The Gross-Zagier Formula on Shimura Curves: (AMS-184) [Course Book ed.] 9781400845644

Table of contents :
Contents
Preface
Chapter One. Introduction and Statement of Main Results
Chapter Two. Weil Representation and Waldspurger Formula
Chapter Three. Mordell-Weil Groups and Generating Series
Chapter Four. Trace of the Generating Series
Chapter Five. Assumptions on the Schwartz Function
Chapter Six. Derivative of the Analytic Kernel
Chapter Seven. Decomposition of the Geometric Kernel
Chapter Eight. Local Heights of CM Points
Bibliography
Index

Citation preview

Annals of Mathematics Studies Number 184

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The Gross–Zagier Formula on Shimura Curves

Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2013

c 2013 by Princeton University Press Copyright Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data Yuan, Xinyi, 1981The Gross–Zagier formula on Shimura curves / Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang. p. cm. – (Annals of mathematics studies ; no. 184) Includes bibliographical references and index. ISBN 978-0-691-15591-3 (hardcover : alk. paper) – ISBN 978-0-691-15592-0 (pbk. : alk. paper) 1. Shimura varieties. 2. Arithmetical algebraic geometry. 3. Automorphic forms. 4. Quaternions. I. Zhang, Shou-Wu. II. Zhang, Wei, 1981- III. Title. QA242.5.Y83 2013 516.3’52–dc23 2012010981 British Library Cataloging-in-Publication Data is available This book has been composed in LATEX. The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper ∞ Printed in the United States of America

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Contents

Preface

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1 Introduction and Statement of Main Results 1.1 Gross–Zagier formula on modular curves . . . 1.2 Shimura curves and abelian varieties . . . . . 1.3 CM points and Gross–Zagier formula . . . . . 1.4 Waldspurger formula . . . . . . . . . . . . . . 1.5 Plan of the proof . . . . . . . . . . . . . . . . 1.6 Notation and terminology . . . . . . . . . . .

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1 1 2 6 9 12 20

2 Weil Representation and Waldspurger Formula 2.1 Weil representation . . . . . . . . . . . . . . . . . 2.2 Shimizu lifting . . . . . . . . . . . . . . . . . . . 2.3 Integral representations of the L-function . . . . 2.4 Proof of Waldspurger formula . . . . . . . . . . . 2.5 Incoherent Eisenstein series . . . . . . . . . . . .

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3 Mordell–Weil Groups and Generating Series 3.1 Basics on Shimura curves . . . . . . . . . . . . . 3.2 Abelian varieties parametrized by Shimura curves 3.3 Main theorem in terms of projectors . . . . . . . 3.4 The generating series . . . . . . . . . . . . . . . . 3.5 Geometric kernel . . . . . . . . . . . . . . . . . . 3.6 Analytic kernel and kernel identity . . . . . . . .

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106 106 110 117 122 128 138 153

4 Trace of the Generating Series 4.1 Discrete series at infinite places . . . 4.2 Modularity of the generating series . 4.3 Degree of the generating series . . . 4.4 The trace identity . . . . . . . . . . 4.5 Pull-back formula: compact case . . 4.6 Pull-back formula: non-compact case 4.7 Interpretation: non-compact case . .

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CONTENTS

5 Assumptions on the Schwartz Function 5.1 Restating the kernel identity . . . . . . 5.2 The assumptions and basic properties . 5.3 Degenerate Schwartz functions I . . . . 5.4 Degenerate Schwartz functions II . . . .

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171 171 174 178 181

6 Derivative of the Analytic Kernel 6.1 Decomposition of the derivative . 6.2 Non-archimedean components . . 6.3 Archimedean components . . . . 6.4 Holomorphic projection . . . . . 6.5 Holomorphic kernel function . . .

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184 184 191 196 197 202

7 Decomposition of the Geometric Kernel 7.1 N´eron–Tate height . . . . . . . . . . . . . . . . . . 7.2 Decomposition of the height series . . . . . . . . . 7.3 Vanishing of the contribution of the Hodge classes 7.4 The goal of the next chapter . . . . . . . . . . . . .

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206 207 216 219 223

8 Local Heights of CM Points 8.1 Archimedean case . . . . . . 8.2 Supersingular case . . . . . 8.3 Superspecial case . . . . . . 8.4 Ordinary case . . . . . . . . 8.5 The j -part . . . . . . . . . .

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Bibliography

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Index

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Preface

The aim of this book is to prove a complete Gross–Zagier formula on quaternionic Shimura curves over totally real fields. The original formula proved by Benedict Gross and Don Zagier in 1983 relates the N´eron–Tate heights of Heegner points on X0 (N ) to the central derivatives of some Rankin–Selberg L-functions under the Heegner condition, which is an assumption of mild ramification. Since then, some generalizations were given in various works by ShouWu Zhang. The proofs of Gross–Zagier and Shou-Wu Zhang depend on some newform theories. There are essential difficulties to removing all ramification assumptions by these methods. This book is a completion of those generalizations in which all ramification restrictions are removed. The Gross–Zagier formula in this book is an analogue of the central value formula of Jean-Loup Waldspurger, and has been speculated by Benedict Gross in terms of representation theory in a lecture at MSRI in 2002. In fact, the Waldspurger formula concerns periods of automorphic forms on quaternion algebras over number fields, while the Gross–Zagier formula may be viewed as a formula of periods of “automorphic forms” on incoherent quaternion algebras. These incoherent automorphic forms are functions on Shimura curves with values in some abelian varieties. Besides many ideas of Gross–Zagier and Shou-Wu Zhang, one main new ingredient of this book is to construct the analytic kernel and the geometric kernel systematically using Weil representations and the generating series of Hecke correspondences of Stephen S. Kudla constructed in 1997, though we do not use his program on the arithmetic Siegel–Weil formula. The construction is inspired by the Waldspurger formula mentioned above. To simplify many computations in both automorphic forms and arithmetic geometry, we take advantage of representation theory and make use of the concepts of degenerate Schwartz functions, coherence of pseudo-theta series, and modularity of generating functions. Acknowledgments. The authors are extremely grateful to Benedict Gross for his MSRI lecture, and for his constant help and support. This MSRI lecture is the main motivation of this book, and many new ideas in this book are derived from discussions with him. The book would have been impossible without the generating series of Hecke operators constructed by Stephen S. Kudla in 1997. We thank him for explaining his work to us during a joint FRG project supported by the NSF. We would also like to acknowledge our debt to the work of Jean-

viii

PREFACE

Loup Waldspurger. We have directly adapted his strategy of proving special value formula to our incoherent situation. The authors are also indebted to many important discussions with and crucial comments of Pierre Deligne, Dorian Goldfeld, Ming-Lun Hsieh, Herv´e Jacquet, Dihua Jiang, Jian-Shu Li, Yifeng Liu, Peter Sarnak, Richard Taylor, Ye Tian, Andrew Wiles, and Tonghai Yang. The authors would like to thank the Morningside Center of Mathematics at the Chinese Academy of Sciences for its hospitality and constant support. Xinyi Yuan was supported by a research fellowship from the Clay Mathematics Institute. Shou-Wu Zhang is very grateful to the Institute for Advanced Studies at Tsinghua University and the Institute for Advanced Studies at the Hong Kong University of Science and Technology for their hospitality and support during the major revision of this book. He was also partially supported by a Guggenheim Fellowship and NSF grants DMS-0354436, DMS-0700322, DMS0970100, and DMS-1065839. Wei Zhang is partially supported by NSF Grant DMS-1001631.

The Gross–Zagier Formula on Shimura Curves

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Chapter One Introduction and Statement of Main Results In this chapter, we will state the main result (Theorem 1.2) of this book and describe the main idea of our proof. Let us start with the original work of Gross and Zagier. 1.1 1.1.1

GROSS–ZAGIER FORMULA ON MODULAR CURVES The original formula

Let N be a positive integer and f ∈ S2 (Γ0 (N )) a newform of weight 2. Let K ⊂ C be an imaginary quadratic field and χ a character of Pic(OK ). Form the L-series L(f, χ, s) as the Rankin–Selberg convolution of the L-series L(f, s) and the L-series L(χ, s). This L-series L(f, χ, s) has a holomorphic continuation to the whole complex plane and satisfies a functional equation relating s to 2 − s. Assume that K has an odd fundamental discriminant D, and satisfies the following Heegner condition: every prime factor of N is split in K. Then the sign of the functional equation of the L-series L(f, χ, s) is −1 and hence L(f, χ, 1) = 0. Let X0 (N ) be the modular curve over Q, whose C-points parametrize isogenies E1 → E2 of elliptic curves over C with kernel isomorphic to Z/N Z. By the Heegner condition, there exists an ideal N of OK such that OK /N ' Z/N Z. For every nonzero ideal I of OK , let PI denote the point on X0 (N )(C) representing the isogeny C/I−→C/IN −1 . Then PI is defined over the Hilbert class field H and depends only on the class of I in Pic(OK ). Form a point in the Jacobian J0 (N ) of X0 (N ) using the cusp ∞ by X Pχ = [PI − ∞] ⊗ χ([I]) ∈ J0 (N )(H) ⊗Z C. [I]∈Pic(OK )

Denote by Pχ (f ) the f -isotypical component of Pχ in J0 (N )(H) ⊗Z C under the action of the Hecke operators. The seminal Gross–Zagier formula proved in [GZ] is as follows: Theorem 1.1 (Gross–Zagier). Denote by h the class number of K, and u half of the number of units of OK . Then hPχ (f ), Pχ (f )iH NT =

hu2 |D|1/2 · L0 (f, χ, 1). 8π 2 (f, f )

2

CHAPTER 1

Here (f, f ) is the Peterson inner product of f , and hPχ (f ), Pχ (f )iH NT denotes the N´eron–Tate height over H. The construction of Heegner points Pχ (f ) involves the idempotent in the Hecke algebra for Γ0 (N ), thus involves some denominators. For application to the BSD conjecture, one should construct such a point on abelian varieties without denominators. This has been achieved when f corresponds to an elliptic curve over Q which we are going to explain as follows. 1.1.2

Application to elliptic curves

Let E be an elliptic curve defined over Q with conductor N . By the landmark modularity theorem of Wiles [Wi] completed by Taylor–Wiles [TW], and the generalization of Breuil–Conrad–Diamond–Taylor [BCDT], E is modular in the sense that L(E, s) = L(f, s) for some newform f of weight 2 for Γ0 (N ). By the isogeny theorem of Faltings [Fa1], there is a finite Q-morphism φ : X0 (N )−→E which takes the cusp ∞ on X0 (N ) to the identity 0 on E. Denote X Pχ (φ) = φ(PI ) ⊗ χ([I]) ∈ E(H) ⊗Z C. [I]∈Pic(OK )

Then the above formula gives hPχ (φ), Pχ (φ)iH NT = deg φ ·

hu2 |D|1/2 · L0 (f, χ, 1). 8π 2 (f, f )

The generalization of the Gross–Zagier formula in this book will be written similar to the above form. We will replace X0 (N ) by a Shimura curve X over a totally real field F , and replace φ by a pair of parametrizations f1 : X → A and f2 : X → A∨ for a dual pair of abelian varieties (A, A∨ ) over F . Then we consider a totally imaginary quadratic extension E of F . Let P ∈ X be a fixed point under the Hecke action of E × . For a finite character χ on Gal(E ab /E), define Pχ (f1 ) ∈ A and Pχ−1 (f2 ) ∈ A∨ as the twisted integral of f1 (P ) and f2 (P ) by χ or χ−1 . The formula will relate the N´eron–Tate height pairing hPχ (f1 ), Pχ−1 (f2 )i with the derivative of L(A, χ, s) at s = 1. 1.2 1.2.1

SHIMURA CURVES AND ABELIAN VARIETIES Incoherent quaternion algebras and Shimura curves

Let F be a number field with adele ring A = AF and let Af be the ring of finite adeles. Let Σ be a finite set of places of F . Up to isomorphism, let B be the unique A-algebra, free of rank 4 as an A-module, whose localization Bv := B ⊗A Fv is isomorphic to M2 (Fv ) if v 6∈ Σ and to the unique division

INTRODUCTION AND STATEMENT OF MAIN RESULTS

3

quaternion algebra over Fv if v ∈ Σ. We call B the quaternion algebra over A with ramification set Σ(B) := Σ. If #Σ is even then B = B ⊗F A for a quaternion algebra B over F unique up to an F -isomorphism. In this case, we call B a coherent quaternion algebra. If #Σ is odd, then B is not the base change of any quaternion algebra over F . In this case, we call B an incoherent quaternion algebra. This terminology is inspired by Kudla’s notion of incoherent collections of quadratic spaces (cf. [Ku2]). Now assume that F is a totally real number field and that B is an incoherent quaternion algebra over A, totally definite at infinity in the sense that Bτ is the Hamiltonian algebra for every archimedean place τ of F . × For each open compact subgroup U of B× f := (B ⊗A Af ) , we have a (compactified) Shimura curve XU over F . For any embedding τ : F ,→ C, the complex points of XU at τ forms a Riemann surface as follows: XU,τ (C) ' B(τ )× \H± × B× f /U ∪ {cusps}. Here B(τ ) is the unique quaternion algebra over F with ramification set Σ \ {τ }, Bf is identified with B(τ )Af as an Af -algebra, and B(τ )× acts on H± through an isomorphism B(τ )τ ' M2 (R). The set {cusps} is non-empty if and only if F = Q and Σ = {∞}. For any two open compact subgroups U1 ⊂ U2 of B× f , one has a natural surjective morphism πU1 ,U2 : XU1 → XU2 . Let X be the projective limit of the system {XU }U . It is a regular scheme over F , locally noetherian but not of finite type. In terms of the notation above, it has a uniformization Xτ (C) ' B(τ )× \H± × B× f /D ∪ {cusps}. × Here D denotes the closure of F × in A× f . If F = Q, then D = F . In general, × D is much larger than F . The Shimura curve X is endowed with an action Tx of x ∈ B× given by “right multiplication by xf .” The action Tx is trivial if and only if xf ∈ D. Each XU is just the quotient of X by the action of U . In terms of the system {XU }U , the action gives an isomorphism Tx : XxU x−1 → XU for each U . The induced action of B× f on the set π0 (XU,F ) of geometrically connected × components of XU factors through the norm map q : B× f → Af and makes × × π0 (XU,F ) a principal homogeneous space over F+ \Af /q(U ). There is a similar description for X.

1.2.2

Hodge classes

The curve XU has a Hodge class LU ∈ Pic(XU )Q . It is the line bundle whose global sections are holomorphic modular forms of weight two. The system L =

4

CHAPTER 1

{LU }U is a direct system in the sense that it is compatible under the pull-back via the projection πU1 ,U2 : XU1 → XU2 . See §3.1.3 for a precise definition. Here are some basic explicit descriptions. If XU is a modular curve, which happens exactly when F = Q and Σ = {∞}, then LU is linearly equivalent to some linear combination of cusps on XU . If F 6= Q or Σ 6= {∞}, then XU has no cusps and LU is isomorphic to the canonical bundle of XU over F for sufficiently small U . For each component α ∈ π0 (XU,F ), denote by LU,α = LU |XU,α the restriction to the connected component XU,α of XU,F corresponding to α. It is also viewed as a divisor class on XU via push-forward under XU,α → XU . Denote by P 1 LU,α the normalized Hodge class on XU,α , and by ξU = α ξU,α ξU,α = deg LU,α the normalized Hodge class on XU . We remark that deg LU,α is independent of α since all geometrically connected components are Galois conjugate to each other. It follows that deg LU,α = deg LU /|F+× \A× f /q(U )|. The degree of LU can be further expressed as the volume of XU . For any open compact subgroup U of B× f , define Z dxdy . vol(XU ) := 2 XU,τ (C) 2πy dxdy on H descends naturally to a measure on XU,τ (C) 2πy 2 via the complex uniformization for any τ : F ,→ C. Then Lemma 3.1 asserts that deg LU = vol(XU ). In particular, the volume is always a positive rational number. For any U1 ⊂ U2 , the projection πU1 ,U2 : XU1 → XU2 has degree Here the measure

deg(πU1 ,U2 ) = vol(XU1 )/vol(XU2 ). It follows from the definition. Because of this, we will often use vol(XU ) as a normalizing factor. 1.2.3

Abelian varieties parametrized by Shimura curves

Let A be a simple abelian variety defined over F . We say that A is parametrized by X if there is a non-constant morphism XU → A over F for some U . By the Eichler–Shimura theory, if A is parametrized by X, then A is of strict GL(2)type in the sense that M = End0 (A) := EndF (A) ⊗Z Q is a field and Lie(A) is a free module of rank one over M ⊗Q F by the induced action. See §3.2 for more details. Define Hom0ξU (XU , A), πA = Hom0ξ (X, A) := lim −→ U

INTRODUCTION AND STATEMENT OF MAIN RESULTS

5

where Hom0ξU (XU , A) denotes the morphisms in HomF (XU , A)⊗ PZ Q using ξU as a base point. More precisely, if ξU is represented by a divisor i ai xi on XU,F , P then f ∈ HomF (XU , A) ⊗Z Q is in πA if and only if i ai f (xi ) = 0 in A(F )Q . Since any morphism XU → A factors through the Jacobian variety JU of XU , we also have πA = Hom0 (J, A) := lim Hom0 (JU , A). −→ U

Here Hom0 (JU , A) = HomF (JU , A)⊗Z Q. The direct limit of Hom(JU , A) defines an integral structure on πA but we will not use this. The space πA admits a natural B× -module structure. It is an automorphic representation of B× over Q. We will see the natural identity EndB× (πA ) = M and that πA has a decomposition π = ⊗M πv where πv is an absolutely irreducible representation of B× v over M . Using the Jacquet–Langlands correspondence, one can define L-series Y L(s, π) = Lv (s, πv ) ∈ M ⊗Q C v

as an entire function of s ∈ C. Let Y L(s, A, M ) = Lv (s, A, M ) ∈ M ⊗Q C be the L-series defined using `-adic representations with coefficients in M ⊗Q Q` , completed at archimedean places using the Γ-function. Then L(s, A, M ) converges absolutely in M ⊗ C for Re(s) > 3/2. The Eichler–Shimura theory asserts that, for almost all finite places v of F , the local L-function of A is given by 1 Lv (s, A, M ) = L(s − , πv ). 2 Conversely, by the Eichler–Shimura theory and the isogeny theorem of Faltings [Fa1], if A is of strict GL(2)-type, and if for some automorphic representation π of B× over Q, Lv (s, A, M ) is equal to L(s − 1/2, πv ) for almost all finite places v, then A is parametrized by the Shimura curve X. If A is parametrized by X, then the dual abelian variety A∨ is also parametrized by X. Denote by M ∨ = End0 (A∨ ). There is a canonical isomorphism M → M ∨ sending a homomorphism m : A → A to its dual m∨ : A∨ → A∨ . There is a perfect B× -invariant pairing πA × πA∨ −→M given by ∨ (f1 , f2 ) = vol(XU )−1 (f1,U ◦ f2,U ),

f1,U ∈ Hom(JU , A), f2,U ∈ Hom(JU , A∨ )

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CHAPTER 1

∨ where f2,U : A → JU is the dual of f2,U composed with the canonical isomorphism JU∨ ' JU . It follows that πA∨ is dual to πA as representations of B× over M. In the case that A is an elliptic curve, we have M = Q and πA is self-dual. For any morphism f ∈ πA represented by a direct system {fU }U , we have

(f, f ) = vol(XU )−1 deg fU . Here deg fU denotes the degree of the finite morphism fU : XU → A. 1.2.4

Height pairing

The usual theory of N´eron–Tate height gives a Q-bilinear non-degenerate pairing h·, ·iNT : A(F )Q × A∨ (F )Q −→R. We refer to §7.1.1 for a quick review. Recall that the field M = End0 (A) acts on A(F )Q by definition, and acts on ∨ A (F )Q through the duality. By the adjoint property of the height pairing in Proposition 7.3, the pairing h·, ·iNT descends to a Q-linear map h·, ·iNT : A(F )Q ⊗M A∨ (F )Q −→R. For any fixed (x, y) ∈ A(F )Q ⊗M A∨ (F )Q , the correspondence a ∈ M 7−→ hax, yiNT ∈ R define an element in Hom(M, R). One has an isomorphism Hom(M, R) ∼ = M ⊗Q R using the trace map. Let hx, yiM denote the corresponding element in M ⊗Q R. Then we have just defined an M -bilinear pairing h·, ·iM : A(F )Q ⊗M A∨ (F )Q −→M ⊗Q R such that h·, ·iNT = trM ⊗R/R h·, ·iM . We call this new pairing an M -linear N´eron–Tate height pairing. 1.3 1.3.1

CM POINTS AND GROSS–ZAGIER FORMULA CM points

Let E/F be a totally imaginary quadratic extension, with a fixed embedding EA ,→ B over A. Then EA× acts on X by the right multiplication via EA× ,→ × B× . Let X E be the subscheme of X of fixed points of X under E × . Up to

7

INTRODUCTION AND STATEMENT OF MAIN RESULTS ×

translation by B× , the subscheme X E does not depend on the choice of the × embedding EA ,→ B. The scheme X E is defined over F . The theory of complex × multiplication asserts that every point in X E (F ) is defined over E ab and that the Galois action is given by the Hecke action under the reciprocity law. × Fix a point P ∈ X E (E ab ) throughout this book. It induces a point PU ∈ XU (E ab ) for every U . We can normalize the complex uniformization XU,τ (C) = B(τ )× \H± × B× f /U ∪ {cusps} so that the point PU is exactly represented by the double coset of [z0 , 1]U . Here z0 ∈ H is the unique fixed point of E × in H via the action induced by the embedding E ,→ B(τ ). A similar description can be made on X. Let A be an abelian variety over F parametrized by X with M = End0 (A) and let χ : Gal(E ab /E) → L× be a character of finite order, where L is a finite field extension of M . For any f ∈ πA , the image f (P ) is a well-defined point in A(E ab )Q . Consider the integration Z Pχ (f ) = f (P τ ) ⊗M χ(τ )dτ ∈ A(E ab )Q ⊗M L, Gal(E/E)

where we use the Haar measure on Gal(E/E) of total volume 1. It is essentially a finite sum, and it is easy to see that Pχ (f ) ∈ A(χ) := (A(E ab )Q ⊗M Lχ )Gal(E

ab

/E)

.

Here Lχ denotes the M -vector space L with the action of Gal(E ab /E) given by the multiplication by the character χ. It is also clear that Pχ (f ) 6= 0 only if the central character ωA of πA is compatible with χ in the sense that ωA · χ|A× = 1. F

Let L(s, AE , χ) ∈ L ⊗Q C be the L-series which is obtained by `-adic representation twisted by χ. Define L(s, πA , χ) = L(s, πA,E ⊗ χ). Here πA,E denotes the base change of πA to E, and χ is considered as a character of E × \A× E via the reciprocity law ab E × \A× E −→Gal(E /E)

which maps uniformizers to geometric Frobenii. As an identity in L ⊗Q C, we have 1 L(s, AE , χ) = L(s − , πA , χ). 2 As a refinement of the Birch and Swinnerton-Dyer conjecture, it is conjectured that the leading term of L(s, AE , χ) is invertible in L ⊗Q C and that ords=1 L(s, AE , χ) = dimL A(χ).

8 1.3.2

CHAPTER 1

Gross–Zagier formula

Assume that ωA · χ|A× = 1. Define a linear space over L by F

P(πA , χ) := HomE × (πA ⊗ χ, L). A

Then the correspondence f 7→ Pχ (f ) defines an element Pχ ∈ P(πA , χ) ⊗L A(χ). Thus Pχ (f ) 6= 0 for some f only if P(πA , χ) 6= 0. By the following Theorem 1.3 of Saito–Tunnel, P(πA , χ) is at most onedimensional, and it is one-dimensional if and only if the ramification set Σ(B) of B is equal to   1 Σ(A, χ) = places v of F : ( , πA,v , χv ) 6= χv (−1)ηv (−1) . 2 In that case, (1/2, πA , χ) = −1 and thus L(1/2, πA , χ) = 0. The next problem is to find a nonzero element of P(πA , χ) if it is onee = πA∨ by the dimensional. Denote π = πA for simplicity. The contragredient π duality map πA × πA∨ −→M. π , χ−1 ). Here P(e π , χ−1 ) has It is more convenient to work with P(π, χ) ⊗L P(e the same dimension as P(π, χ) by Theorem 1.3. When P(π, χ) ⊗ P(e π , χ−1 ) is nonzero, we would like to define a canonical generator denoted by α. Decompose π = ⊗πv and χ = ⊗χv . Then we have a decomposition P(π, χ) = ⊗P(πv , χv ), where the space P(πv , χv ) is defined analogously. Then α will have a decomposition α = ⊗αv for some αv ∈ P(πv , χv ) ⊗ P(e πv , χ−1 v ) to be defined. × × Fix Haar measures dtv on Ev /Fv such that the product measure over all v gives the Tamagawa measure on EA× /A× . We can further assume that the × /OF×v has a volume in Q for all non-archimedean maximal compact subgroup OE v place v. Then αv is defined formally by Z L(1, ηv )L(1, πv , ad) αv (f1 ⊗ f2 ) = (πv (t)f1 , f2 )v χv (t)dt, ζFv (2)L( 12 , πv , χv ) Ev× /Fv× f1 ∈ πv , f2 ∈ π ev . More precisely, we may take an embedding ι : L,→C and define the above integral with value in C. It turns out that, for all places v, the value of αv (f1 ⊗ f2 ) lies in L, and does not depend on the choice of the embedding ι. It is worth mentioning that in the definition of αv , the local equivariant pairing (·, ·)v : πv × π ev → M satisfies the compatibility condition that (·, ·) = ⊗v (·, ·)v . Here the global pairing (·, ·) : πA × π eA∨ → M is the duality map introduced above. The following is the main theorem of this book.

INTRODUCTION AND STATEMENT OF MAIN RESULTS

9

Theorem 1.2. Assume ωA · χ|A× = 1. For any f1 ∈ πA and f2 ∈ πA∨ , F

hPχ (f1 ), Pχ−1 (f2 )iL =

ζF (2)L0 (1/2, πA , χ) α(f1 , f2 ) 4L(1, η)2 L(1, πA , ad)

as an identity in L ⊗Q C. Here h·, ·iL : A(χ) × A∨ (χ−1 ) → L ⊗Q R is the L-linear N´eron–Tate height pairing induced by the M -linear N´eron–Tate height pairing h·, ·iM between A(F ) and A∨ (F ). The theorem compares two elements α,

hPχ (·), Pχ−1 (·)iL

of the L-linear space P(πA , χ) ⊗L P(πA∨ , χ−1 ). The space is at most one-dimensional. If Σ(B) 6= Σ(A, χ), the linear space is zero and thus both sides of the formula are zero. The theorem is vacuous in this case. In the essential case Σ(B) = Σ(A, χ), the linear space is one-dimensional and α is a generator. It follows that hPχ (·), Pχ−1 (·)iL must be a constant multiple of α. Then the theorem can be viewed as an explicit expression of the multiple in terms of special values and special derivatives of L-functions. Note that (1/2, πA , χ) = −1 in this case. In the original Gross–Zagier formula, the Heegner condition implies that Σ(f, χ) = {∞}, so that the Heegner points are constructed from the modular curve. Here Σ(f, χ) is similarly defined in terms of local root numbers. 1.4

WALDSPURGER FORMULA

1.4.1

Linear forms over local fields

Let F be a local field and B a quaternion algebra over F . Then B is isomorphic to either M2 (F ) or the unique division quaternion algebra over F . The Hasse invariant (B) = 1 if B ' M2 (F ), and (B) = −1 if B is the division algebra. Let E be either F ⊕ F or a quadratic field extension over F , with a fixed embedding E ,→ B of algebras over F . Let η : F × → C× be the quadratic character associated to the extension E/F . Let π be an irreducible admissible representation of B × with central character ωπ , and let χ : E × → C× be a character of E × such that ωπ · χ|F × = 1. Define P(π, χ) := HomE × (π ⊗ χ, C).

10

CHAPTER 1

The following result asserts that the dimension of this space is determined by the local root number 1 1 1 ( , π, χ) = ( , σ, χ) = ( , σE ⊗ χ). 2 2 2 Here σ is the Jacquet–Langlands correspondence of π on GL2 (F ), and σE is the base change to E. Theorem 1.3 (Tunnell [Tu], Saito [Sa]). The space P(π, χ) is at most onedimensional, and it is one-dimensional if and only if 1 ( , π, χ) = χ(−1)η(−1)(B). 2 The next problem is to find a nonzero element of P(π, χ) if it is onedimensional. It is more convenient to work with P(π, χ) ⊗ P(e π , χ−1 ). Here π e denotes the contragredient of π, and P(e π , χ−1 ) has the same dimension as P(π, χ) by Theorem 1.3. Fix a Haar measure dt on E × /F × . Define a bilinear form α:π⊗π e−→C by L(1, η)L(1, π, ad) α(f1 ⊗ f2 ) = ζF (2)L( 12 , π, χ)

Z (π(t)f1 , f2 ) χ(t)dt, E × /F ×

f1 ∈ π, f2 ∈ π e.

Here (·, ·) : π × π e → C is a fixed B × -invariant pairing. The integral converges absolutely if both π and χ are unitary. The normalizing factor before the integration is nonzero, and makes α(f1 ⊗ f2 ) = 1 in the following spherical case: • B = M2 (F ), • E is an unramified extension of F , • π and χ are both unramified, × • dt is normalized such that vol(OE /OF× ) = 1,

• f1 ∈ π GL2 (OF ) and f2 ∈ π eGL2 (OF ) satisfy (f1 , f2 ) = 1. It is easy to see that α defines an element of P(π, χ)⊗P(e π , χ−1 ). It is actually a generator of the space. In other words, α 6= 0 if and only if P(π, χ) 6= 0. See [Wa, §III-2, Lemme 10].

11

INTRODUCTION AND STATEMENT OF MAIN RESULTS

1.4.2

Waldspurger formula

We start with the following notations and assumptions: • F is a number field with adele ring A = AF . • B is a quaternion algebra over F . • E is a quadratic field extension of F , with a fixed embedding E ,→ B over F. • π is an irreducible cuspidal automorphic representation of BA× with central character ωπ : F × \A× → C× . • χ : E × \EA× → C× is a character with ωπ · χ|A× = 1. • η : F × \A× → C× is the quadratic character associated to the extension E/F . Define a period integral Pχ : π → C by Z f (t)χ(t)dt, Pχ (f ) =

f ∈ π.

E × \EA× /A×

The integral uses the Haar measure with total volume 1 on E × \EA× /A× . Define e → C in the same way. Pχ−1 : π Theorem 1.4 (Waldspurger [Wa]). For any f1 ∈ π and f2 ∈ π e, we have Pχ (f1 ) · Pχ−1 (f2 ) =

ζF (2)L( 12 , π, χ) α(f1 ⊗ f2 ). 8L(1, η)2 L(1, π, ad)

Here the L-function is L(s, π, χ) = L(s, σ, χ) = L(s, σE ⊗ χ), where σ denotes the Jacquet-Langlands lifting of π, and σE denotes the base change of π to E. The global bilinear form α:π×π e−→C is defined to be the tensor product of the local bilinear forms α : πv ⊗ π ev −→C introduced above. The definition depends on the choice of a local invariant ev → C at every place v of F . Normalize the local pairings pairing (·, ·)v : πv × π by the compatibility (·, ·)Pet = ⊗v (·, ·)v . Here the Petersson pairing (·, ·)Pet : π × π e → C is defined by Z (f1 , f2 )Pet = f1 (h)f2 (h)dh, f1 ∈ π, f2 ∈ π e. B × \B× /A×

12

CHAPTER 1

The integration uses the Tamagawa measure, which has volume 2 on B × \B× /A×. Note that the pair (e π , χ−1 ) is equal to the complex conjugate (π, χ). Take ¯ f2 = f1 in the formula. Then the left-hand side becomes Pχ (f1 ) · Pχ−1 (f2 ) = Pχ (f1 ) · Pχ (f¯1 ) = |Pχ (f1 )|2 . This form is widely used for period formulae in the literature. Remark. The formula was proved by Waldspurger with a slightly different setting (cf. [Wa, Proposition 7]). Note that we use the probability measure in the period integral, while Waldspurger used the Tamagawa measure. Waldspurger assumed that the central character ωπ is trivial, but his proof goes through to the general case. There is an interpretation of the theorem similar to that of Theorem 1.2. In fact, Theorem 1.4 compares the two bilinear forms α, Pχ ⊗ Pχ−1 ∈ P(π, χ) ⊗ P(e π , χ−1 ). The linear space on the right-hand side is nonzero only if Σ(B) = Σ(π, χ). In that case, the space is one-dimensional and generated by α. The formula can be viewed as an explicit expression of the multiple in terms of special values of L-functions. One may start with the data (F, E, σ, χ) satisfying (1/2, σ, χ) = 1, and find the pair (B, π) by the condition Σ(B) = Σ(π, χ). By the construction of B, the Jacquet–Langlands lifting π of σ to BA× always exists, and E can be embedded into B. By this way, the formula is non-trivial. There is a similar point of view for Theorem 1.2. We omit it here. 1.5

PLAN OF THE PROOF

Our proof of Theorem 1.2 is still based on the idea of Gross–Zagier [GZ], namely, to compare an analytic kernel function representing the central derivative of the L-function with a geometric kernel function formed by a generating series of height pairings of CM points. Many ideas of [Zh1, Zh2] are also used in this book. In the following, we give an outline of our proof along the order of this book. It also gives the structure of this book. 1.5.1

Proof of Waldspurger formula

In Chapter 2, we review some basic results on Weil representations, theta liftings and Eisenstein series. In particular, a proof of the Waldspurger formula is contained in the chapter, which has inspired our proof of the Gross–Zagier formula in the book. Assume the notations in Theorem 1.4. Recall that B is a quaternion algebra over F . Write B = BA for simplicity. The Weil representation here is an action

INTRODUCTION AND STATEMENT OF MAIN RESULTS

13

r of GL2 (A) × B× × B× on the space S(B × A× ) of Schwartz functions. For any Φ ∈ S(B × A× ), the theta series is defined by X r(g, (h1 , h2 ))Φ(x, u). θ(g, (h1 , h2 ), Φ) = u∈F ×

By integrating against σ, it gives the theta lifting e. θ : S(B × A× ) ⊗ σ−→π ⊗ π It is also called the Shimizu lifting. We start with the period integrals Pχ (f1 ) and Pχ−1 (f2 ). Assume that under the Shimizu lifting Φ ∈ S(B × A× ), ϕ ∈ σ.

f1 ⊗ f2 = θ(Φ ⊗ ϕ), Then it is easy to have


Pχ (f1 ) Pχ−1 (f2 ) = θ(·, (χ, χ−1 ), Φ), ϕ Pet . Here θ(g, (χ, χ−1 ), Φ) =

Z

Z E × \EA×

θ(g, (tt0 , t0 ), Φ)χ(t)dt0 dt.

E × \EA× /A×

On the other hand, the L-function has an integral representation (I(s, ·, χ, Φ), ϕ)Pet = (∗) L(

s+1 , σ, χ). 2

Here the analytic kernel is Z I(s, g, χ, Φ) =

I(s, g, r(t)Φ)χ(t)dt, E × \EA×

where I(s, g, Φ) is a mixed theta-Eisenstein series defined by X X X δ(γg)s r(γg)Φ(x). I(s, g, Φ) = u∈F × x1 ∈E γ∈P 1 (F )\P 1 (A)

The Waldspurger formula follows from the equalities I(0, g, χ, Φ)

=

(I(0, ·, χ, Φ), ϕ)Pet

=

2 θ(g, (χ, χ−1 ), Φ), 1 (∗) L( , σ, χ) α(f1 ⊗ f2 ). 2

(1.5.1) (1.5.2)

Here (∗) denotes a nonzero explicit constant. Both of the equalities are implied by the Siegel–Weil formula with some minor extra work. The first equality is given by the Siegel–Weil formula on the quadratic space (E, q), and the second equality is given by the Siegel–Weil formula on the quadratic space (B0 , q). Here B0 is the space of elements of B with traces equal to zero.

14 1.5.2

CHAPTER 1

Projectors

Assume that F, E, B and X are as in Theorem 1.2 from now on. Fix an embedding τ : F ,→C. The cohomology H 1,0 (Xτ ) = lim H 1,0 (XU,τ ) −→ U

has a decomposition H 1,0 (Xτ ) =

M

π.

π∈A(B× )

Here A(B× ) denotes the set of irreducible admissible representations π of B× such that the Jacquet–Langlands correspondence σ of π on GL2 (A) is a cuspidal automorphic representation of GL2 (A), discrete of parallel weight two at infinity. e acts on H 1,0 (Xτ ) by f 7−→ (f, f2 ) f1 . This action can Any f1 ⊗ f2 ∈ π ⊗ π be represented by a Hecke operator, and thus it is algebraic. Hence, it lifts to a map T:π⊗π e → Hom0 (J, J ∨ )C . Here Hom0 (J, J ∨ ) = lim Hom0 (JU , JU ). −→ U

We call T(f1 ⊗ f2 ) a projector. 1.5.3

Main result in terms of projectors

Resume the above notations. Assume that χ : E × \EA× −→C× is a character of finite order with the compatibility that χ|A× · ωπ = 1. Theorem 3.15 asserts e, that, for any f1 ⊗ f2 ∈ π ⊗ π hT(f1 ⊗ f2 )Pχ , Pχ−1 iNT =

ζF (2)L0 (1/2, π, χ) α(f1 ⊗ f2 ). 4L(1, η)2 L(1, π, ad)

(1.5.3)

Here P is the CM point on X given by E, and Pχ is the χ-eigencomponent on X. The height pairing is on X. It is easy to show that the theorem is equivalent to Theorem 1.2. The key ingredient is another interpretation of the projector. Let f1 ∈ πA and f2 ∈ πA∨ be as in Theorem 1.2. We introduce Talg (f1 , f2 ) = f2∨ ◦ f1 ∈ Hom0 (J, J ∨ ). One may compare it with the duality maps (f1 , f2 ) = f1 ◦ f2∨ ∈ M. Then it is easy to derive, for any embedding ι : M ,→C, T(f1ι ⊗ f2ι ) = Talg (f1 ⊗ f2 )ι . Here Talg (f1 ⊗ f2 )ι is the ι-eigencomponent of Talg (f1 ⊗ f2 ) by the action of M . By this identity, the equivalence of the two theorems follows from the projection formula of height pairings.

15

INTRODUCTION AND STATEMENT OF MAIN RESULTS

1.5.4

Generating function

Let Φ ∈ S(B × A× ) be a Schwartz function bi-invariant under an open compact subgroup U of B× f . The generating function on XU , for any g ∈ GL2 (A), is defined by X X r(g)Φ(x, aq(x)−1 ) Z(x)U . Z(g, Φ)U = Z0 (φ)U + wU a∈F × x∈U \B× /U f

Here φ = Φ ∈ S(B × A× ) is obtained by averaging on B1∞ . It only changes the archimedean parts. The constant term Z0 (φ)U and the constant wU are less important here. The key part Z(x)U is the Hecke correspondence on XU corresponding to the double coset U xU . Based on the work of many people, we will show that Z(g, Φ)U is an automorphic form of g ∈ GL2 (A) with coefficients in Pic(XU × XU )C . If XU = X0 (N ) is the modular curve, by properly choosing Φ, the series Z(g, Φ)U recovers the classical series X Tn e2πinz , z ∈ H. n≥0

This classical form is used in the proof of Gross–Zagier and S. Zhang. The generating function is the counterpart of the theta series in the incoherent case. In fact, we compute its intersection or arithmetic intersection with other cycles and it is easy to express the result in the form of a usual theta series. 1.5.5

Geometric kernel

e Φ) = Multiplying Z(g, Φ)U by a suitable constant, we obtain a system Z(g, e e {Z(g, Φ)U }U compatible with pull-back maps. Then Z(g, Φ) is an element of Pic(X × X)C = lim Pic(XU × XU )C . −→ U

For any h1 , h2 ∈ B× , define the height series e (h1 , h2 ), Φ) := hZ(g, e Φ) [h1 ]◦ , [h2 ]◦ iNT , Z(g,

g ∈ GL2 (A).

Here [h] denotes the point Th (P ) obtained by multiplication by h, and [h2 ]◦ denotes the degree zero cycle [h] − ξq(h) . The geometric kernel is Z ∗ e (t, 1), Φ) χ(t)dt. e χ, Φ) = Z(g, Z(g, T (F )\T (A)/Z(A)

Here the integral is a regularized integral. It takes the place of the kernel function θ(g, (χ, χ−1 ), Φ) in the incoherent case.

16

CHAPTER 1

1.5.6

Kernel identity

The analytic kernel I(s, g, χ, Φ) is defined by exactly the same formula as in the coherent case. By the integral representation, the derivative I 0 (0, g, χ, Φ) represents L0 (1/2, π, χ). The kernel identity here, analogous to equation (1.5.1), is   e χ, Φ), ϕ (I 0 (0, ·, χ, Φ), ϕ)Pet = 2 Z(·, , ∀ ϕ ∈ σ. (1.5.4) Pet

See Theorem 3.21. We will discuss later the impossibility of the exact equality e χ, Φ). I 0 (0, g, χ, Φ) = 2 Z(g, For any Φ ∈ S(V × A× ) and ϕ ∈ σ, define an “arithmetic theta lifting” by   e ⊗ ϕ) = Z(g, e Φ), ϕ Z(Φ . Pet

It lies in Pic(X × X)C , and thus induces an element of Hom0 (J, J ∨ ) by acting on JU as push-forward of correspondences. The main result for arithmetic theta lifting is e ⊗ ϕ) = L(1, π, ad) T(θ(Φ ⊗ ϕ)), Z(Φ 2ζF (2)

Φ ∈ S(V × A× ), ϕ ∈ σ.

(1.5.5)

Note that the Shimizu lifting e θ : S(B × A× ) ⊗ σ−→π ⊗ π can be defined locally in the incoherent case. By this formula, it is easy to prove the equivalence between the formulation in (1.5.3) and the kernel identity in (1.5.4). 1.5.7

Arithmetic theta lifting

The left-hand side of (1.5.5) can be viewed as an “arithmetic theta lifting.” The proof of equation (1.5.5) takes up most of Chapter 4, although its coherent counterpart is trivial. We give a rough idea of the proof here. The first step is to show that the equation is true up to a constant. In fact, it suffices to prove the equation for their induced actions on the cohomology H 1,0 (Xτ ). By the decomposition M H 1,0 (Xτ ) = π1 , π1 ∈A(B× )

each side of (1.5.5) induces an element in M π1 ⊗ π e2 . π1 ,π2 ∈A(B× )

17

INTRODUCTION AND STATEMENT OF MAIN RESULTS

As functionals on Φ and ϕ, we obtain two elements in the one-dimensional space e). HomGL2 (A)×B× ×B× (S(V × A× ) ⊗ σ, π ⊗ π So they must be equal up to a constant. It remains to check that the constant is exactly given by (1.5.5). For that, fixing a level U , it suffices to prove that traces of two sides, as an operator on H 0,1 (XU,τ ), are equal. It is easy to see that the trace of T(f1 ⊗f2 ) on H 0,1 (XU,τ ) e ⊗ ϕ) on is exactly (f1 , f2 ). The hard part is to figure out the trace of Z(Φ 0,1 H (XU,τ ). e ⊗ ϕ) is reduced to By the Lefschetz fixed point theorem, the trace of Z(Φ e the intersection number ∆U · Z(Φ ⊗ ϕ). Here ∆U : XU → XU × XU is the diagonal embedding. It is further reduced to compute the pull-back ∆∗U Z(g, Φ)U . Roughly speaking, the pull-back is given by an explicit generating function whose coefficients are CM points, Hodge classes or cusps on XU . Then the trace will be given by a mixed theta-Eisenstein series different from our analytic kernel. The computation is very complicated if XU contains cusps. Similar computations on Hilbert modular surfaces were done by Hirzebruch–Zagier. 1.5.8

Degenerate Schwartz functions

In Chapter 5, we introduce two classes of degenerate Schwartz functions. They simplify the computations and arguments of both kernel functions significantly. Of course, we also prove that these assumptions can recover (1.5.4) in the full case. For any non-archimedean place v, define S 1 (Bv × Fv× ) := {Φv ∈ S(Bv × Fv× ) : Φv (x, u) = 0 if v(uq(x)) ≥ −v(dv ) or v(uq(x2 )) ≥ −v(dv )}. Here dv is the local different of F at v, and x = x1 + x2 according to the orthogonal decomposition Bv = Ev + Ev jv (cf. §1.6.4). Define S 2 (Bv × Fv× ) := {Φv ∈ S(Bv × Fv× ) : r(g)Φv (0, u) = 0,

∀ g ∈ GL2 (Fv ), u ∈ Fv× }.

Assume that Φv lies in S 1 (Bv × Fv× ) for all ramified non-archimedean places vnonsplit in E, and assume that Φv lies in S 1 (Bv × Fv× ) for at least two nonarchimedean places v split in E. Here are some major effects of the assumptions: • Kill the self-intersections of CM points in the height series Z(g, (t1 , t2 ), Φ). • Kill the logarithmic singularities coming from both the derivatives and the local heights at v. • Kill the constant term of the generating series Z(g, Φ)U .

18

CHAPTER 1

• Kill the arithmetic intersections coming for the Hodge classes in the height series Z(g, (t1 , t2 ), Φ). The key to recover (1.5.4) in the full case from these assumptions is Theorem 1.3. In fact, like the interpretation of Theorem 1.2, we interpret (1.5.3) as an identity of two vectors in the complex vector space P(π, χ) ⊗ P(e π , χ−1 ). It is at most one-dimensional by Theorem 1.3. Assume that it is one-dimensional. It follows that the ramification set of B agrees with the set Σ of places determined by the local root numbers. It suffices to prove (1.5.3) for some (f1 , f2 ) with α(f1 ⊗ f2 ) 6= 0. Hence, it suffices to prove (1.5.4) for some (Φ, ϕ) with α(θ(Φ ⊗ ϕ)) 6= 0. Go back to the sufficiency of the assumptions. We only need to prove that there exists a Schwartz function Φ satisfying the degeneracy assumptions and the condition that α(θ(Φ ⊗ ϕ)) 6= 0 for some ϕ ∈ σ. It is a local problem and solved in Chapter 5. 1.5.9

Components of the kernel functions

To prove (1.5.4), we need to compute the difference e χ, Φ). PrI 0 (0, g, χ, Φ) − 2 Z(g, Here Pr denotes the projection to the space of cusp forms, holomorphic of parallel weight two. It is easy to write X I 0 (0, g, χ, Φ)(v). PrI 0 (0, g, χ, Φ) = v nonsplit

Roughly speaking, we write I(s, g, χ, Φ) as a Fourier series, which is a sum of Whittaker functions. The Whittaker functions are local products, and the derivative falls into the local Whittaker functions at every place. Then I 0 (0, g, χ, Φ)(v) collects all the terms with the derivative taken at v. We also need to apply Pr, which only changes I 0 (0, g, χ, Φ)(v) for archimedean v. This is done in Chapter 6. On the other hand, we also have X e χ, Φ) = e χ, Φ)(v). Z(g, Z(g, v nonsplit

It essentially follows from the decomposition of a global arithmetic intersection number to the sum of local intersection numbers. Note that there is no self-intersection by the degenerate Schwartz functions. We also prove that the arithmetic intersection with the Hodge class are zero by the degenerate Schwartz functions. This is done in Chapter 7. For archimedean v and good non-archimedean v, we prove I 0 (0, g, χ, Φ)(v) = 2Z(g, χ, Φ)(v)

INTRODUCTION AND STATEMENT OF MAIN RESULTS

19

by explicit computations in Chapter 6 and Chapter 8. It is routine after the work of Gross–Zagier and S. Zhang. For the remaining bad primes, the local components are impossible to compute. This was the essential difficulty to remove the assumptions of mild ramifications in the work of Gross–Zagier and S. Zhang. Our solution is the following approximation argument. 1.5.10

Approximation

We say a function Ψ : GL2 (A) → C is approximated by an automorphic form Ψ0 on GL2 (A) if there exists a finite set S of places of F such that Ψ(g) = Ψ0 (g) for all g ∈ 1S GL2 (AS ). Here is a simple fact. If furthermore Ψ is automorphic, then Ψ = Ψ0 identically. It is true since GL2 (F )GL2 (AS ) is dense in GL2 (A). Come back to the comparison of the kernel functions. Take advantage of the degenerate Schwartz functions. In Chapter 6, we prove that, for bad v, I 0 (0, g, χ, Φ)(v) can be approximated by a coherent kernel function of the form I(0, g, χ, Φ(v)), where Φ(v) = Φv ⊗ Φ0v ∈ S(B(v)A × A× ) is a Schwartz function based on the nearby quaternion algebra B(v) over F obtained by changing the Hasse invariant of B at v. In Chapter 8, we prove e χ, Φ)(v). similar approximation results for Z(g, It follows that the difference e χ, Φ) PrI 0 (0, g, χ, Φ) − 2 Z(g, is approximated by a finite sum of functions of the form I(0, g, χ, Φ(v)). By the above simple consequence of modularity, we conclude that X e χ, Φ) = I(0, g, χ, Φ(v)). (1.5.6) PrI 0 (0, g, χ, Φ) − 2 Z(g, v

The right-hand side is a priori not zero, but it is perpendicular to σ by Theorem 1.3. In fact, the ramification set of each B(v) does not agree with Σ since we have assumed that the ramification set of B agrees with Σ. It proves (1.5.4). Note that (1.5.4) is a weak analogue of (1.5.1). However, the exact analogue e χ, Φ) I 0 (0, g, χ, Φ) = 2 Z(g, does not hold in general. In fact, the right-hand sides satisfies the transfer e χ, Φ) = Z(1, e χ, r(g)Φ), but the left-hand side does not satisfy property Z(g, bF ). such a property unless Φ is invariant under the action of g ∈ GL2 (O On the other hand, (1.5.6) can be viewed as an equality of the two kernel functions modulo coherent kernel functions. In this sense, it is an appropriate analogue of (1.5.1).

20

CHAPTER 1

1.5.11

Pseudo-theta series

In the end, we explain why these local components can be approximated by the coherent kernel functions easily. We mainly look at I 0 (0, g, χ, Φ)(v). By a local version of the Siegel–Weil formula, it is easy to write I 0 (0, g, Φ)(v) as X X kΦv (gv , y, u)r(g v )Φv (y, u), u y∈B(v)

where kΦv (gv , y, u) is a function on gv ∈ GL2 (Fv ) and (y, u) ∈ B(v)v × Fv× determined explicitly by Φv . It looks like a theta series except that at v the function kΦv (g, y, u) is not given by Weil representation on Schwartz function on B(v)v , so we call it a pseudo-theta series. The key is to show that kΦv (1, y, u) is a Schwartz function of (y, u) ∈ B(v)v × Fv× if Φv is degenerate. Then we form the “authentic” theta series for Schwartz function kΦv ⊗ Φv as follows: X X r(g)kΦv (1, y, u)r(g)Φv (y, u). u y∈B(v)

It approximates the original series since they are the same for g ∈ GL2 (A) with gv = 1v . Hence, I 0 (0, g, χ, Φ)(v) is approximated by the coherent kernel function I(0, g, χ, Φv ⊗ kΦv ) on the nearby quaternion algebra. e χ, Φ)(v), we can also write it as a series over As for the local height Z(g, B(v). Roughly speaking, the local formal neighborhoods of the integral model of Shimura curve XU can be uniformized as the quotient of some universal deformation space by the action of B(v)× . Then the local height pairing on the Shimura curve is a summation of intersections of points in the corresponding orbit indexed by B(v)× . 1.6

NOTATION AND TERMINOLOGY

We always denote by F the base number field, and by E a quadratic field extension of F . Except in Chapter 1 and Chapter 2, it is assumed that F is totally real and E is totally imaginary. We normalize the absolute values, additive characters, and measures following Tate’s thesis. To be precise, we start with the local case. 1.6.1

Local fields

In the following, k denotes a local field of a number field. • Normalize the absolute value | · | on k as follows: It is the usual one if k = R. It is the square of the usual one if k = C.

21

INTRODUCTION AND STATEMENT OF MAIN RESULTS

If k is non-archimedean, it maps the uniformizer to N −1 . Here N is the cardinality of the residue field. • Normalize the additive character ψ : k → C× as follows: If k = R, then ψ(x) = e2πix . If k = C, then ψ(x) = e4πiRe(x) . If k is non-archimedean, then it is a finite extension of Qp for some prime p. Take ψ = ψQp ◦ trk/Qp . Here the additive character ψQp of Qp is defined by ψQp (x) = e−2πiι(x) , where ι : Qp /Zp ,→ Q/Z is the natural embedding. • We take the measure dx on k to be the unique Haar measure on k self-dual with respect to ψ in the sense that the Fourier transform Z b Φ(y) := Φ(x)ψ(xy)dx k

b b satisfies the inversion formula Φ(x) = Φ(−x). The measures are determined explicitly as follows: If k = R, then dx is the usual Lebesgue measure. If k = C, then dx is twice of the usual Lebesgue measure. 1

If k is non-archimedean, then vol(Ok ) = |dk | 2 . Here Ok is the ring of integers and dk ∈ k is the different of k over Qp . • We take the Haar measure d× x on k × by d× x = ζk (1)|x|−1 dx. Recall that ζk (s) = (1−N −s )−1 if k is non-archimedean whose residue field has N elements, ζR (s) = π −s/2 Γ(s/2), and ζC (s) = 2(2π)−s Γ(s). With this normalization, if k is non-archimedean, then vol(Ok× , d× x) = vol(Ok , dx). 1.6.2

Haar measures for algebraic groups

Algebraic groups in this book are mainly associated to quadratic extensions and quaternion algebras over the base field. We are going to normalize the Haar measures on them based on the normalization above. Let k be a local field as above. If (V, q) is a quadratic space over k, the self-dual measure on V with respect to (V, q) (and ψ) is the unique Haar measure dx on V such that the Fourier transform Z b Φ(x)ψ(hx, yi)dx Φ(y) := V

b b satisfies the inversion formula Φ(x) = Φ(−x). Here hx, yi = q(x+y)−q(x)−q(y) is the corresponding bilinear pairing.

22

CHAPTER 1

Let K be a quadratic etale algebra extension over k or a quaternion algebra over k, and let q : K → k be the reduced norm. One has one of the following: • K = k ⊕ k and q(x1 , x2 ) = x1 x2 , in this case we say K is a split quadratic extension; • K is a quadratic field extension over k and q = NK/k ; • K is equal to a (either split ornonsplit) quaternion algebra B over k and q is the reduced norm. In all cases we can write q(x) = xx where x is the main involution of x. The pair (K, q) forms a quadratic space over k. Endow K with the self-dual Haar measure dx with respect to (K, q). Endow the multiplicative group K × with a Haar measure d× x as follows. If K is a quadratic etale algebra extension, then d× x = ζK (1) |q(x)|−1 dx. Here ζK (s) = ζk (s)2 if K = k ⊕ k, and ζK (s) is the zeta function of the local field K if K isnonsplit. If K is a quaternion algebra, then d× x = ζk (1) |q(x)|−2 dx. Endow K × /k × with the quotient Haar measure. Endow the subgroup K 1 := {h ∈ K × : q(h) = 1} with the Haar measure dh determined by the exact sequence q

1 −→ K 1 −→ K × −→ q(K × ) −→ 1. In other words, it makes the Haar measure on q(K × ), obtained by the restriction of the Haar measure d× x on k × , equal to the quotient of the Haar measure d× x on K × by dh on K 1 . Next, we consider some explicit forms of these measures. If K is a quadratic, then we have a homomorphism K × /k × −→ K 1 ,

t 7−→ t/t.

It is an isomorphism by Hilbert’s Theorem 90. The Haar measures are compatible under this isomorphism. If K is a quadratic field extension, then the Haar measures dx and d× x respectively on K and K × are the same as the measures normalized in the last subsection by viewing K as a local field. If K = k ⊕ k is split, then the Haar measure on K = k ⊕ k (resp. K × = k × × k × ) is compatible with the Haar measure dx (resp. d× x).

INTRODUCTION AND STATEMENT OF MAIN RESULTS

23

Still, consider the case of quadratic extension. If k is non-archimedean, denote by d ∈ k the different of k over Qp , and by D ∈ Ok the discriminant of K in k. Then 1 × , d× x) = |D| 2 |d|. vol(OK , dx) = vol(OK Furthermore,   2 1 1 vol(K ) = |d| 2  1 1  2|D| 2 |d| 2

if k = R and K = C, if K/k is nonsplit and unramified, if K/k is ramified.

Now assume that K is a quaternion algebra over k. Note that by the split case K = M2 (k), we have normalized Haar measures on GL2 (k), SL2 (k) and PGL2 (k). If k is non-archimedean, then the normalization gives vol(GL2 (Ok )) = ζk (2)−1 vol(Ok )4 ,

vol(SL2 (Ok )) = ζk (2)−1 vol(Ok )3 .

If k = R and K is the Hamiltonian algebra, then the self-dual measure dx on K is four times the usual Lebesgue measure under the natural isometry K = R4 . In this case vol(K 1 ) = 4π 2 . 1.6.3

Global case

Throughout this book we fix a global field F and denote by A = AF the ring of adeles of F . We also fix a quadratic field extension E of F , and denote by η : F × \A× → C× the quadratic character determined by this extension. Except in Chapter 1 and Chapter 2, F is assumed to be totally real and E is assumed to be totally imaginary. We always use v to denote a place of F . For each place v of F , we choose | · |v , ψv , dxv , d× xv as above. By tensor products, they induce global | · |A , ψ, dx, d× x. The absolute values satisfy the product formula, the sum ψ = ⊗v ψv is actually a character on A/F , and the volume of A/F is exactly one under the product measure. For non-archimedean v, we usually denote by OFv the integer ring of Fv , by pv the corresponding prime ideal, by Nv the cardinality of its residue field OFv /pv , by $v a uniformizer of Fv , and by dv ∈ Fv the local different of F over Q. Denote by Dv ∈ Fv the discriminant of the quadratic extension Ev in Fv . We use the convention that Dv = dv = 1 if v is archimedean. Denote by Z = GL1 , viewed as an algebraic group over F . We denote by T = E × the algebraic group over F , and E 1 = {y ∈ E × : q(y) = 1} also defines an algebraic group over F . At every place v, the Haar measures on Z(Fv ), T (Fv ) and E 1 (Fv ) are defined and thus give product Haar measures on Z(A), T (A) and E 1 (A). All of them are Tamagawa measures. The total volume vol(E 1 \E 1 (A)) = 2L(1, η).

24

CHAPTER 1

View V1 = (E, q = NE/F ) as a two-dimensional vector space over F . Let E 1 act on V1 by multiplication. It induces an isomorphism SO(V1 ) ' E 1 of algebraic groups over F . Hilbert Theorem 90 gives an isomorphism SO(V1 ) ' T /Z by E × /F × −→ E 1 ,

t 7−→ t/t.

Let B be a quaternion algebra over F with reduced norm q. Denote B 1 = {y ∈ B × : q(y) = 1}. We have defined Haar measures on Bv× and Bv1 , and they give product Haar measures on BA× and BA1 . Both of them are Tamagawa measures. It follows that vol(B 1 \BA1 ) = 1,

vol(B × \BA× /Z(A)) = 2.

In particular, we have measures on SL2 and GL2 by taking B to be the matrix algebra. More generally, let G be an algebraic group over F . If G is semisimple, endow G(A) with the Tamagawa measure. If Z = GL1 is contained in the center of G such that G/Z is semisimple, endow G(A) with the Haar measure which induces the Tamagawa measure on G(A)/Z(A) via the quotient by (A× , d× x). 1.6.4

Notation on quaternion algebras

Throughout this book, Σ denotes a finite set of places of F . It comes from local root numbers of the Rankin-Selberg L-function. It is assumed to contain all the archimedean places except in Chapter 1 and Chapter 2. We also denote by B the unique quaternion algebra over A such that for every place v of F , the quaternion algebra Bv := B ⊗A Fv over Fv is isomorphic to the matrix algebra if and only if v ∈ / Σ. Alternatively, one can define Bv according to Σ, and B as a restricted product of Bv . We say B is coherent if it is a base change of a quaternion algebra over F ; otherwise, we say B is incoherent. It follows that B is coherent if and only if the cardinality of Σ is even. The reduced norm q makes B a quadratic space V = (B, q) over A. Fix an embedding EA ,→ B if it exists. It gives an orthogonal decomposition B = EA + EA j,

j2 ∈ A × .

Then we get two induced subspaces V1 = (EA , q) and V2 = (EA j, q). Apparently V1 is the base change of the F -space V1 = (E, q). We usually write x = x1 + x2 for the corresponding orthogonal decomposition of x ∈ V. Assume that the cardinality of Σ is odd. We will keep this assumption throughout Chapters 3–8 in this book. Then B is incoherent. But we will get a coherent one by increasing or decreasing Σ by one element. For any place v of F , denote by B(v) the quaternion algebra over F obtained from B by switching the Hasse invariant at v. We call B(v) the nearby quaternion

25

INTRODUCTION AND STATEMENT OF MAIN RESULTS

algebra corresponding to v. Throughout this book, we will fix an identification B(v) ⊗F Av ∼ = Bv . Fix an embedding E ,→ B(v) if v isnonsplit in E. In this case, such an embedding always exists. Then we also have an orthogonal decomposition B(v) = V1 ⊕ V2 (v). For any quaternion algebra B over Fv with a fixed embedding Ev ,→ B, we define q(x2 ) λ : B × −→Fv , x 7−→ q(x) where x = x1 + x2 is the orthogonal decomposition induced by Ev ,→ B. This definition applies to all the quaternion algebras above locally and globally. 1 × The measures on B× v and Bv normalized above define Haar measures on B and B1 := {x ∈ B : q(x) = 1}. They can be viewed as analogues of Tamagawa measures. 1.6.5

L-functions

All global L-functions in this book are the complete L-functions containing the archimedean parts. These include ζF (s),

L(s, η),

L(s, π, χ),

L(s, π, ad).

For example, the Dedekind zeta function Y ζF (s) = ζFv (s) v

is a product over all places v of F . The local L-function ζFv (s) has been introduced in the definition of Haar measures over local fields above. 1.6.6

Subgroups of GL(2)

We introduce the matrix notation:       a a 1 ∗ (a) = m(a) = , d(a) = , d 1 a−1 a       cos θ sin θ 1 1 b , w= . n(b) = , kθ = − sin θ cos θ −1 1 We denote by P ⊂ GL2 and P 1 ⊂ SL2 the subgroups of upper triangular matrices, and by N the standard unipotent subgroup of them. Denote by A ⊂ GL2 the subgroup of diagonal matrices. We have canonical isomorphisms N ' Ga and A ' G2m and thus they are endowed with the Haar measures induced from Fv and Fv× . For any local field Fv , the character   a 12 a b δv : P (Fv ) −→ R× , 7−→ d d v

26

CHAPTER 1

extends to a function δv : GL2 (Fv ) → R× by Q Iwasawa decomposition. For the global field F , the product δ = v δv gives a function on GL2 (A). 1.6.7

Averages and regularized integrations

Let G be a topological group. Assume that G has a left Haar measure dg with finite total volume vol(G). For any function f on G, define Z Z 1 f (g)dg := f (g)dg. vol(G) G G It is independent of the choice of the measure dg. If G is a finite group, then Z 1 X f (g)dg = f (g). |G| G g∈G

Let F be a totally real field. For a function f on F × \A× which is invariant × under the archimedean part F∞ , denote the regularized average of f on A× by Z Z f (z)dz := f (z)dz. F × \A× /Fτ×

A×

Here τ is any archimedean place of F . The definition is independent of the choice of τ . The quotient group F × \A× /Fτ× is a compact with total volume 1 Ress=1 ζF (s) 2 if we use the Haar measure normalized in §1.6.1. If f is further invariant under some open compact subgroup U of A× f , then we have Z X 1 f (z)dz = × × × f (z). |F \A /F∞ U | A× × × × vol(F × \A× /Fτ× ) =

z∈F \A /F∞ ·U

The right-hand side is just a finite sum and the average does not depend on choice of U . Let G be a reductive group over F with an embedding of Z = Gm into the center of G. Assume that the volume of G(F )\G(A)/Z(A) is finite under some Haar measure dg of G(A)/Z(A). Let f be an automorphic function on G(A) which is invariant under the action of Z(F∞ ). We define the regularized integration Z Z Z ∗ f (g)dg := f (zg)dzdg. (1.6.1) G(F )\G(A)/Z(A)

G(F )\G(A)/Z(A)

Z(A)

We also define the regularized average Z f (g)dg G(F )\G(A)/Z(A)

:=

1 vol(G(F )\G(A)/Z(A))

Z

Z f (zg)dzdg.

G(F )\G(A)/Z(A)

Z(A)

27

INTRODUCTION AND STATEMENT OF MAIN RESULTS

In these definitions, f is not required to be invariant under the whole Z(A). We sometimes abbreviate [G] := G(F )\G(A)/Z(A) in the domain of the integrations. For the application in this book, the definition applies to the algebraic groups GL2 and T = E × over F . In the case T = E × , if f is further invariant under T (F∞ ), then Z X 1 f (t)dt = f (t). |T (F )\T (A)/T (F∞ )U | T (F )\T (A)/Z(A) t∈T (F )\T (A)/T (F∞ )U

Here U is an open compact subgroup of T (Af ) which acts trivially on f , and the right-hand side does not depend on the choice of U . 1.6.8

Vector spaces over number fields

Let M be a number field, and V be a vector space over M . The canonical isomorphism M M ⊗Q C = C ι∈Hom(M,C)

induces a canonical decomposition M

V ⊗Q C =

V ι.

ι∈Hom(M,C)

Here we denote V ι = V ⊗(M,ι) C for any embedding ι : M ,→C. In particular, there is a canonical embedding V ι ,→V ⊗Q C. For any v ∈ V , denote by v ι = v ⊗(M,ι) 1 the corresponding element of V ι . Then we have a decomposition X v= v ι ∈ V ⊗Q C. ι∈Hom(M,C)

It is exactly the spectral decomposition of the complex vector space V ⊗Q C under the action of M .

Chapter Two Weil Representation and Waldspurger Formula In this chapter, we will review the theory of Weil representation and its applications to an integral representation of the Rankin–Selberg L-function L(s, π, χ) and to a proof of Waldspurger’s central value formula. We will mostly follow Waldspurger’s treatment with some modifications including Kudla’s construction of incoherent Eisenstein series. We will start with the classical theory of Weil representation of O(F ) × SL2 (F ) on S(V ) for an orthogonal space V over a local field F and its extension to GO(F ) × GL2 (F ) on S(V × F × ) by Waldspurger. We then define theta functions, state the Siegel–Weil formula, and define normalized local Shimizu lifting. The main result of this chapter is an integral formula for the L-series L(s, π, χ) using a kernel function I(s, g, χ, Φ). This kernel function is a mixed Eisenstein and theta series attached to each Φ ∈ S(V×A× ) for V, an orthogonal space obtained from a quaternion algebra over A. The Waldspurger formula is a direct consequence of the Siegel–Weil formula. After the proof of Waldspurger formula, we list some computational results on three types of incoherent Eisenstein series in §2.5. These Eisenstein series will be used in the remaining chapters of this book (for different purposes). 2.1

WEIL REPRESENTATION

Let us start with some basic setup on Weil representation. We follow closely Waldspurger [Wa]. 2.1.1

Non-archimedean case

Let k be a non-archimedean local field and (V, q) a quadratic space over k. Let f 2 (k) the metaplectic O = O(V, q) denote the orthogonal group of (V, q) and SL double cover of SL2 (k). f 2 (k) is a central extension of SL2 (k) by {±1}. One can write Recall that SL f SL2 (k) = SL2 (k) × {±1} with group law given by (g1 , 1 ) · (g2 , 2 ) = (g1 g2 , 1 2 β(g1 , g2 )). Here β : SL2 (k) × SL2 (k) → {±1} is a cocycle in H 2 (SL2 (k), {±1}). Denote by S(V ) the space of locally constant and compactly supported complex-valued functions on V . Such functions are also called Schwartz–Bruhat

29

WEIL REPRESENTATION AND WALDSPURGER FORMULA

f 2 (k)×O(k) functions on V . The Weil representation is an action r of the group SL on V . For any Φ ∈ S(V ), the action is given as follows: • r(h)Φ(x) = Φ(h−1 x),

h ∈ O(k);

• r(m(a))Φ(x) = χ(V,q) (a)|a|dim V /2 Φ(ax), • r(n(b))Φ(x) = ψ(bq(x))Φ(x), • r(w, )Φ = 

dim V

b ∈ k;  w= −1

b γ(V, q)Φ,

a ∈ k× ;

1

 ,  ∈ {±1}.

Several notations need to be explained. The representation depends on the choice of a character ψ : k → C× . We always take ψ to be the standard one described in §1.6. As usual, m(a) and n(b) denote the element (m(a), 1) and f 2 (k). The character χ(V,q) : k × → C× takes values in {±1, ±i}. (n(b), 1) in SL b denotes The constant γ(V, q), an 8-th root of unity, is the Weil index. Finally, Φ the Fourier transform given by Z b Φ(x) := Φ(y)ψ(hx, yi)dy. V

Here hx, yi := q(x + y) − q(x) − q(y) is the corresponding inner product, and the integral uses the self-dual measure. There are many simplifications if dim V is even. In that case, r is trivial on f 2 (k), and thus the Weil representation descends to a the subgroup {±1} of SL representation of SL2 (k) × O(k) on S(V ). The Weil index γ(V, q) is a 4-th root of unity, and the character χ(V,q) becomes the quadratic character associated to the quadratic space (V, q). Namely, χ(V,q) (a) = (a, (−1)

dim V 2

det(V, q)),

a ∈ k× .

Here the right-hand side denotes the Hilbert symbol, and det(V, q) ∈ k × /(k × )2 denotes the image in k × /(k × )2 of the determinant of the moment matrix {(xi , xj )}1≤i,j≤dim V for any basis {xi }1≤i≤dim V of V over k. It is independent of the choice of the basis. 2.1.2

Archimedean case

Now let (V, q) be a quadratic space over R. A Schwartz function Φ on V is an infinitely differentiable complex-valued function on V such that all the partial derivatives of any order are of rapid decay. We explain it as follows. Choose a basis of V , and view V as a Euclidean space of dimension d = dimR V . Then we have coordinate functions t1 , · · · , td . The requirement for Φ is that, for any d-tuples (e1 , · · · , ed ) and (e01 , · · · , e0d ) of non-negative integers, the partial derivative ∂Φ/(∂ e1 t1 · · · ∂ ed td ) exists everywhere, and e0 ∂Φ e0d 1 < ∞. sup t1 · · · td e1 ∂ t1 · · · ∂ ed td (t1 ,··· ,td )

30

CHAPTER 2

The definition does not depend on the choice of the coordinate functions. Denote by S(V ) the space of Schwartz functions on V . The Weil represenf 2 (R) × O(R) on V is defined by the same formulae as tation r of the group SL in the non-archimedean case. It is usually more convenient to consider a smaller space called the Fock model. The theory depends on an orthogonal decomposition V = V + +V − such that the restrictions of q on V ± are positive and negative definite respectively, which we fix once for all. The Fock model S(V ) is the space of functions on V of the form Φ(x) = P (x)e−2π(q(x

+

)−q(x− ))

,

x = x+ + x− , x± ∈ V ± .

Here P can be any polynomial function on V (via any coordinate functions). It is a linear subspace of S(V ), and it is obviously not stable under the action of SL2 (R). However, it has an action of (G, K) in the sense of Harish-Chandra. Recall the Lie algebra G = sl2 (R) × o(V ) and the maximal compact subf 2 (R) is the preimage of SO2 (R) in f 2 (R) × K 0 . The group SO group K = SO 0 + f SL2 (R), and the group K = O(V ) × O(V − ) is the stabilizer of the orthogonal decomposition V = V + + V − . By the formulae of the Weil representation above, for any Φ ∈ S(V ) and f 2 (R) × O(V ), the action r(g, h)Φ gives a smooth (rapid-decay) func(g, h) ∈ SL tion on V . This function lies in S(V ) for (g, h) ∈ K, but it is not necessarily true for general (g, h). However, it is easy to check that the action of any (∂1 , ∂2 ) ∈ sl2 (R) × o(V ) defined by r(∂1 , ∂2 )Φ :=

d d |t1 =0 |t =0 r(et1 ∂1 , et2 ∂2 )Φ dt1 dt2 2

still gives an element of S(V ). Therefore, we get a well-defined action of (G, K) on S(V ). If (V, q) is (positive or negative) definite, then O(V ) is compact and K 0 = O(V ), and the Weil representation gives a full action of O(V ) on S(V ). Functions in S(V ) are of the form Φ(x) = P (x)e−2π|q(x)| ,

x ∈ V.

The distinguished element given by P (x) = 1 is just the usual Gaussian, and also called the standard Schwartz function. The indefinite case is used at few places in this book. We will omit the case k ' C since it is also only used in Chapter 2. Readers interested in that case may find details in [Wa]. 2.1.3

Extension to groups of similitudes

Assume that dim V is even for simplicity. Following Waldspurger [Wa], we extend this action to an action r of GL2 (k) × GO(V ) in the non-archimedean

WEIL REPRESENTATION AND WALDSPURGER FORMULA

31

case and an action of (G, K) in the real case. In §4.1, we will see a slightly different space of Schwartz functions in the archimedean case. If k is non-archimedean, let S(V × k × ) be the space of Schwartz–Bruhat functions, i.e., complex-valued locally constant and compactly supported functions on V × k × . We also write it as S(V × k × ) in the non-archimedean case. The Weil representation is extended by the following formulae: • r(h)Φ(x, u) = Φ(h−1 x, ν(h)u), • r(g)Φ(x, u) = ru (g)Φ(x, u),

h ∈ GO(k); g ∈ SL2 (k);

• r(d(a))Φ(x, u) = Φ(x, a−1 u)|a|− dim V /4 ,

a ∈ k× .

Here ν : GO(k) → k × denotes the similitude map. In the right-hand side of the second formula Φ(x, u) is viewed as a function of x by fixing u, and ru is the Weil representation on V with new norm uq. If k = R is real, let S(V × R× ) be the space of Schwartz functions on V × R× , i.e., infinitely differentiable complex-valued functions Φ(x, u) which have compact support for u in R× and give Schwartz functions on x ∈ V for any fixed u. The Weil representation r of GL2 (k) × GO(V ) on S(V × R× ) is defined by similar formulae. The Fock model S(V × R× ) is the space of finite linear combinations of functions of the form H(u)P (x)e−2π|u|(q(x

+

)−q(x− ))

where P : V → C is any polynomial function on V , and H is any compactly supported smooth function on R× . Similar to the case of S(V ), the formulae give a smooth function r(g, h)Φ on V × R× for any Φ ∈ S(V × R× ) and (g, h) ∈ GL2 (R) × GO(R), and induce an action of (gl2 (R), go(R)) × (O2 (R), K 0 ) on S(V × R× ). In this book we always consider the Fock model instead of the whole space of Schwartz functions. By the action of GO(k) above, we introduce an action of GO(k) on V × k × given by h ◦ (x, u) := (hx, ν(h)−1 u). This action stabilizes the subset (V × k × )a := {(x, u) ∈ V × k × : uq(x) = a}. 2.1.4

Global case

Now we assume that F is a number field and that (V, q) is a quadratic space over F . Then we can define a Weil representation r on S(VA ) (which actually f 2 (A) × O(VA ). When dim V is even, we can define an action depends on ψ) of SL r of GL2 (A) × GO(VA ) on S(VA × A× ) which is the restricted tensor product of S(Vv × Fv ) with spherical element given by the characteristic function of VOFv × OF×v once a global lattice is chosen.

32

CHAPTER 2

Notice that the representation r depends only on the quadratic space (VA , q) over A. We may define representations directly for a pair (V, q) of a free Amodule V with non-degenerate quadratic form q : V → A. It still makes sense to define S(V × A× ) to be the restricted tensor product of S(Vv × Fv× ). The Weil representation extends in this case. If (V, q) is a base change of an orthogonal space over F , then we call this Weil representation coherent; otherwise it is called incoherent. 2.1.5

Siegel–Weil formula

Let F be a number field, and (V, q) a quadratic space over F . Then for any f 2 (A) × Φ ∈ S(VA ), we can form a theta series as a function on SL2 (F )\SL O(V )\O(VA ): X f 2 (A) × O(VA ). θ(g, h, Φ) = r(g, h)Φ(x), (g, h) ∈ SL x∈V

Similarly, when V has even dimension we can define theta series for Φ ∈ S(VA × A× ) as an automorphic form on GL2 (F )\GL2 (A) × GO(V )\GO(VA ): X θ(g, h, Φ) = r(g, h)Φ(x, u). (x,u)∈V ×F ×

Now we introduce the Siegel Eisenstein series. For Φ ∈ S(VA ) and s ∈ C, we have a section g 7−→ δ(g)s r(g)Φ(0) dim V

SL2 s+ 2 in the induced representation IndP ) defined by 1 (χV | · |      V a b s+ dim f 2 (A) → C f 2 f : SL χ (a)f (g) . g = |a| V a−1 f

Here δ is the modulo function explained in the introduction. Thus we can form an Eisenstein series X δ(γg)s r(γg)Φ(0). E(s, g, Φ) = γ∈P 1 (F )\SL2 (F )

It has a meromorphic continuation to s ∈ C and a functional equation with center s = 1 − m 2 . Here we denote m = dim V . Denote by r the Witt index of V , i.e., the maximal dimension of F -subspaces of V consisting of elements of norms zero. Then we always have r ≤ m 2 . We call V anisotropic if r = 0, i.e., q(x) = 0 for x ∈ V if and only if x = 0. Theorem 2.1 (Siegel–Weil formula). Assume that (V, q) is anisotropic or m − r > 2. Then Z E(0, g, Φ) = κ θ(g, h, Φ)dh. SO(V )\SO(VA )

WEIL REPRESENTATION AND WALDSPURGER FORMULA

33

Here the integration uses the Haar measure of total volume one, and ( 2 if m = 1, 2; κ= 1 if m > 2. The theorem holds for more general reductive pairs. For the current situation, it was first treated by Siegel [Si] and Weil [We1, We2] for the case m > 4, and completed by Rallis [Ra] and Kudla–Rallis [KR1, KR2] (for more general symplectic groups). For a more detailed history, we refer to Kudla [Ku4]. Usually the integration is stated over O(V )\O(VA ), but it is easy to see that the integration over SO(V )\SO(VA ) gives the same result. The theorem implicitly states that the Eisenstein series E(s, g, Φ) is analytic at s = 0, and the integration on the right-hand side converges absolutely. Notice that both sides of the equality in the theorem define elements in the space ∞ f HomSO(VA )×SL f 2 (A) (S(V (A)), C (SL2 (F )\SL2 (A))).

One can show that this space is one-dimensional. Thus the most important part of the theorem is the constant κ. Fix an element u ∈ F × . We are interested in the Siegel–Weil formula in the following cases: (1) (V, qV ) = (E, uq) with E a nontrivial quadratic field extension of F and q = NE/F ; (2) (V, qV ) = (B0 , uq) where B0 is the subspace of trace-free elements of anonsplit quaternion algebra B over F and q is induced from the reduced norm on B; (3) (V, qV ) = (B, uq) with B anonsplit quaternion algebra over F and q the reduced norm. The Siegel–Weil formula is always valid in (1), and it is valid in (2) and (3) if B is not the matrix algebra. The Tamagawa number of SO(V ) in the above cases are respectively 2L(1, η), 2, 2. Then it is easy to convert the integration in terms of the integration using the Tamagawa measure. 2.1.6

Local Siegel–Weil formula

Consider the Fourier expansion of both sides of the Siegel–Weil formula of an anisotropic orthogonal space (V, q) over F . Recall that the Siegel–Weil formula asserts that Z κ E(0, g, Φ) = θ(g, h, Φ)dh. vol(SO(F )\SO(A)) SO(F )\SO(A) Now we write both sides in terms of Fourier series.

34

CHAPTER 2

The Eisenstein series has a Fourier expansion E(s, g, Φ) = δ(g)s r(g)Φ(0) + W0 (s, g, Φ) +

X

Wa (s, g, Φ).

a∈F ×

Here the a-th Whittaker function for any a ∈ F is defined by Z Wa (s, g, Φ) = δ(wn(b)g)s r(wn(b)g)Φ(0) ψ(−ab)db. A

One can define the local Whittaker integrals Z δv (wn(b)g)s r(wn(b)g)Φv (0) ψ(−ab)db. Wa,v (s, g, Φv ) = Fv

It has a meromorphic continuation to all s ∈ C. Denote by SOx the stabilizer of any x ∈ V in SO as an algebraic group. The right-hand side of the Siegel–Weil formula is equal to Z X r(g, h)Φ(x)dh SO(F )\SO(A) x∈V

Z

X

=

X

r(g, h)Φ(h0−1 x)dh

SO(F )\SO(A) x∈SO(F )\V h0 ∈SO (F )\SO(F ) x

=

X x∈SO(F )\V

=

X x∈SO(F )\V

Z r(g, h)Φ(x)dh SOx (F )\SO(A)

Z vol(SOx (F )\SOx (A))

r(g, h)Φ(x)dh. SOx (A)\SO(A)

For any a ∈ F × , the Siegel–Weil formula yields an identity of the a-th Fourier coefficients of the two sides as follows: Z vol(SOxa (F )\SOxa (A)) Wa (0, g, Φ) = κ r(g, h)Φ(xa )dh. vol(SO(F )\SO(A)) SOxa (A)\SO(A) Here xa ∈ V is any fixed element of norm a. If such an xa does not exist, the left-hand side is considered to be zero. Note that both integrals above are products of local integrals. It follows that the global identity induces the local identity Z r(g, h)Φv (xa )dh Wa,v (0, g, Φv ) = cv SOxa (Fv )\SO(Fv )

at every place v of F . Here cv 6= 0 is a constant independent of g, Φv . Actually Weil [We2] proved the Siegel–Weil formula by first showing such a local version. The delicate part is to determine the constant cv (after normalizing the Haar

35

WEIL REPRESENTATION AND WALDSPURGER FORMULA

measure on the right-hand side). Note that even if the global quadratic space is anisotropic, the local quadratic space can be isotropic. We state the precise result for any quadratic space (V, q) over any local field k. For any a ∈ k and Φ ∈ S(V ), denote Z Wa (s, g, Φ) = δ(wn(b)g)s r(wn(b)g)Φ(0) ψ(−ab)db. k

Denote by V (a) the set of elements of V with norm a. If it is non-empty, then any xa ∈ V (a) gives a bijection V (a) ∼ = SOxa (k)\SO(k). Under this identity, SO(k)invariant measures of V (a) correspond to Haar measures of SOxa (k)\SO(k). They are unique up to scalar multiples. Theorem 2.2 ([We2], local Siegel–Weil). Let (V, q) be a quadratic space over a local field k. Then the following are true: (1) There is a unique SO(k)-invariant measure da x of V (a) for every a ∈ k × such that Z Ψ(a) := Φ(x)da x V (a)

gives a continuous function for a ∈ k × , and that Z Z Ψ(a)da = Φ(x)dx. k

V

Here da and dx are respectively the self-dual measures on k and V with respect to ψ. (2) With the above measure, Z Wa (0, g, Φ) = γ(V, q)

r(g)Φ(x)da x,

∀a ∈ k × , Φ ∈ S(V ).

V (a)

The right-hand side is considered to be zero if V (a) is empty. The measure da x is easy to determine for small groups in practice. The case a = 0 may be obtained by carefully taking the limit a → 0. The result is similar but more complicated. We omit it here. Remark. For each a ∈ k × representable by V , both sides of the equality in the second part of the theorem define nonzero elements in the space ∞ f HomSO(V )×SL f 2 (F ) (S(V ), C (N (F )\SL2 (F ), ψa )),

f 2 (F ), ψa ) is the space of smooth functions on SL f 2 (F ) with where C ∞ (N (F )\SL character ψa (x) := ψ(ax) under left action of N ' k. It can be shown that this space is one-dimensional. Thus the significance is still the ratio of these two elements. See [Ra, Proposition 4.2].

36 2.2

CHAPTER 2

SHIMIZU LIFTING

Let F be a local or global field and B a quaternion algebra over F . Write V = (B, q) as an orthogonal space with quadratic form q defined by the reduced norm on B. Let B × × B × act on V by x 7→ h1 xh−1 2 ,

x ∈ V,

hi ∈ B × .

Then we have an exact sequence: 1−→F × −→(B × × B × ) o {1, ι}−→GO(V )−→1. Here ι : x 7→ x is the main involution on V = B, and on B × × B × by (h1 , h2 ) 7→ ¯ −1 ), and F × is embedded into the group in the middle by x 7→ (x, x) o 1. ¯ −1 , h (h 2 1 The theta lifting of any representation σ of GL2 (F ) to GO(V ) is induced by the representation π e ⊗ π on B × × B × , where π is the Jacquet–Langlands correspondence of σ. Recall that π 6= 0 only if B = M2 (F ) or σ is discrete. In that case, e) = 1. dimC HomGL2 (F )×B × ×B × (σ ⊗ S(V × F × ), π ⊗ π In the global case, there is a canonical element in this one-dimensional space given by the theta lifting. We will introduce a normalized form in the local case compatible with the global case. 2.2.1

Global Shimizu lifting

Let F be a number field with A the adele ring, σ be a cuspidal automorphic representation of GL2 (A), and B be a quaternion algebra over F . Denote V = (B, q) as above. For any Φ ∈ S(V (A) × A× ), we have the theta function θ(g, h, Φ) =

X X

r(g, h)Φ(x, u),

g ∈ GL2 (A), h ∈ BA× × BA× .

u∈F × x∈V

For any ϕ ∈ σ, define the normalized global Shimizu lifting Z ζF (2) ϕ(g)θ(g, h, Φ)dg, θ(Φ ⊗ ϕ)(h) := 2L(1, π, ad) GL2 (F )\GL2 (A)

h ∈ BA× × BA× .

(2.2.1) It defines an automorphic form θ(Φ ⊗ ϕ) ∈ π ⊗ π e. We will see why we use the normalizing factor. e are If ωσ is the central character of σ, then the central character of π and π respectively ωσ and ωσ−1 . Let F :π⊗π e→C

37

WEIL REPRESENTATION AND WALDSPURGER FORMULA

be the canonical map defined by the Petersson bilinear pairing Z f1 (g)f2 (g)dg, f1 ∈ π, f2 ∈ π e. (f1 , f2 )pet = B × \BA× /Z(A)

The integration uses the Tamagawa measure, which gives vol(B × \BA× /Z(A)) = 2. In the following, let W−1 : σ → W (σ, ψ) be the canonical map from σ to its Whittaker model with respect to the additive character ψ. It is given by the Fourier coefficient. Fix a decomposition W−1 = ⊗v W−1,v . Proposition 2.3 (Waldspurger). For any ϕ ∈ σ and Φ = ⊗v Φv ∈ S(V (A)× A× ), one has Z Y ζv (2) Wϕ,−1,v (g)r(g)Φv (1, 1)dg. Fθ(Φ ⊗ ϕ) = L(1, πv , ad) N (Fv )\GL2 (Fv ) v Here the local factor is 1 for almost all places v. Proof. Here we only repeat Waldspurger’s proof in the case that B isnonsplit so that the Siegel–Weil formula is applicable. For the split case we refer to the original paper [Wa], which takes a different method without the Siegel–Weil formula. The pairing between π and π e is given by integration on the diagonal of (B × \BA× /Z(A))2 . It follows that Z Z ζF (2) Fθ(Φ ⊗ ϕ) = ϕ(g)θ(g, (h, h), Φ)dgdh. 2L(1, π, ad) B × \BA× /Z(A) GL2 (F )\GL2 (A) Let B0 be the subspace of trace-free elements of B. Then B = F ⊕ B0 gives an orthogonal decomposition. The diagonal can be identified with SO0 = SO(B0 ). Thus we have globally Z Z dh θ(g, h, Φ)ϕ(g)dg. Fθ(Φ ⊗ ϕ) = SO0 (F )\SO0 (A)

GL2 (F )\GL2 (A)

The Siegel–Weil formula for the orthogonal space B0 gives Z θ(g, h, Φ)dh = 2J(0, g, Φ).

(2.2.2)

SO0 (F )\SO0 (A)

Here J is a mixed theta-Eisenstein series X δ(γg)s J(s, g, Φ) = γ∈P (F )\GL2 (F )

X

r(γg)Φ(x, u).

(x,u)∈F ×F ×

To check the truth of (2.2.2), we can assume that g ∈ SL2 (A) because both sides depend only on r(g)Φ. By linearity it suffices to consider the case Φ = ΦF ⊗ Φ0 for ΦF ∈ S(A × A× ) and Φ0 ∈ S(B0 (A) × A× ) in the sense that Φ(x, u) = ΦF (z, u)Φ0 (x0 , u),

x = z ⊕ x0 .

38

CHAPTER 2

f 2 (A), Then for any h ∈ SO0 (A) and g ∈ SL2 (A) ⊂ SL X θ(g, u, ΦF )θ(g, h, u, Φ0 ), θ(g, h, Φ) = u∈F ×

X

J(s, g, Φ) =

θ(g, u, ΦF )E(s, g, u, Φ0 ).

u∈F ×

Here θ(g, u, ΦF ) =

X

r(g)ΦF (z, u),

z∈F

θ(g, h, u, Φ0 ) =

X

r(g, h)Φ0 (x0 , u),

x0 ∈B0

X

E(s, g, u, Φ0 ) =

δ(γg)s r(γg)Φ0 (0, u).

γ∈P 1 (F )\SL2 (F )

The (2.2.2) is reduced to the Siegel-Weil formula Z θ(g, h, u, Φ0 )dh = 2E(0, g, u, Φ0 ). SO0 (F )\SO0 (A)

Go back to Fθ(Φ ⊗ ϕ). We have Fθ(Φ ⊗ ϕ) =

ζF (2) L(1, π, ad)

Denote

Z ϕ(g)J(0, g, Φ)dg. GL2 (F )\GL2 (A)

Z A(s) =

ϕ(g)J(s, g, Φ)dg. GL2 (F )\GL2 (A)

By definition, Z A(s) = GL2 (F )\GL2 (A)

Z

δ(γg)s r(γg)Φ(z, u)dg

γ∈P (F )\GL2 (F ) (z,u)∈F ×F ×

δ(g)s ϕ(g)

=

X

X

ϕ(g)

P (F )\GL2 (A)

X

r(g)Φ(z, u)dg

(z,u)∈F ×F ×

=A1 (s) + A2 (s). Here Z A1 (s) = P (F )\GL2 (A)

Z A2 (s) = P (F )\GL2 (A)

X

δ(g)s ϕ(g)

r(g)Φ(z, u)dg,

(z,u)∈F × ×F ×

δ(g)s ϕ(g)

X u∈F ×

r(g)Φ(0, u)dg.

WEIL REPRESENTATION AND WALDSPURGER FORMULA

39

It is easy to see that A2 (s) = 0. In fact, since δ(g)s r(g)Φ(0, u) is invariant under the left action of N (A), the vanishing of A2 (s) is a consequence of the cuspidality property Z ϕ(ng)dn = 0. N (F )\N (A)

The summation in A1 (s) is a single orbit over the diagonal group of GL2 (F ). Thus Z δ(g)s ϕ(g)r(g)Φ(1, 1)dg A(s) = N (F )\GL2 (A) Z = δ(g)s Wϕ,−1 (g)r(g)Φ(1, 1)dg. N (A)\GL2 (A)

It is easy to check that for almost all v, Z L(s + 1, πv , ad) . δ(g)s Wϕ,−1,v (g)r(g)Φv (1, 1)dg = ζv (2s + 2) N (Fv )\GL2 (Fv ) Thus the result follows. 2.2.2



Normalization of the local Shimizu lifting

Now we go back to local situation. Let F be a local field and σ be an irreducible infinite representation of GL2 (F ). We realize σ as a subspace W(σ, ψ) of smooth functions on GL2 (F ) with character ψ under the left translation of N (F ). Let B be any quaternion algebra over F which gives a quadratic space. Now we normalize the local theta lifting θ : W(σ, ψ) ⊗ S(B ⊗ F × )−→π ⊗ π e by Fθ(W ⊗ Φ) =

ζF (2) L(1, π, ad)

Z W (g) r(g)Φ(1, 1)dg.

(2.2.3)

N (F )\GL2 (F )

Here F : π ⊗ π e → C is the canonical contraction. In the global case V = B over a number field, we will have a compatibility of the local and global Shimizu liftings. Note that globally W−1 : σ → W(σ, ψ) is uniquely determined as taking the first Fourier coefficients and F : π ⊗ π e→C is defined in terms of the Petersson pairing. Fix compatible decompositions σ = ⊗v σv and W−1 = ⊗v W−1,v . Fix a decomposition F = ⊗v Fv compatible e = ⊗v π ev . Normalize the global θ by (2.2.1), and normalize with π = ⊗v πv and π the local θv by (2.2.3). Then we get the following compatibility. Corollary 2.4. We have a decomposition θ = ⊗θv in HomGL2 (A)×B × ×B × (σ ⊗ S(V (A) × A× ), π ⊗ π e). A

A

40

CHAPTER 2

2.3

INTEGRAL REPRESENTATIONS OF THE L-FUNCTION

In the following we want to describe an integral representation of the L-function L(s, π, χ). Let F be a number field with ring of adeles A. Let B be a quaternion algebra with ramification set Σ. Fix an embedding EA ,→ B. We have an orthogonal decomposition j2 ∈ A × .

B = EA + EA j,

Write V for the orthogonal space B with reduced norm q, and V1 = EA and V2 = EA j as subspaces of V. Then V1 is coherent and it is the adelization of the F -space V1 := (E, q), and V2 is coherent if and only if Σ is even. For Φ ∈ S(V × A× ), we can form a mixed Eisenstein–theta series X X I(s, g, Φ) = δ(γg)s r(γg)Φ(x1 , u). (x1 ,u)∈V1 ×F ×

γ∈P (F )\GL2 (F )

Define its χ-component: Z I(s, g, χ, Φ) =

χ(t)I(s, g, r(t, 1)Φ)dt. T (F )\T (A)

For any ϕ ∈ σ, we introduce the Petersson pairing Z ϕ(g)I(s, g, χ, Φ)dg. P (s, χ, Φ, ϕ) =

(2.3.1)

Z(A)GL2 (F )\GL2 (A)

Here the integration uses the Tamagawa measure. Proposition 2.5 (Waldspurger). If Φ = ⊗Φv and ϕ = ⊗ϕv are decomposable, then Y P (s, χ, Φ, ϕ) = Pv (s, χv , Φv , ϕv ) v

where Pv (s, χv , Φv , ϕv ) Z Z = χ(t)dt Z(Fv )\T (Fv )

δ(g)s W−1,v (g)r(g)Φv (t−1 , q(t))dg.

N (Fv )\GL2 (Fv )

¯ Here W−1 denotes the Whittaker function of ϕ with respect to ψ. Proof. Bring the definition formula of I(s, g, χ, Φ) to obtain an expression for P (s, χ, Φ, ϕ): Z Z X s ϕ(g)δ(g) χ(t) r(g, (t, 1))Φ(x, u)dtdg. Z(A)P (F )\GL2 (A)

T (F )\T (A)

(x,u)∈V1 ×F ×

41

WEIL REPRESENTATION AND WALDSPURGER FORMULA

We decompose the first integral as a double integral Z Z Z dg = Z(A)P (F )\GL2 (A)

Z(A)N (A)P (F )\GL2 (A)

dndg,

N (F )\N (A)

and perform the integral on N (F )\N (A) to obtain Z Z s δ(g) dg χ(t) Z(A)N (A)P (F )\GL2 (A)

T (F )\T (A)

X

W−q(x)u (g)r(g, (t, 1))Φ(x, u)dt.

(x,u)∈V1 ×F ×

Here, as ϕ is cuspidal, the term x = 0 has no contribution to the integral. In this way, we may change variable (x, u) 7→ (x, q(x−1 )u) to obtain the following expression of the sum: X W−u (g)r(g, (t, 1))Φ(x, q(x−1 )u) (x,u)∈E × ×F ×

=

X

W−u (g)r(g, (tx, 1))Φ(1, u).

(x,u)∈E × ×F ×

The sum over x ∈ E × collapses with quotient T (F ) = E × . Thus the integral becomes Z Z X s δ(g) dg χ(t) W−u (g)r(g, (t, 1))Φ(1, u)dt. Z(A)N (A)P (F )\GL2 (A)

T (A)

u∈F ×

The expression does not change if we make the substitution (g, au) 7→ (gd(a)−1 , u). Thus we have Z

s

Z

δ(g) dg Z(A)N (A)P (F )\GL2 (A)

χ(t) T (A)

X

W−u (d(u−1 )g)r(d(u−1 )g, (t, 1))Φ(1, u)dt.

u∈F ×

The sum over u ∈ F × collapses with quotient P (F ), so we obtain the following expression: Z Z s P (s, χ, Φ, ϕ) = δ(g) dg χ(t)W−1 (g)r(g)Φ(t−1 , q(t))dt. Z(A)N (A)\GL2 (A)

T (A)

We may decompose the inside integral as Z Z Z(A)\T (A)

Z(A)

42

CHAPTER 2

and move the integral ωσ · χ|A× = 1 to obtain Z P (s, χ, Φ, ϕ) =

R Z(A)\T (A)

to the outside. Then we use the fact that Z

χ(t)dt

Z(A)\T (A)

δ(g)s W−1 (g)r(g)Φ(t−1 , q(t))dg.

N (A)\GL2 (A)

 When everything is unramified, Waldspurger has computed these integrals and gotten L((s + 1)/2, πv , χv ) . Pv (s, χv , Φv , ϕv ) = L(s + 1, ηv ) Thus we may define a normalized integral Pv◦ by Pv (s, χv , Φv , ϕv ) =

L((s + 1)/2, πv , χv ) ◦ Pv (s, χv , Φv , ϕv ). L(s + 1, ηv )

This normalized local integral Pv◦ will be regular at s = 0 and equal to Z L(1, ηv )L(1, πv , ad) χv (t)F (π(t)θ(Φv ⊗ ϕv )) dt. ζv (2)L(1/2, πv , χv ) Z(Fv )\T (Fv ) We may write this as αv (θ(Φv ⊗ ϕv )) with αv ∈ Hom(πv ⊗ π ev , C) given by the integration of matrix coefficients: Z L(1, ηv )L(1, πv , ad) χv (t)(π(t)f1 , f2 )dt. αv (f1 ⊗ f2 ) = ζv (2)L(1/2, πv , χv ) Z(Fv )\T (Fv ) e, C). We define the global element α := ⊗v αv in Hom(π ⊗ π We now take value or derivative at s = 0 to obtain Proposition 2.6. P (0, χ, Φ, ϕ) =

L(1/2, π, χ) Y αv (θ(Φv ⊗ ϕv )). L(1, η) v

If Σ is odd, then L(1/2, π, χ) = 0, and P 0 (0, χ, Φ, ϕ) =

L0 (1/2, π, χ) Y αv (θ(Φv ⊗ ϕv )). 2L(1, η) v

Remark. Let AΣ (GL2 , χ) denote the direct sum of cuspidal automorphic representations σ on GL2 (A) such that Σ(σ, χ) = Σ. If Σ is even, let I(g, χ, Φ) be the projection of I(0, g, χ, Φ) on AΣ (GL2 , χ−1 ). If Σ is odd, let I 0 (g, χ, Φ)

43

WEIL REPRESENTATION AND WALDSPURGER FORMULA

denote the projection of I 0 (0, g, χ, Φ) on AΣ (GL2 , χ−1 ). Then we have shown that I(g, χ, Φ) and I 0 (g, χ, Φ) represent the functionals ϕ 7→

L(1/2, π, χ) α(θ(Φ ⊗ ϕ)) L(1, η)

if Σ is even,

ϕ 7→

L0 (1/2, π, χ) α(θ(Φ ⊗ ϕ)) 2L(1, η)

if Σ is odd,

on σ respectively.

2.4

PROOF OF WALDSPURGER FORMULA

Assume now Σ is even. We are going to sketch Waldspurger’s proof of his central value formula in Theorem 1.4. Now the space V = V (A) is coming from a global V = B over F . Recall that the Shimizu lifting θ(Φ ⊗ ϕ) ∈ π ⊗ π e, for any Φ ∈ S(V × A× ) and ϕ ∈ σ, is defined as Z ζF (2) θ(Φ ⊗ ϕ)(h) = ϕ(g)θ(g, h, Φ)dg, h ∈ BA× × BA× . 2L(1, π, ad) GL2 (F )\GL2 (A) We are going to compute

=

4L(1, η)2 · (Pπ,χ ⊗ Pπe,χ−1 )(θ(Φ ⊗ ϕ)) Z θ(Φ ⊗ ϕ)(t1 , t2 )χ(t1 )χ−1 (t2 )dt1 dt2 . (T (F )Z(A)\T (A))2

Here the factor 4L(1, η)2 comes from the Tamagawa measure vol(T (F )Z(A)\T (A)) = 2L(1, η). By definition, the right-hand side equals Z ζF (2) ϕ(g)θ(g, Φ, χ)dg 2L(1, π, ad) Z(A)GL2 (F )\GL2 (A) where Z θ(g, χ, Φ) = Z ∆ (A)T (F )2 \T (A)2

θ(g, (t1 , t2 ), Φ)χ(t1 t−1 2 )dt1 dt2 .

Here Z ∆ is the image of the diagonal embedding F × ,→ (B × )2 . We change the variables by t1 = tt2 to get a double integral Z Z χ(t)dt θ(g, (tt2 , t2 ), Φ)dt2 . θ(g, χ, Φ) = T (F )\T (A)

T (F )Z(A)\T (A)

44

CHAPTER 2

Notice that the diagonal embedding of Z\T in B × × B × can be realized as SO(Ej, q) in the decomposition B = E + Ej. See §1.6 for more details. Apply the Siegel–Weil formula in Theorem 2.1 to the space (Ej, uq) to obtain θ(g, χ, Φ) = L(1, η)I(0, g, χ, Φ). Here we recall that Z I(s, g, χ, Φ) =

χ(t)I(s, g, r(t, 1)Φ)dt T (F )\T (A)

with X

I(s, g, Φ) =

X

δ(γg)s

r(γg)Φ(x, u).

(x,u)∈E×F ×

γ∈P (F )\GL2 (F )

Combining with Proposition 2.6, we have Z

θ(Φ ⊗ ϕ)(t1 , t2 )χ(t1 )χ−1 (t2 )dt1 dt2 (T (F )Z(A)\T (A))2

=

ζF (2)L(1/2, π, χ) α(θ(Φ ⊗ ϕ)). 2L(1, π, ad)

Thus we have obtained 4L(1, η)2 · (Pπ,χ ⊗ Pπe,χ−1 )(θ(Φ ⊗ ϕ)) =

ζF (2)L(1/2, π, χ) α(θ(Φ ⊗ ϕ)). 2L(1, π, ad)

In terms of functionals on π ⊗ π e, the above identity can be written as 4L(1, η)2 · Pπ,χ ⊗ Pπe,χ−1 =

ζF (2)L(1/2, π, χ) · α. 2L(1, π, ad)

It proves Theorem 1.4. 2.5

INCOHERENT EISENSTEIN SERIES

Let (V, q) be an orthogonal space over A, and let Φ ∈ S(V) be a Schwartz function. Recall the associated Siegel–Eisenstein series X f 2 (A). δ(γg)s r(γg)Φ(0), g ∈ SL E(s, g, Φ) = γ∈P 1 (F )\SL2 (F )

It has a meromorphic continuation to s ∈ C and a functional equation with center s = 1 − dim2 V . By the standard theory, we have X Wa (s, g, Φ). E(s, g, Φ) = δ(g)s r(g)Φ(0) + W0 (s, g, Φ) + a∈F ×

WEIL REPRESENTATION AND WALDSPURGER FORMULA

45

Here the a-th Whittaker function for any a ∈ F is defined by Z

δ(wn(b)g)s r(wn(b)g)Φ(0) ψ(−ab)db.

Wa (s, g, Φ) = A

One can define the local Whittaker integrals Z

δv (wn(b)g)s r(wn(b)g)Φv (0) ψ(−ab)db.

Wa,v (s, g, Φv ) = Fv

It has a meromorphic continuation to all s ∈ C. If (V, q) is coherent, then the Siegel–Weil formula (in most convergent cases) gives an expression of E(0, g, Φ) in terms of the integral of the corresponding theta series. If (V, q) is incoherent, then there is no theta series available. However, we can still compute (the Fourier coefficients of) E(0, g, Φ) in terms of the local Siegel– Weil formula. In fact, Theorem 2.2 asserts a way to write the local Whittaker functions Wa,v (0, g, Φv ) in terms of averages of the Schwartz functions. Parallel to Theorem 2.1, we are particularly interested in certain subspaces of quaternion algebras. Let B be an incoherent quaternion algebra over a totally real number field F . Assume that B is totally definite at infinity. As usual let q be the reduced norm on B. Fix a number u ∈ F × . We have the following three cases: (1) (V, qV ) = (EA j, uq) where EA j is the orthogonal complement of EA in B for some quadratic field extension E of F with a fixed embedding EA ,→ B of A-algebras; (2) (V, qV ) = (B0 , uq) where B0 is the subspace of trace-free elements of B; (3) (V, qV ) = (B, uq). We will see that the Eisenstein series E(s, g, Φ) are always holomorphic at s = 0 in these cases. If we take Φ∞ to be the standard Gaussian, the related Eisenstein series in these cases have weights 1, 3/2 and 2 respectively. So we refer these cases as weight 1, weight 3/2 and weight 2, though our results are true for general Φ∞ and the weights are different from these numbers in general. The global Weil index γ(V, qV ) = −1 in all three cases. In fact, we first observe that it does not depend on u since it lies in F × . Then we see that γ(B, uq) = ε(B) = −1 by definition. Here ε(B) is the Hasse invariant. In cases (1) and (2), the orthogonal complement (V⊥ , qV ) of (V, qV ) in (B, uq) is coherent, and thus γ(V⊥ , qV ) = 1. It follows that γ(V, qV ) = γ(B, uq)/γ(V⊥ , qV ) = −1.

46

CHAPTER 2

2.5.1

Eisenstein series of weight one

Consider the first case (V, qV ) = (EA j, uq). The quadratic space is isomorphic to (EA , uq(j)q), but we will still write everything based on the space V2 = EA j. The center of symmetry of the Eisenstein series E(s, g, Φ) is exactly the point s = 0. We consider the (local) Siegel–Weil formula. It is the most important case in this book. We will see that E(0, g, Φ) = 0 by some essentially local reason. Then it will be more interesting for us to take the derivative E 0 (0, g, Φ). See §6.1 for the formula of the derivative, and the context after that for the relation of the derivative with height pairings of CM points. Recall that Z δv (wn(b)g)s r(wn(b)g)Φv (0) ψ(−ab)db, a ∈ Fv . Wa,v (s, g, Φv ) = Fv

For the case a = 0, we use the normalization ◦ W0,v (s, g, Φv )

=

L(s + 1, ηv ) W0,v (s, g, Φv ). L(s, ηv )

L(s + 1, ηv ) has a zero at s = 0 when Ev L(s, ηv ) is split, and is equal to π −1 at s = 0 when v is archimedean. Now we list the precise local Siegel–Weil formula for these local Whittaker functions. We use the convention that |Dv | = |dv | = 1 if v is archimedean. Notice that the normalizing factor

Proposition 2.7.

(1) In the sense of analytic continuation for s ∈ C, 1

1

◦ W0,v (0, g, Φv ) = |Dv | 2 |dv | 2 γ(V2,v , uq) r(g)Φv (0).

Therefore, W0 (0, g, Φ) = −r(g)Φ(0). Furthermore, for almost all places v, ◦ (s, g, Φv ) = δv (g)−s r(g)Φv (0). W0,v

(2) Assume a ∈ Fv× . (a) If a is not represented by (V2,v , uq), then Wa,v (0, g, Φv ) = 0. (b) Assume that there exists xa ∈ V2,v satisfying uq(xa ) = a. Then Z γ(V2,v , uq) Wa,v (0, g, Φv ) = r(g, h)Φv (xa )dh. L(1, ηv ) Ev1 Here the integration uses the Haar measure on Ev1 normalized in §1.6.2.

WEIL REPRESENTATION AND WALDSPURGER FORMULA

47

Proof. We immediately know that all the results at s = 0 are true up to constant multiples independent of g and Φv by Theorem 2.2. To determine the constant, one can trace back the measures in Theorem 2.2. Alternatively, we only need to compute the case g = 1 for some well-chosen function Φv . See [KRY1] for an example. ◦ As for W0,v (s, g, Φv ), it is the image of δ(g)s r2 (g)Φv (0) under the normalized intertwining operator for g ∈ SL2 (Fv ). Hence we know the equality for almost all places.  Proposition 2.8. For any Φ ∈ S(V2 ), one has E(0, g, Φ) = 0. Proof. The incoherence theory of Kudla–Rallis [KR3] shows E(0, g, Φ) = 0 in a more general incoherent case. We can also check it directly by our expression of Whittaker functions above. It is immediate that the constant term E(0, g, Φ) = 0. Now we consider the Whittaker function Y Wa (0, g, Φ) = Wa,v (0, g, Φv ) v

for a ∈ F × . We know that Wa,v (0, g, Φv ) 6= 0 only if au−1 is represented by (V2,v , q). We claim that au−1 cannot be represented by (V2,v , q) for all places v. Denote by B the quaternion algebra over F generated by E and j with relations j 2 = −au−1 , jt = tj, ∀ t ∈ E. If au−1 is represented by some element xv of (V2,v , q) for all places v, then the map j 7→ xv gives an isomorphism BA ∼ = B. It contradicts the incoherence assumption on B.  2.5.2

Eisenstein series of weight two

Now we consider the third case where the Eisenstein series E(s, g, Φ) is defined based on the quadratic space (V, qV ) = (B, uq). For the intertwining part, we still need a normalization ◦ (s, g, Φv ) W0,v

=

1 W0,v (s, g, Φv ). ζv (s + 1)

The following result uses the Haar measures on Fv and B1v normalized in §1.6.1 and §1.6.2. Proposition 2.9. The following are true at s = 0: ◦ (0, g, Φv ) = 0 identically. (1) (a) If Bv isnon-split, then W0,v

48

CHAPTER 2

(b) If Bv is split, then under the identification Bv = M2 (Fv ), ◦ (0, g, Φv ) W0,v Z = |ud−1 | v v

   y x dxdydh r(g)Φv h 0 0 SL2 (OFv ) Fv Fv    Z Z Z 0 x −1 r(g)Φv h dxdydh. |udv |v 0 y SL2 (OF ) Fv Fv

=

Z

Z

v

(2) Assume a ∈ Fv× . (a) If a is not represented by (Bv , uq), then Wa,v (0, g, Φv ) = 0. (b) Assume that there exists xa ∈ Bv satisfying uq(xa ) = a. Then Z r(g)Φv (hxa )dh. Wa,v (0, g, Φv ) = γ(Bv , uq) |a|v B1v

(c) If furthermore B2 = M2 (Fv ), then (b) has the simplification

=

Wa,v (0, g, Φv ) Z |ud−1 | ζ (1) v v v

Z SL2 (OFv )

=

|ud−1 v |v ζv (1)

Z

Fv

Fv

  y r(g)Φv h Z Z Z SL2 (OFv )

Fv

 r(g)Φv



x au−1 y −1

dxdydh

Fv

au−1 y −1

x y

  h dxdydh.

Here both the measures dx and dy are the Haar measure on Fv . (3) The Eisenstein series E(s, g, Φ) is holomorphic at s = 0 with critical value X Z E(0, g, Φ) = r(g)Φ(0) + W0 (0, g, Φ) − r(g)Φ(hxa )dh. a∈F ×

B1

Here xa ∈ B is any element with uq(xa ) = a, and the integration is considered to be zero if such xa does not exist. Furthermore, W0 (0, g, Φ) 6= 0 only if F = Q and Σ = {∞}. In that case, Y 0 ◦ (0, g, Φ∞ ) W0,v (0, g, Φv ). W0 (0, g, Φ) = W0,∞ v6=∞

Proof. The results of (1) follow from (2) by W0,v (0, g, Φv ) = lim Wa,v (0, g, Φv ). a→0

49

WEIL REPRESENTATION AND WALDSPURGER FORMULA

In particular, it is immediate that W0,v (0, g, Φv ) = 0 if Bv is non-split. The proof of (2) is similar to Proposition 2.7. See [KY] for some explicit results which would give (2)(a) and (2)(b). Here we deduce (2)(c) from (2)(b). If Bv = M2 (Fv ), then γ(Bv , uq) = 1. We have    Z 1 Wa,v (0, g, Φv ) = |a|v dh. r(g)Φv h au−1 B1 v

The Iwasawa decomposition gives the general Haar measure identity Z Ψ(h)dh =

|dv |−1 v

Z

SL2 (Fv )

Z Fv×

SL2 (OFv )

Z Fv

  1 Ψ h

x 1





1



y y −1

dxd× ydh.

It yields that Wa,v (0, g, Φv ) Z = |ad−1 | v v

Z Fv×

SL2 (OFv )

=

Z

  1 r(g)Φv h Z Z −1 |adv |v SL2 (OFv )

Fv

x 1 Z

Fv×



y

y −1   y r(g)Φv h

Fv



dxd× ydh  au−1 y −1 x dxd× ydh. au−1 y −1 au−1

By a change of variable x → a−1 uyx, we have Wa,v (0, g, Φv ) = |ud−1 v |v ζv (1)

Z

Z SL2 (OFv )

Z

Fv

Fv

  y r(g)Φv h



x au−1 y −1

dxdydh.

It proves the first identity of (2)(c). The second identity is proved in a similar way by    Z au−1 Wa,v (0, g, Φv ) = |a|v h dh r(g)Φv 1 B1 v

and the Haar measure identity Z SL2 (Fv )

Ψ(h)dh = |dv |−1 v

Z

Z

SL2 (OFv )

Ψ

Z

Fv×

Fv



y −1

 y

1

x 1

  h dxd× ydh.

50

CHAPTER 2

As for (3), we only need to verify the related results on W0 (s, g, Φ). By definition, Y ◦ W0 (s, g, Φ) = ζF (s + 1) W0,v (s, g, Φv ). v ◦ The zeta functions contribute a simple pole at s = 0, and the product of W0,v contribute a zero at s = 0 of multiplicity at least #Σ by (1). It follows that W0 (s, g, Φ) is always holomorphic at s = 0, and vanishes at s = 0 if #Σ > 1. If #Σ = 1, then F = Q and Σ = {∞}. One has Y ◦ 0 ◦ (0, g, Φ∞ ) W0,v (0, g, Φv ). W0 (0, g, Φ) = W0,∞ v6=∞ 0

◦ 0 It is easy to see W0,∞ (0, g, Φ∞ ) = W0,∞ (0, g, Φ∞ ) by ζ∞ (1) = 1.



Remark. The intertwining part W0 (0, g, Φ) (if nonzero) is the only nonholomorphic part in E(0, g, Φ). 2.5.3

Eisenstein series of weight 3/2

Now we consider the second case (V, qV ) = (B0 , uq). Let Φ ∈ S(B0 ) and thus we have the Eisenstein series E(s, g, Φ) and the Whittaker function Wa (s, g, Φ) and its local component. To have a better description of the non-holomorphic part of E(0, g, Φ), we introduce a notation. Let v be a non-archimedean place, and define a linear map `0 : S(M2 (Fv )0 )

−→

by sending any Φ ∈ S(M2 (Fv )0 ) to   Z Z 1 z Φv h−1 (`0 Φ)(z) = |u|v2 GL2 (OFv )

Fv

S(Fv )

x −z

  h dxdh,

z ∈ Fv .

Endow Fv with the quadratic norm uq − (z) = −uz 2 to get a quadratic space (Fv , uq − ). It is standard that the map `0 is equivariant under the action of f 2 (Fv ) via the Weil representation. For a proof we refer to Proposition 4.13. By SL f 2 (Af )-equivariant map `0 : S(M2 (Af )0 ) → taking the product, we obtain an SL S(Af ). Go back to the Eisenstein series E(s, g, Φ). For each place v of F , normalize ◦ (s, gv , Φv ) = W0,v

1 W0,v (s, gv , Φv ). ζv (2s + 1)

For each a ∈ F × , normalize ◦ Wa,v (s, gv , Φv ) =

1 Wa,v (s, gv , Φv ). Lv (s + 1, η−ua )

51

WEIL REPRESENTATION AND WALDSPURGER FORMULA

× → {±1} denotes the quadratic field associated to the quadratic Here η−ua : A√ −ua) of F . If −ua is a square in F , take the convention that extension F ( √ F ( −ua) = F ⊕ F and η−ua = 1. Now we normalize some Haar measures. For each element y ∈ Bv with q(y) ∈ Fv× , denote by Bv,y the centralizer of y in Bv . Then Bv,y = Fv [y] = Fv + Fv y × is a (possibly split) quadratic extension of Fv . Endow B× v,y and Bv with the × × Haar measures normalized in §1.6.2. Endow Bv,y \Bv with the quotient Haar × 2 measure. It is easy to see that vol(B× v,y \Bv ) = 2π if v is archimedean.

Proposition 2.10. Write γv = γ(B0,v , uq). The following are true at s = 0: ◦ (1) (a) If Bv isnon-split, then W0,v (0, g, Φv ) = 0 identically.

(b) If Bv is split, then under the identification Bv = M2 (Fv ), 1

◦ 2 (0, g, Φv ) = γv |2d−3 W0,v v |v r(g)(`0 Φ)(0).

(2) Assume a ∈ Fv× . (a) If a is not represented by (B0,v , uq), then Wa,v (0, g, Φv ) = 0. (b) Assume that there exists xa ∈ B0,v satisfying uq(xa ) = a. Then Z 1 ◦ Wa,v (0, g, Φv ) = γv |8a|v2 r(g, (h, h))Φv (xa )dh. × B× v,xa \Bv

(c) If furthermore in (b) one has Bv = M2 (Fv ) and a = −uz 2 for some z ∈ Fv× , then we have a simplification 1

◦ 2 (0, g, Φv ) = γv |2d−3 Wa,v v |v r(g)(`0 Φv )(z).

(3) The Eisenstein series E(s, g, Φ) is holomorphic at s = 0 with critical value as follows: (a) If #Σ > 1, then E(0, g, Φ) = r(g)Φ(0) Z X − L(1, η−ua ) a∈F ×

× B× xa \B

r(g, (h, h))Φ(xa )dh.

Here xa ∈ B is any element with uq(xa ) = a, and the integration is considered to be zero if such xa does not exist. (b) If F = Q and Σ = {∞}, identify Bf = M2 (Af ) and thus B× f = GL2 (Af ). Then E(0, g, Φ)

=

E hol (0, g, Φ) + E nhol (0, g, Φ),

52

CHAPTER 2

where the holomorphic part and the non-holomorphic part are respectively E hol (0, g, Φ) =

r(g)Φ(0) −

Z

X

L(1, η−ua )

a∈Q×

× B× xa \B

r(g, (h, h))Φ(xa )dh,

E nhol (0, g, Φ) X 1 0 √ W−uz = − 2 ,∞ (0, g∞ , Φ∞ )r(gf )(`0 Φf )(z). 2 2γ ∞ z∈Q Proof. The proof is similar to Proposition 2.7 and Proposition 2.9. It is easy to see that (1) is a consequence of (2)(a) and (2)(c) by the limit W0,v (0, g, Φv ) = lim Wa,v (0, g, Φv ). a→0

See [KRY3] for some explicit results on (2), which could be used to check the explicit constants in (2). Here we only  sketch how (2)(b) implies (2)(c). z 2 whose norm represents a. The By a = −uz , we can take xa = −z centralizer Bv,xa of xa in B× v is exactly A(Fv ), the group of diagonal matrices. It follows that Z 1 2 r(g, (h, h))Φv (xa )dh. Wa,v (0, g, Φv ) = γv |8a|v A(Fv )\GL2 (Fv )

By the Iwasawa decomposition, one has a Haar measure identity Z Z Z −3 Ψ(h)dh = |dv |v 2 Ψ(n(b)h)dbdh. A(Fv )\GL2 (Fv )

GL2 (OFv )

Fv

It follows that Wa,v (0, g, Φv ) Z 1 2 = γv |8ad−3 | v v

Z

GL2 (OFv )

 r(g, (h, h))Φv Z 1 −3 2 = γv |8adv |v

1

1

2 γv |2ud−3 v |v





z

Z

 

Z Fv

x 1

 dxdh

z

2xz −z



z

x −z



Fv

r(g, (h, h))Φv GL2 (OFv )

1

−z r(g, (h, h))Φv

GL2 (OFv )

=

Fv

−x 1 Z

dxdh dxdh.

Now we consider (3). It is similar to the result for W0 (s, g, Φ) in Proposition 2.9, but slightly more complicated. We remark that in both summations on

53

WEIL REPRESENTATION AND WALDSPURGER FORMULA

a, the existence of xa implies that ua > 0 at archimedean places, so η−ua is a nontrivial character and L(1, η−ua ) is a well-defined number. By definition, Y ◦ W0,v (s, g, Φv ). W0 (s, g, Φ) = ζF (2s + 1) v ◦ The zeta functions contribute a simple pole at s = 0, and the product of W0,v contributes a zero at s = 0 of multiplicity at least #Σ. It follows that W0 (s, g, Φ) is always holomorphic at s = 0, and it vanishes at s = 0 if #Σ > 1. Now let a ∈ F × and consider Y ◦ Wa (s, g, Φ) = L(s + 1, η−ua ) Wa,v (s, gv , Φv ). v ◦ is holomorphic at s = 0. If −ua is not a square in F × , The product of Wa,v then Y ◦ Wa,v (0, gv , Φv ). Wa (0, g, Φ) = L(1, η−ua ) v

It can be written as integrals by (2). Assume in the following that −ua is a square in F × , and thus a = −uz 2 for some z ∈ F × . Then Y ◦ Wa (s, g, Φ) = ζF (s + 1) Wa,v (s, gv , Φv ). v

We claim that a is not represented by (B0,v , uq) at any v ∈ Σ. In fact, if a = uq(x) for some x ∈ B0,v , then q(z + x) = z 2 + q(x) = 0, which is impossible ◦ contributes a zero at s = 0 for z + x ∈ Bv . It follows that the product of Wa,v of multiplicity at least #Σ. Then Wa (s, g, Φ) is always holomorphic at s = 0, and it vanishes at s = 0 if #Σ > 1. Now assume that #Σ = 1, i.e., F = Q and Σ = {∞}. Still assume that a = −uz 2 for some z ∈ Q× . Then Y ◦ 0 ◦ Wa (0, g, Φ) = Ress=0 ζQ (s + 1) · Wa,∞ (0, g∞ , Φ∞ ) Wa,v (0, gv , Φv ). v-∞ ◦ By Wa,∞ (0, g∞ , Φ∞ ) = 0, we have 0

◦ 0 0 Wa,∞ (0, g∞ , Φ∞ ) = ζ∞ (1)−1 Wa,∞ (0, g∞ , Φ∞ ) = Wa,∞ (0, g∞ , Φ∞ ).

It follows that Wa (0, g, Φ)

Y

1

=

0 Wa,∞ (0, g∞ , Φ∞ )

=

1 0 Wa,∞ (0, g∞ , Φ∞ ) r(g)(`0 Φf )(z). −√ 2γ∞

2 γv |2d−3 v |v r(g)(`0 Φv )(z)

v-∞

54

CHAPTER 2

The summation over all such a becomes half of the summation over all z ∈ Q× , since each a corresponds to two z. Similarly, Y ◦ 0 ◦ W0 (0, g, Φ) = Ress=0 ζQ (2s + 1) · W0,∞ (0, g∞ , Φ∞ ) W0,v (0, gv , Φv ) v-∞

1 0 W0,∞ (0, g∞ , Φ∞ ) r(g)(`0 Φf )(0). − √ 2 2γ∞

=

The extra factor 2 in the denominator comes from the residue of ζQ (2s + 1). It proves the result.  Remark. In case (3)(b), if setting u = 1, taking φ∞ to be the standard Gaussian, and properly choosing φf , the Eisenstein series E(0, g, Φ) recovers Zagier’s Eisenstein series F(z) =

∞ X

1

X

H(n)q n + y − 2

n=0

2

β(4πm2 y) q −m ,

z ∈ H.

m∈Z

Here y = Im(z), q = e2πiz , and H(n) is Hurwitz class number, and Z ∞ 3 1 β(x) = e−xt t− 2 dt, Re(x) ≥ 0. 16π 1 We refer to [HZ] for more details. See also Corollary 2.12 for an expression of 0 (0, g∞ , Φ∞ ) in terms of β(x). the archimedean part Wa,∞ 2.5.4

Local Whittaker functions at archimedean places

Assume that (V, q) is a positive definite quadratic space over R of dimension d ≥ 1, and let Φ = e−2πq ∈ S(V ) be the standard Gaussian. Here we are going to compute the value at s = 0 of the (local) Whittaker function Z f 2 (R). Wa (s, g, Φ) = δ(wn(b)g)s r(wn(b)g)Φ(0) ψ(−ab)db, g ∈ SL R

It suffices to treat the case g = 1, and the general case can be derived by Iwasawa decomposition. The following result is in the literature, and is expressed in terms of confluent hypergeometric functions. See [Shi, KRY1, KRY3, KY] for example. 2πid In the following, the Weil index γd = γ(V, q) = e 8 . Proposition 2.11. The following are true: (1) For any a ∈ R, d

Wa (s, 1, Φ) = γd

2π s+ 2 s Γ( 2 )Γ( s+d 2 )

Z

e−2π(t−a) t t>0, t>2a

s+d 2 −1

s

(t − 2a) 2 −1 dt.

55

WEIL REPRESENTATION AND WALDSPURGER FORMULA

(2) For any a > 0, d

(2π) 2 d −1 −2πa a2 e . Wa (0, 1, Φ) = γd Γ( d2 ) (3) For any a < 0, Wa (0, 1, Φ)

=

0,

Wa0 (0, 1, Φ)

=

γd

d

π2 Γ( d2 )

∞

Z

d

e−2π(t−a) t 2 −1 (t − 2a)−1 dt. 0

(4) For a = 0, one has ( 0 W0 (0, 1, Φ) = γd π

if d 6= 2, if d = 2.

If d 6= 2, then d

W00 (0, 1, Φ) = γd π

22− 2 . d−2

Proof. We first consider (1). By definition, Z Z r(g)Φ(x)ψ(bq(x))dx ψ(−ab)db. Wa (s, g, Φ) = γd δ(wn(b)g)s V

R

Take an isometry of quadratic space (V, q) ' (Rd , k · k2 ). Here k · k denotes the standard quadratic norm on Rd . We get Z Z 2 2 Wa (s, 1, Φ) = γd δ(wn(b))s e−2πkxk e2πibkxk dx e−2πiab db. Rd

R

The integral in x is just Z e

−2π(1−ib)kxk2

Z dx =

e

Rd

−2π(1−ib)x2

d dx

d

= (1 − ib)− 2 .

R 1

Note that δ(wn(b)) = (1 + b2 )− 2 . We have Z s d Wa (s, 1, Φ) = γd (1 + b2 )− 2 (1 − ib)− 2 e−2πiab db ZR s+d s = γd (1 + ib)− 2 (1 − ib)− 2 e−2πiab db. R

After a standard computation as in [Shi, KRY1, KRY3, KY], we have d

Wa (s, 1, Φ)

=

2π s+ 2 γd s Γ( 2 )Γ( s+d 2 )

Z

e−2π(t−a) t t>0, t>2a

s+d 2 −1

s

(t − 2a) 2 −1 dt.

56

CHAPTER 2

Now we consider (2) and (3). We first assume a > 0. Then a change of variable gives Z ∞ d s+d s 2π s+ 2 −2πa e e−2πt (t + 2a) 2 −1 t 2 −1 dt. Wa (s, 1, Φ) = γd s s+d Γ( 2 )Γ( 2 ) 0 Note that Γ( 2s ) has a pole at s = 0 which contributes a zero of the factor before the integral, while the integral is not convergent at s = 0 due to the singularity s of t 2 −1 at t = 0. The difference Z ∞ Z ∞ s+d s+d s s e−2πt (t + 2a) 2 −1 t 2 −1 dt − e−2πt (2a) 2 −1 t 2 −1 dt 0

0 ∞

Z

e−2πt

=

(t + 2a)

s+d 2 −1

− (2a)

s+d 2 −1

t

0

s

t 2 dt

is convergent and holomorphic at s = 0. It follows that Z ∞ d s+d s 2π s+ 2 −2πa e Wa (s, 1, Φ) = γd lim e−2πt (2a) 2 −1 t 2 −1 dt. s s+d s→0 Γ( )Γ( 0 2 2 ) The integral is a Gamma function, and the result is easily obtained. Now we assume that a < 0. Then Z ∞ d s+d s π s+ 2 s Wa (s, 1, Φ) = γd s e−2π(t−a) t 2 −1 (t − 2a) 2 −1 dt. s+d Γ( 2 + 1)Γ( 2 ) 0 The integral is holomorphic at s = 0. It follows that Wa (s, 1, Φ) = 0 by the zero coming from the factors before the integral. We further have Z ∞ d d π2 e−2π(t−a) t 2 −1 (t − 2a)−1 dt. Wa0 (0, 1, Φ) = γd d Γ( 2 ) 0 The case a = 0 has an explicit expression for all s ∈ C. In fact, Z ∞ d d 2π s+ 2 e−2πt ts+ 2 −2 dt W0 (s, 1, Φ) = γd s s+d Γ( 2 )Γ( 2 ) 0 =

d

γd π2−(s+ 2 −2)

Γ(s +

d 2

− 1)

Γ( 2s )Γ( s+d 2 )

.

It is easy to obtain the result. In the case d = 3, to match the notation in [HZ], we recall the function Z ∞ 3 1 β(x) = e−xt t− 2 dt, Re(x) ≥ 0. 16π 1 It will give the derivative Wa0 (0, 1, Φ) a slightly different expression.



WEIL REPRESENTATION AND WALDSPURGER FORMULA

57

Corollary 2.12. Assume d = 3. For a ≤ 0, √ 0 (0, 1, Φ) = γ3 8 2π 2 e−2πa β(−4πa). Wa,∞ Proof. By Proposition 2.11 (3) for the case d = 3, we have Z ∞ 1 0 Wa,∞ (0, 1, Φ) = γ3 · 2π e−2π(t−a) t 2 (t − 2a)−1 dt 0 Z ∞ 1 1 2πa = γ3 · 2πe (−2a) 2 e4πat t 2 (t + 1)−1 dt Z ∞0 3 − 12 −2πa = γ3 · 2 πe e4πat t− 2 dt. 1

The result is also true for a = 0. Here the last equality uses the identity Z ∞ Z ∞ 1 3 4πat 12 −1 2 4|a| e t (t + 1) dt = e4πat (t + 1)− 2 dt. 0

0

It is the special case p = q =

3 2

of the functional equation

U (p, q, z) = z 1−q U (1 + p − q, 2 − q, z) of the confluent hypergeometric function Z ∞ 1 e−zt tp−1 (t + 1)q−p−1 dt. U (p, q, z) = Γ(p) 0 See [Le], p. 265.



Chapter Three Mordell–Weil Groups and Generating Series The major goal of this chapter is to introduce Theorem 3.21, an identity between the analytic kernel and the geometric kernel, and describe how it is equivalent to Theorem 1.2. We first define the generating series, and then use it to define the geometric kernel. The analytic kernel is the same as that in the Waldspurger formula, except that we take derivative here. As a bridge between these two theorems, we also introduce Theorem 3.15, an identity formulated in terms of projectors. In §3.1, we review some basic notations and results on Shimura curves. In §3.2, we will review the Eichler–Shimura theory for abelian varieties parametrized by Shimura curves and give more details on the main result in Theorem 1.2. In §3.3, we state our main result using the projector T(f1 ⊗ f2 ) in Theorem 3.15, and prove that it is equivalent to Theorem 1.2. In §3.4, we will define Z(g, Φ), a generating series with coefficients in Pic(X × X)C . It acts on X as an algebraic correspondence. This series is an extension of Kudla’s generating series for Shimura varieties of orthogonal type [Ku1]. In §3.5, we define the geometric kernel Z(g, Φ, χ) and its normalization. It is essentially a linear combination of height pairings of CM points on X with the image of these points under the action of Z(g, Φ). In §3.6, we recall the analytic kernel function and state a kernel identity, and we show how this identity implies the main theorem in the introduction. It is based on an expression of the projector T (f1 ⊗ f2 ) in terms of the generating series (Theorem 3.22), which will be proved in the next chapter. 3.1

BASICS ON SHIMURA CURVES

Let F be a totally real number field, and Σ be a finite set of places of F with an odd cardinality and including all the infinite places. It gives a totally definite incoherent quaternion algebra B over A with ramification set Σ. 3.1.1

Shimura curves

For each open compact subgroup U of B× f , we have a (compactified) Shimura curve XU . It is a projective curve over F such that, for any embedding τ : F ,→ C, the complex points of XU at τ have the uniformization XU,τ (C) ' B(τ )× \H± × B× f /U ∪ {cusps}.

59

MORDELL–WEIL GROUPS AND GENERATING SERIES

Here B(τ ) is the unique quaternion algebra over F with ramification set Σ \ {τ }, Bf is identified with B(τ )Af as an Af -algebra, and B(τ )× acts on H± through an isomorphism B(τ )τ ' M2 (R). The set {cusps} is non-empty if and only if F = Q and Σ = {∞}. We usually use [z, β]U to denote the image of (z, β) ∈ H± × B× f in XU,τ (C). It is smooth when U is small enough. For any two open compact subsets U1 ⊂ U2 of B× f , one has a natural surjective morphism πU1 ,U2 : XU1 → XU2 . Let X be the projective limit of the system {XU }U . It is a regular scheme over F , locally noetherian but not of finite type. In terms of the notation above, it has a uniformization Xτ (C) ' B(τ )× \H± × B× f /D ∪ {cusps}. × Here D denotes the closure of F × in A× f . If F = Q, then D = Q . In general, D is much larger than F × . For any U , the curve XU is connected but not geometrically connected in general. For any α ∈ π0 (XU,F ), we usually denote by XU,α the corresponding connected component of XU,F . It gives a decomposition

a

XU,F =

XU,α .

× α∈F+ \A× f /q(U )

The action of Gal(F /F ) on π0 (XU,F ) factors through the reciprocity law Gal(F /F )−→F+× \A× f /q(U ). It actually makes π0 (XU,F ) a principal homogeneous space under F+× \A× f /q(U ). In terms of the complex uniformization, one has × × × π0 (XU,τ (C)) ' B(τ )× + \Bf /U = F+ \Af /q(U ). × Here B(τ )× with totally positive norms. The de+ denotes elements of B(τ ) composition into connected components is given by a −1 Γh \H∗ , Γh = B(τ )× . XU,τ (C) ' + ∩ hU h × h∈B(τ )× + \Bf /U

Here H∗ = H ∪ P1 (Q) if F = Q and Σ = {∞}; otherwise, H∗ = H. A point of XU,τ (C) represented by [z, β]U with z ∈ H and β ∈ B× f lies in XU,q(β) . 3.1.2

CM points

Let E/F be a totally imaginary quadratic extension. Fix an embedding EA ,→ B. Then EA× acts on X by the right multiplication via the embedding EA× ,→ B× .

60

CHAPTER 3 ×

Let X E be the subscheme of X of fixed points of X under E × . Up to the × translation by B× , the subscheme X E does not depend on the choice of the × embedding EA ,→ B. The scheme X E is defined over F . Let H be the normalizer of E in B× . A simple calculation shows that H is a dihedral group generated by EA× and an element j ∈ B× such that jxj−1 = × × x for all x ∈ EA× . Then H acts on X E , which makes X E (F ) a principal × × homogeneous space over H/DE , where DE is the closure of E∞ E in EA× . The × theory of complex multiplication asserts that every point in X E (F ) is defined over E ab and that Galois action is given by the reciprocity map Gal(E ab /F ) ' H/DE . ×

Fix a point P ∈ X E (E ab ) throughout this book. It induces a point PU ∈ XU (E ab ) for every U . We can normalize the complex uniformization XU,τ (C) = B(τ )× \H± × B× f /U ∪ {cusps} so that the point PU is exactly represented by the double coset of [z0 , 1]U . Here z0 ∈ H is the unique fixed point of E × in H via the action induced by the embedding E ,→ B(τ ). Similar description can be made on X. If we take the geometrically connected component of PU as the neutral component, we obtain an identification π0 (XU,F ) = F+× \A× f /q(U ). A point reprelies in the geometrically connected sented by [z, h]U with z ∈ H and h ∈ B× f component represented by q(h). 3.1.3

Hodge classes

The curve XU has a Hodge class LU ∈ Pic(XU )Q . It is the line bundle for holomorphic modular forms of weight two, and it is essentially the canonical bundle modified by the ramification points. The system L = {LU }U is compatible with pull-back maps. The following definition is close to that in [Zh1]. Let ωXU /F be the canonical bundle of XU . Define X LU = ωXU /F + (1 − e−1 x ) x. x∈XU (F )

Here group operation in Pic(XU )Q is written additively, and for each x ∈ XU (F ), the ramification index is described as follows. In terms of the above complex uniformization, the index ex is just the ramification index of any preimage of x in the quotient map H∗ → Γh \H∗ . We also give an algebraic description here. If x is a cusp, then define ex = ∞ so that the multiplicity 1 − e−1 x = 1. If x is not a cusp, ex is defined to be the ramification index of any preimage of x in the map XU 0 → XU for any sufficiently small open compact subgroup U 0 of U . Here U 0 is said to be sufficiently small if each geometrically connected component of XU 0 is a free quotient of H under

MORDELL–WEIL GROUPS AND GENERATING SERIES

61

the complex uniformization (for any embedding τ : F ,→ C). The index ex is independent of the choice of U 0 and the preimage of x, and the Hodge bundle LU is defined over F . For each component α ∈ π0 (XU,F ), denote the restriction LU,α = LU |XU,α to the connected component XU,α of XU,F corresponding to α. It is also viewed as a divisor class on XU via push-forward under XU,α → XU . Denote by P 1 LU,α the normalized Hodge class on XU,α , and by ξU = α ξU,α ξU,α = deg LU,α the normalized Hodge class on XU . We remark that deg LU,α is independent of α since all geometrically connected components are Galois conjugate to each other. It follows that deg LU,α = deg LU /|F+× \A× f /q(U )|. The degree of LU can be further expressed as the voldxdy on XU,τ (C) for any embedding ume vol(XU ), defined as the integral of 2πy 2 τ : F ,→ C. Lemma 3.1. The following are true: (1) One always has deg LU = vol(XU ). Hence, vol(XU ) is a rational number independent of the choice of τ : F ,→ C. (2) For any inclusion U1 ⊂ U2 , the projection πU1 ,U2 : XU1 → XU2 has degree deg(πU1 ,U2 ) = vol(XU1 )/vol(XU2 ). Proof. It is easy to obtain (2) by the definition. To prove (1), we first introduce the Petersson metric k·kPet of LU,τ (C). On any connected component Γ\H∗ of XU,τ (C), a section of LU,τ (C) is of the form f (z)dz, where f : H → C is a holomorphic modular form of weight two with respect to Γ. The Petersson metric on this connected component is given by kf (z)dzkPet = 4π · Imz · |f (z)|. The result of the lemma follows from Chern’s integration formula Z c1 (LU,τ (C), k · kPet ). deg LU = XU,τ (C)

Here the Chern form (on each connected component) is exactly c1 (LU,τ (C), k · kPet ) =

dx ∧ dy ∂∂ log(4πy) = . πi 2πy 2

The Petersson metric has a logarithmic singularity at the cusps, but Chern’s integration formula still applies here by a simple local computation. 

62 3.1.4

CHAPTER 3

Hecke correspondences

Infinite level The Shimura curve X is endowed with an action Tx of x ∈ B× given by “the right multiplication by xf .” The action Tx is trivial if and only if xf ∈ D. Each XU is just the quotient of X by the action of U under this action. In terms of the system {XU }U , the action gives an isomorphism Tx : XxU x−1 → XU for each U . Recall that the Hecke algebra H := Cc∞ (B× f ) consists of smooth and com→ C. Its multiplication is given by the pactly supported functions φ : B× f convolution Z (φ1 ∗ φ2 )(x) := φ1 (x0 )φ2 (x0−1 x)dx0 . B× f

It has no multiplicative unit. For any admissible representation (V, ρ) of B× f , there is a standard action of H on V by Z φ(x)ρ(x)v. H−→End(V ), ρ(φ) : v 7−→ B× f

It actually gives an equivalence between the category of admissible representations of B× f and the category of admissible representations of H. For each φ ∈ H, define a “correspondence” on X with complex coefficients by Z T(φ) := B× f

φ(x)Tx .

The algebra HU has an involution φ 7−→ (φt : x 7→ φ(x−1 )), which is compatible with the transpose of the algebraic correspondence T(φ). Finite level Fix an open compact subgroup U of B× f . Let Z(x) ⊂ X × X be the graph of Tx . Let Z(x)U denote the image of Z(x) in XU × XU . Then Z(x)U can also be defined as the image of the morphism (πUx ,U , πUx ,U ◦ Tx ) :

XUx −→XU × XU .

Here Ux = U ∩xU x−1 is an open and compact subgroup of B× f . It is an algebraic correspondence on XU , and also viewed as an element of Pic(XU × XU ). In terms of complex uniformization above, the push-forward gives X [z, βy]U . (Z(x)U )∗ : [z, β]U 7−→ y∈U xU/U

MORDELL–WEIL GROUPS AND GENERATING SERIES

63

The transpose Z(x)tU of Z(x)U is simply equal to Z(x−1 )U . We can extend the action to the Hecke algebra × HU = Cc∞ (U \B× f /U ) = {f ∈ H : f (U xU ) = x, ∀x ∈ Bf }

of compactly supported functions φ : B× f → C that are bi-invariant under the action of U . It is a sub-algebra of H, with multiplicative unit vol(U )−1 1U . For each φ ∈ HU , the linear combination X φ(x) Z(x)U T(φ)U := x∈U \B× f /U

gives an element of Pic(XU × XU )C . It is an algebraic correspondence on XU with complex coefficients. The algebra HU has an involution φ 7−→ (φt : x 7→ φ(x−1 )), which is compatible with the transpose of algebraic correspondences. Now we vary U . It is easy to check that, for any φ ∈ H, the system e T(φ) = {vol(XU ) T(φ)U }U is compatible with pull-back maps and thus defines an element in Pic(X × X)C := lim Pic(XU × XU )C . −→ U

In this book, we take the convention that algebraic correspondences act on cohomology groups via pull-back, and act on divisors via push-forward. There may be a few exceptions, which we will specify. We sometimes abbreviate the push-forward D∗ as D for a correspondence D. 3.1.5

Differential forms

Denote by A(B× ) the set of (equivalence classes of) irreducible admissible representations π of B× such that the Jacquet–Langlands correspondence JL(π) of π on GL2 (A) is a cuspidal automorphic representation of GL2 (A), discrete of parallel weight two at infinity. Note the condition that JL(π) is discrete of parallel weight two at infinity is equivalent to the condition that π∞ is the trivial representation of B× ∞. Fix an embedding τ : F ,→C. Then we have a cohomological group H 1 (XU,τ (C), Q) with Hodge structure H 1 (XU,τ (C), Q) ⊗ C = H 1,0 (XU,τ (C)) ⊕ H 0,1 (XU,τ (C)),

64

CHAPTER 3

where H 1,0 (XU,τ )

=

Γ(XU,τ (C), Ω1XU,τ (C)/C ),

H 0,1 (XU,τ )

=

Γ(XU,τ (C), ΩXU,τ (C)/C ).

1

By the complex uniformization, these spaces can be identified with the spaces of cusp forms of B(τ )× A with weight (2, 0, · · · , 0) and (−2, 0, · · · , 0) invariant under U . By the Jacquet–Langlands correspondence, we exactly have M M πU , H 1,0 (XU,τ ) ' π ¯U . H 1,0 (XU,τ ) ' π∈A(B× )

π∈A(B× )

Here π U is the space of vector of π fixed by U , and it is nonzero only for finitely many π. Each π appears with multiplicity one in the above decomposition. In particular, the decomposition is compatible with the action of the Hecke algebra HU . Here HU acts on H 1,0 (XU,τ ) via the pull-back maps of algebraic correspondences described above, and acts on π U by the restriction of the action of H on π. Taking the direct limit, we obtain decompositions M M π, H 0,1 (Xτ ) ' π ¯ H 1,0 (Xτ ) ' π∈A(B× )

π∈A(B× )

as representations of B× . Here H 1,0 (Xτ ) := lim H 1,0 (XU,τ ), −→ U

3.1.6

H 0,1 (Xτ ) := lim H 0,1 (XU,τ ). −→ U

Jacobian variety and its dual

For each level U , denote by JU the Jacobian variety of XU . There is a canonical isomorphism JU → JU∨ , and JU can be considered either as the Albanese variety or as the Picard variety of XU of line bundles of degree 0 by canonical duality. In this book, we take the convention that JU denotes the Albanese variety and JU∨ denotes the Picard variety. It seems to be superfluous for fixed U , but makes an essential difference when varying U . For any natural map πU 0 ,U : XU 0 → XU given by an inclusion U 0 ⊂ U , we have an induced algebraic homomorphism (πU 0 ,U )∗ : JU 0 → JU given by push∗ ∨ ∨ forward of divisors, and an algebraic homomorphism πU 0 ,U : JU → JU 0 given by pull-back of line bundles. Thus we obtain a projective system J := {JU }U and a direct system J ∨ := {JU∨ }U . We may think of J as a projective limit and J ∨ as a direct limit, but they are obviously not algebraic varieties. We will only consider their realizations in terms of algebraic points, cohomologies, and homomorphisms.

65

MORDELL–WEIL GROUPS AND GENERATING SERIES

Algebraic points For a ring R = Z, Q, R, C, and any field extension F 0 of F , define J(F 0 )R :

=

lim J (F 0 ) ⊗Z R = lim Cl0 (XU,F 0 ) ⊗Z R, ←− U ←− U

∨

0

J (F )R :

=

lim J ∨ (F 0 ) −→ U U

U

⊗Z R = lim Pic0 (XU,F 0 ) ⊗Z R. −→ U

Similarly, we also define Cl(XF 0 )R := lim Cl(XU,F 0 ) ⊗Z R, ←− U

Pic(XF 0 )R := lim Pic(XU,F 0 ) ⊗Z R. −→ U

When R = Z, we omit the subscript R. The following are some examples we have just constructed: • The Hodge bundle L = {LU }U is an element of Pic(X)Q . • The CM point P fixed above gives an element P = {PU }U ∈ Cl(XF ). • The normalized Hodge bundle ξP = {ξU,P }U gives an element of Cl(XF )Q . Here ξU,P = ξU,α(P ) where α(P ) ∈ π0 (XU,F ) denotes the component containing PU . • The difference P ◦ := P − ξP lies in J(F )Q . There is a natural height pairing h·, ·iNT : J(F )Q × J ∨ (F )Q −→R. We extend this pairing to a bilinear pairing h·, ·iNT : J(F )C × J ∨ (F )C −→C. In fact, the usual theory of N´eron–Tate heights gives a pairing h·, ·iNT : JU (F )Q × JU∨ (F )Q −→R over F . See §7.1 for example. Now we vary U . Let U 0 ⊂ U be any inclusion, and πU 0 ,U : XU 0 → XU be the natural map as above. By the projection formula, ∗ 0 0 hπU 0 ,U α, β iNT = hα, (πU 0 ,U )∗ β iNT ,

α ∈ JU∨ (F )Q , β 0 ∈ JU (F )Q .

It follows from Proposition 7.3 or the projection formula for the arithmetic intersection theory. Then we can take limits to define the height pairing between J and J ∨ .

66

CHAPTER 3

Cohomology groups Fix an embedding τ : F ,→C. Consider the cohomology with Hodge structure H 1 (YU,τ ) := H 1 (YU,τ (C), Q),

H 1 (YU,τ ) ⊗Q C = H 1,0 (YU,τ ) ⊕ H 0,1 (YU,τ ).

Here Y represents X, J or J ∨ . Taking limit, we obtain H 1 (Jτ ) :

lim H 1 (JU,τ ), −→

=

U

H

1

(Jτ∨ )

:

∨ lim H 1 (JU,τ ). ←−

=

U

They are endowed with rational Hodge structures of weight 1 by the limit construction. There is a canonical isomorphism H 1 (JU,τ ) = H 1 (XU,τ ) keeping the direct system. Thus H 1 (Jτ ) is canonically isomorphic to H 1 (Xτ ) = lim H 1 (XU,τ ). −→ U

By definition, H 1 (Xτ ) ⊗Q C = H 1,0 (Xτ ) ⊕ H 0,1 (Xτ ). Here H 1,0 (Xτ ) has been considered in §3.1.5, and H 0,1 (Xτ ) has a similar decomposition. Write H 1 (XU,τ )0 = H 1 (XU,τ ) for each U , but view {H 1 (XU,τ )0 }U as a projective system by the transition map (πU 0 ,U )∗ . Denote H 1 (Xτ )0 := lim H 1 (XU,τ )0 . ←− U

It is canonically isomorphic to H 1 (Jτ∨ ). Algebraic homomorphisms By abstract non-sense, we get a direct system {Hom(JU , JU∨ )}U . Here each term denotes the group of algebraic homomorphisms between the two abelian varieties over F . Hence, it is reasonable to define Hom(J, J ∨ ) := lim Hom(JU , JU∨ ). −→ U

Similarly, define Hom0 (J, J ∨ ) := lim Hom(JU , JU∨ ) ⊗Z Q. −→ U

It is the group of homomorphisms up to isogeny. Define Hom0 (J, J ∨ )C to be the base change of Hom0 (J, J ∨ ).

MORDELL–WEIL GROUPS AND GENERATING SERIES

67

Consider the cohomology groups above. We have a natural injection Hom0 (JU , JU∨ )C ,−→ Hom(H 1 (XU,τ )0 , H 1 (XU,τ )) induced by the pull-back map of H 0,1 . Here the right-hand side denotes the group of homomorphisms between Hodge structures. The limit gives an injection Hom0 (J, J ∨ )C ,−→ Hom(H 1 (Xτ )0 , H 1 (Xτ )). The image is actually contained in the smaller space Homcont (H 1 (Xτ )0 , H 1 (Xτ )) := lim Hom(H 1 (XU,τ )0 , H 1 (XU,τ )). −→ U

Conversely, we can define Hom0 (J ∨ , J) := lim Hom(JU∨ , JU ) ⊗Z Q. ←− U

It has a canonical element vol(X)−1 = {vol(XU )−1 }U as the limit of the scalar vol(XU )−1 ∈ Hom(JU∨ , JU ) ⊗Z Q. This space is less used in this book. Direct system of algebraic correspondences Consider the group Pic(X × X) := lim Pic(XU × XU ). −→ U

On each level U , we obtain a map Pic(XU × XU )−→Hom(JU , JU∨ ) given by push-forward maps of correspondences. Both sides are direct systems when varying U . We claim that these two direct systems are compatible. Then we obtain a well-defined map Pic(X × X)−→Hom(J, J ∨ ). The compatibility follows from the following basic result. Lemma 3.2. Let ϕ : Y 0 → Y be a finite morphism of two projective curves over any field, and let D ∈ Pic(Y × Y ) be a correspondence on Y . View D0 = (ϕ × ϕ)∗ D ∈ Pic(Y 0 × Y 0 ) as a correspondence on Y 0 . Then D0 = Γtϕ ◦ D ◦ Γϕ .

68

CHAPTER 3

Here Γϕ denotes the graph of ϕ in Y × Y 0 viewed as a correspondence from Y to Y 0 , and Γtϕ denotes the transpose of Γϕ viewed as a correspondence from Y 0 to Y . Here ◦ denotes the usual composition of correspondences. For a proof of the lemma, we refer to [Li, Proposition 1.13]. Here we explain it in a more direct language. Let α0 ∈ Pic(Y 0 ) be a divisor on Y 0 . Then the lemma says D∗0 α0 = ϕ∗ (D∗ ϕ∗ α0 ). Go back to the map Pic(XU × XU )−→Hom0 (JU , JU∨ ). Let D = {DU }U be an element of Pic(X × X), and α = {αU }U be an element of J(F ). Namely, {DU }U is compatible with pull-back and {αU }U is compatible with push-forward. By Lemma 3.2 above, we see that the new system D∗ α = {(DU )∗ αU }U is compatible with pull-back. It gives the compatibility. 3.2

ABELIAN VARIETIES PARAMETRIZED BY SHIMURA CURVES

In this section, we will explain the background of Theorem 1.2. 3.2.1

Abelian varieties of GL(2)-type

Let K be a field of characteristic zero. We will work on the category AV 0 of abelian varieties over K up to isogeny. Namely, the objects of AV 0 are abelian varieties over K, and the morphism group of two objects A and B in AV 0 is defined by Hom0 (A, B) := HomK (A, B) ⊗Z Q. We also denote End0 (A) = Hom0 (A, A) conventionally. An abelian variety A over K is said to be of GL(2)-type over K, if there exists a commutative subring M of End0 (A), such that the induced action of M on Lie(A) makes Lie(A) a free module of rank one over M ⊗Q K. In that case, we also say that the pair (A, M ) is of GL(2)-type. Note that the choice M may not be unique. The notion of abelian varieties of GL(2)-type is the simplest generalization of elliptic curves. We say that a simple abelian variety A over K is of strict GL(2)-type over K if (A, End0 (A)) is an abelian variety of GL(2)-type over K. Equivalently, the algebra End0 (A) is a field of degree equal to dim A. In that case, the only choice for M is End0 (A). By the positivity of the Rosati involution, M is either a totally real field or a CM field. See [Mu] for example. In the next subsection, we will restrict to the case that K is a totally real number field. Then GL(2)-type over K is actually equivalent to strict GL(2)type. In fact, we have the following result:

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69

Lemma 3.3. Let A be a simple abelian variety of dimension g over a subfield K of R. Then the degree [D : Q] of the endomorphism ring D = End0 (A) divides g. Proof. Consider the representation of D on V = H 1 (A(C), Q) induced by the endomorphisms. The complex conjugation gives an action c : A(C) → A(C), viewed as an automorphism of the real Lie group A(C). Then it induces an action c∗ on V , which commutes with the action of D on V . Therefore, the decomposition V = V + ⊕ V − of V into the two eigenspaces of c∗ gives two subrepresentations of V . We claim that both subspaces have dimension g. Once this is true, then V + is a vector space over the division algebra D gives [D : Q] | g. For the claim, consider the Hodge decomposition ¯ V ⊗Q C = Γ(A(C), Ω1 ) ⊕ Γ(A(C), Ω). The action c∗ switches the factors on the right-hand side due to the condition that A is defined over a field invariant under c. It follows that dim V + =  dim V − = g. In the following, assume that K is a number field, and (A, M ) is a simple abelian variety of GL(2)-type over K. We will define an L-series L(s, A, M ) taking values in M ⊗Q C such that the Hasse–Weil L-function is given by L(s, A) = NM ⊗C/C L(s, A, M ). We only need to decompose the local L-function. Fix a non-archimedean place v of K and a prime number ` not divisible by v such that M` := M ⊗ Q` is still a field. Then the local L-series at v is defined by Lv (s, A) = Pv (Nv−s )−1 , where Nv is the cardinality of the residue field k(v) of v, and the characteristic polynomial Pv ∈ Z[T ] is defined by Pv (T ) = detQ` (1 − Frob(v)T |V` (A)Iv ). Here V` (A) is the `-adic Tate module over Q` , Iv is the inertia subgroup of the decomposition group in Dv := Gal(K v /Kv ), and Frob(v) is the geometric ¯ Frobenius in Dv /Iv = Gal(k(v)/k(v)). The polynomial Pv (T ) has coefficients in Z and does not depend on the choice of `. Since V` (A) has a module structure over M` , we have a decomposition Pv (T ) = NM` /Q` Pv (T, M ) where NM` /Q` : M` [T ] → Q` [T ] denotes the norm map, and Pv (T, M ) := detM` (1 − Frob(v)|V` (A)Iv ).

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Here detM` denotes the determinant of the operator on the M` -space V` (A)Iv . The polynomial Pv (T, M ) has coefficients in M and does not depend on the choice of `. Its degree is 2 if A has good reduction at v. Otherwise, its degree is 0 or 1. We define a formal Dirichlet series with coefficients in M as follows: Y Lv (s, A, M ), Lv (s, A, M ) := Pv (Nv−s , M )−1 . L(s, A, M ) := v-∞

For any embedding ι : M ,→C, we obtain the usual (complex) L-functions L(s, A, ι) and Lv (s, A, ι) by base change. By Weil’s result on the Riemann hypothesis for abelian varieties in positive characteristics, L(s, A, ι) is absolutely convergent for Res > 3/2. We have the desired decomposition Y L(s, A, ι). L(s, A) = ι

In the next subsection we will discuss the modularity conjecture of A when F = K is totally real in the sense that there is an absolutely irreducible automorphic representation σf = ⊗σv of GL2 (Af ) over M such that Lv (s, A, M ) = L(s − 1/2, σv ). 3.2.2

Rational representations on GL(2) (2)

Let F be a totally real number field with adeles A. Let σ∞ be the complex admissible representation of GL2 (F∞ ), discrete of (parallel) weight 2 with trivial central character. Let σf be an irreducible automorphic representation of GL2 (Af ) over a Qvector space. We say that σf is automorphic (resp. cuspidal) of weight 2 if (2) σ∞ ⊗Q σf is a direct sum of irreducible automorphic (resp. cuspidal) complex representations σi of GL2 (A). We would like to study how σf is decomposed into a product of local representations of GL2 (Fv ) and how σf ⊗Q C is decomposed into irreducible representations. To do this, it will be more convenient to use the quaternion algebra B0 over A whose ramification set is the set Σ0 of all archimedean places. In this way, we can view σf as a representation of B× 0,f and extend it to a representation π acts trivially. on B× over Q so that B× 0,∞ More generally, we will work on an arbitrary quaternion algebra B over A whose ramification set Σ contains all archimedean places. It can be either coherent or incoherent. Recall that A(B× ) denotes the set of isomorphism classes of irreducible automorphic representations π of B× of weight 0, i.e., irreducible admissible complex representations π of B× whose Jacquet–Langlands correspondence σ on GL2 (A) is automorphic cuspidal and discrete of parallel weight 2. Let A(B× , Q) be the set of isomorphism classes of irreducible representations π of B× over Q such that π ⊗Q C is a direct sum of representations in A(B× ). Theorem 3.4. The following are true:

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71

(1) For every π ∈ A(B× ) and α ∈ Aut(C), the α-conjugate representation π α := π⊗(C,α) C is still in A(B× ). The correspondence (α, π) 7→ π α defines an action of Aut(C) on A(B× ). The stabilizer of every π is Gal(C/M ) for some number field M , and the orbit of π is indexed by the set Hom(M, C) of embeddings. Moreover, M is the number field generated by eigenvalues of spherical Hecke operators of π. (2) Let {π ι : ι ∈ Hom(M, C)} be an orbit of A(B× ) under the action of Aut(C) as above. Then there is a unique π ∈ A(B× , Q) with EndB× (π) = M such that π ι = π ⊗(M,ι) C. It follows that M πι . π ⊗Q C = ι∈Hom(M,C)

Moreover, the correspondence {π ι : ι} 7−→ π gives a bijection ∼

A(B× )/Aut(C)−→A(B× , Q). (3) For every π ∈ A(B× , Q) with EndB× (π) = M , we have a unique decomposition O πv . π= v

Here πv is an irreducible admissible representation of B× v over M , and the tensor product is a restricted tensor product over M . Moreover, M is the number field generated by the spherical Hecke eigenvalues of π. (4) (Jacquet–Langlands correspondence) For any π ∈ A(B× , Q) there is a (σ) = EndB× (π), such that πv ' σv unique σ ∈ A(B× 0 , Q) with EndB× 0 at every place v ∈ / Σ(B). The correspondence π 7→ σ gives a bijection between A(B× , Q) and the subset of A(B× 0 , Q) of elements σ such that σvι = σv ⊗ι C is square-integrable for every finite place v ∈ Σ(B) and every embedding ι : EndB× (σ),→C. 0

Let σ = ⊗v σv ∈

A(B× 0 , Q)

with M = EndB× (σ). Define the local L-function 0

L(s, σv ) = Pv (qv−s )−1 ∈ M ⊗Q C by the classification of the representation σv . Here Pv (T ) ∈ M (T ) is a polynomial over M of degree at most two. By the above theorem, L(s, σv ) is the unique function of s ∈ C valued in M ⊗ C whose component for ι : M ,→C is given by L(s, σv , ι) = L(s, σvι ). We may also define the global L-function Y L(s, σ) = L(s, σv ) ∈ M ⊗Q C v

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by putting the usual Γ-factors at infinite places. Then L(s, σ) has a holomorphic continuation to whole complex plane with a functional equation L(s, σ) = (s, σ) L(1 − s, σ e) where (s, σ) is a function of s valued in (M ⊗Q C)× . If π ∈ A(B× , Q) with Jacquet–Langlands correspondence σ ∈ A(B× 0 , Q), define L(s, π) = L(s, σ). L(s, πv ) = L(s, σv ), The strong multiplicity one theorem on A(B× 0 ) induces the strong multiplicity one theorem on A(B× ). Since a local unramified representation is determined by its local L-series, two representations in A(B× , Q) are isomorphic if and only if they have the same local L-series at all but finitely many places. Let us return to the proof of Theorem 3.4. Recall that we have considered the complex Hecke algebras H

=

{locally constant and compactly supported φ : B× f → C},

HU

=

{φ ∈ H : φ(U xU ) = φ(x), ∀x ∈ B× f }.

Now we introduce the rational Hecke sub-algebras HQ

=

{locally constant and compactly supported φ : B× f → Q},

HU,Q

=

{φ ∈ H : φ(U xU ) = φ(x), ∀x ∈ B× f }.

We further introduce the spherical subalgebras TU

=

{φ ∈ HU : φ = 1US ⊗ φS , φS : BS,× → C}, f

TU,Q

=

{φ ∈ HU,Q : φ = 1US ⊗ φS , φS : BS,× → Q}. f

Here S is the set of finite places v of F outside Σ(B) such that Uv is not maximal. It is well-known that TU and TU,Q are commutative. Case where B is coherent Assume first that B is coherent. Then B = B ⊗F A for a totally definite quater× , Q) the space of locally nion algebra B over F . Denote by C ∞ (B × \B× /B∞ × × constant functions f : B → Q left invariant under B × . Since B∞ is connected, ∞ × × × it acts trivially on this space. Define C (B \B /B∞ , C) similarly. By the spectral decomposition, M × , C) = π. C ∞ (B × \B× /B∞ π∈A(B)

Since the left-hand side has a Q-structure, Aut(C) acts on A(B). We need to filter the above spaces in terms of open compact subgroups U of B× f . In fact, × C ∞ (B × \B× /B∞ , Q) = lim C(B × \B× f /U, Q), −→ U

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MORDELL–WEIL GROUPS AND GENERATING SERIES

× , C) = lim C(B × \B× C ∞ (B × \B× /B∞ f /U, C). −→ U

× × Here C(B \B× f /U, Q) (resp. C(B \Bf /U, C)) denotes the space of maps from the finite set B × \B× f /U to Q (resp. C). Fix such a U . Let HU,Q (resp. TU,Q ) be the image of HU,Q (resp. TU,Q ) in End(C(B × \B× f /U, Q)). Let HU and TU be their base changes to C. We have the following decomposition of HU -modules ×

M

C(B × \B× f /U, C) =

πU .

π∈A(B× )

By the strong multiplicity one theorem, {π U 6= 0} is a finite set of distinct finite-dimensional irreducible HU -modules. By the density theorem of Jacobson and Chevalley for semisimple modules, the map M HU −→ EndC (π U ) π U 6=0

is surjective. See Theorem 11.16 of [La] for example. Hence, we have obtained M M TU = C, HU = End(π U ). π U 6=0

π U 6=0

This shows that HU,Q is semisimple with center TU,Q , and that the set of π that appears in the sum is indexed by (SpecTU,Q )(C) with a compatible action by Aut(C). Thus the orbits of {π U 6= 0} under Aut(C) are indexed by closed points in SpecTU,Q . More precisely, decompose TU,Q as a direct sum of fields M in the form M TU,Q = M. Then each M represents a conjugacy class {(π U )ι : ι ∈ Hom(M, C)} such that the action of TU,Q on (π U )ι is given by the composition ι

TU,Q −→M −→C. Fix an M and define π U = C(B × \Bf /U, Q) ⊗TU,Q M. Then π U is a module over HU,Q such that π U ⊗(M,ι) C = (π U )ι ,

π U ⊗Q C =

M (π U )ι .

As (π U )ι are non-isomorphic to each other, π U is irreducible over Q and geometrically irreducible over M . Thus EndHU,Q (π U ) = M .

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If U 0 is an open compact subgroup of U , then we have a morphism TU 0 ,Q → TU,Q . The strong multiplicity one theorem shows that this is surjective. If M 0 appears in SpecTU 0 ,Q , we can define the HU,Q -module π U which includes π U . The direct limit of these spaces forms a representation π such that π ⊗(M,ι) C = π ι ,

π ⊗Q C = ⊕π ι .

This proves part (1). The remaining part of part (2) follows from the following: Lemma 3.5. For two irreducible representations π1 and π2 in A(B× , Q), HomB× (π1 , π2 ) ⊗Q C = HomB× (π1 ⊗Q C, π2 ⊗Q C). Thus π1 and π2 are isomorphic if and only if π1 ⊗Q C and π2 ⊗Q C have a common irreducible component in A(B× ). Proof. It is clear that the left-hand side is included in the right-hand side. We need to prove the other direction. Let φ be one element in the right-hand side. Decomposing πi ⊗ C into irreducible representations over C, there is an irreducible complex representation σ and surjective morphisms αi : πi ⊗ C−→σ such that α1 = α2 ◦ φ. Let π0 be a representation satisfying the first part of Theorem 3.4 (2) for the conjugacy class of σ. Thus we have another projection α0 : π0 ⊗ C−→σ. It is clear that the restriction of αi on each πi ⊗ 1 is injective with images αi (πi ⊗ 1) generating σ respectively. Let U be an open compact subgroup of B× such that σ U is one-dimensional. Applying the element 1U ∈ HU , we have an σ U = Cαi (πiU ). This implies that πiU 6= 0 for each i. Let vi ∈ πiU be any nonzero elements. Then there is a ci ∈ C× such that ci αi (vi ) = α0 (v0 ) for i = 1, 2. We see that ci αi (πi ) = α0 (π0 ) since both sides are irreducible over Q with a common element α0 (v0 ). This shows that there is an isomorphism f : πi −→π0 bringing vi to v0 . Thus we may assume in the lemma that both πi are equal to π0 . Let M be the field of spherical eigenvalues of σ. Then the right hand is given by: End(π0 ⊗Q C) = End(⊕ι:M →C σ ι ) = ⊕ι:M −→C C = M ⊗ C. Since M acts on π0 , the left-hand includes M ⊗ C. Thus we must have the equality.  Part (3) and part (4) of the theorem are clear as π is absolutely irreducible over M . The following is a by-product of the proof. Theorem 3.6. Assume that B is coherent with rational structure B over F . Then M × , Q) = π. C ∞ (B × \B× /B∞ π∈A(B× ,Q)

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Case when B is incoherent Assume B is incoherent. Then B defines a Shimura curve X = limU XU . Fix an ←− open and compact subgroup U of B× f . Fix an embedding τ : F → C. Then we have an action of B× on the Hodge structure H 1 (Xτ , Q). Let c be the involution on H 1 (Xτ , Q) induced by complex conjugation on Xτ (C). Let HU,Q and TU,Q be the images of HU,Q and TU,Q in End(H 1 (XU,τ , Q)) respectively. Both of them are finite-dimensional since H 1 (XU,τ , Q) is finitedimensional. The complex conjugation c commutes with the action of HU,C . Consider the C-linear actions of TU,C := TU,Q ⊗Q C,

HU,C := HU,Q ⊗Q C,

c

on the complex Hodge structure H 1 (XU,τ , C) = H 1,0 (XU,τ ) ⊕ H 0,1 (Xτ ). The action c switches the last two factors. We call a rational Hodge structure (V, c) of weight 1 with an involution on the underlying Q vector space a rational Hodge structure over R if c switches the Hodge factors V 1,0 and V 0,1 . By the Jacquet–Langlands correspondence, we have an isomorphism of HU,C modules: M (π U ⊕ π ¯ U ), H 1 (XU,τ , C) = π∈A(B× )

¯U where π ¯ := π ⊗(C,c) C is the complex conjugation of π. The direct sum π U ⊕ π gives a Hodge structure over R. Since the left-hand has a rational structure, this shows that A(B× ) is closed under Aut(C). Under this isomorphism, we have M M TU,C = C, HU,C = End(π U ). π∈A(B× )

π U 6=0

Note that the direct sums have only finitely many nonzero terms. This shows that HU,Q is semisimple with center TU,Q and the representations appearing in the sum are indexed by C-points of the scheme SpecTU,Q . Then we can decompose TU,Q as a direct sum of fields M in the form TU,Q = ⊕M. Each M represents a conjugacy class {π ι : ι ∈ Hom(M, C)} with each π ι ∈ A(B× ) such that the action on TU,Q on (π U )ι is given by composition ι

TU,Q −→M −→C. Fix M . Define V U := H 1 (XU,τ , Q) ⊗TU,Q M. Then V U is a rational Hodge structure over R with an action by HU,Q . It further has decompositions: M V U ⊗(M,ι) C = (π U )ι ⊕ (π U )ι , V U ⊗Q C = ((π U )ι ⊕ (π U )¯ι ). ι:M −→C

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Here ¯ι means the composition of ι and the complex conjugation of C. If U 0 is an open compact subgroup of U , then SpecTU,Q is a subscheme of SpecTU 0 ,Q . 0 Thus we can similarly construct the space V U , which includes V U . The direct limit of V U forms a rational Hodge structure V over R with an action by M . The Hodge decompositions are V ⊗(M,ι) C = π ι ⊕ π ι ,

V ⊗Q C =

M

(π ι ⊕ π¯ι ).

ι:M −→C

Choose a U such that (π U )ι is one-dimensional for all ι. Then V U has dimension 2 over M , and dimension 2 dim M over Q. It follows that V U is simple as an M -Hodge structure with EndM −Hodge/R (V U ) = M. Now we define π := HomM −Hodge/R (V U , V ) as a representation of B× . By construction, we have the following properties: (a) π has coefficient M , and dimM π U = 1; (b) π ⊗ C is included into HomHodge/R (VCU , VC ) ' VCdim M . By these two properties, we have isomorphisms of B× -modules: π ⊗Q C '

M

πι ,

π ⊗(M,ι) C ' π ι .

ι:M ,→C

This proved part (1) and the first half of part (2). The other parts can be proved using the same argument as in the coherent case. The following is a by-product of the proof. Theorem 3.7. Assume that B is incoherent and defines a Shimura curve X over F . Fix an embedding τ : F ,→C. Then we have the following decomposition of rational Hodge structures over R with an B× -action: H 1 (Xτ (C), Q) =

M

π ⊗M V (π).

π∈A(B× ,Q)

Here M = EndB× (π) is a number field, and V (π) is an irreducible rational Hodge structure over R defined by V (π) := HomB× (π, H 1 (Xτ (C), Q)).

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3.2.3

Abelian varieties parametrized by Shimura curves

Let B be a totally definite incoherent quaternion algebra and let X = limU XU be the associated Shimura curve. Let A be a simple abelian variety over F . We say that A is parametrized by X if there is a non-constant morphism f : XU → A over F for some compact and open subgroup U of B× f . Using the normalized Hodge class ξU as a base point, it is equivalent to the existence of a nonzero homomorphism JU → A where JU = Cl0 (XU ) is the Jacobian variety. Recall in Chapter 1 we have introduced the key definition Hom0ξU (XU , A). πA = Hom0ξ (X, A) = lim −→ U

Here

Hom0ξU (XU , A)

denotes the Q-vector space of morphisms in HomF (XU , A) ⊗Z Q

which maps the Hodge class ξU of XU to zero in A. More precisely, Hom0ξU (XU , A) consists of elements a ⊗ f for any rational number a ∈ Q and any morphism f : XU → A such that ξU is mapped to zero in the composition f∗

α

Pic(XU )Q −→ CH0 (A)Q −→ A(F )Q . Here CH0 (A) is the Chow group of cycles of dimension zero on A, and α : CH0 (A) → A(F ) is the canonical map sending a formal summation of points (up to linear equivalence) to the same summation of points with respect to the group law on A. The collection {Hom0ξU (XU , A)}U forms a direct system naturally, and thus the direct limit makes sense. Since any morphism XU → A factors through the Jacobian JU of XU , we have Hom0ξU (XU , A) = Hom0 (JU , A), and Hom0 (JU , A). πA = Hom0 (J, A) = lim −→ U

It is easy to see that representation πA is admissible. Theorem 3.8. Let A be a simple abelian variety over F parametrized by X with M = End0 (A). Then A is of strict GL(2)-type, πA ∈ A(B× , Q), End(πA ) = M and Lv (s, A, M ) = L(s, πA,v ) for each finite place v of F . Moreover, the map A 7→ πA defines a bijection ∼

{abelian varieties parametrized by X up to isogeny}−→A(B× , Q).

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Proof. Fix an embedding τ : F ,→R. Theorem 3.7 gives a decomposition of the Hodge structure on H 1 (Xτ (C), Q) over R into a direct sum of π ⊗M V (π) over π ∈ A(B× ), where M = End(π),

V (π) = HomB× (π, H 1 (Xτ (C), Q)).

By Riemann’s theorem, V (π) defines a simple abelian variety Aπ,τ over R such that H 1 (Aπ,τ (C), Q) = V (π) as a rational Hodge structure over R. Moreover, the embedding π−→HomHodge/R (V (π), H 1 (Xτ (C), Q)) defines an embedding π−→Hom0R (Jτ , Aπ,τ ). Lemma 3.9. The following are true: (1) Every Aπ,τ has a unique model Aπ over F such that the image of π in Hom0 (Jτ , Aπ,τ ) is in Hom0 (J, Aπ ); (2) Each Aπ has the endomorphism ring equal to End(π) and the local L-series of Aπ is given by the local L-series of π. The lemma shows that each JU has a decomposition in the category of abelian varieties up to isogeny by M JU ' π U ⊗End(π) Aπ . π∈A(B× ,Q)

Here Aπ is simple and its local L-series is equal to those of π. This implies in particular that Aπ is not isogenous to each other. Thus we have Hom0 (JU , Aπ ) ' π U . Consequently, every abelian varieties parametrized by X must be isogenous to one of Aπ and then πA ' π. The theorem follows. It remains to prove the lemma. We need only check that in the proof of Theorem 3.7, the action of HU on H 1 (J, Q) can be realized as algebraic correspondences on J. In fact, the characteristic function of the double coset U xU is given by the Hecke correspondence Z(x)U on the Jacobian JU of XU via push-forward. It gives an action of HU,Q on JU (in the category of abelian varieties up to isogeny). Then HU,Q is generated by the image of Z(x)U in End0 (JU ), and TU,Q is generated by the image of Z(x)U in End0 (JU ) such that xv = 1 at all places v such that Uv is not maximal. We can then define Aπ to be JU ⊗TU End(π) when π U is one-dimensional. Clearly Aπ has an action by End(π) with the Betti cohomology V (π) at place τ . This implies that End(π) ⊂ End0 (Aπ ). Thus A is of GL(2)-type. By the Eichler–Shimura theory, the local L-series of Aπ , given by L-series of forms appearing in the cotangent space of Aπ , is exactly the L-series of π (up to a translation on s).

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79

It remains to show that Aπ is simple with End0 (Aπ ) = End(π) =: M . Write A ' ⊕Bini up to isogeny, with Bi simple and non-isogenous to each other. Then we have an embedding M −→ ⊕ Mni (Di ),

Di := End0 (Bi ).

By Lemma 3.3, deg Di | dim Bi . Thus deg M | ni dim Bi . On the other hand X deg M = dim Aπ = ni dim Bi . i

It follows that there is only one term in the decomposition. Thus we may write Aπ ' B n with B simple and D = End0 (B). Then deg M = n deg D. It follows that D is commutative and included into M , and that for almost all places v, Lv (s, Aπ , M ) = Lv (s, B, D)n . Thus almost all the local L-series of Aπ have coefficients in D. It follows that the spherical eigenvalues on π take values in D. So we must have M = D,  Aπ = B. This shows that Aπ is simple with End0 (Aπ ) = End(π). Let F be a totally real field. Let A be a simple abelian variety of GL(2)-type over F with M = End0 (A). We say that A is automorphic if there is a rational automorphic representation σ ∈ A(B× 0 ) with End(σ) = M such that Lv (s, A, M ) = L(s − 1/2, σv ) for all finite places v of F . By the work of Carayol, it suffices to have equality for unramified primes. By Theorem 3.4, it is equivalent to the existence of a conjugacy class {σ ι : ι} in A(B× 0 ) such that Lv (s, A, ι) = L(s − 1/2, σvι ). The automorphy of an abelian variety of GL(2)-type depends only on its isogeny class, by Faltings’s isogeny theorem in [Fa1]. We say that an automorphic representation σ ∈ A(B× 0 , Q) is geometric if it corresponds to an automorphic abelian variety of GL(2)-type over F . Denote by Ageom (B0 , Q) the subset of geometric representation. Then we have defined a bijection between Ageom (B0 , Q) and {Automorphic abelian varieties of GL(2)-type over F up to isogeny}. Conjecture 3.10. The following are true: (1) Every abelian varieties of GL(2)-type over F is automorphic. (2) Every rational cuspidal representation in A(B× 0 , Q) is geometric.

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3.2.4

Duality

Let A be an abelian variety over F parametrized by X, and denote M = End0 (A). Then the dual A∨ of A is also parametrized by X, and denote M ∨ := End(A∨ ). For any endomorphism M : A → A, the pull-back map m∗ : Pic0 (A) → Pic0 (A) gives a homomorphism m∨ : A∨ → A∨ . Thus we get a canonical isomophism M −→M ∨ , m 7−→ m∨ . Identify M ∨ with M by this isomorphism. The goal of this subsection is to consider the duality between Hom0 (JU , A) πA = Hom0 (J, A) = lim −→ U

and Hom0 (JU , A∨ ) πA∨ = Hom0 (J, A∨ ) = lim −→ U

×

as representations of B

over M .

Lemma 3.11. The map (·, ·)U : Hom0 (JU , A) × Hom0 (JU , A∨ )−→M defined by (f1 , f2 ) 7−→ f1 ◦ f2∨ is a perfect pairing of vector spaces over M . Here for any f2 : JU → A∨ , the homomorphism f2∨ : A → JU represents the homomorphism f2∗ : Pic0 (A∨ ) → Pic0 (JU ) under the canonical isomorphisms (A∨ )∨ = A and JU∨ = JU . Moreover, the pairing is Hermitian in the sense that (tx, y)U = (x, t∨ y)U ,

t ∈ HU,Q .

Here the involution t 7→ t∨ on HU,Q is induced by the transpose Z(x)U 7→ Z(x−1 )U . Proof. It is easy to see that the map Hom0 (JU , A∨ )−→Hom0 (A, JU ),

f2 7−→ f2∨

is an isomorphism. Via this isomorphism, the pairing (·, ·)U becomes the composition map Hom0 (A, JU ) × Hom0 (JU , A)−→M. It is verified to be perfect by writing JU as a product of simple abelian varieties up to isogeny. The second result follows from the fact that the dual of the endomorphism Z(x)U : JU → JU is exactly Z(x−1 )U : JU → JU . It can be checked by definition. 

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Now we extend the pairing in the lemma to the direct limit. Define a pairing (·, ·) : πA × πA∨ −→M by setting (f1 , f2 ) :=

1 1 ∨ (f1,U , f2,U )U = f1,U ◦ f2,U . vol(XU ) vol(XU )

Here f1 = {f1,U }U ∈ πA and f2 = {f2,U }U ∈ πA∨ , and U is any compact open subgroup of B× f such that f1,U and f2,U are defined. Theorem 3.12. The above definition does not depend on the choice of U and gives a perfect M -bilinear pairing (·, ·) : πA × πA∨ −→M. It is B× -invariant in the sense that (πA (h)f1 , πA∨ (h)f2 ) = (f1 , f2 ),

∀ h ∈ B× , f1 ∈ πA , f2 ∈ πA∨ .

Proof. We need only check the independence of U . Everything else follows from the lemma. For two compact open subgroups U1 ⊂ U2 of B× f , the projection φ : XU1 → XU2 induces two morphisms φ∗ : JU2 −→JU1 ,

φ∗ : JU1 −→JU2 .

The composition is φ∗ ◦ φ∗ = deg φ = vol(XU1 )/vol(XU2 ). Then the definition is independent of U .



Remark. In the limit level, the pairing is just the composition f∨

2 A −→ J∨

vol(X)−1

−→

f2

J −→ A.

Here the canonical element vol(X)−1 = {vol(XU )−1 }U ∈ Hom0 (J ∨ , J) is as in §3.1.6. Let λ : A → A∨ be a polarization, and τ : M → M be the Rosati involution induced by λ. Then we have an isomorphism πA ' πA∨ which is τ -linear under the action of M . In this way, we get a τ -Hermitian pairing (·, ·)λ :

πA × πA −→M.

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For any nonzero j ∈ Hom0 (JU , A), we can define a polarization by λ := (j ◦ j ∨ )−1 . It can be shown that the induced Hermitian pairing on πA is positive definite in the sense that, for nonzero element f ∈ πA , the value (f, f )λ lies in the subgroup × of totally positive elements of M × . Especially, (j, j)λ = 1. M+ Remark. If A is an elliptic curve, then M = Q and the canonical polarization gives an isomorphism A ' A∨ . The resulting pairing πA × πA → Q on πA = Hom0ξ (X, A) is defined by (f1 , f2 ) =

1 ∨ f1,U ◦ f2,U . vol(XU )

In particular, (f, f ) =

3.2.5

1 deg(fU : XU → A). vol(XU )

The main theorem in complex coefficients

In the rest of the book, we will write the main theorem (Theorem 1.2) in complex coefficients. In fact, M L ⊗Q C = C. ι:L,→C

Theorem 3.13. For any embedding ι : L ,→ C, we have hPχ (f1 )ι , Pχ−1 (f2 )ι iNT =

ι , χι ) ζF (2)L0 (1/2, πA α(f1ι , f2ι ). ι 4L(1, η)2 L(1, πA , ad)

Here the basic setting is as in this section. Namely, X is a Shimura curve over a totally real field F , and A is an abelian variety parametrized by X. Some extra notations are as follows: • L is a finite extension of M = End0 (A). • χ : Gal(E ab /E) → L× be a character of finite order, also viewed as a character of E × \EA× by the reciprocity law. • P be the CM point on X given by a CM extension E of F . • The point Z

f1 (P τ ) ⊗M χ(τ )dτ

Pχ (f1 ) = Gal(E/E) ab

lies in A(E )Q ⊗M L, and is χ-invariant under the action of Gal(E ab /E).

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83

• The height hPχ (f1 )ι , Pχ−1 (f2 )ι iNT ∈ C is the projection of the L-linear height hPχ (f1 ), Pχ−1 (f2 )iL ∈ L ⊗Q C to the component C indexed by ι. There is a more direct interpretation of the height pairing. The usual theory of N´eron–Tate height gives a Q-bilinear non-degenerate pairing h·, ·iNT : A(F )Q × A∨ (F )Q −→R. We refer to §7.1.1 for a quick review. Recall that the field M = End0 (A) acts on A(F )Q by definition, and acts on A∨ (F )Q through the duality. By the adjoint property of the height pairing in Proposition 7.3, the pairing h·, ·iNT descends to a Q-linear map h·, ·iNT : A(F )Q ⊗M A∨ (F )Q −→R. For simplicity, denote V = A(F )Q ⊗M A∨ (F )Q . It is an M -module. The Q-linear map h·, ·iNT : V −→R induces a C-linear map h·, ·iNT : V ⊗Q C−→C. By linear algebra, V ⊗Q C = V ⊗M (M ⊗Q C) = V ⊗M (⊕ι:M ,→C C) = ⊕ι:M ,→C V ι . Here we denote W ι = W ⊗(M,ι) C for any vector space W over M , and denote by wι the image of w in W ι for any w ∈ W . The C-linear map induces, for each ι : M ,→C, a C-linear map h·, ·iιNT : V ι −→C. In other words, we obtain a C-bilinear pairing h·, ·iNT : A(F )ιQ × A∨ (F )ιQ −→C. 3.3

MAIN THEOREM IN TERMS OF PROJECTORS

By the basic definitions and basic properties in §3.1.5 and §3.1.6, we define the projector T(f1 ⊗f2 ), and state an equivalent form of the main theorem (Theorem 1.2) in Theorem 3.15. In §3.3.4 we prove the equivalence between these two theorems, based on an expression of the projector in terms of parametrizations of modular abelian varieties in §3.3.3.

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3.3.1

Projector I: Cohomological definition

The goal of this subsection is, for any π ∈ A(B× ), to introduce a homomorphism T: π⊗π e ,−→ Hom0 (J, J ∨ )C . It will be given by Hecke correspondences. Fix a decomposition M

H 1,0 (Xτ ) =

π.

π∈A(B× )

For any open compact subgroup U of B× f , the decomposition induces a decomposition M H 1,0 (XU,τ ) = πU , π∈A(B× )

and a decomposition of the dual space M

H 1,0 (XU,τ )∨ =

π eU .

π∈A(B× )

It follows that M

Hom(H 1,0 (XU,τ ), H 1,0 (XU,τ )) =

π1 ,π2

π1U ⊗ π e2U .

∈A(B× )

It gives an injection iU : Π∆,U ,−→ Hom(H 1,0 (XU,τ ), H 1,0 (XU,τ )), where the “diagonal” Π∆,U :=

M

πU ⊗ π eU .

π∈A(B× )

Note that the choice of the isomorphism M H 1,0 (Xτ )−→

π

π∈A(B× )

as an H-modules is not unique. One has the freedom of multiplying every component on the right-hand side by a constant. Once such an isomorphism is chosen, the corresponding isomorphisms of H 1,0 (XU,τ )∨ and H 1,0 (XU,τ )∨ are uniquely determined. In particular, the map iU does not depend on the choice of the isomorphism for H 1,0 (Xτ ). We will prove in Proposition 3.14 that the image of iU is actually contained in the image of the inclusion Hom0 (JU , JU∨ )C ,−→ Hom(H 1,0 (XU,τ ), H 1,0 (XU,τ )).

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85

Hence, it induces a well-defined map TU : Π∆,U ,−→ Hom0 (JU , JU∨ )C . Now vary U . Denote Π∆ :=

M

π⊗π e.

π∈A(B× )

We will prove that the system T = {vol(XU )TU }U gives a well-defined map T : Π∆ ,−→ Hom0 (J, J ∨ )C . It is the definition of the projector we propose in this section. Next, we prove the algebraicity of the image of iU , which is crucial for the definition of TU and T. We also sketch the reason for the compatibility of the system T = {vol(XU )TU }U . Proposition 3.14. (1) Let U be an open compact subgroup of B× f . For any α ∈ Π∆,U , there is a function φ ∈ HU such that iU (α) = T(φ)∗U in Hom(H 1,0 (XU,τ ), H 1,0 (XU,τ )). Hence, the map TU : Π∆,U ,−→ Hom0 (JU , JU∨ )C is well-defined and independent of the choice of the embedding τ : F ,→C. (2) The system T = {vol(XU )TU }U is a direct system and defines a map T : Π∆ ,−→ Hom0 (J, J ∨ )C . Proof. We first prove (1), which is the essential part of the proposition. Consider the natural map M EndC (π U ) = Π∆,U . R : HU −→ π∈A(B× )

Here the direct sums are actually finite sums of finite-dimensional representations of HU since π U is nonzero for only finitely many π. The map R is surjective, and takes φ to be a preimage of α in HU . It satisfies the requirement. The surjectivity is also used in the proof of Theorem 3.4. It follows from the density theorem of Jacobson and Chevalley for semisimple modules (cf. Theorem 11.16 of [La]), and the property that {π U 6= 0} is a finite set of distinct finite-dimensional irreducible HU -modules. Now we consider (2). It suffices to show that the system i := {vol(XU )iU }U

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gives a well-defined map i : Π∆ ,−→ Homcont (H 1,0 (Xτ )0 , H 1,0 (Xτ )). ∗ 1,0 For any U 0 ⊂ U , the pull-back map πU (XU,τ ) → H 1,0 (XU 0 ,τ ) 0 ,U : H 0 0 ∗ U → π U exactly equal to the natural inclusion π U ,→π U . induces a map πU 0 ,U : π The push-forward map (πU 0 ,U )∗ : H 1,0 (XU 0 ,τ ) → H 1,0 (XU,τ ) induces a map 0 (πU 0 ,U )∗ : π U → π U which is more complicated than the pull-back. However, ∗ we always have (πU 0 ,U )∗ πU 0 ,U = deg πU 0 ,U . 0 eU and f 0 ∈ π U , It suffices to verify that, for any f1 ⊗ f2 ∈ π U ⊗ π ∗ 0 0 vol(XU 0 ) · T0U 0 (f1 ⊗ f2 )(f 0 ) = vol(XU ) · πU 0 ,U (TU (f1 ⊗ f2 )((πU 0 ,U )∗ f )) .

By definition, T0U 0 (f1 ⊗ f2 )(f 0 ) = (f 0 , f2 ) f1 . and T0U (f1 ⊗ f2 )((πU 0 ,U )∗ f 0 ) = ((πU 0 ,U )∗ f 0 , f2 ) f1 . It is reduced to check vol(XU 0 ) · (f 0 , f2 ) = vol(XU ) · ((πU 0 ,U )∗ f 0 , f2 ). Notice that f2 ∈ π eU is invariant under U , so (f 0 , f2 ) = (f 00 , f2 ),

((πU 0 ,U )∗ f 0 , f2 ) = ((πU 0 ,U )∗ f 00 , f2 ),

where f 00 ∈ π U is the average Z f =

π(h)f 0 dh.

00

U

Then we have ∗ 00 = deg(πU 0 ,U )f 00 . (πU 0 ,U )∗ f 00 = (πU 0 ,U )∗ πU 0 ,U f

It gives vol(XU 0 ) · (f 00 , f2 ) = vol(XU ) · ((πU 0 ,U )∗ f 00 , f2 ). The result follows. 3.3.2

Formulation in terms of projectors

Assume the following notations and assumptions: • F is a totally real field with adele ring A = AF . • B is a totally definite incoherent quaternion algebra over A.



MORDELL–WEIL GROUPS AND GENERATING SERIES

87

• E is a totally imaginary quadratic extension of F , with a fixed embedding EA ,→ B over A. • π ∈ A(B× ), i.e., π is an irreducible admissible complex representation of B× such that its Jacquet–Langlands correspondence σ is a cuspidal automorphic representation of GL2 (A), discrete of weight two at all infinite places. • χ : E × \EA× → C× is a character of finite order with ωπ · χ|A× = 1. • Denote by η : F × \A× → C× the quadratic character associated to the extension E/F . ×

Recall that P is an element in X E (E ab ) which we fix once for all. Introduce the χ-eigencomponent Z Tt (P − ξP ) χ(t)dt ∈ J(F )C . Pχ = T (F )\T (A)/Z(A)

Here Tx denotes the Hecke action given by right-multiplication, acting as pushforward on the divisor. For the regularized integral, we refer to §1.6. The following is an equivalent form of Theorem 1.2. Theorem 3.15. For any f1 ⊗ f2 ∈ π ⊗ π e, hT(f1 ⊗ f2 )Pχ , Pχ−1 iNT =

ζF (2)L0 (1/2, π, χ) α(f1 ⊗ f2 ). 4L(1, η)2 L(1, π, ad)

We explain the height pairing in the theorem. By definition, T(f1 ⊗ f2 ) ∈ Hom0 (J, J ∨ )C ⊂ Hom(J(F )C , J ∨ (F )C ). Thus we have T(f1 ⊗ f2 )Pχ ∈ J ∨ (F )C . The height pairing in the theorem is just the natural height pairing h·, ·iNT : J(F )C × J ∨ (F )C −→ C. It is obtained as the limit of the usual N´eron–Tate height pairing. See §3.1.6 for more details. In the following, we prove the equivalence between Theorem 3.15 and Theorem 1.2. 3.3.3

Projector II: Algebraic interpretation

Assume the notation of Theorem 1.2. parametrized by X, and

Recall that A is an abelian variety

Hom0 (JU , A). πA = Hom0 (J, A) = lim −→ U

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It is an irreducible admissible representation of B× over M = End0 (A). By the Eichler–Shimura construction, there is a canonical action of M on JU , compatible with both push-forward and pull-back when varying U . In this sense, M commutes with πA . Recall that we have introduced the duality (·, ·) : πA × πA∨ −→M by 1 1 ∨ (f1,U , f2,U )U = f1,U ◦ f2,U . vol(XU ) vol(XU ) Now we consider the composition in the opposite order. Denote ∨ Talg (f1 , f2 )U := f2,U ◦ f1,U ∈ Hom0 (JU , JU ). (f1 , f2 ) =

It is easy to see that Talg (f1 , f2 ) := {Talg (f1 , f2 )U }U is a direct system. It defines an element Talg (f1 , f2 ) ∈ Hom0 (J, J ∨ ). It follows that we have obtained a map Talg : πA × πA∨ −→Hom0 (J, J ∨ ). Since M commutes with all the related maps here, the map descends to a map Talg : πA ⊗M πA∨ −→Hom0 (J, J ∨ ). For any embedding ι : M ,→C, the base change by ι gives 0 ι ι ∨ ι ⊗C πA Tιalg : πA ∨ −→Hom (J, J ) .

Note that Hom0 (J, J ∨ )ι embeds naturally into Hom0 (J, J ∨ ) ⊗Q C. See §1.6.8 for example. Therefore, we can also write the map as 0 ι ι ∨ ⊗C πA Tιalg : πA ∨ −→Hom (J, J ) ⊗Q C.

It actually gives a decomposition Talg =

X

Tιalg

ι∈Hom(M,C)

in Hom0 (J, J ∨ ) ⊗Q C. It is just the decomposition induced by the spectral decomposition under the action of M . Recall that, by the cohomological method, we have also defined a projector 0 ι ι ∨ ⊗C πA T : πA ∨ −→Hom (J, J ) ⊗Q C.

The definition employs the duality map ι ι × πA (·, ·) : πA ∨ −→C

obtained from the duality between πA and πA∨ over M . It is reasonable to expect that these two definitions agree.

MORDELL–WEIL GROUPS AND GENERATING SERIES

89

Proposition 3.16. For any f1 ∈ πA and f2 ∈ πA∨ , and embedding ι : M ,→C, one has T(f1ι ⊗ f2ι ) = Talg (f1 ⊗ f2 )ι in Hom0 (J, J ∨ ) ⊗Q C. Proof. It suffices to prove the identity on the level of any open compact subgroup U of B× f . Let U = Hom0 (JU , A), f1 ∈ π A

0 U ∨ f2 ∈ πA ∨ = Hom (JU , A ).

We need to prove vol(XU ) · T(f1ι ⊗ f2ι )U = Talg (f1 ⊗ f2 )ιU . Here the right-hand side is defined as an element of Hom0 (JU , JU∨ ) ⊗Q C by a similar method. Since we are in the case of characteristic zero, it suffices to prove that they induce the same action on the tangent space, namely H 1,0 (JU,τ ). By the property of GL(2)-type, H 1,0 (Aτ ) is a free module of rank one under the pull-back action of M M ⊗Q C = C. ι∈Hom(M,C)

Thus we can decompose H 1,0 (Aτ ) =

M

H 1,0 (Aτ )(ι)

ι∈Hom(M,C)

by the idempotents. Each piece H 1,0 (Aτ )(ι) is just a C-vector space of dimension one, on which M acts by ι. On the other hand, the action of M on H 1,0 (JU,τ ) via pull-back induces a decomposition M H 1,0 (JU,τ ) = H 1,0 (JU,τ )(0) ⊕ H 1,0 (JU,τ )(ι). ι∈Hom(M,C)

Here M kills H 1,0 (JU,τ )(0), and acts on H 1,0 (JU,τ )(ι) by ι. Fix a nonzero element ω ι ∈ H 1,0 (Aτ )(ι). Then we obtain an isomorphism U,ι −→H 1,0 (JU,τ )(ι), πA

g 7−→ g ∗ ω ι .

It is equivariant under the action of HU . The pull-back action of Talg (f1ι ⊗ f2ι )ιU on H 1,0 (XU,τ ) is given by (f2∨ ◦ f1 )∗ : H 1,0 (XU,τ )(ι)−→H 1,0 (XU,τ )(ι). We need to prove that (f2∨ ◦ f1 )∗ α = vol(XU ) · i(f1ι ⊗ f2ι )U α

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for any α ∈ H 1,0 (XU,τ )(ι). Here i(f1ι ⊗ f2ι )U is the induced action of T(f1ι ⊗ f2ι )U on H 1,0 (XU,τ ). U and c ∈ C. Note that We can always write α = c · f10∗ ω ι for some f10 ∈ πA U,ι U,ι the pairing πA × πA∨ → C is induced by the pairing U U πA × πA ∨ −→M,

(g1 , g2 ) 7−→ vol(XU )−1 · g1 ◦ g2∨ .

It follows that i(f1ι ⊗ f2ι )U f10∗ ω ι = vol(XU )−1 · (f10 ◦ f2∨ )ι · f1∗ ω ι . Hence, the desired equality becomes (f2∨ ◦ f1 )∗ f10∗ ω ι = (f10 ◦ f2∨ )ι · f1∗ ω ι . Note that (f10 ◦ f2∨ ) ∈ M behaves as a scalar. We have (f2∨ ◦ f1 )∗ f10∗ = (f10 ◦ f2∨ ◦ f1 )∗ = (f10 ◦ f2∨ )ι · f1∗ . The equality follows. 3.3.4



The equivalence

Now we consider the equivalence between Theorem 3.15 and Theorem 1.2. Resume the notation of Theorem 1.2. It suffices to prove the height identity hT(f1ι ⊗ f2ι )Pχι , P(χι )−1 iNT = hPχ (f1 )ι , Pχ−1 (f2 )ι iNT . Under the reciprocity law rec : E × \EA× −→Gal(E ab /E), both sides depend on (χ, χ−1 ) in the same manner. So it suffices to prove that, for any P1 , P2 ∈ J(F )Q , hT(f1ι ⊗ f2ι )P1 , P2 iNT = hP1 (f1 )ι , P2 (f2 )ι iNT . Here P1 (f1 ) = f1∗ P1 ∈ A(F )Q ,

P2 (f2 ) = f2∗ P2 ∈ A∨ (F )Q .

By Proposition 3.16, T(f1ι ⊗ f2ι ) = Talg (f1 ⊗ f2 )ι . It follows that, in J ∨ (F )Q ⊗Q C, T(f1ι ⊗ f2ι )P1 = Talg (f1 ⊗ f2 )ι P1 = Talg (f1 ⊗ f2 )ι P1ι = Talg (f1 ⊗ f2 )P1ι . It is reduced to check hTalg (f1 ⊗ f2 )P1ι , P2ι iNT = hP1 (f1 )ι , P2 (f2 )ι iNT .

MORDELL–WEIL GROUPS AND GENERATING SERIES

91

U U and f1 ∈ πA This follows from the projection formula. Assume that f1 ∈ πA ∨ ∨ for some U . Then they are realized as f1 : JU → A and f2 : JU → A . By definition, we can view Talg (f1 ⊗ f2 ) = f2∨ ◦ f1 as an endomorphism of JU . Realize P1 and P2 as points in JU (F ). It follows that

hTalg (f1 ⊗ f2 )P1ι , P2ι iNT = hf2∨ (f1 (P1ι )), P2ι iNT = hf1 (P1ι ), f2 (P2ι )iNT . Here the last identity follows from Proposition 7.3. This finishes the proof. Both Theorem 3.15 and Theorem 1.2 are not in the forms for which we can perform computations. By expressing the projector T(f1 ⊗ f2 ) as an arithmetic theta lifting, we will obtain another equivalent form in Theorem 3.21. The statement of Theorem 3.21 is the major goal for the remaining sections of this chapter. 3.4

THE GENERATING SERIES

Let V be the orthogonal space B with reduced norm q. We consider the space S(V×A× ) with an action of B× ×B× ×GL2 (A) given by the Weil representation, and the space Pic(XU × XU ) Pic(X × X) := lim −→ U

with an action of B× × B× by right multiplications. In this section we want to construct an element Ze ∈ HomB× ×B× ×GL2 (A) (S(V × A× ),

C ∞ (GL2 (F )\GL2 (A)) ⊗ Pic(X × X))

using Kudla’s generating series and modularity proved in [YZZ]. 3.4.1

New class of Schwartz functions

× Our first observation is that Pic(X × X) is invariant under B× ∞ × B∞ . Thus the × × element Z must factor through the maximal B∞ × B∞ quotient of S(V × A× ). Such a quotient is identified with the space S(V × A× ) of functions on V × A× obtained as integrals Z r(h∞ )Φ dh∞ , Φ ∈ S(V × A× ). × × F∞ \(B× ∞ ×B∞ )

× × is embedded diagonally into B× Here F∞ ∞ × B∞ and dh∞ is any fixed Haar measure of the quotient. Since × × × 1 1 \(B× F∞ ∞ × B∞ ) = F∞ B∞ × B∞ /{±1}, × the integral is just averages on F∞ and B1∞ × B1∞ . It is easy to see that the 1 1 average on B∞ × B∞ is the same as the average on one component of B1∞ × B1∞ . Thus we introduce the simplified integration Z Z Φ := r(ch)Φ dhdc, Φ ∈ S(V × A× ). × F∞

B1∞

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CHAPTER 3

Here the integral on B1∞ uses the Haar measure of total volume one, and the × uses the usual Haar measure introduced in §1.6. We normalize integral on F∞ the quotient map by S(V × A× ) −→ S(V × A× ),

Φ 7−→ Φ.

It is easy to see that S(V × A× ) has a decomposition S(V × A× ) = ⊗v S(Vv × Fv× ). Here S(Vv × Fv× ) = S(Vv × Fv× ) if v is non-archimedean. Assume that v is archimedean. We will describe S(Vv × Fv× ) in more details. It consists of all functions of the form Z Z Φv := r(ch)Φv dhdc, Φv ∈ S(Vv × Fv× ). Fv×

B1v

Recall that S(Vv × Fv× ) is the space of finite linear combinations of functions of the form H(u)P (x)e−2π|u|q(x) where P is any polynomial function on Vv , and H is any smooth and compactly supported function on Fv× . Then it is easy to verify that S(Vv × Fv× ) is the space of functions on Vv × Fv× of the form (P1 (uq(x)) + sgn(u)P2 (uq(x))) e−2π|u|q(x) where P1 and P2 are polynomials with complex coefficients. Here sgn(u) = u/|u| denotes the sign of u ∈ R× . × The Weil representation descends to an action of GL2 (Fv ) × B× v × Bv on × × × S(Vv × Fv ). Here Bv × Bv acts trivially, and GL2 (Fv ) acts by the same formula as S(Vv × Fv× ). By the tensor product, we have the Weil representation of GL2 (A) × B× × B× on S(V × A× ). 3.4.2

Constant term of Eisenstein series

The constant term of the generating series is defined in terms of the constant term of the corresponding Siegel Eisenstein series. So we recall some results on the Eisenstein series in §2.5.2 in slightly different notations. Fix φ ∈ S(V × A× ) and u ∈ F × . We have a Siegel Eisenstein series X E(s, g, u, φ) = δ(γg)s r(γg)φ(0, u), g ∈ GL2 (A). γ∈P 1 (F )\SL2 (F )

In the case g ∈ SL2 (A), in terms of the notation in §2.5.2, it is just E(s, g, φ(·, u)) defined by the quadratic space (B, uq). It is standard to have the Fourier expansion X Wa (s, g, u, φ) E(s, g, u, φ) = δ(g)s r(g)φ(0, u) + W0 (s, g, u, φ) + a∈F ×

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MORDELL–WEIL GROUPS AND GENERATING SERIES

where Z Wa (s, g, u, φ) :=

δ(wn(b)g)s r(wn(b)g)φ(0, u) ψ(−ab)db,

a ∈ F.

A

We are particularly interested in the value at s = 0 of the constant term E0 (s, g, u, φ) := δ(g)s r(g)φ(0, u) + W0 (s, g, u, φ). The definition is naturally extended to all u ∈ A× . Analytic properties of the intertwining part W0 (s, g, u, φ) are discussed in §2.5.2. For the definition of the generating series, we need the following three basic properties: • W0 (s, g, u, φ) has an analytic continuation to s = 0, and thus W0 (0, g, u, φ) is well-defined. • W0 (s, g, u, φ) 6= 0 only if F = Q and Σ = {∞}. • W0 (0, g, u, φ) = W0 (0, 1, u, r(g)φ) and thus E0 (0, g, u, φ) = E0 (0, 1, u, r(g)φ). At s = 0, we get E0 (0, g, u, φ) = r(g)φ(0, u) + W0 (0, g, u, φ). Here the W0 (0, g, u, φ) 6= 0 only if F = Q and Σ = {∞}. For convenience, we sometimes write E0 (g, u, φ) := E0 (0, g, u, φ),

E0 (u, φ) := E0 (0, 1, u, φ),

W0 (g, u, φ) := W0 (0, g, u, φ),

W0 (u, φ) := W0 (0, 1, u, φ).

and

3.4.3

Hecke correspondences

Fix an open and compact group U of B× f . Recall that we have defined Z(x)U to be the image of the morphism (πUx ,U , πUx ,U ◦ Tx ) :

XUx −→XU × XU .

Here Ux = U ∩ xU x−1 is an open and compact subgroup of B× f . In terms of complex uniformization above, the push-forward map by the Hecke correspondence Z(x)U gives X [z, βy]U . Z(x)U : [z, β]U 7−→ y∈U xU/U

Here [z, β]U represents the image of (z, β) ∈ H± × B× f in XU,τ (C). On the other hand, we can view Z(x)U as an element of Pic(XU × XU ). It is the view point we will take here to define the generating series. For convenience, for any x = xf x∞ ∈ B× , by Z(x)U we mean Z(xf )U .

94 3.4.4

CHAPTER 3

Hodge classes

On MK := XU × XU , one has a Hodge class LK ∈ Pic(MK ) ⊗ Q defined as LK :=

1 ∗ (p LU + p∗2 LU ). 2 1

Here the Hodge class LU of XU is introduced in §3.1.3. Next, we introduce some notations for components of LK . After fixing a base point of X, the geometrically connected components of XU are indexed by F+× \A× f /q(U ), and we use XU,α to denote the corresponding × component for α ∈ F+ \A× f /q(U ). Then the geometrically connected compo2 nents of MK = XU × XU are naturally indexed by (α1 , α2 ) ∈ (F+× \A× f /q(U )) . × × For any α ∈ F+ \Af /q(U ), denote a

MK,α =

XU,β × XU,αβ

× β∈F+ \A× f /q(U )

as a subvariety of MK . It is still defined over F by the reciprocity law which describes the Galois action on the components. Then a MK,α . MK = × α∈F+ \A× f /q(U )

View the Hodge bundle LK,α := LK |MK,α of MK,α as a line bundle of MK by trivial extension outside MK,α . 3.4.5

Generating series

For any φ ∈ S(V × A× ) invariant under K = U × U , form a generating series Z(φ)U := Z0 (φ)U + Z∗ (φ)U . Here the constant term and the non-constant part are respectively X X Z0 (φ)U := − E0 (α−1 u, φ) LK,α ,

(3.4.1)

(3.4.2)

× u∈µ2U \F × α∈F+ \A× f /q(U )

Z∗ (φ)U

:=

wU

X

X

φ(x, aq(x)−1 ) Z(x)U .

(3.4.3)

a∈F × x∈K\B× f

Here µU = F × ∩ U , µ2U = {α2 : α ∈ µU } and wU = |{1, −1} ∩ U | is equal to 1 or 2. It is easy to see that both µU and µ2U are subgroups of the unit group OF× of finite indexes, and that wU is 1 for U small enough. Furthermore, µ2U = {1} if F = Q.

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MORDELL–WEIL GROUPS AND GENERATING SERIES

The constant term E0 (u, φ) = φ(0, u) + W0 (u, φ) is introduced in §3.4.2. The intertwining part W0 (u, φ) is “the only nonholomorphic part” of the generating series. It is nonzero only when F = Q and Σ = {∞}, which happens exactly when the Shimura curve has cusps. We will see the reason for W0 (u, φ) to appear in the generating series in §4.2 and §4.3. For any g ∈ GL2 (A), we write Z(g, φ)U = Z(r(g)φ)U ,

Z0 (g, φ)U = Z0 (r(g)φ)U ,

Z∗ (g, φ)U = Z∗ (r(g)φ)U .

They are considered as functions on GL2 (A) with vector values in Pic(XU × XU )C . For any Φ ∈ S(V × A× ) invariant under K = U × U , it is routine to define Z(Φ)U = Z(Φ)U ,

Z0 (Φ)U = Z0 (Φ)U ,

Z∗ (Φ)U = Z∗ (Φ)U .

Here the averaging Φ ∈ S(V × A× ) of Φ is defined in §3.4.1. Theorem 3.17. The series Z(r(g)Φ)U is absolutely convergent and defines an automorphic form on g ∈ GL2 (A) with coefficients in Pic(XU × XU )C . By the modularity, we mean that `(Z(r(g)Φ)U ) is absolutely convergent and defines an automorphic form for any linear functional ` : Pic(XU × XU )C → C. The theorem will be proved in §4.2. It is essentially the modularity result proved in [YZZ]. But here let us consider some of its functorial properties. 3.4.6

Normalization of the generating series

We normalize the generating series as follows: 1

2[F :Q]−1 hF |DF |− 2 e Z(Φ) Z(Φ)U . U := [OF× : µ2U ]

(3.4.4)

Here hF denotes the class number of F and DF denotes the discriminant of F . The whole normalizing factor is always 2 if F = Q. The denominator is finite since µ2U is a subgroup of OF× with finite index, and it makes the right-hand side compatible under pull-back from different levels. Only the numerator is independent of U , and we choose it for reasons we will see later. The normalizing factor can also be written as 1

2[F :Q]−1 hF |DF |− 2 1 Ress=1 ζF (s). = 2Rµ2U [OF× : µ2U ]

(3.4.5)

Here Rµ2U is the “regulator” of µ2U in R[F :Q]−1 , whose definition is similar to the regulator of F . The equality follows from the class number formula. This normalization makes the system e e Z(Φ) := {Z(Φ) U }U a well-defined element of Pic(X × X)C .

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e Lemma 3.18. The system Z(Φ) is compatible with pull-back and thus defines an element of Pic(X × X)C . The map e (Φ, g) 7→ Z(r(g)Φ) defines an element Ze ∈ HomB× ×B× ×GL2 (A) (S(V × A× ),

C ∞ (GL2 (F )\GL2 (A)) ⊗ Pic(X × X))C .

Proof. By the modularity in Theorem 3.17, it suffices to prove the compatibility and the equivariance under B× × B× . It suffices to check that, for any × open compact subgroups U 0 and U of B× f and any elements h1 and h2 of Bf , 0 −1 0 −1 such that h1 U h1 ⊂ U and h2 U h2 ⊂ U and that U acts trivially on Φ, one has 1 Z(r(h1 , h2 )Φ)U 0 , Π∗ Z(Φ)U = 2 [µU : µ2U 0 ] where Π : XU 0 × XU 0 −→ XU × XU is the morphism given by the right multiplication by (h1 , h2 ). The case h1 = h2 = 1 gives the independence of Ze on U , and the general case gives the equivariance. We check the corresponding identity for the constant term X X E0 (α−1 u, Φ) LK,α Z0 (Φ)U = − × u∈µ2U \F × α∈F+ \A× f /q(U )

and the a-th Fourier coefficient X

Za (Φ)U = wU

Φ(x, aq(x)−1 ) Z(x)U

x∈U \B× f /U

for all a ∈ F × . Since the pull-backs of the Hodge bundles are still the Hodge bundles and Π maps MK 0 ,α0 to MK 0 ,α0 q(h−1 h2 ) , one has 1

X

Π∗ LK,α =

LK 0 ,α0 .

× × 0 α0 ∈F+ \(F+ αq(h1 h−1 2 )q(U ))/q(U )

It follows that Π∗ Z0 (Φ)U

= =

− −

X

X

× 0 α0 ∈F+ \A× f /q(U )

u∈µ2U \F ×

X

X

× 0 u∈µ2 \F × α0 ∈F+ \A× U f /q(U )

E0 (α0−1 q(h1 h−1 2 )u, Φ) LK 0 ,α0 E0 (α0−1 u, r(h1 , h2 )Φ) LK 0 ,α0 .

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MORDELL–WEIL GROUPS AND GENERATING SERIES

Replace µ2U \F × by µ2U 0 \F × in the inner summation. We get Π∗ Z0 (Φ)U =

1 Z0 (r(h1 , h2 )Φ)U 0 . [µ2U : µ2U 0 ]

To compute Π∗ Za (Φ)U , the key is to express Π∗ Z(x)U as a linear combination of Z(x0 )U 0 . Note that two subvarieties Z(x01 )U 0 = Z(x02 )U 0 if and only if the cosets F × U 0 x01 U 0 = F × U 0 x02 U 0 . And Z(x0 )U 0 is a component of Π∗ Z(x)U if 0 × and only if h−1 1 x h2 ⊂ F U xU . It follows that X

Π∗ Z(x)U =

Z(x0 )U 0 .

0 × 0 × 0 h−1 1 x h2 ∈(F U )\F U xU/U

Arrange Π∗ Za (Φ)U in terms of Z(x0 )U 0 . We have X X Π∗ Za (Φ)U = wU Φ(x, aq(x)−1 ) Z(x0 )U 0 . −1 0 0 × x0 ∈(F × U 0 )\B× f /U x∈U \F U h1 x h2 U/U

0 × −1 0 It is easy to see tha U \F × U h−1 1 x h2 U/U = µU \F h1 x h2 . Thus

X

Π∗ Za (Φ)U = wU

X

0 x0 ∈(F × U 0 )\B× f /U

b∈µU

r(h1 , h2 )Φ(bx0 , aq(bx0 )−1 ) Z(x0 )U 0 .

\F ×

Replace µU \F × by µU 0 \F × in the inner summation. We get Π∗ Za (Φ)U =

wU [µU : µU 0 ]

X

X

0 b∈µ 0 \F × x0 ∈(F × U 0 )\B× U f /U

r(h1 , h2 )Φ(bx0 , aq(bx0 )−1 ) Z(x0 )U 0 . Note that Z(x0 )U 0 = Z(bx0 )U 0 and [µU : µ2U ] = 2[F :Q]−1 wU . The above becomes Π∗ Za (Φ)U =

wU 0 : µ2U 0 ]

[µ2U

X

r(h1 , h2 )Φ(x0 , aq(x0 )−1 ) Z(x0 )U 0 .

0 x0 ∈U 0 \B× f /U

It is just Π∗ Za (Φ)U =

[µ2U

1 Za (r(h1 , h2 )Φ)U 0 . : µ2U 0 ] 

3.5

GEOMETRIC KERNEL

Let χ be a finite character of E × \EA× . In this section we define our geometric kernel function Z(g, χ, Φ) by means of height pairing of CM points for each

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CHAPTER 3

e χ, ·) in the Φ ∈ S(V × A× ). More precisely we will construct an element Z(·, space HomGL2 (A)×E × ×E × (S(V × A× )  χ  χ−1 , C0∞ (GL2 (F )\GL2 (A), χ|A× )). A

A

×

Here S(V × A )  χ  χ−1 means the space S(V × A× ) with the action of GL2 (A) × EA× × EA× by the Weil representation twisted by the character (χ, χ−1 ) on EA× × EA× ; and C0∞ (GL2 (F )\GL2 (A)), χ|A× ) means the space of cuspidal and smooth functions on GL2 (F )\GL2 (A) with the character χ|A× under translation by Z(A) and with trivial action by EA× × EA× . The element is automatically cuspidal though the original generating series Z(Φ) does not need to be without some assumption (cf. Assumption 5.4). 3.5.1

Height series

Let Φ ∈ S(V × A× ) be a Schwartz function as above. Recall that we have a e generating series Z(Φ) whose coefficients lie in Pic(X × X)C . By 3.1.6, we see e that the push-forward action of Z(Φ) U on XU gives a natural map e Z(Φ) :

J(F )C −→ J ∨ (F )C . ×

As in the introduction, fix an element P ∈ X E (E ab ). For any h ∈ B× , denote [h] = T (h)P and [h]◦ = [h]−ξq(h) . Here we have identified π0 (XU,F ) with ◦ F+× \A× f /q(U ) so that PU is indexed by 1 for each U . By definition, [h] ∈ J(F )C . × For any h1 , h2 ∈ B , define the height series e Φ) [h1 ]◦ , [h2 ]◦ iNT , e (h1 , h2 ), Φ) := hZ(g, Z(g,

g ∈ GL2 (A).

(3.5.1)

e Φ) [h1 ]◦ ∈ J ∨ (F )C and [h2 ]◦ ∈ J(F )C , and the N´eron–Tate height Here Z(g, pairing h·, ·iNT : J(F )C × J ∨ (F )C −→C is obtained naturally from the N´eron–Tate height pairing on each level U . See §3.1.6 for more details. It is clear that the definition does not depend on the infinite parts of h1 , h2 . And below we have more basic properties. e (h1 , h2 ), Φ) is independent of the choice Lemma 3.19. The definition of Z(g, of P , and it is invariant under the left action of T (F ) × T (F ) on (h1 , h2 ). Furthermore, it is always a cusp form on g ∈ GL2 (A). Proof. Take any t1 , t2 ∈ T (A). By definition, e (t1 h1 , t2 h2 ), Φ) = hZ(g, e Φ) T (h1 )T (t1 )P ◦ , T (h2 )T (t2 )P ◦ iNT . Z(g, View T (t1 )P ◦ and T (t2 )P ◦ as Galois conjugates of P ◦ by the reciprocity law. If t1 = t2 , then the height pairing does not change since the height pairing is

MORDELL–WEIL GROUPS AND GENERATING SERIES

99

invariant under the Galois action. If t1 , t2 ∈ T (F ), then by the definition of P we get T (t1 )P ◦ = T (t2 )P ◦ , and the pairing still does not change. e (h1 , h2 ), Φ) follows from the cuspidality of The cuspidality of Z(g, e Φ) [h1 ]◦ . In other words, the constant term Ze0 (g, Φ) [h1 ]◦ = 0. It is equivZ(g, alent to check Z0 (g, Φ) [h1 ]◦U = 0 for any open compact subgroup U of B× f acting trivially on Φ. We will postpone this to §4.3.1. Roughly speaking, as a correspondence, Z0 (g, Φ)U is a linear combination of Hodge classes LK,α,β , the (α, β)-component of the total Hodge bundle LK =

1 ∗ (p LU + p∗2 LU ). 2 1

It is very easy to see that they map Div0 (XU,F ) to 0. 3.5.2



Geometric kernel

Recall that χ is a character on T (F )\T (A) which is trivial at infinity. In terms of the regularized integrations introduced in §1.6, we define Z ∗ e χ, Φ) := e (t, 1), Φ) χ(t)dt. Z(g, Z(g, (3.5.2) T (F )\T (A)/Z(A)

It is called the geometric kernel. e χ, Φ) is cuspidal and of central Lemma 3.20. The automorphic form Z(g, character χ|A× on g ∈ GL2 (A). Moreover, for any t1 , t2 ∈ EA× , e χ, Φ). e χ, r(t1 , t2 )Φ) = χ(t−1 t2 )Z(g, Z(g, 1 e χ, Φ) defines an element Thus the map Φ 7→ Z(g, × −1 e χ, ·) ∈ Hom Z(·, , GL2 (A)×E × ×E × (S(V × A )  χ  χ A

A

C0∞ (GL2 (F )Z(A)\GL2 (A), χ|A× )). Proof. Only the central character needs justification. Let c ∈ A× be in the e e χ, Φ). By definition, χ, Φ) = χ(c) Z(g, center of GL2 (A). We need to show Z(cg, Z

∗

e Z(cg, (t, 1), Φ) χ(t)dt.

e Z(cg, χ, Φ) = T (F )\T (A)/Z(A)

e e (c−1 t, 1), Φ). It is easy to see that the claim We claim that Z(cg, (t, 1), Φ) = Z(g, implies the desired result. It suffices to check e e Φ) [c−1 t]◦ . Z(cg, Φ) [t]◦ = Z(g,

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CHAPTER 3

And we only need to check Z(cg, Φ)U [t]◦U = Z(g, Φ)U [c−1 t]◦U on the level U . By the cuspidality, we only need to show Za (cg, Φ)U [t]◦U = Za (g, Φ)U [c−1 t]◦U for any a ∈ F × . By definition, X

Za (cg, Φ)U =

r(cg)Φ(x, aq(x)−1 ) Z(x)U .

x∈U \B× f /U

Note that r(cg)Φ(x, aq(x)−1 ) = r(g)Φ(cx, aq(cx)−1 ). The above becomes X r(g)Φ(x, aq(x)−1 ) Z(c−1 x)U . Za (cg, Φ)U = x∈U \B× f /U

Since c−1 is in the center, right multiplication by c−1 gives an automorphism of XU which switches the geometrically connected components of XU by c−2 . This automorphism is exactly the Hecke correspondence Z(c−1 )U . Then we have Z(c−1 x)U = Z(x)U ◦ Z(c−1 )U as operators, which gives Za (cg, Φ)U = Za (g, Φ)U ◦ Z(c−1 )U . It finishes the proof. 3.6



ANALYTIC KERNEL AND KERNEL IDENTITY

We recall the analytic kernel, and state the main result on a relation between the analytic kernel and the geometric kernel. 3.6.1

The analytic kernel

Fix a Schwartz function Φ ∈ S(V × A× ). Recall the associated series X X δ(γg)s r(γg)Φ(x1 , u) I(s, g, Φ) = γ∈P 1 (F )\SL2 (F )

(x1 ,u)∈E×F ×

and the twisted average Z I(s, g, χ, Φ) =

I(s, g, r(t, 1)Φ) χ(t)dt. T (F )\T (A)

Up to some simple factors, we have L(

s+1 , π, χ) ≈ (I(s, g, χ, Φ), ϕ(g))Pet 2

101

MORDELL–WEIL GROUPS AND GENERATING SERIES

for ϕ ∈ σ. Taking derivative, we get 1 L0 ( , π, χ) ≈ (I 0 (0, g, χ, Φ), ϕ(g))Pet . 2 It follows that I 0 (0, g, χ, Φ) is the kernel function representing L0 ( 12 , π, χ). We call it the analytic kernel. 3.6.2

Kernel identity

Theorem 3.21. Let Φ ∈ S(V × A× ) be any Schwartz function. Then   e χ, Φ), ϕ (I 0 (0, ·, χ, Φ), ϕ)Pet = 2 Z(·, , ∀ ϕ ∈ σ. Pet

Recall that the Petersson bilinear pairing Z (ϕ1 , ϕ2 )Pet :=

ϕ1 (g)ϕ2 (g)dg.

Z(A)GL2 (F )\GL2 (A)

It only makes sense if ϕ1 , ϕ2 are automorphic with inverse central characters. Note that the pairing is not Hermitian, but bilinear in our definition. The integration uses the Tamagawa measure, though it does not matter in the current theorem. e χ, Φ) The pairings in the theorem make sense. The central character of Z(g, −1 is χ|A× = ωσ by Lemma 3.20, and it is easy to check that the same result is true for I 0 (0, g, χ, Φ). 3.6.3

Projector III: Arithmetic theta lifting

Let Φ ∈ S(V × A× ) be a Schwartz function. Recall that we have a generating e Φ), which is a modular form in g with coefficients in Pic(X × X)C . series Z(g, Consider the “arithmetic theta lifting” Z ∗ e Φ) dg, ϕ ∈ σ. e ⊗ ϕ) := ϕ(g) Z(g, Z(Φ GL2 (F )\GL2 (A)/Z(A)

e Φ) has no central character, so the integral is a regularized inteNote that Z(g, gral introduced in §1.6. By definition, we have e ⊗ ϕ) ∈ Pic(X × X)C . Z(Φ By §3.1.6, via the push-forward action, we can view e ⊗ ϕ) ∈ Hom0 (J, J ∨ )C . Z(Φ On the other hand, we have a theta lifting θ : S(V × A× ) ⊗ σ−→π ⊗ π e.

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CHAPTER 3

It is normalized in (2.2.3) place by place. By §3.3.1, one has a map T:π⊗π e −→ Hom0 (J, J ∨ )C . The composition gives T ◦ θ : S(V × A× ) ⊗ σ−→Hom0 (J, J ∨ )C . Theorem 3.22. As an identity in Hom0 (J, J ∨ )C , one has e ⊗ ϕ) = L(1, π, ad) T(θ(Φ ⊗ ϕ)), Z(Φ 2ζF (2)

Φ ∈ S(V × A× ), ϕ ∈ σ.

It is not surprising that the theorem is true up to a constant. Namely, there is a constant c(π) independent of Φ and ϕ such that e ⊗ ϕ) = c(π)T(θ(Φ ⊗ ϕ)). Z(Φ To see the existence of c(π), it suffices to check this in Homcont (H 1,0 (Xτ )0 , H 1,0 (Xτ )) by the natural embedding Hom0 (J, J ∨ )C ,−→ Homcont (H 1,0 (Xτ )0 , H 1,0 (Xτ )). Fix an embedding τ : F → C, and fix a decomposition M π1 . H 1,0 (Xτ ) = π1 ∈A(B× )

It gives a decomposition M

H 1,0 (XU,τ ) =

π1U ,

π1 ∈A(B× )

and a decomposition of the dual space H 1,0 (XU,τ )0∨ =

M

π e2U .

π2 ∈A(B× )

Then Hom(H 1,0 (XU,τ )0 , H 1,0 (XU,τ )) =

M π1 ,π2

π1U ⊗ π e2U .

∈A(B× )

Taking direct limit, we obtain Homcont (H 1,0 (Xτ )0 , H 1,0 (Xτ )) '

M π1 ,π2 ∈A(B× )

π1 ⊗ π e2 .

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MORDELL–WEIL GROUPS AND GENERATING SERIES

It follows that both Ze and T ◦ θ can be viewed as elements of M π1 ⊗ π e2 ). Hom(S(V × A× ) ⊗ σ, π1 ,π2 ∈A(B× )

The key point now is that both of them are invariant under the diagonal action of GL2 (A) on Φ ⊗ ϕ, and the action of B× × B× on Φ. So both of them must be multiples of the Shimizu lifting. Namely, they belong to the one-dimensional space e). HomGL2 (A)×B× ×B× (S(V × A× ) ⊗ σ, π ⊗ π We already know that T ◦ θ is a nonzero element in this space. So there is a constant c(π) such that Ze = c(π) · T ◦ θ. (3.6.1) In the next chapter, we will use the Lefschetz fixed point theorem to prove that the constant c(π) is the same as in Theorem 3.22. 3.6.4

Proof of Theorem 3.15 by the kernel identity

Here we deduce Theorem 3.15 from Theorem 3.21 and Theorem 3.22. Let f1 and f2 be as Theorem 3.15. Let θ = ⊗v θv : S(B × A× ) ⊗ σ−→π ⊗ π e be the Shimizu lifting normalized in (2.2.3). The map is surjective since π ⊗ π e is an irreducible representation of B× × B× . Then f1 ⊗ f2 must be in the image of θ. By linearity, we can assume that f1 ⊗ f2 = θ(ϕ ⊗ Φ),

ϕ ∈ σ, Φ ∈ S(B × A× ).

Apply Theorem 3.21 to (ϕ, Φ) above. We have   e χ, Φ), ϕ(g) (I 0 (0, g, χ, Φ), ϕ(g))Pet = 2 Z(g,

. Pet

Recall in (2.3.1) that we defined that P (s, χ, Φ, ϕ) = (I(s, g, χ, Φ), ϕ(g))Pet . By Proposition 2.6, P 0 (0, χ, Φ, ϕ) =

L0 (1/2, π, χ) α(θ(Φ ⊗ ϕ)). 2L(1, η)

It remains to convert the right-hand side. We claim that   e χ, Φ), ϕ(g) Z(g,

= Pet

L(1, π, ad)L(1, η) hT(f1 ⊗ f2 )Pχ , Pχ−1 iNT . ζF (2)

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CHAPTER 3

It obviously implies the theorem. To prove the claim, recall that Z ∗ e χ, Φ) = e Φ)[t]◦ , 1iNT χ(t)dt Z(g, hZ(g, T (F )\T (A)/Z(A) Z e Φ)[t]◦ , 1iNT χ(t)dt. = 2L(1, η) hZ(g, T (F )\T (A)/Z(A) 0

Recall that for any t ∈ T (A), e Φ)[t0 t]◦ , [t0 ]◦ iNT = hZ(g, e Φ)[t]◦ , [1]◦ iNT . hZ(g, The identity is true by viewing the action of t0 on ([t]◦ , [1]◦ ) as a Galois action which does not change the height pairing. Therefore, we obtain Z Z e χ, Φ) = 2L(1, η) Z(g, T (F )\T (A)/Z(A)

T (F )\T (A)/Z(A)

e Φ)[t0 t]◦ , [t0 ]◦ iNT χ(t)dt0 dt. hZ(g, By a change of variable, it becomes Z e χ, Φ) = 2L(1, η) Z(g,

Z

T (F )\T (A)/Z(A)

T (F )\T (A)/Z(A)

e Φ)[t1 ]◦ , [t2 ]◦ iNT χ(t1 t−1 )dt1 dt2 . hZ(g, 2 It is just e χ, Φ) = 2L(1, η) hZ(g, e Φ)Pχ , Pχ−1 iNT . Z(g, Here

Z Pχ =

χ(t)[t]◦ dt. T (F )\T (A)/Z(A)

It follows that   e χ, Φ), ϕ(g) Z(g,

Pet

e ⊗ ϕ)Pχ , Pχ−1 iNT . = 2L(1, η) hZ(Φ

Here the arithmetic theta lifting Z ∗ e ⊗ ϕ) = Z(Φ

e Φ) dg ϕ(g) Z(g,

GL2 (F )\GL2 (A)/Z(A)

is defined above. Notice that the Petersson pairing is the usual integration, but it is equal to our regularized integration. By Theorem 3.22, e ⊗ ϕ) = L(1, π, ad) T(θ(Φ ⊗ ϕ)). Z(Φ 2ζF (2) Hence,   e χ, Φ), ϕ(g) Z(g, It finishes the proof.

= Pet

L(1, π, ad)L(1, η) hT(f1 ⊗ f2 )Pχ , Pχ−1 iNT . ζF (2)

MORDELL–WEIL GROUPS AND GENERATING SERIES

3.6.5

105

Interpretation by linear functionals

Similar to the interpretation of Theorem 1.2, Theorem 3.15 is an identity of two vectors in the complex vector space P(π, χ) ⊗ P(e π , χ−1 ). Here we recall that P(π, χ) = HomT (A) (π ⊗ χ, C). It is at most one-dimensional by Theorem 1.3. Note that α is a generator of P(π, χ) ⊗ P(e π , χ−1 ), and hT(·)Pχ , Pχ−1 iNT must be a multiple of α. To get an essential case of the theorems, we need to assume that B is determined by the local root numbers as in Theorem 1.3. To be precise, we list all the notations and assumptions in the following: (1) F is a totally real number field and E is a totally imaginary quadratic extension of F . (2) σ is a cuspidal automorphic representation of GL2 (A), discrete of weight 2 at all infinite places of F . (3) χ is a character on E × \EA× of finite order.  (4) Σ = v : ( 12 , πv , χv ) 6= χv ηv (−1) has an odd cardinality. (5) B is an incoherent quaternion algebra over A with ramification set Σ, endowed with a fixed embedding EA ,→B over A. (6) π is the admissible representation of B× whose Jacquel–Langlands correspondence is σ. By (2) and (3), the set Σ contains all the archimedean places of F . It follows that B is totally definite quaternion algebra over A. The embedding EA ,→B and the representation π exist by the construction of B, and π is trivial under the action of the infinite part B× ∞ . We refer to the assumptions as “geometric assumptions” since they are the basic assumptions to obtain a non-trivial formula for the height pairing of CM cycles on Shimura curves. As a consequence of the multiplicity one property, we immediately obtain the following result under the geometric assumptions. Lemma 3.23. The following are true: (1) Theorem 3.15 holds for all (f1 , f2 ) if and only if it holds for some (f1 , f2 ) with α(f1 , f2 ) 6= 0. (2) Theorem 3.21 holds for all (Φ, ϕ) if and only if it holds for some (Φ, ϕ) with α(θ(Φ ⊗ ϕ)) 6= 0. Note that (2) is just a consequence of (1) by the relation of these two theorems described above. Under assumption (5), we can also interpret Theorem 3.21 as an identity in HomGL2 (A)×E × ×E × (σ  S(B × A× )  (χ, χ−1 ), C). A

A

It is one-dimensional by its relation to P(π, χ) ⊗ P(e π , χ−1 ) via the theta lifting.

Chapter Four Trace of the Generating Series

The goal of this chapter is to prove Theorem 3.17 and Theorem 3.22 in the last chapter. Before going to the proofs, in §4.1 we give more details on the new space S(V × R× ) of Schwartz functions including the formation of theta series and Eisenstein series by them. Theorem 3.17 asserts the modularity of the generating series. The major part of its proof is in §4.2, where we reduce the problem to the results in [YZZ]. In this way, the modularity is proved on the open Shimura variety. To extend to the compactification, it suffices to prove the degree (as a correspondence) is modular. The degree is proved to be given by Eisenstein series in §4.3, and thus the modularity follows. Theorem 3.22 is an identity between T (f1 ⊗ f2 ) and (Z(φ), ϕ). We know the identity is true up to constant. To determine the constant, it suffices to compare the traces of both operators. By the Lefschetz trace formula, the trace of (Z(φ), ϕ) is reduced to the degree of the pull-back of (Z(φ), ϕ) by the diagonal map X → X × X in §4.4. The computation of the degree takes up §4.5, §4.6 and §4.7. The compact case is treated in §4.5, while the non-compact case takes up the last two sections.

4.1

DISCRETE SERIES AT INFINITE PLACES

In §2.1, we reviewed Waldspurger’s extension of the Weil representation to the Schwartz function space S(V × k × ). Following §3.4.1, we will consider a new class S(V × k × ) of Schwartz functions. It is the same as the original one in the non-archimedean case, but different in the archimedean case. We will also construct theta series and Eisenstein series from such functions.

4.1.1

Discrete series at infinity

Let V be a positive definite quadratic space over R. Let S(V × R× ) denote the space of functions on V × R× of the form φ(x, u) = (P1 (uq(x)) + sgn(u)P2 (uq(x))) e−2π|u|q(x)

107

TRACE OF THE GENERATING SERIES

with polynomials Pi of complex coefficients. Here sgn(u) = u/|u| denotes the sign of u ∈ R× . As in §3.4.1, we have a (surjective) quotient map Z Z r(ch)Φ dhdc. S(V × R× ) −→ S(V × R× ), Φ 7−→ Φ := R×

O(V )

By the formulae for the Weil representation on S(V × R× ) in §2.1, one has a smooth function r(g, h)φ on V ×R× for any φ ∈ S(V ×R× ) and (g, h) ∈ GL2 (R)× GO(R). It induces an action r of (gl2 (R), O2 (R)) × GO(V ) on S(V × R× ). The action of GO(V ) is trivial by definition. The standard Schwartz function φ ∈ S(V × R× ) is the Gaussian φ(x, u) = e−2πuq(x) 1R+ (u). Here 1R+ is the characteristic function of the set R+ of positive real numbers. Assume that dim V = 2d is even for simplicity. Then one verifies that, for any g ∈ GL2 (R), ( (d) if x 6= 0, Wuq(x) (g) r(g)φ(x, u) = (d) W0 (g, u) if x = 0. Here the standard holomorphic Whittaker function W (d) is as follows: d

Wa(d) (g)

=

|y0 | 2 e2πia(x0 +iy0 ) ediθ 1R+ (a det(g)),

(d)

=

|y0 | 2 ediθ 1R+ (u det(g)),

W0 (g, u)

d

a ∈ R× ;

u ∈ R× .

Here we express g ∈ GL2 (R) by the Iwasawa decomposition as follows:     cos θ sin θ y0 x0 z0 . g= z0 1 − sin θ cos θ The space S(V × R× ) actually gives the discrete series of GL2 (R) of weight d. The standard Schwartz function gives an element of the lowest weight. Furthermore, the action of gl2 (R) on the standard function generates the whole space S(V × R× ). By this fact, all our major computations will be reduced to the standard function. 4.1.2

Some notations

For convenience, we introduce some global notations. Let F be a totally real field with adele ring A. For any (coherent or incoherent) quadratic space V over A which is positive definite at infinity, denote restricted tensor product S(V × A× ) := ⊗v S(Vv × Fv× ). Here S(Vv × Fv× ) = S(Vv × Fv× ) if v is non-archimedean. The same integration × × O(F∞ ) gives a surjection S(V × A× ) → S(V × A× ). on F∞

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CHAPTER 4

For any a ∈ A× and x ∈ V with q(x) 6= 0, denote r(g, h)φ(x)a := r(g, h)φ(x, aq(x)−1 ).

(4.1.1)

Use similar notations in the local case. Note that the infinite part r(g∞ , h∞ )φ∞ (x∞ , a∞ q(x∞ )−1 ) is independent of x∞ , h∞ . Thus we use the notation r(g, h)φ(x)a := r(gf , hf )φf (x, aq(x)−1 ) r(g∞ )φ∞ (x∞ , a∞ q(x∞ )−1 )

(4.1.2)

for (g, h) ∈ GL2 (A) × GO(Vf ), x ∈ Vf and a ∈ A× . It is independent of the choice of x∞ ∈ V∞ . × Similarly, for (g, h) ∈ GL2 (A) × GO(Vf ), (x, u) ∈ Vf × A× f with uq(x) ∈ F , it makes sense to define r(g, h)φ(x, u) := r(gf , hf )φf (x, u) r(g∞ )φ∞ (x∞ , u∞ )

(4.1.3)

× with u∞ q(x∞ ) ∈ F × and equal to uq(x) ∈ F × . for any (x∞ , u∞ ) ∈ V∞ × F∞

4.1.3

Theta series

First let us consider the definition of theta series. Let V be a positive definite even-dimensional quadratic space over a totally real field F . Let φ ∈ S(V (A) × A× ). Similar to the case for S(V (A) × A× ), we have the theta series θ(g, u, φ) :=

X

r(g)φ(x, u),

g ∈ GL2 (A), u ∈ A× .

x∈V

If u ∈ F × , it is invariant under the left action of SL2 (F ) on g. To get an automorphic form on GL2 (A), we need to sum over u ∈ F × . This time the definition is slightly different due to a convergence issue here. There is an open compact subgroup K ⊂ GO(Af ) such that φf is invariant under the action of K by Weil representation. Denote µK = F × ∩ K. Then µK is a subgroup of the unit group OF× , and thus is a finitely generated abelian group. Our theta series is of the following form: X

θ(g, φ)K :=

X

r(g)φ(x, u).

(4.1.4)

u∈µ2K \F × x∈V

If F = Q, then µ2K = 1 and thus the definition has the same form as the original definition. In terms of θ(g, u, φ), the theta series is just θ(g, φ)K =

X u∈µ2K \F ×

θ(g, u, φ).

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TRACE OF THE GENERATING SERIES

The summation is well-defined since for any α ∈ µK , θ(g, α2 u, φ) =

X

r(g)φ(x, α2 u) =

x∈V

X

r(g)φ(αx, u)

x∈V

=

X

r(g)φ(x, u) = θ(g, u, φ).

x∈V

We will show that the summations are absolutely convergent. Then θ(g, φ)K is an automorphic form on g ∈ GL2 (A), and θ(g, r(h)φ)K is an automorphic form on (g, h) ∈ GL2 (A) × GO(A). Furthermore, if φ∞ is standard, then θ(g, φ)K is holomorphic of weight 12 dim V . It is the major reason for introducing this new class of functions. By choosing fundamental domains, we can rewrite the sum as X

θ(g, φ)K =

r(g)φ(0, u) + wK

u∈µ2K \F ×

X

r(g)φ(x, u).

(x,u)∈µK \((V −{0})×F × )

Here the natural action of µK on V × F × is just α ◦ (x, u) 7→ (αx, α−2 u). The summation over u is well-defined since φ(αx, α−2 u) = r(α−1 )φ(x, u) = φ(x, u) for any α ∈ µK . The factor wK = |{1, −1} ∩ K| ∈ {1, 2}, and it is 1 for K small enough. Now we check the convergence of (4.1.4). We claim that the summation over u is actually a finite sum depending on (g, h). For fixed (g, h), there is a compact subset A ⊂ A× f such that r(g, h)φf (x, u) 6= 0 only if u ∈ A. Thus the summation is taken over u ∈ µ2K \(F × ∩ A), which is a finite set since µ2K is a finite-index subgroup of the unit group OF× , and F × ∩ A is included in a finite union of cosets of OF× . The definition depends on the choice of K. In fact, if K 0 ⊂ K is another open compact subgroup, then θ(g, φ)K 0 = [µ2K : µ2K 0 ] θ(g, φ)K . 4.1.4

Eisenstein series

We can also define Eisenstein series for such Schwartz functions. The definition is valid for both coherent or incoherent quadratic space. Let φ ∈ S(V × A× ), where V is a quadratic space over A which is positive definite of even dimension at infinity. First, as in the case of S(V × A× ), we introduce E(s, g, u, φ) :=

X

δ(γg)s r(γg)φ(0, u),

γ∈P 1 (F )\SL2 (F )

It is invariant under the left action of SL2 (F ) on g.

g ∈ GL2 (A), u ∈ F × .

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Second, take an open compact subgroup K of GO(A) such that φ is invariant under K as above. Define X E(s, g, φ)K := E(s, g, u, φ). u∈µ2K \F ×

Similar to the argument above, the summation over u is well-defined, and it is essentially a finite sum. It follows that E(s, g, φ)K is an automorphic form on g ∈ GL2 (A). 4.1.5

Relation with the old definitions

In the following we want to describe a connection from the definitions of theta series and Eisenstein here to those in §2.1. Take theta series as an example. Let V be a positive definite even dimensional quadratic space over a totally real field F . Let Φ ∈ S(V (A) × A× ). The theta series in §2.1 is simply X r(g)Φ(x, u). θ(g, Φ) = (x,u)∈V ×F ×

By the quotient map, we set Z Z φ=Φ= × F∞

r(ch)Φ dhdc ∈ S(V × R× ).

O(V∞ )

Assume that K ⊂ GO(Af ) is an open compact subgroup acting trivially on φ. Then we have X X θ(g, φ)K = r(g)φ(x, u). u∈µ2K \F × x∈V

The relation is that Z Z θ(zg, r(h)Φ) dhdz = × µK \F∞

O(V∞ )

1 θ(g, φ)K . wK

(4.1.5)

By this identity, we can switch between these two kinds of Schwartz functions. The same relation holds for the Eisenstein series. To check the identity, it suffices to assume that Φ is invariant under the action of O(V∞ ). In fact, since O(V∞ ) is compact, we can replace Φ by its average on O(V∞ ). Then the identity comes from a coset identity. 4.2

MODULARITY OF THE GENERATING SERIES

In this section, we prove Theorem 3.17 on the modularity of the generating series. It is essentially an example of the modularity theorem in [YZZ].

111

TRACE OF THE GENERATING SERIES

Let φ ∈ S(V × A× ) be a Schwartz function invariant under the action of K = U × U for some open compact subgroup U of B× f . We need to prove the modularity of Z(g, φ)U = Z0 (g, φ)U + Z∗ (g, φ)U ,

g ∈ GL2 (A).

Here the constant term Z0 (g, φ)U = −

X

X

E0 (α−1 u, r(g)φ) LK,α ,

(4.2.1)

× u∈µ2U \F × α∈F+ \A× f /q(U )

and the non-constant part Z∗ (g, φ)U = wU

X

X

r(g)φ(x)a Z(x)U .

(4.2.2)

a∈F × x∈K\B× f

Here the intertwining part W0 (u, φ) is introduced in §3.4.2, and the notation r(g)φ(x)a is introduced in (4.1.2). We also write Z(φ)U = Z(1, φ)U and thus Z(g, φ)U = Z(r(g)φ)U . Use similar notations for Z0 (1, φ)U , Z∗ (1, φ)U , Za (1, φ)U . 4.2.1

Reduction to the standard Gaussian

By linearity we can assume that φ = φf ⊗φ∞ . It suffices to prove the modularity of Z(g, φ)U in the case that φ∞ is the standard Schwartz function. In fact, (2) denote by φ∞ the standard Schwartz function at infinity, and denote φ0 = (2) (2) × ) under the action of gl2 (F∞ ), we can φf ⊗ φ∞ . Since φ∞ generates S(V∞ × F∞ (2) assume that φ∞ = r(∂)φ∞ from some element ∂ ∈ gl2 (F∞ ). The modularity of Z(g, φ0 )U = Z(r(g)φ0 )U implies the modularity of Z(r(get∂ )φ0 )U for any t ∈ R. Taking the derivative at t = 0, we get the modularity of Z(r(g)φ)U as desired. In the following, we assume that φ = φf ⊗ φ∞ with φ∞ standard. 4.2.2

Result for the general spin group

Now we state the main result of [YZZ], which can be seen as a continuation of the previous works [HZ, KM1, KM2, KM3, GKZ, Bor, Zha]. Before going to the details, we point out that the result in [YZZ] is modularity on open Shimura varieties. To get a modularity on the compactified Shimura varieties, we also need to consider the boundary behavior. It is reduced to check the modularity of the degree of the generating series at the end of this section. The degree is computed in the next section. The result is stated in terms of the general spin group of (V, q). Recall that GSpin(V) = {(h1 , h2 ) ∈ B× × B× : q(h1 ) = q(h2 )} is naturally a subgroup of B× × B× which is compatible with the action on V. 0 Let MK 0 be the compactified Shimura surface associated to GSpin(Vf ) for any

112

CHAPTER 4

open compact subgroup K 0 ⊂ GSpin(Vf ). For any Schwartz function φ0 ∈ S(V) invariant under K 0 × O(V∞ ), and any g ∈ SL2 (A), one has a generating series X X Z(g, φ0 )K 0 = −r(g)φ0 (0) LK 0 + wK 0 r(g)φ0 (y) Z(y)K 0 . × y∈K 0 \V (a) f a∈F+

0 Here wK 0 = |{1, −1} ∩ K 0 |, LK 0 is the Hodge bundle on MK 0 , and Vf (a) is the set of elements of Vf with norm a. The special curve Z(y)K 0 is defined for × 0 0 y ∈ B× f if q(y) ∈ F+ . In the case K = K ∩ GSpin(Vf ), the image of Z(y)K 0 under the natural map MK 0 → MK is just Z(y)U . 0◦ 0 Denote by MK 0 the open part of MK 0 before compactification. They are different unless F = Q and Bf = M2 (Af ). In [YZZ], we have proved that if φ0 = φ0f ⊗ φ0∞ with φ0∞ equal to the standard Gaussian, then the restriction Z(g, φ0 )K 0 |MK0◦0 to the open part is absolutely convergent and defines an au0◦ tomorphic form on SL2 (A) with coefficients in Pic(MK 0 ). We claim that the 0 modularity can easily be extended to any φ ∈ S(V). In fact, we can assume φ0 = φ0f ⊗ φ0∞ by linearity. Denote by φ00∞ the standard Gaussian in S(V), and denote φ00 = φ0f ⊗ φ00∞ . The key is that φ00∞ generates S(V∞ ) under the action of the Lie algebra sl2 (F∞ ). Thus we can assume that φ0∞ = r(∂)φ00∞ for some element ∂ ∈ sl2 (F∞ ). The modularity of 0◦ t∂ Z(g, φ00 )K 0 = Z(r(g)φ00 )K 0 on MK )φ00 )K 0 for 0 implies the modularity of Z(r(ge any t ∈ R. Taking the derivative at t = 0, we get the modularity of Z(r(g)φ0 )U 0◦ on MK 0 as desired. More generally, we can change the quadratic form by a fixed constant u ∈ F × and get a new series X X ru (g)φ0 (y) Z(y)K 0 Z(g, φ0 , uq)K 0 = −ru (g)φ0 (0) LK 0 + wK 0 × y∈K 0 \V (a) f a∈F+

for any g ∈ SL2 (A). Here ru is the Weil representation with respect to the quadratic space (Vf , uq). Then the result of [YZZ] also gives the modularity of Z(g, φ0 , uq)|MK0◦0 . We will come back to discuss the boundary later. When K 0 is sufficiently small, one has wK 0 = 1. In general one needs the factor wK 0 in the definition of the generating series. The modularity result for big level K 0 can be obtained by push-forward from small levels. 4.2.3

Map between the Shimura varieties

× × × For any h ∈ B× f × Bf and any open compact subgroup K = U × U of Bf × Bf as above, the group K h := GSpin(Vf ) ∩ hKh−1 is an open compact subgroup of 0 GSpin(Vf ). Let ih : MK h → MK be the finite map given by right multiplication by h. 0 Lemma 4.1. The image of ih : MK h → MK is exactly MK,ν(h)−1 , and 0 the degree of ih : MK h → MK,ν(h) is equal to [µ0U : µ2U ] everywhere. Here µU = F × ∩ U and µ0U = F+× ∩ q(U ).

113

TRACE OF THE GENERATING SERIES

Proof. Fix an archimedean place τ , denote by B = B(τ ) the nearby quaternion algebra, and by G = GSpin(B, q). Then we have uniformizations 0 MK h ,τ (C)

=

G(F )+ \(H × H) × G(Af )/K h ,

MK,τ (C)

=

× × × (B+ × B+ )\(H × H) × B× f × Bf /K.

× × ×B+ . If F = Q and B = M2 (Q) Here G(F )+ is the intersection of G(F ) with B+ we need to add two boundary divisors H × {cusps} and {cusps} × H. 0 The image of ih : MK h → MK is represented by × × (B+ × B+ ) ((H × H) × G(Af )hK) . × Note that G(Af ) is a normal subgroup of B× f × Bf . It is easy the see the image 0 is exactly MK,ν(h)−1 . Both the geometrically connected components of MK h × × and MK,ν(h)−1 are parametrized by F+ \Af /q(U ). So ih is one-to-one on the geometrically connected components. Now we figure out the degree. Any geometrically connected component of 0 MK h ,τ (C) is of the form

Γ0 \H × H,

Γ0 = G(F )+ ∩ (gK h g −1 )

for some g ∈ G(Af ). Its image in MK,τ (C) is just Γ\H × H,

× × × B+ ) ∩ (ghKh−1 g −1 ). Γ = (B+

Note that the stabilizer of a general point (i.e., not a CM point) of H × H in × × × B+ is just the center F × × F × . It follows that the degree of ih on that B+ component is equal to [Γ : ZΓ Γ0 ], where ZΓ = (F × × F × ) ∩ Γ = µU × µU . Then the degree is just [Γ : (µU × µU )(Γ ∩ G(F ))]. Write Γ = (Γ1 , Γ2 ) in terms of components. The action of Γ ∩ G(F ) on Γ can always transfer the first component in Γ1 into 1. Then it is easy to see that [Γ : (µU × µU )(Γ ∩ G(F ))] = [Γ2 : µU Γ12 ] = [q(Γ2 ) : µ2U ]. Here Γ12 is the subgroup of elements of Γ2 with norm 1. The second equality is induced by the bijective norm map q : Γ2 /(µU Γ12 ) −→ q(Γ2 )/µ2U . It remains to prove q(Γ2 ) = µ0U , or equivalently × ∩ U 0 ) = F+× ∩ q(U 0 ) q(B+

(4.2.3)

−1 where U 0 = g2 h2 U h−1 if we write h = (h1 , h2 ) and g = (g1 , g2 ). Only the 2 g2 × × 0 ∩ U 0 ) needs justification. Let a ∈ F+× ∩ q(U 0 ) be implication F+ ∩ q(U ) ⊂ q(B+

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× any element. Then we can find b ∈ B+ and u ∈ U 0 with q(b) = q(u) = a. The × ∩ U 0 ), it existence of b follows from the Hasse principle. To show that a ∈ q(B+ 1 01 1 −1 01 is equivalent to showing that bB ∩ uU = b(B ∩ b uU ) is non-empty. Here B 1 and U 01 still denote the subgroups of elements with norm 1. By the strong approximation theorem, B 1 is dense in B1f . Thus B 1 ∩ b−1 uU 01 is non-empty since b−1 uU 01 is open in B1f . 

4.2.4

Components of the generating series

Let φ ∈ S(V × A× ) be a Schwartz function invariant under K = U × U for some open compact subgroup U of B× f . We need to prove the modularity of Z(g, φ)U = Z0 (g, φ)U + Z∗ (g, φ)U ,

g ∈ GL2 (A).

It suffices to prove the modularity of the component Z(g, φ)U,α := Z(g, φ)U |MK,α for all α ∈ F+× \A× f /q(U ). We further decompose Z(g, φ)U,α = Z0 (g, φ)U,α + Z∗ (g, φ)U,α into the sum of the constant term and the non-constant part. × −1 × F+ q(U ). It is easy to have Fix an h ∈ B× f × Bf such that ν(h) ∈ α X

Z0 (g, φ)U,α = −

(r(g, h)φ(0, u) + W0 (u, r(g, h)φ)) LK,α .

u∈µ2U \F ×

Now we treat Z∗ (g, φ)U,α . For any x ∈ B× f , the Hecke operator Z(x)U = U xU is completely contained in MK,q(x) . It has contribution to Z(g, φ)U,α if and only if q(x) ∈ αF+× q(U ). In that case we have x ∈ Kh−1 y for some y ∈ B× f with norm in q(y) ∈ F+× . A different y 0 yields Kh−1 y 0 = Kh−1 y if and only if y 0 = hKh−1 y. It follows that q(y 0 ) ∈ q(y)µ0U with µ0U = F+× ∩ q(U ) as above. Taking these into account, we have Z∗ (g, φ)U,α

=

wU

X

X

X

r(g, h)φ(y, aq(y)−1 ) Z(h−1 y)U .

a∈F × b∈µ0 \F × y∈K h \Vf (b) + U

Here Vf (b) = {x ∈ V : q(x) = b}. Therefore, Z∗ (g, φ)U,α = wU

X

X

u∈µ0U \F ×

× a∈F+

X y∈K h \V

r(g, h)φ(y, u) Z(h−1 y)U . f (a)

(4.2.4)

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TRACE OF THE GENERATING SERIES

4.2.5

Modularity on open Shimura varieties

Fix one component α ∈ F+× \A× f /q(U ). We need to prove the modularity of × −1 × F+ q(U ) as above. Z(g, φ)U,α above. Let h ∈ Bf × B× f such that ν(h) ∈ α 0 Consider the finite map ih : MK h → MK,α . We want to express Z(g, φ)U,α as 0 the push-forward of some generating series on MK h. Note that ih∗ LK h = deg(ih )LK,α with deg(ih ) = [µ0U : µ2U ] by the above lemma. Then X r(g, h)φ(0, u) LK h + Z0,int (g, φ)U,α . Z0 (g, φ)U,α = −ih∗ u∈µ0U \F ×

Here Z0,int (g, φ)U,α = Z0,int (g, φ)U |MK,α is the α-component of the intertwining part X X Z0,int (g, φ)U = − W0 (β −1 u, r(g)φ) LK,β . × u∈µ2U \F × β∈F+ \A× f /q(U )

By ih∗ Z(y)K h = Z(h−1 y)U , we see that X X Z∗ (g, φ)U,α = wU ih∗ u∈µ0U \F ×

X

r(g, h)φ(y, u) Z(y)K h .

× y∈K h \V(a) a∈F+

In summary, we have Z(g, φ)U,α = Z0,int (g, φ)U,α +

X

ih∗ Z(1, r(g, h)φ(·, u), uq)K h .

u∈µ0U \F ×

Here we view r(g, h)φ(·, u) as a Schwartz function on S(V). The summation on u is a finite sum depending on (g, h), by the reason similar to the convergence of (4.1.4). Denote by 0◦ ◦ ◦ MK , MK,α MK 0, respectively the open parts of 0 MK 0,

MK ,

MK,α

before completion. ◦ = 0 as a divisor class on the open part We claim that Z0,int (g, φ)U,α |MK,α ◦ MK,α for all α. In fact, Z0,int (g, φ)U,α is a linear combination of Lβ,β 0 =

1 ∗ (p LU,β + p∗2 LU,β 0 ) 2 1

where p1 , p2 denote the projections of XU,β × XU,β 0 to the two components, and LU,β , LU,β 0 denote the Hodge bundles on these two components. A well-known

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result asserts that LU,β is linearly equivalent to a divisor supported on cusps on ◦ . The same is XU,β . Thus it is linearly equivalent to zero on the open part XU,β true for LU,β 0 . It proves the claimed result. 0◦ By the modularity of the generating series on the open part MK 0 , we see ◦ that Z(g, φ)U,α |MK,α is absolutely convergent and invariant under the left action by SL2 (F ). It is easy to check that it is also invariant under the left action by the diagonal group d(F × ). Thus it is invariant under the left action of ◦ , and thus the modularity GL2 (F ). It proves the modularity of Z(g, φ)U,α |MK,α of Z(g, φ)U |MK◦ . ◦ ◦ . Note that MK = MK unless It gives Theorem 3.17 on the open part MK F = Q and Bf = M2 (Af ). 4.2.6

Extending to the boundary

◦ . It happens if and only if F = Q and Here we consider the case MK 6= MK Bf = M2 (Af ). We have already proved the modularity of Z(g, φ)U |MK◦ , and we will extend it to the modularity of Z(g, φ)U . Recall that the modularity of Z(g, φ)U |MK◦ means that `(Z(g, Φ)U |MK◦ ) is an automorphic form for any linear functional ` : SC → C. Here SC denotes the ◦ )C generated by the coefficients of Z(g, φ)U |MK◦ over C. subspace of Pic(MK We first claim that SC is finite-dimensional. For that it suffices to assume that φ = φf × φ∞ with φ∞ equal to the standard Schwartz function. It is easy to see from the definition that Z(g, φ)U is invariant under the right action of some open compact subgroup W of GL2 (Af ) on g. Then we have a injection

SC∨ ,→ Hilb2 (GL2 (F ), W ),

` 7−→ `(Z(g, Φ)U |MK◦ ).

Here Hilb2 (GL2 (F ), W ) is the space of holomorphic Hilbert modular forms on GL2 (A) with parallel weight two and level W . The space is finite-dimensional, and thus SC is finite-dimensional. Go back to the modularity of Z(g, φ)U for a general φ. Pick a basis {D1 , · · · , Dr } of SC, and write Z(g, φ)U |MK◦ =

r X

fi (g)Di .

i=1

Here every fi is a complex-valued function of g. By the modularity, every fi is a usual holomorphic Hilbert modular forms of parallel weight two. It suffices to prove the modularity of the difference A(g) := Z(g, φ)U −

r X

fi (g)Di .

i=1

Here Di denotes the Zariski closure of Di in MK . Since the restriction A(g)|MK◦ is zero, the coefficients of A(g) (after possibly moving by linear equivalence)

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TRACE OF THE GENERATING SERIES

must be supported on the boundary ◦ = (XU × {cusps}) ∪ (XU × {cusps}). MK − MK

By a result of Manin [Ma] and Drinfel’d [Dr], the difference of any two cusps on the same geometrically connected component of XU is a torsion divisor in Pic(XU ). Fix a cusp Pα on XU,α for any component α ∈ F+× \A× f /q(U ). Then ◦ is a linear combination of XU,α × Pβ and any divisor supported on MK − MK Pα × XU,β . Therefore, we can always write X (Bα,β (g)(XU,α × Pβ ) + Cα,β (g)(Pα × XU,β )) . A(g) = × α,β∈F+ \A× f /q(U )

Here Bα,β and Cα,β are complex-valued functions of g. It suffices to prove that Bα,β and Cα,β are automorphic. Fix (α, β) and consider Bα,β (g). By intersection on MK , Bα,β (g) = A(g)·(Pα ×XU,β ) = Z(g, φ)U ·(Pα ×XU,β )−

r X

fi (g)Di ·(Pα ×XU,β ).

i=1

Therefore, it suffices to prove the modularity of Z(g, φ)U · (Pα × XU,β ) = Z(g, φ)U,α−1 β · (Pα × XU,β ) = deg Z(g, φ)U,α−1 β . Here deg Z(g, φ)U,α−1 β denotes the degree as a correspondence. Namely, it is the degree of Z(g, φ)U,α−1 β D for any divisor D on XU of degree one. A similar result is true for Cα,β (g). Indeed, it is reduced to the modularity of the degree of the transpose correspondence Z(g, φ)tU,α−1 β . The coefficients of Z(g, φ)U,α−1 β are the Hodge bundles and divisors of the form Z(x)U . By this, it is easy to check that deg Z(g, φ)tU,α−1 β = deg Z(g, φ)U,α−1 β . Therefore, Theorem 3.17 is reduced to the modularity of deg Z(g, φ)U,α for any α ∈ F+× \A× f /q(U ). In Proposition 4.2, we will prove that the degree is an Eisenstein series. It finishes the proof of Theorem 3.17. 4.3

DEGREE OF THE GENERATING SERIES

Here we describe the actions of the generating series Z(g, φ)U and its components on a divisor, and compute the degrees of the components as correspondences. 4.3.1

Action of the generating series

We consider the action of Z(g, φ)U = Z0 (g, φ)U + Z∗ (g, φ)U

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on a point [z, β]U of XU represented by (z, β) ∈ H × B× f . The result is simple for the constant term X X E0 (α−1 u, r(g)φ) LK,α . Z0 (g, φ)U = − × u∈µ2U \F × α∈F+ \A× f /q(U )

By definition X

LK,α =

LK,α0 ,αα0 ,

LK,α0 ,αα0 =

× α0 ∈F+ \A× f /q(U )

1 ∗ (p LU,α0 + p∗2 LU,αα0 ). 2 1

Here p1 , p2 are the projections of XU,α × XU,αα0 to its components. It follows that 1 LK,α ◦ [z, β]U = LU,αq(β) . 2 Therefore, Z0 (g, φ)U [z, β]U = −

1 2

X

X

× α∈F+ \A× f /q(U )

u∈µ2U \F ×

E0 (α−1 u, r(g)φ) LU,αq(β) .

Now we write down the action of the non-constant part X X r(g)φ(x)a Z(x)U . Z∗ (g, φ)U = wU a∈F × x∈U \B× /U f

By definition, Z(x)U [z, β] =

X [z, βαj ],

if U xU =

j

a

αj U.

j

It follows from (4.2.2) that Z∗ (g, φ)U [z, β]

X

= wU

X

r(g)φ(x)a [z, βx].

a∈F × x∈B× /U f

For any α ∈ F+× \A× f /q(U ), we also consider the action of the generating series Z(g, φ)U,α = Z(g, φ)|MK,α . Note that the constant term X

Z0 (g, φ)U,α = −

E0 (α−1 u, r(g)φ) LK,α .

u∈µ2U \F ×

Then Z0 (g, φ)U,α [z, β]U = −

1 2

X u∈µ2U \F ×

E0 (α−1 u, r(g)φ) LU,αq(β) .

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TRACE OF THE GENERATING SERIES

By (4.2.4), the non-constant part is given by X X r(g, (h, 1))φ(y, u)Z(h−1 y)U . Z∗ (g, φ)U,α = wU u∈µ0U \F × y∈K h \Bad f −1 × F+ q(U ), and Here h is any element of B× f such that q(h) ∈ α h −1 ad K = GSpin(Vf ) ∩ hKh . Here Bf is defined by: [ Bad = {x ∈ Bf : q(x) ∈ F+× } = Bf (a), f × a∈F+

Baf

=

Bf (a) = {x ∈ Bf : q(x) = a},

a ∈ A× f .

Note that the second notation is also valid in the local case. And the infinite (2) component of r(g, (h, 1))φ(y, u) is understood to be Wuq(y) (g∞ ), which makes sense for q(y) ∈ F+× . We are going to write down the action of Z∗ (g, φ)U,α on XU . Assume that h is an element of B× f for simplicity. That is, the second component is trivial. The action of Z(h−1 y)U is given by the coset U h−1 yU/U . We have identities U h−1 yU/U = Kh−1 y/U = h−1 (hKh−1 y/U ) = h−1 (K h y/U 1 ). Here U 1 = U ∩ B1f = {b ∈ U : q(b) = 1}. By this it is easy to see that X X Z∗ (g, φ)U,α [z, β] = wU r(g, (h, 1))φ(y, u) [z, βh−1 y]. 1 u∈µ0U \F × y∈Bad f /U

Equivalently, Z∗ (g, φ)U,α [z, β] = wU

X

X

u∈µ0U \F ×

× a∈F+

X y∈Bf

r(g, (h, 1))φ(y, u) [z, βh−1 y].

(a)/U 1

(4.3.1) −1 × F+ q(U ). Here h is any element of B× f such that q(h) ∈ α

4.3.2

Degree of the generating series

Now we compute the degree of Z(g, φ)U,α for any α ∈ F+× \A× f /q(U ). It is a × × correspondence from XU,β to XU,αβ for any β ∈ F+ \Af /q(U ). The degree of this correspondence is just the degree of Z(g, φ)U,α D for any degree-one divisor D on XU . It is independent of D. By definition, we see that deg Z(x)U = [U xU : U ]. Recall that in §4.1 the weight two Eisenstein series are defined as follows: X δ(γg)s r(γg)φ(0, u), E(s, g, u, φ) = γ∈P 1 (F )\SL2 (F )

E(s, g, φ)U =

X

E(s, g, u, φ).

u∈µ2U \F ×

The result can be viewed as a geometric variant of the Siegel–Weil formula.

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Proposition 4.2. For any α ∈ F+× \A× f /q(U ), 1 deg Z(g, φ)U,α = − (deg LU,α ) E(0, g, r(h)φ)U . 2 × −1 × Here h is any element of B× F+ q(U ), and deg LU,α f × Bf such that ν(h) ∈ α is independent of α.

Proof. Note that XU is connected, but not geometrically connected. Then the Galois group acts transitively on the set of geometrically connected components, and the action switches the Hodge bundles LU,α between components. Thus the degree κ◦U = deg LU,α is independent of α. By the action described above, we immediately see that X 1 deg Z0 (g, φ)U,α = − κ◦U E0 (α−1 u, r(g)φ), 2 u∈µ2U \F × X X X deg Z∗ (g, φ)U,α = wU r(g, h)φ(x, u). u∈µ0U \F × a∈F × x∈Bf (a)/U 1 +

Write E(0, g, φ)U = E0 (0, g, φ)U + E∗ (0, g, φ)U . Here E0 (0, g, φ)U is the constant term, and X E∗ (0, g, φ)U = Ea (0, g, φ)U a∈F ×

is the non-constant part. We are going to compare the constant terms and the non-constant parts of these two series. By definition, E0 (u, r(g)φ) = E0 (0, g, u, φ). Thus we get the identity for constant terms: 1 deg Z0 (g, φ)U,α = − κ◦U E0 (0, g, r(h)φ)U . 2 The corresponding identity for the non-constant parts follows from the local results in Proposition 2.9. Consider X X X r(g, h)φ(x, u). deg Z∗ (g, φ)U,α = wU u∈µ0U \F × a∈F × x∈Bf (a)/U 1 +

Let xa be any fixed element in B× f with norm a. The last summation above equals Z 1 r(g, h)φ(bxa , u)db vol(U 1 )vol(B1∞ ) B1 1 Wau (0, g, u, r(h)φ). =− 1 vol(U )vol(B1∞ )

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TRACE OF THE GENERATING SERIES

The equality follows from Proposition 2.9. Both sides are zero if a lies in F × − F+× . It follows that X X wU Wau (0, g, u, r(h)φ). deg Z∗ (g, φ)U,α = − 1 1 vol(U )vol(B∞ ) 0 × × u∈µU \F

a∈F

Thus, deg Z∗ (g, φ)U,α = −

wU E∗ (0, g, r(h)φ). [µ0U : µ2U ]vol(U 1 )vol(B1∞ )

It remains to check κ◦U =

2 wU . [µ0U : µ2U ]vol(U 1 )vol(B1∞ )

(4.3.2)

Here the Haar measure on B1v for each place v is as in §1.6.2. In particular, vol(B1v ) = 4π 2 for any archimedean place v. Fix an archimedean place τ , and denote by B = B(τ ) the nearby quaternion algebra. The factor [µ0U : µ2U ] is exactly the degree of the natural map × (B 1 ∩ U 1 )\H −→ (B+ ∩ U )\H.

In fact, the degree is just × × [(B+ ∩ U ) : (B 1 ∩ U 1 )µU ] = [q(B+ ∩ U ) : µ2U ] = [µ0U : µ2U ].

The last identity follows from (4.2.3). Denote by κ◦U 1 the degree of the Hodge dxdy bundle of (B 1 ∩U 1 )\H. It is also the integration of on (B 1 ∩U 1 )\H. Then 2πy 2 we have the relation κ◦U 1 = [µ0U : µ2U ]κ◦U . Now (4.3.2) is equivalent to κ◦U 1 =

2 wU . vol(U 1 )vol(B1∞ )

(4.3.3)

We will interpret the equality as the fact that the Tamagawa number of B 1 is 1. Endow the Haar measure on Bv1 for every place v as in §1.6.2. It gives a product measure on BA1 , which is exactly the Tamagawa measure. Since B 1 is simply connected, we have vol(B 1 \BA1 ) = 1. By the strong approximation 1 U 1 . It follows that theorem, BA1 = B 1 B∞ 1 1,τ 1 B 1 \BA1 = B 1 \B 1 B∞ U 1 = (Γ\Bτ1 )B∞ U .

Here we denote Γ = B 1 ∩ U 1 . It follows that 1,τ vol(Γ\Bτ1 )vol(B∞ )vol(U 1 ) = 1

(4.3.4)

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Consider the volume of Γ\Bτ1 . Note that Bτ1 ' SL2 (R). By the Iwasawa decomposition, any element is uniquely of the form     1 cos θ sin θ 1 x y2 , x ∈ R, y ∈ R+ , θ ∈ [0, 2π). 1 − sin θ cos θ 1 y− 2 dxdy dθ. It follows that 2y 2 Z dxdy 2π 1 . vol(Γ\Bτ ) = wU Γ\H 2y 2

The measure on Bτ1 is just

Here H is the space of (x, y), and the factor  wU appears due  to the action of cos θ sin θ . −1 ∈ Γ (in the case wU = 2) on the matrix − sin θ cos θ On the other hand, the degree of the Hodge bundle Z dxdy κ◦U 1 = . 2 Γ\H 2πy It yields that vol(Γ\Bτ1 ) =

2π 2 ◦ κ 1. wU U

Then (4.3.4) becomes 2π 2 ◦ 1,τ κ 1 vol(B∞ )vol(U 1 ) = 1. wU U It is equivalent to (4.3.3) because vol(B1τ ) = 4π 2 . It finishes the proof.



Remark. If U is maximal, by computing the volumes, we can obtain an explicit formula Y 3 (Nv − 1). κ◦U 1 = 4(4π)−[F :Q] |DF | 2 ζF (2) v∈Σ, v-∞

See also Vign´eras [Vi].

4.4

THE TRACE IDENTITY

The goal for the rest of this chapter is to prove Theorem 3.22. We first recall the content of the theorem. Let Φ ∈ S(V × A× ) be a Schwartz function. We have defined the “arithmetic theta lifting” Z ∗ e Φ) dg, ϕ ∈ σ. e ϕ(g) Z(g, Z(Φ ⊗ ϕ) = GL2 (F )\GL2 (A)/Z(A)

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TRACE OF THE GENERATING SERIES

As a correspondence, it gives a map e ⊗ ϕ) : Z(Φ

J(F )C −→J ∨ (F )C .

On the other hand, we have a theta lifting θ:

S(V × A× ) ⊗ σ−→π ⊗ π e.

It is normalized in (2.2.3) place by place. By §3.3.1, one has a map T:π⊗π e −→ Hom0 (J, J ∨ )C . Theorem 3.22 asserts e ⊗ ϕ) = L(1, π, ad) T(θ(Φ ⊗ ϕ)) Z(Φ 2ζF (2) as an identity in Hom0 (J, J ∨ )C . By the argument right after the theorem, there is a constant c(π) independent of Φ and ϕ such that e ⊗ ϕ) = c(π)T(θ(Φ ⊗ ϕ)). Z(Φ

(4.4.1)

The task is to prove that c(π) is the given one. 4.4.1

Lefschetz trace formula

Fix an open compact subgroup U of B× f acting trivially on Φ. Fix an emi bedding τ : F ,→ C. Write H (XU ) = H i (XU,τ (C), C) and H p,q (XU ) = H p,q (XU,τ (C), C) for simplicity. View (4.4.1) as an identity of operators on H 1,0 (XU ). Taking traces, we obtain e ⊗ ϕ) |H 1,0 (XU ) ) = c(π) tr( T(θ(Φ ⊗ ϕ)) |H 1,0 (XU ) ). tr( Z(Φ To compute c(π), it suffices to compute the traces in both sides. Trace of projectors Let f1 ⊗ f2 ∈ π U ⊗ π eU be any vector. By definition, T(f1 ⊗ f2 ) = vol(XU ) T(f1 ⊗ f2 )U . Fix a decomposition H 1,0 (XU ) =

M

π 0U .

π 0 ∈A(B× )

The action of T(f1 ⊗ f2 )U on H 1,0 (XU ) is simply f 7−→ (f, f2 ) f1 .

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CHAPTER 4

Its trace is simply the pairing (f1 , f2 ). It follows that tr( T(θ(Φ ⊗ ϕ)) |H 1,0 (XU ) ) = vol(XU ) Fθ(Φ ⊗ ϕ). Here F = (·, ·) is the canonical form F : π ⊗ π e → C. Recall that the Shimizu lifting is normalized in (2.2.3) so that Z Y ζv (2) Wϕv ,−1 (g)r(g)Φv (1, 1)dg. Fθ(Φ ⊗ ϕ) = L(1, πv , ad) N (Fv )\GL2 (Fv ) v Trace of the arithmetic theta lifting e ⊗ ϕ) on XU × XU . The e ⊗ ϕ)U for the corresponding divisor of Z(Φ Write Z(Φ key to compute the trace is the following Lefschetz trace formula X e Φ)U |H i (XU )). e ⊗ ϕ)U = (−1)i tr(Z(g, deg ∆∗ Z(Φ i=0,1,2

Here ∆ : XU → XU × XU is the diagonal embedding, and the left-hand side is e ⊗ ϕ)U . just the number of fixed points of the correspondence Z(Φ It is easy to see that e ⊗ ϕ)U |H 0 (XU )) = tr(Z(Φ e ⊗ ϕ)U |H 2 (XU )) = deg(Z(Φ e ⊗ ϕ)U ). tr(Z(Φ e ⊗ ϕ)U ) is in the sense of correspondence. Namely, it is the degree Here deg(Z(Φ e ⊗ ϕ)U D for any divisor D on XU of degree one. of Z(Φ As for H 1 (XU ), we have the decomposition H 1 (XU ) = H 1,0 (XU ) ⊕ H 0,1 (XU ) = H 1,0 (XU ) ⊕ H 1,0 (XU ). Here H 1,0 (XU ) = Γ(XU , Ω1XU ) ⊗F C. The tensor product is through τ : F ,→ C. The complex conjugation is taken on e ⊗ ϕ)U is the C part, which fixes Γ(XU , Ω1XU ) because τ (F ) ⊂ R. Note that Z(Φ 1,0 e defined over F since so is Z(g, Φ)U . So their traces on H (XU ) are the same as on H 1,0 (XU ). It follows that e ⊗ ϕ)U |H 1,0 (XU )) = tr(Z(Φ e ⊗ ϕ)U |H 1,0 (XU )). tr(Z(Φ In summary, we have e ⊗ ϕ)|H 1,0 (XU )) = − 1 deg ∆∗ Z(Φ e ⊗ ϕ)U . e ⊗ ϕ)U + deg Z(Φ tr(Z(Φ 2 Then Theorem 3.22 is implied by the following two results: Proposition 4.3. Z ∗ e Φ)U dg ϕ(g) deg Z(g, GL2 (F )\GL2 (A)/Z(A)

=

0.

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TRACE OF THE GENERATING SERIES

Proposition 4.4. Z

∗

e Φ)U dg ϕ(g) deg ∆∗ Z(g,

GL2 (F )\GL2 (A)/Z(A)

=

−vol(XU )

L(1, π, ad) Fθ(Φ ⊗ ϕ). ζF (2)

Both propositions are proved by the explicit expressions of the degrees. We e Φ)U is essentially an Eisenwill see that Proposition 4.3 is true because deg Z(g, stein series, and Proposition 4.4 is the geometric case of Proposition 2.3. 4.4.2

Degrees of correspondences

Here we prove Proposition 4.3. Denote by φ = Φ ∈ S(V × A× ) the average of Φ at infinity. It suffices to prove the same result for deg Z(g, φ)U . By definition, X deg Z(g, φ)U = deg Z(g, φ)U,α . × α∈F+ \A× f /q(U )

By Proposition 4.2, each component 1 deg Z(g, φ)U,α = − κ◦U E(0, g, r(h, 1)φ)U . 2 −1 × F+ q(U ), and the Eisenstein Here h is any element of B× f such that q(h) ∈ α series are given by X E(s, g, u, φ) = δ(γg)s r(γg)φ(0, u), γ∈P 1 (F )\SL2 (F )

E(s, g, φ)U =

X

E(s, g, u, φ).

u∈µ2U \F ×

It suffices to show that Z ∗ ϕ(g) E(0, g, φ)U dg = 0. GL2 (F )\GL2 (A)/Z(A)

The general case E(0, g, r(h, 1)φ)U is obtained by replacing φ by r(h, 1)φ. By definition, the integral equals Z Z ϕ(zg) E(0, zg, φ)U dz dg. GL2 (F )\GL2 (A)/Z(A)

F × \A× f /UZ

Here UZ = U ∩ Z(A) acts trivially on ϕ and E. It further becomes (ϕ(g), E(0, g, ωσ , φ)U )pet .

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Here ωσ is the central character of σ, and Z E(s, g, ωσ , φ)U = E(s, zg, φ)U ωσ (z)dz. F × \A× f /UZ

By definition,

=

E(s, g, ωσ , φ)U Z X γ∈P (F )\GL2 (F )

X F × \A× f /UZ

It is a P-series defined by Z B(s, g) =

X

F × \A× f /UZ

δ(γg)s r(γzg)φ(0, u)ωσ (z)dz.

u∈µ2U \F ×

δ(g)s r(zg)φ(0, u) ωσ (z)dz,

u∈µ2U \F ×

which transfers as B(s, pnzg) = ωσ−1 (z)B(s, g),

p ∈ P (F ), n ∈ N (A), z ∈ Z(A).

By the standard theory, we see that E(s, g, ωσ , φ)U is orthogonal to σ. 4.4.3

Degree of pull-back

The proof of Proposition 4.4 will take up the rest of this chapter. It is implied by an expression of deg ∆∗ Z(g, φ)U in terms of the incoherent Eisenstein series of weight 3/2 in §2.5.3. We call this formula the pull-back formula. The treatment is naturally divided into two cases: (1) Compact case #Σ > 1. In this case, the Shimura curve XU◦ is compact and the related Eisenstein series of weight 3/2 is holomorphic. (2) Non-compact case #Σ = 1. It happens if and only if F = Q and Σ = {∞}. The Shimura curve XU is just a classical modular curve (if choosing suitable U ). In this case, the Shimura curve XU◦ is non-compact and the related Eisenstein series of weight 3/2 is non-holomorphic. Then our computation is complicated by extra terms coming from the cusp of XU and the non-holomorphic terms of the Eisenstein series. To illustrate the idea, we list the pull-back formula in the compact case in the following. Proposition 4.5. Let φ ∈ S(A × A× ) be invariant under U × U . Assume that #Σ > 1. Then deg ∆∗ Z(g, φ)U

=

−vol(XU ) J(0, g, φ)U ,

Here the mixed Eisenstein–theta series X δ(γg)s J(s, g, φ)U = γ∈P (F )\GL2 (F )

X

X

u∈µ2U \F × x∈F

g ∈ GL2 (A).

r(γg)φ(x, u).

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TRACE OF THE GENERATING SERIES

We will prove this formula in the next section. Its counterpart in the noncompact case is Theorem 4.15 in §4.6, where the difference of two sides is nonzero and given by some extra terms. In §4.7, we obtain a description of the extra terms in terms of a Poincar´e series in Theorem 4.20. It finishes Proposition 4.4 (in the non-compact case). Now let us see how Proposition 4.5 deduces Proposition 4.4 (in the compact case). Assume that #Σ > 1. By taking φ = Φ, we need to show: Z ∗ e g, φ)U dg = L(1, π, ad) Fθ(Φ ⊗ ϕ). (4.4.2) ϕ(g) J(0, ζF (2) GL2 (F )\GL2 (A)/Z(A) Here we normalize 1

[F :Q]−1 hF |DF |− 2 e g, φ)U := 2 J(s, g, φ)U . J(s, [OF× : µ2U ]

e The normalizing factor is the same as in the definition of Z(Φ) U in (3.4.4). We also define X X X J(s, g, Φ) = δ(γg)s r(γg)Φ(x, u). u∈F × x∈F

γ∈P (F )\GL2 (F )

We can assume that Φ is invariant under the action of B1∞ as usual. We prove it by a few steps as follows: Z J(s, zg, Φ)dz, J(s, g, φ)U =wU

(4.4.3)

× µU \F∞

Z

∗

Z e g, φ)U dg = ϕ(g) J(s,

GL2 (F )\GL2 (A)/Z(A)

ϕ(g)J(s, g, Φ)dg, GL2 (F )\GL2 (A)

(4.4.4) Z

L(1, π, ad) Fθ(Φ ⊗ ϕ). ϕ(g) J(0, g, Φ)dg = ζF (2) GL2 (F )\GL2 (A)

(4.4.5)

First verify (4.4.3). Similar to (4.1.5), it follows from the relation φ = r(z)Φ dz. To check it, we can assume that −1 ∈ / U since both sides change by the same multiple when varying U . Then it suffices to check Z X X X X r(zγg)Φ(x, u)dz = r(γg)φ(x, u).

R

× F∞

× µU \F∞ u∈F × x∈F

u∈µ2U \F × x∈F

The key is that r(α)Φ = Φ for α ∈ µU . So the left-hand side becomes Z X X r(zγg)Φ(x, u)dz = r(γg)φ(x, u). (x,u)∈µU \(F ×F × )

It gives the result.

× F∞

(x,u)∈µU \(F ×F × )

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Now we verify (4.4.4). The left-hand side is equal to Z 1 Z 2[F :Q]−1 hF |DF |− 2 ∗ ϕ(g) J(s, g, r(z)Φ)dzdg. wU × [OF× : µ2U ] µU \F∞ GL2 (F )\GL2 (A)/Z(A) By the definition in (1.6.1), the above double integral equals Z Z Z 2 ϕ(z 0 zg)J(s, z 0 zg, Φ)dzdz 0 dg. × Ress=1 ζF (s) GL2 (F )\GL2 (A)/Z(A) F × \A× /Fτ× µU \F∞ Here τ is any archimedean place of F . Use the identity × τ,× ≈ (µU \F∞ ) · Fτ× , µU \F∞

and split the inner integration. Then the integrations on Fτ× and on F × \A× /Fτ× collapse to an integration on F × \A× . The above becomes Z τ,× Z 2 vol(µU \F∞ ) ϕ(z 0 g)J(s, z 0 g, Φ)dz 0 dg Ress=1 ζF (s) GL2 (F )\GL2 (A)/Z(A) F × \A× τ,× Z ) 2 vol(µU \F∞ ϕ(g)J(s, g, Φ)dg. = Ress=1 ζF (s) GL2 (F )\GL2 (A) τ,× ∼ Note that µU is a subgroup of rank n − 1 in F∞ = (R× )n−1 . Here n = [F : Q]. The volume 1 vol(µU \(R× )n−1 ) = vol(µ2U \(R× )n−1 ) 2n−1 wU 1 1 n−1 vol(µ2U \(R× )= R 2. = +) wU wU µU

Here Rµ2U is the regulator of µ2U defined similar to the regulator of OF× . So we only need to check wU

1 2 Rµ2U /wU 2[F :Q]−1 hF |DF |− 2 = 1. · × 2 Ress=1 ζF (s) [OF : µU ]

It is equivalent to (3.4.5), another description of the normalizing factor. It finishes (4.4.4). The proof of (4.4.5) is just the standard folding-unfolding process. It is not different from the computation of A(s) in Proposition 2.3. The proof there also applies to the incoherent case. 4.5

PULL-BACK FORMULA: COMPACT CASE

In this section, we prove the formula for the degree of ∆∗ Z(g, φ)U in Proposition 4.5 when the Shimura curve XU has no cusps (or equivalently the open part XU◦ is compact). It happens if and only if #Σ > 1. It includes “almost all” Σ, but is much simpler than the non-compact case. It has the same flavor as [YZZ, Proposition 3.1]. For the convenience of readers, we give a detailed proof here.

129

TRACE OF THE GENERATING SERIES

4.5.1

CM cycles on the Shimura curve

Denote by ∆ : XU → XU × XU the diagonal embedding. Then ∆∗ Z(g, φ)U is an automorphic form on GL2 (A) with coefficients in Pic(XU )C . Our goal is to obtain a formula for it. Let B0 be the set of elements in B with trace zero. It gives an orthogonal decomposition B = A ⊕ B0 of quadratic spaces. Denote the conjugation action of B× on B0 by Ad, i.e., Ad(h) ◦ x = hxh−1 for any h ∈ B× and x ∈ B0 . The action keeps the norm. Fix an archimedean place τ of F , denote by B = B(τ ) the nearby quaternion algebra, and identify BA×f = B× f as usual. Similar to B, we have an orthogonal decomposition B = F ⊕ B0 , and a conjugation action Ad of B × on B0 . For any element y ∈ B0 with q(y) 6= 0, denote by By (resp. Bf,y ) the centralizer of y in B (resp. Bf ). Then By = F [y] = F ⊕ F y is a quadratic extension of F . In particular, By is a CM extension of F if and only if y lies in the subset B0,+ of elements of B0 with totally positive norms. Assume y ∈ B0,+ so that By is a CM extension. Let τy be the unique point in H fixed by the action of By× through the embedding By× ⊂ Bτ (R). Then τ y is the unique fixed point of By× in H− . Write τy± = {τy , τ y } ⊂ H± . The (zero-dimensional) Shimura variety associated to By is just Sh(By× , W ) = τy± × By× \B× f,y /W, where W is an open compact subgroup of B× f,y . , denote by C(y, h) the push-forward of the map For any h ∈ B× U f × Sh(By , Uh ) → XU given by × × ± τy± × By× \B× f,y /Uh −→ B \H × Bf /U,

(τ, b) 7−→ (τ, bh).

−1 and the map is always an embedding if U is sufficiently Here Uh = B× f,y ∩ hU h small. Then C(y, h)U = B × \B × (τy± × B× f,y hU/U ). × It depends only on the coset of h in B× f,y \Bf /U . By the reciprocity law, C(y, h)U is a divisor of XU defined over F .

Lemma 4.6. Let y, y 0 ∈ B0,+ , and h, h0 ∈ B× f . Then C(y, h)U = C(y 0 , h0 )U ⇐⇒ F × · Ad(U ) ◦ (h−1 yh) = F × · Ad(U ) ◦ (h0−1 y 0 h0 ). Proof. We first prove “⇐=.” Assume that the right-hand side is true. The subvariety C(y, h)U is represented by the set B × (τy± × B× f,y hU ). It is easy to see that it does not change if we multiply y by an element in F × or multiply h by an element in U . Thus we can assume that h−1 yh = h0−1 y 0 h0 .

130

CHAPTER 4

Then q(y) = q(y 0 ) and we can find γ ∈ B × such that y 0 = γ −1 yγ. It follows that τy±0 = γ −1 τy± , and Bf,y0 = γ −1 Bf,y γ. Furthermore, h−1 yh = h0−1 γ −1 yγh0 implies that γh0 h−1 commutes with y, and thus lies in B× f,y . Hence, 0 B × (τy±0 × B× f,y 0 h U )

=

0 B × (γ −1 τy± × γ −1 B× f,y γh U )

=

× 0 × ± B × (τy± × B× f,y γh U ) = B (τy × Bf,y hU ).

It proves that C(y, h)U = C(y 0 , h0 )U . Now we prove “=⇒.” Assume that C(y, h)U = C(y 0 , h0 )U . Then there is a γ ∈ B × such that τy0 = γ −1 τy or τy0 = γ −1 τ y . The stabilizer of τy0 in B × is By×0 , and the stabilizer of γ −1 τy (or γ −1 τy ) in B × is exactly Bγ×−1 yγ . It follows that By0 = Bγ −1 yγ in both cases. Note that By0 is a CM extension over F , and both y 0 and γ −1 yγ have trace zero. It follows that y 0 = cγ −1 yγ for some c ∈ F × . By the direction we have proved, C(y, h)U = C(y 0 , h0 )U = C(γ −1 yγ, h0 )U = C(y, γh0 )U . × 0 It follows that B× f,y hU = Bf,y γh U . The result follows.



With the lemma, we can define C(y)U for any y in × Bad f,0 := {x ∈ Bf,0 : q(x) ∈ F+ }.

In fact, write y = h−1 y0 h for any y0 ∈ B0,+ and h ∈ B× f . Then define C(y)U := C(y0 , h)U . It is independent of the choice of (y0 , h). We further know that C(y)U = C(y 0 )U if and only if F × · Ad(U ) ◦ y = F × · Ad(U ) ◦ y 0 . 4.5.2

Pull-back as cycles

Lemma 4.7. The following are true for x ∈ B× f : (1) If x ∈ F × U , then ∆∗ Z(x)U = −ωXU . Here ωXU denotes the canonical bundle of XU . (2) If x ∈ / F × U , then ∆∗ Z(x)U =

X

[F [y]× U ∩ U xU : U ] C(y)U .

× y∈Ad(U )\Bad f,0 /F

Here F [y] = F + F y is the totally imaginary quadratic field over F generated by y in Bf , and F [y]× denotes its multiplicative group.

131

TRACE OF THE GENERATING SERIES

Proof. If x ∈ F × U , then Z(x)U = Z(1)U = ∆. The result follows from the definition of the canonical bundle. Next, we assume that x ∈ / F × U . Then ∆ · Z(x)U is a proper intersection. Let τ be an archimedean place and B = B(τ ) be the nearby quaternion algebra. Recall the uniformization × × ± XU,τ (C) × XU,τ (C) = (B × \H± × B× f /U ) × (B \H × Bf /U ).

The divisor Z(x)U of XU × XU is represented by {(τ, hU ) × (τ, hU xU ) : τ ∈ H± , h ∈ B× f }. Assume that some point (τ, h)U lies in the intersection ∆ · Z(x)U . Then × such that there exist z ∈ U xU and γ ∈ B+ γτ = τ,

γhU = hzU.

/ F × U . It follows that τ is a CM point, and F [γ] = We have γ ∈ / F × since x ∈ F + F γ = F + F y is a CM extension of F . Here y ∈ B0 is the trace-free part of γ, and q(y) is totally positive. Note that y is determined by τ . The condition on z is equivalent to z ∈ h−1 γhU . Such a γ exists for z if and only if z ∈ h−1 F [y]× hU = F [h−1 yh]× U . It follows that the number of such z ∈ U xU/U , which is just the the multiplicity of (τ, h)U in ∆ · Z(x)U , is exactly equal to   F [h−1 yh]× U ∩ U xU : U . In particular, it depends only on h−1 yh. So this number is exactly the multiplicity of each point of C(h−1 yh)U in ∆ · Z(x)U . A geometric reason for these multiplicities to be equal is that all these points lie in a Galois orbit.  Now we can give a formula for the pull-back ∆∗ Z(g, φ)U . Recall from (4.2.1) and (4.2.2) that Z(g, φ)U = Z0 (g, φ)U + Z∗ (g, φ)U , where Z0 (g, φ)U

=

X

−

X

r(g)φ(0, α−1 u) LK,α ,

× u∈µ2U \F × α∈F+ \A× f /q(U )

Z∗ (g, φ)U

=

wU

X

X

r(g)φ(x)a Z(x)U .

a∈F × x∈U \B× /U f

Here the intertwining part vanishes since we assume that XU◦ is compact. Proposition 4.8 (Pull-back as cycles). Assume that #Σ > 1 and φ = φ0 ⊗ φ0 under the orthogonal decomposition B = A ⊕ B0 . Then X f 2 (A). ∆∗ Z(g, φ)U = θ(g, u, φ0 )C(g, u, φ0 )U , g ∈ SL u∈µ2U \F ×

132

CHAPTER 4

Here the generating series on XU is defined by C(g, u, φ0 )U

−r(g)φ0 (0, u) LU X 1 r(g)φ0 (y, u) C(y)U . + ×∩U :µ ] [F [y] U ad

=

y∈Ad(U )\Bf,0

f 2 (A) with coefficients in Pic(XU )C . Here It is an automorphic form on g ∈ SL F [y] = F + F y is the CM extension over F generated by y, and r(g)φ0 (y, u) = r(g)φ0 (1, uq(y)) is understood in the sense of (4.1.3). Proof. The computation below can be simplified in the case where U is small enough so that all the ramification indices [F [y]× ∩ U : µU ] = 1. But we include all cases to see the matching of the ramifications. / F × U . We get Divide the summation in Z∗ (φ) into x ∈ F × U and x ∈ ∆∗ Z(φ)U = P + Q + R with P

=

X

−

X

φ(0, α−1 u) ∆∗ LK,α ,

× u∈µ2U \F × α∈F+ \A× f /q(U )

Q

=

wU

X

X

φ(x)a ∆∗ Z(x)U ,

a∈F × x∈U \(F × U )/U

R

=

wU

X

X

a∈F ×

× x∈U \(B× f −F U )/U

φ(x)a ∆∗ Z(x)U .

We first consider P . The pull-back ∆∗ LK,α is nontrivial if and only if LK,α lies in the same component as ∆. Then α = 1. It is easy to see that ∆∗ LK,1 equals the Hodge bundle LU of XU . Then X φ(0, u) LU . P =− u∈µ2U \F ×

By Lemma 4.7, X Q = −wU a∈F ×

X x∈µU

φ(x)a ωXU = −

\F ×

X

X

u∈µ2U \F ×

x∈F ×

φ(x, u)ωXU .

It remains to treat R. Consider the a-th coefficient X Ra := wU φ(x)a ∆∗ Z(x)U . × x∈U \(B× f −F U )/U

The lemma also implies X Ra = wU

X

× × y∈Ad(U )\Bad x∈U \(B× f,0 /F f −F U )/U



(F [y]× )U ∩ U xU : U



φ(x)a C(y)U .

133

TRACE OF THE GENERATING SERIES

Write 

X

 (F [y]× )U ∩ U xU : U =

[zU ∩ U xU : U ] .

z∈(F [y]× U −F × U )/U

In order to make [zU ∩ U xU : U ] nonzero we need x ∈ U zU . It follows that X X φ(z)a C(y)U . Ra = wU × z∈(F [y]× U −F × U )/U y∈Ad(U )\Bad f,0 /F

The summation over z transforms as X X − = z∈F [y]× U/U

z∈F × U/U

X

X

−

z∈F [y]× /(F [y]× ∩U )

.

z∈F × /µU

Note that y 2 = −q(y) ∈ F is totally negative. The algebra F [y] = F + F y is a CM extension over F . By Dirichlet’s unit theorem, the index ey := [F [y]× ∩ U : µU ] is finite. Then we can replace the sum over F [y]× /(F [y]× ∩ U ) by that over F [y]× /µU divided by ey . It follows that Ra

=

X

wU

× y∈Ad(U )\Bad f,0 /F

1 ey

X

X

−wU

φ(z)a C(y)U

z∈F [y]× /µU

X

φ(z)a C(y)U .

× z∈F × /µU y∈Ad(U )\Bad f,0 /F

Using F [y]× = (F + F × y) Ra

=

`

F × , it becomes

X 1 φ(z)a C(y)U ey × z∈(F +F × y)/µU y∈Ad(U )\Bad f,0 /F   X X 1 −wU φ(z)a C(y)U . 1− ey × ad × X

wU

z∈F /µU

y∈Ad(U )\Bf,0 /F

Go back to R =

P

a∈F ×

Ra . It is easy to get R = R1 + R2

with R1

X

=

X

× u∈µ2U \F × y∈Ad(U )\Bad f,0 /F

R2

=

−

X

X

u∈µ2U \F × z∈F ×

φ(z, u)

1 ey

X

φ(z, u) C(y)U ,

z∈F +F × y

X y∈Ad(U )\Bad f,0 /F

  1 1− C(y)U . ey ×

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CHAPTER 4

Using the splitting φ = φ0 ⊗ φ0 , one has R1

X

X

=

× u∈µ2U \F × y∈Ad(U )\Bad f,0 /F

X

=

z∈F

X

θ(g, u, φ0 )

u∈µ2U \F ×

X 1 X 0 φ (z, u) φ0 (x, u) C(y)U ey ×

y∈Ad(U )\Bad f,0

x∈F y

1 φ0 (y, u) C(y)U . ey

It is the major part of the pull-back formula. As for R2 , use the formula X

LU = ωXU +

y∈Ad(U )\Bad f,0 /F

  1 1− C(y)U . ey ×

It is easy to get R2 + Q + P = −

X

X

φ(x, u)LU = −

u∈µ2U \F × x∈F

X

θ(g, u, φ0 ) φ0 (0, u)LU .

u∈µ2U \F ×

The pull-back formula follows. The formula between the Hodge bundle and the canonical bundle can be proved by the definition of the Hodge bundle. The modularity of C(g, u, φ0 )U can be derived from that of Z(g, φ)U using the pull-back formula. For fixed φ0 we can vary φ0 to get information on a single C(g, u, φ0 )U . It is actually an example of the modularity in [YZZ], and the proof in [YZZ] uses this pull-back method. When U is small enough, the factor [F [y]× ∩ U : µU ]−1 = 1 and the series is the same as that in [YZZ]. In general, the factor should be there and the modularity result for big level U can be obtained by push-forward from small levels.  4.5.3

Degree of the pull-back

By Proposition 4.8, the proof of Proposition 4.5 (in the compact case) is reduced to the following result. Note that we can first obtain Proposition 4.5 for g ∈ SL2 (A), and extend it to GL2 (A) for the action of an element d(a) with a ∈ A× . Proposition 4.9. Assume that #Σ > 1. Let C(g, u, φ0 )U be the generating function on XU as in Proposition 4.8. Then deg C(g, u, φ0 )U = −vol(XU ) E(0, g, u, φ0 ),

f 2 (A). g ∈ SL

Here the Eisenstein series E(s, g, u, φ0 ) =

X γ∈P 1 (F )\SL2 (F )

δ(γg)s r(γg)φ0 (0, u),

f 2 (A). g ∈ SL

135

TRACE OF THE GENERATING SERIES

Proof. It follows from the local Siegel–Weil formula in §2.5.3. The proof is very similar to Proposition 4.2, except that the computation is more complicated. Recall that C(g, u, φ0 )U

=

−r(g)φ0 (0, u) LU X 1 r(g)φ0 (y, u) C(y)U . + × [F [y] ∩ U : µU ] ad y∈Ad(U )\Bf,0

Recall from Proposition 2.10 that E(0, g, u, φ0 ) = r(g)φ0 (0, u) −

X

Z L(1, η−ua )

a∈F ×

× B× ya \B

r(g, (h, h))φ0 (ya , u)dh.

Here ya ∈ B is any element with uq(ya ) = a, and the integration is considered to be zero if such ya does not exist. By Lemma 3.1, it is immediate to have the identity on the constant terms: deg C0 (g, u, φ0 )U = −vol(XU ) E0 (0, g, u, φ0 ). It implies the result without much computation. In fact, we first verify that both E(0, g, u, φ0 ) and vol(XU )−1 deg C(g, u, φ0 )U define elements in the onedimensional space ∞ f HomSO(B0 )×SL f 2 (A) (S(B0 ), C (SL2 (F )\SL2 (A))).

Then they must be proportional, and their quotient is given by the identity between the constant terms. The argument has used the modularity of C0 (g, u, φ0 )U , which is not valid if #Σ = 1. In the following we propose a computation which can (and will) be used in the case #Σ = 1. Fix an a ∈ F × . In the following we verify deg Ca (g, u, φ0 )U = −vol(XU ) Ea (0, g, u, φ0 ). It is easy to have Ca (g, u, φ0 )U

X

=

y∈Ad(U )\Bf,0 (a)

1 r(g)φ0 (y, u) C(y)U . [F [y]× ∩ U : µU ]

By the choice of ya , we can rewrite it as

=

Ca (g, u, φ0 )U X × h∈B× f,ya \Bf /U

[By×a

1 r(g, (h, h))φ0 (ya , u) C(ya , h)U . ∩ (hU h−1 ) : µU ]

136

CHAPTER 4

In the following we abbreviate ya as y for simplicity. Let B = B(τ ) be the nearby quaternion algebra for some archimedean place τ . Identifying B(Aτ ) = Bτ , the y is an element of B0 (Aτ ). We can assume that y actually lies in B0 since the result depends only on the norm of y. The divisor C(y, h)U is represented by the set × ± × B × \(B × τy± × B× f,y hU/U ) = τy × By \Bf,y hU/U.

Its degree is given by twice of the order of × × × −1 By× \B× ). f,y hU/U = By \Bf,y /(Bf,y ∩ hU h

It follows that

=

deg Ca (g, u, φ0 )U × −1 X |By× \B× )| f,y /(Bf,y ∩ hU h 2 r(g, (h, h))φ0 (y, u). × −1 [By ∩ (hU h ) : µU ] × × h∈Bf,y \Bf /U

On the other hand, Z Ea (0, g, u, φ0 )

= =

−L(1, η−ua )

r(g, (h, h))φ0 (y, u)dh

× B× y \B × vol(B× ∞,y \B∞ )

−L(1, η−ua ) X × · vol(B× f,y \Bf,y hU ) r(g, (h, h))φ0 (y, u). × h∈B× f,y \Bf /U

Hence, it is reduced to verify 2 =

× −1 |By× \B× )| f,y /(Bf,y ∩ hU h

[By× ∩ (hU h−1 ) : µU ]

× × × vol(XU ) L(1, η−ua ) vol(B× ∞,y \B∞ ) vol(Bf,y \Bf,y hU ).

× 2 d Note vol(B× ∞,y \B∞ ) = (2π ) with d = [F : Q], and

  × × × −1 −1 B× h = (B× )\(hU h−1 )h. f,y \Bf,y hU = Bf,y \Bf,y hU h f,y ∩ hU h The desired equality becomes vol(XU ) =

× × × × −1 −1 2(2π 2 )−d |By \Bf,y /(Bf,y ∩ hU h )| vol(Bf,y ∩ hU h ) · . · L(1, η−ua ) vol(U ) [By× ∩ (hU h−1 ) : µU ]

For simplicity, denote × −1 , UZ = A× L = By0 , Lf = AL,f , Uh = B× f,y ∩ hU h f ∩ Uh = Af ∩ U.

137

TRACE OF THE GENERATING SERIES

Then η−ua is exactly the quadratic character ηL associated to the CM extension L/F . What we need to show is just vol(XU ) =

× × 2(2π 2 )−d |L \Lf /Uh | vol(Uh ) · × · . L(1, ηL ) [L ∩ Uh : µU ] vol(U )

We are going to need the following identities: (1) 2L(1, ηL ) = vol(L× /L× A× ), × −1 × (2) vol(L× /L× A× ) = |A× vol(L× /L× F∞ UZ ), f /F UZ | ×

×

(3) vol(L /L

× F∞ UZ )

|L× \L× f /Uh | vol(Uh ) . =2 [L× ∩ Uh : µU ] vol(UZ ) d

The first identity is just the result for the Tamagawa number of SO(L, q) ∼ = L× /F × . For the second identity, we have × UZ ) vol(L× /L× F∞ × × = [L× A× : L× F∞ UZ ] = |A× /F × F∞ UZ |. × × × vol(L /L A )

For the third identity, × UZ ) vol(L× /L× F∞ = vol × |L× \Lf /Uh |



L× L× ∞ Uh × L× F∞ UZ



 = vol

L× ∞ Uh × F∞ UZ



1 . [L× ∩ Uh : µU ]

× d Use vol(L× ∞ /F∞ ) = 2 . Applying (1), (2) and (3), the identity we need to prove is equivalent to × vol(XU ) = 4(4π 2 )−d |A× f /F UZ |

vol(UZ ) . vol(U )

(4.5.1)

This clean expression does not depend on L. Recall from Proposition 4.2 that the Hodge bundle LU has the same degree κ◦U on all geometrically connected components of XU . It follows that × vol(XU ) = κ◦U · |A× f /F+ q(U )|, × where |A× f /F+ q(U )| is just the number of geometrically connected components. Furthermore, by equation (4.3.2) in the proof of Proposition 4.2,

κ◦U =

2wU . : µ2U ]vol(U 1 )

(4π 2 )d [µ0U

Thus (4.5.1) becomes [µ0U

wU vol(UZ ) × × × |A× . f /F+ q(U )| = 2|Af /F UZ | 2 1 : µU ]vol(U ) vol(U )

138

CHAPTER 4

Note that × |A× f /F+ q(U )| × |A× f /F UZ |

= [F × UZ : F+× q(U )] = [F × : F+× ]

vol(UZ ) 0 [µ : µU ]. vol(q(U )) U

The result follows from the following simple identities: • [F × : F+× ] = 2d , • [µU : µ2U ] = 2d−1 wU , • vol(U ) = vol(U 1 ) · vol(q(U )).  4.6

PULL-BACK FORMULA: NON-COMPACT CASE

In this section, we give a formula for ∆∗ Z(g, φ)U and its degree in the case where XU contains cusps. In other words, the open part XU◦ is non-compact. Then we have F = Q, Σ = {∞} and Bf = M2 (Af ). The extra computation for the pull-back comes from the cusps, and the extra computation for the degree comes from the non-holomorphic terms of the related Eisenstein series. The pull-back formula (4.4.3) can be modified in (4.4.6) with an extra term: deg ∆∗ Z(g, φ)U = −vol(XU ) J(0, g, φ)U + vol(XU ) B(g, φ). 4.6.1

Pull-back as cycles

There is only one archimedean place, and the nearby quaternion algebra B = M2 (Q). Recall that cusps on the upper half plane H form the set P1 (Q) = × = GL2 (Q)+ acts transitively. Then the set of cusps on Q ∪ {∞} on which B+ XU (C) is just the finite set × × ∼ \P1 (Q) × B× B+ f /U = {∞} × P (Q)+ \Bf /U.

Here P (Q)+ is the set of upper triangular matrices with positive determinants, × . For any h ∈ P (Q)+ \B× which is exactly the stabilizer of ∞ in B+ f /U , we abbreviate the corresponding cusp [∞, h]U as hhiU . Resume the notations for CM points in the last section. In particular, C(y)U denotes a finite set of CM points for any y in Bad f,0 . Note that Lemma 4.6 is still true in the current setting without any change. Lemma 4.7 becomes the following statement. Lemma 4.10. The following are true for x ∈ B× f : (1) If x ∈ F × U , then ∆∗ Z(x)U = −ωXU .

139

TRACE OF THE GENERATING SERIES

(2) If x ∈ / F × U , then ∆∗ Z(x)U

X

=

[F [y]× U ∩ U xU : U ] C(y)U

× y∈Ad(U )\Bad f,0 /F

+

X

X

τ (γ) hhiU .

γ∈P (Q)+ /(P (Q)+ ∩hU h−1 ) h∈P (Q)+ \B× f /U h−1 γh∈U xU

Here τ : P (R)+ → R is the continuous function defined by   a b τ = min(1, d/a). d Proof. The result in (1) still follows from the definition of the canonical bundle as in the compact case. For (2), the contribution of non-cusps in ∆·Z(x)U is still given by Lemma 4.7. It remains to treat the contribution of cusps in ∆ · Z(x)U . Let hhiU be a cusp, represented by h ∈ P (Q)+ \B× f /U , in the intersection ∆ · Z(x)U . We need to compute the multiplicity at hhiU . Write X Z(x)U hhiU = hhyiU . y∈U xU/U

We only need to consider cosets y ∈ U xU/U with hhyiU = hhiU , which happens if and only if hyU = γhU for some γ ∈ P (Q)+ . It happens if and only if y ∈ h−1 P (Q)+ hU ∩ U xU/U. By the relation y = h−1 γh, it is equivalent to γ ∈ P (Q)+ /(P (Q)+ ∩ hU h−1 ) and h−1 γh ∈ U xU. Now we compute the multiplicity. At a neighborhood of hhiU in XU (C), we have X Z(x)U [z, h]U '

[z, hy]U .

y∈h−1 P (Q)+ hU ∩U xU/U

We only sum over the set of y fixing hhiU , since other y does not contribute to the multiplicity at hhiU . Using y = h−1 γh, we have [z, hy]U = [γ −1 z, h]U . It follows that X [γ −1 z, h]U . Z(x)U [z, h]U ' γ

Consider the behavior of z → ∞. In terms of the complex uniformization, a uniformizer at hhiU is given by q = e(z/r) where r = [N (Z) : N (Q) ∩ hU h−1 ].

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CHAPTER 4

A local coordinate of (hhiU , hhiU ) in XU (C) × XU (C) is just a pair (q1 , q2 ) = (e(z1 /r), e(z2 /r)).  −1    a a b −ba−1 d−1 Write γ = , and thus γ −1 = . Then d−1 d e(γ −1 z/r) = e(

a−1 z − ba−1 d−1 b ) = q d/a e(− ). d−1 r ar

It follows that a local equation of Z(x)U at (hhiU , hhiU ) is just f (q1 , q2 ) =

Y

(q2 − e(−

γ

b d/a ) q ). ar 1

It is a polynomial. The intersection multiplicity with the diagonal q1 = q2 is exactly X min(1, d/a). γ

The result is proved.



By the lemma, Proposition 4.8 is modified as follows. Proposition 4.11. Assume that F = Q and Σ = {∞}. Assume that φ = φ0 ⊗ φ0 under the orthogonal decomposition B = A ⊕ B0 . Then for any g ∈ f 2 (A), SL ∆∗ Z(g, φ)U

=

X

θ(g, u, φ0 )C(g, u, φ0 )U +

u∈Q×

X

W0 (g, u, φ) LU + D(g, φ)U .

u∈Q×

Here the generating function C(g, u, φ0 )U

=

−r(g)φ0 (0, u) LU X 1 r(g)φ0 (y, u) C(y)U + ×∩U :µ ] [Q[y] U ad y∈Ad(U )\Bf,0

is as in Proposition 4.8, and

=

D(g, φ)U X wU a∈Q×

X

X

h∈P (Q)+ \B× f /U

γ∈P (Q)+ /(P (Q)+

r(g)φ(h−1 γh)a τ (γ) hhiU . ∩hU h−1 )

Proof. Note that µ2U = 1 by F = Q. We have a simplification X X = . u∈µ2U \F ×

u∈Q×

Similar to the compact case, write ∆∗ Z(φ)U = P + Q + R + S

141

TRACE OF THE GENERATING SERIES

with P

=

X

X

−

φ(0, α−1 u) ∆∗ LK,α ,

× u∈Q× α∈Q× + \Af /q(U )

Q

=

wU

X

X

φ(x)a ∆∗ Z(x)U ,

a∈Q× x∈U \(Q× U )/U

R S

=

wU

X

X

a∈Q×

× x∈U \(B× f −Q U )/U

X

X

=

φ(x)a ∆∗ Z(x)U , W0 (uα−1 , φ) ∆∗ LK,α .

× u∈Q× α∈Q× + \Af /q(U )

Here the extra part S comes from the constant part of Z(g, φ)U . Similar to the compact case, it is easy to have X P = − φ(0, u) LU , u∈Q×

Q

−

=

X X

φ(x, u) ωXU ,

u∈Q× x∈Q×

S

X

=

W0 (u, φ) LU .

u∈Q×

The expression of R involves cusps. Recall that X ∆∗ Z(x)U = [Q[y]× U ∩ U xU : U ] C(y)U × y∈Ad(U )\Bad f,0 /Q

+

X

X

τ (γ) hhiU .

γ∈P (Q)+ /(P (Q)+ ∩hU h−1 ) h∈P (Q)+ \B× f /U h−1 γh∈U xU

Accordingly, write R = Rcm + Rcusp where Rcm (resp. Rcusp ) denotes the contribution of CM points (resp. cusps) in R. The expression for Rcm is still as before. Now we consider X X Rcusp = wU φ(x)a a∈Q× x∈U \(B× −Q× U )/U f

X h∈P (Q)+ \B× f /U

By the substitution x = h−1 γh, X X Rcusp = wU

X

τ (γ) hhiU .

γ∈P (Q)+ /(P (Q)+ ∩hU h h−1 γh∈U xU

X

a∈Q× h∈P (Q)+ \B× /U γ∈P (Q)+ /(P (Q)+ ∩hU h−1 ) f / ×U h−1 γh∈Q

−1

)

τ (γ) φ(h−1 γh)a hhiU .

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CHAPTER 4

It is easy to have Rcusp = D(1, φ)U + Rcusp,0 with D(1, φ)U given in the statement of the proposition and X X X Rcusp,0 = −wU a∈Q× h∈P (Q)+ \B× /U γ∈P (Q)+ /(P (Q)+ ∩hU h−1 ) f h−1 γh∈Q× U

τ (γ) φ(h−1 γh)a hhiU . The summation on γ is just γ ∈ Q× /(Q× ∩ hU h−1 ) = Q× /µU . Here µU ⊂ {±1} with wU . We further have τ (γ) = 1 since γ is a scalar matrix. Thus X X X φ(z, u)a hhiU . Rcusp,0 = − u∈Q× z∈Q×

h∈P (Q)+ \B× f /U

The formula of the Hodge bundle in the non-compact case is   X X 1 LU = ωXU + 1− C(y)U + ey ad ×

hhiU .

h∈P (Q)+ \B× f /U

y∈Ad(U )\Bf,0 /Q

The extra part is the cusps, each of which has multiplicity one in the formula. It is easy to see that Rcusp,0 exactly serves as this extra part for the constant term in the original pull-back formula. More precisely, we have X P + Q + Rcm + Rcusp,0 = θ(1, u, φ0 ) C(1, u, φ0 )U . u∈Q×

 Remark. The generating function C(g, u, φ0 )U is automorphic on the open part XU◦ , but not automorphic on the compactification XU any more. In particular, its degree is no longer automorphic. 4.6.2

Some coset identities

The degree of C(g, u, φ0 )U is still given by Proposition 4.9. It remains to compute the degree of the cusp part D(g, φ)U . We first prove some expressions related to the set P (Q)+ \B× f /U of cusps on XU . The result even gives a way to rewrite the cusp part of Lemma 4.10. Lemma 4.12. The following are true over F = Q: (1) For any open compact subgroup C of Af , Af = Q + C,

Af /C = Q/(Q ∩ C).

143

TRACE OF THE GENERATING SERIES

(2) Let U be an open compact subgroup of GL2 (Af ) contained in the maximal b Then compact subgroup U 0 = GL2 (Z). 0 b P (Q)+ \GL2 (Af )/U = N (Z)\U /(U · {±1}).

(3) Let Ψ : GL2 (Af ) → C be a compactly supported function bi-invariant under U . Then X X τ (γ) Ψ(h−1 γh) hhiU h∈P (Q)+ \GL2 (Af )/U γ∈P (Q)+ /(P (Q)+ ∩hU h−1 )

1 2

=

X h∈U 0 /U

hhiU

X x=(x1 ,x2 )∈A(Q)+

  x1 −1 Ψ h

Z min(|x1 |∞ , |x2 |∞ ) Af

b x2

  h db.

Here U 0 is as in (2), and τ is as in Lemma 4.10. Proof. The first identity in (1) is the standard strong approximation. The second identity in (1) follows from the first one. Now we consider (2). We first claim that P (Q)+ \GL2 (Af )/U = (P (Q)+ N (Af ))\GL2 (Af )/U. It is equivalent to P (Q)+ N (Af )hU = P (Q)+ hU,

∀h ∈ GL2 (Af ).

Here both sides are viewed as subsets of GL2 (Af ). It suffices to prove N (Af )hU h−1 = N (Q)hU h−1 ,

h ∈ GL2 (Af ).

In fact, N (Q) · hU h−1 = N (Q) · (N (Af ) ∩ hU h−1 ) · hU h−1 = N (Af ) · hU h−1 . Here the second equality follows from (1) by identifying N ∼ = Ga . It proves the claimed result. Go back to P (Q)+ \GL2 (Af )/U . We will apply the Iwasawa decomposition GL2 (Af ) = P (Af )U 0 . It is easy to see P (Q)+ N (Af )U 0 = N (Af )A(Q)+ U 0 = N (Af )A(Af )U 0 = P (Af )U 0 . It follows that P (Q)+ \GL2 (Af )/U = (P (Q)+ N (Af ))\P (Af )U 0 /U = (P (Q)+ N (Af ))\P (Q)+ N (Af )U 0 /U = (P (Q)+ N (Af ) ∩ U 0 )\U 0 /U.

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CHAPTER 4

b The result is proved. It is easy to verify P (Q)+ N (Af ) ∩ U 0 = {±1} · N (Z). Now we prove (3). We start with the left-hand side. First consider the summation on γ. For any matrix in P (Q)+ ∩ hU h−1 , the coefficients on its diagonal are units of Z. They must be (simultaneously) 1 or −1. It follows that P (Q)+ ∩ hU h−1 = µU · (N (Q) ∩ hU h−1 ). It follows that P (Q)+ /(P (Q)+ ∩ hU h−1 ) =


(A(Q)+ /µU ) × N (Q)/(N (Q) ∩ hU h−1 ) .

Apply also the identity in (2). The left-hand side of (3) is equal to X X X τ (x) Ψ(h−1 xnh) hhiU 0 /(U ·{±1}) x∈A(Q)+ /µU n∈N (Q)/(N (Q)∩hU h−1 ) b h∈N (Z)\U

=

=

1 2 1 2

X

X

X

0 /U x∈A(Q)+ b h∈N (Z)\U

X

τ (x) Ψ(h−1 xnh) hhiU

n∈N (Q)/(N (Q)∩hU h−1 )

X

X

τ (x) Ψ(h−1 xnh) hhiU .

0 /U x∈A(Q)+ n∈N (Af )/(N (Af )∩hU h−1 ) b h∈N (Z)\U

Here the second identity uses the second result in (1). On the other hand, the right-hand side of (3) is equal to X 1 X hhiU min(|x1 |∞ , |x2 |∞ ) 2 h∈U 0 /U x=(x1 ,x2 )∈A(Q)+     Z x1 bx1 −1 Ψ h h db ·|x1 |Af x2 Af Z X 1 X hhiU min(1, x2 /x1 ) Ψ(h−1 xnh)dn = 2 N (A ) f 0 h∈U /U

=

1 2

X

x=(x1 ,x2 )∈A(Q)+

hhiU

h∈U 0 /U

X

τ (x)

x∈A(Q)+

X

Ψ(h−1 xnh).

b n∈N (Af )/N (Z)

It remains to verify that, for any x ∈ A(Q)+ , X X

Ψ(h−1 xnh) hhiU

0 /U n∈N (Af )/(N (Af )∩hU h−1 ) b h∈N (Z)\U

=

X

X

h∈U 0 /U

b n∈N (Af )/N (Z)

Ψ(h−1 xnh) hhiU .

It is mentally easier to start with the right-hand side. In fact, the right-hand side is equal to X X X Ψ(h0−1 xnh0 ) hh0 iU . 0 /U h0 ∈N (Z)hU/U b b b h∈N (Z)\U n∈N (Af )/N (Z)

145

TRACE OF THE GENERATING SERIES

Note that b b b ∩ hU h−1 )) · h. N (Z)hU/U = (N (Z)/(N (Z) The triple sum becomes X

X

X

Ψ(h−1 n0−1 xnn0 h) hn0 hiU

0 /U n0 ∈N (Z)/(N b b b b h∈N (Z)\U (Z)∩hU h−1 ) n∈N (Af )/N (Z)

=

X 0 /U b h∈N (Z)\U

=

X

b : (N (Z) b ∩ hU h−1 )] [N (Z)

Ψ(h−1 xnh) hhiU

b n∈N (Af )/N (Z)

X

X

Ψ(h−1 xnh) hhiU .

0 /U n∈N (A )/(N (Z)∩hU b b h∈N (Z)\U h−1 ) f

It finishes the proof since b ∩ hU h−1 = N (Af ) ∩ hU h−1 N (Z) by the fact that h ∈ U 0 .



Remark. The result in Lemma 4.12 (3) “simplifies” the expression of Lemma 4.10 (2). In fact, set Ψ to be the characteristic function of U xU in Lemma 4.12 (3). We obtain X [F [y]× U ∩ U xU : U ] C(y)U ∆∗ Z(x)U = × y∈Ad(U )\Bad f,0 /F

+

1 2

X

hhiU

h∈U 0 /U

Z ·

1U xU Af

X

min(|y1 |∞ , |y2 |∞ )

y=(y1 ,y2 )∈A(Q)+

  y1 h−1

b y2

  h db.

For example, let Tn be the standard Hecke correspondence on XU , where n is a positive integer such that U is maximal at any prime factor p of n. Assume that n is not a perfect square so that ∆ · Tn is a proper intersection. Then the multiplicity of h1iU in ∆∗ Tn is exactly X min(d, n/d). d|n, d>0

4.6.3

Degree of the pull-back

Denote by Vhyp = (Q2 , q) with q(x1 , x2 ) = x1 x2 the hyperbolic quadratic space over Q. It is naturally embedded as a subspace of the matrix algebra (M2 , q) with q = det as diagonal matrices. There is a natural equivariant map from S(M2 (Af ) × Af × ) to S(Vhyp (Af ) × Af × ), which already appears in Proposition 2.10. The degree of the pull-back is best described using this map. Here we review it in more detail.

146

CHAPTER 4

Equivariant maps Let k be a non-archimedean local field. Define a map ` : S(M2 (k) × k × )

−→

S(Vhyp (k) × k × )

by sending any φ ∈ S(M2 (k) × k × ) to  Z Z  1 x1 −1 2 (`φ)((x1 , x2 ), u) = |u| φ h GL2 (Ok )

k

b x2



 h, u dbdh.

In the case x1 6= x2 , a “more intrinsic” expression is   Z x1 3 12 −1 (`φ)((x1 , x2 ), u) = |udk | |x1 − x2 | φ h A(k)\GL2 (k)

 x2

 h, u dh.

1

It occurs in the proof of Proposition 2.10. Here |dk | 2 appears as vol(Ok ). In fact, by the Iwasawa decomposition, the second expression is equal to    Z  Z 1 x1 (x1 − x2 )b −1 2 |u| |x1 − x2 | h, u dbdh φ h x2 GL2 (Ok ) k    Z Z  1 x1 b φ h−1 = |u| 2 h, u dbdh. x2 GL2 (Ok ) k Using the same formulae, we also have a map `0 : S(M2 (k)0 × k × )

−→

S(Vhyp (k)0 × k × ).

Here M2 (k)0 is the subspace of trace-free elements, and Vhyp (k)0 is the subspace of elements (x1 , x2 ) of Vhyp (k) with trace x1 + x2 = 0. We have Vhyp (k)0 ∼ = (k, q − ) with q − (z) = −z 2 . If φ = φ0 ⊗φ0 is a decomposition respecting M2 (k) = k ⊕ M2 (k)0 , then `φ = φ0 ⊗ `0 φ0 . Proposition 4.13. (1) The map ` (resp. `0 ) is equivariant under the acf 2 (k)) as the symplectic similitude group via the tion of GL2 (k) (resp. SL Weil representation. (2) The images of the maps are Im(`0 ) Im(`)

{φ0 ∈ S(Vhyp (k)0 × k × ) : φ0 ((z, −z), u) = φ0 ((−z, z), u)}, = {φ0 ∈ S(Vhyp (k) × k × ) : φ0 ((x1 , x2 ), u) = φ0 ((x2 , x1 ), u)}.

=

Proof. We first consider (1). The following result is standard. Take ` for example. In the definition of `, the action of h commutes with the action of GL2 (k) (as the symplectic group). Thus we only need to check `0 (r(g)φ) = r(g)(`0 φ),

g ∈ GL2 (k)

147

TRACE OF THE GENERATING SERIES

for 0

(` φ)((x1 , x2 ), u) = |u|

1 2



Z φ k

x1

b x2



 , u db.

By definition of the Weil representation, it is easy to check the results for g = m(a), d(a) and n(b). It remains to consider the action of the Weyl element 0 where w, which is essentially the Fourier transform. Write (M2 , q) = Vhyp ⊕Vhyp   y 1 0 2 Vhyp = (k , −q) is realized as the space of matrices of the form . It y2 suffices to check the result for the case φ = φ(1) ⊗φ(2) with respect to the decom0 . The action of w respects this decomposition. position (M2 , q) = Vhyp ⊕ Vhyp Thus it remains to check       Z Z b b (2) (2) φ , u db = r(w)φ , u db. 0 0 k k The right-hand side is equal to  Z Z (2) φ k

Z =

0 (k) Vhyp

φ(2)

 0

k

y1



y2   y1 , u dy1 .

 , u ψ(−u(by2 ))dy1 dy2 db

Here we used the inversion formula to collapse a double integration. Now we consider (2). It suffices to prove that for `0 . Note that S(M2 (k)0 × k × ) S(Vhyp (k)0 × k × )

S(M − 2(k)0 ) ⊗ S(k × ), S(Vhyp (k)0 ) ⊗ S(k × ).

= =

It is reduced to prove that `0 (S(M2 (k)0 )) = {φ0 ∈ S(Vhyp (k)0 ) : φ0 (z, −z) = φ0 (−z, z)}. Here by abuse of notation, `0 : S(M2 (k)0 ) → S(Vhyp (k)0 ) denotes the map given by    Z  Z z b φ h−1 h dbdh. (`φ0 )(z, −z) = −z GL2 (Ok ) k Use the alternative definition Z 1 (`φ0 )(z, −z) = |d3k | 2 |2z|

  z −1 φ h

A(k)\GL2 (k)

 By a change of variable h →

1 1

 h, we obtain

(`φ0 )(z, −z) = (`φ0 )(−z, z).

−z

  h dh.

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CHAPTER 4

It follows that `0 (S(M2 (k)0 )) ⊂ {φ0 ∈ S(Vhyp (k)0 ) : φ0 (z, −z) = φ0 (−z, z)}. The other direction is implied by the fact that the right-hand side is an irref 2 (k) under the Weil representation. In fact, it is the ducible representation of SL f 2 (k). One Howe lifting of the trivial representation of O(Vhyp (k)0 ) = {±1} to SL can also check the irreducibility explicitly.  Go back to the global case. Define ` = ⊗p `p : S(M2 (Af ) × Af × )

−→

S(Vhyp (Af ) × Af × ).

It is a well-defined equivariant map since 1 × 1 ζp (2) Vhyp (Zp )×Zp

`p (1M2 (Zp )×Z× )= p and the product converges absolutely. Degree of the cusps

b be the standard maximal compact Proposition 4.14. Let U 0 = GL2 (Z) × subgroup of Bf = GL2 (Af ) containing U . Then for any g ∈ GL2 (A), deg D(g, φ)U =

X

π 2 vol(XU )

1

X

2 r(g)`φ(x, u) |u|∞ min(|x1 |∞ , |x2 |∞ ).

u∈Q× x=(x1 ,x2 )∈A(Q)+

Here r(g)`φ(x, u) := r(gf )`φf (x, u) · r(g∞ )φ∞ (1, uq(x)). Proof. Recall from Proposition 4.11 that D(g, φ)U X = wU

X

X

r(g)φ(h−1 γh)a τ (γ) hhiU .

a∈Q× h∈P (Q)+ \B× /U γ∈P (Q)+ /(P (Q)+ ∩hU h−1 ) f

It suffices to compute the degree of the a-th coefficient

=

Da (g, φ)U X wU

X

h∈P (Q)+ \B× f /U

γ∈P (Q)+ /(P (Q)+

r(g)φ(h−1 γh)a τ (γ) hhiU . ∩hU h−1 )

By Lemma 4.12 (3), Da (g, φ)U

=

wU 2

X h∈U 0 /U

Z · Af

hhiU

X

min(|x1 |∞ , |x2 |∞ )

x=(x1 ,x2 )∈A(Q)+

  x1 −1 r(g)φ h

b x2

  h db. a

149

TRACE OF THE GENERATING SERIES

Taking degree, hhiU becomes 1, and the summation over h becomes an integration on U 0 . Therefore, deg Da (g, φ)U X wU = 2 vol(U )

1 2 |u(x)|∞ min(|x1 |∞ , |x2 |∞ ) r(g)`φ(x)a .

x=(x1 ,x2 )∈A(Q)+

Here u(x) = a/(x1 x2 ). Now we want to convert the coefficient in terms of vol(XU ). We claim that (for F = Q) wU . (4.6.1) vol(XU ) = 2π 2 vol(U ) Once it is true, then deg Da (g, φ)U =

1

X

π 2 vol(XU )

2 |u(x)|∞ min(|x1 |∞ , |x2 |∞ ) r(g)`φ(x)a .

x=(x1 ,x2 )∈A(Q)+

It is an expression as in the proposition. It remains to verify (4.6.1). One can first verify it for maximal U , and derive the general case by comparing different U . Alternatively, it can be derived from (4.5.1) in the proof of Proposition 4.9 which asserts that vol(XU ) = π −2 |Q× \A× f /UZ |

vol(UZ ) . vol(U )

It gives (4.6.1) since |Q

×

\A× f /UZ |

=

vol(Q× \A× f ) vol((UZ ∩ Q× )\UZ )

=

1/2 wU . = vol(UZ )/wU 2 vol(UZ )

It finishes the proof.



Degree of the pull-back Recall that in the compact case, Proposition 4.5 gives deg ∆∗ Z(g, φ)U

=

−vol(XU ) J(0, g, φ)U ,

Here the mixed Eisenstein–theta series X J(s, g, φ)U = δ(γg)s γ∈P (F )\GL2 (F )

X

X

u∈µ2U \F ×

x∈F

g ∈ GL2 (A).

r(γg)φ(x, u).

In the case φ = φ0 ⊗ φ0 under the orthogonal decomposition B = A ⊕ B0 , one has X J(s, g, u, φ)U = θ(g, u, φ0 )E(s, g, u, φ0 ). u∈µ2U \F ×

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CHAPTER 4

In the non-compact case here, the equality is not true, and the difference of the two sides will be given by some summation on the hyperbolic space Vhyp . Theorem 4.15. Assume that F = Q and Σ = {∞}. For any g ∈ GL2 (A), deg ∆∗ Z(g, φ)U

=

−vol(XU ) J(0, g, φ)U + vol(XU ) B(g, φ),

where B(g, φ) :

=

X

X

r(gf )`φf (x, u) β φ∞ (g∞ , x, u)

u∈Q× x∈Vhyp

−

X

1

0 W0,∞ (0, g∞ , u, φ∞ ) |u|A2 f

Z r(gf )`φf ((0, x2 ), u)dx2 . Af

u∈Q×

Here in the case φ∞ = φ0∞ ⊗ φ0,∞ , the archimedean part β φ∞ (g, x, u)

1 0 = − √ W−u(x (0, g, u, φ0,∞ ) r(g)φ0∞ (1, u(x1 + x2 )2 /4) 2 1 −x2 ) /4 2 2γu 1

+π 2 |u| 2 min(|x1 |, |x2 |) 1R+ (x1 x2 ) r(g)φ∞ (1, ux1 x2 ), ∀ x = (x1 , x2 ) ∈ R2 , u ∈ R× , g ∈ GL2 (R). Here γu denotes the Weil index γ(B0,∞ , uq) = e

3πi sgn(u) 4

.

Proof. It is easy to reduce the problem to the case g ∈ SL2 (A). Or equivf 2 (A) since the quadratic spaces in both sides are alently, assume that g ∈ SL even-dimensional. Of course, in the proof we will meet odd-dimensional space f 2 (A) really makes a difference. By linearity, it suffices to prove the since SL result for the split case φ = φ0 ⊗ φ0 . Recall that Proposition 4.11 gives ∆∗ Z(g, φ)U X X = θ(g, u, φ0 )C(g, u, φ0 )U + W0 (0, g, u, φ) LU + D(g, φ)U . u∈Q×

u∈Q×

The three terms on the right-hand side are the contributions respectively from the CM points, the Hodge bundle and the cusps. The generating function C(g, u, φ0 )U = −r(g)φ0 (0, u) LU X + y∈Ad(U )\Bad f,0

1 r(g)φ0 (y, u) C(y)U [F [y]× ∩ U : µU ]

has the same expression as in Proposition 4.8. In particular, the proof of Proposition 4.9 applies to deg C(g, u, φ0 )U and gives deg C(g, u, φ0 )U = −vol(XU ) r(g)φ0 (0, u) Z X + vol(XU ) L(1, η−ua ) a∈Q×

× B× xa \B

r(g, (h, h))φ0 (xa , u)dh.

151

TRACE OF THE GENERATING SERIES

Then Proposition 2.10 exactly gives deg C(g, u, φ0 )U = −vol(XU ) E hol (0, g, u, φ0 ). Here E hol (0, g, u, φ0 ) is the holomorphic part of E(0, g, u, φ0 ). It follows that deg ∆∗ Z(g, φ)U + vol(XU ) J(0, g, φ)U X = deg D(g, φ) − vol(XU ) W0 (0, g, u, φ) u∈Q×

+vol(XU )

X

θ(g, u, φ0 )E nhol (0, g, u, φ0 ).

u∈Q×

We are going to rewrite these three terms on the right-hand side. By Proposition 4.14, deg D(g, φ)U =

1

X

X

π 2 vol(XU )

2 |u|∞ min(|x1 |∞ , |x2 |∞ ) r(g)`φ(x, u).

u∈Q× x=(x1 ,x2 )∈A(Q)+

On the other hand, by Proposition 2.10, E nhol (0, g, u, φ0 )

−

=

1 0 √ W−uz 2 ,∞ (0, g∞ , u, φ0,∞ ) r(gf )`0 φ0,f (z, u). 2 2γ u z∈Q X

It follows that X

θ(g, u, φ0 )E nhol (0, g, u, φ)

u∈Q×

=

=

−

1 0 √ W−uz 2 ,∞ (0, g∞ , u, φ0,∞ ) 2 2γ u × x∈Q z∈Q u∈Q X XX

r(g∞ )φ0∞ (x, u) r(gf )`φf (x + z, x − z, u) X X 1 0 √ − W−u(x (0, g∞ , u, φ0,∞ ) 2 1 −x2 ) /4,∞ 2 2γ u u∈Q× x ,x ∈Q 1

2

r(g∞ )φ0∞ (

x1 + x2 , u) r(gf )`φf (x1 , x2 , u). 2

It remains to treat W0 (0, g, u, φ). By Proposition 2.9, Y 0 ◦ W0 (0, g, u, φ) = W0,v (0, g, u, φv ) W0,v (0, g, u, φv ) v-∞

where for any v 6= ∞, ◦ (0, g, u, φv ) W0,v

Z

Z



Z

= |u|v

r(g)φv SL2 (Zv )

Qv

Qv

0 0

x y



 h1 , u dxdydh1 .

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CHAPTER 4

We claim that 1 2

◦ W0,v (0, g, u, φv )

Z

= |u|v

r(g)`φv ((0, y), u)dy. Qv

For that it suffices to verify Z Z Z



0 0

r(g)φv SL2 (Zv )

Qv

Z

Z

Qv

Start with the right-hand side. Write h

h

 h1 , u dxdydh1

    0 x −1 r(g)φv h h, u dxdydh. 0 y Qv −1

−1



Z

= GL2 (Zv )

Qv

x y



0 x 0 y



 =

 =

a c

0 ax + by 0 cx + dy

b d 

 , then we have

.

A change of variable (x0 , y 0 ) = (ax + by, cx + dy) transfers the right-hand side to    Z Z Z 0 x r(g)φv h, u dxdydh. 0 y GL2 (Zv ) Qv Qv It only remains to change the domain of the integration from GL2 (Zv ) to 0 SL2 (Zv ). Write h ∈ GL2 (Zv ) as d(a)h0 with a ∈ Z× v and h ∈ SL2 (Zv ). The last triple integral becomes    Z Z Z Z 0 ax 0 h , u dxdydadh r(g)φv 0 ay SL2 (Zv ) Z× Qv Qv v    Z Z Z 0 x r(g)φv = h0 , u dxdydh. 0 y SL2 (Zv ) Qv Qv It finishes the proof.



What is needed for Proposition 4.4 Theorem 4.15 asserts deg ∆∗ Z(g, φ)U

=

−vol(XU ) J(0, g, φ)U + vol(XU ) B(g, φ),

with the “extra series” B(g, φ) a new part in the non-compact case. Recall that Proposition 4.4 is proved in the compact case by Proposition 4.5. To finish it in the non-compact case, it remains to prove Z ∗ ϕ(g) B(g, φ) dg = 0, ∀ ϕ ∈ σ, φ ∈ S(B × A× ). (4.6.2) GL2 (Q)\GL2 (A)/Z(A)

153

TRACE OF THE GENERATING SERIES

We can assume that φ∞ is the standard Gaussian since S(B∞ ×R× ) is generated by the standard one under the action of the Lie algebra gl2 (R). Before considering the integration, a basic question is to explain the modularity of X X B(g, φ) = r(gf )`φf (x, u) β φ∞ (g∞ , x, u) u∈Q× x∈Vhyp

−

X

1

0 W0,∞ (0, g∞ , u, φ∞ ) |u|A2 f

Z r(gf )`φf ((0, x2 ), u)dx2 . Af

u∈Q×

It is automorphic since it is the difference of two automorphic forms by Theorem 4.15. It is reasonable to expect a more direct interpretation, and such an interpretation should lead to a proof of (4.6.2). The double sum in the series B(g, φ) looks like a theta series (on Vhyp ), except that the archimedean part β φ∞ (g∞ , x, u) is not defined in terms of the Weil representation. However, as we will see in the following section, the archimedean part surprisingly transfers according to the Weil representation by the work of Hirzebruch–Zagier [HZ] on Hilbert modular surfaces. It makes the double sum in B(g, φ) appear as a theta series. The theta series is “singular” in that the archimedean part, even transferring according to the Weil representation, is much worse than a Schwartz function. It is also different from pseudo-theta series coming from derivatives and local heights we will see in this book. Our alternative approach is to use partial Fourier transform to rewrite the series in a simpler form. Then we eventually write the whole series B(g, φ), including the sum with the constant part, as a Poincar´e series up to analytic continuation. Then the modularity is proved and equation (4.6.2) follows immediately. 4.7

INTERPRETATION: NON-COMPACT CASE

As described at the end of the last section, the goal of this section is to reinterpret the series B(g, φ) and finish the proof of Theorem 3.22. 4.7.1

The archimedean part

Here we rewrite the archimedean part β φ∞ (g∞ , x, u) of B(g, φ) by translating a result of Hirzebruch–Zagier [HZ]. The work of Hirzebruch–Zagier Hirzebruch–Zagier [HZ] studied the intersection numbers of Hecke operators on Hilbert modular surfaces. Though written in a different language, their treatment is very similar to our case. In terms of the language of this book, the main result of [HZ] is roughly as follows. Fix a real quadratic field K over Q (of prime discriminant), and denote by M ◦ = SL2 (OK )\H2

154

CHAPTER 4

the Hilbert modular surface associated to K. There is a generating function X ZHZ (z) = TN e2πiN τ , z ∈ H. N ≥0

Each coefficient TN is an explicit special curve in M analogous to the classical Hecke correspondence on X(1) × X(1). In particular, T1 is just the image of the modular curve X(1) diagonally embedded in M . Denote by M the minimal desingularization of the compactification M ◦ ∪ {cusps}. There is a canonical extension TNc of TN to a divisor in M , and thus c (z) of ZHZ (z) to M . The paper computed there is a canonical extension ZHZ c c (z). One main result of the paper is the the intersection number of T1 with ZHZ expression c (z) ≈ θ(z)F(z) + W(z). T1c · ZHZ Here θ is the classical holomorphic theta series of weight 1/2, F(z) is Zagier’s almost holomorphic Eisenstein series of weight 3/2, and W(z) is a (nonholomorphic) theta function defined by the real quadratic space K. Here we write “≈” since the equality is up to certain linear operations. c (z) is a modular form of weight two in Then [HZ] concluded that T1c · ZHZ c (z). At that time, the z ∈ H. It also obtained the modularity of TNc · ZHZ c modularity of ZHZ (z) or ZHZ (z) (even as a cohomology class) was not known. The result of [HZ] is a landmark for the succeeding works [KM1, KM2, KM3, GKZ, Bor, Zha, YZZ] on modularity of generating series of special cycles on Shimura varieties. The counterparts of K,

M,

T1c ,

c ZHZ (z),

θ(z),

F(z),

W(z)

in this book are respectively Q2 ,

XU × XU ,

∆,

Z(g, φ)U ,

θ(g, φ0 ),

E(0, g, φ0 ),

B(g, φ).

The archimedean part of B(g, φ) is essentially the same as that of W(z). So we can apply the corresponding result of [HZ] to it. The archimedean part Assume that the archimedean parts φ∞ , φ0,∞ and φ0∞ are standard. Here we treat the archimedean part

=

β φ∞ (g, x, u) 1 0 − √ W−u(x (0, g, u, φ0,∞ ) r(g)φ0∞ (1, u(x1 + x2 )2 /4) 2 1 −x2 ) /4 2 2γu 1

+π 2 |u| 2 min(|x1 |, |x2 |) 1R+ (x1 x2 ) r(g)φ∞ (1, ux1 x2 ).

155

TRACE OF THE GENERATING SERIES

Recall that in Corollary 2.12 we have obtained √ 0 (0, 1, u, φ0 ) = γu 8 2π 2 e−2πa β(−4πa), Wa,∞ Here β, as introduced in [HZ], is given by Z ∞ 3 1 β(z) = e−zt t− 2 dt, 16π 1

u > 0, a ≤ 0.

Re(z) ≥ 0.

It follows that for g = 1, u > 0, β φ∞ (1, x, u) =

 1  π 2 e−2πux1 x2 u 2 min(|x1 |, |x2 |) 1R+ (x1 x2 ) − 4β(πu(x1 − x2 )2 ) .

If u < 0, then β φ∞ (1, x, u) = 0 automatically. For simplicity, denote β(x, u) := β φ∞ (1, x, u),

(x, u) ∈ Vhyp (R) × R× .

It is a continuous function on Vhyp (R) × R× , integrable in x for any fixed u. It is not a Schwartz function in the usual sense because it is not differentiable around x1 x2 = 0. Nonetheless, by the formulae of the Weil representation, the function r(g)β is a well-defined continuous function on Vhyp (R) × R× for any g ∈ GL2 (R). The following is a surprising result of [HZ] which makes their series W(z) a theta series. We state it in the current language. Proposition 4.16. Assume that φ∞ is the standard Gaussian. For any g ∈ GL2 (R) and (x, u) ∈ Vhyp (R) × R× , β φ∞ (g, x, u) = r(g)β(x, u). Proof. It is essentially Proposition 1 in §2.3 of [HZ]. The equality holds for g = 1 by definition and we need to extend it to the general case. It is easy to verify that β φ∞ (n(b)g, x, u) β φ∞ (m(a)g, x, u) β φ∞ (d(a)g, x, u)

= β φ∞ (g, x, u) ψ(buq(x)), = β φ∞ (g, ax, u) |a|, = β φ∞ (g, x, a

−1

u) |a|

b ∈ R, ×

a∈R , − 12

,

a ∈ R× .

The formulae match the following formula r(n(b)g)β(x, u) = r(g)β(x, u) ψ(buq(x)), b ∈ R, × r(m(a)g)β(x, u) = r(g)β(ax, u) |a|, a∈R , r(d(a)g)β(x, u)

=

1

r(g)β(x, a−1 u) |a|− 2 ,

a ∈ R× .

156

CHAPTER 4

It follows that if the proposition is true for some g, then it is true for all elements of P (R)g. By the Bruhat decomposition GL2 (R) = P (R) ∪ P (R)wN (R), we only need to verify ∀ b ∈ R.

β φ∞ (wn(b), x, u) = r(wn(b))β(x, u),

We can assume that u > 0 since both sides are zero for u < 0. We need to prove that the Fourier transform of r(n(b))β(x, u) is exactly equal to β φ∞ (wn(b), x, u). It is easy to have r(n(b))β(x, u) β(x, u) ψ(buq(x))  1  = π 2 e2πiu(b+i)x1 x2 u 2 min(|x1 |, |x2 |) 1R+ (x1 x2 ) − 4β(πu(x1 − x2 )2 ) . =

Now we compute β φ∞ (wn(b), x, u). The function β φ∞ (g, x, u) is of weight two under the right action of SO(2, R) on g. Use the Iwasawa decomposition 

1 −1

with r =



1

b 1



 =

1

−b/r2 1





1/r r

−b/r −1/r

1/r −b/r



√ b2 + 1. We obtain β φ∞ (wn(b), x, u) = =

(−b + i)2 β φ∞ (n(−b/r2 )m(1/r), x, u) r2 (−b + i)2 2πiur−2 (−b+i)x1 x2 π2 e r4 

 1 · u 2 min(|x1 |, |x2 |) 1R+ (x1 x2 ) − 4r β(πur−2 (x1 − x2 )2 ) . By a simple change of variable, it suffices to assume that u = 1. Denoting z = b + i, in terms of the notation of Proposition 1 in [HZ, §2.3], we simply have r(n(b))β(x, u) β φ∞ (wn(b), x, u)

−2π 2 Wz (x1 , x2 ), = −2π 2 z −2 W−1/z (x1 , x2 ).

=

Then the result we need is exactly the result of Proposition 1 in [HZ, §2.3].



157

TRACE OF THE GENERATING SERIES

Poisson summation formula Go back to the series B(g, φ). Assume that φ∞ is standard. By Proposition 4.16, we have X X B(g, φ) = r(gf )`φf (x, u) r(g∞ )β(x, u) u∈Q× x∈Vhyp

−

X

0 W0,∞ (0, g∞ , u, φ∞ )

1 2

Z

|u|Af

r(gf )`φf ((0, x2 ), u)dx2 . Af

u∈Q×

A rough thought is that the double sum X X r(gf )`φf (x, u) r(g∞ )β(x, u) u∈Q× x∈Vhyp

is a theta series on Vhyp . Then the proof of (4.6.2) would follow the general fact that the (global) theta lifting of any cuspidal automorphic representation of GL2 (A) to GSO(Vhyp (A)) = A× × A× is zero. However, this argument is NOT true since it simply ignores the singularity of β(x, u) and the presence of the second term in B(g, φ). In fact, it is very easy to see that the second term Z X 1 0 W0,∞ (0, g∞ , u, φ∞ ) |u|A2 f r(gf )`φf ((0, x2 ), u)dx2 , Af

u∈Q×

an expression for the intertwining part W0 (0, g, φ) of the Eisenstein series E(0, g, φ), is not automorphic in g (unless it is identically zero). We know that B(g, φ) is automorphic. So the double sum is not automorphic in general. It cannot be treated as a theta series in the usual way. The problem is caused by the slow decay of β and the singularity of β around x1 x2 = 0. They violate the Poisson summation formula used in the proof of the modularity of theta series. Next, we recall a version of the Poisson summation formula. Let (V, q) be a quadratic space over Q. For an integrable function Φ : V (A) → C, recall its Fourier transform Z b Φ(y)ψ(hx, yi)dy. Φ(x) = V (A)

Here the integration uses the self-dual Haar measure, and the pairing hx, yi = q(x + y) − q(x) − q(y) is as usual. The Poisson summation formula, which only holds for suitable Φ, is X X b Φ(x) = Φ(x). x∈V

x∈V

In particular, it holds for the usual Schwartz–Bruhat functions on V (A), i.e., linear combinations of pure tensors ⊗v Φv where Φp ∈ S(V (Qp )) is locally constant and compactly supported for every finite place p, and Φ∞ ∈ S(V (R)) is

158

CHAPTER 4

infinitely differentiable and all its partial derivatives (of any orders) have the property of rapid decay. In the Poisson summation formula, the condition of being Schwartz–Bruhat is very strong. One can relax them in many ways. The following result improves it at the archimedean part. Lemma 4.17. The Poisson summation formula holds for Φ if Φ = ⊗v Φv where Φp ∈ S(V (Qp )) for every finite place p, and Φ∞ is a continuous and integrable function on V (R) satisfying the decay condition b ∞ (x)| < C(1 + |x|)− dim V − |Φ∞ (x)| + |Φ for some C > 0 and  > 0. Here | · | is the norm induced from the usual norm on Rdim V by any choice of an R-linear identification V (R) ' Rdim V . Proof. It can be easily reduced to the classical setting. Let VZ be any Z-lattice of V on which ψ∞ ◦ q is trivial. By linearity, we can assume that Φp = 1ap +pep VZp for every finite place p. Of course, ap = 0 and ep = 0 for almost all p. By a translation of the form x → x + a for some a ∈ V , we can assume that all ap = 0. By a contraction of the form x → kx for some k ∈ Q× , we can assume that all k = 0. Then Φp = 1VZp for every finite place p. Thus X

Φ(x) =

x∈V

X

Φ∞ (x).

x∈VZ

It is reduced to the Poisson summation for the lattice VZ in the real vector space V (R). See [Gra, Theorem 3.1.17] for example. It is easy to see that the proof goes through a general (non-degenerate) norm q on V (R).  Go back to the function β(x, u). Write β(x, u)

=

β 1 (x, u) − β 2 (x, u)

where β 1 (x, u) β 2 (x, u)

= =

1

π 2 e−2πux1 x2 u 2 min(|x1 |, |x2 |) 1R+ (u) 1R+ (x1 x2 ), 4π 2 e−2πux1 x2 β(πu(x1 − x2 )2 ) 1R+ (u).

Fix u > 0. It is easy to see that 2

2

β 2 (x, u) < 4π 2 e−πu(x1 +x2 ) . It decays very fast. The function β 1 (x, u) does not satisfy the growth condition in the lemma. In fact, taking x = (t, t−1 ) with t → ∞, then β 1 (x, u)

=

1

π 2 e−2πu u 2 t−1 ∼ |x|−1 .

It explains in some extent the failure of the Poisson summation formula for β(·, u).

159

TRACE OF THE GENERATING SERIES

In [HZ], the series W(z) is essentially a multiple of X

r(g∞ )β(x, 1).

x∈OK

Here g∞ is the unique element of the parabolic subgroup P 1 (R) with g∞ (i) = z. It does not have the counterpart of the constant part appearing in B(g, φ). Then [HZ] uses the Poisson summation formula on OK to conclude the modularity of W(z). One can verify that the Poisson summation formula holds in that case, even though the function β does not satisify the decay condition in Lemma 4.17. These different properties reflect the differences between the anisotropic K and the isotropic Q2 . 4.7.2

Partial Fourier transforms

Our idea is to use partial Fourier transforms to write B(g, φ) in a simpler form to make the modularity transparent. Partial Fourier transforms work here by two reasons: • The function β((x1 , x2 ), u), viewed as a function in x2 by fixing (x1 , u), satisfies the decay condition in Lemma 4.17. Thus the Poisson summation formula for x2 is valid, which gives a partial Poisson summation formula for x. • Under the partial Fourier transform, the action of GL2 via the Weil representation is transferred to the regular action. This intertwining property greatly simplifies the series. We will illustrate the idea by the example of the usual theta series (defined by Schwartz–Bruhat functions). Schwartz–Bruhat case Let k be a local field, and Vhyp be the hyperbolic quadratic space over k. Let S(Vhyp (k) × k × ) be the space of Schwartz–Bruhat functions. If k is nonarchimedean, it is just S(Vhyp (k) × k × ). If k is archimedean, it is much bigger than the space S(Vhyp (k) × k × ) we use more often. See §2.1 for more details. We only use the space S(Vhyp (k) × k × ) in this subsection. For any Φ ∈ S(Vhyp (k) × k × ), its partial Fourier transform (on the second variable) is defined by Z 1 e 2 Φ((x1 , y), u)ψ(ux2 y)dy. Φ((x1 , x2 ), u) = |u| k 1

Here the measure dy is the Haar measure on k in §1.6 and the factor |u| 2 makes it self-dual.

160

CHAPTER 4

Recall that GL2 (k) acts on S(Vhyp (k) × k × ) by the Weil representation r. The regular action ρ of GL2 (k) on S(Vhyp (k) × k × ) is given by ρ(g)Φ(x, u) = Φ(xg, det(g)−1 u). Here the action xg is the usual right multiplication of the matrix g on the row vector x = (x1 , x2 ). The following result is classical. Lemma 4.18. For any g ∈ GL2 (k) and Φ ∈ S(Vhyp (k) × k × ), ^ = ρ(g)Φ. e r(g)Φ In other words, the partial Fourier transform defines a GL2 (k)-intertwining map (S(Vhyp (k) × k × ), r)−→(S(Vhyp (k) × k × ), ρ). Proof. Use the definition of Weil representation to check the identity for m(a), n(b), d(a) and w. See also [JL, Proposition 1.6] for the case of S(Vhyp (k)).  The partial Fourier transform can be used to write a theta series on Vhyp as the sum of a residue form with a Poincar´e series. Let F be a number field and Φ ∈ S(Vhyp (A) × A× ) be a Schwartz–Bruhat function. We are going to rewrite the theta series X X θ(g, Φ) = r(g)Φ((x1 , x2 ), u), g ∈ GL2 (A). u∈F × (x1 ,x2 )∈Vhyp (F )

Use the Poisson summation on x2 by fixing (x1 , u). We get X x2 ∈F

r(g)Φ((x1 , x2 ), u) =

X

^ 1 , x2 ), u). r(g)Φ((x

x2 ∈F

^ to ρ(g)Φ. e It follows that Then the lemma permits us to change r(g)Φ X X e 1 , x2 ), u). θ(g, Φ) = ρ(g)Φ((x u∈F × (x1 ,x2 )∈F 2

Consider the right action of GL2 (F ) on F 2 × F × defined by g : ((x1 , x2 ), u) 7−→ ((x1 , x2 )g, det(g)−1 u). There are two orbits, represented respectively by ((0, 0), 1) and ((0, 1), 1). The stabilizers of these two representatives are respectively SL2 (F ) and N (F ). It follows that X X e e θ(g, Φ) = ρ(γg)Φ((0, 1), 1). Φ((0, 0), det(g)−1 u) + u∈F ×

γ∈N (F )\GL2 (F )

161

TRACE OF THE GENERATING SERIES

Both terms on the right-hand side are automorphic for g ∈ GL2 (A). The first term is a residue form in the sense that it is an automorphic form for det(g) ∈ GL1 (A). The second term is a Poincar´e series in that e e ρ(ng)Φ((0, 1), 1) = ρ(ng)Φ((0, 1), 1),

n ∈ N (A).

It is well-known that both of them are orthogonal to cusp forms. Archimedean part Go back to the function β((x1 , x2 ), u). For any g ∈ GL2 (R), the function r(g)β is continuous and integrable in x2 . So we can still define the partial Fourier transform Z ^ 1 , x2 ), u) = |u| 12 r(g)β((x1 , y), u)ψ(ux2 y)dy. r(g)β((x R

^ is defined as in the Schwartz case. The action ρ of GL2 (R) on r(g)β ^ u) We will see that the only discontinuous points of the function r(g)β(x, are given by x = 0. Thus Lemma 4.18 is true for β as long as x 6= 0. Lemma 4.19. The following are true for any g ∈ GL2 (R) and u ∈ R× . (1) For any (x1 , x2 ) 6= (0, 0), 2

2

−πu(x1 +x2 ) e 1 , x2 ), u) = 1 − e β((x 1R+ (u). 4u(x1 − ix2 )2

^ u) is continuous at all points (x, u) with x 6= 0. (2) The function r(g)β(x, (3) The function r(g)β((x1 , x2 ), u), viewed as a function of x2 ∈ R for any fixed (x1 , u), satisfies the decay condition of Φ∞ in Lemma 4.17. (4) If x 6= 0, then ^ u) = ρ(g)β(x, e u). r(g)β(x, (5) If x = 0, then ^ u) r(g)β(0,

=

W00 (0, g, u, φ∞ )

=

 Here we write g =

a c

b d

 .

π(ci − d) 1R (u det(g)), 4(ci + d) + π(ci − d) 1R (u det(g)). 2(ci + d) +

162

CHAPTER 4

Proof. Result (1) is obtained by explicit computations. We only sketch it here. Write β(x, u) = β 1 (x, u) − β 2 (x, u) where β 1 (x, u) β 2 (x, u)

= =

1

π 2 e−2πux1 x2 u 2 min(|x1 |, |x2 |) 1R+ (u) 1R+ (x1 x2 ), 4π 2 e−2πux1 x2 β(πu(x1 − x2 )2 ) 1R+ (u).

Explicit computation gives f ((x1 , x2 ), u) β 1

=

1 − e−2πux1 (x1 −ix2 ) 1R+ (u), 4u(x1 − ix2 )2

f ((x1 , x2 ), u) β 2

=

e−πu(x1 +x2 ) − e−2πux1 (x1 −ix2 ) 1R+ (u). 4u(x1 − ix2 )2

2

2

Here the second identity is obtained by replacing β by its definition Z ∞ 3 1 β(z) = e−zt t− 2 dt 16π 1 and changing the order of the integration. It gives 2

2

−πu(x1 +x2 ) e 1 , x2 ), u) = 1 − e 1R+ (u). β((x 4u(x1 − ix2 )2

It proves (1). e is continuous away from (0, 0). Expand the exponential The function β function. We have e 1 , x2 ), u) = π x1 + ix2 1R (u) + O(x2 + x2 ), β((x 1 2 4 x1 − ix2 +

(x1 , x2 ) → (0, 0).

(4.7.1)

It follows that all the directional limits exist. Then it verifies (2), (3) and (4) for g = 1. We first prove (2) and (3) for general g. By Proposition 4.16, β has weight two under the action r of O(2, R). Thus by the Iwasawa decomposition, we only need to consider the case g ∈ P (R). It is routine to verify that ^ = ρ(g)β, e r(g)β

∀g ∈ P (R).

e with g ∈ P (R). They It follows that we only need to prove the results for ρ(g)β can be easily reduced to g = 1. ^ u) and ρ(g)β(x, e u) Now we prove (4) for general g. Note that both r(g)β(x, are continuous at x 6= 0. It suffices to prove that, for each g and u, ^ u) = ρ(g)β(x, e u) r(g)β(x,

163

TRACE OF THE GENERATING SERIES

for almost all x ∈ R2 . The proof is similar to Lemma 4.18. To see the appearance of “almost all,” we sketch it here. The result for g = 1 is trivial. It is easy to see that if the result is true for g, then it is also true for all elements in the coset P (R)g. So it suffices to prove the result for the Weyl element g = w. Assume that u > 0 since both sides are zero if u < 0. By definition, ^ r(w)β((x 1 , x2 ), u) Z 1 = |u| 2 r(w)β((x1 , y), u) ψ(ux2 y)dy ZR Z Z 3 = |u| 2 β((z, t), u) ψ(ux1 t) ψ(uyz) ψ(ux2 y)dzdtdy R R R Z Z e = |u| β((z, x1 ), u) ψ(uyz) ψ(ux2 y)dzdy. R

R

Here the powers of |u| come out by different ways of normalizing the Haar measures. Using the inversion formula to contract the double integral, we obtain ^ e e r(w)β((x 1 , x2 ), u) = β((−x2 , x1 ), u) = ρ(w)β((x1 , x2 ), u). It holds at all (continuous points) (x1 , x2 ) 6= (0, 0). ^ u). By definition, r(g)β ^ It remains to prove (5). We first compute r(g)β(0, is the partial Fourier transform for x2 , so it is continuous in x2 . It follows that ^ u) = lim r(g)β((0, ^ r(g)β(0, x2 ), u). x2 →0

By (4), for x2 6= 0, we have −1 ^ e e x2 ), u) = β((cx u). r(g)β((0, x2 ), u) = ρ(g)β((0, 2 , dx2 ), det(g)

Taking the limit x2 → 0, by equation (4.7.1), we have ^ u) = π · c + id 1R (u det(g)) = π · ci − d 1R (u det(g)). r(g)β(0, 4 c − id + 4 ci + d + On the other hand, Proposition 2.11 gives π W00 (0, 1, u, φ∞ ) = − 1R+ (u). 2 It is easy to verify that W00 (0, γg, u, φ∞ ) = W00 (0, g, det(γ)−1 u, φ∞ ),

∀γ ∈ P (R).

Therefore, the weight-two property yields that π W00 (0, g, u, φ∞ ) = − e2iθ 1R+ (u det(g)). 2

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CHAPTER 4

Here θ comes from the Iwasawa decomposition    0  a b a b0 cos(θ) = c d d0 − sin(θ)

sin(θ) cos(θ)

 .

But it is easy to verify e2iθ = −

ci − d . ci + d

It finishes the proof. 4.7.3



Interpretation of the extra series

In this subsection we will write B(g, φ) as a Poincar´e series, and finish the proof of Theorem 3.22 in the non-compact case. Partial Poisson summation Recall from Theorem 4.15 that X X r(gf )`φf (x, u) r(g∞ )β(x, u) B(g, φ) = u∈Q× x∈Vhyp

−

1

X

0 W0,∞ (0, g∞ , u, φ∞ ) |u|A2 f

Z r(gf )`φf ((0, x2 ), u)dx2 . Af

u∈Q×

^f (0, u) by In the second summation, the non-archimedean part is exactly r(g)`φ ^ definition, and the archimedean part is equal to 2 r(g ∞ )β(0, u) by Lemma 4.19. It follows that X X r(gf )`φf (x, u) r(g∞ )β(x, u) B(g, φ) = u∈Q× x∈Vhyp

−2

X

^ ^ r(g f )`φf (0, u) r(g∞ )β(0, u).

u∈Q×

We consider a more general situation. For any ηf ∈ S(Vhyp (Af ) × A× f ), denote η = ηf ⊗ β and B(g, η) :

=

X

X

u∈Q× x∈Vhyp 0

r(g)η(x, u) −

X u∈Q×

] r(g)η(0, u) −

X

0

] (0, u). r(g)η

u∈Q×

] denotes the partial Fourier transform with respect to x1 . We will Here r(g)η prove that B(g, η) is automorphic and “orthogonal” to all cusp forms with trivial central character at infinity. The modularity explains the reason for changing 0 ] to r(g)η ] . half of r(g)η

165

TRACE OF THE GENERATING SERIES

For clarification, recall that for ηp ∈ S(Vhyp (Qp )×Q× p ) we have the following three Fourier transforms: Z 1 ηep ((x1 , x2 ), u) = |u|p2 ηp ((x1 , y), u) ψp (x2 y) dy, Qp

ηep 0 ((x1 , x2 ), u)

=

ηbp ((x1 , x2 ), u)

=

1 2

Z

|u|p

ηp ((y, x2 ), u) ψp (x1 y) dy, Qp

Z

Z

|u|p

ηp ((y1 , y2 ), u) ψp (x2 y1 + x1 y2 ) dy1 dy2 . Qp

Qp

The powers of |u|p are to change the measures to the self-dual measures. The composition of any two transforms above is equal to the third one composed with a transformation of the form (x1 , x2 ) → (±x1 , ±x2 ) or (x1 , x2 ) → (±x2 , ±x1 ). The definitions are also valid for β in the archimedean case. The special case B(g, `φf ⊗ β) is exactly the original series B(g, φ). In fact, 0 ^ ^ r(gf )`φf (0, u) = r(g f )`φf (0, u) follows from the symmetry `(r(gf )φf )((x1 , x2 ), u) = `(r(gf )φf )((x2 , x1 ), u). See Proposition 4.13 for the symmetry. The identity 0 ^ ^ r(g ∞ )β (0, u) = r(g∞ )β(0, u) also follows from the symmetry of r(g∞ )β. We 0 ^ ^ will see that the equality r(g f )`φf (0, u) = r(gf )`φf (0, u) is the only special property we need in the consideration. Go back to the study of the general B(g, η). To ease the notation, consider X 0 ] ] (0, u). r(g)η(x, u) − r(g)η(0, u) − r(g)η B(g, u, η) : = x∈Vhyp

Here u ∈ Q× . Write the summation x into x1 ∈ Q and x2 ∈ Q. Apply the Poisson summation formula in Lemma 4.17 to the sum x2 ∈ Q. We obtain

=

B(g, u, η) X X

0

] ] ] r(g)η((x 1 , x2 ), u) − r(g)η((0, 0), u) − r(g)η ((0, 0), u)

x1 ∈Q x2 ∈Q

=

X X0

0

] ] r(g)η((x 1 , x2 ), u) − r(g)η ((0, 0), u).

x1 ∈Q x2 ∈Q

Here the summation

X0 x2 ∈Q

=

X

.

x2 ∈Q−({x1 }∩{0})

The double summation excludes the point (x1 , x2 ) = (0, 0). By Lemma 4.18 and Lemma 4.19, change the action r to the action ρ in the double sum. As for the (0, 0)-term, it is easy to have 0 \ ^ ((0, 0), u) = r(g ^ \ r(g)η ηf ((0, 0), u). f )ηf ((0, 0), u) = ρ(gf )e f

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CHAPTER 4

Therefore,

=

B(g, u, η) X X0

\ ^ ρ(g)e η ((x1 , x2 ), u) − ρ(g ηf ((0, 0), u) r(g ∞ )β((0, 0), u). f )e

x1 ∈Q x2 ∈Q

Interpretation by Poincar´e series The last double sum for B(g, u, η) is not absolutely convergent, while each single sum is absolutely convergent in the sense that X0 |ρ(g)e η ((x1 , x2 ), u)| < ∞ x2 ∈Q

and

X X 0 ρ(g)e η ((x1 , x2 ), u) < ∞. x1 ∈Q x2 ∈Q

As we will see, the situation resembles the classical almost holomorphic weighttwo Eisenstein series X X0 1 π G2 (z) = . − (mz + n)2 Im(z) m∈Z n∈Z

We will actually reduce the problem to the classical case. Ignore the convergence for the moment. Then the summation (x1 , x2 ) ∈ Q2 − {(0, 0)} is a single orbit of (0, 1) under the right action of SL2 (Q). The stabilizer of (0, 1) in SL2 (Q) is exactly N (Q). It follows that the double sum in B(g, u, η) can be formally transformed to X ρ(γg)e η ((0, 1), u). γ∈N (Q)\SL2 (Q)

The standard way to rigorize the above idea is to introduce an s-variable and consider the analytic continuation. Define X B(s, g, u, η) := δ∞ (γg)s ρ(γg)e η ((0, 1), u), g ∈ GL2 (A). γ∈N (Q)\SL2 (Q)

Here δ∞ only picks up the archimedean part. The series is a Poincar´e series in that δ∞ (ng)s ρ(ng)e η ((0, 1), u) = δ∞ (g)s ρ(g)e η ((0, 1), u), It is not exactly an Eisenstein series in the sense that δ∞ (g)s ρ(g)e η ((0, 1), u)

n ∈ N (A).

167

TRACE OF THE GENERATING SERIES

is not a principal series. We will see that the series B(s, g, u, η) is absolutely convergent for Re(s) > 0, and thus defines an automorphic form for g ∈ SL2 (A). In the absolute convergence range Re(s) > 0, it is easy to recover the expression X X0 B(s, g, u, η) = δ∞ (γ(x1 ,x2 ) g)s ρ(g)e η ((x1 , x2 ), u). x1 ∈Q x2 ∈Q



 ∗ ∗ , and x 1 x2 δ∞ (γ(x1 ,x2 ) g) does not depend on the choice of the matrix. In fact, explicit computation gives |Im(z)|1/2 . δ∞ (γ(x1 ,x2 ) g) = |x1 z + x2 |   ai + b a b Here we denote g∞ = . and z = c d ci + d Now we introduce the counterpart B(s, g, η) of the original series B(g, η). We might define it to be X X −s B(s, g, u, η) = | det(γ)|∞2 δ∞ (γg)s ρ(γg)e η ((0, 1), 1). Here γ(x1 ,x2 ) denotes any matrix in SL2 (Q) of the form

u∈Q×

γ∈N (Q)\GL2 (Q)

But it would not be automorphic on g ∈ GL2 (A) due to the extra power of | det(γ)|∞ on the right-hand side. To solve the problem, we multiple the sum by a similar power of | det(g)|∞ . To keep the central character of the sum trivial at infinity, we divide it by the same power of | det(g)|A . Therefore, define X s B(s, g, u, η). B(s, g, η) := | det(g)|A2 f u∈Q×

Here | det(g)|Af =

Y

| det(g)|p .

p 0, and have meromorphic continuations to all s ∈ C. In particular, both of them are holomorphic at s = 0 with B(0, g, u, η) = B(g, u, η),

B(0, g, η) = B(g, η).

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CHAPTER 4

Proof. It suffices to prove the results for B(s, g, u, η). We reduce it to the standard Eisenstein series of weight two for congruence subgroups in the classical language. Start with the archimedean part. By Lemma 4.19, 2

2

−πu(x1 +x2 ) e 1 , x2 ), u) = 1 − e β((x 1R+ (u). 4u(x1 − ix2 )2

It has two parts α1 (x1 , x2 )

=

1 1R (u), 4u(x1 − ix2 )2 +

=

e−πu(x1 +x2 ) − 1R (u). 4u(x1 − ix2 )2 +

2

α2 (x1 , x2 )

2

We accordingly have B(s, g, u, η) = B 1 (s, g) + B 2 (s, g) where B 1 (s, g)

=

X X0

δ∞ (γ(x1 ,x2 ) g)s ρ(g)(ηef ⊗ α2 )((x1 , x2 ), u),

x1 ∈Q x2 ∈Q 2

B (s, g)

=

X X0

δ∞ (γ(x1 ,x2 ) g)s ρ(g)(ηef ⊗ α2 )((x1 , x2 ), u).

x1 ∈Q x2 ∈Q

For the theorem, it suffices to assume u∞ > 0.

Then we also  assume  a b det(g∞ ) > 0 since every term is zero if det(g∞ ) < 0. Denote g∞ = c d ai + b . Then and z = ci + d δ∞ (γ(x1 ,x2 ) g) =

Im(z)1/2 . |x1 z + x2 |

By this, it is easy to see that B 2 (s, g) is absolutely convergent (and thus holomorphic) for any s ∈ C. In particular, X X0 ρ(g)(ηef ⊗ α2 )((x1 , x2 ), u) B 2 (0, g) = x1 ∈Q x2 ∈Q

is the contribution of α2 in B(g, u, η). Now consider B 1 (s, g), which has the convergence trouble. We first have ρ(g∞ )α1 (x1 , x2 )

=

^ r(g ∞ )β((0, 0), u)

=

(ad − bc) , 4u(ci + d)2 (x1 z + x2 )2 π(ci − d) . 4(ci + d)

−

169

TRACE OF THE GENERATING SERIES

The second identity is still from Lemma 4.19. It follows that B 1 (s, g) =

−

Im(z)s/2 (ad − bc) X X 0 ρ(gf )ηef ((x1 , x2 ), u). 2 4u(ci + d) (x1 z + x2 )2 |x1 z + x2 |s x1 ∈Q x2 ∈Q

Note that g and u are fixed. Write Φf (x1 , x2 ) = ρ(gf )ηef ((x1 , x2 ), u) for simplicity. It is a Schwartz function on S(Vhyp (Af ) × A× f ). Then B 1 (s, g) = − where B 11 (s, g) =

X X0 x1 ∈Q x2 ∈Q

(ad − bc) 11 B (s, g) 4u(ci + d)2

Im(z)s/2 Φf (x1 , x2 ). (x1 z + x2 )2 |x1 z + x2 |s

It suffices to prove similar results for B 11 (s, g), i.e., B 11 (s, g) is absolutely convergent for Re(s) > 0 and has a meromorphic continuation to s ∈ C with

=

B 11 (0, g) X X0 x1 ∈Q x2 ∈Q

1 π Φf (x1 , x2 ) − (x1 z + x2 )2 Im(z)

Z

Z Φf (x1 , x2 )dx1 dx2 . Af

Af

Here the double integral comes from the identity Z Z \ ρ(gf )e ηf ((0, 0), u) = |u|Af ρ(gf )e ηf ((x1 , x2 ), u)dx1 dx2 . Af

Af

By additivity, we can assume that Φf = ⊗p Φp with Φp = 1rp +pep Z2p for some rp ∈ Q2p at every finite place p. Of course, rp = 0 and ep = 0 for almost all p. By a change of variable of the form (x1 , x2 ) → c(x1 , x2 ) with c ∈ Q× , we can assume that all ep ≥ 0 and rp ∈ Z2p . It follows that there are r ∈ Z2 and N ∈ Z+ such that Φp = 1r+N Z2p for every p. In this case, we simply have B 11 (s, g) =

X

0

X

x1 ∈r1 +N Z x2 ∈r2 +N Z

Im(z)s/2 . (x1 z + x2 )2 |x1 z + x2 |s

Here we write r = (r1 , r2 ). We need to prove the meromorphic continuation and the identity B 11 (0, g) =

X

X

x1 ∈r1 +N Z x2 ∈r2 +N Z

0

1 π . − 2 2 (x1 z + x2 ) N Im(z)

It is exactly the result for the classical Eisenstein series of weight two for the principal congruence subgroup Γ(N ). See [Sch, §III-2, §V-3] for example. 

170

CHAPTER 4

Proof of Proposition 4.4 Now it is easy to finish the proof of Proposition 4.4. Recall that it is reduced to equation (4.6.2), which asserts that Z ∗ ϕ(g) B(g, φ) dg = 0, ∀ ϕ ∈ σ, φ ∈ S(B × A× ). GL2 (Q)\GL2 (A)/Z(A)

We can assume that φ∞ is standard by the action of the Lie algebra sl2 (R). By Theorem 4.20, it suffices to prove that, for Re(s) > 0, Z ∗ ϕ(g) B(s, g, η) dg = 0. (4.7.2) GL2 (Q)\GL2 (A)/Z(A)

Here for any g ∈ GL2 (A), B(s, g, η)

s

X

=

| det(γg)|A2 f δ∞ (γg)s ρ(γg)e η ((0, 1), 1).

γ∈N (Q)\GL2 (Q)

The proof is similar to the proof of the orthogonality between Eisenstein series and cusp forms. By definition of regularized integration in §1.6, the left-hand side of (4.7.2) is equal to Z Z ϕ(zg) B(s, zg, η) dz dg GL2 (Q)\GL2 (A)/Z(A)

Z(Q)\Z(Af )

Z =

ϕ(g) B(s, g, η) dg.

2 GL2 (Q)\GL2 (A)/Z(R)

By the expression of B(s, g, η), the last integral is transformed to Z s ϕ(g) | det(g)|A2 f δ∞ (g)s ρ(g)e η ((0, 1), 1) dg. N (Q)\GL2 (A)/Z(R)

Like a principle series, s

| det(g)|A2 f δ∞ (g)s ρ(g)e η ((0, 1), 1) is invariant under the left action of N (A). The integration is zero by the cuspidality condition Z ϕ(ng)dn = 0. N (Q)\N (A)

It finishes the proof.

Chapter Five Assumptions on the Schwartz Function In this chapter and the rest of this book, we assume all the geometric assumptions in §3.6.5. In this chapter, we impose some assumptions on the Schwartz function φ ∈ S(V × A× ), which we will keep from the rest of this book. These assumptions greatly simplify the computations, but imply the kernel identity for all φ. In §5.1, we restate the kernel identity in terms of un-normalized kernel functions Z(g, φ, χ)U and I 0 (0, g, φ, χ)U . It depends on U , but we always fix a U from now on. The rest of this book is to work on this version. In §5.2, we state the assumptions. It is the key section of this chapter. We also claim in Theorem 5.7 that we can “add” these assumptions to the kernel identity without losing the generality. In §5.3 and §5.4 we study two different classes of degenerate Schwartz functions in the assumptions. The goals are to prove Theorem 5.7. 5.1

RESTATING THE KERNEL IDENTITY

Let φ = φf ⊗ φ∞ ∈ S(V × A× ) be a Schwartz function. Assume that the archimedean part φ∞ is standard, and that the finite part φf is invariant under the action of K = U × U for some open compact subgroup U of B× f . 5.1.1

Rewriting the analytic kernel

Since we change the space from S(B × A× ) to S(B × A× ), the definition of the kernel function needs modification. The definition of I(s, g, φ)U below are in the same flavor of holomorphic theta series and Eisenstein series in §4.1. Define a mixed theta-Eisenstein series X X X I(s, g, φ)U := δ(γg)s r(γg)φ(x1 , u). (5.1.1) u∈µ2U \F × γ∈P 1 (F )\SL2 (F )

x1 ∈E

Here µU = F × ∩ U as before. The series converges by the same reason as that for (4.1.4). It is well-defined and gives an automorphic form of GL2 (A). As in the case of S(B × A× ), it is a finite linear combination of products of theta series and Eisenstein series. We do not need this fact for the moment. If wU = 1, we can rewrite it as X X δ(γg)s r(γg)φ(x1 , u). I(s, g, φ)U = γ∈P (F )\GL2 (F )

(x1 ,u)∈µU \(E×F × )

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CHAPTER 5

Here the action of µU on E × F × is still α ◦ (x1 , u) 7→ (αx1 , α−2 u). In general, we need a factor wU = |U ∩ {1, −1}| before r(γg)φ(x1 , u) for any (x1 , u) ∈ µU \(E × × F × ). We further define the twisted average Z ∗ I(s, g, χ, φ)U := I(s, g, r(t, 1)φ)U χ(t)dt. (5.1.2) T (F )\T (A)/Z(A)

Here T = E × denotes the torus over F , Z = F × denotes the sub-torus, and the regularized integration is introduced in (1.6.1). Note that both χ(t) and I(s, g, r(t, 1)φ)U are invariant under both T (F )T (F∞ ) as functions of t ∈ T (A), so that the regularized integration is well-defined. The definitions of I(s, g, φ)U and I(s, g, χ, φ)U depend on the choice of U . In fact, if we have a different choice U 0 ⊂ U , then I(s, g, φ)U 0 = [µ2U : µ2U 0 ]I(s, g, φ)U ,

I(s, g, χ, φ)U 0 = [µ2U : µ2U 0 ]I(s, g, χ, φ)U .

In most of the computations, we will fix a level U . 5.1.2

Recall the height series

We first recall the definition of the generating function and the height series. Recall that the generating series Z(g, φ)U = Z0 (g, φ)U + Z∗ (g, φ)U ,

g ∈ GL2 (A).

The constant term and the non-constant part are respectively X X E0 (0, g, α−1 u, φ) LK,α , Z0 (g, φ)U = − × u∈µ2U \F × α∈F+ \A× f /q(U )

Z∗ (g, φ)U

=

wU

X

X

r(g)φ(x)a Z(x)U .

a∈F × x∈K\B× f

Recall that the height series are as follows: Z(g, (h1 , h2 ), φ)U

=

Z(g, χ, φ)U

=

hZ(g, φ)U [h1 ]◦U , [h2 ]◦U iNT , h1 , h2 ∈ B× ; Z ∗ Z(g, (t, 1), φ)U χ(t)dt. T (F )\T (A)/Z(A)

Their normalizations are given by e (h1 , h2 ), φ) Z(g,

=

e χ, φ) Z(g,

=

2[F :Q]−1 hF p Z(g, (h1 , h2 ), φ)U , [OF× : µ2U ] |DF | 2[F :Q]−1 hF p Z(g, χ, φ)U . : µ2U ] |DF |

[OF×

The normalized series are independent of U .

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ASSUMPTIONS ON THE SCHWARTZ FUNCTION

5.1.3

Rewriting the kernel identity

In terms of the kernel functions depending on U , Theorem 3.21 becomes the following result. Theorem 5.1. Let φ ∈ S(V×A× ) be a Schwartz function bi-invariant under the action of U for some open compact subgroup U of B× f . Then for any ϕ ∈ σ, (I 0 (0, ·, χ, φ)U , ϕ)Pet = 2 (Z(·, χ, φ)U , ϕ)Pet . Remark. In general, the ideal identity I 0 (0, g, χ, φ)U = 2Z(g, χ, φ)U is not true. By a philosophy of Kudla, an identity of the form I 0 (0, g, r(t1 , t2 )φ)U ≈ 2Z(g, (t1 , t2 ), φ)U ,

t1 , t2 ∈ T (A)

is called an arithmetic Siegel–Weil formula. However, such an equality in terms of our definitions is too good to be true even up to holomorphic projection. To have such an identity, one has to modify the definition of the right-hand side by some “natural” way. For more examples of arithmetic Siegel–Weil formulae, we refer to [KRY1, KRY2, Ku3]. Here we check that Theorem 5.1 is equivalent to Theorem 3.21. In fact, they are just stated under different normalizations. Let Φ and φ be as in both theorems. We will show the equalities in two theorems are equivalent assuming φ = Φ. It suffices to show that I(s, g, χ, Φ) =

2[F :Q]−1 hF p I(s, g, χ, φ)U . : µ2U ] |DF |

[OF×

(5.1.3)

Recall that Z I(s, g, χ, Φ)

=

I(s, g, r(t, 1)Φ) χ(t)dt, T (F )\T (A) Z ∗

I(s, g, χ, φ)U

=

I(s, g, r(t, 1)φ)U χ(t)dt. T (F )\T (A)/Z(A)

Here I(s, g, Φ)

=

X

X

δ(γg)s

γ∈P 1 (F )\SL2 (F ) u∈F ×

I(s, g, φ)U

=

X

X

γ∈P 1 (F )\SL2 (F ) u∈µ2U \F ×

X

r(γg)Φ(x1 , u),

x1 ∈E

δ(γg)s

X

r(γg)φ(x1 , u).

x1 ∈E

We see that the proof of (5.1.3) is exactly the same as (4.4.4). We will not repeat it here.

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CHAPTER 5

5.2

THE ASSUMPTIONS AND BASIC PROPERTIES

5.2.1

Assumptions on the Schwartz function

Let SF be the set of all places of F . We first have a disjoint union SF = S∞ ∪ Snonsplit ∪ Ssplit , where • S∞ is the set of archimedean places of F . • Snonsplit is the set of non-archimedean places of F nonsplit in E. • Ssplit is the set of non-archimedean places of F split in E. We further decompose it as a disjoint union SF = S∞ ∪ S1 ∪ S2 ∪ (Snonsplit − S1 ) ∪ (Ssplit − S2 ), where we pick S1 and S2 as follows: • S1 is a finite subset of Snonsplit , containing all places in Snonsplit that are either ramified over Q, or ramified in E, or ramified in B, or ramified in σ, or ramified in χ. Assume that S1 has at least two elements. • S2 consists of two places in Ssplit at which σ and χ are unramified. Assumption 5.2. The Schwartz function φ = ⊗φv ∈ S(B × A× ) is a pure tensor, and φv is standard for any v ∈ S∞ . See §4.1 for details of standard Schwartz functions. Assumption 5.3. For any v ∈ S1 , φv lies in the space 1

S (Bv × Fv× ) := {φv ∈ S(Bv × Fv× ) : φv (x, u) = 0 if v(uq(x)) ≥ −v(dv ) or v(uq(x2 )) ≥ −v(dv )}. Here dv is the local different of F at v, and x2 denotes the orthogonal projection of x in V2,v = Ev jv . Assumption 5.4. For any v ∈ S2 , φv lies in the space 2

S (Bv × Fv× ) := {φv ∈ S(Bv × Fv× ) : r(g)φv (0, u) = 0,

∀ g ∈ GL2 (Fv ), u ∈ Fv× }.

Assumption 5.5. For any v ∈ Snonsplit − S1 , assume that φv is the standard characteristic function of OBv × OF×v . Q Assumption 5.6. Let U = v Uv be an open compact subgroup of B× f satisfying

ASSUMPTIONS ON THE SCHWARTZ FUNCTION

175

• φ is invariant under the action of K = U × U . • χ is invariant under the action of UT := U ∩ T (Af ). • Uv is of the form (1 + $vr OBv )× for some r ≥ 0 for every finite place v. • Uv is maximal for all v ∈ Snonsplit − S1 and v ∈ S2 . • U does not contain −1. • U is sufficiently small such that each connected component of the complex points of XU is an unramified quotient of H by the complex uniformization. We explain the last condition. Let τ : F ,→ C be an embedding. Then a XU,τ (C) = Γh \H ∪ {cusps}. × h∈B(τ )× + \Bf /U

−1 is a discrete subgroup of B(τ )× Here Γh = B(τ )× + ∩ hU h + . Now we require that the quotient map H → Γh \H is unramified everywhere for every τ and every h. Notice that we do not impose any condition at places in Ssplit − S2 . The crucial assumptions that simplify the computations are Assumption 5.3 and Assumption 5.4. In Theorem 5.7 below, we will see that those assumptions do not lose the generality of the derivative formula. The global argument using the theorem to extend the derivative formula to the general case is in the last chapter. Assumption 5.3 will always be combined with the assumption g ∈ 1S1 GL2 (AS1 ). Its major effects are as follows:

• Kill the constant term of E 0 (0, g, u, φ1 ). • Kill the self-intersections of CM points in the height series Z(g, (t1 , t2 ), φ). • Kill the iv -part of the local height pairings of CM points in Z(g, (t1 , t2 ), φ) for v ∈ Ssplit . • Kill the logarithmic singularities coming from both the derivatives and the local heights at v. Consequently, both kφv and mφv are non-singular for v ∈ S1 so that the v-part kernel functions can be approximated by theta series. The major effects of Assumption 5.4 are as follows: • Kill the absolute constant term of the analytic kernel I 0 (0, g, χ, φ)U so that it satisfies the growth of the holomorphic projection. • Kill the constant term of the generating series Z(g, φ).

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• Kill the arithmetic intersections coming for the Hodge classes in the height series Z(g, (t1 , t2 ), φ). • Kill the jv -part of the local height pairing of CM points in Z(g, (t1 , t2 ), φ) for v ∈ Snonsplit . 5.2.2

Sufficiency of the assumptions

The main result in this chapter asserts that it suffices to prove the formula under the above assumptions. Theorem 5.7. Theorem 5.1 is true for all (φ, U ) if and only if it is true for all (φ, U ) satisfying Assumptions 5.2–5.6. Let (φ, U ) be as in Theorem 5.1. Take Φ ∈ S(B × A× ) with Φ = φ. Then Theorem 5.1 for (φ, U ) is equivalent to Theorem 3.21 for Φ. It follows that we can always shrink U to meet Assumption 5.6. By Lemma 3.23 (2), Theorem 5.7 is a consequence of the following result. Proposition 5.8. There exists a Schwartz function Φ = ⊗v Φv ∈ S(B × A× ) satisfying the following two conditions: • The Schwartz function φ = Φ ∈ S(B × A× ) satisfies Assumptions 5.2–5.5. • The pairing α(θ(φf ⊗ ϕf )) 6= 0 for some ϕ ∈ σ. Here θ = ⊗v θv is the product of the local theta lifting θv : σv ⊗ S(Bv ⊗ Fv× )−→πv ⊗ π ev normalized in (2.2.3). The bilinear form α = ⊗v αv is a product of local bilinear ev → C given by forms αv : πv ⊗ π Z L(1, ηv )L(1, πv , ad) (πv (t)f1,v , f2,v ) χv (t)dt. αv (f1,v , f2,v ) = ζv (2)L( 12 , πv , χv ) Fv× \Ev× The proposition is actually local. Namely, we only need to find (Φv , ϕv ) for each place v such that φv = Φv satisfies the relevant assumption and α(θv (Φv ⊗ ϕv )) 6= 0. Here we take the convention that Φv = Φv if v is non-archimedean. The archimedean case is trivial. In fact, for any archimedean place v, one can take Φv to be any Schwartz function with Φv standard, and ϕv ∈ σv such that θv (Φv ⊗ ϕv ) 6= 0. Since π is one-dimensional, this implies α(θv (Φv ⊗ ϕv )) 6= 0. In the case v ∈ Snonsplit − S1 , by Assumption 5.5 φv has to be standard. In this unramified case, α(θv (φv ⊗ ϕv )) = 1 if we take ϕv to be the spherical vector. In the case v ∈ Ssplit − S2 , there is no restriction on φv so the proposition is also true. It remains to treat the case v ∈ S1 and the case v ∈ S2 . They are exactly given in the next two sections.

177

ASSUMPTIONS ON THE SCHWARTZ FUNCTION

5.2.3

Some simple properties of the assumptions 1

In the following, we state some simple results on the descriptions of S (Bv ×Fv× ) 2 and S (Bv × Fv× ). They will be useful when applying the assumptions above. The first result is related to Assumption 5.3. Lemma 5.9. Let v ∈ Snonsplit and φv ∈ S(Bv × Fv× ). Assume that there is a constant c > 0 such that φv (x, u) = 0 for all (x, u) with v(uq(x2 )) ≥ c. Then we can write r X φi1,v ⊗ φi2,v . φv = i=1

Here for each i, φi1,v ∈ S(Ev ×Fv× ) and φi2,v ∈ S(Ev j×Fv× ) satisfy φi2,v (x2 , u) = 0 for all (x2 , u) with v(uq(x2 )) ≥ c. × Proof. We can find non-trivial open subgroups of A1 ⊂ Ev , A2 ⊂ OE , v A3 ⊂ OF×v such that φv is constant on any coset of A1 \Ev × A2 \Ev j × A3 \Fv× . Here A1 acts by addition, and A2 , A3 act by multiplication. It follows that we have linear combination X X X φv (a1 + a2 , b) 1(a1 +A1 )×(A3 b) ⊗ 1(A2 a2 )×(A3 b) . φv = a1 ∈A1 \Ev a2 ∈A2 \Ev j b∈A3 \Fv×

It is essentially a finite sum. The coefficient of 1(a1 +A1 )×(A3 b) ⊗ 1(A2 a2 )×(A3 b) is nonzero only if v(bq(a2 )) < c. It proves the result since v(A3 b q(A2 a2 )) =  v(bq(a2 )). The following result is related to Assumption 5.4. Indeed it gives an alter2 native description of S (Bv × Fv× ). Lemma 5.10. Let v be a non-archimedean place of F and φv ∈ S(Bv × Fv× ). The following are equivalent: (1) The value r(g)φv (0, u) = 0,

∀ g ∈ GL2 (Fv ), u ∈ Fv× .

(2) The average Z r(g, h)φv (x, u)dx = 0, Bv (a) × × × ∀ g ∈ GL2 (Fv ), h ∈ B× v × Bv , a ∈ Fv , u ∈ Fv .

Proof. Recall that Bv (a) = {x ∈ Bv : q(x) = a} is a homogeneous space of B1v = Bv (1), and endowed with the measure induced from the Haar measure of

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B1v . We first observe that we can assume h = 1 in (2). In fact, Z Z r(g, h)φv (x, u)dx = r(g)φv (h−1 x, ν(h)u)dx Bv (a) B (a) Z v = r(g)φv (x, u0 )dx Bv (a0 )

with a0 = aν(h)−1 and u0 = uν(h). The local Siegel-Weil gives Z Z r(g)φv (x, u)dx = Bv (a)

r(wn(b)g)φv (0, u)ψv (−ab)db.

Fv

It follows that (1) implies (2). Conversely, (2) implies (1) by viewing the righthand side as the Fourier transform of r(wn(b)g)φv (0, u) as a function of b ∈  Fv . Remark. Consider the case φv is a local component of a global φ. The Siegel Eisenstein series E(0, g, u, φ) is introduced before. Then (1) would imply the vanishing of its constant term, and (2) would imply the vanishing of all other terms.

5.3

DEGENERATE SCHWARTZ FUNCTIONS I

The goal in this section is to prove Proposition 5.8 for v ∈ S1 . We will prove slightly more general results below. Let v be a non-archimedean place of F nonsplit in E. Recall that we have introduced 1

S (Bv × Fv× ) = {φv ∈ S(Bv × Fv× ) : φv (x, u) = 0 if v(uq(x)) ≥ −v(dv ) or v(uq(x2 )) ≥ −v(dv )}. The goal here is to show that this space “generates” the whole space in the consideration of the final result. The following is the main result of this section. Proposition 5.11. Let σv be an infinite dimensional irreducible representation of GL2 (Fv ). Then for any nonzero GL2 (Fv )-equivariant homomorphism 1 S(Bv × Fv× ) → σv , the image of S (Bv × Fv× ) in σv is nonzero. It implies Proposition 5.8 for v ∈ S1 . In fact, fix any f1,v ⊗ f2,v ∈ πv ⊗ π ev with α(f1,v ⊗ f2,v ) 6= 0. Consider the nonzero homomorphism α : S(Bv × Fv× ) −→ σv ,

φv 7−→ θ0 (φv ⊗ f1,v ⊗ f2,v ).

Here θ0 : S(Bv × Fv× ) ⊗ πv ⊗ π ev −→ σv

179

ASSUMPTIONS ON THE SCHWARTZ FUNCTION

is the theta lifting in the opposite direction. It is unique up to scalars. By the above proposition, we can find ϕv lies in the image of α. In other words, we have θ(φv ⊗ ϕv ) = cf1,v ⊗ f2,v for some c 6= 0. Then (φv , ϕv ) satisfies Proposition 5.8. In the following, we prove Proposition 5.11. For convenience, we introduce 1

S weak (Bv × Fv× ) := {φv ∈ S(Bv × Fv× ) : φv |(Bv sing ∪Ev )×Fv× = 0}. 1

Here Bv sing = {x ∈ Bv : q(x) = 0}. By compactness, for any φv ∈ S weak (Bv × Fv× ), there exists a constant c such that φv (x, u) = 0 if v(uq(x)) > c. The same 1 result holds for uq(x2 ) in thenonsplit case. Then it is easy to see that S (Bv × 1 × × Fv× ) generates  v ×Fv ) under the action of the group m(Fv ) ⊂ GL2 (Fv )  S weak (B a . Thus the former can be viewed as an effective version of of elements a−1 1

the latter. In particular, it suffices to prove the proposition for S weak (Bv × Fv× ). We will prove a more general result. For simplicity, let F be a nonarchimedean local field and let (V, q) be a non-degenerate quadratic space over F of even dimension. Then we have the Weil representation of GL2 (F ) on S(V × F × ), the space of Schwartz–Bruhat functions on V × F × . Let α : S(V × F × ) → σ be a surjective morphism to an irreducible and admissible representation σ of GL2 (F ). We will prove the following result which obviously implies Proposition 5.11. Proposition 5.12. Let W be a proper subspace of V of even dimension. Assume that σ is not one dimensional, and that in the case W 6= 0, W is nondegenerate, and that its orthogonal complement W 0 is anisotropic. Then there is a function φ ∈ S(V × F × ) with a nonzero image in σ such that the support supp(φ) of φ contains only elements (x, u) such that q(x) 6= 0 and that W (x) := W + F x is non-degenerate of dimension dim W + 1. Let us start with the following proposition, which allows us to modify any test function to a function with support at points (x, u) ∈ V × F × with components x of nonzero norm q(x) 6= 0. Proposition 5.13. Let φ ∈ S(V × F × ) be an element with a nonzero image in σ. Then there is a function φe ∈ S(V × F × ) with a nonzero image in σ such that
e ⊂ supp(φ) ∩ Vq6=0 × F × . supp(φ) The key to prove this proposition is the following lemma. It is well-known but we give a proof for readers’ convenience.

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Lemma 5.14. Let σ be an irreducible admissible representation of GL2 (F ) whose dimension is greater than one. Then the only vector in σ invariant under the action of the unipotent group N (F ) is zero. Proof. Let v be such an invariant vector. By smoothness, it is also fixed by some compact open subgroup U of SL2 (F ). Then v is invariant under the subgroup generated by N (F ) and U . It is easy to see that U − P 1 (F ) is nonempty, and let γ ∈ U − P 1 (F ) be one element. A basic fact asserts that SL2 (F ) is generated by N (F ) and γ as long as γ is not in P 1 (F ). It follows that v is invariant under SL2 (F ). If v 6= 0, then by irreducibility σ is generated by v under the action of GL2 (F ). It follows that all elements of σ are invariant under SL2 (F ). Thus the representation σ factors through the determinant map, which implies it must be one-dimensional. Contradiction!  Proof of Proposition 5.13. Applying the lemma above, we obtain an element b ∈ F such that σ(n(b))α(φ) − α(φ) 6= 0. e with The left-hand side is equal to α(φ) φe = r(n(b))φ − φ. By definition, we have e u) = (ψ(buq(x)) − 1)φ(x). φ(x, Thus such a φe has support
e ⊂ supp(φ) ∩ Vq6=0 × F × . supp(φ)  Proof of Proposition 5.12. If W = 0, then the result is implied by Proposition 5.13. Thus we assume that W 6= 0. We have an orthogonal decomposition V = W ⊕ W 0 , and an identification S(V × F × ) = S(W × F × ) ⊗ S(W 0 × F × ). The action of GL2 (F ) is given by actions on S(W × F × ) and S(W 0 × F × ) respectively. Choose any φ ∈ S(V × F × ) such that α(φ) 6= 0. We may assume that φ = f ⊗ f 0 is a pure tensor. Since W 0 is anisotropic, S(Wq60=0 × F × ) is a subspace in S(W 0 × F × ) with quotient S(F × ). The quotient map is given by evaluation at (0, u). Thus S(W 0 × F × ) = S(Wq60=0 × F × ) + r(w)S(Wq60=0 × F × )

181

ASSUMPTIONS ON THE SCHWARTZ FUNCTION

as w acts as the Fourier transform up to a scale multiple. In this way, we may write fi0 ∈ S(Wq60=0 × F × ). f 0 = f10 + r(w)f20 , Then we have a decomposition φ = φ1 + r(w)φ2 ,

φ1 := f ⊗ f10 , φ2 := r(w−1 )f ⊗ f20 .

One of α(φi ) 6= 0, and the support of this φi consists of points (x, u) such that W (x) is non-degenerate. Applying Proposition 5.13 to this φi , we get the function we want.  5.4

DEGENERATE SCHWARTZ FUNCTIONS II

The goal in this section is to prove Proposition 5.8 for v ∈ S2 . It is a consequence of the following Proposition 5.15. Let v be a non-archimedean place of F split in B. The space we are considering is 2

S (Bv × Fv× ) = {φv ∈ S(Bv × Fv× ) : r(g)φv (0, u) = 0,

∀ g ∈ GL2 (Fv ), u ∈ Fv× }.

For simplicity, we abbreviate (Fv , Ev , Bv , σv , πv ) as (F, E, B, σ, π). In the following result, we only need the assumptions that B is a matrix algebra, both σ and χ are unramified, and that σ is unitary. We do not need E to be split over F in this local case. Proposition 5.15. Then there exists a degenerate Schwartz function φ ∈ 2 S (B × F × ) such that the following are true: × × × OB . (1) The function φ is invariant under GL2 (OF ) × OB

(2) The pairing α(θ(φ ⊗ ϕ)) 6= 0 for any nonzero ϕ in the one-dimensional space π GL2 (OF ) . Proof. Since all the data are unramified, the lemma can be verified by explicit computations. However, here we provide an explicit φ and check it without involved computations. We first take φ0 to be the standard characteristic function of OB × OF× . Then (1) and (2) are satisfied. The truth of (2) follows from the unramified theta lifting θ(φ0 ⊗ ϕ) = f1 ⊗ f2 where f1 and f2 are respectively nonzero vectors in the one-dimensional spaces × × 2 eOB . The problem is that φ0 does not lie in S (B × F × ). We solve π OB and π this problem by the action of the spherical Hecke algebra.

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Let L ∈ Cc∞ (B × ) be a locally constant and compactly supported function on B . Then L acts on π by the usual Hecke actions, and it acts on φ0 ∈ S(B ×F × ) by Z ×

(Lφ0 )(x, u) :=

φ0 (h−1 x, q(h)u) L(h)dh.

B× × The integration uses the Haar measure with vol(OB ) = 1. Note that the action of L commutes with the action of g ∈ GL2 (F ). It is immediate that Z 0 0 r(g)(Lφ )(0, u) = (L r(g)φ )(0, u) = r(g)φ0 (0, q(h)u) L(h)dh. B× × Assume that φ0 is invariant under the left (or right) action of OB , and that × 1 × 1 × 0 the support of L is contained in OB B OB = B OB . Then φ (0, q(h)u) = φ0 (0, u) for any h in the support of L. In that case,

r(g)(Lφ0 )(0, u) = deg(L) r(g)φ0 (0, u), where

Z L(x)dx.

deg(L) = B×

2

In summary, the difference Lφ0 − deg(L)φ0 lies in S (B × F × ) as long as L is × × and φ0 is invariant under the left (or right) action of OB . supported on B 1 OB Go back to the standard φ0 . We will find some L such that φ := Lφ0 − deg(L)φ0 satisfies the conditions of the proposition. In fact, assume that L is in the spherical Hecke algebra × × \B × /OB ). HO× = Cc∞ (OB B

It makes Lf computable and φ satisfy (1). × The algebra HO× acts on the one-dimensional space π OB via a character B

λπ : HO× → C. B

As usual, denote by T℘ the characteristic function of   $ × × OB OB . 1 Then we have

1

λπ (T℘ ) = N 2 (α + β). Here N is the cardinality of the residue field of F , and α and β are the Satake parameters of σ.

ASSUMPTIONS ON THE SCHWARTZ FUNCTION

183

Therefore, α(θ(Lφ ⊗ ϕ)) = α(Lf1 ⊗ f2 ) = λσ (L) α(f1 ⊗ f2 ). It follows that α(θ(φ ⊗ ϕ)) = (λσ (L) − deg(L)) α(f1 ⊗ f2 ). It suffices to find L ∈ HO× such that λσ (L) 6= deg(L). Set L(x) := (T℘ )2 ($x) B

× , and the above inequality becomes for any x ∈ B × . It is supported on B 1 OB 1 1 (N 2 (α + β))2 6= (N + 1)2 . αβ

Equivalently, we want α 1 6= N, . β N It is true because σ is unramified, unitary and infinite-dimensional. In fact, any such representation arises from the obvious case |α| = |β| = 1 or from a complementary series. In the latter case, the absolute values |α|, |β| are exactly 1 1  in the open interval (N − 2 , N 2 ).

Chapter Six Derivative of the Analytic Kernel Let φ = φf ⊗ φ∞ ∈ S(V × A× ) be a Schwartz function. Assume that the archimedean part φ∞ is standard, and that the finite part φf is invariant under the action of K = U × U for some open compact subgroup U of B× f . Recall that in §5.1 we have introduced the series Z ∗ I(s, g, χ, φ)U = I(s, g, r(t, 1)φ)U χ(t)dt, T (F )\T (A)/Z(A)

where I(s, g, φ)U =

X

X

u∈µ2U \F × γ∈P 1 (F )\SL2 (F )

δ(γg)s

X

r(γg)φ(x1 , u).

x1 ∈E

In this chapter, we compute the derivative I 0 (0, g, χ, φ)U and its holomorphic projection PrI 0 (0, g, χ, φ)U . We assume all the assumptions in §5.2, which significantly simplify the results. The main content of this section is various local formulae. We usually fix U and abbreviate I(s, g, φ)U and I(s, g, χ, φ)U as I(s, g, φ) and I(s, g, χ, φ), or even as I(s, g) and I(s, g, χ). In §6.1, we decompose the kernel function I 0 (0, g, φ) into a sum of infinitely many local terms I 0 (0, g, φ)(v) indexed by places v of F nonsplit in E. Each local term is a period integral of some kernel function K(v) (g, (t1 , t2 )). In §6.2, we deal with the v-part I 0 (0, g, φ)(v) for non-archimedean v. An explicit formula is given in the unramified case, and an approximation is given in the ramified case assuming the Schwartz function is degenerate. In §6.2, we show an explicit result of the v-part I 0 (0, g, φ)(v) for archimedean v. In §6.4, we review a general formula of holomorphic projection, and estimate the growth of the kernel function in order to apply the formula. In §6.5, we compute the holomorphic projection PrI 0 (0, g, χ) of the kernel I 0 (0, g, χ). When compared with the geometric kernel function, we use PrI 0 (0, g, χ) instead of I 0 (0, g, χ). 6.1

DECOMPOSITION OF THE DERIVATIVE

Let φ ∈ S(V×A× ) be bi-invariant under the action of an open compact subgroup U of B× f . Here V = (B, q) is determined by a set Σ of places odd cardinality

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DERIVATIVE OF THE ANALYTIC KERNEL

containing all the archimedean places. Recall that X X X δ(γg)s r(γg)φ2 (0, u) r(γg)φ1 (x1 , u). I(s, g, φ)U = x1 ∈E

u∈µ2U \F × γ∈P 1 (F )\SL2 (F )

In this section we decompose the derivative I 0 (0, g, φ)U into a sum over places of F where the derivative is taken. As in the case of S(V×A× ), the series I(s, g, φ)U is a finite linear combination of products of theta series and Eisenstein series. So the derivative is moved to the Eisenstein series. Recall that we have the orthogonal decomposition V = V1 ⊕ V2 , where V1 = EA and V2 = EA j. It yields a decomposition S(V × A× ) = S(V1 × A× ) ⊗ S(V2 × A× ). More precisely, any φ1 ∈ S(V1 × A× ) and φ2 ∈ S(V2 × A× ) gives φ1 ⊗ φ2 ∈ S(V × A× ) defined by (φ1 ⊗ φ2 )(x1 + x2 , u) := φ1 (x1 , u)φ2 (x2 , u). Any element of S(V × A× ) is a finite linear combination of functions of the form φ1 ⊗ φ2 . More importantly, the decomposition preserves Weil representation in the sense that r(g, (t1 , t2 ))(φ1 ⊗ φ2 )(x, u) = r1 (g, (t1 , t2 ))φ1 (x1 , u) r2 (g, (t1 , t2 ))φ2 (x2 , u) for any (g, (t1 , t2 )) ∈ GL2 (A) × EA× × EA× . Here we write r1 , r2 for the Weil representation associated to the vector spaces V1 , V2 . The group EA× × EA× acts on V` by (t1 , t2 ) ◦ x` = t1 x` t−1 2 . It is compatible with the action on V. By linearity, we may reduce the computation to the decomposable case φ = φ1 ⊗ φ2 and we further assume that φ2 = ⊗v φ2,v is a pure tensor. It follows that X θ(g, u, φ1 ) E(s, g, u, φ2 ), I(s, g, φ)U = u∈µ2U \F ×

where for any g ∈ GL2 (A), the theta series and the Eisenstein series are given by X r(g)φ1 (x1 , u), θ(g, u, φ1 ) = x1 ∈E

E(s, g, u, φ2 )

=

X

δ(γg)s r(γg)φ2 (0, u).

γ∈P 1 (F )\SL2 (F )

It suffices to study the behavior of Eisenstein series at s = 0. Let us start with the standard Fourier expansion X Wa (s, g, u, φ2 ). E(s, g, u, φ2 ) = δ(g)s r(g)φ2 (0, u) + a∈F

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Here the Whittaker function for a ∈ F, u ∈ F × is given by Z δ(wn(b)g)s r(wn(b)g)φ2 (0, u)ψ(−ab)db. Wa (s, g, u, φ2 ) = A

For each place v of F , we also introduce the local Whittaker function for a ∈ Fv , u ∈ Fv× by Z δ(wn(b)g)s r(wn(b)g)φ2,v (0, u)ψv (−ab)db. Wa,v (s, g, u, φ2,v ) = Fv

In the following we will suppress the dependence of the series on φ, φ1 , φ2 and U. 6.1.1

Vanishing of the central value

Before taking the derivative, we recall the fact that E(0, g, u) = 0 and examine the local reason for that. It will inspire us to arrange the derivative according to Kudla’s philosophy. The results below are considered in §2.5.1 in a slightly different setting. We use slightly different normalizations of the local Whittaker functions by the Weil index γu,v = γ(V2,v , uq). For a ∈ Fv× , denote ◦ Wa,v (s, g, u)

=

−1 γu,v Wa,v (s, g, u).

Normalize the intertwining part by ◦ W0,v (s, g, u)

=

−1 γu,v

1 1 L(s + 1, ηv ) |Dv |− 2 |dv |− 2 W0,v (s, g, u). L(s, ηv )

Here we use the convention that Dv = dv = 1 if v is archimedean. The normalL(s + 1, ηv ) izing factor has a zero at s = 0 when Ev is split, and is equal to L(s, ηv ) π −1 at s = 0 when v is archimedean. Globally, denote Y ◦ Wa◦ (s, g, u) = Wa,v (s, gv , u). v

Then Wa◦ (s, g, u) = −Wa (s, g, u), It follows from the incoherence condition tion for L(s, η), we also have

a ∈ F ×, u ∈ F ×. Q

v

W0◦ (0, g, u) = −W0 (0, g, u),

γu,v = 1. By the functional equau ∈ F ×.

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DERIVATIVE OF THE ANALYTIC KERNEL

Proposition 6.1.

(1) In the sense of analytic continuation for s ∈ C, ◦ (0, g, u) = r(g)φ2,v (0, u). W0,v

Therefore, the global W0◦ (0, g, φ) = r(g)φ(0, u). Furthermore, for almost all places v, ◦ (s, g, u) = δv (g)−s r(g)φ2,v (0, u). W0,v

(2) Assume a ∈ Fv× . ◦ (a) If a is not represented by (V2,v , uq), then Wa,v (0, g, u) = 0.

(b) Assume that there exists xa ∈ V2,v satisfying uq(xa ) = a. Then Z 1 ◦ (0, g, u) = r(g, h)φ2,v (xa , u)dh. Wa,v L(1, ηv ) Ev1 Here the integration uses the Haar measure on Ev1 normalized in §1.6.2. If g ∈ SL2 (A), it follows from Proposition 2.7 for the quadratic space (V2 , uq). The general case is reduced to SL2 (A) by an action of d(c). Proposition 6.2. For any φ2 ∈ S(V2 × A× ), one has E(0, g, u) = 0 for any g ∈ GL2 (A) and u ∈ A× . It follows that I(0, g, φ) = 0 identically. This is Proposition 2.8. We recall the proof a little bit. The vanishing of the constant term E0 (0, g, u) is immediate. Let a ∈ F × , and consider the Whittaker function Y Wa,v (0, g, u). Wa (0, g, u) = v

The local result asserts that Wa,v (0, g, u) 6= 0 only if a is represented by (V2,v , uq). Then the key is the following simple result. Lemma 6.3. For any a, u ∈ F × , there is a place v of F such that a is not represented by (V2,v , uq). This result is very important to us, so we repeat its proof here. Denote by B the (global) quaternion algebra over F generated by E and j with relations j 2 = −au−1 ,

jt = tj, ∀ t ∈ E.

If au−1 is represented by some element xv of (V2,v , q) for all places v, then the map j 7→ xv gives an isomorphism BA ∼ = B. It contradicts the incoherence assumption on B.

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CHAPTER 6

6.1.2

Decomposition of the derivative

For an element u ∈ F × and a place v of F , denote by Fu (v) the set of a ∈ F × represented by (V2 (Av ), uq). Then Fu (v) is non-empty only if E isnonsplit at v. Note that Lemma 6.3 implies that such an a is not represented by (V2,v , uq). ◦ By Proposition 6.1, Wa,v (0, g, u) = 0 for any a ∈ Fu (v). Fix a place v of F nonsplit in E. For any a ∈ Fu (v), taking the derivative yields ◦ 0 (0, g, u)Wa◦,v (0, g, u). Wa◦ 0 (0, g, u) = Wa,v It follows that E 0 (0, g, u) = E00 (0, g, u) −

X

◦ 0 Wa,v (0, g, u)Wa◦,v (0, g, u).

X

v nonsplit a∈Fu (v)

Here the constant term E0 (s, g, u) = δ(g)s r(g)φ2 (0, u) + W0 (s, g, u). ◦ (s, g, u) vanishes In fact, if a ∈ F × does not belong to Fu (v) for any v, then Wa,w ◦0 at s = 0 for at least two places w, which implies Wa (0, g, u) = 0.

Notation 6.4. Let v be a placenonsplit in E. If φ = φ1 ⊗ φ2 and φ2 = ⊗v φ2,v , denote the v-part by X

E 0 (0, g, u, φ2 )(v) :=

◦ 0 Wa,v (0, g, u, φ2 )Wa◦,v (0, g, u, φ2 ),

a∈Fu (v)

X

I 0 (0, g, φ)(v) :=

θ(g, u, φ1 )E 0 (0, g, u, φ2 )(v).

u∈µ2U \F ×

The definition for I 0 (0, g, φ)(v) extends to general φ by linearity. By definition, we have a decomposition X I 0 (0, g)(v) + I 0 (0, g) = − v nonsplit

X

θ(g, u)E00 (0, g, u).

u∈µ2U \F ×

We will show later that the “extra part,” the second sum, is essentially zero by the assumptions in §5.2. But we first take care of I 0 (0, g)(v) for any fixed non-split v. Fix a place v of F non-split in E. Denote by B = B(v) the nearby quaternion algebra. Then we have a splitting B = E + Ej. Let V = (B, q) be the corresponding quadratic space with the reduced norm q, and V = V1 ⊕ V2 be the corresponding orthogonal decomposition. We identify the quadratic spaces V2,w = V2,w unless w = v. Then we have Fu (v) = uq(V2 ) − {0}.

189

DERIVATIVE OF THE ANALYTIC KERNEL

In fact, any a ∈ Fu (v) is represented by (V2,v , uq) since uq(V2,v ) − {0} and uq(V2,v ) − {0} are two (only) different cosets of Fv× /q(Ev× ). On the other hand, a is represented by (V2 (Av ), uq) by definition. Then the Hasse principle implies that a is represented by (V2 , uq). Z uses the Haar measure of total volume one In the following, the integral as introduced in §1.6. Proposition 6.5. For any place vnonsplit in E, Z I 0 (0, g, φ)(v) = 2

(v)

Z(A)T (F )\T (A)

Kφ (g, (t, t))dt,

where for t1 , t2 ∈ T (A) and g ∈ GL2 (A), (v)

Kφ (g, (t1 , t2 ))

(v)

= Kr(t1 ,t2 )φ (g) X X = kr(t1 ,t2 )φv (g, y, u)r(g, (t1 , t2 ))φv (y, u). u∈µ2U \F × y∈V −V1

Here kφv (g, y, u) is linear in φv . In the case φv = φ1,v ⊗φ2,v under the orthogonal decomposition, it is given by L(1, ηv ) 0 ◦ r(g)φ1,v (y1 , u)Wuq(y (0, g, u, φ2,v ) 2 ),v vol(Ev1 )

kφv (g, y, u) =

for any y = y1 + y2 ∈ Vv with y2 6= 0 under the orthogonal decomposition V = V1 + V2 .

Proof. By linearity, it suffices to treat the case φv = φ1,v ⊗ φ2,v . By Proposition 6.1,

=

E 0 (0, g, u)(v) X

0 ◦,v ◦ Wuq(y (0, g, u)Wuq(y (0, g, u) 2 ),v 2)

y2 ∈E 1 \(V2 −{0})

= = =

1 Lv (1, η)

0 ◦ Wuq(y (0, g, u) 2 ),v

X y2 ∈E 1 \(V2 −{0})

1 vol(Ev1 )Lv (1, η) 1 vol(Ev1 )Lv (1, η)

Z

X y2 ∈E 1 \(V2 −{0})

Z E 1 \E 1 (A)

E 1 (A)

X y2 ∈V2 −{0}

Z E 1 (Av )

r(g)φv2 (y2 τ, u)dτ

0 ◦ Wuq(y (0, g, u)r(g)φv2 (y2 τ, u)dτ 2 τ ),v

0 ◦ Wuq(y (0, g, u)r(g)φv2 (y2 τ, u)dτ. 2 τ ),v

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CHAPTER 6

Therefore, we have the following expression for I 0 (0, g)(v): X X 1 I 0 (0, g)(v) = r(g)φ1 (y1 , u) vol(Ev1 )Lv (1, η) u∈µ2U \F × y1 ∈V1 Z X 0 ◦ Wuq(y (0, g, u)r(g)φv2 (y2 τ, u)dτ. · 2 τ ),v E 1 \E 1 (A) y ∈V −{0} 2 2

Combine the two sums for y1 and y2 to obtain Z X X Z(A)T (F )\T (A)

u∈µ2U \F × y=y1 +y2 ∈V y2 6=0 0 ◦ v −1 r(g)φ1,v (y1 , u)Wuq(y yt, u)dt. −1 t),v (0, g, u)r(g)φ (t 2t (v)

By definition of kφv and Kφ , we have

= =

I 0 (0, g)(v) Z X X 1 kφv (g, t−1 yt, u)r(g)φv (t−1 yt, u)dt L(1, η) Z(A)T (F )\T (A) u∈µ2U \F × x∈V −V1 Z 1 (v) K (g, (t, t))dt. L(1, η) Z(A)T (F )\T (A) φ

Since vol(Z(A)T (F )\T (A)) = 2L(1, η), we get the result. Here we have used the relation kφv (g, t−1 yt, u) = kr(t,t)φv (g, y, u) which follows from the definition. See also the lemma below.  Lemma 6.6. The function kφv (g, y, u) behaves like Weil representation under the action of P (Fv ) and Ev× × Ev× . Namely, kφv (m(a)g, y, u)

= kφv (n(b)g, y, u) = kφv (d(c)g, y, u) = kr(t1 ,t2 )φv (g, y, u) =

|a|2 kφv (g, ay, u), a ∈ Fv× , ψ(buq(y))kφv (g, y, u), b ∈ Fv , |c|−1 kφv (g, y, c−1 u), c ∈ Fv× , −1 kφv (g, t−1 (t1 , t2 ) ∈ Ev× × Ev× . 1 yt2 , q(t1 t2 )u),

Proof. These identities follow from the definition of Weil representation and some simple transformation of integrals. It suffices to assume that φv = φ1,v ⊗ φ2,v by linearity. Then it follows from similar results for the Whittaker  function Wa,v (s, g, u). We omit the proof. Now we take care of the contribution from the constant term E00 (0, g, u). It will simply vanish in some degenerate cases. Proposition 6.7. Under Assumption 5.3, X I 0 (0, g, φ) = − I 0 (0, g, φ)(v), ∀g ∈ P (FS1 )GL2 (AS1 ). v nonsplit

191

DERIVATIVE OF THE ANALYTIC KERNEL

Proof. By the assumption, φv (Ev , Fv× ) = 0 for any v ∈ S1 . By Lemma 5.9, we can simply assume that φ = φ1 ⊗ φ2 with φ2,v (0, Fv× ) = 0 for any v ∈ S1 . We will check that E00 (0, g, u) = log δ(g)r2 (g)φ2 (0, u) + W00 (0, g, u) vanishes at g ∈ P (FS1 )GL2 (AS1 ). It is immediate that r2 (g)φ2 (0, u) = 0 for g ∈ P (FS1 )GL2 (AS1 ). It remains to treat W00 (0, g, u). Take the derivative on Y 1 1 L(s, η) W0◦ (s, g, u) |Dv | 2 |dv | 2 L(s + 1, η) v L(s, η)/L(0, η) Y ◦ =− W0,v (s, g, u). L(s + 1, η)/L(1, η) v

W0 (s, g, u) = −

We obtain W00 (0, g, u) = −

  L(s, η) d |s=0 log W0◦ (0, g, u) ds L(s + 1, η) Y X ◦ 0 ◦ W0,v (0, g, u) W0,v − 0 (0, g, u). v 0 6=v

v

By Proposition 6.1, we get W00 (0, g, u) = −

  L(s, η) d |s=0 log r(g)φ2 (0, u) ds L(s + 1, η) X ◦ 0 W0,v (0, g, u)r(g v )φv2 (0, u). − v

Then r(g)φ2 (0, u) = 0 as above, and r(g v )φv2 (0, u) = 0 for any v since it has a  factor r(gv0 )φ2,v0 (0, u) = 0 for any v 0 ∈ S1 − {v}. 6.2

NON-ARCHIMEDEAN COMPONENTS

Assume that v is a non-archimedean placenonsplit in E. Resume the notations in the last section. We now consider the local kernel function kφv (g, y, u), which has the expression kφv (g, y, u) =

L(1, ηv ) 0 ◦ r(g)φ1,v (y1 , u)Wuq(y (0, g, u, φ2,v ), 2 ),v vol(Ev1 ) y = y1 + y2 ∈ Vv − V1v ,

if φv = φ1,v ⊗ φ2,v .

192 6.2.1

CHAPTER 6

Main results

Let v be a non-archimedean placenonsplit in E, and Bv be the quaternion division algebra over Fv non-isomorphic to Bv . Proposition 6.8. The following results are true: (1) Assume that v ∈ Snonsplit − S1 is unramified as in Assumption 5.5. Then kφv (1, y, u) = 1OB

× v ×OFv

(y, u)

v(q(y2 )) + 1 log Nv . 2

1

(2) Assume that v ∈ S1 so that φv ∈ S (Bv × Fv× ) satisfies Assumption 5.3. Then kφv (1, y, u) extends to a Schwartz function of (y, u) ∈ Bv × Fv× . We consider its consequences. We first look at (1). In that unramified case, it is easy to see that kφv (1, y, u) = kφv (g, y, u),

∀ g ∈ GL2 (OFv ).

Then by Iwasawa decomposition and Lemma 6.6, we know kr(t1 ,t2 )φv (g, y, u) explicitly for all (g, (t1 , t2 )). It will cancel the local height of CM points at v. Now we consider a place v that does not satisfy the conditions in (1). Then the computation of kφv may be very complicated or useless. It is better to consider the whole series X X (v) kr(t1 ,t2 )φv (g, y, u) r(g, (t1 , t2 ))φv (y, u). Kφ (g, (t1 , t2 )) = u∈µ2U \F × y∈V −V1

It looks like a theta series. We call it a pseudo-theta series. It has a strong connection with the usual theta series. In (2), we have shown that kφv (y, u) := kφv (1, y, u) extends to a Schwartz function for (y, u) ∈ Vv × Fv× under Assumption 5.3. We did this because we want to compare the above pseudo-theta series with the usual theta series

=

θ(g, (t1 , t2 ), kφv ⊗ φv ) X X r(g, (t1 , t2 ))kφv (y, u) r(g, (t1 , t2 ))φv (y, u). u∈µ2U \F × y∈V

It seems that these two series have a good chance to equal if gv = 1. In fact, it is supported by the equality r(t1 , t2 )kφv (y, u) = kr(t1 ,t2 )φv (1, y, u) shown in Lemma 6.6. Another difficulty for them to be equal is that the summations of y are over slightly different spaces. This problem is also solved by Assumption 5.3. In fact, the assumption implies that r(t1 , t2 )φv (y, u) = 0 for all y ∈ E, since it has a factor r(t1 , t2 )φv0 (y, u) = 0 for any v 0 ∈ S1 − {v}. Therefore, the two series are equal if gS1 = 1. We can also extend the equality to P (FS1 )GL2 (AS1 ) by Lemma 6.6.

193

DERIVATIVE OF THE ANALYTIC KERNEL 1

Corollary 6.9. Let v ∈ S1 so that φv ∈ S (Bv × Fv× ) satisfies Assumption 5.3. Then (v) Kφ (g, (t1 , t2 )) = θ(g, (t1 , t2 ), kφv ⊗ φv ) for all (g, (t1 , t2 )) ∈ P (FS1 )GL2 (AS1 ) × T (A) × T (A). (v)

In that situation, we say Kφ is approximated by θ(kφv ⊗ φv ). They are usually not equal, unless we know the modularity of the pseudo-theta series. 6.2.2

The computation

To prove Proposition 6.8, we first show a formula for the Whittaker function ◦ (s, 1, u) in the most general case. Wa,v Proposition 6.10. Let v be any non-archimedean place of F . (1) For any a ∈ Fv , ◦ (s, 1, u) Wa,v

=

1

|dv | 2 (1 − Nv−s )

∞ X n=0

Nv−ns+n

Z φ2,v (x2 , u)du x2 , Dn (a)

where du x2 is the self-dual measure of (V2,v , uq) and Dn (a) = {x2 ∈ V2,v : uq(x2 ) ∈ a + pnv d−1 v }. (2) Assume that φ2,v (x2 , u) = 0 if v(uq(x2 )) > −v(dv ). Then there is a con◦ (s, 1, u) = 0 identically for all a ∈ Fv satisfying stant c > 0 such that Wa,v v(a) > c or v(a) < −c. Proof. We first compute (1). Recall that Z δ(wn(b)g)s r(wn(b)g)φ2,v (0, u)ψv (−ab)db. Wa,v (s, g, u) = Fv

Expand the action of w in terms of the Fourier transform. We obtain Z Z Wa,v (s, g, u) = γu,v δ(wn(b)g)s r(g)φ2,v (x2 , u)ψv (b(uq(x2 ) − a))du x2 db. Fv

V2,v

Here du x2 is the self-dual measure for (V2,v , uq). It follows that Z Z ◦ s δ(wn(b)) φ2,v (x2 , u)ψv (b(uq(x2 ) − a))du x2 db. Wa,v (s, 1, u) = Fv

V2,v

It suffices to verify the formulae for Whittaker functions under the condition that u = 1. The general case is obtained by replacing q by uq and φ2,v (x2 ) by

194

CHAPTER 6

φ2,v (x2 , u). We will drop the dependence on u to simplify the notation. Then we write Z Z ◦ (s, 1) = δ(wn(b))s φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db. Wa,v Fv

V2,v

By 

if b ∈ OFv , otherwise,

1 |b|−1

δ(wn(b)) =

we will split the integral over Fv into the sum of an integral over OFv and an integral over Fv − OFv . Then Z Z ◦ φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db Wa,v (s, 1) = OFv

V2,v

Z

|b|−s

+

Z φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db.

Fv −OFv

V2,v

The second integral can be decomposed as Z ∞ Z X Nv−ns φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db −n −n+1 V2,v n=1 pv −pv Z Z ∞ X = Nv−ns φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db −n p V2,v v n=1 Z ∞ Z X − Nv−ns φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db. −(n−1) V2,v n=1 pv

Combine with the first integral to obtain Z ∞ Z X ◦ −ns Nv φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db Wa,v (s, 1) = −n

n=0 pv ∞ Z X

V2,v

−

n=0

p−n v

=(1 − Nv−s )

Nv−(n+1)s

Z φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db V2,v

∞ X n=0

Nv−ns

Z

Z

p−n v

φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db. V2,v

As for the last double integral, change the order of the integration. The integral on b is nonzero if and only if q(x2 ) − a ∈ pnv d−1 v . Here dv is the local different of F over Q, and also the conductor of ψv . Then we have Z ∞ X ◦ (s, 1) = (1 − Nv−s ) Nv−ns vol(p−n ) φ2,v (x2 )dx2 Wa,v v Dn (a)

n=0

=

1

|dv | 2 (1 − Nv−s )

∞ X n=0

Nv−ns+n

Z φ2,v (x2 )dx2 . Dn (a)

195

DERIVATIVE OF THE ANALYTIC KERNEL

It proves (1). Now we show (2) using (1). The key is that only those Dn (a) with n ≥ 0 are involved in the formula. Recall that Dn (a) = {x2 ∈ V2,v : uq(x2 ) ∈ a + pnv d−1 v }. If v(a) < −v(dv ), then for every x2 ∈ Dn (a), we have v(uq(x2 )) = v(a). ◦ (s, 1, u) = 0 if v(a) Then φ2,v (x2 , u) = 0 if v(a) is too small. It follows that Wa,v is too small. This is apparently true for all Schwartz function φv . If v(a) ≥ −v(dv ), then for every x2 ∈ Dn (a), we have v(uq(x2 )) ≥ −v(dv ). ◦ By the assumption, φ2,v (·, u) is zero on Dn (a). In that case, Wa,v (s, 1, u) = 0 identically. It proves the result.  Proof of Proposition 6.8. Both results are obtained as a consequence of Proposition 6.10. We first look at (2). By linearity and Lemma 5.9, it suffices to consider the case that φv = φ1,v ⊗ φ2,v with φ2,v satisfies the condition of Proposition 6.10 (2). Set kφv (1, y, u) to be zero if y2 = 0. It is easy to see that it gives a Schwartz function by Proposition 6.10 (2). Now we consider (1). It suffices to show that for any a ∈ Fu (v), ◦ 0 (0, 1, u) = 1OFv (a)1O× (u) Wa,v Fv

v(a) + 1 (1 + Nv−1 ) log Nv . 2

Use the formula in Proposition 6.10. We need to simplify Dn (a) = {x2 ∈ V2,v : uq(x2 ) − a ∈ pnv }. We first have v(a) 6= v(q(x2 )) because a is not represented by uq(x2 ). Actually v(q(x2 )) is always even and v(a) must be odd. Then v(q(x2 ) − a) = min{v(a), v(q(x2 ))},

∀ x2 ∈ V2,v .

We see that Dn (a) is empty if v(a) < n. Otherwise, it is equal to Dn := {x2 ∈ V2,v : uq(x2 ) ∈ pnv }. It follows that v(a) ◦ Wa,v (s, 1, u)

=

(1 −

Nv−s )

X

Nv−ns+n

Z φ2,v (x2 , u)du x2 . Dn

n=0

It is a finite sum and we do not have any convergence problem. Then v(a) ◦ 0 (0, 1, u) Wa,v

=

log Nv

X n=0

Nvn

Z φ2,v (x2 , u)du x2 . Dn

It is nonzero only if u ∈ OF×v and a ∈ OFv . Identify V2,v with Ev . Then [ n+1 2 ]

Dn = {x2 ∈ Ev : q(x2 ) ∈ pnv } = pv

OEv ,

196

CHAPTER 6 −2[ n+1 ]

2 and vol(Dn ) = Nv . Note that v(a) is odd since it is not represented by q2 . Then it is easy to have

v(a) ◦ 0 (0, 1, u) = log Nv Wa,v

X

n−2[ n+1 2 ]

Nv

v(a) + 1 (1 + Nv−1 ). 2

=

n=0

 6.3

ARCHIMEDEAN COMPONENTS

For an archimedean place v, the quaternion algebra Bv is isomorphic to the Hamiltonian quaternion. We will compute kφv (g, y, u) for standard φv introduced in §4.1. The computation here is done by Proposition 2.11. The result involves the exponential integral Ei defined by Z z t e dt, z ∈ C. Ei(z) = −∞ t Another expression is z

Z Ei(z) = γ + log(−z) + 0

et − 1 dt, t

where γ is the Euler constant. It follows that it has a logarithmic singularity near 0. This fact is useful when we compare the result here with the archimedean local height, since we know that Green’s functions have a logarithmic singularity. Proposition 6.11.  − 1 Ei(4πuq(y )y ) |y |e2πiuq(y)(x0 +iy0 ) e2iθ 2 0 0 kφv (g, y, u) = 2 0

if uy0 > 0, if uy0 < 0,

for any  g=



z0 z0

y0

x0 1



cos θ − sin θ

sin θ cos θ

 ∈ GL2 (Fv )

in the form of the Iwasawa decomposition. Proof. It suffices to show the formula in the case g = 1: ( − 12 Ei(4πuq(y2 ))e−2πuq(y) if u > 0; kφv (1, y, u) = 0 if u < 0. The general case is obtained by Proposition 6.6 and the fact that r(kθ )φv = e2iθ φv .

197

DERIVATIVE OF THE ANALYTIC KERNEL

It amounts to show that, for any a ∈ Fu (v) (i.e., ua < 0), ( −πe−2πa Ei(4πa) if u > 0; ◦ 0 Wa,v (0, 1, u) = 0 if u < 0. The case u < 0 is trivial. The essential case is u > 0 and a < 0. It is just the case of d = 2 in Proposition 2.11 (3).  6.4

HOLOMORPHIC PROJECTION

In this section we consider the general theory of holomorphic projection which we will apply to the form I 0 (0, g, χ) in the next section. Denote by A(GL2 (A), ω) the space of automorphic forms of central character ω, by A0 (GL2 (A), ω) the (2) subspace of cusp forms, and by A0 (GL2 (A), ω) the subspace of holomorphic cusp forms of parallel weight two. The usual Petersson inner product is just Z (f1 , f2 )pet = f1 (g)f2 (g)dg, f1 , f2 ∈ A(GL2 (A), ω). Z(A)GL2 (F )\GL2 (A) (2)

Denote by Pr : A(GL2 (A), ω) → A0 (GL2 (A), ω) the orthogonal projection. Namely, for any f ∈ A(GL2 (A), ω), the image Pr(f ) is the unique form in (2) A0 (GL2 (A), ω) such that (Pr(f ), ϕ)pet = (f, ϕ)pet ,

(2)

∀ϕ ∈ A0 (GL2 (A), ω).

We simply call Pr(f ) the holomorphic projection of f . Apparently Pr(f ) = 0 if f is an Eisenstein series. 6.4.1

A general formula

For any automorphic form f for GL2 (A) we define a Whittaker function Z [F :Q] (2) W (g∞ ) δ(h)s fψ (gh)W (2) (h)dh. fψ,s (g) = (4π) Z(F∞ )N (F∞ )\GL2 (F∞ )

Here W (2) is the standard holomorphic Whittaker function of weight two at infinity, and fψ denotes the Whittaker function of f with respect to the character ψ : F \A → C× . ior

Proposition 6.12. Let f ∈ A(GL2 (A), ω) be a form with asymptotic behav   a 0 f g = Og (|a|1− A ) 0 1

as a ∈ A× , |a|A → ∞ for some  > 0. Then the holomorphic projection Pr(f ) has the Whittaker function Pr(f )ψ (g) = lim fψ,s (g). s→0

198

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Proof. For any Whittaker function W of GL2 (A) with decomposition W (g) = W (2) (g∞ )Wf (g) such that W (2) (g∞ ) is standard holomorphic of weight 2 and that Wf (g) is compactly supported modulo Z(Af )N (Af ), the Poincar´e series is defined as X ϕW (g) := lim W (γg)δ∞ (γg)s , s→0+

γ∈Z(F )N (F )\G(F )

where 1 2

δ∞ (g) = |a∞ /d∞ | ,

g∞

 a∞ = 0

 b∞ k d∞ ∞

with k∞ the maximal compact subgroup of GL2 (F∞ ). Assume that W and f have the same central character. Since f has asymptotic behavior as in the proposition, their inner product can be computed as follows: Z (f, ϕW )pet = f (g)ϕW (g)dg Z(A)GL2 (F )\GL2 (A) Z = lim f (g)W (g)δ∞ (g)s dg s→0 Z(A)N (F )\GL (A) 2 Z = lim fψ (g)W (g)δ∞ (g)s dg. s→0

Z(A)N (A)\GL2 (A)

We may apply this formula to Pr(f ) which has the same inner product with ϕW as f . Write Pr(f )ψ (g) = W (2) (g∞ )Pr(f )ψ (g). Then the above integral is a product of integrals over finite places and integrals at infinite places: Z Z ∞ |W (2) (g)|2 dg = y 2 e−4πy dy/y 2 = (4π)−1 . Z(R)N (R)\GL2 (R)

0

In other words, we have (f, ϕW )pet = (4π)

−[F :Q]

Z Pr(f )ψ (g)Wf (g)dg. Z(Af )N (Af )\GL2 (Af )

As W f can be any Whittaker function with compact support modulo Z(Af )N (Af ), the combinations of the above formulae give the proposition.  We introduce an operator Pr0 formally defined on the function space of N (F )\GL2 (A). For any function f : N (F )\GL2 (A) → C, denote as above Z [F :Q] (2) fψ,s (g) = (4π) W (g∞ ) δ(h)s fψ (gh)W (2) (h)dh Z(F∞ )N (F∞ )\GL2 (F∞ )

199

DERIVATIVE OF THE ANALYTIC KERNEL

if it has meromorphic continuation around s = 0. Here fψ denotes the first Fourier coefficient of f . Denote f s→0 fψ,s (g), Pr0 (f )ψ (g) = lim f s→0 denotes the constant term of the Laurent exwhere the “quasi-limit” lim pansion at s = 0. Finally, we write X Pr0 (f )ψ (d∗ (a)g). Pr0 (f )(g) = a∈F ×

The above result is just Pr(f ) = Pr0 (f ) under the growth condition. In general, Pr0 (f ) is not automorphic when f is automorphic but fails the growth condition of Proposition 6.12. 6.4.2

Growth of the kernel function

Now we want to consider the growth of I 0 (0, g, χ, φ) so that we can apply the formula. The growth condition in Proposition 6.12 is not satisfied for general φ ∈ S(B × A× ). However, we will prove that Assumption 5.4 miraculously fits the growth condition! Recall from (5.1.2) that Z ∗ I(s, g, r(t, 1)φ) χ(t)dt. I(s, g, χ, φ) = T (F )\T (A)/Z(A)

It has central character χ|A× = ωσ−1 . The integral is essentially a finite sum. So F the growth of I 0 (0, g, χ, φ) is reduced to the growth of I 0 (0, g, φ). If φ = φ1 ⊗ φ2 , we have X θ(g, u, φ1 )E(s, g, u, φ2 ). I(s, g, φ) = u∈µ2U \F ×

Denote the “absolute constant term” I00 (s, g, φ) :=

X

I00 (s, g, u, φ)

u∈µ2U \F ×

where I00 (s, g, u, φ) := θ0 (g, u, φ1 )E0 (s, g, u, φ2 ) is the product of the constant terms of θ(g, u, φ1 ) and E(s, g, u, φ2 ). Equivalently, I00 (s, g, u, φ) = δ(g)s r(g)φ(0, u) + r1 (g)φ1 (0, u)W0 (s, g, u, φ2 ). Taking integration, we obtain Z ∗ I00 (s, g, χ, φ) = T (F )\T (A)/Z(A)

I00 (s, g, r(t, 1)φ) χ(t)dt.

200

CHAPTER 6

Note that we use the notation I00 since it is only a part of the constant term I0 of I. For general φ ∈ S(B × A× ), extend the definition of I00 (s, g, u, φ) and I00 (s, g, φ) by linearity. Alternatively, we can use the formula I00 (s, g, u, φ) = δ(g)s r(g)φ(0, u) Z Z − δ(wn(b)g)s A

r(g)φ(x2 , u)ψ(buq(x2 ))du x2 db.

V2

Lemma 6.13. For any φ ∈ S(B × A× ), the difference 0 I 0 (0, g, χ, φ) − I00 (0, g, χ, φ)

decays exponentially in the sense that there exists g > 0 depending on g such that 0 I 0 (0, d∗ (a)g, χ, φ) − I00 (0, d∗ (a)g, χ, φ) = O(e−g |a|A ),

a ∈ A× .

Proof. It suffices to consider the case φ = φ1 ⊗ φ2 . Fix an archimedean place τ of F . First, we reduce the problem to the case that a ∈ Fτ× . Fix g and view 0 Ig (a) := I 0 (0, d∗ (a)g, χ, φ) − I00 (0, d∗ (a)g, χ, φ)

as a function of a ∈ A× . It is easy to verify that Ig (a) is invariant under the action of F × on a. By smoothness, there exists a nontrivial open and compact subgroup Q of A× f (depending on g) such that Ig (a) is invariant under the action of Q on a. The quotient F × \A× /Fτ× Q is finite. Fix a set of representatives ai ∈ A× . It suffices to prove the result for a in each single coset ai F × Fτ× Q. By the invariance of Ig (a) and |a|A under F × Q, we can assume that a ∈ ai Fτ× . The problem in this case becomes Ig (ai b) = O(e−g |ai b|A ),

b ∈ Fτ× .

It is just Id∗ (ai )g (b) = O(e−g |b|τ ),

b ∈ Fτ× .

The problem is reduced to Fτ× . Second, reduce the problem to θ(d∗ (a)g, u, φ1 )E 0 (0, d∗ (a)g, u, φ2 ) − θ0 (d∗ (a)g, u, φ1 )E00 (0, d∗ (a)g, u, φ2 ) = O(e−g,u |a|τ ),

a ∈ Fτ× .

It follows from the fact that I 0 (0, g, χ, φ) is essentially a finite sum of I 0 (0, g, r(t, 1)φ), and the fact that I 0 (0, g, φ) is essentially a finite sum of

201

DERIVATIVE OF THE ANALYTIC KERNEL

θ(g, u, φ1 )E 0 (0, g, u, φ2 ). The first finite sum is independent of g, and the second finite sum is independent of g∞ . Finally, it is classical that θ(d∗ (a)g, u, φ1 ) − θ0 (d∗ (a)g, u, φ1 ) and 0 E (0, d∗ (a)g, u, φ2 )−E00 (0, d∗ (a)g, u, φ2 ) decay exponentially. It yields the result combining with 1

θ0 (d∗ (a)g, u, φ1 )

=

Og,u (|a|τ2 ),

a ∈ Fτ× ,

E00 (0, d∗ (a)g, u, φ2 )

=

Og,u (|a|τ2 log |a|τ ),

1

a ∈ Fτ× . 

Proposition 6.14. Under Assumption 5.4, 0 I00 (0, g, u, φ) = 0,

∀ g ∈ GL2 (A), u ∈ F × .

Therefore, there exists g > 0 depending on g such that I 0 (0, d∗ (a)g, χ, φ) = O(e−g |a|A ),

a ∈ A× .

Proof. It suffices to prove the first statement. It amounts to get a more 0 explicit expression of I00 (0, g, u, φ). For any φ ∈ S(B × A× ) of the form φ = φ1 ⊗ φ2 , 0 I00 (0, g, u, φ) = log δ(g) r(g)φ(0, u) + r1 (g)φ1 (0, u)W00 (0, g, u, φ2 ).

Recall that the computation in Proposition 6.7 gives X ◦ 0 W00 (0, g, u) = −c0 r(g)φ2 (0, u) − W0,v (0, gv , u)r2 (g v )φ2,v (0, u) v

with the constant

  L(s, η) d |s=0 log . c0 = ds L(s + 1, η)

It follows that 0 I00 (0, g, u, φ) = (log δ(g) − c0 )r(g)φ(0, u) X ◦ 0 r(g v )φv (0, u) · r1 (gv )φ1,v (0, u)W0,v (0, gv , u, φ2,v ). − v

By linearity, ◦ 0 κ◦φv (gv , u) := r1 (gv )φ1,v (0, u)W0,v (0, gv , u, φ2,v )

makes sense even if φv is not of the form φ1,v ⊗ φ2,v . Thus we always have X 0 (0, g, u, φ) = (log δ(g) − c0 )r(g)φ(0, u) − r(g v )φv (0, u) κ◦φv (g, u). I00 v 0 Now I00 (0, g, u, φ) = 0 follows easily from Assumption 5.4.



202 6.4.3

CHAPTER 6

The work of Gross–Zagier and S. Zhang

Here we recall the treatment of the holomorphic projection of I 0 (0, g, χ, φ) in [GZ] and [Zh1, Zh2, Zh3]. Without Assumption 5.4, the best bound is 0 I00 (0, d∗ (a)g, χ, φ) = Og (|a|A log |a|A ).

Then Lemma 6.13 gives I 0 (0, d∗ (a)g, χ, φ) = Og (|a|A log |a|A ). Thus I 0 (0, g, χ, φ) fails the growth condition of Proposition 6.12. To apply the proposition, let J (s, g, u, φ) be the Eisenstein series formed by I00 (s, g, u, φ), i.e., X J (s, g, u, φ) := I00 (s, γg, u, φ). γ∈P (F )\GL2 (F )

Take integrations to get J (s, g, φ) :=

X

J (s, g, u, φ),

u∈µ2U \F ×

Z

∗

J (s, g, χ, φ) :=

J (s, g, r(t, 1)φ)χ(t)dt. T (F )\T (A)/Z(A)

Then I 0 (0, g, χ, φ) − J 0 (0, g, χ, φ) satisfies the growth condition, and we can apply the proposition to it. Since J 0 (0, g, χ) is orthogonal to all cusp forms, the definition of holomorphic projection yields PrI 0 (0, g, χ) = Pr(I 0 (0, g, χ) − J 0 (0, g, χ)). The right-hand side can be computed as Pr0 (I 0 (0, g, χ) − J 0 (0, g, χ)) = Pr0 I 0 (0, g, χ) − Pr0 J 0 (0, g, χ). Then one has to treat the “extra term” Pr0 J 0 (0, g, χ). This extra term essentially “cancels” the height pairings involving the Hodge class. In [GZ], the cancellation is verified by explicit computation. In [Zh1, Zh2, Zh3], the cancellation is proved by a theory of derivation after detailed analysis of the structure of Pr0 J 0 (0, g, χ). 6.5

HOLOMORPHIC KERNEL FUNCTION

By Proposition 6.14, we can apply Proposition 6.12 to I 0 (0, g, χ). We get Z ∗ 0 Pr0 I 0 (0, g, r(t, 1)φ) χ(t)dt. PrI (0, g, χ) = T (F )\T (A)/Z(A)

203

DERIVATIVE OF THE ANALYTIC KERNEL

It suffices to compute Pr0 I 0 (0, g, r(t, 1)φ) or just Pr0 I 0 (0, g, φ), where the operator Pr0 is defined after Proposition 6.12. Recall that in Proposition 6.7, we have the simple decomposition X I 0 (0, g, φ)(v), ∀g ∈ P (FS1 )GL2 (AS1 ). I 0 (0, g, φ) = − v nonsplit

It is true under Assumption 5.3. Proposition 6.15. Under Assumption 5.3, Pr0 (I 0 (0, g, φ)) = −

X

X

I 0 (0, g, φ)(v) −

v|∞

I 0 (0, g, φ)(v),

v-∞ nonsplit

∀g ∈ P (FS1 )GL2 (AS1 ). Here I 0 (0, g, φ)(v) is the same as in Proposition 6.5, and for any archimedean v, Z (v) 0 I (0, g, φ)(v) = 2 Kφ (g, (t, t))dt, Z(A)T (F )\T (A) (v) Kφ (g, (t1 , t2 ))

=

X a∈F ×

kv,s (y) =

X

f s→0 lim

Γ(s + 1) 2(4π)s

r(g, (t1 , t2 ))φ(y)a kv,s (y),

× y∈µU \(B(v)× + −E )

Z 1

∞

1 dt. t(1 − λ(y)t)s+1

Here λ(y) = q(y2 )/q(y) is viewed as an element of Fv , and φ(y)a = φ(y, aq(y)−1 ) is as before. Proof. It suffices to check that Pr0 (I 0 (0, g)(v)) = I 0 (0, g)(v) for finite v, and Pr0 (I 0 (0, g)(v)) = I 0 (0, g)(v) for infinite v. They actually hold for all g ∈ GL2 (A). By Proposition 6.5, Z (v) Kφ (g, (t, t))dt I 0 (0, g)(v) = 2 Z(A)T (F )\T (A)

with (v)

Kφ (g, (t1 , t2 )) =

X

X

kr(t1 ,t2 )φv (g, y, u)r(g, (t1 , t2 ))φv (y, u).

u∈µ2U \F × y∈B(v)−E

Note that the integral above is just a finite sum. We have a simple rule r(n(b)g, (t1 , t2 ))φv (y, u) = ψ(uq(y)b) r(g, (t1 , t2 ))φv (y, u),

204

CHAPTER 6

and its analogue kr(t1 ,t2 )φv (n(b)g, y, u) = ψ(uq(y)b)kr(t1 ,t2 )φv (g, y, u) shown in Proposition 6.6. By these rules it is easy to see that the first Fourier coefficient is given by (v)

=

Kφ (g, (t1 , t2 ))ψ X

kr(t1 ,t2 )φv (gv , yv , uv )r(g, (t1 , t2 ))φv (y, u).

(y,u)∈µU \((B(v)−E)×F × )1

If v is non-archimedean, all the infinite components are already holomorphic (v) of weight two. So the operator Pr0 does not change Kφ (g, (t1 , t2 ))ψ at all. Thus X (v) (v) Kφ (d∗ (a)g, (t1 , t2 ))ψ . Pr0 (Kφ (g, (t1 , t2 ))) = a∈F × (v)

It is easy to check that it is exactly equal to Kφ (g, (t1 , t2 )) by Proposition 6.6 that kr(t1 ,t2 )φv transforms according to the Weil representation under upper (v) triangular matrices. We conclude that Pr0 does not change Kφ (g, (t1 , t2 )), and thus we have Pr0 (I 0 (0, g)(v)) = I 0 (0, g)(v). Now we look at the case that v is archimedean. The only difference is that we kφv ,s (g, y, u), and then take a “quasi-limit” need to replace kφv (g, y, u) by some e f It suffices to consider the case that uq(y) = 1. It is given by lim. Z dy0 (2) e y0s e−2πy0 kφv (d∗ (y0 ), y, u) . kφv ,s (g, y, u) = 4πW (gv ) y0 Fv,+ Then e kφv ,s (g, y, u) 6= 0 only if u > 0, since kφv (d∗ (y0 ), y, u) 6= 0 only if u > 0. Assume that u > 0, which is equivalent to q(y) > 0 since we assume uq(y) = 1 for the moment. By Proposition 6.11, Z dy0 y0s e−2πy0 kφv (d∗ (y0 ), y, u) y0 Fv,+ Z dy0 1 = − y s e−2πy0 Ei(4πuq(y2 )y0 ) y0 e−2πy0 2 Fv,+ 0 y0 Z ∞ 1 dy0 q(y2 ) > 0) = − y0s+1 e−4πy0 Ei(−4παy0 ) (α = −uq(y2 ) = − 2 0 y0 q(y) Z ∞ Z ∞ 1 dy0 = y0s+1 e−4πy0 t−1 e−4παy0 t dt 2 0 y0 1 Z ∞ Z ∞ dy0 1 = t−1 y0s+1 e−4π(1+αt)y0 dt 2 1 y0 0 Z 1 Γ(s + 1) ∞ dt. = s+1 2(4π) t(1 + αt)s+1 1

205

DERIVATIVE OF THE ANALYTIC KERNEL

Hence, Γ(s + 1) e kφv ,s (g, y, u) = W (2) (gv ) 2(4π)s

Z

∞

1

1 t(1 −

q(y2 ) s+1 q(y) t)

dt = W (2) (gv )kv,s (y).

This matches the result in the proposition. Since kv,s (y) is invariant under the multiplication action of F × on y, it is easy to get (v)

Pr0 (Kφ (g, (t1 , t2 ))) Pr0 (I 0 (0, g)(v))

(v)

= Kφ (g, (t1 , t2 )), =

I 0 (0, g)(v). 

Chapter Seven Decomposition of the Geometric Kernel Let φ = φf ⊗ φ∞ ∈ S(V × A× )U ×U be a Schwartz function with standard φ∞ . Assume that −1 ∈ / U to simply notations. Recall that in §5.1 we have introduced the generating series Z(g, φ)U = Z0 (g, φ)U + Z∗ (g, φ)U ,

g ∈ GL2 (A).

Here the non-constant part Z∗ (g, φ)U =

X

X

r(g)φ(x)a Z(x)U .

a∈F × x∈K\B× f

We further have the height series as follows: Z(g, (h1 , h2 ), φ)U

=

Z(g, χ, φ)U

=

hZ(g, φ)U [h1 ]◦U , [h2 ]◦U iNT , h1 , h2 ∈ B× ; Z ∗ Z(g, (t, 1), φ)U χ(t)dt. T (F )\T (A)/Z(A)

By Lemma 3.19, Z(g, (h1 , h2 ), φ)U is cuspidal in g. So we can replace Z(g, φ)U by Z∗ (g, φ)U in the definition of Z(g, (h1 , h2 ), φ)U . The constant term Z0 (g, φ)U will be ignored in the rest of this book. The goal of this chapter is to decompose the height series Z(g, (t1 , t2 ), φ)U = hZ∗ (g, φ)U [t1 ]◦U , [t2 ]◦U iNT ,

t1 , t2 ∈ B× .

We presume the assumptions in §5.2. Then there is not horizontal self intersection in the above pairing, and the intersections with the Hodge bundles are zero. Then we have a decomposition to a sum of local heights by standard results in Arakelov theory. In §7.1, we review the definition of the N´eron–Tate height, and how to compute it by the arithmetic Hodge index theorem. When there is no horizontal self-intersection, the height pairing automatically decomposes to a summation of local pairings. In §7.2, we decompose the height series Z(g, (t1 , t2 ), φ) into local heights. To do the decomposition, we describe the arithmetic models we are using. In §7.3, we prove that the contribution of the Hodge bundles in the height series is zero. It is true under the assumptions in §5.2. In §7.4, we compare two kernel functions and state the computational result (Theorem 7.8) from the next chapter. We deduce the kernel identity from Theorem 7.8.

207

DECOMPOSITION OF THE GEOMETRIC KERNEL

7.1

´ NERON–TATE HEIGHT

In this section we review some basic theory of the N´eron–Tate heights, the arithmetic intersection theory on arithmetic surfaces by Arakelov [Ar] and GilletSoul´e [GS], the arithmetic Hodge index theorem by Faltings [Fa2] and Hriljac [Hr], and the notion of admissible arithmetic extension by Zhang [Zh2]. The Hodge index theorem gives a way to “compute” the N´eron–Tate height by flat arithmetic extensions. Flat arithmetic extensions are only for divisors of degree zero, so it is not enough in the computation. It is the reason to use admissible arithmetic extensions. If the divisors are disjoint on the generic fiber, we have a natural decomposition of the N´eron–Tate height into local heights. 7.1.1

N´ eron–Tate height on abelian varieties

All the results here can be found in Serre’s book [Se]. In the following, fix a number field F . Our normalization of heights depends on F . The standard height on the projective space PnF is a map h : Pn (F ) → R defined by X 1 h(x) := log max{|x0 |w , |x1 |w , · · · , |xn |w }. [K : F ] w Here we represent x = (x0 , · · · , xn ) in terms of the homogeneous coordinate, the field K is any finite extension of F containing all x0 , · · · , xn , and the summation is over all places w of K with normalized absolute value | · |w . The choices of x0 , · · · , xn and K are not unique, but the definition is independent of the choices. Let Y be a projective variety over F , and L be a line bundle on Y . There is a Weil height hL : Y (F ) → R associated to L. It is uniquely determined by L up to a bounded function on Y (F ). To review the definition, we first assume that L is very ample. In that case, choose an embedding i : Y ,→ Pn such that i∗ O(1) ' L, and set hL to be the composition i

h

Y (K) −→ Pn (K) −→ R. The function hL differs by a bounded function for a different choice of the ⊗(−1) for two very embedding i. For a general line bundle L, write L = L1 ⊗ L2 ample line bundles L1 and L2 on Y . Set hL = hL1 − hL2 . Similarly, hL differs by a bounded function for a different choice of (L1 , L2 ) and (hL1 , hL2 ). Let A be an abelian variety over F . Let L be a line bundle on A which is symmetric in the sense that [−1]∗ L ' L. Here [−1] denotes the endomorphism of A given by the inverse in the group law. Define the N´eron–Tate height b hL : A(F )−→R by ˆ L (x) := lim 1 hL (N x). h N →∞ N 2 Here hL : A(F ) → R is any Weil height associated to L defined above. The limit exists and does not depend on the choice of hL , so it is also called the canonical height.

208

CHAPTER 7

Theorem 7.1. Assume that L is ample and symmetric on A. Then the following are true: (1) One has b hL (x) ≥ 0 for any x ∈ A(F ). The equality holds if and only if x is a torsion point. (2) The function b hL : A(F )−→R is quadratic in the sense that b hL (x + y) + b hL (x − y) = 2b hL (x) + 2b hL (y),

∀x, y ∈ A(F ).

By the theorem, we can define a bilinear pairing on A(F ) by hx, yiL := b hL (x + y) − b hL (x) − b hL (y),

∀x, y ∈ A(F ).

It is positive definite up to torsion points. It is called the N´eron–Tate height pairing. By linearity, it extends to a unique positive definite Hermitian pairing on A(F )C . There is a refinement of the N´eron–Tate height pairing. Let A be as above, and let P be the Poincar´e line bundle on A × A∨ . Then P is symmetric and its N´eron–Tate height on A × A∨ gives a map b hP : A(F ) × A∨ (F )−→R. It is a bilinear pairing, and we also denote it by h·, ·iNT : A(F ) × A∨ (F )−→R. It is called the N´eron–Tate height pairing between A and A∨ . Go back to the situation of Theorem 7.1. The ample line bundle L gives a polarization φL : A−→A∨ , x 7−→ Tx∗ L ⊗ L⊗(−1) . The following result asserts the compatibility of the two pairings we have just defined. Theorem 7.2. Assume that L is ample and symmetric on A. Then hx, yiL = hx, φL (y)iNT ,

∀x, y ∈ A(F ).

Let f : A → B be a homomorphism of abelian varieties over F . The pullback f ∗ : Pic0 (B) → Pic0 (A) induces a homomorphism f ∨ : B ∨ → A∨ . The following projection formula asserts that f ∨ is the adjoint of f under the height pairing. Proposition 7.3. For any homomorphism f : A → B as above, hf (x), yiNT = hx, f ∨ (y)iNT ,

∀x ∈ A(F ), y ∈ B ∨ (F ).

Proof. It follows from the interpretation of the pairing in the theorem on page 37 of [Se], and the functoriality in the first theorem on page 35 of [Se]. 

DECOMPOSITION OF THE GEOMETRIC KERNEL

7.1.2

209

N´ eron–Tate height on curves

Let X be a smooth algebraic curve over F of genus g > 0. Let J = J(X) be the Jacobian variety of X. Let j : X ,→ J be the usual embedding sending x to the divisor class of x − a. Here a ∈ Div(XF ) is a fixed divisor of degree 1. Recall that the theta divisor Θ on J is the image of the finite map X g−1 −→J,

(x1 , · · · , xg−1 ) 7−→ x1 + · · · + xg−1 − (g − 1)a.

The theta divisor Θ is ample. It depends on the choice of a. The N´eron–Tate height pairing h·, ·iNT : J(F ) × J(F )−→R is defined to be the composition of the isomorphism (id, φΘ ) : J(F ) × J(F )−→J(F ) × J ∨ (F ) with the pairing h·, ·iNT : J(F ) × J ∨ (F )−→R introduced above. Here the polarization φΘ : J → J ∨ is the canonical isomorphism, which is independent of the choice of a. b = Θ + [−1]∗ Θ on J. It is symmetric Alternatively, define a new divisor Θ by definition. One can check that it is independent of the choice of a. It gives the N´eron–Tate height ˆ b : J(F )−→R. h Θ Then the N´eron–Tate height pairing h·, ·iNT : J(F ) × J(F )−→C is equal to the pairing 1 h·, ·iΘ b : J(F ) × J(F )−→C. 2 By the identification J(F ) = Pic0 (XF ), obtain h·, ·iNT : Pic0 (XF ) × Pic0 (XF )−→C. This is the pairing described by the arithmetic Hodge index theorem. If X is a disjoint union of projective smooth curves X1 , · · · , Xr over F , the above theory can be extended to X. Recall that Div0 (X) (resp. Pic0 (X)) are the subgroup of elements of Div(X) (resp. Pic(X)) with degree zero on every connected component of X. Then we have canonicalQdecompositions Div0 (X) = ⊕i Div0 (Xi ), Pic0 (X) = ⊕i Pic0 (Xi ) and J(X) = i J(Xi ). The N´eron–Tate height pairings on Pic0 (Xi ) extend uniquely to Pic0 (X) such that the direct sum Pic0 (X) = ⊕i Pic0 (Xi ) gives an orthogonal decomposition.

210 7.1.3

CHAPTER 7

Intersection theory on arithmetic surfaces

Let X be an arithmetic surface over OF . Namely, X is a projective and flat scheme over Spec(OF ) with absolute dimension two. Assume that X is regular, but allow it to be disconnected. Now we recall the arithmetic intersection theory defined by Gillet-Soul´e [GS]. b = (D, gD ) where D is a divisor on X , An arithmetic divisor on X is a pair D and gD : X (C) − |D(C)| → R is a Green’s function of D. Namely, gD is smooth with logarithmic singularity along |D(C)| such that the current ωD =

∂∂ gD + δD(C) πi

on X (C)` is actually a smooth (1, 1)-form on X (C). Note that X (C) = σ:F ,→C Xσ (C). So gD is just a collection of functions gD,σ : Xσ (C) − |Dσ (C)| satisfying similar properties. We call ωD the curvature b form of D. An arithmetic vertical divisor on X is a divisor of the form (V, f ) where f is a smooth function on X (C) and V is a vertical divisor on X in the sense that V is supported in finitely many fibers of X over Spec(OF ). In particular, (V, 0) itself is an arithmetic vertical divisor. We sometimes write V for (V, 0), and call it a finite vertical divisor. Similarly, we call (0, f ) an infinite vertical divisor. d ) the group of arithmetic divisors on X . For any rational Denote by Div(X function f on X , the divisor c ) := (div(f ), − log |f |∞ ) div(f is an arithmetic divisor. Here − log |f |∞ restricted to each Xσ (C) is just c ) is called an arithmetic − log |f |σ . An arithmetic divisor of the form div(f c ). The principle divisor. The group of principle divisors is denoted by Pr(X group of arithmetic divisor classes can be identified with the Picard group of isomorphism classes of Hermitian line bundles on X via the Chern class map c )−→Div(X d )/Pr(X c ), Pic(X

L 7→ c1 (L).

More precisely, for any Hermitian line bundle L = (L, k · k) on X , its arithmetic Chern class cˆ1 (L) is defined by cˆ1 (L) := (div(s), − log ksk)

c )). (mod Pr(X

c ) does not Here s is any nonzero rational section of L, and the class in Pic(X depend on the choice of s. In this way, we can view any Hermitian line bundle as an arithmetic divisor class on X . The arithmetic intersection is a symmetric pairing c ) × Pic(X c )−→R. Pic(X

211

DECOMPOSITION OF THE GEOMETRIC KERNEL

b 1 = (D1 , gD ) and D b 2 = (D2 , gD ) in Div(X d ) In the case that two divisors D 1 2 have no horizontal self-intersection in the sense that |D1,F | ∩ |D2,F | is empty, the intersection is defined by Z b 2 = (D1 · D2 ) + gD (D2 (C)) + b1 · D gD2 ωD1 . (7.1.1) D 1 X (C)

Here gD1 acts on (D2 (C)) by linearity, and the finite part X mv (D1 · D2 ) log Nv (D1 · D2 ) =

(7.1.2)

v-∞

sums over all finite primes v of F . For archimedean v, we take the convention that log Nv equals 1 (resp. 2) if v is real (resp. complex). For each v, the multiplicity X mv (D1 · D2 ) = mx (D1 · D2 ), x∈π −1 (v)

where π : X → Spec(OF ) denotes the structure morphism, and π −1 (v) is just the set of closed points of X lying above v. Here mx (D1 · D2 ) is the usual intersection multiplicity in algebraic geometry. In this case, we have a natural decomposition X b2 = b1 · D b 2 )v . b1 · D (D D v

If v is non-archimedean, (D1 · D2 )v = mv (D1 · D2 ) log Nv . If v is archimedean, Z b 2 )v = gD ,v (D2,v (C)) + b1 · D (D 1

gD2 ,v ωD1 ,v . Xv (C)

If there are horizontal self-intersections, the above formula does not work and we can use principal divisors to “move” D2 to convert to the above situation. We further remark that the definitions still make sense if we allow the finite part D1 , D2 to be divisors on X with coefficients in Q, R or C. In the case C, we make the intersection to be Hermitian. 7.1.4

Arithmetic Hodge index theorem

Let F be any number field, and X be a complete smooth curve over F . Note that X may not be connected. Recall that Div0 (X) is the group of divisors on X that has degree zero on every connected component of X, and Pic0 (X) is the

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group of rational equivalence classes in Div0 (X). The N´eron–Tate height gives a bilinear pairing h·, ·iNT on Pic0 (X). Let X be an integral model of X over OF , i.e., X is an arithmetic surface over OF with XF = X. Assume that X is regular with semistable reduction. An b on X is called flat if its intersection with any vertical divisor arithmetic divisor D on X is zero. Equivalently, its curvature form ωD is zero and its intersection with any finite vertical divisor on X is zero. b = (D, gD ) forces the generic fiber DF It is easy to see that the flatness of D 0 to lie in Div (X). Then it makes sense to talk about the N´eron–Tate height of DF . The following is the arithmetic Hodge index theorem of Faltings [Fa2] and Hriljac [Hr]. b 2 = (D2 , gD ) be two flat arithb 1 = (D1 , gD ) and D Theorem 7.4. Let D 1 2 metic divisors on X . Then b 2 = −hD1,F , D2,F iNT . b1 · D D To recover the N´eron–Tate height on Pic0 (XF ), it suffices to consider arithmetic intersections on integral models of X over all finite extensions L of F . The equality above will be modified by a factor [L : F ] due to our normalization. 7.1.5

Admissible arithmetic divisors

Let X be an integral model of X as above, and fix an arithmetic divisor class c ) whose generic fiber has degree one on any connected component of ξˆ ∈ Pic(X X. b = (D, gD ) be an arithmetic divisor on X . We can always write Let D D = H + V where H is the horizontal part of D, and V is the vertical part of D. Namely, every irreducible component of V is contained in a fiber of X over Spec(OF ), but every irreducible component of H is mapped surjectively to Spec(OF ) via the structure morphism X → Spec(OF ). The arithmetic divisor b b = (D, gD ) is called ξ-admissible D if the following conditions hold on each connected component of X : b − deg D · ξˆ is flat; • The difference D Z ˆ = 0 at any archimedean place v of F ; gD c1 (ξ) • The integral Xv (C)

ˆ v = 0 at any non-archimedean place v of F . • The intersection (V · ξ) b Any divisor D in Div(X) has a unique ξ-admissible extension, which we b b denote by D. It depends on ξ. If D belongs to Div0 (X), then the extension b b is apparently flat. It is easy to check that for two ξ-admissible D divisors b 2 = (D2 , gD ), D b1 = D b 2 if and only if D1 = D2 . b 1 = (D1 , gD ) and D D 1 2

DECOMPOSITION OF THE GEOMETRIC KERNEL

213

Our admissibility on the Green’s function is the same as what Arakelov [Ar] originally required, but we also put conditions on the finite part. By complex analysis, there is a unique symmetric function g : X(C) × X(C) − ∆−→R b depending on the curvature form of ξb which gives all the ξ-admissible Green’s functions. Here ∆ denotes the diagonal of X(C) × X(C). Namely, for any b b = (D, gD ), ξ-admissible divisor D gD (x) = g(D(C), z),

∀z ∈ X(C) − |D(C)|.

Here g(D(C), z) makes sense by linearity of divisors. See [Ar] for more information. b b 2 = (D2 , gD ), the b 1 = (D1 , gD ) and D For any two ξ-admissible divisors D 1 2 infinite part b1 · D b 2 )∞ = g(D1 (C), D2 (C)). (D In fact, it suffices to check Z gD2 ωD1 = 0. X (C)

By definition, ωD1 is a multiple of the curvature form ωξˆ of ξ on every connected b 2. component of X (C). Then the integration is zero by the admissibility of D 7.1.6

Admissible extension

c For the rest of this section, fix an arithmetic class ξˆ ∈ lim Pic(Y) whose generic −→ fiber has degree one on any geometrically connected component. Here the inverse limit is taking over all integral models Y of XL over OL for all extensions L/F , and the arrows between different models are the pull-back maps on arithmetic divisors. Let D ∈ Div(XF ). Assume that D is defined over a finite extension L of F and Y is an integral model of XL over OL . Assume further that ξˆ is represented ˆ Any arithmetic extension by an arithmetic divisor on Y. We still denote it by ξ. b = (D + V, gD ), where D is the Zariski closure of D, of D on Y is of the form D V is some finite vertical divisor, and gD is some Green’s function of D. There is b b is an ξ-admissible arithmetic divisor, a unique choice of V and gD such that D b and we call it the ξ-admissible extension of D on Y. Assume that D1 , D2 ∈ Div(XF ) are defined over L, and Y is an integral model over OL as above. Define a pairing hD1 , D2 i := −

1 b1 · D b 2. D [L : F ]

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b b 1 and D b 2 are the ξ-admissible Here D extensions on Y. When D1 , D2 ∈ 0 Div (XF¯ ), the pairing is just the N´eron–Tate height pairing by the arithmetic Hodge index theorem. b When varying L and Y, the ξ-admissibility is preserved by pull-backs. Hence b The pairing does not the pairing depends only on D1 , D2 and the choice of ξ. b c factor through Pic(XF ), since rational equivalence does not keep ξ-admissibility. 7.1.7

Decomposition of the pairing

There is a non-canonical decomposition h·, ·i = −i − j depending on the choice of integral models. Let X, ξb be as above. Fix a regular and semistable integral model Y0 of XL0 over OL0 for some field extension L0 of F with a fixed embedding L0 ,→ F . Assume that ξb is realized as a divisor class on Y0 . Let D1 , D2 ∈ Div(XF ) be two divisors with disjoint supports. Assume that D2 is defined over L0 . Let L be any field extension of L0 such that D1 is defined over L. Then we can decompose hD1 , D2 i according to the model Y0,OL . b i = (Di + Vi , gi ) be We first consider the case that Y0,OL is regular. Let D b b 2 = 0 since V1 is the ξ-admissible extensions on the model. Note that V1 · D ˆ b b orthogonal to both D2 − deg D2 · ξ and ξ. It follows that
1 hD1 , D2 i = − D1 · D2 + D1 · V2 + g(D1 (C), D2 (C)) . [L : F ] Here g is introduced above. Define 1 (D1 · D2 + g(D1 (C), D2 (C))), [L : F ] 1 D1 · V2 . j(D1 , D2 ) := [L : F ] i(D1 , D2 ) :=

Then we have a decomposition hD1 , D2 i = −i(D1 , D2 ) − j(D1 , D2 ). The decomposition is still valid even if Y0,OL is not regular. We have the b ξ-admissible extension of D2 on the regular model Y0 . Pull it back to Y0,OL . We b 2 = (D2 + V2 , g2 ) on Y0,O . All divisors on Y0 are Cartier get the extension D L divisors since it is regular. It follows that D2 and V2 are Cartier divisors on Y0,OL since they are pull-backs of Cartier divisors. Hence the intersections i(D1 , D2 ) and j(D1 , D2 ) are well-defined. To verify the equality, we first decompose the pairing on any desingularization of Y0,OL , and then use the projection formula. By the definition of arithmetic intersection theory above, we have further decompositions to local heights: X i(D1 , D2 ) = iw (D1 , D2 ) log Nw , w∈SL

j(D1 , D2 )

=

X w∈SL

jw (D1 , D2 ) log Nw .

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DECOMPOSITION OF THE GEOMETRIC KERNEL

Here SL denote the set of all places of F . The local multiplicities are defined according to (7.1.2). More precisely,  1  (D1 · D2 )w , w - ∞,  [L : F ] iw (D1 , D2 ) = 1   g(D1,w (C), D2,w (C)), w | ∞, [L : F ] and

1 (D1 · V2 )w , jw (D1 , D2 ) = [L : F ]  0,  

w - ∞, w | ∞.

Note that iw and jw are local intersection multiplicities of i and j over the model Y0,OL . It is convenient to rearrange the decomposition in terms of places of F . Namely, X X i(D1 , D2 ) = iv (D1 , D2 ) log Nv , j(D1 , D2 ) = jv (D1 , D2 ) log Nv v∈SF

v∈SF

with iv (D1 , D2 ) =

X 1 iw (D1 , D2 ), #SLv w∈SLv

X 1 jv (D1 , D2 ) = jw (D1 , D2 ). #SLv w∈SLv

Here SF denotes the set of all places of F , and SLv denotes the set of places of L lying over v. Fix an embedding F ,→ F v , or equivalently fix an extension v of the valuation v to F . By varying the fields as in the global case, we have well-defined It is the same as pairings iv and jv on Div(XF v ) for proper intersections. considering intersections on the model Y0,OF v . The formulae above have the following equivalent forms: Z iv (D1σ , D2σ )dσ, iv (D1 , D2 ) = Gal(F /F )

Z jv (D1 , D2 ) =

jv (D1σ , D2σ )dσ. Gal(F /F )

Here the integral on the Galois group takes the Haar measure with total volume one. If D1 , D2 have common irreducible components, we can define i(D1 , D2 ) and j(D1 , D2 ) by a slightly different method. The pairing i(D1 , D2 ) cannot be decomposed to a sum of local heights any more while the decomposition for j(D1 , D2 ) is still valid. This case does not happen in the computation of this book. In any case, jv is identically zero if v is archimedean or the model Y0 is smooth over all primes of L0 dividing v.

216 7.2

CHAPTER 7

DECOMPOSITION OF THE HEIGHT SERIES

Go back to the setting of Shimura curve XU . Our goal in the geometric side is to compute Z(g, (t1 , t2 )) = hZ∗ (g)(t1 − ξt1 ), t2 − ξt2 iNT ,

t1 , t2 ∈ CU .

Here we write ξt = ξU,q(t) as above. In this section, we decompose the above pairing into a sum of local heights and some global pairings with ξ. 7.2.1

Arithmetic model of the Shimura curve

Recall that we have a Shimura curve XU over F . For each archimedean place τ of F , the set of complex points under τ : F → C forms a Riemann surface uniformized by an = B(τ )× \H± × B× XU,τ f /U ∪ {cusps}. Here B(τ ) is the nearby quaternion algebra over F . We assume that U satisfies the conditions in Assumption 5.6. In particular, the last condition assures that the Hodge bundle LU = ωXU /F + {cusps}. Here ωXU /F is the canonical bundle, and each cusp has exactly multiplicity one. Recall that XU has a canonical regular and semistable integral model XU over OF . At each finite place v of F , the base change XU,OFv to OFv parametrizes p-divisible groups with level structures. Locally at a geometric point, XU,OFv is the universal deformation space of certain p-divisible groups. See [Car, Zh1] for example. The Hodge bundle LU can be canonically extended to a Hermitian line bundle on XU . Take advantage of the fact that LU is isomorphic to the sum of the canonical bundle ωXU /F with the cusps. Define LU to be the sum of the relative dualizing sheaf ωXU /OF with the Zariski closure of each cusp of XU in XU . It is an integral model of LU . Define b U := (LU , k · kPet ). L Here the Petersson metric k · kPet on LU`is defined such that its restriction to every connected component of XU (C) = τ :F ,→C XU,τ (C) takes the form kf (z)dzkPet = 4π · Imz · |f (z)|. See Lemma 3.1 for more details. Then we define 1 b ξbU = ◦ L U. κU Here κ◦U is the degree of LU on any geometrically connected component of XU as before.

217

DECOMPOSITION OF THE GEOMETRIC KERNEL

We will denote by h·, ·i the ξbU -admissible pairing explained in the last section. To consider the decomposition i + j and its corresponding local components, we need an integral model over a field where the CM points are rational. Let H = HU be the minimal field extension of E which contains the fields of definition of all t ∈ CU . Then H is an abelian extension over E given by the reciprocity law. We will use the regular integral model YU of XU over OH introduced in the following to get the decomposition i + j. Recall that we have assumed in Assumption 5.6 that Uv is of the form (1 + $vr OBv )× for every finite place v, where OBv is a maximal order of Bv . To describe YU , it suffices to describe the corresponding local model YU,w = YU × OHw for any finite place w of H. Let v be the place of F induced from w. Then YU,w is defined by the following process: • Let U 0 = U v Uv0 with Uv0 = OB×v the maximal compact subgroup. Then XU 0 has a canonical regular model XU 0 ,v over OFv . It is smooth if Bv is the matrix algebra. Let XU0 be the normalization of XU 0 ,v in the function field of XU,Hw . • Make a minimal desingularization of XU0 to get YU,w . With the model YU , we can always write hβ, ti = −i(β, t) − j(β, t),

β ∈ CMU , t ∈ CU .

We can further write i, j in terms of their corresponding local components if β 6= t. 7.2.2

Decomposition of the kernel function

Go back to Z(g, (t1 , t2 )) = hZ∗ (g)(t1 − ξt1 ), t2 − ξt2 iNT ,

t1 , t2 ∈ CU .

We first write Z(g, (t1 , t2 )) = hZ∗ (g)t1 , t2 i − hZ∗ (g)t1 , ξt2 i − hZ∗ (g)ξt1 , t2 i + hZ∗ (g)ξt1 , ξt2 i. We will prove that the last three terms are zero under Assumption 5.4. But we first look at the first term hZ∗ (g)t1 , t2 i. We claim that under Assumption 5.3, the pairing hZ∗ (g)t1 , t2 i does not involve any self-intersection for any g ∈ 1S1 GL2 (AS1 ). In other words, the multiplicity of [t2 ] in Z∗ (g)t1 is zero. Recall that X X r(g)φ(x)a [t1 x]. Z∗ (g)t1 = a∈F × x∈B× /U f

Note that the factor wU disappears since we assume −1 ∈ / U in Assumption 5.3 to simplify the notations.

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× Apparently [t1 x] = [t2 ] as CM points on XU if and only if x ∈ t−1 1 t2 E U . It follows that the coefficient of [t2 ] in Z∗ (g)t1 is equal to

X

X

r(g)φ(x)a

× a∈F × x∈t−1 1 t2 E U/U

=

X

X

r(g)φ(t−1 1 t2 y)a .

a∈F × y∈E × /(E × ∩U )

As in the analytic side, under Assumption 5.3, r(g)φ(t−1 1 t2 y)a = 0 for all y ∈ E(A) and g ∈ 1S1 GL2 (AS1 ). So there is no self-intersection under the assumption. Now the decomposition of hZ∗ (g)t1 , t2 i is routine. By the model Y over OH , we can decompose hZ∗ (g)t1 , t2 i = −i(Z∗ (g)t1 , t2 ) − j(Z∗ (g)t1 , t2 ). The j-part is always a sum of local pairings over places and Galois orbits, and so is the i-part if there are no self-intersections occurring. In the following we list decomposition of i(Z∗ (g)t1 , t2 ) as a sum of local heights under Assumption 5.3 and for g ∈ 1S1 GL2 (AS1 ). All the notations and decompositions apply to j(Z∗ (g)t1 , t2 ) even without these assumptions. Note that Galois conjugates of points in CU over E are described easily by multiplication by elements of T (F )\T (Af ) via the reciprocity law. It is convenient to group local intersections according to places of E. We first write i(Z∗ (g)t1 , t2 ) iν (Z∗ (g)t1 , t2 )

1 X iν (Z∗ (g)t1 , t2 ) log Nν , 2 ν∈SE Z = iν (Z∗ (g)tσ1 , tσ2 )dσ. =

Gal(E ab /E)

Here SE denotes the set of all places of E, and E ab denotes the maximal abelian extension of E. In the integration, we have replaced E by E ab since all CM points and their conjugates are defined over E ab . The definition of iν depends on fixed embeddings H ,→ E and E ,→ E ν , and can be viewed as intersections on YU ×OH OE ν . By class field theory, we have a continuous surjective homomorphism T (F )\T (A) −→ Gal(E ab /E) whose kernel contains T (F∞ ). It follows that we can further write Z iν (Z∗ (g)t1 , t2 )

=

iν (Z∗ (g)tt1 , tt2 )dt. T (F )\T (A)/Z(A)

The regularized averaging integral on T (F )\T (A) is introduced in §1.6.

219

DECOMPOSITION OF THE GEOMETRIC KERNEL

To compare with the analytic kernel, we also need to group the pairing in terms of places of F . We have: X iv (Z∗ (g)t1 , t2 ) log Nv , i(Z∗ (g)t1 , t2 ) = v∈SF

Z iv (Z∗ (g)t1 , t2 )

=

iv (Z∗ (g)tt1 , tt2 )dt, T (F )\T (A)/Z(A)

iv (Z∗ (g)t1 , t2 )

=

X 1 iν (Z∗ (g)t1 , t2 ). #SEv ν∈SEv

Here SEv denotes the set of places of E lying over v. It has one or two elements. The local pairing X X X r(g)φ(x)a iv (t1 x, t2 ) iv (Z∗ (g)t1 , t2 ) = a∈F × x∈B× /U f

v

is our main goal for the next chapter. We will divide it into a few cases and discuss them in different sections. We will have explicit expressions for iv in the case that v is archimedean or the Shimura curve has good reduction at v. 7.3

VANISHING OF THE CONTRIBUTION OF THE HODGE CLASSES

Recall that Z(g, (t1 , t2 )) = hZ∗ (g)t1 , t2 i − hZ∗ (g)ξt1 , t2 i + hZ∗ (g)ξt1 , ξt2 i − hZ∗ (g)t1 , ξt2 i. ˆ Here the pairings on the right-hand side are ξ-admissible pairings. We have considered a decomposition of the first term on the right-hand side in the last section. The main result here asserts the vanishing of the other terms. Proposition 7.5. Assuming Assumption 5.4, then hZ∗ (g)ξt1 , t2 i = hZ∗ (g)ξt1 , ξt2 i = hZ∗ (g)t1 , ξt2 i = 0,

∀ t1 , t2 ∈ T (Af ).

The vanishing follows from the vanishing of the degree of the components of Z∗ (g). 7.3.1

Vanishing of the degree of the generating series

For any α ∈ F+× \A× f /q(U ), the α-component of the generating series is given by Z(g, φ)U,α = Z(g, φ)|MK,α . Recall that in Proposition 4.2 we have 1 deg Z(g, φ)U,α = − κ◦U E(0, g, r(h)φ)U . 2

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−1 × Here h is any element of B× F+ q(U ). The weight two f such that q(h) ∈ α Eisenstein series are defined as follows: X δ(γg)s r(γg)φ(0, u), E(s, g, u, φ) = γ∈P 1 (F )\SL2 (F )

E(s, g, φ)U =

X

E(s, g, u, φ).

u∈µ2U \F ×

These series vanish immediately by Assumption 5.4. Thus we have the following lemma. Lemma 7.6. Under Assumption 5.4, the constant term Z0 (g, φ)U,α = 0 and the degree deg Z(g, φ)U,α = 0 for any α ∈ F+× \A× f /q(U ). By the lemma, it is easy to get the vanishing of the first two terms in Proposition 7.5. They actually follow from the vanishing of Z∗ (g)ξt1 . The correspondences Z(x) are ´etale on the generic fiber; it keeps the canonical bundle up to a multiple under pull-back and push-forward. Then the Hodge bundles are eigenvectors of all Hecke operators up to translation of components. More precisely, one has Z(x)ξt1 = (deg Z(x))ξt1 x , ∀x ∈ B× f . It follows that for any α ∈ F+× \A× f /q(U ), Z∗ (g)U,α ξt1 = deg Z∗ (g)U,α ξαt1 = 0. It remains to treat hZ∗ (g)t1 , ξt2 i. Our idea is to “move” the action of Z∗ (g) to ξt2 so that we can apply the above argument. It will be achieved by the projection formula on the integral models. So it depends on extensions of Hecke operators to the integral model. 7.3.2

Integral models of Hecke operators

For convenience, we denote Z(x) := Z(1P x) for any x ∈ B× P . Here P can be any subset of SF such that either P or SF − P is a finite set. Then it is easy to verify that Z(x) = Z(xP )Z(xP ) = Z(xP )Z(xP ) as correspondences for any x ∈ B. Let S be a finite subset of non-archimedean places of F containing all v such that Uv is not maximal or that E is ramified at v. Then the models XU and YU are smooth away from S. We will have “good extension” of Z(x) away from S. For any x ∈ B× P , let Z(x) be the Zariski closure of Z(x) in XU ×OF XU . If x ∈ (BSf )× , then many good properties of Z(x) are obtained in [Zh1]. For example, it has a canonical moduli interpretation, and satisfies (1) Z(x1 ) commutes with Z(x2 ) for any x1 , x2 ∈ (BSf )× ; Q S × (2) Z(x) = v∈S / Z(xv ) for any x ∈ (Bf ) ;

DECOMPOSITION OF THE GEOMETRIC KERNEL

221

(3) for any x ∈ (BSf )× , both structure projections from Z(x) to XU are finite everywhere, and ´etale above the set of places v with xv ∈ Uv . Fix x ∈ (BSf )× . Define an arithmetic class D(x) on XU by \ = Z(x)ξˆ − deg Z(x) ξ. b D(x) := Z(x)ξˆ − Z(x)ξ Then D(x) is a vertical divisor since it is zero on the generic fiber. We claim that D(x) is a constant divisor, i.e., the pull-back of an arithmetic divisor D on d and sometimes we identify it with Spec(OF ). Then it only depends on deg(D), this number. Now we explain why D(x) is a constant divisor. First, D(x) is constant at archimedean places because the Petersson metric on the upper half plane is invariant under the action of GL2 (R)+ . Now we look at non-archimedean places. Both structure morphisms of Z(x) are ´etale above S, so the pull-back and push-forward keep LU above S since they keep both the relative dualizing sheaf and the Zariski closure of the cusps. Then the finite part of D(x) is lying above primes not in S. Note that XU is smooth outside S; its special fibers outside S are irreducible. Hence the finite part of D(x) is a linear combination of these special fibers which are constant. Lemma 7.7. Let v be a non-archimedean place outside S. For any x ∈ B× v and any D ∈ Div(XU,F ), hZ(x)D, ξi = deg Z(x) hD, ξi − deg(D) D(x). Here D(x) is viewed as a constant, and deg(D) is the sum of the degrees of D on all geometrically connected components. Proof. We first reduce the problem to the case that D is defined over F . Indeed, since ξ and Z(x) are defined over F , both sides of the equality are invariant under Galois actions on D. So it suffices to show the result for the sum of all Galois conjugates of D. Assume that D is defined over F . By §7.1, ˆ hZ(x)D, ξi = hZ(x)D, ξi. Here Z(x)D is the Zariski closure in XU , and we denote the normalized intersection hD1 , D2 i := −D1 · D2 for any arithmetic divisors D1 , D2 on XU . The correspondence Z(x) keeps Zariski closure by finiteness of its structure morphisms. It follows that Z(x)D = Z(x)D. By the projection formula, we get ˆ = hD, Z(x)t ξi ˆ ˆ = hD, Z(x−1 ) ξi. hZ(x)D, ξi = hZ(x)D, ξi Here Z(x)t denotes the transpose of Z(x) as correspondences.

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c U )Q . We first have We claim that Z(x)t ξˆ = Z(x)ξˆ as elements in Pic(X t −1 t −1 Z(x) = Z(x ) by Z(x) = Z(x ). Since Uv is maximal, we know that Uv x−1 Uv = q(x)−1 Uv xUv = q(x)−1 Uv xUv . Then Z(x) is the composition of Z(x)t with Z(q(x)−1 ). Note that Z(q(x)−1 ) acts by right multiplication by q(x), which gives a Galois automorphism on XU ˆ or by the reciprocity law. It suffices to show Z(q(x)−1 ) acts trivially on ξ, −1 equivalently the constant D(q(x) ) = 0. It is true since the automorphism has a finite order. ˆ By the definition of D(x), Go back to hZ(x)D, ξi = hD, Z(x) ξi. ˆ + hD, D(x)i. hZ(x)D, ξi = deg Z(x)hD, ξi The second pairing hD, D(x)i = − deg(D) D(x), because D(x) is a constant divisor.



There is an explicit way to compute D(x) for x ∈ (BSf )× by [Zh1]. We briefly mention it here, since we do not need it in this book. First, D(x) satisfies a “product rule”: X D(x) = deg Z(xv ) D(xv ), x ∈ (BSf )× . v ∈S /

It follows from the definition, and is called a “derivation” in the paper. Then the computation of D(x) amounts to each single D(xv ). It can be computed by the deformation method in Proposition 4.3.2 in [Zh1]. 7.3.3

Vanishing of the third term

Now we are ready to prove hZ∗ (g)t1 , ξt2 i = 0. By Galois action of t1 via the reciprocity law, we have hZ∗ (g)t1 , ξt2 i = hZ∗ (g)[t1 t−1 2 ], ξ1 i. So it suffices to consider hZ∗ (g)t, ξ1 i for any t ∈ T (Af ). After separating components, we get hZ∗ (g)t, ξ1 i = hZ∗ (g)q(1/t) t, ξ1 i = hZ∗ (g)q(1/t) t, ξi. It is also equal to hZ∗ (g)q(1/t) [1], ξi by the Galois action. Hence it only depends on the geometrically connected component of t in XU . By (4.2.4), X X X Wa(2) (g∞ ) r(g, (t, 1))φf (y, u)Z(t−1 y). Z∗ (g)q(1/t) = u∈µ0U \F × a∈F × +

y∈K t \Bf (a)

For fixed (a, u), denote the correspondence X r(g, (t, 1))φ(y, u)Z(t−1 y). A := y∈K t \Bf (a)

223

DECOMPOSITION OF THE GEOMETRIC KERNEL

We will prove that each hA[1], ξi = 0. For each non-archimedean place v, we have the correspondence X r(g, (t, 1))φv (y, u)Z(t−1 Av := v y). y∈Kvt \Bv (a)

Q Then Av equals the identity at almost all v. Furthermore, A = v-∞ Av as a composition of correspondences. −1 t 1 Use the identity Uv t−1 v yUv /Uv = tv (Kv y/Uv ) as before. We obtain X deg Av = r(g, (t, 1))φv (y, u). y∈Bv (a)/Uv1

It follows that deg Av = 0 for any v ∈ S2 by Assumption 5.4 and Lemma 5.10. Fix a place v ∈ S2 . Then deg Av = 0 and deg Av = 0 since S2 has two elements. We will see that these are the reason for hA[1], ξi = 0. Write X v hA[1], ξi = hAv Av [1], ξi = r(g, (t, 1))φv (yv , u)hZ(t−1 v yv )A [1], ξi. yv ∈Kvt \Bv (a)

Apply Lemma 7.7 to Z(t−1 v yv ). We have X v hA[1], ξi = r(g, (t, 1))φv (yv , u) deg Z(t−1 v yv ) hA [1], ξi yv ∈Kvt \Bv (a)

X

−

r(g, (t, 1))φv (yv , u) deg(Av [1]) D(t−1 v yv ).

yv ∈Kvt \Bv (a)

In a manner similar to Lemma 7.7, it is just hA[1], ξi

=

(deg Av ) hAv [1], ξi − (deg Av ) D(Av ),

with D(Av ) =

X

r(g, (t, 1))φv (yv , u) D(t−1 v yv ).

yv ∈Kvt \Bv (a)

The result follows since deg Av = deg Av = 0. 7.4 7.4.1

THE GOAL OF THE NEXT CHAPTER Difference of the kernel functions

Now let us try to prove Theorem 5.1. We need to prove (I 0 (0, g, χ, φ)U , ϕ(g))Pet = 2 (Z(g, χ, φ)U , ϕ(g))Pet ,

∀ϕ ∈ σ.

By Theorem 5.7, we can assume that (φ, U ) satisfies all the assumptions in §5.2.

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CHAPTER 7

Recall that in (5.1.1) and (5.1.2) we have introduced X X X I(s, g, φ)U = δ(γg)s r(γg)φ(x1 , u), x1 ∈E

u∈µ2U \F × γ∈P 1 (F )\SL2 (F )

Z I(s, g, χ, φ)U

∗

=

I(s, g, r(t, 1)φ)U χ(t)dt. T (F )\T (A)/Z(A)

On the other hand, we have introduced a generating series X X r(g)φ(x)a Z(x)U . Z(g, φ)U = Z0 (g, φ)U + a∈F × x∈U \B× /U f

It is an automorphic form of GL2 (A) holomorphic of parallel weight two. The coefficients of Z(g, φ)U are Hecke operators on the Shimura curve XU . Then we have the height series Z(g, (t1 , t2 ), φ)U

=

Z(g, χ, φ)U

=

hZ(g, φ)U t◦1 , t◦2 iNT , Z ∗ Z(g, (t, 1), φ)U χ(t)dt. T (F )\T (A)/Z(A)

For simplicity, we will omit the dependence of the series on U as before. By holomorphic projection, it suffices to prove (PrI 0 (0, g, χ, φ) − 2Z(g, χ, φ), ϕ(g))Pet = 0.

(7.4.1)

0

The holomorphic projection functors Pr and Pr are introduced in §6.4. Here Pr denotes the holomorphic projection defined by functoriality, and Pr0 is just an algorithm to compute Pr. Proposition 6.12 asserts Pr = Pr0 under certain growth conditions. In particular, I 0 (0, g, χ, φ) satisfies the growth condition by Proposition 6.14. It follows that PrI 0 (0, g, χ, φ)

=

Pr0 I 0 (0, g, χ, φ) Z ∗

=

Pr0 I 0 (0, g, r(t, 1)φ) χ(t)dt.

T (F )\T (A)/Z(A)

Note that the integration is actually a finite summation. As for Pr0 I 0 (0, g, r(t, 1)φ), Proposition 6.15 gives X X Pr0 I 0 (0, g, φ) = − I 0 (0, g, φ)(v) − I 0 (0, g, φ)(v), v|∞

v-∞ nonsplit

∀g ∈ P (FS1 )GL2 (AS1 ). Here I 0 (0, g, φ)(v) I 0 (0, g, φ)(v)

= =

Z 2 Z 2

(v)

Kφ (g, (t, t))dt,

v | ∞;

Z(A)T (F )\T (A) (v)

Z(A)T (F )\T (A)

Kφ (g, (t, t))dt,

v - ∞.

225

DECOMPOSITION OF THE GEOMETRIC KERNEL

(v)

The series Kφ (g, (t1 , t2 )) is a pseudo-theta series on the nearby quaternion algebra B(v). Note that the integrals are essentially finite sums. On the other hand, Proposition 7.5 asserts Z(g, (t1 , t2 ), φ) = hZ∗ (g, φ)t1 , t2 i. b The right-hand side is the ξ-admissible pairing on the integral model YU . By §7.2, it has a decomposition in terms of local arithmetic intersections: hZ∗ (g, φ)t1 , t2 i X X iv (Z∗ (g, φ)t1 , t2 ) log Nv − jv (Z∗ (g, φ)t1 , t2 ) log Nv . = − v

v

Furthermore, Z iv (Z∗ (g, φ)t1 , t2 )

=

iv (Z∗ (g, φ)tt1 , tt2 )dt, T (F )\T (A)/Z(A)

Z jv (Z∗ (g, φ)t1 , t2 )

=

jv (Z∗ (g, φ)tt1 , tt2 )dt. T (F )\T (A)/Z(A)

All these identities are valid for g ∈ 1S1 GL2 (AS1 ). Putting these two parts together, we have Z ∗Z D(g, (tt1 , t), φ)χ(t1 )dt1 dt, PrI 0 (0, g, χ, φ) − 2Z(g, χ, φ) = 2 [T ]

(7.4.2)

[T ]

where

=

D(g, (t1 , t2 ), φ) X (v) Kφ (g, (t1 , t2 )) + v|∞

−

(v)

X

Kφ (g, (t1 , t2 ))

v-∞ nonsplit

X

iv (Z∗ (g, φ)t1 , t2 ) log Nv −

X

v

jv (Z∗ (g, φ)t1 , t2 ) log Nv .

v

Here we write [T ] for T (F )\T (A)/Z(A) for simplicity. 7.4.2

Main result on local computations of the next chapter

We have already seen that the computation of Pr0 (I 0 (0, g, r(t1 , t2 )φ)) − 2hZ(g, φ) t◦1 , t◦2 iNT amounts to the computation of the difference (v)

Kφ (g, (t1 , t2 )) − iv (Z∗ (g, φ)t1 , t2 ) log Nv − jv (Z∗ (g, φ)t1 , t2 ) log Nv . (v)

(v)

In the archimedean case, Kφ should be replaced by Kφ . The following is the main result of the next chapter.

226

CHAPTER 7

Theorem 7.8. Assume all the assumptions in §5.2. Then the following are true for all t1 , t2 ∈ T (Af ) and all g ∈ 1S1 GL2 (AS1 ): (1) If v ∈ Ssplit , then iv (Z∗ (g, φ)t1 , t2 ) = jv (Z∗ (g, φ)t1 , t2 ) = 0. (2) If v ∈ S∞ , then (v)

Kφ (g, (t1 , t2 )) = iv (Z∗ (g, φ)t1 , t2 ),

jv (Z∗ (g, φ)t1 , t2 ) = 0.

(3) If v ∈ Snonsplit − S1 , then (v)

Kφ (g, (t1 , t2 )) = iv (Z∗ (g, φ)t1 , t2 ) log Nv ,

jv (Z∗ (g, φ)t1 , t2 ) = 0.

(4) If v ∈ S1 , then there exist Schwartz functions kφv , mφv , lφv ∈ S(B(v)v × Fv× ) depending on φv and Uv such that: (v)

Kφ (g, (t1 , t2 )) = θ(g, (t1 , t2 ), kφv ⊗ φv ), iv (Z∗ (g, φ)t1 , t2 ) = θ(g, (t1 , t2 ), mφv ⊗ φv ), jv (Z∗ (g, φ)t1 , t2 ) = θ(g, (t1 , t2 ), lφv ⊗ φv ). We recall that B(v) is the nearby quaternion algebra of B. The theta series X X θ(g, (t1 , t2 ), φ0 ) = r(g, (t1 , t2 ))φ0 (y, u) u∈µ2U \F × y∈B(v)

for any φ0 ∈ S(B(v)A × A× ). It is automorphic for g ∈ GL2 (A). Some results of the theorem are known. By definition, jv = 0 in (2) and (3). (v) In (4), the result for Kφ has been proved in Corollary 6.9. We know explicit (v)

(v)

expressions of Kφ in (2) and Kφ in (3) before, so the result in these cases are proved by explicit computation of iv (Z∗ (g, φ)t1 , t2 ). 7.4.3

Completion of the proof

Recall that we only need to prove Theorem 5.1 under the assumption that the ramification set of B is equal to Σ. It is a consequence of Theorem 1.3. See §3.6.5 for more details. Go back to (7.4.2) for the proof of Theorem 5.1. By Theorem 7.8, almost all terms in the summation defining D(g, (t1 , t2 ), φ) are canceled. The remaining terms are just some theta series. More precisely, X D(g, (t1 , t2 ), φ) = θ(g, (t1 , t2 ), φv ⊗ dφv ), ∀g ∈ 1S1 GL2 (AS1 ). v∈S1

227

DECOMPOSITION OF THE GEOMETRIC KERNEL

Here dφv = kφv − (log Nv )mφv − (log Nv )lφv is a Schwartz function in S(B(v)v × Fv× ). In summary, we have proved that Z ∗Z X 0 PrI (0, g, χ, φ) − 2Z(g, χ, φ) = 2 θ(g, (tt1 , t), φv ⊗ dφv )χ(t1 )dt1 dt [T ] v∈S 1

[T ]

is true for g ∈ 1S1 GL2 (AS1 ). The simple argument mentioned in the introduction asserts that it is true for all g ∈ GL2 (A). In fact, the equality is true for all g ∈ GL2 (F )GL2 (AS ) since both sides are automorphic. Then it is true for all g ∈ GL2 (A) because GL2 (F )GL2 (AS ) is dense in GL2 (A). To prove (7.4.1), it suffices to prove for all v ∈ S1 , (I(0, g, χ, φv ⊗ dφv ), ϕ(g))Pet = 0. Here I(0, g, χ, φv ⊗ dφv ) =

Z ∗Z [T ]

θ(g, (tt1 , t), φv ⊗ dφv )χ(t1 )dt1 dt. [T ]

To make use of the Shimizu lifting, we change the space S(B(v)A × A× ) to × ) such that Φ∞ S(B(v)A × A× ). Let Φ∞ be a Schwartz function in S(B∞ × F∞ v is standard. Then Φ(v) := Φ∞ ⊗ φf ⊗ dφv lies in S(B(v)A × A× ). Introduce Z Z θ(g, (tt1 , t), Φ(v))χ(t1 )dt1 dt, I(0, g, χ, Φ(v)) = T (F )\T (A)

T (F )\T (A)/Z(A)

where X

θ(g, (t1 , t2 ), Φ(v)) =

X

r(g, (t1 , t2 ))Φ(v)(y, u).

u∈F × y∈B(v)

Similar to (4.4.4), we can show that there is an explicit positive constant c such that I(0, g, χ, Φ(v)) = c I(0, g, χ, φv ⊗ dφv ). Thus it is reduced to prove (I(0, g, χ, Φ(v)), ϕ(g))Pet = 0. Now the result follows from Theorem 1.3, the result of Tunnell [Tu] and Saito [Sa]. The key is that the theta series θ(g, (t1 , t2 ), Φ(v)) is defined on the nearby quaternion algebra B(v). In fact, (I(0, g, χ, Φ(v)), ϕ(g))Pet Z Z = T (F )\T (A)/Z(A)

T (F )\T (A)/Z(A)

θ0 (Φ(v) ⊗ ϕ)(t1 , t2 )χ(t1 )χ−1 (t2 )dt1 dt2

228

CHAPTER 7

where for h1 , h2 ∈ B(v)× A , the Shimizu lifting Z 0 θ(g, (h1 , h2 ), Φ(v))ϕ(g)dg. θ (Φ(v) ⊗ ϕ)(h1 , h2 ) = GL2 (F )\GL2 (A)

Note here we use θ0 because our normalization is different from (2.2.1). Then we must have θ0 (Φ(v) ⊗ ϕ) =

r X

fi ⊗ fi0 ,

fi ∈ π(v), fi0 ∈ π e(v).

i=1

Here π(v) denotes the automorphic representation of B(v)× A obtained from σ by the Jacquet-Langland correspondence. It follows that (I(0, g, χ, Φ(v)), ϕ(g))Pet =

r X

Pχ (fi )Pχ−1 (fi0 ).

i=1

Here

Z Pχ (fi ) =

fi (t)dt T (F )\T (A)/Z(A)

is the period integral considered in Theorem 1.4. Note that the ramification set of B(v) is not equal to Σ, since we have assumed that B matches Σ. As a consequence, Theorem 1.3 implies P(π(v), χ) = HomEA× (π(v) ⊗ χ, C) = 0. It follows that Pχ (fi ) = 0 for all f ∈ π(v). The result is proved. 7.4.4

Rough idea of our proof of Theorem 7.8

We will follow the work of Gross–Zagier [GZ] and its extension in [Zh2] to compute the local intersection numbers. The first step of the proof is to write iv (Z∗ (g, φ)t1 , t2 ) and jv (Z∗ (g, φ)t1 , t2 ) in the form of a pseudo-theta series. It is achieved by the v-adic uniformization of the integral model or reduction of the Shimura curve. To illustrate the idea, consider iv (Z∗ (g, φ)t1 , t2 ) for a place v split in B butnonsplit in E. Denote by B = B(v) the nearby quaternion algebra. The height pairing is based on the natural isomorphism v× × × × v v : CMU = E × \B× f /U −→ B \(B ×E × Bv /Uv ) × Bf /U .

There is a local multiplicity function m on (Bv× − Ev× ) × B× v /Uv such that iv (β1 , t2 ) =

X γ∈µU

\B ×

−1 m(γt2v , β1v )1U v ((β1v )−1 γtv2 )

229

DECOMPOSITION OF THE GEOMETRIC KERNEL

for any two distinct CM points β1 ∈ CMU and t2 ∈ CU . By this formula, a simple computation gives X X iv (Z∗ (g)t1 , t2 ) = r(g, (t1 , t2 ))φv (y, u) mr(g,(t1 ,t2 ))φv (y, u) u∈µ2U \F × y∈B−E

where the desired mφv is given by Z mφv (y, u) = m(y, x−1 )φv (x, uq(y)/q(x))dx, B× v

y ∈ Bv − Ev , u ∈ Fv× .

To prove Theorem 7.8 (3) for the i-part, we compute mφv (y, u) and verify that it is the same as the result for kφv (1, y, u) given in Corollary 6.8. The key in this unramified case is the explicit expression of the multiplicity function m of Zhang [Zh2], which was obtained by the technique of Gross [Gro1]. To prove Theorem 7.8 (4) for the i-part, it suffices to prove that mφv extends to a Schwartz function on Bv × Fv× . A priori, mφv is locally constant in (Bv − Ev ) × Fv× with logarithmic singularity along Ev × Fv× . The singularity comes from the singularity of m, which corresponds to the case of self-intersection. However, the singularity of mφv is killed by Assumption 5.3.

Chapter Eight Local Heights of CM Points The goal of this chapter is to prove Theorem 7.8, namely, to compute the local heights and compare them with the derivatives computed before. We check the theorem place by place. We assume all the assumptions in §5.2 throughout this chapter. According to the reduction of the Shimura curve, we divide the situation to the following four cases: • archimedean case: v is archimedean; • supersingular case: v isnonsplit in E but split in B; • superspecial case: v isnonsplit in both E and B; • ordinary case: v is split in both E and B. The treatments in different cases are similar in spirit, except that the fourth case is slightly different. Of course, the supersingular case is divided into two subcases: unramified case (v ∈ / S1 ) and ramified case (v ∈ S1 ). 8.1

ARCHIMEDEAN CASE

In this section we want to describe local heights of CM points at any archimedean place v. Denote B = B(v) and fix an identification B(Af ) = Bf . We will use the uniformization × \H × B× XU,v (C) = B+ f /U. We follow the treatment of Gross–Zagier [GZ]. See also [Zh2]. 8.1.1

Multiplicity function

For any two points z1 , z2 ∈ H, the hyperbolic cosine of the hyperbolic distance between them is given by d(z1 , z2 ) = 1 +

|z1 − z2 |2 . 2Im(z1 )Im(z2 )

It is invariant under the action of GL2 (R). For any s ∈ C with Re(s) > 0, denote ms (z1 , z2 ) = Qs (d(z1 , z2 )),

231

LOCAL HEIGHTS OF CM POINTS

where

∞

Z Qs (t) =

−1−s  p du t + t2 − 1 cosh u

0

is the Legendre function of the second kind. Note that Q0 (1 + 2λ) =

1 1 log(1 + ), λ > 0. 2 λ

We see that m0 (z1 , z2 ) has the right logarithmic singularity. For any two distinct points of × \H × BA×f /U XU,v (C) = B+

represented by (z1 , β1 ), (z2 , β2 ) ∈ H × BA×f , we denote X ms (z1 , γz2 ) 1U (β1−1 γβ2 ). gs ((z1 , β1 ), (z2 , β2 )) = × γ∈µU \B+

It is easy to see that the sum is independent of the choice of the representatives (z1 , β1 ), (z2 , β2 ), and hence defines a pairing on XU,v (C). Then the local height is given by f s→0 gs ((z1 , β1 ), (z2 , β2 )). iv ((z1 , β1 ), (z2 , β2 )) = lim f s→0 denotes the constant term at s = 0 of gs ((z1 , β1 ), (z2 , β2 )), which Here lim converges for Re(s) > 0 and has meromorphic continuation to s = 0 with a simple pole. The definition above uses adelic language, but it is not hard to convert it to the classical language. Observe that gs ((z1 , β1 ), (z2 , β2 )) 6= 0 only if there is a × such that β1 ∈ γ0 β2 U , which just means that (z1 , β1 ), (z2 , β2 ) are in γ0 ∈ B+ the same connected component. Assuming this, then (z2 , β2 ) = (z20 , β1 ) where z20 = γ0 z2 . We have gs ((z1 , β1 ), (z2 , β2 )) =gs ((z1 , β1 ), (z20 , β1 )) X ms (z1 , γz20 ) 1U (β1−1 γβ1 ) = = × γ∈µU \B+

X

ms (z1 , γz20 ).

γ∈µU \Γ

× Here we denote Γ = B+ ∩ β1 U β1−1 . The connected component of these two points is exactly × × B+ \H × B+ β1 U/U ≈ Γ\H,

(z, bβ1 U ) 7→ b−1 z.

The stabilizer of H in Γ is exactly Γ ∩ F × = µU . Now we see that the formula is the same as those in [GZ] and [Zh2]. × − Ev× , Next, we consider the special case of CM points. For any γ ∈ Bv,+ we have |z0 − γz0 |2 1+ = 1 − 2λ(γ). 2Im(z0 )Im(γz0 )

232

CHAPTER 8

Here λ(γ) = q(γ2 )/q(γ) is introduced at the end of the introduction. Thus it is convenient to denote ms (γ) = Qs (1 − 2λ(γ)),

γ ∈ Bv× − Ev× .

For any two distinct CM points β1 , β2 ∈ CMU , we obtain X ms (γ) 1U (β1−1 γβ2 ), gs (β1 , β2 ) = × γ∈µU \B+

and f s→0 gs (β1 , β2 ). iv (β1 , β2 ) = lim Note that ms (γ) is not well-defined for γ ∈ E × . The above summation is understood to be X gs (β1 , β2 ) = ms (γ). × γ∈µU \B+ , β1−1 γβ2 ∈U

/ U for all Then it still makes sense because β1 6= β2 implies that β1−1 γβ2 ∈ γ ∈ E × . Anyway, it is safer to write X ms (γ) 1U (β1−1 γβ2 ). gs (β1 , β2 ) = × γ∈µU \(B+ −E × )

Now the right-hand side is well-defined even for β1 = β2 . In this case it can be interpreted as a contribution of some local height by an arithmetic adjunction formula, but we do not need this fact here since there is no self-intersection under our assumptions in §5.2. 8.1.2

Kernel function

We are going to compute the local height X X iv (Z∗ (g)t1 , t2 ) = r(g)φ(x)a iv (t1 x, t2 ). a∈F × x∈B× /U f

It is well-defined under Assumption 5.3 which kills the self-intersections. The (v) goal is to show that it is equal to Kφ (g, (t1 , t2 )) obtained in Proposition 6.15. We still assume the assumption. Proposition 8.1. For any t1 , t2 ∈ CU , X X f s→0 lim iv (Z∗ (g)t1 , t2 ) := a∈F ×

× y∈µU \(B+ −E × )

(v)

In particular, iv (Z∗ (g)t1 , t2 ) = Kφ (g, (t1 , t2 )).

r(g, (t1 , t2 ))φ(y)a ms (y).

233

LOCAL HEIGHTS OF CM POINTS

Proof. By the above formula,

= =

iv (Z∗ (g)t1 , t2 ) X X f s→0 r(g)φ(x)a lim a∈F ×

x∈B× f /U

X

f s→0 lim

a∈F ×

=

X

X

ms (γ) 1U (x−1 t−1 1 γt2 )

× γ∈µU \(B+ −E × )

X

r(g)φ(t−1 1 γt2 )a ms (γ)

× γ∈µU \(B+ −E × )

X

f s→0 lim

a∈F ×

r(g, (t1 , t2 ))φ(γ)a ms (γ).

× γ∈µU \(B+ −E × )

Here the second equality is obtained by replacing x by t−1 1 γt2 .



We want to compare the above result with the holomorphic projection (v)

Kφ (g, (t1 , t2 )) =

X

X

f s→0 lim

a∈F ×

r(g, (t1 , t2 ))φ(y)a kv,s (y)

× y∈µU \(B+ −E × )

computed in Proposition 6.15. It amounts to compare ms (y) = Qs (1 − 2λ(y)) with Γ(s + 1) kv,s (y) = 2(4π)s

Z 1

∞

1 dt. t(1 − λv (y)t)s+1

By the result of Gross–Zagier, Z ∞ 1 dt = 2Qs (1 − 2λ) + O(|λ|−s−2 ), t(1 − λt)s+1 1

λ → −∞,

and the error term vanishes at s = 0. We conclude that (v)

Kφ (g, (t1 , t2 )) = iv (Z∗ (g)t1 , t2 ). 8.2

SUPERSINGULAR CASE

Let v be a finite prime of F nonsplit in E but split in B. We consider the local pairing iv , which depends on the fixed embeddings H ⊂ E ⊂ E v and the model YU over OH . It actually depends only on the local integral model YU,w = YU ×OH OHw where w is the place of H induced by the embeddings. We will use the local multiplicity functions treated in Zhang [Zh2]. For more details, we refer to that paper.

234

CHAPTER 8

8.2.1

Multiplicity function

Let B = B(v) be the nearby quaternion algebra over F . Make an identification B(Av ) = Bv . Then the set of supersingular points on XK over v is parametrized by v SU = B × \(Fv× / det(Uv )) × (Bv× f /U ).

We have a natural isomorphism v× × × × v v : CMU = E × \B× f /U −→ B \(B ×E × Bv /Uv ) × Bf /U

sending β to (1, βv , β v ). The reduction map CMU → SU is given by taking norm on the first factor: × q : B × ×E × B× v −→ Fv ,

(b, β) 7−→ q(b)q(β).

The intersection pairing is given by a multiplicity function m on HUv := Bv× ×Ev× B× v /Uv . More precisely, the intersection of two points (b1 , β1 ), (b2 , β2 ) ∈ HUv is given by −1 gv ((b1 , β1 ), (b2 , β2 )) = m(b−1 1 b2 , β1 β2 ).

The multiplicity function m is defined everywhere in HUv except at the image of (1, 1). It satisfies the property m(b, β) = m(b−1 , β −1 ). Lemma 8.2. For any two distinct CM points β1 ∈ CMU and t2 ∈ CU , their local height is given by iv (β1 , t2 ) =

X γ∈µU

−1 m(γt2v , β1v )1U v ((β1v )−1 γtv2 ).

\B ×

Proof. Like the archimedean case, we compute the height by pulling back × of the intersection of to HUv × Bv× f . The height is the sum over γ ∈ µU \B v v v v  (1, β1v , β1 ) with γ(1, t2v , t2 ) = (γ, t2v , γt2 ) = (γt2v , 1, γt2 ) on HUv × Bv× f . Analogous to the archimedean case, the summation is well-defined for all β1 6= t2 . Indeed, assume that there is a γ ∈ E × such that (β1v )−1 γtv2 ∈ U v and −1 −1 ) is not well-defined. Then we must have γt2v ∈ Ev× and β1v γt2v ∈ m(γt2v , β1v v × U . It forces γ ∈ E and γt2 ∈ β1 U , which implies that β1 = t2 ∈ CMU .

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LOCAL HEIGHTS OF CM POINTS

8.2.2

The kernel function

Now we compute iv (Z∗ (g)t1 , t2 ) =

X

X

r(g)φ(x)a iv (t1 x, t2 ).

a∈F × x∈B× /U f

As in the archimedean case, we assume that φ is degenerate at two different finite places v1 , v2 of F which arenonsplit in E, and only consider g in P (Fv1 ,v2 )GL2 (Av1 ,v2 ). By the above formula, we have iv (Z∗ (g)t1 , t2 ) X X r(g)φ(x)a = a∈F ×

x∈B× f /U

X γ∈µU

Replace xv by t−1 1 γt2 and then get X X r(g)φv (t−1 1 γt2 )a a∈F × γ∈µU \B ×

X

X

u∈µ2U \F ×

γ∈B ×

= ·

X

−1 −1 m(γt2 , x−1 t−1 t1 γt2 ). 1 )1U v (x

\B ×

X

−1 r(g)φv (xv )a m(t−1 ) 1 γt2 , x

xv ∈B× v /Uv

r(g, (t1 , t2 ))φv (γ, u)

−1 r(g)φv (xv , uq(γ)/q(xv ))m(t−1 ). 1 γt2 , x

xv ∈B× v /Uv

For convenience, we introduce the following notation. Notation 8.3. X mφv (y, u) =

m(y, x−1 )φv (x, uq(y)/q(x))

x∈B× v /Uv

=

X

m(y −1 , x)φv (x, uq(y)/q(x)),

(y, u) ∈ (Bv − Ev ) × Fv× .

x∈B× v /Uv

/ Ev since m(y, x−1 ) has no Notice that mφv (y, u) is well-defined for y ∈ singularity for such y. By this notation, we obtain the following result. Proposition 8.4. For g ∈ P (FS1 )GL2 (AS1 ), X X iv (Z∗ (g)t1 , t2 ) = r(g, (t1 , t2 ))φv (y, u) mr(g,(t1 ,t2 ))φv (y, u). u∈µ2U \F × y∈B−E

Here we can change the summation to y ∈ B − E since the contribution of y ∈ E is zero by Assumption 5.3. We should compare the following result with Lemma 6.6 for kφv (g, y, u).

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Lemma 8.5. The function mφv (y, u) behaves like Weil representation under the action of P (Fv ) × (Ev× × Ev× ) on (y, u). Namely, mr(g,(t1 ,t2 ))φv (y, u) = r(g, (t1 , t2 ))mφv (y, u),

(g, (t1 , t2 )) ∈ P (Fv ) × (Ev× × Ev× ).

More precisely, mr(m(a))φv (y, u) mr(n(b))φv (y, u)

= =

|a|2 mφv (ay, u), a ∈ Fv× , ψ(buq(y))mφv (y, u), b ∈ Fv ,

mr(d(c))φv (y, u)

=

|c|−1 mφv (y, c−1 u),

mr(t1 ,t2 )φv (g, y, u)

=

−1 mφv (t−1 1 yt2 , q(t1 t2 )u),

c ∈ Fv× , (t1 , t2 ) ∈ Ev× × Ev× .

Proof. They follow from some basic properties of the multiplicity function m(x, y). We only verify the first identity:

=

mr(m(a))φv (y, u) X m(y −1 , x)r(gv )φv (ax, uq(y)/q(x))|a|2 x∈B× v /Uv

=

|a|2

X

m(y −1 , a−1 x)r(gv )φv (x, uq(y)/q(a−1 x))

x∈B× v /Uv

=

|a|2

X

m((ay)−1 , x)r(gv )φv (x, uq(ay)/q(x))

x∈B× v /Uv

=

|a|2 mφv (ay, u). 

8.2.3

Unramified case

Fix an isomorphism Bv = M2 (Fv ). In this subsection we compute mφv (y, u) in the following unramified case: (1) φv is the characteristic function of M2 (OFv ) × OF×v ; (2) Uv is the maximal compact subgroup GL2 (OFv ). By [Zh2], there is a decomposition GL2 (Fv ) =

∞ a c=0

Ev× hc GL2 (OFv ),

 1 hc = 0

 0 . $c

(8.2.1)

The following result is Lemma 5.5.2 in [Zh2]. There is a small mistake in the original statement. Here is the corrected one.

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LOCAL HEIGHTS OF CM POINTS

Lemma 8.6. The multiplicity function m(b, β) 6= 0 only if q(b)q(β) ∈ OF×v . In this case, assume that β ∈ Ev× hc GL2 (OFv ). Then  1   2 (ordv λ(b) + 1) if c = 0; m(b, β) = Nv1−c (Nv + 1)−1 if c > 0, Ev /Fv is unramified;   1 −c if c > 0, Ev /Fv is ramified. 2 Nv Here λ(b) = q(b2 )/q(b) and b2 is given by the orthogonal decomposition b = b1 + b2 with respect to Bv = Ev + Ev jv . Proposition 8.7. The function mφv (y, u) 6= 0 only if (y, u) ∈ OBv × OF×v . In this case, 1 mφv (y, u) = (ordv q(y2 ) + 1). 2 Proof. We will use Lemma 8.6. Recall that X mφv (y, u) = m(y −1 , x)φv (x, uq(y)/q(x)). x∈GL2 (Fv )/Uv

Note that m(y −1 , x) 6= 0 only if ordv (q(x)/q(y)) = 0. Under this condition, φv (x, uq(y)/q(x)) 6= 0 if and only if u ∈ OF×v and x ∈ M2 (OFv ). It follows that mφv (y, u) 6= 0 only if u ∈ OF×v and n = ord(q(y)) ≥ 0. Assuming these two conditions, we have X m(y −1 , x), mφv (y, u) = x∈M2 (OFv )n /Uv

where M2 (OFv )n denotes the set of integral matrices whose determinants have valuation n. Using decomposition (8.2.1), we obtain mφv (y, u) =

∞ X

m(y −1 , hc )vol(Ev× hc GL2 (OFv ) ∩ M2 (OFv )n ).

c=0

We first consider the case that Ev /Fv is unramified. The set in the righthand side is non-empty only if n − c is even and non-negative. In this case it is given by × h U . $(n−c)/2 OE v c v The volume of this set is 1 if c = 0 and Nvc−1 (Nv +1) if c > 0 by the computation of [Zh2, p. 101]. It follows that, for c > 0 with 2 | (n − c), m(y −1 , hc )vol(Ev× hc GL2 (OFv ) ∩ M2 (OFv )n ) = 1. If n is even, mφv (y, u) =

1 1 n (ordv λ(y) + 1) + = (ordv q(y2 ) + 1). 2 2 2

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1 (n + 1). It is easy to see that ordv q(y1 ) is even and 2 ordv q(y2 ) is odd. Then n = ordv q(y2 ), since If n is odd, mφv (y, u) =

n = ordv q(y) = min{ordv q(y1 ), ordv q(y2 )} is odd. We still get mφv (y, u) = 12 (ordv q(y2 ) + 1) in this case. Now assume that Ev /Fv is ramified. Then the condition that 2 | (n − c) is unnecessary, and vol(Ev× hc GL2 (OFv ) ∩ M2 (OFv )n ) = Nvc . Thus mφv (y, u) =

1 1 1 (ordv λ(y) + 1) + n · = (ordv q(y2 ) + 1). 2 2 2 

We immediately see that in the unramified case, mφv matches the analytic kernel kφv computed in Proposition 6.8. Proposition 8.8. Let v ∈ Snonsplit − S1 be as in Assumption 5.5. Then kr(t1 ,t2 )φv (g, y, u) = mr(g,(t1 ,t2 ))φv (y, u) log Nv , and thus (v)

Kφ (g, (t1 , t2 )) = iv (Z∗ (g)t1 , t2 ) log Nv . Proof. The case (g, t1 , t2 ) = (1, 1, 1) follows from the above result and Corollary 6.8. It is also true for g ∈ GL2 (OFv ) since it is easy to see that such g has the same kernel functions as 1 for standard φv . For the general case, apply the action of P (Fv ) and Ev× × Ev× . The equality follows from Proposition 6.6 and Lemma 8.5.  8.2.4

Ramified case

Now we consider general Uv . By Proposition 8.7 for the unramified case, we know that mφv may have logarithmic singularity around the boundary Ev × Fv× . The singularity is caused by self-intersections in the computation of local 1 multiplicity. However, we will see that there is no singularity if φv ∈ S (Bv ×Fv× ) is degenerate. 1

Proposition 8.9. Assume that φv ∈ S (Bv × Fv× ) and it is invariant under the right action of Uv . Then mφv (y, u) can be extended to a Schwartz function for (y, u) ∈ Bv × Fv× . Proof. By the choice of φv , there is a constant c > 0 such that φv (x, u) 6= 0 only if −c < v(q(x)) < c and −c < v(u) < c. Recall that Z 1 mφv (y, u) = m(y −1 , x)φv (x, uq(y)/q(x))dx. vol(Uv ) B× v

239

LOCAL HEIGHTS OF CM POINTS

In order that m(y −1 , x) 6= 0, we have to make q(y)/q(x) ∈ q(Uv ) and thus v(q(y)) = v(q(x)). It follows that φv (x, uq(y)/q(x)) 6= 0 only if −c < v(u) < c. The same is true for mφv (y, u) by looking at the integral above. Then it is easy to see that mφv (y, u) is Schwartz for u ∈ Fv× . On the other hand, mφv (y, u) 6= 0 only if −c < v(q(y)) < c, which follows from the fact that φv (x, uq(y)/q(x)) 6= 0 only if −c < v(q(x)) < c. Extend mφv to Bv × F × by taking zero outside Bv× × F × . We only need to show that it is locally constant in Bv× × F × . We have φv (Ev Uv , Fv× ) = 0 by the Assumption 5.3 and the invariance of φv under Uv . Thus Z 1 mφv (y, u) = m(y −1 , x)(1 − 1Ev× Uv (x))φv (x, uq(y)/q(x))dx. vol(Uv ) B× v It is locally constant in Bv× × Fv× , since m(y −1 , x)(1 − 1Ev× Uv (x)) is locally  constant as a function on Bv× × B× v . This completes the proof. As in the analytic case, we want to approximate the above pseudo-theta series for iv (Z∗ (g)t1 , t2 ) by the usual theta series θ(g, (t1 , t2 ), mφv ⊗ φv ) X X = r(g, (t1 , t2 ))mφv (y, u) r(g, (t1 , t2 ))φv (y, u). u∈µ2U \F × y∈V

The following result is parallel to Corollary 6.9. Corollary 8.10. Let v ∈ S1 − Σ satisfy Assumption 5.3. Then iv (Z∗ (g, φ)t1 , t2 ) = θ(g, (t1 , t2 ), mφv ⊗ φv ), 8.3

∀g ∈ 1S1 GL2 (AS1 ).

SUPERSPECIAL CASE

Let v be a finite prime of F nonsplit in both B and E, and we consider the local height iv (Z∗ (g)t1 , t2 ). The Shimura curve always has a bad reduction at v due to the ramification of the quaternion algebra. We only control the singularities as in Proposition 8.9. It is enough for approximation since there are only finitely many placesnonsplit in B. In fact, all such places lie in S1 and we can use Assumption 5.3 as above. Most of the definitions and computations are similar to the supersingular case and will be mentioned briefly. Meanwhile, we will pay special attention to the parts that are different to the supersingular case. 8.3.1

Kernel function

Denote by B = B(v) the nearby quaternion algebra. We fix identifications Bv ' M2 (Fv ) and B(Af v ) ' Bvf . The intersection pairing is given by a multiplicity function m on HUv := Bv× ×Ev× B× v /Uv .

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CHAPTER 8

More precisely, the intersection of two points (b1 , β1 ), (b2 , β2 ) ∈ HUv is given by −1 gv ((b1 , β1 ), (b2 , β2 )) = m(b−1 1 b2 , β1 β2 ).

The multiplicity function m is defined everywhere on HUv except at the image of (1, 1). It satisfies the property m(b, β) = m(b−1 , β −1 ). For any two distinct CM points β1 ∈ CMU and t2 ∈ CU , their local height is given by X −1 iv (β1 , t2 ) = m(γt2v , β1v )1U v ((β1v )−1 γtv2 ). γ∈µU \B ×

Analogous to Proposition 8.4, we have the following result. Proposition 8.11. For g ∈ P (FS1 )GL2 (AS1 ), X X r(g, (t1 , t2 ))φv (y, u) mr(g,(t1 ,t2 ))φv (y, u). iv (Z∗ (g, φ)t1 , t2 ) = u∈µ2U \F × y∈B−E

Here we use the same notation: X m(y −1 , x)φv (x, uq(y)/q(x)), mφv (y, u) =

(y, u) ∈ (Bv − Ev ) × Fv× .

x∈B× v /Uv

Lemma 8.5 is still true. It says that the action of P (Fv ) × (Ev× × Ev× ) on mr(g,(t1 ,t2 ))φv (y, u) behaves like Weil representation. The following is a basic result used to control the singularity of the series. Its proof is given in the next two subsections. Lemma 8.12.

(1) If v is unramified in E, then m(b, β) 6= 0 only if ordv (q(b)q(β)) = 0,

b ∈ Fv× GL2 (OFv ).

(2) If v is ramified in E, then m(b, β) 6= 0 only if [ × ordv (q(b)q(β)) = 0, b ∈ Fv GL2 (OFv )

1 $v



Fv× GL2 (OFv ).

The main result below is parallel to Proposition 8.9. 1

Proposition 8.13. Assume φv ∈ S (Bv × Fv× ) is invariant under the right action of Uv . Then mφv (y, u) can be extended to a Schwartz function for (y, u) ∈ Bv × Fv× . Proof. The proof is very similar to Proposition 8.9. By the argument of Proposition 8.9, there is a constant C > 0 such that mφv (y, u) 6= 0 only if −C < v(q(y)) < C and −C < v(u) < C. Extend mφv to Bv × F × by taking zero outside Bv× × F × . The same method shows that it is locally constant on Bv× × F × . It is compactly supported in y by Lemma 8.12 since v(q(y)) is bounded. 

241

LOCAL HEIGHTS OF CM POINTS

As in the analytic case and the supersingular case, denote X X r(g, (t1 , t2 ))mφv (y, u) r(g, (t1 , t2 ))φv (y, u). θ(g, (t1 , t2 ), mφv ⊗φv ) = u∈µ2U \F × y∈V

Then it approximates the original series as in Corollary 6.9 and Corollary 8.10. Corollary 8.14. Let v ∈ S1 ∩ Σ satisfy Assumption 5.3. Then iv (Z∗ (g, φ)t1 , t2 ) = θ(g, (t1 , t2 ), mφv ⊗ φv ), 8.3.2

∀g ∈ 1S1 GL2 (AS1 ).

Support of the multiplicity function: unramified quadratic extension

Here we verify Lemma 8.12 assuming that v is unramified in E. The case that v is ramified in E is slightly different and considered in the next subsection. The idea is very simple: two points in HUv = Bv× ×E × B× v /Uv have a nonzero intersection only if they specialize to the same irreducible component of the special fiber in the related formal neighborhood. We first look at the case of full level, for which the integral model is very clear by the Cherednik–Drinfeld uniformization. We can easily have an explicit expression for the multiplicity function, but we do not need it in this book. Assume that Uv = OB×v is maximal and U v is sufficiently small. By the b ur , the completion of the reciprocity law, all points in CMU are defined over E v b ur = Fbur since Ev is unramified maximal unramified extension of Ev . We have E v v over Fv . In particular, the field Hw is unramified over Fv , and the model YU,w = XU ×OF OHw since the latter is still regular. It suffices to compute the intersections over XU ×OF OFbur . v The rigid analytic space XUan has the uniformization v b Fbvur ) × Z × Bv× b Fbvur = B × \(Ω⊗ XUan ⊗ f /U .

Here Ω is the rigid analytic upper half plane over Fv . More importantly, it has the integral version v b ⊗O b Fbur ) × Z × Bv× b Fbur = B × \(Ω XbU ⊗O f /U . v

v

Here XbU denotes the formal completion of the integral model XU along the b is the formal model of Ω over OF obtained by special fiber over v, and Ω v successive blowing-ups of rational points on the special fibers of the scheme P1OFv . b is regular and semistable. Its special fiber is a union The formal model Ω of P1 ’s indexed by scalar equivalence class of OFv -lattices of Fv2 , and acted transitively by GL2 (Fv ). Hence these irreducible components are parametrized by GL2 (Fv )/Fv× GL2 (OFv ).

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CHAPTER 8

It follows that the set VU of irreducible components of XU ×OFbur can be indexed v as v VU = B × \(GL2 (Fv )/Fv× GL2 (OFv )) × Z × Bv× f /U . Consider the set of CM points v× v CMU = E × \BA×f /U = B × \(B × ×E × B× v /Uv ) × Bf /U .

The embedding CMU → XUan is given by b Fbvur ) × Z, HUv −→(Ω⊗

(b, β) 7−→ (bz0 , ordv (q(b)q(β))).

Here HUv = Bv× ×E × B× v /Uv is the space where the multiplicity function is defined, and z0 is a point in Ω(Ev ) fixed by Ev× . Since all CM points are defined over Fbvur , their reductions are smooth points. The reduction map CMU → VU is given by the Bv× -equivariant map HUv −→(GL2 (Fv )/Fv× GL2 (OFv )) × Z,

(b, β) 7−→ (bb0 , ordv (q(b)q(β))).

Here b0 represents the irreducible component of the reduction of z0 . Consider the multiplicity function m(b, β). It is equal to the intersection b ⊗O b Fbur ) × Z. If their of Zariski closures of the points (b, β) and (1, 1) in (Ω v intersection is nonzero, they have to lie in the same irreducible component on the special fiber. It is true if and only if b ∈ Fv× GL2 (OFv ) and ordv (q(b)q(β)) = 0. It gives the lemma. Next, we assume that Uv = 1 + prv OBv is general. Denote U 0 = OB×v × U v . By the construction in §7.2, there is a morphism YU,w → XU 0 ,v . It induces a map VU → VU 0 on the sets of irreducible components on the special fibers. Composing with the reduction map CMU → VU , we obtain a map CMU → VU 0 which is also the composition of CMU → CMU 0 and CMU 0 → VU 0 . Hence it is induced by the Bv× -equivariant map HUv (b, β)

−→ 7−→

(GL2 (Fv )/Fv× GL2 (OFv )) × Z, (bb0 , ordv (q(b)q(β))).

It has the same form as above. Then m(b, β) is nonzero only if (b, β) and (1, 1) have the same image in the map, which still implies ordv (q(b)q(β)) = 0 and b ∈ Fv× GL2 (OFv ). 8.3.3

Support of the multiplicity function: ramified quadratic extension

Assume that v is ramified in E. We consider Lemma 8.12. Similar to the unramified case, the general case essentially follows from the case of full level. We first assume that Uv = OB×v is maximal. Then all points in CMU are b ur . But E b ur is a quadratic extension of Fb ur b ur , and Hw ⊂ E still defined over E v v v v

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LOCAL HEIGHTS OF CM POINTS

this time. The model YU,w is obtained from X ×OF OHw by blowing-up all the ordinary double points on the special fiber. b ur . The uniformization on the generic We consider uniformizations over E v fiber does not change: bvur = B × \(Ω0 ⊗ bvur ) × Z × Bv× /U v . bE bE XUan ⊗ f

0

b 0 be the blowing-up of all double points on the special b v . Let Ω Here Ω = Ω⊗E b b Ev . It is regular and semistable. Then the formal completion of fiber of Ω⊗O YbU,w along its special fiber is uniformized by v b 0 ⊗O b Ebur = B × \(Ω b Ebur ) × Z × Bv× YbU,w ⊗O f /U . v

v

b0

The special fiber of Ω consists of the strict transforms of the irreducible b and exceptional components coming from components on the special fiber of Ω the blowing-up. The reduction map sends CMU to the set VU0 of exceptional b components. The exceptional components are indexed by double points in Ω, 2 and each double point corresponds to a pair of adjacent lattices in Fv . The action of GL2 (Fv ) on the double points is transitive. Then VU0 ∼ = GL2 (Fv )/Sv where Sv is the stabilizer of any double point. Similar to the unramified case, the reduction map CMU → VU0 is given by the GL2 (Fv )-equivariant map HUv −→(GL2 (Fv )/Sv ) × Z,

(b, β) 7−→ (bb0 , ordv (q(b)q(β))).

Use the same argument that two points intersect on the special fiber if and only if they reduce to the same irreducible component. We see that m(b, β) 6= 0 only if b ∈ Sv and ordv (q(b)q(β)) = 0. It suffices to bound Sv . the edge beTake Sv to be the stabilizer of the double point corresponding to   0 1 tween the lattices OFv ⊕OFv and OFv ⊕$v OFv . The action of hv = $v 0 switch these two lattices. Then it is easy to see that Sv ⊂ Fv× GL2 (OFv ) ∪ hv Fv× GL2 (OFv ). The result is verified in this case. We remark that the group Sv is generated by the center Fv× , the element hv , and the subgroup    a b ∈ GL2 (OFv ) : c ∈ pv . Γ0 (pv ) = c d Now we consider the general case Uv = 1 + prv OBv . It is similar to the unramified case. Still compare it with U 0 = OB×v × U v . The reduction map CMU → VU0 0 is given by the Bv× -equivariant map HUv −→(GL2 (Fv )/Sv ) × Z,

(b, β) 7−→ (bb0 , ordv (q(b)q(β))).

The multiplicity function m(b, β) is nonzero only if ordv (q(b)q(β)) = 0 and b ∈ Fv× GL2 (OFv ).

244 8.4

CHAPTER 8

ORDINARY CASE

In this section we consider the case that v is a finite prime of F split in E. The local height is expected to vanish because there is no corresponding v-part in the analytic kernel in this case. Proposition 8.15. Under Assumption 5.3, iv (Z∗ (g)t1 , t2 ) = 0,

∀ g ∈ 1S1 GL2 (AS1 ).

Let ν1 and ν2 be the two primes of E lying over v. They correspond to two places w1 and w2 of H via our fixed embedding E ,→ F v . For ` = 1, 2, the intersection multiplicity iν ` is computed on the model YU ×OH OE ν where the `

fiber product is taken according to the fixed inclusions H ⊂ E ⊂ E ν` . It is actually a base change of the local integral model YU,w = YU ×OH OHw for w a place of H induced by ν` . we see thatBv is split. Fix an identification By the embedding Ev → Bv  Fv Bv ∼ . Assume that ν1 corresponds to = M2 (Fv ) under which Ev = Fv     0 Fv of Ev . It suffices to and ν2 corresponds to the ideal 0 Fv show that iν 1 (Z∗ (g)t1 , t2 ) = 0. We still make use of the results of [Zh2]. The reduction map of CM points to ordinary points at ν1 is given by v× × E × \B× f /U −→ E \(N (Fv )\GL2 (Fv )) × Bf /U.

The intersection multiplicity is a function gν1 : (N (Fv )Uv /Uv )2 → R. An explicit expression of gν1 for general Uv are proved in [Zh2, Lemma 6.3.2]. But we do not need it here. The local height pairing of two distinct CM points β1 , β2 ∈ E × \B× f /U is given by X gν1 (β1 , γβ2 )1U v (β1−1 γβ2 ). iν 1 (β1 , β2 ) = γ∈µU \E ×

Unlike other cases, the above summation has nothing to do with the nearby quaternion algebra. It is only a “small” sum for γ ∈ E × . This is the key for the vanishing of the local height series. Now we can look at X X X iν 1 (Z∗ (g)t1 , t2 ) = r(g)φ(x)a gν1 (t1 x, γt2 )1U v (x−1 t−1 1 γt2 ). a∈F × x∈B× /U f

γ∈µU \E ×

v v A nonzero term has to satisfy xv ∈ t−1 1 γt2 U . For such x , Assumption 5.3 gives φS1 (x)a = φS1 (t−1 1 γt2 )a = 0.

It follows that iν 1 (Z∗ (g)t1 , t2 ) = 0.

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LOCAL HEIGHTS OF CM POINTS

8.5

THE J -PART

Now we consider jv (Z∗ (g)t1 , t2 ) for a non-archimedean place v of F . It is nonzero only if XU has a bad reduction at v. Similar to the i-part, we will show that it can be approximated by theta series under Assumption 5.3. The pairing jv (Z∗ (g)t1 , t2 ) is the average of jν (Z∗ (g)t1 , t2 ) for each nonarchimedean place ν of E lying over v. Similar to the computation of the i-part, the intersection is computed on the model YU,w introduced in §7.2. Here w is the place of H induced by ν. By the definition in §7.1, jν (Z∗ (g)t1 , t2 ) =

1 Z∗ (g)t1 · Vt2 . [H : F ]

Here Z∗ (g)t1 is the Zariski closure in YU,w and Vt2 is a linear combination of ˆ irreducible components in the special fibers of YU,w which gives the ξ-admissible arithmetic extension of t2 . It suffices to treat (Z∗ (g)t, C) for any t ∈ CU and any irreducible component 1 D · C for C in special fiber of YU,w . Here we use the notation (D, C) = [H : F ] convenience. It is essentially a question about the reduction of CM points in CMU to irreducible components on the special fiber on the model YU,w × OE ν . Many cases have been described explicitly in [Zh2]. By the construction in §7.2, there is a morphism YU,w → XU 0 ,v , where U 0 = OB×v × U v . By this map, we classify the irreducible component C on the special fiber into the following three categories: • C is ordinary if the image of C in XU 0 ,v is not a point. • C is supersingular if v is split B and C maps to a point in XU 0 ,v . Notice this point must be supersingular. • C is superspecial if v isnonsplit in B. Let VUord , VUsing , and VUspe denote the set of these components. 8.5.1

Ordinary Components

We first consider (Z∗ (g)t, C) in the case that C is an ordinary component. It is nonzero only if points in CMU reduce to ordinary components, which happens exactly when E is split at v. Let ν1 , ν2 be the two places of E above v, and we use the convention of the splitting Ev = Fv ⊕ Fv at the beginning of §8.4. It suffices to consider jν 1 . The treatment is very similar to Proposition 4.2 by separating geometrically connected components. By Lemma 5.4.2 in [Zh2], the ordinary components are parametrized by VUord = F+× \A× f /q(U ) × P (Fv )\GL2 (Fv )/Uv .

246

CHAPTER 8

Note that the first double coset is exactly the set of geometrically connected components. The reduction CMU −→VUord is given by the natural map: E × \B× f /U g

−→ F+× \A× f /q(U ) × P (Fv )\GL2 (Fv )/Uv , 7−→ (det g, gv ).

For any β ∈ CMU , the intersection (β, C) 6= 0 only if β and C are in the same geometrically connected component. Once this is true, it is given by a locally constant function lC for βv ∈ B× v . Moreover, the function lC factors through P (Fv )\GL2 (Fv )/Uv . In summary, we have (β, C) = lC (βv )1F × q(βC )q(U ) (q(β)). +

Here βC ∈ B× f is any element such that q(βC ) gives the geometrically connected component containing C. Therefore, we have (Z∗ (g)t, C) = (Z∗ (g)α t, C) where α = q(t)−1 q(βC ). By (4.3.1), X X X Z∗ (g)α t =

−1 r(g, (βC t, 1))φ(y, u) [βC y].

u∈µ0U \F × a∈F × y∈Bf (a)/U 1 +

Thus (Z∗ (g)t, C)

=

X

X u∈µ0U \F ×

X

−1 r(g, (βC t, 1))φ(y, u)lC (βC y).

× y∈B (a)/U 1 f a∈F+

Similar to Lemma 7.6, it is zero by Assumption 5.4. In fact, fixing a place v 0 ∈ S2 − {v}, the inside sum has a factor Z X 1 r(g, (t, 1))φv0 (y, u) = r(g, (t, 1))φv0 (y, u)dy = 0. vol(Uv10 ) Bv0 (a) 1 y∈Bv0 (a)/Uv0

8.5.2

Supersingular components

Now we consider (Z∗ (g)t, C) in the case that C is a supersingular component. It is nonzero only if points in CMU reduce to supersingular components, which happens exactly when both B and E arenonsplit at v. The treatment is similar to §8.2. Denote B = B(v) and fix an isomorphism B(Avf ) ' Bvf as usual. The key is to characterize the reduction map CMU −→VUsing −→SU . Here SU is the set of supersingular points in XU,v . Recall that × × × v × × v × CMU = E × \B× f /U = B \(B ×E × Bv ) × (Bf ) /U ,→ B \Hv × (Bf ) /U

247

LOCAL HEIGHTS OF CM POINTS

where Hv = Bv× ×Ev× B× v . We also have the parametrization × × v × SU = B × \B× f /U = B \Fv × (Bf ) /U.

Then the reduction map CMU → SU is given by the product of determinants: Hv −→Fv× ,

(b, β) 7−→ q(b)q(β).

Fix one supersingular point z on the special fiber of XU,v . Denote by NUv the formal completion of XU,OF along z. It depends only on Uv , and represents v the deformation of the formal OFv -module of height two with Drinfeld Uv -level structures. The formal completion XbU,OF of XU,v along its supersingular locus v SU is given by XbU,OF = B × \(NUv × Z) × (Bvf )× /U v . v

To check this identity, note that both sides are formal schemes supported at finite sets, then it essentially follows from the coset expression of SU . eU be the minimal desingularization of NU ⊗O b Hw . Then the formal Let N v v sing completion of YU,w along the union of fibers in VU can be described as eU × Z) × (Bv )× /U. YbU = B × \(N f v e be the set of irreducible components on the special fiber of N eU × Z. It Let V v × also admits an action by Bv × Uv . Then we have a description e × (Bv )× /U. VUsing = B × \V f Our conclusion is that the map CMU −→VUsing −→SU is given by (Bv× × Uv )-equivariant maps e Hv −→V−→Z. We are now applying the above result to compute the intersection pairing (Z∗ (g)t, C). It is very similar to our treatment of the i-part. Let (C0 , βC ) ∈ e × (Bv )× be a couple representing C ∈ V sing . The intersection with C0 defines V f U a locally constant function lC0 on HUv = Bv× ×Ev× B× v /Uv . Unlike the multiplicity function m, lC0 has no singularity on HUv . For any CM point β ∈ CMU , the intersection pairing is given by X −1 lC0 (γ, βv )1U v (βC γβ v ). (β, C) = γ∈µU \B ×

Hence, we obtain (Z∗ (g)t, C) =

X

X

a∈F × x∈B× /U f

r(g)φ(x)a

X γ∈µU \B ×

−1 v v lC0 (γ, tv xv )1U v (βC γt x ).

248

CHAPTER 8

Now we convert the above to a pseudo-theta series. The process is the same as the i-part. We sketch it here. Change the order of the summations. Note that −1 v v γt x ) 6= 0 if and only if xv ∈ t−1 γ −1 βC U v . Put it into the sum. We 1U v (βC have X X X r(g)φv (t−1 γ −1 βC )a r(g)φv (x)a lC0 (γ, tv xv ). (Z∗ (g)t, C) = a∈F × γ∈µU \B ×

x∈B× v /Uv

Denote   uq(y) lC0 (y −1 , x)dx. φv x, × q(x) Bv

Z lφv (y, u) = lC0 ,φv (y, u) := Then (Z∗ (g)t, C) =

X

X

r(g, (t, βC ))φv (y)a lr(g,(t,1))φv (y)a

a∈F × y∈µU \B ×

=

X

X

r(g, (t, βC ))φv (y, u)lr(g,(t,1))φv (y, u).

u∈µ2U \F × y∈B ×

It is a pseudo-theta series. 1 We claim that if φv ∈ S (Bv × Fv× ), then lφv extends to a Schwartz function × for (y, u) ∈ Bv ×Fv . The proof of Proposition 8.9 applies here. We only explain that v(q(y)−1 q(x)) is a constant on the support of lC0 (y −1 , x), which is needed for lφv to be compactly supported. In fact, lC0 (y −1 , x) 6= 0 only if the point (y −1 , x) ∈ Hv and C0 ∈ VH have the same image in Fv× . It determines the coset q(y)−1 q(x)q(Uv ) in Fv× uniquely. Similar to all the pseudo-theta series we treated before, our conclusion is (Z∗ (g, φ)t, C) = θ(g, (t, βC ), r(βc−1 , 1)lφv ⊗ φv ),

∀g ∈ P (FS1 )GL2 (AS1 ).

In our particular case, C is in the geometrically connected components of t2 , and thus we can take βC = t2 . 8.5.3

Superspecial components

Now we consider (Z∗ (g)t, C) in the case that C is a superspecial component. It happens when v isnonsplit in B. Resume the notations in the treatment of the i-part. It is similar to the supersingular case. The curve XU 0 ,v has the explicit uniformization as formal schemes: b ⊗O b Fbur ) × Z × (Bvf )× /U v . b Fbur = B × \(Ω XbU 0 ,v ⊗O v

v

For general levels, the uniformization is easily done in the level of rigid spaces: v b Fbvur = B × \Σr × Bv× XUan ⊗ f /U .

249

LOCAL HEIGHTS OF CM POINTS

b Fbvur ) × Z depending on r. Here Σr is some etale rigid-analytic cover of (Ω⊗ b bF ur ) × Z in the rigid space bO Take the normalization of the formal scheme (Ω⊗ v b ur , and make a minimal resolution of singularities. We obtain a regular b Fbur H Σr ⊗ w v b r over OH ur . The construction is compatible with the algebraic formal scheme Σ w construction of YU,w , i.e., b r × Bv× /U v . b Hb ur = B × \Σ YbU,w ⊗O f w

Here YbU,w is the formal completion of the YU,w along its special fiber. The uniformizations here are not explicit at all, but we only need some group-theoretical properties. e be the set of irreducible components of Σ b r . Then the reduction Let V spe × e Under Assumption is given by a Bv -equivariant map HUv → V. CMU → V 5.3, the same calculation as in supersingular case will show that (Z∗ (g)t, C) can be approximated by a theta series on the quadratic space B.

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Index A(B× ), 63 A(B× ), 70 A(B× , Q), 70

H 1,0 , 64, 66 hhiU , 138 Hom0 (J, J ∨ ), 66

analytic kernel, 101 approximate, 19, 193 arithmetic theta lifting, 16

Hecke algebra, 72 Hodge class, 60, 94 I 0 (0, g, φ)(v), 188 I(s, g, Φ), 40 I(s, g, χ, Φ), 40 I(s, g, χ, φ)U , 172 I(s, g, φ)U , 171 i(D1 , D2 ), 214 iv (D1 , D2 ), 215 iRv (D1 , D2 ), 215 ∗ R , 26 , 26

B 1 , 24 B, 2, 24 Bad f , 119 Baf , Bf (a), 119 Bv , 24 B1 , 25 coherent, incoherent, 24, 32 degenerate Schwartz function, 17

J, 64 J ∨ , 64 JU , 64 j, 24 j(D1 , D2 ), 214 jv (D1 , D2 ), 215 jv (D1 , D2 ), 215

E(s, g, φ)K , 110 E(s, g, u, φ), 92, 109 E 1 , 23 E0 (g, u, φ), 93 E0 (s, g, u, φ), 93 E0 (u, φ), 93 η, 9 Ei, 196

geometric kernel, 99

K h , 112 (v) Kφ , 203 kφv , 189 kv,s , 203 (v) Kφ , 189

H, 217 H 1 , 66 H 1 (XU,τ ), 66 H 0,1 , 64

LK , 94 LU , 60 LK,α , 94 `φ, 146

[G], 27 γ(V, q), 29

255

256 `0 φ, 146 λ, 25 µK , 108 µU , 94 µ0U , 112 mφv , 235 ν, 31 nearby quaternion algebra, 25 P ◦ , 65 Pr, 197 Pr0 , 198 Pic(X × X), 67 Φ, 91, 92, 107 φ(x)a , 108 r, 29 r1 , r2 , 185 Σ, 8 SU , 234 S, 31, 91, 92, 106 S, 31 standard Schwartz function, 107 T , 23 θ(g, h, φ), 108 T(f1 ⊗ f2 ), 85 U 1 , 119 (V × k × )a , 31 V1 , 24 V, 24, 40 V1 , 24, 40 V2 , 24, 40 vol(XU ), 61 W0 (g, u, φ), 93 W0 (u, φ), 93 Wa (s, g, u, φ), 93 wU , 94

INDEX

X, 59 ξP , 65 ξU , 4, 61 ξα , 4, 61 ξt , 216 ξU,α , 4, 61 XU , 3, 58, 59 XU,α , 59 XU , 216 YU , 217 YU,w , 217 Z(φ)U , 94 Z(g, φ)U,α , 114 Z(g, φ)U , 95, 111 Z(x), Z(x)U , 62 Z(x)U , 93 Z∗ (φ), 94 Z∗ (g, φ), 95, 118 Z∗ (g, φ)U , 111 Z0 (φ), 94 Z0 (g, φ), 95 Z0 (g, φ)U , 111