord,grp N , 393, 420 κ ~ ord,tor N , 401 κ ~ ord,tor N , 420 κ ~ ord N κ e , 408 ~ ord,grp , 408 N κ e ~ ord,tor , 409 N κ e N∨ , 96, 391 Next , 392 Ngrp , 95, 96, 390 Ngrp,∨ , 96, 391 Nκ , 97, 116 Ngrp κ , 97, 116 Nκe , 107 Ntor κ e , 107 Ngrp κ e , 107 Ntor κ , 98, 116 Nσ˘ ,τ , 454 e 496 N, b e σ˘ ,τ , 458 N

principal, 172 naive ordinary level structure of type (L/pr L, h · , · i, Dpr ), 179 principal, 178 of type φ0D−1,pr : Gr0D−1,pr → Gr0D#

`˘

b e σ˘ ,τ , 458 N

page 553

σ ˘ ,τ

˘ e `,+ N σ ˘ ,τ , 458 (N )∧ x ¯ , 504, 507 ∇, 25, 97, 101, 121, 391, 398, 470, 485 naive level structure of type (GrZ−1,n0 , h · , · i11,n0 ) principal, 231 ˆ h · , · i), 173 of type (L ⊗ Z, Z

−1,pr

principal, 221, 232 symplectic condition, 221 of type φ0D,pr : Gr0D,pr → Gr0D# ,pr principal, 178 naive ordinary level-n structure datum of type (L/nL, h · , · i, Dpr ), 233 orbit of ´etale-locally-defined, 240 naive ordinary pre-level-n structure datum of type (L/nL, h · , · i, Dpr ), 231 symplectic condition, 233 NBl · ( · ), 64 ~ min NBlJ~H,d pol (M H ), 147 0 ~ ord,min ), 360 NBl ~ (M JH,d polord 0

H

NBlJ~

H ord (~ H ), 360 ord

NBlJ~

~ min (M H0 ), 148

H,d0 pol

H0 ,d0 pol0

NBlJH,d0 pol (Mmin ), 63 H H NBlJH,d0 pol ( H ), 63 NBlJHp ,d0 polp (Mmin p ), 309 H H NBlJHp ,d0 polp ( Hp ), 309 NBlJH0 ,d pol0 (Mmin H0 ), 64 0 neat, 3, 24 N´eron model, 127, 145, 161, 202 N´eron–Severi group, 472 nerve, 454, 496, 497 spectral sequence, 461, 492

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554

Compactifications with Ordinary Loci

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page 554

Compactifications with Ordinary Loci

non-ord

, 365 nondegenerate cone, 34 nonemptiness of ordinary locus, 2, 13, 17, 18, 204, 369–370 nonordinary locus, 365 normal morphism, 350 ~ ord,grp /M ~ ord NS(N H ), 472 ord ord ~ /M ~ H ), 472 NS(N ν, 22, 170 ν−1,n0 , 231 ord r , 221, 232, 239 ν−1,p ord ν−1,p r ,S , 222 ν(αHn0 ), 173 ord ν(αH ), 180 p ord ν(αH ord ), 179 r p

ν(αn0 ), 173 r ), 178 ν(αpord ν(g), 22, 170 ν(Hn0 ), 173 ord , 180 νH p ord , 179 νH ord r p

νˆ, 165 νˆ−1,¯s , 71 p νˆ−1,¯ s , 317 νˆs¯\ , 71 νˆs¯ord , 182, 193, 199 νˆs¯p , 174 νˆs¯p,\ , 316 νn , 78, 79 νn0 , 173, 324 νn\ 0 , 313 νpr , 193 νp\,ord , 314 r r , 178, 179, 193, 206, 217, 229, 230 νpord νpord,O , 200 r ord νpr ,S , 206, 221 νe, 371, 374 O, 22 O(1), 61, 337, 345 Oaux , 128 0 Oaux , 128 (Oaux , ?aux , Laux , h · , · iaux , h0,aux ), 9, 128, 198 O-lattice, 22 O-multi-rank, 28, 62, 346 O0 , 27, 43, 169, 371, 478, 484 (O, ?, L, h · , · i, h0 ), 2, 21, 22

e h · , · ie, e (O, ?, L, h0 ), 44 OUσ , 493–495 OU+σ , 494–495 OUτ˘ , 448, 450 OUτ˘ , 448, 450 OU0+ , 448, 450 τ ˘ OU+τ˘ , 449–450 OU++ , 449–450 τ ˘ OX0+ , 445 OX+ , 445 OX+ , 494 OX++ , 445 i, 23 ωAgaux ,daux ,n , 127 ωAg,d,n,n , 127 ωB~ , 339 1 ΩM ~ ord,tor /~ S0,r , 469 H

H

1

Ω~Nord,tor /M ~ ord,tor , 396 H

•

Ω~Nord,tor /M ~ ord,tor , 397 H

1

Ω~Nord,tor /~S0,r , 469 H

1

, 100 ΩNtor /Mtor H •

, 100 ΩNtor /Mtor H 1

Ω~b ord ΞΦ ˘ 1

ord ~ /Ξ ˘ ,Σ b ΦH ,δH ,ΣΦ ,rκ ˘ c,δH c Φ H H c H

, 439

Ω~b ord ΞΦ ˘

ord ~ × /Ξ ˘ ,Σ b ΦH ,δH ,ΣΦ ˘ c,δH c Φ H H C ~ ord c H ΦH ,δH

439 1 ΩΞb 1

/ΞΦ ,δ ,Σ ˘ ,Σ ˘ ,δ b Φ ΦH H H ˘ c H c Φ H c H

ΩΞb

0,rH

, 207, 281

b1 ~ Ω [d log ∞], S/S0,rH ♥

281

ω, 339 ωM ~ H , 141, 149, 511 ωM ~ ord , 213, 214 Haux ωM , 333, 336 ~ ord,tor Haux ,Σord aux

ωM ~ tor

H,d0 pol

, 147

ωM ~ min , 143, 149, 336, 337, 517 H ωM ~ ord , 213, 214, 345, 483 H ωM ~ ord,min , 337, 345, 347, 348 H ωM ~ ord,tor , 346, see also ωM ~ ord,tor H

H,Σord

Ω1M ~ ord,tor /~ S H

0,rH

,

˘ c,δH c H

, 94

× /(ΞΦ ,δ ,Σ ˘ ,Σ ˘ ,δ b Φ ΦH C H H ˘ c H c Φ H ΦH ,δH c H

94 b1 ~ Ω S/S

ord

~b C ˘ Φ

[d log ∞], 296, 396

b˘ C Φ

˘ c,δH c H

)

,

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Index

ωM ~ min , 347 H,rH

ωM ~ ord,tor , 11, 306, 331, 333, 336, 337, 429 H,Σord

ωMtor

p Gaux (ˆ Zp ),Σaux

, 336

ωMH , 61, 120, 139 ωMHaux , 139 ωMmin , 140 Haux ωMHp , 139 aux ωMmin , 139, 140, 363 p Haux

ωMtorp

p Haux ,Σaux Mtorp Haux Mtor Haux ,Σaux Mmin H Mtor Hp ,Σp

ω

, 139

, 363, see also ωMtorp

ω , 138 ω , 61, 140 ω , 306 ωMmin , 347

p Haux ,Σaux

Z

H,Σ

0 ωM tor , 139 H,Σ

ωMtor , 61, 123, see also ωMtor

H,Σ

Ω1Mtor /S0 [d log ∞], 59, 100 H ω~Nord,tor , 429 κ

0,rH

[d log ∞], 396

Ω1Ntor /S0 [d log ∞], 100 e 1R/O , 286 Ω F ,(p) [ζprH ] 0

e1 ~ Ω S/S e1 ~ Ω S/S

0,rH 0,rH

by P0Z (A∞,p ) × Pord,0 Z,D (Qp ), 328 by power of p, 184, 190 by Up operator, 184, 190, 304 generalized, 184 ordinary Kuga family, 8, 13, 390 extended, 392 generalized, 8, 13, 390 partial toroidal compactification, 8, 13, 14, 392–407 ordinary level structure, 6, 10, 177–183 naive, see naive ordinary level structure symplectic-liftability condition, 179, 180 of type (L ⊗ Zp , h · , · i, D), 180 Z

H,rH

Ω~1Nord,tor /~S

555

lifting, 182 principal, 179 of type (L ⊗ Zp , h · , · i, Z ⊗ Zp , D), 315

ωMtor , 138

H

page 555

, 286 [d log ∞], 286

openness of versality, 287–288 ord, 376 ordinary abelian scheme, 10, 155, 180, 198, 206, 267, 362 ordinary cusp, 7, 347 ordinary cusp label, 172, 373 ordinary good algebraic model, 286–287 ordinary good formal model, 283–284 ordinary Hecke twist, 183, 184 by G1,Z (A∞,p ) × Pord 1,Z,D (Qp ), 321 by G(A∞,p ) × Pord D (Qp ), 190–192, 300, 301, 399 by G(A∞,p ) × Pord D (Qp ), 399 by Gh,Z (A∞,p ) × Pord h,Z,D (Qp ), 311 by GZ (A∞,p ) × Pord DZ (Qp ), 311 by Pord D (Qp ), 183–190 by Pord D (Zp ), 183, 190 by PZ (A∞,p ) × Pord Z,D (Qp ), 328 ∞ by (PZ (A∞,p ) × Pord Z,D (Qp ))/U2,Z (A ), 322

ˆ Z

principal, 314 ordinary level-H structure datum ˆ h · , · i, D), 241 of type (L ⊗ Z, Z

ordinary level-n structure datum of type (L/nL, h · , · i, Dpr ) naive, see naive ordinary level-n structure datum ˆ h · , · i, D), 233 of type (L ⊗ Z, Z

orbit of ´etale-locally-defined, 240 ordinary locus, v, 2, 6–8, 10, 203–205 full, 14, 365 nonemptiness, 2, 13, 17, 18, 204, 369–370 ordinary pre-level-n structure datum of type (L/nL, h · , · i, Dpr ) naive, see naive ordinary pre-level-n structure datum ˆ h · , · i, D), 233 of type (L ⊗ Z, Z

ordinary semi-abelian scheme, 155, 240, 362 [p−r0 gp,−1 ]ord , 184, 191 [p−r0 gp,0 ]ord , 185 −1 [p−r0 gp,#,0 ]ord , 186 P0 , 118 P0 (R), 118 Pord 1,Z,D (R), 172, 378 PA , 24, 164 ~pord ΦH ,δH , 339, 495 PB , 77 Pord D,0 , 479 Pord D,0 (R), 479

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556

Compactifications with Ordinary Loci

Compactifications with Ordinary Loci

Pord D , 6, 170 Pord D (R), 169 Pess Dpr , 237 Pord DZ , 311 Pord DZ (R), 311 p-divisible group attached to abelian variety, 164 connected part, 155, 164 Dieudonn´e–Manin classification, 156 ´etale, 156 ´etale part, 164 multiplicative-type, 156 multiplicative-type part, 164 Pord,0 h,Z,D (R), 172, 378 Pord h,Z,D (R), 172, 378 (Pord h,Z,D n U1,Z )(R), 172 b ord P 1,˘ Z,D (R), 378 ord b PD (R), 377 b ord,0 (R), 378 P

PΦ e , 48 f

b ord˘ (R), 378 P h,Z,D b 0˘ , 459 P

eeord P (R), 377 Z,e D ee0 (R), 45, 377 P Z ee(R), 45, 377 P Z Pord,0 Z,D (R), 172, 378 Pord Z,D (R), 172, 378 Pess Zn , 235 Pess Zpr ,Dpr , 237

h,˘ Z,D

`