Arithmetic Moduli of Elliptic Curves. (AM-108), Volume 108 9781400881710

Table of contents :
TABLE OF CONTENTS
INTRODUCTION
Chapter 1: GENERALITIES ON "A-STRUCTURES" AND "A-GENERATORS"
1.1 Review of relative Cartier divisors
1.2 Relative Cartier divisors in curves
1.3 Existence of incidence schemes
1.4 Points of "exact order N" and cyclic subgroups
1.5 A mild generalization: A -structures and A -generators
1.6 General representability theorems for A-structures and A-generators
1.7 Factorization into prime powers of A-structures and A-generators
1.8 Full sets of sections
1.9 Intrinsic A-structures and A-generators
1.10 Relation to Cartier divisors
1.11 Extensions of an etale group
1.12 Roots of unity
1.13 Some open problems
Chapter 2: REVIEW OF ELLIPTIC CURVES
2.1 The group structure
2.2 Generalized Weierstrass equations, and some elementary universal families
2.3 The structure of [N]
2.4 Rigidity
2.5 Manifestations of autoduality
2.6 Hasse's theory
2.7 Applications to rigidity
2.8 Pairings
2.9 Deformation theory
Chapter 3: THE FOUR BASIC MODULI PROBLEMS FOR ELLIPTIC CURVES: SORITES
3.1 Γ(N)-structures
3.2 Γ1(N)-structures
3.3 Balanced Γ1(N)-structures
3.4 Γ0(N)-structures
3.5 Factorization into prime powers
3.6 Relative representability
3.7 The situation when N is invertible
Chapter 4: THE FORMALISM OF MODULI PROBLEMS
4.1 The category (Ell)
4.2 Moduli problems
4.3 Representable moduli problems
4.4 Rigid moduli problems
4.5 Geometric properties of moduli problems
4.6 Some examples
4.7 A basic result: representability and rigidity
4.8 Another example
4.9 Yet another example
4.10 Lemmas on group-schemes
4.11 Modular families
4.12 More geometric properties of moduli problems
4.13 The category (Ell/R)
4.14 Moduli problems of finite level
APPENDIX: MORE ON RIGIDITY AND REPRESENTABILITY
Chapter 5: Regularity theorems
5.1 First main theorem
5.2 Axiomatics
5.3 End of the proof
5.4 Summary of parameters at supersingular points
5.5 First applications
5.6 Pairings
Chapter 6: CYCLICITY
6.1 The main theorem
6.2 Axiomatics
6.3 End of the proof
6.4 Cyclicity as a closed condition
6.5 The moduli problem [N–Isog]
6.6 The moduli problem [Γ0(N)]: proof of the First Main Theorem
6.7 Detailed theory of cyclic isogenies and cyclic subgroups; standard factorizations
6.8 More on [N-Isog]
Chapter 7: QUOTIENTS BY FINITE GROUPS
7.1 The general situation
7.2 A descent situation
7.3 Quotients of product problems
7.4 Applications to the four basic moduli problems
7.5 Axiomatics
7.6 Applications to regularity
7.7 Summary of parameters at supersingular points
7.8 More parameters for [Γ0(p^n)] at supersingular points
7.9 Detailed study of the congruence quotients [Γ0(p^n); a,b] of [bal. Γ1p^n)]
APPENDIX: BASE CHANGE FOR RINGS OF INVARIANTS
Chapter 8: COARSE MODULI SCHEMES, CUSPS, AND COMPACTIFICATIONS
8.1 Coarse moduli schemes
8.2 The j-line as a coarse moduli scheme
8.3 Localization of moduli problems over the j-line
8.4 The j-invariant as a fine modulus, coarse moduli schemes as fine moduli schemes (!)
8.5 Base change for coarse moduli schemes
8.6 Cusps by normalization near infinity; compactified coarse moduli schemes
8.7 Interlude: The groups T[N] and T
8.8 Relation to the Tate curve
8.9 Relation with ordinary elliptic curves via the Serre-Tate parameter
8.10 Other universality properties of the groups T[N]
8.11 Computation of Cusps(P) via the Tate curve
Chapter 9: MODULI PROBLEMS VIEWED OVER CYCLOTOMIC INTEGER RINGS
9.1 Generalities
9.2 A descent situation
9.3 The situation near infinity
9.4 Applications to the basic moduli problems
Chapter 10: THE CALCULUS OF CUSPS AND COMPONENTS VIA THE GROUPS T[N] AND THE GLOBAL STRUCTURE OF THE BASIC MODULI PROBLEMS
10.1 Motivation
10.2 Analysis of (Γ(N)]
10.3 Group action
10.4 Canonical problems
10.5 Explication for T[N]
10.6 Cusp-labels and component-labels
10.7 Some combinatorial lemmas
10.8 Application to structure near infinity
10.9 Applications to the four basic moduli problems
10.10 Detailed analysis at a prime p, balanced subgroups
10.11 Basic examples of balanced subgroups
10.12 Application to the moduli problem [ΓQ(P^n); a,a]
10.13 The numerology of moduli schemes, via the line bundle ω
Chapter 11 : INTERLUDE: EXOTIC MODULAR MORPHISMS AND ISOMORPHISMS
11.1 Motivation
11.2 "Abstract" morphisms
11.3 Some basic examples
Chapter 12: NEW MODULI PROBLEMS IN CHARACTERISTIC p; IGUSA CURVES
12.1 Frobenius
12.2: Basic lemmas
12.3 Igusa structures
12.4 The Hasse invariant
12.5 Ordinary curves
12.6 First analysis of the Igusa curve
12.7 Analysis of cusps
12.8 The equation defining Ig(p), and a theorem of Serre
12.9 Numerology of Igusa curves
12.10 "Exotic" projections from Igusa curves; "exotic" Igusa structures
Chapter 13: REDUCTIONS mod p OF THE BASIC MODULI PROBLEMS
13.1 Some general considerations on crossings at supersingular points
13.2 Modular schemes as examples
13.3 Analysis of p-power isogenies between elliptic curves
13.4 Global structure of the moduli spaces W(P,[Γ0,(p^n)]), M(P,[p^n-Isog])
13.5 Analysis of [Γ1(p^n)] in characteristic p
13.6 Explicit calculations via the groups T[p^n] of [Γ0(p^n)], [Γ1(p^n)]
13.7 The reduction mod p of [Γ(p^n)]^can
13.8 Complete local ring of [Γ(p^n)]^can at supersingular points; intersection numbers
13.9 Distribution of the cusps on [Γ(p^n)]^can
13.10 The reduction mod p of a general p-power level moduli problem
13.11 The reduction mod p of [bal. Γ1(p^n)]^can
13.12 The reduction mod p of quotients of [bal. Γ1(p^n)] by subgroups of (Z/p^nZ)^× × (Z/p^nZ)^×
Chapter 14: APPLICATION TO THEOREMS OF GOOD REDUCTION
14.1 General review of vanishing cycles
14.2 Application to curves
14.3 Application to modular curves: explicitation of the numerical criterion
14.4 Characters and conductors
14.5 The Good Reduction Theorem
14.6 Explicitation of the Good Reduction Theorem
14.7 Application to Jacobians
NOTES ADDED IN PROOF
REFERENCES

Citation preview

Annals o f Mathematics Studies Number 108

ARITHMETIC MODULI OF ELLIPTIC CURVES

BY

NICHOLAS M. KATZ AND

BARRY MAZUR

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1985

Copyright © 1985 by Princeton University Press ALL RIGHTS RESERVED

The Annals of Mathematics Studies are edited by William Browder, Robert P. Langlands, John Milnor, and Elias M. Stein Corresponding editors: Stefan Hildebrandt, H. Blaine Lawson, Louis Nirenberg, and David Vogan

Clothbound editions of Princeton University Press books are printed on acid-free paper, and binding materials are chosen for strength and durability. Pa­ perbacks, while satisfactory for personal collections, are not usually suitable for library rebinding

ISBN 0-691-08349-5 (cloth) ISBN 0-691-08352-5 (paper)

Printed in the United States of America by Princeton University Press, 41 William Street Princeton, New Jersey

☆

Library of Congress Cataloging in Publication data will be found on the last printed page of this book

TAB LE OF CONTENTS INTRODUCTION

xi

Chapter 1:

GENERALITIES ON “ A-STRU CT UR ES” AND “ A-GENERATORS” 1 .1 R ev ie w of re la tiv e C a rtie r d iv is o rs 1.2 R e la tiv e C a rtie r d iv is o rs in c u rv e s 1.3 E x iste n ce of in cid en ce schem es 1.4 P o in ts of (ie x a c t order N ” and c y c lic subgroups 1 .5 A m ild g e n era liz a tio n : A -stru c tu re s an d A -generators 1.6 G en eral re p re s e n ta b ility theorem s fo r A -stru ctu res and A -g en erato rs 1 .7 F actorization into prime pow ers of A -stru ctu res and A -g en erato rs 1.8 Full s e ts of s e c tio n s 1.9 In trin sic A -stru c tu re s and A -generators 1 . 1 0 R elatio n to C a rtie r d iv is o rs 1 . 1 1 E xten sion s of an e ta le group 1 . 1 2 R oots o f unity 1 . 1 3 Some open problem s

Chapter 2: 2 .1 2 .2 2 .3 2 .4 2 .5 2 .6 2 .7 2 .8 2 .9 Chapter 3: 3 .1 3 .2 3.3 3 .4 3 .5 3 .6 3 .7

3 3 7 12 17 20 22 26 32 38 40 48 55 61

REVIEW OF ELLIPTIC CURVES The group stru ctu re G e n eraliz ed W e ie rstra ss eq u atio n s, and some elem entary u n iv ersa l fa m ilie s The stru ctu re of [N] R ig id ity M an ifestation s o f au to d u ality H a s s e ’s theory A p p lic a tio n s to rig id ity P airin g s D eform ation theory

67 73 75 77 81 85 87 91

THE FOUR BASIC MODULI PROBLEMS FOR ELLIPTIC CURVES: SORITES T(N) -stru ctu res re s tru c tu re s B a la n c ed F^(N) -stru ctu res re s tru c tu re s F actorization into prime pow ers R e la tiv e re p re se n ta b ility The situ atio n when N is in vertib le

98 98 99 100 100 101 102 104

v

63 63

vi

CONTENTS

Chapter 4: 4 .1 4.2 4 .3 4.4 4 .5 4 .6 4.7 4.8 4 .9 4 .10 4 .1 1 4 .1 2 4.13 4.14

THE FORMALISM OF MODULI PROBLEMS The catego ry (Ell) Moduli problem s R ep resen tab le moduli problems R igid moduli problems Geom etric p roperties of moduli problems Some exam ples A b asic re s u lt: re p re se n ta b ility and rig id ity A nother exam ple Y et another exam ple Lemmas on group-schem es Modular fa m ilie s More geom etric p roperties o f moduli problems The category (Ell/R) Moduli problem s of fin ite le v e l

107 107 107 108 10 9 10 9 110 111 117 117 118 120 12 1 122 123

APPENDIX: MORE ON RIGIDITY AND REPRESENTABILITY

12 5

Chapter 5: 5.1 5.2 5.3 5.4 5 .5 5 .6

R eg u la rity theorem s F irst main theorem A xio m atics End of the proof Summary of param eters a t su p ersin g u lar points F irst a p p lic a tio n s P airin g s

129 129 12 9 13 5 14 3 14 3 150

Chapter 6: 6.1 6.2 6.3 6.4 6.5 6 .6

CYCLICITY The main theorem A xio m atics End o f the proof C y c lic ity a s a c lo se d condition The moduli problem [N-Isog] The moduli problem [ r Q(N)]: proof of the F irst Main Theorem D e ta ile d theory of c y c lic iso g e n ies and c y c lic subgroups; stan d ard fa c to riz a tio n s More on [N-Isog]

15 2 152 15 5 15 6 162 16 4

QUOTIENTS BY FINITE GROUPS The g en eral situ atio n A d e sc e n t situ atio n Q uotients of product problems A p p lic a tio n s to the four b asic moduli problems A xio m a tics A p p lic a tio n s to reg u larity Summary of param eters a t su p ersin gu lar points More param eters for [I^(pn)] a t su p ersin gu lar points D e ta ile d study of the congruence quotients [ l^ ( p n) ;a ,b ] o f [bal. r ^ p 11)]

186 186 19 5 196 197 201 207 208 208

6.7 6 .8 Chapter 7: 7 .1 7.2 7.3 7.4 7.5 7.6 7 .7 7 .8 7.9 APPENDIX:

BASE CHANGE FOR RINGS OF INVARIANTS

166 16 7 17 8

210 2 15

CONTENTS

Chapter 8: 8 .1 8.2 8 .3 8.4 8 .5 8 .6 8 .7 8.8 8 .9 8.10 8 .1 1 Chapter 9: 9.1 9.2 9 .3 9.4

COARSE MODULI SCHEMES, CUSPS, AND COMPACTIFICATIONS C o arse moduli sch em es The j -lin e a s a c o a rse moduli schem e L o c a liz a tio n of moduli problem s o ver the j-/ine The j -in varian t a s a fine modulus, c o a rse moduli sch em es a s fin e moduli schem es (!) B a se change for c o a rs e moduli sch em es C u sp s by norm alization near in fin ity ; com p actified c o a rse moduli schem es Interlude: The groups T[N] and T R elatio n to the Tate curve R elatio n w ith ordinary e llip tic c u rv e s via the S erre -T ate param eter Other u n iv e rsa lity p rop erties of the groups T[N] Com putation of C u s p s ^ ) via the Tate cu rve MODULI PROBLEMS VIEWED OVER CYCLOTOMIC INTEGER RINGS G e n e ra litie s A d e sc e n t situ atio n The situ atio n near in fin ity A p p lic a tio n s to the b asic moduli problem s

Chapter 10: THE CALCULUS OF CUSPS AND COMPONENTS VIA THE GROUPS T[N] AND THE GLOBAL STRUCTURE OF THE BASIC MODULI PROBLEMS 1 0 .1 M otivation 10 .2 A n a ly s is of [T(N)] 1 0 .3 Group a c tio n 1 0 .4 C an o n ical problem s 1 0 .5 E xp lica tio n for T[N] 1 0 .6 C u sp -la b e ls and com ponent-labels 1 0 . 7 Some com binatorial lemmas 1 0 .8 A p p lic a tio n to stru ctu re near in fin ity 1 0 .9 A p p lic a tio n s to the four b asic moduli problems 1 0 . 1 0 D e ta ile d a n a ly s is a t a prime p , b alan ced subgroups 1 0 . 1 1 B a s ic exam ples of b alan ced subgroups 1 0 . 1 2 A p p lic a tio n to the moduli problem [ I ^ ( P n) ; a , a ] 1 0 . 1 3 The numerology of moduli schem es, v ia the line bundle co

v ii

224 2 24 2 28 232 234 243 2 46 251 2 58 260 261 2 66 271 2 71 2 77 2 78 281

2 86 2 86 287 2 90 293 2 95 295 2 96 299 301 309 324 326 328

Chapter 1 1 : INTERLUDE: EXOTIC MODULAR MORPHISMS AND ISOMORPHISMS 1 1 . 1 M otivation 1 1 . 2 “ A b s tr a c t” morphisms 1 1 . 3 Some b asic exam ples

3 39 3 39 339 3 40

Chapter 12 : NEW MODULI PROBLEMS IN CHARACTERISTIC p; IGUSA CURVES 1 2 .1 Frobenius

344 344

viii

CONTENTS

1 2 .2 : B a sic lemmas 1 2 .3 Igusa stru ctu res 1 2 .4 The H asse in varian t 1 2 .5 O rdinary c u rves 1 2 .6 F irst a n a ly s is of the Igusa cu rve 1 2 .7 A n a ly s is of cu sp s 1 2 .8 The equation defining Ig(p) , and a theorem of S erre 1 2 .9 Numerology o f Igusa cu rves 1 2 . 1 0 “ E x o tic ” p ro jectio n s from Igusa c u rv e s ; “ e x o tic ” Igusa stru ctu res Chapter 13: 1 3 .1 1 3 .2 1 3 .3 1 3 .4 1 3 .5 1 3 .6

REDUCTIONS mod p OF THE BASIC MODULI PROBLEMS Some gen eral con sid eratio n s on c ro ssin g s at su p ersin gu lar points Modular schem es a s exam ples A n a ly s is of p -power iso g e n ie s betw een e llip tic cu rves G lobal stru ctu re o f the moduli sp a c es » , [ r 9 (pn)]),5H (y,[pn- i s o g ] ) A n a ly s is of [ F ^ p 11)] in c h a ra c te ristic p E x p lic it c a lc u la tio n s via the groups T[pn] Of

1 3 .7 13 .8 13 .9 13 .10 13 .11 1 3 .1 2 Chapter 14: 14 .1 14 .2 1 4 .3 1 4 .4 1 4 .5 1 4 .6 1 4 .7

[r^ (p n)l, [TjCp")]

The reduction modpof [r(pn)]can C om plete lo c a l ring of [ F (pn)]can a t su p ersin gu lar p o in ts; in te rsec tio n numbers D istrib u tion of the cu sp s on [F (p n)]can The reduction mod p o f a g en eral p-power le v e l moduli problem The reduction modpof [bal. T ^ p 11)]0311 The reduction modpof quotients o f [bal. F ^ p 11)] by subgroups of (Z/pnZ)x x (Z/pn Z)x

345 3 49 3 53 359 361 366 3 68 3 76 381 389 389 3 96 3 99 407

414

418 424 429 4 33 435 441 4 50

APPLICATION TO THEOREMS OF GOOD REDUCTION 4 57 G eneral review o f vanishing c y c le s 457 A p p lica tio n to cu rves 467 A p p lica tio n to modular c u rv e s: e x p licita tio n of the num erical c riterio n 471 C h aracters and conductors 479 The Good R eduction Theorem 480 E x p lic itatio n of the Good R eduction Theorem 5 00 A p p lica tio n to Jac o b ia n s 502

NOTES ADDED IN PROOF

505

REFERENCES

5 11

INTRODUCTION This book is devoted to giving an account of the arithmetic theory of the moduli s p a c e s of e llip tic c u rv e s. The main emphasis is on under­ standing the behavior of these moduli sp a c es at primes dividing the “ l e v e l ” of the moduli problem being considered.

Until recently, this

seemed a very difficult problem, b e ca u se one had no a priori construction of these s p a c e s at the “ b a d ” primes. One defined them as schemes over, say,

Z[l/N] a s the solution to some w ell-p o sed moduli problem which

only made se n s e for e llip tic cu rves over rings in which N was in v e rtib le , and then one used a p rocess of norm alization to extend them to schemes over Z , e .g ., one took the “ p ro j” of the graded subring of the ring of all modular forms of the type in question con sisting of those with in teg ral Fourier (“ q -exp a nsio n ” ) c o e ffic ie n ts a t the c u sp s.

This procedure pro­

duced a scheme over Z , but one had no idea of what moduli interpretation this scheme had, nor a fortiori did one have any idea of the modular inter­ pretation of its reduction modulo p , for p a prime dividing the le v e l. H istorically, the only c a s e where this question was in any way s a t i s ­ factorily understood w as the c a s e of I^(p), which made u n iversa l s e n s e as the moduli problem “ p - is o g e n ie s ” , or “ finite flat subgroup-schemes of rank p . ” This modular interpretation was implicitly known to Kronecker, for whom it appears a s the statement that the “ modular equation of degree

p , reduced mod p , is the curve in the ( j 1 , j 2)-plane

( j 1 - ( j 2)P) ( ( j 1 )P- j 2) = 0 One knows the crucial role that the reinterpretation of this Kronecker congruence by Eichler-Shimura as the “ congruence re la tio n ”

ix

X

INTRODUCTION

Tp = F + V mod p played in the reduction of the Ramanujan conjecture to W eil’s “ Riemann Hypothesis for v a r ie tie s over finite f i e l d s . ” Eichler-Shimura made use of this relation to prove that for a l l but fin ite ly many p the Ramanujan conjecture held for any given cusp form of weight two and le v e l N which is a simultaneous eigenfunction of a ll Tp with p not dividing N. The method w as in trin sica lly incapable of sp ecifyin g the excep tion al p , which were b e lie v e d to c o n s is t only of primes dividing N . P artly in order to s ettle d e fin itiv e ly this question of excep tion al p for weight two forms, Igusa, in a brilliant s e r ie s of papers ([ig 2, 3, 4, 5 ]), gave a complete and d efin itive account of the le v e l N moduli scheme over Z[l/N ].

Except for a difference of mathematical language, and the

modular interpretation of the cusps by Deligne-Rapoport, there have been no “ improvements” to Igusa’s account of what happens over Z[l/N]. Although Igusa’s papers contain many stimulating speculations about the situation mod p for “ b ad ”

p (e.g., the footnote ([Ig 2], p. 472) where he

points out that the genus of I^(p) is c lo s e ly related to the number of [super] singular points in ch ara c teristic

p ), there w as to be no significant

progress in understanding the situation at “ b a d ”

p for another decade.

In 1 9 6 8 , Deligne completed the general reduction of Ramanujan’s con­ jecture, for forms of arbitrary weight, and in particular for A , to W e il’s Riemann Hypothesis for v a rie tie s over finite fie ld s .

In his artic le [De 1],

he mentions that in fact the L^(p) moduli scheme, (with su itab le au xiliary prime-to-p rigidification) is actu a lly a regu lar scheme, and in a letter of Ju ly 10 , 19 7 0 to Parshin he proves this regularity by checking what happens at the supersingular points in c h ara c teristic Simultaneously, another theme w as developing.

p. Shimura conjectured

and Casselm an [Cmn 1] proved that for p = 2 9, 53, 6 1 , 73, 89, 97, the Jacobian of (the modular curve for) ^ ( p ) , modulo the Jacob ian of I^(p), acquired good reduction over the field

Q(£p) • C asselm an [Cmn 2]

INTRODUCTION

xi

explained that such theorems of good reduction could be predicted by the “ Langlands ph ilosop h y” , which related, c on je c tu ra lly , such q uestions to qu estion s in representation theory which w ere already well-understood. The paper of Deligne-Rapoport [De-Ra] in 1972 provided an e x h au stive account of what w as then known about arithmetic moduli of e llip tic c u rves. It gave a complete account of le v e l N moduli problems over Z[l/N ], in­ cluding a modular interpretation of the compactified moduli scheme (i.e ., including the cu sp s) as the moduli sp ace of “ generalized e llip tic curves with au x ilia ry s tru c tu re .” It a l s o gave a modular interpretation over

Z

to the r\(p) moduli problem, and with it a proof that the good reduction phenomenon of Shim ura-Casselm an held for any p . Another innovation w as the system atic use of algebraic s ta c k s , as developed by Mumford [Mum 1] and Deligne-Mumford [De-Mu]. The next sign ifica n t progress came in 1 9 7 4 , with D rinfeld’s introduc­ tion (in the context of his theory of ‘ ‘ e llip tic m odules” ) of the notion of a “ fu ll le v e l N-structure” on an e llip tic curve over an arbitrary scheme, where N need not be invertible, a s a pair of points

P,Q of order N

such that the group-scheme E[N] of points of order N is equal to the sum

2

fep +bQ]

a,b mod N

a s a C a rtier d iv iso r inside E . Drinfeld showed that with this definition, the corresponding fu ll le v e l N moduli problem for his “ e llip tic m odules” was regular.

It was clear to the experts, although never published, that

with D rin fe ld ’s definition applied to usual e llip tic cu rves, one obtained a moduli problem over Z which was regular, and which, over coincided with the usual “ fu ll le v e l N ” moduli problem.

Z [l/N ],

In particular,

one now had a modular interpretation of its reduction modulo any p , as the moduli space of e llip tic c u rves, together with Drinfeld le v e l N s tru c ­ tures, over Fp-algebras.

With this modular interpretation, it became a

pleasan t e x e rc is e to c a lc u l a t e e x p lic itly the reduction modulo any prime p.

xii

INTRODUCTION

In a letter to Drinfeld of January 2 1 , 1 9 7 5 , Deligne explained how the Drinfeld idea of using Cartier d iv is o rs allowed one to define u n iv ersa lly the r\(N) problem a s w e ll as a “ b a la n c e d ” v e rsio n of it by sayin g that a point P in an e llip tic curve has “ exac t order N ” if it is killed by N and if the Cartier d iv is o r inside E defined by

2

fep]

a mod N

is ac tu a lly a subgroup-scheme of E . Deligne a ls o explained that the resulting moduli problem was regular. In June 1 9 7 9 , the present authors rediscovered D e lig n e ’s

T^N) idea,

and they formulated a Drinfeldian v ersion of ^ ( N ) by defining a finite lo c ally free subgroup-scheme G of rank N inside an e llip tic curve to be c yc lic if lo c ally f.p.p.f. on the base one could find a point P in it which generated it, in the s en se that G =

2

[aP]

a mod N

a s Cartier d iviso rs inside E .

Using this definition of IT^(N) a s the

moduli problem of “ e llip tic curves together with c y c lic subgroup-schemes which are finite lo c a lly free of rank N , ” they proved that the ^ ( N ) problem was regular and worked out e xp licitly the reductions

mod p of

a ll the “ stan d ard ” moduli problems. These calcu lation s of s p e c ia l fib ers, together with some intricate (due to wild ramification) calcu lation s of the topological invariants of the s p e c ia l fibers, led to a direct geometric verification of a rather general theorem of good reduction which includes the Shimura-Casselm an-DeligneRapoport theorem as a s p e c ia l c ase .

For the most part, this good reduction

theorem is a ls o a consequence of the above-mentioned Langlands philosophy, which reduces it to a known question in representation theory (cf. [La] and [M W], proof of Prop. 2, §2, Chapter 3). In writing this book, we have tried sim ultaneously to be self-contained and to be a s general a s p ossib le.

INTRODUCTION

x iii

In the first chapter, we develop the D rinfeldian notions of le v e l structure through the notion of eq uality of Cartier d iv iso rs in the ambient e llip tic curve.

With an e ye to future applications to the moduli of higher­

dimensional ab e lian v a r i e ti e s , in which the points of order N c e a s e to be a C artier d iv iso r, we give a reformulation of a l l the Drinfeldian notions in the context of fin ite lo c a lly free commutative group-schemes, w ithout reference to any ambient s p ac e. T his reformulation, and the questions it ra ise s, may prove to be of some independent interest. This chapter is followed by a short “ R ev ie w of Elliptic C u r v e s ” , in which we re c a ll a l l the b asic fa c ts we w ill use about e llip tic cu rv e s. We give either complete proofs or p recise references for a l l of th ese fa c ts . In Chapter 3, we apply the general notions developed in the first chapter to the s p e c ia l c a s e of e llip tic cu rv e s, and we formulate in terms of them the b a s ic moduli problems for e llip tic curves. In Chapter 4, we develo p a rudimentary formalism for speaking about these moduli problems, which amounts to working s y s te m a tic a lly with s ta c k s without ever sayin g so. We speak rather of “ re la tiv e ly rep resen ta­ ble moduli problems” , a notion which seem s admirably suited to our purposes, and which is a throw-back to Mumford’s original exposition [Mum 1]. A fte r th e se preliminary chapters, we turn to the detailed study of the b a s ic moduli problems a s “ open arithmetic s u r f a c e s . ” The basic re su lts on their structure and inter-relations (e.g., which are regular, which are fin ite fla t over which others, which are quotients by finite groups of which o th e rs ,•••) are given in Chapters 5, 6 and 7. The remaining 7 chapters are devoted to the detailed study of th ese same moduli problems a s “ curves over S p e c ( Z ) . ” In Chapter 8, we compactify our moduli problems, re la tiv e to Spec (Z) , by adding the cusps.

In Chapter 9, we explain how to d e a l syste m a tic a lly

with th ese moduli problems which are “ r e a l l y ” defined over cyclotomic integer rings rather than o ver

Z . In Chapter 1 0 we give the b asic resu lts

on the structure of our compactified moduli problems a s re la tiv e cu rves.

x iv

INTRODUCTION

A fter a b r i e f digression concerning “ e x o t i c ’ ’ isomorphisms of moduli problems in Chapter 1 1 , the remaining three chapters are devoted to the d etailed study of the degeneration at bad primes of our moduli problems a s re la tiv e cu rves.

Chapter 12 g ives the theory of the Igusa cu rves,

which are the “ b a s i c ” p-power le v e l moduli problems in c h ara c teristic

p.

In Chapter 1 3 , we give the detailed structure of the reduction mod p of each of our b asic moduli problems a s a “ d isjoint union, with c ro ssin gs at the supersingular p o in ts ” , of su itab le Igusa cu rves. In Chapter 14 , w e apply the s p e c ific c a lc u la tio n s of the previous chapter to prove a general theorem of good reduction for s u itab le “ p i e c e s ” of Jac ob ia n s of modular cu rves. We would like to thank the IHES for providing the congenial atmosphere in which this book w as written.

We warmly thank Ofer Gabber, whose in­

numerable comments and corrections were invaluable to us in preparing the final v ersion of this work.

Lauri Hein and Perry Di Verita of Princeton

U niversity, and Helen Mann of Princeton U niversity P re ss prepared the original and fin a l manuscripts re sp e ctiv e ly .

To them and to our editor,

Barbara Stump of Princeton U niversity P re s s , we extend our thanks for their patience in the fa ce of our numerous and unexpected re v is io n s .

NICHOLAS M. KATZ BARRY MAZUR

Arithmetic Moduli of Elliptic Curves

Chapter 1 GENERALITIES ON “ A-STRU CTURES” AND “ A-GENERATORS” ( 1 . 1 ) R eview o f re la tiv e C a rtie r d iv iso rs (Compare [Mum 2], pp. 6 1 - 7 3 .) (1 .1.1)

L et S be an arbitrary scheme, and let X be an S-scheme. By

an e ffe c tiv e C artier d iv iso r D in X/S we mean a clo se d subscheme D C X such that D is flat over S ( the ideal s h ea f 1(D) C © x

is an invertible 0 x -module, i.e.,

it is a lo c a lly free G x -module of rank one. This notion is lo c a l on S . When S is affine, say S = Spec (R) , it means that we can cover X by affin e opens U- = Spec (A -), A^ an R-algebra, such that D fl U- is defined in

by one equation f • = 0 , where f^ e A^

is an element such that A i/ fiA i is fla t over R

1 f-

is not a z ero-d iviso r in A- .

The tautological exact sequence on X ,

o i(d) ->gx - eD->o , becomes on U- = Spec (A-) the exact sequence

0 --------------------------

. A / f j A j -------- . 0 .

( 1 . 1 . 2 ) Given two e ffe c tiv e Cartier d iv iso rs

D and D' in X / S , their

sum D + D' is the e ffe c tiv e C artier d iviso r in X/S defined lo c a lly by the 3

4

ARITHMETIC MODULI OF ELLIPTIC CURVES

product of the defining equations of D and D'.

E x p lic itly , if S = S p e c (R )

and if on an affine open Spec (A) of X , D and D' are defined re s p e c ­ tiv e ly by equations f = 0 and g = 0 , then D +D ' is defined there by fg = 0 .

To check that fg is not a zero-d iviso r in A , one notes the com­

mutative diagram

To check that A/fgA

is fla t over R , one notes the short exact sequence

0 ----- >A/gA

A /fgA

>A/fA ----- >0 ,

which exhibits A/fgA as an extension of flat R-modules. ( 1 . 1 . 3 ) Given an e ffe c tiv e Cartier d iv iso r D in X / S , we may sp eak of the inverse (as invertible 0 x -module) I- 1 (D) of its ideal sheaf.

We have

a tautological exact sequence O . 0 xY -> r \ D)

The inclusion of ©x

0n D q® I- 1 (D) -> 0 .

in I- 1 (D) allow s us to view the constant function

“ 1 ” as a global sec tio n of I~1(D) , and we may recover D as the scheme of zeroes of this global section of I- 1 (D ) . C o n v erse ly , suppose we are given a pair (£, £) c on sisting of an in­ v e rtib le G x -module £ on X together with a global sectio n I e H ° (X ,£ ) which s i ts in a short exact sequence of 0 x -modules

ex o— ► with £ / 0 x

g — >£ /© x — . o

fla t over S . Then the scheme of z eroes of the section £ of

£ is e a s i ly see n to be an e ffe c tiv e Cartier divisor D in X/S , and there is a unique isomorphism of (£, £) with (I 1 (D ),“ 1 ’ 0 .

1. GENERALITIES ON ‘ ‘A-STRUCTURES” AND ‘ ‘ A-GENERATORS’ ’

5

This construction a llo w s us to interpret e ffe c tiv e Cartier d iv is o rs in X/S as isomorphism c l a s s e s of pairs

(£, £) as above.

From this point

of view , the operation “ sum of e ffe c tiv e Cartier d iviso rs in X/R ” is none other than the operation of tensor product: (£ ,£ ) + (£',£') = ( £ » £ ' , £ ® D

.

The zero element for this addition is the pair (©x , l ) , corresponding to the empty Cartier divisor. ( 1 .1 .4 )

There are two natural situ ations in which one can define the

in v erse image of a re la tiv e Cartier d ivisor.

F ir s t let

T -* S be an arbitrary morphism of schem es. diviso r

D inX/S

scheme

Then for any e ffe c tiv e C artier

, s ay represented by a pair

(£, £), the clo sed sub­

Dx T of X™ = X x T is an e ffe c tiv e C artier d iv iso r in 1

S

1

S

X t /T , represented by the pair ( £ T, £T) on X T . To s e e this, it s u ffic e s to treat the c a s e when S = Spec (R) and T = S p ec (R ') are both affine; then the sequence on X T = X ® R ' ©

® R - J 9}.. >£ ® r ' ---------->£ ® r 7 © A R

® r' A R

is obtained from the short exact sequence on X o — ►© x _ L £ — , £ /© x — . 0 by applying the functor

R ' . B ec au se £ / 0 Y is assumed flat over R , R

A

this sequence s ta y s short e xact after ® R ^ and its la s t term is R -fla t. R

Therefore (£ ®R', £®1) d efines an e ffe c tiv e Cartier d iv is o r in X & R 7 R ' , R

as required.

R

6

ARITHMETIC MODULI OF ELLIPTIC CURVES

Second, let

be a flat morphism of S-schemes.

Then any e ffe c tiv e Cartier d iv iso r D

in X/S gives ris e to an e ffe c tiv e Cartier d iv is o r f*(D) in Y / S .

Indeed,

the c artesian diagram f*(D) C

^Y

f flat

D C

^X

shows that f*(D) is fla t over D , and hence, is flat over S .

D being S-flat, that f*(D)

To s e e that the ideal sh ea f I(f*(D)) is an invertible

GY -module, we remark that this ideal sh e a f is none other than f*(I(D)) , as fo llo w s from the fact that the short exact sequence on X 0 -* 1(D) ^ 0 X - 0 D - 0 remains short e x act after application of the functor f* , thanks to the fla tn e s s of f . ( 1 .1 .5 ) We now turn to the question of recognizing which closed sub­ schemes of X are in fact e ffe c tiv e Cartier d iv iso rs in X/S . PROPOSITION 1 . 1 . 5 . 1 .

Suppose that S IS lo c a lly noetherian, and that

X is an S-schem e of fin ite type which is fla t over S . coherent s h e a f on X which is fla t o ver S .

L e t 3“ be a

Then the n e c e s s a ry and

s u ffic ie n t condition that 3“ be fla t over 0 X is that for ev ery geom etric point of S , i.e ., every morphism S p ec (k )-> S

with k an a lg e b ra ic a lly

1. GENERALITIES ON *‘A-STRUCTURES’ ’ AND ‘ ‘A-GENERATORS”

c lo se d field , the induced coherent s h e a f 3" ® k on the fib re X ® k R over

7

be fla t

R

X ®k * R

P roof. This is ju st the fibre-by-fibre criterion of fla tn e s s [A-K 1, V, 3.6]. Q.E.D. COROLLARY 1.1.5 .2 .

L e t S be lo c a lly noetherian, X a fla t S-schem e

of fin ite type, an d D C X a c lo se d subschem e which is fla t o ver S . Then D is an e ffe c tiv e C a rtie r d iv is o r in X/S i f and only if for a l l geom etric p oints S p e c ( k ) ^ S

o f S , the c lo se d subschem e D®k of X ® k is an R

R

e ffe c tiv e C a rtier d iv is o r in X ® k / k . R

Proof. The n e c e s s ity is a s p e c ia l c a s e of the preservation of e ffe c tiv e C artier d iv iso rs in X/S under arbitrary change of base T -* S . For the s u ffic ie n c y , we apply the proposition to J = 1(D), which is S-flat b ecau se it s i t s in the short exact sequence 0 -> 1(D) -> © x ^ 0 D - 0 , in which both © x and © D are S-flat by h ypothesis.

Tensoring over ©s

with k (for any geometric point Spec (k) -> S ), this sequence s ta y s exact ( S -fla tn e s s of ©D ), so comparing first terms y ield s 1(D) ® k -=-> I(D®k) . R

R

By hypoth esis,

I(D®k) is an invertible © x ^^-module, so in particular it

is © X(g)^-flat.

Therefore 1(D) is © x -flat by the proposition.

1(D) is a l s o coherent, it is a lo c a lly free ©x -module.

B e c au se

That it is lo c ally

free of rank one may then be s ee n by restricting it to the fibres Xk of X/S , on each of which it is lo c a lly free of rank one.

( 1 .2 ) (1.2.1)

Q.E.D.

R e la tiv e C a rtie r d iv is o rs in cu rves Let S be an arbitrary scheme. By a smooth curve C / S , we w ill

8

ARITHMETIC MODULI OF ELLIPTIC CURVES

a lw a y s mean a smooth morphism C -> S of re la tiv e dimension one which is separated and of finite presentation. LEMMA 1 .2 .2 .

If C/S

is a smooth cu rve, then an y sec tio n s e C(S)

d efin es an e ffe c tiv e C a rtie r d iv is o r, noted [ s ] , in C / S. Proof. B e c a u s e C/S is of finite presentation, we are immediately reduced to the c a s e when S = Spec (R) with R noetherian, (EGA IV, 8 . 9 . 1 ) then by Corollary 1 . 1 . 5 . 2 to the c a s e when R is an alg eb raic ally closed field k , and in this c a s e the ass e rtio n is obvious. LEMMA 1 .2 .3 .

L e t C/S

be a smooth curve a s above.

Q.E.D.

L e t D C C be a

c lo se d sub-schem e which is fin ite and fla t over S , and of fin ite p re se n ta­ tion o ver S . Then D is an e ffe c tiv e C a rtie r d iv iso r in C/S , which is proper over S . C o n v erse ly ev ery e ffe c tiv e C a rtier d iv is o r in C/S which is proper over S is of th is form. Proof. To prove the first statement, we are immediately reduced (by EGA IV, 8 .9 .1 and 1 1 . 2 . 6 . 1 ) to the c a s e when S = S p ec (R ) with R noetherian, then by Corollary 1 . 1 . 5 . 2 to the c a s e when R is an alg eb raic ally clo sed fie ld k , in which c a s e it is obvious that D is an e ffe c tiv e Cartier divisor.

That D is proper over S follow s from its being assumed finite

over S . C o n versely, any e ffe c tiv e C artier d iv iso r D in C/S is certainly of finite presentation, so we are reduced to the c a s e when S = Spec (R) with R noetherian (EGA IV, 8 .9 .1 and 1 1 . 2 . 6 . 1 ) .

We must prove that D is

finite o ver R . But D is proper over R , so it su ffic e s to prove that D/R has finite fibres.

Thus we are reduced to the c a s e when R is an

1. GENERALITIES ON “ A-STRUCTURES’ ’ AND ‘ ‘A-GENERATORS’ ’

9

algebraically closed field k, i.e., to the assertion that an effective Cartier divisor in a smooth curve over k is a finite k-scheme, which is obvious. Q.E.D. If C/S is a proper smooth curve, then e v e ry effective

R EM A R K 1 . 2 . 4 .

Cartier divisor in C/S is automatically proper over S . DEFINITION 1.2.5.

Let C/S be a smooth curve, D an effective Cartier

divisor in C/S which is proper over S . Then locally on S , say S = Spec (R ), the affine ring of D is a locally free R-module of finite rank. This rank, which is constant Zariski locally on S , is called the degree of D, noted deg(D).

In terms of an (£, I) presentation of D,

the exact sequence on C

o—► e c - L £ — >£/G — >o has £ / 0 ^ I_ 1(D)/0

**

I_ 1(D) ® 0~ ^ ec

® 0~ = an invertible ©^.-module. 0C

Therefore D is proper over S if and only if the sheaf £ / 0 on C has its support proper over S . If this is the case, then locally on S , say S = S p e c(R ), the R-module h °( c

,£ / G ) = H ^ c . r ^ D ) / © )

is a locally free R-module of rank = degree (D). LEMMA 1.2.6.

L e t C/S

be a smooth curve, D^ and D2 two e ffe c tiv e

C a rtie r d iv is o rs in C/S which are both proper over S . Then Dj + D2 is proper o ver S , and we h ave the e q u a lity

deg(D x +D2) = deg(D1 ) + deg(D2) . Proof. In terms of representatives

and ( £ 2 ,£2) of D1 and D2

respectively, we have a short exact sequence of sheaves on C

10

ARITHMETIC MODULI OF ELLIPTIC CURVES

0 - £ 2 / 0 - , £ 1 ® £ 2/ © ^ £ 1 ® £ 2 / f 2 ^ 0 obtained by applying the snake lemma to the diagram

e

0

xl

0

■0

£ 1®£2 when bottom row i s obtained by applying x

*°

L

■e —

0

^ £ x

.

In any c a s e , the above exact sequence, when rewritten 0

^ 2 pj®D2

( ^ 1 ® ^ 2 ^®®D j +D2

(£ 2)|d s

( £ j ® £ 2)|D i + d 2

shows that 0 ^ 0 2

*" ' ^ 1 ®

( V

M

0 !

is proper over S and shows, taking Euler ch aracter­

i s ti c s , that the degrees add a s asserted . Q.E.D. LEMMA 1 . 2 .7 .

L e t C/S

be a smooth curve.

Then for an y sec tio n

s e C(S) , the a s s o c ia te d e ffe c tiv e C a rtie r d iv iso r [ s ] in C/S

is proper

over S , and has degree one. C o n v e rse ly , an y e ffe c tiv e C a rtie r d iv iso r D in C/S which is proper over S and has degree one is of the form [ s ] for a unique sec tio n s e C(S) . Proof. If s € C ( S ) , then the Cartier d ivisor

[s] s i t s in a diagran

1. GENERALITIES ON “ A-STRUCTURES’ ’ AND “ A-GENERATORS”

11

[s] c _

which shows that [ s ] is proper over S , of degree one. C o n v erse ly , if D is an e ffe c tiv e Cartier d iviso r in C/S , proper over S of degree one, then we have a diagram D Cl­

in which the diagonal arrow is an isomorphism (because lo c a lly on S , say S = Spec (R) , it turns the affine ring of D into an R-algebra which is an in v e rtib le R-module, i.e ., into R i t s e l f ) . LEMMA 1 . 2 .8 .

Q.E.D.

L e t C ', C be two smooth cu rves o ver S , and f

a morphism w hich is fin ite and fla t of degree noted d e g (f ).

If D is an

e ffe c tiv e C a rtie r d iv iso r in C /S, proper over S , then its in v e rse image f*(D)

in C ' is an e ffe c tiv e C a rtie r d iv iso r in C V S , proper o ver S , and

its degree is g iven by deg(f*(D)) = d e g ( f ) d e g (D) .

12

ARITHMETIC MODULI OF ELLIPTIC CURVES

P roof. That f*(D) is an e ffe c tiv e Cartier d iv iso r in C'/S we have already see n resu lts from the fla tn e ss of f. The c artesian diagram of schemes f*(D)

f

I finite flat

shows that f*(D) is finite flat over D of degree = d e g (f) , and the resu lt fo llo w s.

Q.E.D.

LEMMA 1 .2 .9 .

L et C/S be a smooth curve, D an e ffe c tiv e C artier

d iviso r in C/S which is proper over S , and T -> S an arb itra ry morphism of schem es.

Then DT is an e ffe c tiv e C artier d iv iso r in C T , proper over

T , w hose degree is u nchanged : d e g (D T) = deg(D) . Proof. That DT is an e ffe c tiv e Cartier d iv iso r in C T we have already seen.

The as s e rtio n of properness of DT over T , a s w e ll a s the fact

that Dt

is finite lo c a lly free over T of rank = deg(D) , result from the

c artesian diagram

finite lo c ally free of d e g (D ).

Q.E.D.

(1 .3 ) E x iste n ce of Incidence schem es ( 1 . 3 . 1 ) G iven two e ffe c tiv e C artier d iviso rs D ,D ' in X / S , we s ay that D' < D if D ' C D

(inclusion of c lo se d subschemes of X ), i.e ., if

1. GENERALITIES ON “ A-STRUCTURES” AND “ A-GENERATORS”

13

1(D) C I(D0 (inclusion of ideal s h e a v e s inside 0 x ), or equivalently if there e x is ts an e ffe c tiv e C artier d iviso r D" in X/S such that D = D'+ D " . In terms of local equations f = 0 for D and g = 0 for D', the condition D'< D means p re c ise ly that g d iv id e s f , i .e ., that lo c a lly we have f = gh for some function h . This function h is unique (because g is not a zero-d ivisor), and h i t s e l f is not a zero diviso r (because gh = f is not a zero divisor).

The lo c a lly defined subschemes h = 0 of X

patch together to define a subscheme D" of X , whose id eal sh ea f I(D"), lo c a lly hGx , is in vertib le. To s e e that D" is flat over S , we may assume S = S p e c ( R ) ,

X = Spec ( A ),

D is defined by f = 0 ,

D' by

g = 0 , and D" by h = 0 , with f = g h . The exact sequence

0 ------->A/hA then shows that A/hA

A/fA

>A/gA

>0

is R-flat (because A/fA and A/gA are R - f l a t ),

a s required. G lobally, in terms of re p re se n tativ e s

(£, £) for D and

(£ , I ) for

D', the condition D'< D is p re c ise ly that the global section £ of £ van ish id en tic ally in £|D' = £ ® ® d'* ( £ ' Z') with £ / = £ ® ( £ )~1 and I

Then ^ = 1 / 1'.

represented by

The global v e rsion of the

above short exact sequ en ce is (1.3.l . l )

0 -> £" ® ©x

® Gn ^ 0 . 0X

Gx

(1 .3 .2 ) We now turn from a general X/S to a smooth curve C/S (cf. 1 . 2 .1 ) . LEMMA 1 . 3 .3 .

L e t C/S

be a smooth curve,

D and D'

e ffe c tiv e C a rtie r

d iv is o rs in C/S which a re both proper o ver S . If D'< D , then the e ffe c tiv e C a rtie r d iv is o r D " = D - D '

is proper over S, and

deg(D") = deg(D) - deg(D') .

14

ARITHMETIC MODULI OF ELLIPTIC CURVES

Proof. The properness is obvious from the short exact sequence ( 1 . 3 . 1 . 1 ) . The degree a sse rtio n then resu lts from (1.2.6). K E Y LEMMA 1 .3 .4 .

L e t C/S

Q.E.D.

be a smooth curve, D and D' e ffe c tiv e

C a rtier d iv iso rs in C / S , with D ' proper over S . (1)

Then

there e x is ts a unique c lo se d subschem e Z C S which is u n iversa l

for the re la tio n D'< D in the follow in g s e n s e : given any morphism of schem es T ^ S , D^p < Dt (2)

the in v erse im ages D^ and DT in C T s a tis fy

if and only if the morphism T -> S fa c to rs through Z ; the subschem e Z C S is defin ed lo c a lly on S by deg(D')

e q u a tio n s; (3) form ation of the c lo se d subschem e Z C S commutes w ith arb itrary change of b ase S'-> S , in the s e n se that the c lo se d subschem e Z ' of S' “ u n iversa l for the re la tio n D '' < D s -

S

is none other than Z x S ' . s

Proof. The question is c le a rly local on S , which we may assume affine, s a y S = Spec (R) . In terms of a rep resen tative

(£, £) for D , the condi­

tion D'< D is that the global sectio n t of £ van ish id en tically in ^

q

=

B e c au se D' is finite lo c ally -free over S , and £|D

is an invertible 0 D'-module, the module H°(D', £|D') is a lo c a lly free R-module of rank = d eg(D '). e l,

L o ca lly on R , we may choose an R -b asis

of this R-module.

The element I has a unique exp ressio n

deg(D')

I =

r-e- , c o e fficien ts

r- e R .

i =l

The condition “ £ = 0 ” is then represented by the c lo se d subscheme of Sp ec(R ) defined by the simultaneous vanishing of C O R O L L A R Y 1. 3 .5 .

L et C/S

. Q.E.D.

be a smooth curve, D and D' e ffe c tiv e

C a rtier d iv iso rs in C/S which a re both proper o ver S of the sam e degree. Then there e x is ts a unique c lo se d subschem e Z C S which is u n iversa l

1. GENERALITIES ON ‘ 1A-STRUCTURES’ ’ AND *‘ A -GENERATORS ’ ’

for the re la tio n D = D', in the s e n s e that fo r an y T D t = DT In ^ T

^ an(^ on^y

15

S , we h ave

T -> S fa c to rs through Z . The suhschem e

Z C S is d efin ed lo c a lly in S by deg(D) equations, and its form ation commutes w ith a rb itra ry change of b ase S'-> S . Proof. The condition D = D / is eq uivalent to the condition D'< D , because D and D' both have the same degree.

Q.E.D.

( 1 .3 . 6 ) Let C/S be a smooth curve which is a group-scheme over S . Let D be an e ffe c tiv e Cartier d iv is o r in C/S which is proper over S . We s a y that D is a subgroup of C/S if for e very S-scheme T the sub­ s e t D(T) of the group C(T) is in fact a sub-group.

This amounts to the

e x iste n c e of a (n e c e s s a r ily unique) structure of finite flat S-group-scheme on D such that D C >C is an S-group-scheme homomorphism. C O R O L L A R Y 1 . 3 .7 .

L e t C/S be a smooth curve which is a group-schem e

over S . L e t D be an e ffe c tiv e C a rtie r d iv iso r in C/S , proper o ver S . Then there e x is ts a unique c lo se d subschem e Z C S which is u n iv ersa l for the re la tio n “ D is a subgroup, ” in the s e n s e that for an y T ^ S , DT in C t /T is a subgroup i f and only if T -» S fa c to rs through Z . The subschem e Z C S is d efin ed lo c a lly in S by 1 + deg(D) + (deg(D ))2 equations, an d its form ation commutes with a rb itra ry change of b ase S' -> S . Proof. Let us denote by e f C(S) the identity sectio n for the group structure, by

inv : C -> C

the S-automorphism of C exp ressin g inversion, and by m : C x C -» C S

the S-morphism “ m u ltiplicatio n .” In order that D be a subgroup, it is n e c e s s a ry and sufficient that the fo llowing three conditions be s a tis fie d : (1)

Denoting by

[e] the e ffe c tiv e Cartier d iv is o r in C/S attached to

the identity sec tio n e € C ( S ) , we have the inequality of Cartier d iv iso rs

16

ARITHMETIC MODULI OF ELLIPTIC CURVES

in C/S [e] < D . (This is the condition that the subset D(T) C C(T) a lw a y s contain the identity.) (2) We have the equality of e ffe c tiv e Cartier d iv iso rs in C/S D = inv*(D) . (This is the condition that the subset D(T) C C(T) be s tab le by inversio n .) (3) Let W denote the S-scheme D x D , which represents the functor

s

“ ordered pairs of points of D . ”

Let (P1 ,P2) be the u n iversa l pair of

points of D , so that we have a tautological diagram

Then we must have the inequality of e ffe c tiv e Cartier d iv iso rs in C w [m(P1 ,P2>] < Dw . (This is the condition that the su b set D(T) C C(T) a lw a y s be s ta b le by multiplication.) Now condition (1) is represented lo c a lly on S by one equation. dition (2) is represented lo c a lly on S by deg(D) equations. (3)

Con­

Condition

is represented lo c a lly on W by one equation, but b ecau se W is

finite lo c a lly free over S of rank (deg(D))2 , the vanishing on W of a single function is equivalent to the vanishing on S of its (deg(D)) “ c o o rd in a te s .” Q.E.D.

1. GENERALITIES ON “ A-STRUCTURES* ’ AND “ A-GENERATORS”

17

( 1 .4 ) P o in ts of “ e x a c t order N ” and c y c lic subgroups ( 1 . 4 . 1 ) L et C/S be a smooth curve (cf. 1 . 2 .1 ) , given with a structure of commutative S-group-scheme.

We w ill refer to such a C/S a s a “ smooth

commutative S-group-scheme of re la tiv e dimension o n e ” ; in view of 1 . 2 . 1 , C/S is autom atically separated and of finite presentation o ver S . Let N > 1 be an integer. We s a y that a point P e C(S) has “ exact order N ” if the e ffe c tiv e C artier d iv iso r D in C/S of degree N defined by (1 .4 .1.1 )

D=

[P] + [2P] +•••+ [NP]

is a subgroup of C/S . We c a ll this subgroup the c y c lic subgroup of rank N “ g e n e r a te d ” by P .

We s a y that a c lo se d subgroup-scheme G C C

which is fin ite lo c a lly -fr e e of rank N over S is c y c lic if, lo c a lly f.p.p.f. (SGA III, Exp. IV, 6 .3 ) on S , G admits a generator, i .e ., if there e x ists some f.p .l.p .f. (faith fu lly flat, lo c a lly of finite presentation) morphism T -» S and a point

P e C(T) = C t (T) of “ e xact order N ” on C t /T

which generates the subgroup G T of C T , i .e ., such that we have an equality of C artier d iv is o rs in C T : N

(1 .4.1.2 )

gt

= 2

[aP] '

a~l C le arly if a point P e C(S) has “ exact order N ” in C/S , and generates a subgroup G of rank N , then for any change of b ase T -» S the induced point PT 1 the

0 e C(S) has “ exac t order pn

indeed it “ g e n e r a te s ” the

18

ARITHMETIC MODULI OF ELLIPTIC CURVES

subgroup K e r ( F n : C -» C^P

of rank pn . This example show s that a

given point can have many different “ exact o rd e rs.” LEMMA 1 . 4 .4 .

Suppose that N is in v ertib le on S .

L et P e C(S)

be a

point k ille d by N . Then the follow in g conditions are eq uivalent. (1) P has 11 e x ac t order N ” in C /S. (2) For e very geom etric point Spec (k) ^ S

of S , the induced point

Pj^ € C(k) = C^(k) has “ exact order N ” in C^/k. (3) For e v ery geom etric point Spec(k)

S of S , the induced point

Pk e C(k) has e xact order N in the u sual s e n s e that N is the le a s t p o sitiv e in teger which k ills P ^ , i.e ., the N points !a P ^ i , a =1 , - **, N are a ll d istin c t in C ( k ) . (4)

The e ffe c tiv e C a rtie r d iv iso r in C/S N

2 [ap] a-l

in C/S (5)

is fin ite e ta le o ver S . The unique S -group homomorphism Z/NZ ^ C

“ 1 ” e Z/NZ

which maps

to P € C(S) d e fin es a c lo se d S -immersion Z/NZ C _ c

which id e n tifie s the con stant S -schem e Z/NZ with the C a rtie r d iv iso r I [aP] : N

Z/NZ

^

[aP] .

a=l Proof. (1) => (2), because the property of having £‘ exact order N ” is preserved under arbitrary changes of base T -» S . (2)

=> (3).

Let G be the rank N subgroup-scheme of C^ “ generated”

by P ^ . Then G is finite flat of rank N over k .

B e c au se N is in v e rti­

ble in S , it is invertible in k , and therefore G is automatically finite

1. GENERALITIES ON 4‘A-STRUCTURES’ ’ AND ‘ ‘A-GENERATORS”

19

e ta le over k of rank N. Therefore as a C artier d ivisor in C ^ , G con ­ s i s t s of a uniquely determined s e t of N distinct points.

The equality of

Cartier d iv is o rs in Ci k

N g

= £

b P k]

a =l

shows that the N points { a P ^ !, a = l , - - - , N , are a l l distinct in C^, as required. (3) (4).

Let us denote by D the e ffe c tiv e Cartier diviso r N

D = ^

[aP]

a=l

in C/S . A s a scheme o ver S , D is finite and lo c a lly free of rank N over S . It is finite e ta le over S if and only if its discriminant (determi­ nant of the matrix tr(e^ej) ••• ) is invertible on S . This holds if and only if for a l l geometric points Spec(k) -> S , the C artier d ivisor in C^ N

Dk = 2

[aPk]

a-1 is finite e ta le o ver k , i.e ., if and only if (3) holds. (3)

(5). The morphism of S-groups

certain ly factors through the e ffe c tiv e Cartier divisor N

D = ^

[aP] ,

a=l thus defining an S-morphism (Z/NZ)g - D .

Z/NZ

C mapping 1 to P

20

ARITHMETIC MODULI OF ELLIPTIC CURVES

Both source and target of this morphism are finite lo c a lly -fre e over S of the same rank N , so lo c a lly on S , say S = S p e c ( R ) , the map on coordinate rings is given by an NxN matrix over R .

Our map is an

isomorphism if and only if this matrix has determinant a unit in R . T here­ fore our map is an isomorphism if and only if for a l l geometric points Spec(k)-*S

of S , the k-morphism lhP, Z/NZ ----------- L

N Dk = £

[aPk ]

a =l

is an isomorphism, i.e ., if and only if the N points i a P j J , are a ll distinct. (5) =>(1).

Indeed if (5) holds, then

structure of subgroup of C / S. REMA RK 1 . 4 .5 .

a=l,---,N,

N 2 [aP] is endowed with the

a=l

Q.E.D.

If we do not assume N invertible on S , we s t i l l have

the implications (3)

(4 ) < = > (5) *=> (1) = > (2) .

A point P e C(S) which is killed by N and which s a t i s f i e s any of the equivalent conditions (3), (4), (5), might be c alle d an “ e ta le point of exact order N . ”

(1 .5 )

A mild g e n eraliz atio n : A -stru ctu res and A -gen erators

( 1 . 5 . 1 ) A s before, C/S is a smooth commutative group-scheme over S of re la tiv e dimension one (cf. 1 . 4 .1 ) . abelian group. (1 .5.1.1)

Let A be an “ a b s tr a c t” finite

A homomorphism of abstract groups : A

C(S)

is said to be an A-structure on C/S if the e ffe c tiv e Cartier diviso r D in C/S of degree = #(A) defined by

1. GENERALITIES ON “ A-STRUCTURES” AND “ A-GENERATORS”

(1.5 .1.2 )

21

D = S [