Euler Systems. (AM-147), Volume 147 [1 ed.] 9781400865208, 9780691050768

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Annals of Mathematics Studies Number 147

Euler Systems by Karl Rubin

Hermann Weyl Lectures The Institute for Advanced Study

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

2000

Copyright © 2000 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex All Rights Reserved The Annals of Mathematics Studies are edited by John N. Mather and Elias M. Stein Library of Congress Catalog Card Number 99-069141 ISBN 0-691-05075-9 (cloth) ISBN 0-691-05076-7 (pbk.) The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed

The paper used in this publication meets the minimum requirements of ANSIINISO Z39.48-1992 (R 1997) (Permanence of Paper)

www.pup.princeton.edu Printed in the United States of America

3 5 7 9 10 8 6 4 2

3 5 7 9 10 8 6 4 2 (Pbk.)

HERMANN WEYL LECTURES

The Hermann Weyl Lectures are organized and sponsored by the School of Mathematics of the Institute for Advanced Study. Their aim is to provide broad surveys of various topics in mathematics, accessible to nonspecialists, to be published eventually in the Annals of Mathematics Studies.

Contents Acknowledgments

xi

Introduction Notation

3 6

Chapter 1. Galois Cohomology of p-adic Representations 1.1. p-adic Representations 1.2. Galois Cohomology 1.3. Local Cohomology Groups 1.4. Local Duality 1.5. Global Cohomology Groups 1.6. Examples of Selmer Groups 1.7. Global Duality

9 9 11

12

18 21 23 28

Chapter 2. Euler Systems: Definition and Main Results 2.1. Euler Systems 2.2. Results over K 2.3. Results over K 00 2.4. Twisting by Characters of Finite Order

33 33 36

Chapter 3. Examples and Applications 3.1. Preliminaries 3.2. Cyclotomic Units 3.3. Elliptic Units 3.4. Stickelberger Elements 3.5. Elliptic Curves 3.6. Symmetric Square of an Elliptic Curve

47 47

Chapter 4. Derived Cohomology Classes 4.1. Setup 4.2. The Universal Euler System 4.3. Properties of the Universal Euler System 4.4. Kolyvagin's Derivative Construction

75 75 78

vii

40

43

48

55 55 63 73

80 83

CONTENTS

viii

4.5. 4.6. 4.7. 4.8.

Local Properties of the Derivative Classes Local Behavior at Primes Not Dividing pt Local Behavior at Primes Dividing t The Congruence

90 92 98 102

Chapter 5. Bounding the Selmer Group 5.1. Preliminaries 5.2. Bounding the Order of the Selmer Group 5.3. Bounding the Exponent of the Selmer Group

105 105 106 114

Chapter 6. Twisting 6.1. Twisting Representations 6.2. Twisting Cohomology Groups 6.3. Twisting Euler Systems 6.4. Twisting Theorems 6.5. Examples and Applications

119 119 121 122 125 125

Chapter 7. Iwasawa Theory 7.1. Overview 7.2. Galois Groups and the Evaluation Map 7.3. Proof of Theorem 2.3.2 7.4. The Kernel and Cokernel of the Restriction Map 7.5. Galois Equivariance of the Evaluation Maps 7.6. Proof of Proposition 7.1.7 7.7. Proof of Proposition 7.1.9

129 129 135 141 145 147 151 154

Chapter 8. Euler Systems and p-adic £-functions 8.1. The Setting 8.2. Perrin-Riou's p-adic £-function and Related Conjectures 8.3. Connection with Euler Systems when d_ = 1 8.4. Example: Cyclotomic Units 8.5. Connection with Euler Systems when d_ > 1

163 164 166 168 171 173

Chapter 9. Variants 9.1. Rigidity 9.2. Finite Primes Splitting Completely in Koo/ K 9.3. Euler Systems of Finite Depth 9.4. Anticyclotomic Euler Systems 9.5. Additional Local Conditions 9.6. Varying the Euler Factors

175 175 178 179 180 183 185

Appendix A.

189

Linear Algebra

CONTENTS

A.l. Herbrand Quotients A.2. p-adic Representations

ix

189 191

Appendix B. Continuous Cohomology and Inverse Limits B.l. Preliminaries B.2. Continuous Cohomology B.3. Inverse Limits B.4. Induced Modules B.5. Semilocal Galois Cohomology

195 195 195 198 201 202

Appendix C. Cohomology of p-adic Analytic Groups C.l. Irreducible Actions of Compact Groups C.2. Application to Galois Representations

205 205 207

Appendix D. p-adic Calculations in Cyclotomic Fields D.l. Local Units in Cyclotomic Fields D.2. Cyclotomic Units

211 211 216

Bibliography

219

Index of Symbols

223

Subject Index

227

Acknowledgments This book is an outgrowth of the Hermann Weyllectures I gave at the Institute for Advanced Study in October, 1995. Some ofthe work and writing was done while I was in residence at the Institute for Advanced Study and the Institut des Hautes Etudes Scientifiques. I would like to thank both the IAS and the IHES for their hospitality and financial support, and the NSF for additional financial support. I am indebted to many people for numerous helpful conversations during the period that this book was in preparation, especially Avner Ash, Ralph Greenberg, Barry Mazur, Bernadette Perrin-Riou, Alice Silverberg, and Warren Sinnott. I am also very grateful to Christophe Cornut, Alice Silverberg, and Tom Weston for their careful reading of the manuscript and their comments, and to the audiences of graduate courses I gave at Ohio State University and Stanford University for their patience as I was developing this material. Finally, special thanks go to Victor Kolyvagin and Francisco Thaine for their pioneering work. Karl Rubin December, 1999

xi

Euler Systems

Introduction History. In 1986, Francisco Thaine [Th] introduced a remarkable method for bounding ideal class groups of real abelian extensions of Q. Namely, ifF is such a field, he used cyclotomic units in fields F(JLt), for a large class of rational primes £, to construct explicitly a large collection of principal ideals of F. His construction produced enough principal ideals to bound the exponent of the different Galois-eigencomponents of the ideal class group of F, in terms of the cyclotomic units of F. Although Thaine's results were already known as a corollary of lwasawa's "main conjecture", proved by Mazur and Wiles [MW], Thaine's proof was very much simpler. The author [Rul] was able to apply Thaine's method essentially unchanged to bound ideal class groups of abelian extensions of imaginary quadratic fields in terms of elliptic units, with important consequences for the arithmetic of elliptic curves with complex multiplication. Shortly after this, Victor Kolyvagin [Kol] discovered (independently of Thaine) a similar remarkable method, in his case to bound the Selmer group of an elliptic curve. Suppose E is a modular 1 elliptic curve over Q, with sign +1 in the functional equation of its £-function. Kolyvagin's method used Heegner points on E over anticyclotomic extensions of prime conductor of an imaginary quadratic field K (in place of cyclotomic units in abelian extensions of Q) to construct cohomology classes over K (in place of principal ideals). He used these cohomology classes, along with duality theorems from Galois cohomology, to bound the exponent of the Selmer group of E over Q. The overall structure of his proof was very similar to that of Thaine. Inspired by Thaine's work and his own, Kolyvagin then made another fundamental advance. In his paper [Ko2] he introduced what he called "Euler systems." In Thaine's setting (the Euler system of cyclotomic units) 1 C. Breuil, B. Conrad, F. Diamond, and R. Taylor have recently announced that they have succeeded in extending the methods of Wiles to prove that every elliptic curve over Q is modular. Given this result, one can remove the assumption that E is modular, both here and throughout the discussion of elliptic curves in §3.5.

3

4

INTRODUCTION

Kolyvagin showed how to use cyclotomic units in fields F(JLr), for a large class of integers r (no longer just primes), to bound the orders of the different Galois-eigencomponents of the ideal class group of F, rather than just their exponents. Similarly, by using a larger collection of Heegner points in the situation described above, Kolyvagin was able to give a bound for the order of the Selmer group of E. Thanks to the theorem of Gross and Zagier [GZ], which links Heegner points with the £-function of E, Kolyvagin's bound is closely related to the order predicted by the Birch and Swinnerton-Dyer conjecture.

This book. This book describes a general theory of Euler systems for p-adic representations. We start with a finite-dimensional p-adic representation T of the Galois group of a number field K. (Thaine's situation is the case where K = Q and T is \!!.!! JLpn twisted by an even Dirichlet character; in Kolyvagin's case T is the Tate module of a modular elliptic curve.) We define an Euler system for T to be a collection of cohomology classes CF E H 1 (F, T), for a family of abelian extensions F of K, with a relation between CF' and cF whenever F C F'. Our main results show how the existence of an Euler system leads to bounds on the sizes of Selmer groups attached to the Galois module Hom(T, ILp"" ), bounds which depend only on the given Euler system. The proofs of these theorems in our general setting parallel closely (with some additional complications) Kolyvagin's original proof. Results similar to ours have recently been obtained independently by Kato [Ka2] and PerrinRiou [PR5]. What we do not do here is construct new Euler systems. This is the deepest and most difficult part of the theory. Since Kolyvagin's introduction of the concept of an Euler system there have been very few new Euler systems found, but each has been extremely important. Kato [Ka3] has constructed a new Euler system for a modular elliptic curve over Q, very different from Kolyvagin's system of Heegner points (see §3.5). Flach [Fl] has used a collection of cohomology classes (but not a complete Euler system) to bound the exponent but not the order of the Selmer group of the symmetric square of a modular elliptic curve. One common feature of all the Euler systems mentioned above is that they are closely related to special values of £-functions. An important benefit of this connection is that the bounds on Selmer groups that come out of the Euler system machinery are then linked to £-values. Such bounds provide evidence for the Bloch-Kato conjectures [BK], which predict the orders of these Selmer groups in terms of £-values.

INTRODUCTION

5

Our definition of Euler system says nothing about £-values. If there is an Euler system for T then there is a whole family of them (for example, the collection of Euler system cohomology classes is a Zp-module, as well as a Gal(K / K)-module). If one multiplies an Euler system by p, one gets a new Euler system but a worse bound on the associated Selmer groups. The philosophy underlying this book, although it is explicitly discussed only in Chapter 8, is that under certain circumstances, not only should there exist an Euler system for T, but there should exist a "best possible" Euler system, which will be related to (and contain all the information in) the p-adic £-function attached toT. A remark about generality. It is difficult to formulate the "most general" definition of an Euler system, and we do not attempt to do this here. The difficulty is partly due to the fact that the number of examples on which to base a generalization is quite small. In the end, we choose a definition which does not cover the case of Kolyvagin's Heegner points, because to use a more inclusive definition would introduce too many difficulties. (In Chapter 9 we discuss possible modifications of our definition, including one which does include the case of Heegner points.) On the other hand, we do allow the base field K to be an arbitrary number field, instead of requiring K = Q. Although this adds a layer of notation to all proofs, it does not significantly increase the difficulty. A reader wishing to restrict to the simplest (and most interesting) case K = Q should feel free to do so. Organization. In Chapter 1 we introduce the local and global cohomology groups, and state the duality theorems, which will be required to state and prove our main results. Chapter 2 contains the definition of an Euler system, followed by the statements of our main theorems bounding the Selmer group of Hom(T, f.Lpoo) over the base field K (§2.2) and over Z~-extensions Koo of K (§2.3). Chapter 3 contains some concrete applications of the theorems of Chapter 2. We apply those theorems to three different Euler systems. The first is constructed from cyclotomic units, and is used to study ideal class groups of real abelian fields (§3.2). The second is constructed from Stickelberger elements, and is used to study the minus part of ideal class groups of abelian fields (§3.4). The third is constructed by Kato from Beilinson elements in the K -theory of modular curves, and is used to study the Selmer groups of modular elliptic curves (§3.5). The proofs of the theorems of Chapter 2 are given in Chapters 4 through 7. In Chapter 4 we give Kolyvagin's "derivative" construction,

INTRODUCTION

6

taking the Euler system cohomology classes defined over abelian extensions of K and using them to produce cohomology classes over K itself. We then analyze the localizations of these derived classes, information which is crucial to the proofs of our main theorems. In Chapter 5 we bound the Selmer group over K by using the derived classes of Chapter 4 and global duality. Bounding the Selmer group over Koo is similar but more difficult; this is accomplished in Chapter 7 after a digression in Chapter 6 which is used to reduce the proof to a simpler setting. In Chapter 8 we discuss the conjectural connection between Euler systems and p-adic £-functions. This connection relies heavily on conjectures of Perrin-Riou [PR4]. Assuming a strong version of Perrin-Riou's conjectures, and subject to some hypotheses on the representation T, we show that there is an Euler system for T which is closely related to the p-adic £-function. Chapter 9 discusses possible variants of our definition of Euler systems. Finally, there is some material which is used in the text, but which is outside our main themes. Rather than interrupt the exposition with this material, we include it in four appendices (A-D). Notation

If F is a field, F will denote a fixed separable closure of F and GF

= Gal(F/F).

(All fields we deal with will be perfect, so we may as well assume that F is an algebraic closure of F.) Also, pab will denote the maximal abelian extension of F, and if F is a local field pur will denote the maximal unramified extension of F. If F is a global field and :E is a set of places of F, then Fr. will denote the maximal extension ofF which is unramified outside :E. If K C F is an extension of fields, we will write K c, F to indicate that [F : K] is finite. IfF is a field and B is a Grmodule, F(B) will denote the fixed field of the kernel of the map GF --+ Aut(B), i.e., the smallest extension ofF whose absolute Galois group acts trivially on B. If 0 is a ring and B is an 0-module then Anno(B) c 0 will denote the annihilator of B in 0. If M E 0 then B M will denote the kernel of multiplication by M on B, and similarly if M is an ideal of 0. If B is a free 0-module, Tis an 0-linear endomorphism of B, and xis an indeterminate, we will write P(riB; x)

= det(l- rxiB)

E O[x],

NOTATION

7

the determinant of 1 - rx acting on B. Caution: If a is, for example, a Galois automorphism which acts on B, then P(riB;a) means P(riB;x) evaluated at x =a, and not det(1- raiB). The Galois module of n-th roots of unity will be denoted by J.Ln· If p is a fixed rational prime and F is a field of characteristic different from p, the cyclotomic character Ccyc :

Gp

--t

z;

is the character giving the action of G F on J.Lpoo, and the Teichmiiller character w : G F --+ (z; hors is the character giving the action of G F on J.Lp (if p is odd) or on p.4 (if p = 2). Hence w has order at most p- 1 or 2, respectively (with equality ifF= Q) and (e) = w- 1 ecyc takes values in 1 + pZp (resp. 1 + 4Z2)· If B is an abelian group, Bdiv will denote the maximal divisible subgroup of B. If pis a fixed rational prime, we define the p-adic completion of B to be the double dual

BA = Hom(Hom(B, QpfZv), QpfZv) (where Hom always denotes continuous homomorphisms if the groups involved comes with natural topologies). For example, if B is a Zv-module then BA = B; if B is a finitely generated abelian group then BA = B ®z Zp. In general, BA is a Zp-module and there is a canonical map from B to BA. If r is an endomorphism of B then we will often write Br=O for the kernel of r, write Br=l for the subgroup fixed by r, etc. If { Ai : i E I} is an inverse system, we will denote by {ai }i (or sometimes simply {ai}) an element of \!!!! Ai. Finally, if Sis a set, then lSI will denote the cardinality of S. Most of these notations will be recalled when they first occur.

CHAPTER 1

Galois Cohomology of p-adic Representations In this chapter we introduce our basic objects of study: p-adic Galois representations, their cohomology groups, and especially Selmer groups. We begin by recalling basic facts about cohomology groups associated to p-adic representations, material which is mostly well-known but included here for completeness. A Selmer group is a subgroup of a global cohomology group determined by "local conditions". In §1.3 we discuss these local conditions, which are defined in terms of special subgroups of the local cohomology groups. In §1.4 we state without proof the results we need concerning the Tate pairing on local cohomology groups, and we study how our special subgroups behave with respect to this pairing. In §1.5 and §1.6 we define Selmer groups and give the basic examples of ideal class groups and Selmer groups of elliptic curves and abelian varieties. Then in §1.7, using Poitou-Tate global duality and the local orthogonality results from §1.4, we derive our main tool (Theorem 1.7.3) for bounding the size of Selmer groups. 1.1. p-adic Representations

Definition 1.1.1. Suppose K is a field, pis a rational prime, and 0 is the ring of integers of a finite extension

1. Theorems 2.3.2, 2.3.3, and 2.3.4 below will be proved in Chapter 7.

z;

Notation. If Kc,F c K 00 , we will write Ap = O[Gal(F/K)]. Let Gal(Koo/ K) and let A denote the Iwasawa algebra

A :::: O[[r]]

= \!!!!

r=

Ap,

KCrFCKoo

so A is (noncanonically) isomorphic to a power series ring over 0 in d variables. We say that a A-module B is pseudo-null if B is annihilated by an ideal of A of height at least two. A pseudo-isomorphism is a A-module homomorphism with pseudo-null kernel and cokernel, and two A-modules are pseudo-isomorphic if there is a pseudo-isomorphism between them. If B is a finitely generated torsion A-module then there is an injective pseudoisomorphism

with

h

E A, and we define the characteristic ideal of

char( B)

= IT fiA.

B

i

The characteristic ideal is well-defined, although the individual h are not. The individual ideals (elementary divisors) fiA are uniquely determined if we add the extra requirement that fH 1 I fi for every i. If B is a finitely generated A-module which is not torsion, we define char(B) = 0. If 0 --t B' --t B --t B" --t 0 is an exact sequence of finitely generated A-modules, then char(B)

= char(B')char(B").

We will need the following weak assumption to rule out some very special bad cases. In particular it is satisfied if K = Q.

2.3. RESULTS OVER

Koo

41

Hypothesis Hyp(K00 /K). If rankzp(r) = 1 and GKoo acts either trivially or by the cyclotomic character on V, then either K is a totally real field and Leopoldt's conjecture holds forK (i.e., the p-adic completion of 0 ~ injects into (0 K ® Zp) x), or K is an imaginary quadratic field. We will also write Hyp(K00 , T) {resp. Hyp(K00 , V)) for hypotheses Hyp(K, T) {resp. Hyp(K, V)) with GK replaced by G Koo, i.e., Hypotheses Hyp(K00 ,T). {i) There is arE GKoo such that • r acts trivially on /Lpoo, on {0~) 1 /Poo, and on K(l), • Tf(r- 1)T is free of rank one over 0. {ii) T ® lk is an irreducible k[GKoo]-module. Hypotheses Hyp(K00 , V). {i) There is a r E G Koo such that • r acts trivially on /Lpoo, on (0~) 1 /Poo, and on K(l), • dim(V/(r- 1)V) = 1. {ii) V is an irreducible cl1[GK ]-module. 00

There are simple implications Hyp(K00 ,T)

=* Hyp(Koo, V)

.tJ. Hyp(K, T) Definition 2.3.1. Recall that D H!x,(K,T)

.tJ.

=*

Hyp(K, V).

= ci1JO. = ¥!!!

Define A-modules

H 1 (F,T) KCrFCKoo SEp (K 00 , W*) = ~ SEp (F, W*) KCrFCKoo Xoo = Homo(SEp(K00 , W*),D),

limits with respect to corestriction and restriction maps, respectively. If c is an Euler system let CK,oo = {cF}Kc,FcKoo denote the corresponding element of H!x,(K, T) and define an ideal of A by indA{c)

= {¢(cK,oo): ¢ E HomA{H!x,{K,T),A)}

CA.

The ideal indA{c) measures the A-divisibility of CK,oo, just as indo{ c) of Definition 2.2.1 measures the 0-divisibility of CK. Recall that c is an Euler system for (T, Koo) if it is an Euler system for (T, IC, N) with Koo C /C. Theorem 2.3.2. Suppose c is an Euler system for (T, K 00 ), and V satisfies Hyp(K00 , V). If CK,oo does not belong to the A-torsion submodule of H~(K, T), then Xoo is a torsion A-module.

42

2. EULER SYSTEMS: DEFINITION AND MAIN RESULTS

Theorem 2.3.3. Suppose c is an Euler system for (T, K 00 ), and T satisfies hypotheses Hyp(K00 , T) and Hyp(Koo/ K). Then char(Xoo) divides indA(c). Theorem 2.3.4. Suppose c is an Euler system for (T, K 00 ), and V satisfies hypotheses Hyp(K00 , V) andHyp(Koo/K). Then there is a nonnegative integer t such that char(X00 ) divides ptindA(c). Remark 2.3.5. The assertion that Xoo is a torsion A-module is called the weak Leopoldt conjecture forT. See [Gr2] or [PR4] (§1.3 and Appendice B). Remark 2.3.6. As with Theorem 2.2.2, these three theorems all give bounds for the size of Sr;p(K00 , W*) rather than the true Selmer group fu¥S(F, W*). Combining these results with the global duality results from §1.7 gives Theorem 2.3.8 below concerning the true Selmer group. Suppose that for every K CrF C Koo and every prime w dividing p we have subspaces H}(Fw, V) C H 1 (Fw, V) and H}(Fw, V*) C H 1 (Fw, V*) which are orthogonal complements under the pairing (, }Fw, as in §1.7. We suppose further that if F C F' and w' I w then CorF~,/FwH}(F~,, V) C H}(Fw, V), ResF~,/FwH}(Fw, V*) C Hj(F~,, V*).

(In fact, the local pairing and our assumptions about orthogonality show that these two inclusions are equivalent.) These conditions ensure that, if K Cr F Cr F' C JC, the natural restriction and corestriction maps induce maps S(F, W *) ---'H s1 (F'P' T) --+ H s1 ( rp, D T) ~ S(F'' W*)' where we write H 1 (Fp, ·) = EBwlpH 1 (Fw, · ), and similarly for Hj and H! = H 1 JHj. Define S(Koo, W*)

= fu¥

S(F, W*),

KCrFCKoo

H~,s(Kp,T)

= ¥!!!

H!(Fp,T).

KCf FCKoo

Proposition 2.3.7. There is an exact sequence 0 --+ H~,s(Kp,T)jloc!;p(H~(K,T))

--+ Homo(S(Koo, W*), D) --+ Xoo --+ 0

2.4. TWISTING BY CHARACTERS OF FINITE ORDER

43

where loc!;P : H~ (K, T) --+ H~,s (Kp, T) is the localization map. Proof. By Corollary B.3.5, H~(K,T)

=

¥!!!

S"Ep(F,T).

KCrFCKoo

Thus the proposition follows from Corollary 1.7.5 by passing to the (direct) limit and applying Homo(·, D). D Theorem 2.3.8. Suppose that cis an Euler system for (T,K00 ), and V satisfies hypotheses Hyp(K00 , V) and Hyp(K00 /K). Suppose further that loc!;P(cK,oo) ~ H~, 8 (Kp,T)A-tors and H~, 8 (Kp,T)/Aloc!;P(cK,oo) is a torsion A-module. Then Homo(S(K00 , W*), D) is a torsion A-module and (i) there is a nonnegative integer t such that

char(Homo(S(K00 , W*),D)) divides ptchar(H~, 8 (Kp,T)/Aloc!;P(cK,oo)), (ii) if T satisfies Hyp(K00 , T) then

char(Homo(S(K00 , W*),D)) divides char(H~, 8 (Kp,T)/Aloc!;p(cK,oo)). Proof. Since loc!;p (cK,oo) ~ H~, 8 (Kp, T)A-tors, we see that CK,oo cannot belong to H~(K, T)A-tors· Therefore Theorem 2.3.2 and Proposition 2.3.7 show that Homo(S(K00 , W*), D) is a torsion A-module and that

char(Homo(S(K00 , W*), D))

= char(X00 )char(H~, 8 (Kp, T)/loc!;p (H~(K, T))). Our assumptions ensure that loci; p (H~(K, T)) is a rank-one A-module, so there is a map 1/J : lacE p (H~(K, T)) --+ A with pseudo-null cokernel. Then 1/J(loc!;P (cK,oo))A

= char(,P(loc!;p (H~(K, T)))f,P(Ioc!;p (cK,oo))A) :::> char(loc!;p (H~(K, T))f Aloc!;p (cK,oo)),

and by definition indA(c) divides 1/J o loc!;p(cK,oo). The theorem follows easily from these divisibilities and the divisibilities of Theorems 2.3.4 and 2.3.3. D 2.4. Twisting by Characters of Finite Order

Suppose cis an Euler system for (T, JC,N) as defined in Definition 2.1.1. The consequences of the existence of such an Euler system described in §2.2 and §2.3 do not depend on JC (except that, in the case of §2.3, the field JC must contain K 00 ). We could always take JC to be the "minimal" field ICmin described in Remark 2.1.4, and ignore the Euler system classes CF for F C/.. ICmin, and still obtain the results stated above.

44

2. EULER SYSTEMS: DEFINITION AND MAIN RESULTS

However, there is a way to make use of the additional information contained in an Euler system for a non-minimal /C. Namely, in this section we show how to take an Euler system for (T,JC,N) and obtain from it an Euler system for twists T ~ x of T by characters x of finite order of Gal(IC/ K) (see below). For example, if JC is the maximal abelian extension of K, then we will obtain Euler systems for all twists of T by characters of finite order, and the results of this chapter then give (possibly trivial, possibly not) bounds for all the corresponding Selmer groups. Suppose x : GK -t 0 x is a character of finite order. As in Example 1.1.2 we will denote by Ox a free rank-one 0-module on which GK acts via x, and we fix a generator ~x of Ox We will write T~x for the representation T~oOx.

Definition 2.4.1. Suppose c is an Euler system for (T, IC,N) and x is a character of finite order of Gal(IC/ K) with values in ox. Let L = JCker(x) be the field cut out by X· If KerF e IC, define c} E H 1 (F,T ~ x) to be the image of Cp£ under the composition

H 1 (FL,T)

®(,"l

H 1 (FL,T)~Ox ~H 1 (FL,T~x)

cor

H 1 (F,T~x)

(we get the center isomorphism since G F L is in the kernel of x). Proposition 2.4.2. Suppose c is an Euler system for (T, JC,N) and X : Gal(IC/ K) --+

ox

is a character of finite order. If f is the conductor of X then the collection

{c}: KerF e IC} defined above is an Euler system for (T ~ x, IC, fN). Proof. If KerFerF' e IC then using Definition 2.1.1 we have CorF'fF(c},)

= CorF'LfF(cF'L ~~x) = CorFL/F ((CorF'L/FLcF'L) ® ~x) = CorFL/F(( = CorFL/F(

II P(Fr~ 1 IT*;Fr~ 1 )cFL) ~~x)

qE'E(F'L/FL)

II

P(Fr~ 1 IT*; x(Frq)Fr~ 1 )(cFL ~ ~x))

qE'E(F'L/FL)

II P(Fr~ 1 IT*; x(Frq)Fr~ 1 ) CorFLfF(CFL ~ ~x) qE'E(F'L/FL) = II P(Fr~ 1 I(T ~ x)*; Fr~ 1 ) c} =

qE'E(F'L/FL)

2.4. TWISTING BY CHARACTERS OF FINITE ORDER

45

where as usual P(Fr~ 1 I(T ® x)*; x) = det(l- Fr~ 1 xi(T ® x)*), and

E(F'LIFL) = {primes q: q f N, q ramifies in F'L but not in FL}

= {primes q : q f fN, q ramifies in F' but not in F}.

0

This proves the proposition.

Lemma 2.4.3. With notation as in Definition 2.4.1, suppose Kc,F C K 00 and L c, L' C /C. If every prime which ramifies in L' I K is already ramified in L I K, then the image of c} under the composition

H 1 (F,T®x) is

Res

®Cl

H 1 (FL',T®x) ~ H 1 (FL',T)

L:

x(8)8cFL'·

t!EGal(FL' /F)

Proof. Since c is an Euler system, and every prime which ramifies in L' I K ramifies in Ll K, we have CorFL' /FL(cFL') = CFL· Thus the image of c} under the composition above is (ResFL' /FCorFLjF(cFL

®ex))® e;t

= (ResFL'/FCorFL'fF(CFL' ®ex)) ®e;t = ( L 8(CFL'@ ex))® e;t t!EGal(FL' /F)

=

L:

x(8)8cFL'·

t!EGal(FL' /F)

0

CHAPTER 3

Examples and Applications In this chapter we give the basic examples of Euler systems and their applications, using the results of Chapter 2.

3.1. Preliminaries

ox.

Suppose xis a character of GK into As in Example 1.1.2 we will denote by Ox a free rank-one 0-module on which GK acts via X· Recall that D if!/0 0 ® (QpfZp)· We will also write

=

=

Dx

= D ®o Ox = Ox® (Qp/Zp)·

For the first three examples (§§3.2, 3.3, and 3.4) we will assume that x has finite prime-to-p order. As in §1.6.B and §1.6.C we take T = Ox, and we then have W = Dx and T* = 0(1) ® Ox-1 = Ox-1ecyc' where ccyc is the cyclotomic character. Let L = f(kerx be the abelian extension of K corresponding to x, and write 6. = Gal(L/K). Thus 6. is a cyclic group of order prime top. As in Definition 1.6.3, if B is a Z[6.]-module we write BA for the p-adic completion of B and BX for the x-component of BA ®zp 0. We also fix a generator of Ox-1, and this choice determines an isomorphism BX ~ (BA ®zp Ox-1 )~.

Lemma 3.1.1. (i) Ifx "# 1 then H 1 (L(P.poo)/K, W) = 0. (ii) If X is not the character giving the action of G K on J.l.p, then H 1 (L(J.£poo)/K, W*) = 0.

ox

Proof. Write n = L(J.tpoo) as in §2.2. Suppose p: GK-+ is a character. Recall that tJ is the maximal ideal of 0 and k = 0 /tJ is the residue field. Write kp = k ® Op. Since 16.1 is prime top, the inflation-restriction sequence shows that

H 1 (0/K,kp) = Hom(Gal(!l/L),kp)~ = Hom(Gal(!l/L),k~) (note that 6. acts trivially on Gal(!l/L) because 0/K is abelian). Further, if 1r is a generator of tJ, it follows from the exact sequence 0 ---+ kp

---t

Dp 47

..!.t Dp ---+

0

3. EXAMPLES AND APPLICATIONS

48

that

= 0 :::} H 1 (0.JK,Dp) = 0. If pis not congruent to 1 modulo p, then k~ = 0 and so H 1 (0./ K, Dp) = 0. Applying this with p = x proves (i), and with p = x- 1ccyc proves (ii). D H 1 (0.JK,kp)

=0

:::} H 1 (0.JK,Dp)p

3.2. Cyclotomic Units

The Euler system of cyclotomic units is studied in detail in [Ko2] and [Ru3]. An Euler system for Zp(1). Take K = Q. For every extension F of Q, Kummer theory shows as in Example 1.2.1 that

H 1 (F, Zp(1))

= ~H 1 (F,JLp.. ) = ~px J(Fx)p" = (Fxt

(3.1)

r

Where (FX iS the p-adiC COmpletion Of px. Fix a collection {(m: mE z+} such that (m is a primitive m-th root of unity and (;:.n = (m for every m and n. (For example, we could fix an embedding of Q into C and choose (m = e21ri/m .) For every m 2:::: 1 and every prime l we have the relation ( (m - 1) { NQ(~ 2 and x(p) "I 1, then

IAzl

divides [££: CL,x]·

Proof. We will apply Theorem 2.2.2 with the Euler system c constructed from cyclotomic units above. Since rankoT* = 1, we see that Hyp(Q, T*) is satisfied with T = 1. Further, in this case n = L(p,p~ ), and since X is nontrivial and even, Lemma 3.1.1 shows that the error terms nw• and nw. in Theorem 2.2.2 are both zero. By (3.3) we have maps

£f

-

Om

iff.{m, if £1m.

An Euler system for Zp. Again we take K extension F of Q, class field theory shows that

H 1 (F,Zp)

= Q.

(3.8) For every finite

= Hom(Gp,Zp) = Hom(A~/Fx,zp) =

Hom(A~/(Fx Bp),

Zp)

(3.9)

where A~ denotes the group of ideles of F and

Bp =

II F; II{1} II o~,w c X

X

wloo

wlp

A~,

wfpoo

since every (continuous) homomorphism from A~ into Zp must vanish on Bp. FUrther, the map which sends an idele to the corresponding ideal class induces an exact sequence 0 ---+ UF/£F ---+ A~/(Fx BF) ---+ AF ---+ 0

(3.10)

where UF denotes the local units ofF® Qp (i.e., UF = EBvlp0~, 11 ) and tF denotes the closure of the global units of F in UF, and AF is the ideal class group of F. We will write Zp[Jl.m]x = UQ(~m)· Definition 3.4.2. Fix an integer b prime to 2p (a precise choice will be made later), and for every m E z+ prime to b we use the Stickelberger elements Om to define (j(b)

m

=

{(b/b)(}~ (b-')'b)(1-Fr; )0m 1

if pI m E Z[Gal(Q( )/Q)] ifp{m Jl.m

(the two separate cases are to ensure, using (3.8), that O~~~Q(~m) = (j~) for every m). Stickelberger's theorem (see for example [Wa] Theorem 6.10

3.4. STICKELBERGER ELEMENTS

or [Lan) Theorem 1.2.3) shows that iJ~) AQ(J£m) we can view multiplication by iJ~) as a map

= 0.

57

Thus, using (3.10)

AQ(J£m/(Q(JLm)X BQ(J£m)) ---+ Zp(JLm]X /£Q(J£m)• and we define 4>m = 4>~) E Hom(Aq(J£m/(Q(JLm)x BQ(J£m)), Zp) to be the composition X

ij(b)

X

-

AQ{J£m/(Q(JLm) BQ{J£m}) ~ Zp[JLm] /CQ(J£m} 1 -c

Zp [JLmr / (Zp [JLm] x )tors

>.m)

Zp

where c denotes complex conjugation in Gal(Q(JLm)/Q), so (1-c)£Q(J£m) is finite, and Am is the map defined in Appendix D, Definition D.1.2. Finally, we define c~ E H 1 (Q(JLm), Zp) to be the element corresponding to 4>m under (3.9). Proposition 3.4.3. Suppose m is prime to b and l is a prime not dividing b. Then

Proof. It follows from a standard result of class field theory (see for example [T2) §11(13)) that, with the identification (3.9), the map CorQ(J£mtl/Q(J£m) is induced by the inclusion AQ(J£m) " Suppose first that l f mp. By Lemma D.1.4 the restriction of Aml to Zp[JLm]x is Am o (-Frt), and by (3.8) we have iJ~~~Q(J£m) = (1- Fr£ 1 )iJ~>. Therefore

= f/Jmo(-Frt)(1-Fr£ 1 ) = f/>mo(1-Frt) = (1-Fr£ 1 )f/>m and hence CorQ(J£mtl/Q(J£m)(c~l) = (1-Fr£ 1 )c~. Similarly (but more sim= ply), if l divides mp then Lemma D.l.4 and (3.8) show that 4>mtiAx 4>m and then CorQ(~£mtl/Q(~£ml(c~t) = c~. 0 f!>mtiAx

Q(>'ml

Q(,..ml

Remark 3.4.4. Technically we should write c~ 00 instead of c~, since the ray class field of Q modulo m is the real subfield Q(JLm)+. But CorQ(J£m)/Q(J£m)+ (c~)

= 0 E H 1 (Q(JLm)+, Zp)

(because we annihilated all even components in our definition), so we will never need to deal with those classes and there should be no confusion. For every prime l

"# p we have

det(1- Fr£ 1 xjZp(1))

= 1- e- 1 x.

3. EXAMPLES AND APPLICATIONS

58

But Proposition 3.4.3 shows that the collection {e~ E H 1 (Q(J.tm), Zp)} satisfies a distribution relation with Frobenius polynomials 1 - Fri 1 , not 1 -£- 1 Fri 1 , so this collection is not an Euler system for the trivial representation Zp. However, since 1-

r 1 x = 1- X

we can modify the classes produce a new collection

e~

(mod (l- 1)Zp[x])

(see Lemma 9.6.1 and Example 9.6.2) to

{em E H 1 (Q(J.tm),Zv): mE

z+

and (m,b)

= 1}

which is an Euler system for (Zp, Qab,b, bp), where Qab,b denotes the maximal abelian extension of Q unramified outside b. Further (Lemma 9.6.1(ii)), - = cpn _, £or every n. we have Cpn Note that this Euler system depends on the choice of b.

The setting. As in §3.2 let K = Q, let T = Ox for a character X of GK of finite prime-to-p order, and keep the rest of the notation of the beginning of §3.2 as well. For the rest of this section we assume that x is odd, and we let b be a nonzero integer prime to 2p and to the conductor f of X· (A precise choice of b will be made later.) Note that these hypotheses imply that p > 2, because there are no odd characters of odd order. Let .6. = Gal(Q(JL1 )/Q). Since xis nontrivial and of order prime to p, we have Hi(.6., Ox) = 0 for every i ~ 0. Therefore the restriction map gives an isomorphism (compare with (3.9)) H 1 (Qn,T)

= H 1 (Qn(JL,),Ox)~

~ Hom(AQn(,..,/Qn(JL1 )x, Ox)~ C Hom(AQn(#Jo,)' 0),

(3.11)

the inclusion using our fixed generator of Ox· The Euler system e for Zp constructed above gives rise (by Proposition 2.4.2) to an Euler system c =eX for (T, Qab,b, bfp). By Lemmas 2.4.3 and 9.6.1(iii), the image under (3.11) of CQ in Hom(AQ(,..,)' 0) is

I: x(«S)«Se, = I: x(«S)«Se/ = I: x(«S)~.

(3.12)

The Selmer group. We have W* = Dx-lecyc· As in §3.2, let L be the fixed field of the kernel of x, let Ln = LQn, let Qn,p be the completion of Qn above p, let An be the ideal class group of Ln, and let AL = Ao, the ideal class group of L. We take H}(Qn,p, V) and H}(Qn,p, V*) to be as defined in the examples of §1.6.B and §1.6.C, respectively.

3.4. STICKELBERGER ELEMENTS

Proposition 3.4.5. (i) S(Q, W*) ~ (ii) S(Qoo, W*) ~ ~A~.

59

AL

n

Proof. Let £n denote the group of global units of Ln. Since X is odd, £~ is finite, so (£n ® Qp/Zp)'X. = 0. Now the proposition follows from Proposition 1.6.4{ii). 0

The minus part of the ideal class group of L. The following theorem (or more precisely, its Corollary 3.4. 7) was first proved by Mazur and Wiles in [MW]. A proof using Euler systems, but somewhat different from the one here, was given by Kolyvagin in [Ko2], see also [Ru4]. Define the generalized Bernouilli number B1,x.-1

1 I

=f

L:x- 1(a)a

= x(91 ).

a=l

Recall that w : Gq --t (z; )tors is the Teichmiiller character giving the action of Gq on l'p (recall also that p ::j:. 2, since we have assumed that x is an odd character of order prime to p). Theorem 3.4.6. Suppose that x is an odd character of order prime to p, that x(p) ::j:. 1, and that x- 1w(p) ::j:. 1. Then IAzl ~ IO/Bl,x.-101.

ox.

Proof. Since x ::j:. w, we can choose b prime to 2pf so that b-x(b) E Let c be the Euler system for T constructed above from Stickelberger elements, with this choice of b. Since T has rank one over 0, Hyp(Q, T) is satisfied with T = 1, so we can apply Theorem 2.2.10 with this Euler system. As in the proof of Theorem 3.2.3, since xis odd and different from w, Lemma 3.1.1 shows that nw = ntv = 0 in Theorem 2.2.2. Using the definition of H} in §1.6.B and local class field theory, we have identifications (the top row is the local analogue of (3.11))

H 1 (Qp,T) ~Hom(E9wlpGQ(I-',),.•Ox.).6. ~Hom(Qp(~t 1 )x ,Ox.).6.

~ where Qp(~t 1 )

!

~

= Q(~t 1 ) ® Qp and Iw is the inertia group in GL,.·

Hi(Qp,T) ~ Hom(Zp[~t 1 ]x,Ox.)a ~ Hom(Zp[~-t 1 ]x,o)x.- 1 .

Thus (3.13)

3. EXAMPLES AND APPLICATIONS

60

With this identification, using {3.12) and Definition 3.4.2 of BJ6 ), and writing c for complex conjugation, loc{p},T(cQ) =

L x(~)(AJ o {1- c)BJ L x(~)(A 1 o ~- 1 {1- c)8} >) 6)) 6

liE A

=

6

liE A

= Af o L(X(~)~- 1 ){1- c)BJ6) liE A

= 2{b- x{b)){1- x- 1 {p))BI,x-1 2:: x(~)A~. liE A

Since x- 1w{p) =I 1, Lemma D.1.5 shows that EoEA x(~)A~ generates the {free rank-one) 0-module Hom(Zp[IJ.1]x, O)X- 1. We chose b so that b- x(b) E ox, and we assumed that x{p) =I 1 and x has order prime top, so 1- x{p) E ox. Thus (3.13) shows that Oloc{p},T(cQ)

= B 1 ,x-1H~{Qp, T).

Now Theorem 2.2.10 yields IS{Q, W*)l :::; [H~(Qp,T): B1,x-1H~(Qp,T)]

= IO/B1,x-10I.

0

Corollary 3.4.7 (Mazur & Wiles [MW] Theorem 1.10.2). With hypotheses as in Theorem 3.4.6, IAzl

= IO/B1,x-1o1.

Proof. As in Corollary 3.2.4, this follows from Theorem 3.4.6 and the an0 alytic class number formula. See for example [Ru4] Theorem 4.3.

Remarks 3.4.8. If x = w, then Az = 0 and B 1,x-10 = p- 1 0. If x{p) = 1, or x- 1w{p) = 1 but x =F w, the equality of Corollary 3.4.7 can be deduced from Theorem 3.4.13 below (Iwasawa's "main conjecture"). See [MW], §1.10 Theorem 2. See also §9.1. The p-adic £-function. There is a natural map XA : O[(Gal(Qoo(l-'t)/Q)]]

given by

= O[~][[Gal{Qoo/Q)]]

x on ~ and the identity on Gal{Q (c) = w- 1ccyc : Gal{Q /Q)

00

00

~ A

/Q). Let

--+ 1 + pZp,

let Tw (e-) : A -+ A be the twisting map induced by

'Y

1-t

(c)('Yh

for "f E Gal{Qoo/Q), and let 11 1-t 11• denote the involution of A induced by 7- 1 for"( E Gal(Q 00 /Q).

"( 1-t

3.4. STICKELBERGER ELEMENTS

61

Write 9/poo = {0/p"+l}n· H b is prime to 2fp then by (3.7) and (3.8),

(b- "/b)9Jpoo E Zp[[Gal(Q(JL/poo)/Q)]], and so by restriction we have XA((b- 'Yb)9Jpoo) EA. H x ¥- w then we can fix b so that b- x(b) E ox, and then XA(b- 'Yb) E Ax. We will write

XA(OjpOO)

= XA(b- 'Yb)- 1 XA((b- 'Yb)OjpOO)

E A

which is independent of b. Theorem 3.4.9. If X is odd and

x ¥- w,

then

XA(9Jpoo)• = Tw(e)(.Cx-lw) where .Cx-lw is the p-adic L-function defined in Theorem 3.2.9 for the even character x- 1 w. Proof. This was proved by lwasawa; see [Iw2] §6 or [Wa] Theorem 7.10. If p is a character of finite order of Gal( Q 00 / Q), it follows from the definitions

that

P(XA(9Jp t) = p- 1 (XA(9Jpoo)) = (1- X- 1 p{p))Bl,x-lp 00

= (1- x- 1 p(p))L(o,x- 1 p) = (c:)p(.Cx-lw) = p(Tw(e)(.Cx-lw)). Since this is true for every p, the equality of the theorem holds.

D

Direct limit of the ideal class groups. The main result of this section, Theorem 3.4.13 below, is equivalent to Theorem 3.2.10 by standard methods of lwasawa theory (see for example [Ru3] §8), so we will only sketch the proof. Let U denote the direct limit (not the inverse limit) of the local units of Qn(JL 1 ) ® Qp. Recall that~= Gal(Q(JL 1 )/Q) ~ Gal(Qn(JL 1 )/Qn)· Lemma 3.4.10. There .is an isomorphism of A-modules Hom(U, Ox)a

~

{AA E9 o

1,

if x(p) ¥:if x(p) = 1.

Sketch of proof. Let Y 00 denote the inverse limit of the p-adic completions of the multiplicative groups (Q(JLfp") ®Qp)x. There is a natural Kummer pairing U x Yoo - t Zp(1)

which leads to a A-module isomorphism

(Yoo ® Oxw-l)Gal(Q(I-',,)/Q) ~ Hom(U, Ox)A ® O(e)· The lemma then follows from a result of Iwasawa ([Iw3] Theorem 25; see D also [Gi] Proposition 1).

3. EXAMPLES AND APPLICATIONS

62

Corollary 3.4.11. Suppose that X is odd and X =F w. Then we can choose b so that, if c is the Euler system defined above, then char(H~, 8 (Qp, T)/ Aloc{p}( {CQ,. }n))

=

Tw{e)(.Cclw) { Tw{e)(.J.Cx-lw) .JTw{e)(.Cx-lw)

if x- 1w(p) =F 1, x(p) if x- 1w(p) = 1, if x(p) = 1,

=I 1,

where .1 is the augmentation ideal of A. Sketch of proof. For every n, exactly as in (3.13) we have H!(Qn,p,T)

= Hom(Un,Ox)A

and so H;,,s (Qp, T) = Hom(U, Ox )A. Let

Afp"",x

= J~~l:x(6)>.~P"

EHom(U,Ox)A.

6EA

One computes, using Lemma 3.4.10, that there are A-module isomorphisms Hom(U, Ox)A I A>.fp"",x ~

o { O(e)

o

if x- 1w(p) =F 1, x(p) =F 1,

~f x- 1w(p) = 1, 1f x(p) = 1.

(The first case follows from Lemma D.1.5; the others require more work.) Also, by the definition of CQ,. and Lemma 2.4.3 we have loc{p}(CQ,.)

=L

x(6)>.Jpn+l

0

6- 1 (1- c)(b- 'Yb)BJpn+l.

6EA

Thus Aloc{p} ({CQ,. }n) = A>.fp"",x o 2(XA(b- 'Yb)XA(OJp""))

= XA(b- 'YbtXA(Ofp"" t

A>.fp"",x

= XA(Ofp"" )• A>.fp"",x since b was chosen so that XA (b- 'Yb) E Ax . The corollary now follows from Theorem 3.4.9. 0

Theorem 3.4.12. Ifx is an odd character of order prime top and X =F w, then char(Homo(~A~,D)) divides Tw{e)(.Cx-lw)· Sketch of proof. Let c be the Euler system constructed above, with b chosen to satisfy Corollary 3.4.11. Since T has rank one over 0, we see that Hyp(Q, T) is satisfied with r = 1. Thus we can apply Theorem 2.3.8(ii),

3.5. ELLIPTIC CURVES

63

and we conclude (using Proposition 3.4.5{ii) to identify the Selmer group with the direct limit of the ideal class groups) that char(Homo(~A~, D)) divides char(H~, 8 (Qp,T)/ Aloc{p}( {cQ,. }n)).

If X(P) ::j:. 1 and x- 1w(p) ::j:. 1, the theorem now follows immediately from Corollary 3.4.11. The two exceptional cases remain. First suppose that x- 1w(p) 1. In this case we conclude from Corollary 3.4.11 that char{Hom 0 (~ A~, D)) divides Tw(e) {.1.Cx-lw), so to complete the proof it will suffice to show that Tw(e)(.J) cannot divide char{Homo{~A~,D)). Briefly, if Tw(e)(.J) divides char{Homo(~A~,D)) then class field theory and Kummer theory show (see for example [Lan] Chapter 6 or [Wa] §13.5) that there is a divisible subgroup of Q(p. 1P)x ®(QpfZp) which generates an unramified extension of Q(J.I.fpoo ). But this would contradict Leopoldt's conjecture, which holds for Q(p. 1p)· Now suppose x(p) = 1. In this case, if xo denotes the trivial character then the definition {Theorem 3.2.9) of .Cx-lw shows that

=

Xo(Tw(e)(.Cx-lw))

= w- 1 ccyc{.Cx-lw) = {1- x(p))L(O,x) = 0.

In other words, .1 divides Tw(e)(.Cx-lw) so we cannot hope to show in this case that char{Homo (~A~, D)) is not divisible by .1. Instead, one must "improve" the Euler system c to remove this extra zero. We omit the details. 0 Theorem 3.4.13 {Mazur & Wiles [MW]). order prime to p and X ::j:. w then char(Homo(~A~,D))

If x is an odd character of

= Tw(e)(.Cx-lw)·

Proof. This follows from Theorem 3.4.12 by the usual analytic class number

argument. See [MW] §1.6, where this equality is deduced from divisibilities opposite to those of Theorem 3.4.12. 0 3.5. Elliptic Curves The "Heegner point Euler system" for elliptic curves, used by Kolyvagin in [Ko2], does not fit precisely into the framework we have established. We will discuss in §9.4 how to adapt Definition 2.1.1 to include the system of Heegner points. However, Kato ([Ka3], [Scho]) has constructed an Euler system for the Tate module of a modular 2 elliptic curve, using Beilinson elements in the K -theory of modular curves. In this section we describe applications of Kato's Euler system. 2 See

the footnote on page 3.

3. EXAMPLES AND APPLICATIONS

64

The setting. Suppose E is an elliptic curve defined over Q, and take K Q, K 00 Q00 , 0 Zp, and T Tp(E), the p-adic Tate module of E, as in Example 1.1.5. Then V Vp(E) Tp(E) ® Qp and W Epoo. The Weil pairing gives isomorphisms V !::! V*, T !::! T*, and W !::! W*. As in the previous sections, Qn is the subfield of Q 00 with [Qn : Q] = pn, and Qn,p is the completion of Qn at the unique prime above p.

=

=

=

=

=

=

=

The p-adic cohomology groups. As in §1.6, for every n we let H}(Qn,p• V)

= image(E(Qn,p) ® Qp

2 then

= [E(Qp): E1(Qp) + E(Qp)tors]P- 1Zv-

Proof. The diagram {3.14) shows that an element of Hi(Qp, V) belongs to Hi(Qp,T) if and only if its image in Hom(E(Qp),Qp) takes E(Qp) into Zp. Thus by {3.15), we have exp~ 8 (Hi(Qp,T)) = pazP

where AE(E(Qp)) = p-azv- If p > 2 then AE{E1{Qp)) rankzpE(Qp) = 1, we see that

= pZp and, since

= [E(Qp): E1(Qp) + E(Qp)tors]·

[.XE(E(Qp)): AE(E1(Qp)]

D

The £-functions. Definition 3.5.2. Let

= L ann-8 = II lq(q-8)-1 00

L(E, s)

n=1

q

denote the Hasse-Weil £-function of E, where lq(q- 8) is the usual Euler factor at q. If mE z+ we will also write

Lm(E, s)

=

L

ann- 8

= II lq(q-s)- 1 =

(II lq(q-

)L(E, s) qfm qlm for the £-function with the Euler factors dividing m removed. If x is a character of GQ of conductor fx, let 8 )

(n,m)=1

Lm(E, x, s)

=

L

x(n)ann- 8

II

lq(q- 8x(q))- 1.

qtfxm

(n,fxm)=1

When m

=

= 1 we write simply L(E,x,s), and then we have Lm(E,x,s) = (II lq(q-sx(q)))L(E,x,s).

{3.16)

qlm

If E is modular then these functions all have analytic continuations to C.

3. EXAMPLES AND APPLICATIONS

66

The Euler system. Kato has constructed an Euler system in this setting. Let N denote the conductor of E, and let OE be the fundamental real period of E (which corresponds to our choice of differential WE). Theorem 3.5.3 (Kato [Ka3]; see also [Scho]). Suppose that E is modular. Then there is a positive integer r E, independent of p, and an Euler system c for Tp(E) such that for every n 2:: 0 and every character x of Gal(Qn/Q),

L

')'EGal(Qn/Q)

xb) exp:E(loc{p}(c~J) = rELNp(E, x, 1)/0E.

In particular we have

exp:E(loc{p}(cQ))

= rELNp(E, 1)/0E.

See [Scho], especially §5, for the construction of the Euler system and the proof of the identities in the case where E has good reduction at p. (See also [Ru9] Corollary 7.2 to get from [Scho] Theorem 5.2.6 to the statement above.) Consequences of Kato's Euler system. Following Kato, we will apply the results of Chapter 2 to bound the Selmer group of E. Let III( E) be the Tate-Shafarevich group of E. Theorem 3.5.4 (Kato [Ka3]). Suppose E is modular and E does not have complex multiplication. (i) If L(E, 1) =f. 0 then E(Q) and III(E) are finite. (ii) Suppose L is a finite abelian extension of Q and x is a character of Gal{L/Q). If L(E,x, 1) =f. 0 then E(L)" and III(E;L)" are finite. Remarks 3.5.5. We will prove a more precise version of Theorem 3.5.4(i) in Theorem 3.5.11 below. Kato actually constructs an Euler system for (Tp(E),Qab,DD',NpDD') for appropriate auxiliary integers D,D', where Qab,DD' is the maximal abelian extension of Q unramified outside DD'. Thus (for some choice of D and D', depending on x) Proposition 2.4.2 gives an Euler system for Tp(E) ®X for every character x of GQ of finite order, with properties analogous to those of Theorem 3.5.3. These twisted Euler systems are needed to prove Theorem 3.5.4(ii). For simplicity we will not treat this more general setting here, so we will only prove Theorem 3.5.4(i) below. But the method for (ii) is the same. Theorem 3.5.4(i) was first proved by Kolyvagin in [Ko2], using a system of Heegner points, along with work of Gross and Zagier [GZ], Bump, Friedberg, and Hoffstein [BFH], and Murty and Murty [MM]. The Euler

3.5. ELLIPTIC CURVES

67

system proof given here, due to Kato, is self-contained in the sense that it replaces [GZ], [BFH], and [MM] by the calculation of Theorem 3.5.3. Corollary 3.5.6. Suppose E is modular and E does not have complex multiplication. Then E(Q 00 ) is finitely generated. Proof. A theorem of Rohrlich [Ro] shows that L(E, x, 1) =/; 0 for almost all characters x of finite order of Gal(Q 00 /Q). A result of Serre ([Se4] Theoreme 3) shows that E(Qoohors is finite, and the corollary follows without difficulty from Theorem 3.5.4(ii). (See for example [RW], pp. 242-243.) D

Remark 3.5.7. When E has complex multiplication, the representation Tp(E) does not satisfy hypothesis Hyp(Q, V)(i) (see Remark 3.5.10 below), so we cannot apply the results of §2.2 and §2.3 with Kato's Euler system. However, Theorem 3.5.4 and Corollary 3.5.6 are known in that case, as Theorem 3.5.4 for CM curves can be proved using the elliptic unit Euler system of §3.3. See [CW], [Ru5] §11, and [RW]. See also the final example of §6.5. V~rification

of the hypotheses. Fix a Zp-basis ofT and let

PE,p : GQ --+ Aut(T) ~ GL2(Zp) be the p-adic representation of GQ attached to E with respect to this basis. Proposition 3.5.8. (i) Suppose that E does not have complex multiplication. Then H 1 (Q(Epoo )/Q, Epoo) is finite and Tp(E) satisfies hypotheses Hyp(Q 00 , V). (ii) Suppose that PE,p is surjective. Then H 1 (Q(Epoo )/Q, Epoo) = 0 and Tp(E) satisfies hypotheses Hyp(Q 00 , T). Proof. The Weil pairing shows that

GQ{~£poo)

= PE~p(SL2(Zp)).

If E does not have complex multiplication then Serre's theorem ([Se4]

Theoreme 3) says that the image of PE,p is open in GL2(Zp). It follows that Vp(E) is an irreducible GQoo -representation, and if PE,p is surjective then Ep is an irreducible F p[GQoo]-representation. It also follows that we can find T E GQ(~£poo) such that

PE,p(r)

=

G~)

with x =/; 0, and such aT satisfies hypothesis Hyp(Q 00 , V)(i). If PE,p is surjective we can take x = 1, and then T satisfies hypothesis Hyp(Q 00 , T)(i).

3. EXAMPLES AND APPLICATIONS

68

We have H 1 (Q(Epoo )/Q, Epoo)

= H 1 (PE,p(GQ), (QpfZp) 2 )

which is zero if PE,p(GQ) = GL2(Zp), and finite if PE,p(GQ) is open in GL 2(Zp) (see also Corollary C.2.2). This completes the proof of the proposition. 0 Remark 3.5.9. Serre's theorem (see [Se4] Corollaire 1 of Theoreme 3) also shows that if E does not have complex multiplication then PE,p is surjective for all but finitely many p. Remark 3.5.10. The conditions on r in hypothesis Hyp(Q, V)(i) force PE,p(r) to be nontrivial and unipotent. Thus if E has complex multiplication then there is no T satisfying Hyp(Q, V)(i). Bounding S(Q, Epoo ). Recall that N is the conductor of E. Theorem 3.5.11. Suppose E is modular, E does not have complex multiplication, and L(E, 1) -::1 0. (i) E(Q) and III(E)poo are finite. (ii) Suppose in addition that E has good reduction at p, that p does not divide 2rEIE(Fp)l (where E is the reduction of E modulo p andrE is as in Theorem 3.5.3}, and that PE,p is surjective. Then ..d LN(E, 1) IIII(E) poo I dwz es E . 0

Proof. Recall that lq(q- is the Euler factor of L(E, s) at q, and that by Proposition 1.6.8, S(Q, Epoo) is the usual p-power Selmer group of E. Since L(E, 1) -::1 0, and lq(q- 1 ) is easily seen to be nonzero for every q, Theorem 3.5.3 shows that loc{p}(cQ) -::1 0. By Proposition 3.5.8(i) and (3.14) we can apply Theorem 2.2.10(i) to conclude that S(Q, Epoo) is finite. This proves (i), and it follows (see for example Proposition 1.6.8) that S(Q, Epoo) = III(E)poo. If E has good reduction at p then plp(p- 1 ) = IE(Fv)l and 8)

[E(Qp): E1(Qp)

+ E(Qp)tors]

divides IE(Fp)l.

Therefore if p f 2rEIE(F p)l then exp~ 8 (H;(Qp,Tp(E)))

=

exp;:, 8 (Zploc{p}(cQ))

=

u

P-lzp

u p- 1 (LN(E,

1)/0E)Zp

by Proposition 3.5.1 and Theorem 3.5.3. By Proposition 3.5.8(ii), if further PE,p is surjective then we can apply Theorem 2.2.10(ii) (with nw = 0 and = 0) and (ii) follows. 0

nw

3.5. ELLIPTIC CURVES

69

Remarks 3.5.12. In Corollary 3.5.19 below, we will prove using Iwasawa theory that Theorem 3.5.ll(ii) holds for almost all p, even when p divides IE(Fv)l. This is needed to prove Theorem 3.5.4(i), since IE(Fv)l could be divisible by p for infinitely many p. However, since IE(Fv)l < 2p for all primes p > 5, we see that if E(Q)tors "I 0 then IE(Fv)l is prime top for almost all p. Thus for curves E with nontrivial rational torsion points, Theorem 3.5.4(i) follows directly from Theorem 3.5.11. The Euler system techniques we are using give an upper bound for the order of the Selmer group, but no lower bound. In this case there is no analogue of the analytic class number formula that enabled us to go from the Euler system divisibility to equality in Corollaries 3.2.4 and 3.4.7. The p-adic £-function and the Coleman map. Suppose for this section that E has either good ordinary reduction or multiplicative reducand {3 = pfa E pZp be the eigenvalues of Frobenius tion at p. Let a E over F P if E has good ordinary reduction at p, and let (a, {3) = (1,p) (resp. (-1, -p)) if E has split (resp. nonsplit) multiplicative reduction. Write Gn = Gal(Qn/Q) = Gal(Qn,p/Qp), and fix a generator {(pn }n of\!!!! /Lpn. If x is a character of Gal(Q 00 /Q) of conductor pn define the Gauss sum r(x) = x( -y)qn.

z;

2::

-yEGal{Q(#'pn )/Q)

Fix also an embedding of Qp into C so that we can identify complex and p-adic characters of Gq. The following theorem is proved in [MSD] in the case of good ordinary reduction. See [MTT] for the (even more) general statement. Theorem 3.5.13. Suppose E is modular and E has either good ordinary reduction or multiplicative reduction at p. Let a be as above. Then there are a nonzero integer CE independent of p, and a p-adic £-function LE E c'E 1 A, such that for every character X of Gal(Qoo/Q) of finite order, (1 - n- 1 ) 2 L(E, 1)/0E if x = 1 and E has good reduction at p { x(£E) = (1- n- 1 )L(E, 1)/0E if x ~ 1 and E is multiplicative at p a-nr(x)L(E,x- 1 ,1)/0E ifx has conductorpn > 1. If m E z+, define

LE,m

=(

IT

lq(q- 1 Fr; 1 ) )c.E E c'E1 A.

qlm,q#p

Using (3.16) and Theorem 3.5.13 one obtains expressions for x(£E,m) in terms of Lm(E, x- 1 , 1) similar to those in Theorem 3.5.13.

70

3. EXAMPLES AND APPLICATIONS

Proposition 3.5.14. Suppose that E has either good ordinary reduction or multiplicative reduction at p. Then there is a A-module map Coloo : H~.s(Qv, T) "--+ A such that for every z = {zn} E H~, 8 (Qp, T) and every nontrivial character X of Gn, we have

x(Coloo(z))

= a-kr(x)

L

x- 1 ('y) exp:E (z~)

"YEGn

where pk is the conductor of x. If xo is the trivial character then

Further, if E has split multiplicative reduction at p then the image of Col00 is contained in the augmentation ideal of A. Proof. The proof is based on work of Coleman (Co]. See the appendix of (Ru9] for an explicit construction of Col00 in this case, and see §8.1 for a discussion of a generalization due to Perrin-Riou (PR2]. D

Using the Coleman map Col00 described above, we can relate Kato's Euler system to the p-adic £-function. Corollary 3.5.15. With hypotheses as in Theorem 3.5.13, with rE as in Theorem 3.5.3, and with other notation as above, we have Coloo(loc{p}({cqn}))

= rECE,N·

Proof. Fix a character x of Gal(Q 00 /Q) of finite order. Theorem 3.5.3 and Proposition 3.5.14 allow us to compute x(Col 00 (loc{v} ({cqn} ))), the definition (Theorem 3.5.13) of CE and (3.16) allow us to compute x(rECE,N), and these values turn out to be equal. For example, if x is nontrivial then both are equal to rEa-nr(x)LNp(E, X- 1 , 1)/0E.

When x = 1, we need to use the fact that lp{p- 1 ) is (1 - a- 1 )(1 - {3- 1 ) (resp. (1 - /3- 1 )) if E has good (resp. multiplicative) reduction at p. Since x(Coloo(loc{p}({cqn}))) = x(rELE,N) for all x, the corollary follows. D Bounding S(Q 00 , Epoo ). Recall that N is the conductor of E, and let

3.5. ELLIPTIC CURVES

71

Theorem 3.5.16. Suppose E is modular, E does not have complex multiplication, and E has either good ordinary reduction or nonsplit multiplicative reduction at p. Then Zoo is a finitely generated torsion A-module and there is an integer t such that char(Zoo) divides ptCE,NA. If PE,p is surjective and p f TE nqjN,q;Cp lq(q-l ), then char( Zoo) divides LEA. If E has split multiplicative reduction at p, then the same results hold with char(Zoo) replaced by ..7char(Z00 ), where ..1 is the augmentation ideal of A. Proof. Rohrlich [Ro] proved that CE =f. 0. Thus the theorem follows by combining Propositions· 3.5.8 and 3.5.14, Corollary 3.5.15, and Theorem 2.3.8. []

Remark 3.5.17. For a discussion of the "extra zero" (the extra factor of ..1 in Theorem 3.5.16) in the case of split multiplicative reduction, see [MTT]. Corollary 3.5.18. Let E be as in Theorem 3.5.16. If p is a prime where E has good ordinary reduction and P

f II IE(Qq)torsl, qjN

then Zoo has no nonzero finite submodules. Proof. This corollary is due to Greenberg (Gr2], (Gr3]; we sketch a proof here. Let :E be the set of places of Q dividing Npoo, and let Qr; be the maximal extension of Q unramified outside :E. By Lemma 1.5.3 there is an exact sequence

0-+ S(Qoo,Epoo)-+ H 1 (Qr;/Q 00 ,Epoo)-+

ffiqEE

ffivjq H!(Qoo,v,Epoo). (3.17)

It follows from local duality (Theorem 1.4.1 and Proposition 1.4.3) that for every place v of Q00 , we have ijom(H!(Qoo,v,Epoo),QpfZp) ~ \m!E(Qn,vt n

r

where as usual ( denotes p-adic completion. If v I q for some q =f. p, and E(Qq) has no p-torsion, then it is not hard to show that E(Qoo,v) has no p-torsion and so E(Qn,vt = 0 for every

3. EXAMPLES AND APPLICATIONS

72

n. Thus for p as in the statement of the corollary, the Pontryagin dual of (3.17) is ¥!!! E(Qn,pr ----+ Hom(H1(QE/Q 00 , Epoo ), Qp/Zp) ----+ Zoo ----+ 0. n

Since Q 00 /Q is totally ramified at p, we have ¥!!!E(Qn,pr n

= ¥!!!E1(Qn,p) = ¥!!!E(Pn) n

n

and this is free of rank one over A (see for example [PR1] Theoreme 3.1 or [Schn] Lemma 6, §A.1). It now follows, using the fact that Zoo is a torsion A-module (Theorem 3.5.16) and using Propositions 3, 4, and 5 of [Gr2] that Hom(H1(QE/Q 00 , Epoo ), Qp/Zp) has no nonzero finite submodules. 0 By the Lemma on p. 123 of [Gr2] the same is true of Z 00 • Corollary 3.5.19. Suppose that E is modular, that E does not have complex multiplication, that E has good reduction at p, that p does not divide 2rE TiqiN lq(q- 1 )IE(Qq)torsl {where rE is as in Theorem 3.5.3}, and that pE ,p is surjective. Then jiii(E)poo

I

divides

L~~ 1).

Proof. First, if E has good supersingular reduction at p then IE(Fp)l is prime top, so the corollary follows from Theorem 3.5.1l(ii). Thus we may assume that E has good ordinary reduction at p. In this case the corollary is a well-known consequence of Theorem 3.5.16 and Corollary 3.5.18; see for example [PRl] §6 or [Schn] §2 for details. The idea is that if Zoo has no nonzero finite submodules and char(Zoo) divides CEA, then IS(Qoo, Epoo )Gal(Qoo/Q) I divides xo(CE,N ), where

xo denotes the trivial character, and xo(CE,N) = (1- a- 1)2 II lq(q- 1 )(L(E, 1)/0E)· qiN

On the other hand, one can show that the restriction map

S(Q, Epoo) ----+ S(Q 00 , Epoo )Gai(Qoo/Q) is injective with cokernel of order divisible by (1- a- 1) 2 , and the corollary follows. 0 Remark 3.5.20. The Birch and Swinnerton-Dyer conjecture predicts that the divisibility of Corollary 3.5.19 holds for almost all, but not all, primes p.

3.6. SYMMETRIC SQUARE OF AN ELLIPTIC CURVE

73

Proof of Theorem 3.5.4(i). Suppose E is modular, E does not have complex multiplication, and L(E, 1) ::j:. 0. By Theorem 3.5.11, E(Q) is finite and ID(E)p"" is finite for every p. By Corollary 3.5.19 (and using Serre's theorem, see Remark 3.5.9), ID(E)p"" = 0 for almost all p. This proves Theorem 3.5.4(i). D Remark 3.5.21. We can also now prove part of Theorem 3.5.4(ii), in the special case where L is contained in Q 00 and E has either good ordinary or multiplicative reduction at p. For in that case, Theorem 3.5.16 shows that x(char(Hom(S(Q 00 , Ep"" ), QpfZp))) divides a nonzero multiple of L(E, x, 1)/0E. If L(E, x, 1) =I 0 it follows that S(Q 00 , Ep"" )X is finite. The kernel of the restriction map S(L, Ep"") -t S(Q 00 , Ep"") is contained in the finite group H 1 (Q 00 / L, EC:~"" ), and so we conclude that both E(L p: and ID(E;L);oo are finite. 3.6. Symmetric Square of an Elliptic Curve Let E be an elliptic curve over Q and let Tp(E) be the p-adic Tate module of E. Let T be the symmetric square of Tp(E), i.e., the threedimensional Zp-representation of GQ defined by T

= (Tp(E) ® Tp(E)) f{t ® t'- t' ® t: t, t' E Tp(E)}.

Suppose r has eigenvalues a and a- 1 on Tp(E) with a 2 ~ 1 (mod p). Then T E GQ(I'p"") (as in Proposition 3.5.8), and T has eigenvalues a 2 , 1, and a- 2 on T, so r satisfies hypothesis Hyp(Q, T)(i). If the p-adic representation attached to E is surjective and p > 3 then we can always find such a r, and further TjpT is an irreducible GQ-module and

H 1 (0/Q, W)

= H 1 (0/Q, W*) = 0.

Thus if we had an Euler system forT, then we could apply Theorem 2.2.10 to study the Selmer group S(Q, W*). See [Fl] for important progress in this direction.

CHAPTER 4

Derived Cohomology Classes The proofs of the main theorems stated in Chapter 2 consist of two steps. First we use an Euler system to construct auxiliary cohomology classes which Kolyvagin calls "derivative" classes, and second we use these derived classes along with the duality theorems of §1. 7 to bound Selmer groups. In this chapter we carry out the first of these steps. In §4.2 and §4.3 we define and study the "universal Euler system" associated toT and Koof K. In §4.4 we construct the Kolyvagin derivative classes, and in §4.5 we state the local properties of these derivative classes, which will be crucial in all the applications. The remainder of this chapter is devoted to the proofs of these properties. 4.1. Setup

Keep the notation of §2.1. We have a fixed number field K, a p-adic representation T of GK with coefficients in the ring of integers 0 of some finite extension iJ? of Qp, and we assume that T is unramified outside a finite set of primes of K. The letter q will always denote a prime of K. For every prime q of K not dividing p, we let K(q) denote the maximal p-extension of K inside the ray class field of K modulo q. Similarly, let K{l) denote the maximal pextension of K inside the Hilbert class field of K. Class field theory shows that K(q)/ K{l) is unramified outside q, totally ramified above q, and cyclic with Galois group canonically isomorphic to the maximal p-quotient of (C:h/q)x /(0~ {mod q)). Let rq = Gal(K(q)/K(l)). Fix an ideal N of K divisible by p and by all primes where T is ramified, as in Definition 2.1.1. Define

n = n(N) = {squarefree products of primes q of K such that q f N}. If~

En,

say~=

q1 · · · qk, then we define K(~)

K(~)

to be the compositum

= K(qt) · · · K(qk)· 75

4. DERIVED COHOMOLOGY CLASSES

76

Note that K(t) is contained in, but not in general equal to, the maximal p-extension of K inside the ray class field of K modulo t. We define r.

= Gal(K(t)/K(I)).

Ramification considerations show that the fields K(q) are linearly disjoint over K(I), so there is a natural isomorphism r. ~

II

rq

primes qlt

where r q is identified with the inertia group of q in r.. If s 1 t this allows us to view r.& as a subgroup of r., as well as a quotient. Fix a z:-extension Koo/ K in which no finite prime splits completely, as in Definition 2.1.1. If KeF e K 00 , let F(t) = FK(t). As in Chapter 2, we will write KerF to indicate that F is a finite extension of K, and if KerF e Koo we let rF(t)

= Gal(F(t)/K(l)).

Again, we will often identify r. with the subgroup of rF(t) generated by the inertia groups of primes dividing t, and r F(l) with the subgroup generated by the inertia groups of primes dividing p, and then (since Koo/ K is unramified outside p) rF ~ rF(l) x r •. As above, if sl t we can also identify rF(s) with a subgroup of rF(t)· Figure 1 illustrates these fields and Galois groups. For t E R define

=

La

E Z(r.) e Z(Gal(K(t)/K)). uer. If sIt and KerF e Koo we can view Ns E Z(r.) e Z(Gal(F(t)/K)) as above, and then N. = NsNt/s· As in Chapter 2, let Frq denote a Frobenius of q in GK, and N.

P(Fr~ 1 IT*;x)

= det(1- Fr~ 1 xiT*)

E O[x].

Definition 4.1.1. Suppose KerF e K 00 and M E 0 is nonzero. We define RF,M e R to be the set of all t E R such that for every prime q dividing t, • M I [K(q) : K(l)], • M I P(Fr~ 1 IT*; 1), • q splits completely in F(l) / K. As in Definition 1.4.6, if M E 0 is nonzero we let M E z+ denote the smallest power of p which is divisible in 0 by M.

4.1. SETUP

F(t:)

)

77

y

F(t:)

r"'' rF(•l

y/K(t:q)

f/.. K(t:)

r, K(q)

K(l)

I

K FIGURE

1

Lemma 4.1.2. Suppose q E R is a prime of K and ME 0 is nonzero. (i) M I (K(q): K(l)] ¢::::::} q splits completely in K(p.M, (0~) 1 /M). (ii) P(Fr; 1 1T*; N(q)Fr; 1 ) annihilates T. (iii) If M I (K(q) : K(l)] then

P(Fr; 1 IT*;x)

=det(1-

FrqxiWM)

(mod M).

(iv) If M I (K(q): K(l)] then P(Fr; 1 IT*; Fr; 1 ) annihilates WM. Proof. Class field theory identifies Gal(K(q)/ K) with the maximal p-quotient of (OK/q)x f(O~ (mod q)). Thus if q f p, then (K(q): K(l)] divides I(OK/q)xl = (N(q) -1) and

Frq fixes ILM

¢::::::}

M divides

I(OK/qfl

¢::::::}

M divides I(OK/q)xl.

If Frq fixes p.M we have further

Frq fixes (0~) 1 /M

¢::::::}

(0~ (mod q)) C ((OK/q)x)M.

This proves (i). One checks easily that P(Fr; 1 IT*;x)

= det(1- Fr; 1 xiT*) = det(1- N(q)- 1 Frqx1T).

This and the Cayley-Hamilton theorem prove (ii), (iii), and (iv).

0

4. DERIVED COHOMOLOGY CLASSES

78

The following lemma, together with the Tchebotarev density theorem, will give a large supply of primes in RF,M. By F(WM) we mean the smallest extension of F whose absolute Galois group acts trivially on WM (or equivalently, the fixed field of the kernel of the action of G F on W M). Lemma 4.1.3. Suppose KerF e K 00 and ME 0 is nonzero. Suppose further that r E GK acts trivially on K 00 (l)K(Jtpoo, (0~) 1 /Poo) and that rr=l # 0. If q is a prime of K not dividing N such that the Frobenius Frq ofq is a conjugate ofr on F(l)K(WM,Jt.M,(0~) 1 1M), then q E RF,M· Proof. Note that q is unramified in F(l)K(WM,Jt.M, (O~)lfM)JK. Since Frq fixes K(Jt.M, (O~)lfM), it follows from Lemma 4.1.2(i) that M divides [K(q) : K(l)]. Since Frq fixes F(l), we see that q splits completely in F(l)/K. By Lemma 4.1.2(iii) we have

P(Fr; 1 IT*; 1) :: det(1- FrqiWM)

= det(1- riWM) det(1- riT) = 0

=

(mod M),

the first equality because Fr q is a conjugate of r on W M, the second because # 0. Thus q E RF,M. 0

rr=l

4.2. The Universal Euler System

We now define the "universal Euler system". This might also be called a universal distribution for the Euler system distribution relation of Definition 2.1.1. In the special case K = Q and T = Zp(1), it is closely related to the universal ordinary distribution studied by Kubert in [Ku] (see also [Lan) §2.9 or [Wa) §12.3). Definition 4.2.1. Suppose that t E Rand KerF e K 00 • We define an O[Gal(F(t)/ K))-module XF(r) as follows. First suppose that every prime q dividing t satisfies K(q) # K(l). For every s dividing t let XF(s) be an indeterminate. Set XF(r) = YF(r)/ZF(r) where: YF(r)

is the free O[Gal(F(t)/K)]-module on generators

ZF(r)

is the submodule of YF(r) generated by the relations

UXF(s)

= XF(s)

NqXF(qs)

= P(Fr; 1 1T*;Fr; 1 )XF(s)

{xF(s)

if a E Gal(F(t)/F(s)) if qs It.

:sIt},

= rt/tn

For general t, let t' be the product of all primes q dividing t such that K(q) =f. K(l). Then by definition F(t) = F(t'), and further we have {F(s) : s I t} = {F(s) : s I t'}. We simply let XF(r) be XF(r') as defined above.

4.2. THE UNIVERSAL EULER SYSTEM

79

If .sj t and K cfFcfF' C K= then there are natural O[Gal(F'(t)/ K)]module maps XF'(~)

--+ XF(~) induced by

XF(s) --+ XF'(~) induced by

XF'(t) .-.+ XF(t)

fort It,

XF(t) .-.+ NF'(~)/F(~)XF'(t)

(4.1) fort j.s.

(4.2)

The map (4.1) is clearly surjective, and Proposition 4.3.1(v) below will show that the map (4.2) is injective. Definition 4.2.2. The universal Euler system (for (T, N, K=f K)) is

X= X(T,N,K=fK) = fu!?XF(~)· F,r

direct limit with respect to the maps (4.2). Using the maps (4.1), (4.2) we also define X=,~

=

~ XF(~) and X=,'R

= fu!t X=,~·

K Cf FCK 00

) = Hom(r ,T°F) = o 5

and similarly (since 5 is prime, so r.s is cyclic) H2(F(~)/F(~),TGF< .. >)

= TGF(1l /lr.siTGF.

Now pass to the inverse limit over Fin (4.3). Using Corollary B.3.6 and our assumption that the decomposition group in Gal(Koo/ K) of every finite prime is infinite, we obtain an exact sequence 0--+ H~(K(~),T)

- t H~(K(~s),Tl• - t

¥!!!

T°F/!r.siTGF.

KCrFCKoo

By Lemma B.3.2, this inverse limit is zero, so this proves (ii).

D

4.3. Properties of the Universal Euler System Recall that q; is the field of fractions of 0.

Proposition 4.3.1.

(i) (ii) (iii)

XF(~)

Suppose~

En and KerF c K

00 •

Then

is a finitely generated free 0-module,

Xpc~J ® q; is a free rank-one module over q>[Gal(F(~)/K)],

XF(~) is a free O[Gal(F(~)/K(~))J-module of rank [K(~): K], (iv) for every FcrF' C K 00 , the map (4.1) induces an isomorphism XF'(~) ®o[Gal(F'(~)/K)] O[Gal(F(~)/K)] ~ XF(~)• (v) for every sl ~and FCrF' C K 00 , the map (4.2) induces an isomor. X ~ XGal(F'(~)/F(s)) p hlSm F(s) ----r F'(~)

4.3. PROPERTIES OF THE UNIVERSAL EULER SYSTEM

81

Proof. Let r' be the product of all primes q dividing r such that r q =/:- { 1}. Then XF(t') = XF(t)• F(r') = F(r), and K(r') = K(r), so the proposition for r is equivalent to the proposition for r'. Thus without loss of generality we may replace r by r', i.e., we may simplify the proof by assuming that r q =/:- {1} for every q dividing r. We will prove the proposition by constructing a specific 0-basis of XF(t)· Fix a set of representatives A1 C Gal(F(r)/K) of Gal(K(l)/K), and for every prime q dividing r let Aq = rq- {1} C Gal(F(t)/K). For every ideals dividing r, define a subset AF,s C Gal(F(r)/ K) by AF,s

= Gal(F(r)/ K(r)) A1 II Aq primes

qls

= {9F91 II 9q : 9F E Gal(F(r)/ K(r)), 91 E Ab 1 =/:- 9q E r q} qls

and then define a finite subset BF(t)

BF(t)

of XF(t) by

= UAF,sXF(s)

C

XF(t)·

sit

We will show that BF(t) is an 0-basis of XF(t)· Clearly Aq U {Nq} generates O[rq] over 0, so Gal(F(r)/ K(r)) A1

IT

(Aq

U {Nq})

qls

generates O[rF(sJl· For every q dividing s we have a relation NqXF(s)

= P(Fr; 1 IT*;Fr; 1 )XF(s/q)

E O[Gal(F(s/q)/K)]xF(s/q)·

It follows easily, by induction on the number of primes dividing r, that BF(t) generates XF(t) over 0. FUrther, sit

qlt

= [F(l): KJII!rql = [F(r): K]. qlt

On the other hand, we claim that ranko(XF(tJ) ;::: [F(r) : K]. To see this, let YF(t) and ZF(t) be as in Definition 4.2.1 of XF(t)· One can check directly that the assignment xF(s) t-+

IINq II(!rqi+(P(Fr; 1 1T*;Fr; 1 )-lrql)l~ql) ql(t/s) qls q

82

4. DERIVED COHOMOLOGY CLASSES

induces a well-defined homomorphism from YF(t} to O[Gal(F(t)/ K)] which is zero on ZF(t}· Thus we obtain a map

- t 4>[Gal(F(t)/K)].

If xis a character of Gal(F(t)/ K) into an algebraic closure of 4>, say x has conductor exactly s, then X(VJ(XF(a)))

= IJ 1rq1 "#

0.

qJt

It follows that

(XF(t) ® 4>)

?: [F(t): K] ?: IBF(-r}l·

Since BF(t} generates XF(t} over 0, we conclude that equality holds, that BF(t} is an 0-basis of XF(t}• that XF(t} is torsion-free, and that

F

F

-+ \!!!!(WM/WM)°FC•l ~ H:.0(K(t),WM)-+ 0. F

By Lemma B.3.2, \!!!! wZFC•>

= 0, and the proposition follows.

Proposition 4.4.8. Suppose c is an Euler system and family of O[GK]-module maps

t

E

0

R. There is a

{dF: XF(t)-+ (WM/WM)°FC•l: KerF C Koo} lifting c, i.e., such that the following diagrams commute dF'

XF'(t) -(WM/WM) NF'(•)/F(T)

l

1

G

F'(•)

NF'(•)/F(T)

XF(t) ~ (WM /WM)GF(T) where the bottom map on the left sends XF(s) ~--+ cF(s) for all s dividing t as in Lemma 4.2.3, and on the right KCrFCrF' C K 00 • These conditions determine each dF uniquely up to an element of Homo(GKJ(XF(t)• WM ). Proof. We first illustrate the proof in a simplified setting. If wZFC•l

= 0,

then Proposition 4.4.5(i) becomes a short exact sequence which (abbreviating R = (0/MO)[Gal(F(t)/K)J and XF(t)/M = XF(t)/MXF(t)) induces an exact sequence 0-+ HomR(XF(c)/M, W::C•l)-+ HomR(XF(t)/M, (WM /WM)°FC•l) 6 FC•l

HomR(XF(t)/M,H 1 (F(t), WM))-+ Extk(XF(t)/M, w;;c•>).

By Lemma 4.4.6(i) andProposition 4.3.4(i), Extk(XF(t)/M, WJC•l) so we can choose a map dF lifting c in this case.

= 0,

4. DERIVED COHOMOLOGY CLASSES

88

In general, since wZF may be nonzero, we pass to the limit and use the short exact sequence of Proposition 4.4. 7 instead of Proposition 4.4.5(i). Arguing as above, using Lemma 4.4.6(ii) and Propositions 4.4.7 and 4.3.4(ii), and writing A~ = O[[Gal(K00 (t)/K)]], we obtain an exact sequence 0

~ HomA.(X 00 ,r/MXoo,~·\!!!! ~) F

~ HomA.(X 00 ,~/MXoo,~ 1 \!!!! (WM /WM)°F) F

~ HomA.(Xoo,~/MX 00 ,~,H~(K(t), WM)) ~ 0. (4.6) Therefore there is a map d 00 : X 00 ,~ --+ \!!!! (WM

/WM )° F

such that

F

6~ o doo({xF(s)}F)

= {cF(s)}F

for every s dividing t. We define dp to be the composition Xp ~ Xoo.~ ®A. O[Gal(F(t)/K)] doo®l

\!!!!(WM/WM)GF'(•) ®A. O[Gal(F(t)/K)] F'

~ (WM /WM)°F

where the left-hand isomorphism comes from Corollary 4.3.2 and the righthand map is the natural projection. (Explicitly, dF(XF(s)) is the projection of d 00 ({xF'(s)}) to (WM/WM)°F.) It is straightforward to check that these maps have the desired properties. By (4.6), d 00 is unique up to an element ofHomaK (X 00 ,~, \!!!! W/; ), and it follows that dp is well-defined up to an element of Homo[GKJ(XF(~)• WM ). 0 Remark 4.4.9. We will only need to use the existence of the maps dp of Proposition 4.4.8 for individual F. The compatibility as F varies (the righthand diagram of the proposition) is needed in order to get the uniqueness portion of the proposition, i.e., to make the map dp well-defined up to an element of Homo[GKJ(XF(~)• WM ).

Definition 4.4.10. Suppose c is an Euler system, M E 0 is nonzero, KerF C K 00 , and t E RF,M· Fix a map d

= dp:

XF(~) ~WM/WM

in a family lifting c as in Proposition 4.4.8. Lemma 4.4.2 shows that d(D~,FXF(~)) E (WM /WM)GF,

4.4. KOLYVAGIN'S DERIVATIVE CONSTRUCTION

where Dr,F

89

= NF(l)/FDr is as in Definition 4.4.1, and we define II:[F,r,M] = 8p(d(Dr,FXF(r))) E H 1 (F, WM)·

We can describe this definition with the following diagram

II:[F,r,M] where the commutativity of the inner square is part of Proposition 4.4.5(iii).

Remark 4.4.11. The class II:[F,r,M] is independent of the choice of NF(l)/F used to define Dr,F, since by Lemma 4.4.2, Dr,FXF(r) E XF(r)/MXF(r) is independent of this choice. The definition of II:[F,r,M] is also independent the choice of d in Proposition 4.4.8. For if d' is another choice, then d-d' E HomaK(XF(r)• WM ), so by Lemma 4.4.2 and Proposition 4.4.5(i), d(Dr,FXF(r))- d'(Dr,FXF(r)) E image((WM )°F)

= ker(8F ).

Also, note that II:[F,r,M] depends only on the images of the Euler system classes C£ in H 1 (L, WM ), not in H 1 (L, T). However, the extra information in H 1 (L, T) will be used to prove Theorem 4.5.1 below. See §9.3 for a further discussion in this direction. The class II:[F,r,M] does depend (because Dr does) on the choice of generators u q of the groups r q. Making another set of choices will multiply II:[F,r,M] by a unit in (OfMO)x. For the next two lemmas, suppose c is an Euler system, M E 0 is nonzero, K Ct F C Kcx, and ~ E 'RF,M as in Definition 4.4.10.

Lemma 4.4.12. Suppose d : XF(r) -+ WM /WM is a lifting of the Euler system c as in Proposition 4.4.8. Let f E WM be any lifting of d(Dr,FXF(r))· Then II:[F,r,M] is represented by the cocycle 'Y

1--t

("!- 1)f E WM for 'Y E Gp.

Proof. This is a combination of the definition of II:[F,r,M] above with the explicit description of the connecting map 8p (Proposition 4.4.5(ii)). 0

Lemma 4.4.13. (i) The class II:[F,l,M] is the image of cp under the map H 1 (F,T)-+ H 1 (F, WM)· (ii) More generally, the restriction of II:[F,r,M] to H 1 (F(~), WM) is equal to the image of Dr,FCF(r) in H 1 (F(~), WM)·

4. DERIVED COHOMOLOGY CLASSES

90

(iii) If M

I M'

and

t

E nF,M' then under the natural maps we have

K[F,~,M'] t---~ K[F,~,M]

K[F,~,M] t - - - -

(M' /M)K[F.~.M']

Proof. All three assertions follow from Definition 4.4.10. For the first we take t = 1, so D~,F = NF(l)/F• and use Proposition 4.4.5{iii) and the Euler system relation CorF(l)/F(cp(l)) = cp. 0 4.5. Local Properties of the Derivative Classes

Fix an Euler system c for T. In this section we will state the main results describing the local behavior of the derivative classes K[F.~,M] of §4.4. We will see {Theorem 4.5.1) that K[F.~,MJ belongs to the Selmer group sE (F, wM) where :E is the set of primes of K dividing pt (see Remark 1.5.8 for this slight abuse of notation). For our applications it will be crucial to understand {Theorem 4.5.4) the ramification of K[F.~,M] at primes dividing t.

The proofs will be given in the remaining sections of this chapter. Theorem 4.5.1. Suppose thatM E 0 is nonzero, thatKCrF C Kcx, and that t E nF,M. For every place w ofF not dividing pt, (K[F,~,MJ)w E H}{Fw, WM)·

In other words, K[F.~.MJ E sEp•(F, WM)

where

:Ep~

is the set of primes of K dividing pt.

Theorem 4.5.1 will be proved in §4.6. Lemma 4.5.2. Suppose M E 0 is nonzero and q E 'RK,M is prime. Then there is a unique Qq{x) E (0/MO)[x] such that

P(Fr; 1 IT";x)

= (x -1)Qq{x)

{mod M).

Proof. Take 1 1 Q ( ) _ P(Fr; !T*;x)- P(Fr; 1T*; 1)

q

x -

x-1

·

Since q E RK,M, we know that M divides P(Fr; 1 1T*; 1) so this polynomial has the desired property. The uniqueness comes from the fact that x - 1 is not a zero divisor in (0/MO)[x]. 0

4.5. LOCAL PROPERTIES OF THE DERIVATIVE CLASSES

91

Definition 4.5.3. Suppose M E 0 is nonzero and q E 'RK,M is prime. The choices of u q E r q (Definition 4.4.1) and Fr q depend on the choice of a prime .Q of k above q. We use the same choice for both, and we further fix aq in the inertia group of .Q extending u q. By Lemma 1.4.7(i) (which applies thanks to Lemma 4.1.2(i)) there are well-defined isomorphisms ~ WFrq=1 aq : H s1(Kq, W M ) ---+ M

/3q : H}(Kq, WM) ~ WM/(Frq -l)WM

given on cocycles by

aq(c)

= c(Ciq),

/3q(c)

= c(Frq).

If q E 'RK,M, then P(Fr~ 1 IT*;Fr~ 1 ) annihilates WM by Lemma 4.1.2(iv). Thus the polynomial Qq of Lemma 4.5.2 induces a map

Qq(Fr~ 1 )

:

WM/(Frq -l)WM ---+ w.r;q= 1 .

We define the "finite-singular comparison" map

¢{

8

:

H}(Kq, WM) ---+ H;(Kq, WM)

to be the composition Hj(Kq, WM)

~

WM/(Frq -l)WM Qq(Fr;l)

Fr -1

WMq-

a;l

1

--:....t H 8 (Kq, WM)·

= Kq,

and we can view ¢~ 8 as a map from Hj(Fo, WM) to H~(Fo, WM)· We will still write ¢~ 8 in this case, and suppress the dependence on .Q.

If KcfF C Koo and q E 'RF,M, then Fo

Theorem 4.5.4. Suppose M E 0 is nonzero, K cfF C K 00 , q is prime, and tq E 'RF,M· If ¢~ 8 is the map defined above, and (~[F,tq,MJ)~ denotes the image of ~[F,tq,M] in H~(Fo, WM), then

(~[F,tq,Mj)~ = ¢{ 8 (~[F,t,Mj). In other words, the singular part of ~[F,tq,M] at q is controlled by the (finite) localization of ~(F,t,M] at q. Theorem 4.5.4 will be proved in §4.7. Corollary 4.5.5. Suppose M E 0 is nonzero, q is prime, and tq E nK,M. Suppose further that WM/(Frq- l)WM is free of rank one over 0/MO. Then the order of (~[K,tq,MJ)~ in H~(Kq, WM) is equal to the order of (~[K,t,MJ)q in H}(Kq, WM)·

4. DERIVED COHOMOLOGY CLASSES

92

Proof. The maps aq and /3q in Definition 4.5.3 are both isomorphisms, and by Lemma 4.1.2(iii) and Corollary A.2.7 (applied with r = Fr; 1 and

Q(x) = Qq(x}}, so is the map Qq(Fr; 1 ). Thus the corollary follows from Theorem 4.5.4.

¢{

8

is an isomorphism and 0

4.6. Local Behavior at Primes Not Dividing pt

For this section fix an Euler system c for T and a nonzero element ME 0. If Kc,F C K 00 and t E RF,M, we need to show that (K[F,~,MJ)w E Hj ( Fw, W M) for every place w ofF not dividing pt. When w is archimedean (Lemma 4.6.3}, or when w is nonarchimedean and T is unramified at w

(Corollary 4.6.2(ii}}, this is not difficult. We treat those cases first, and then go on to the general case. Proposition 4.6.1. Suppose Kc,F C K 00 and t E R. For every prime Q of F(t} not dividing p, and every 'Y E GK, we have ('YcF(~))Q E H~r(F(t}Q, T},

('YcF(~))Q E Hj(F(t}Q, WM)

where Cp(~) is the image of Cp(~) under H 1 (F(t}, T) ~ H 1 (F(t}, WM ). Proof. Since {'YcF(~)}F E H"to(K(t},T}, the first inclusion follows from Corollary B.3.5 and the second from Lemma 1.3.8(i). 0

Corollary 4.6.2. Suppose Kc,F C Koo and t E RF,M· If Q is a prime ofF not dividing

pt,

then

(i) (K[F,~,Mj)Q E H~r(FQ, WM), (ii) ifT is unramified at Q then (K[F.~,MJ)Q E Hj(FQ, WM)· Proof. Let D~,F be as in Definition 4.4.10 and write I for an inertia group of Q in Gp. Since F(t)/F is unramified at Q we have I C GF(~)• so by Lemma 4.4.13(ii) the restriction of K[F,~,MJ to I is equal to the image of D~,FCF(~) in H 1 (I, WM)· By Proposition 4.6.1, the latter is zero. This shows that (~t[F,~,MJ)Q E H~r(FQ, WM), and if Tis unramified at Q then Lemma 1.3.8(ii) shows that Hj(FQ, WM) = H~r(FQ, WM)· 0

c, F C K 00 and t E RF,M. If w is an infinite place ofF, then (K[F,~,MJ)w E H}(Fw, WM)·

Lemma 4.6.3. Suppose K

Proof. Let w be a place of F(t) above w. Since F(t)/F ramifies only at primes dividing t, the place w splits completely in F(t)/ F. Thus Lemma 4.4.13(ii) shows that (K[F.~.MJ)w is the image of (D~.FCF(~))w under H 1 (F(t),;;,T) = H 1 (Fw,T)

By Remark 1.3.7 we have H}(Fw,T) from Lemma 1.3.8(i).

---+ H 1 (Fw, WM)·

= H 1 (Fw,T),

so the lemma follows 0

4.6. LOCAL BEHAVIOR AT PRIMES NOT DIVIDING

p~

93

Remark 4.6.4. In the nonarchimedean case, if w is a prime of K not dividing pt, then Corollary 4.6.2{i) shows that (t~:[F,~,MJ)w E H~.(Fw, WM). Unfortunately, for primes w where Tis ramified it may not be true that Hj(Fw, WM) = H~.(Fw, WM)· However, we do get immediately the following corollary, with only a slightly stronger assumption on t. Corollary 4.6.5. There is a nonzero mE 0, independent of M, such that for every K CrF C K 00 , every t E nF,Mm, and every prime Q ofF not dividing pt, we have (t~:[F.~,MJ)Q E Hj(FQ, WM)·

Proof. Choose m E 0 such that for every prime q of K not dividing p, m annihilates wzq I (WZq )div, where Iq is an inertia group for q in GK and (WZq )div is the maximal divisible submodule of wzq . Clearly we can take m to be nonzero, since wzq / (WZq )div is always finite and is zero whenever W is unramified at q. Suppose K Cr F C K 00 • If Q is a prime ofF not dividing p, and q is the prime of K below Q, then Iq is also an inertia group of Q in GF. Therefore Lemma 1.3.5{iii) shows that m annihilates H~.(FQ, WMm)/ Hj(FQ, WMm), so by Corollary 4.6.2 we have (mt~:[F,~,MmJ)Q E Hj(FQ, WMm)· Lemma 4.4.13{iii) shows that m~~:[F,~,Mm) is the image of II:[F,~,MJ in H 1 (F, WMm), and the corollary follows. D Corollary 4.6.5 is already strong enough to use in place of Theorem 4.5.1 in proving the theorems of Chapter 2. Thus one could skip the rest of this section if one were so inclined. To prove Theorem 4.5.1 for primes Q where T may be ramified is much more delicate. We will mimic the construction of II:[F,~,MJ locally, and use Proposition 4.6.1 to show that (t~:[F.~,MJ)Q can be constructed inside H 1 (FQ, TIQ. /MTZQ.). The theorem will follow directly from this. Fix for the rest of this section an ideal dividing pt (but not necessarily in R).

t

E n and a prime q of K not

Definition 4.6.6. Fix inertia and decomposition groups I C V C G K of q. If L is a finite extension of K, unramified at q, let SL denote the set of primes of Labove q and abbreviate

Hi(Lq, WM)

=

Hi(Lq,Tz/MTz)

=

E9 Hi(LQ, WM), E9 Hi(LQ,TIQ./MTIQ.)

QESL

where for each Q E SL, we write IQ for the inertia subgroup of GLQ.· (Since L/ K is unramified at q, each IQ is conjugate to I.) Write ( · )q

4. DERIVED COHOMOLOGY CLASSES

94

or resq : Hi(L, WM) -t Hi(Lq, WM) for the sum of the restriction maps. Note that Hi(Lq, WM) and Hi(Lq,TijMTI) are Gal(L/K)-modules: this can be seen directly (every u E Gal(L/K) induces an isomorphism

Hi(LQ, TIQ. /MTIQ.)

~

Hi(LuQ, u(TIQ. /MTIQ.))

= Hi(LuQ, TI-rQ. /MTI-rQ.) for every Q, and summing these maps over Q E S L gives an automorphism of Hi(Lq, TI /MTI) and similarly for Hi(Lq, WM)), or see Corollary B.5.2. Write

and define a GK-submodule wit c w M by

As in §B.4, let Indv(WM) C WM denote the GK-submodule of maps satisfying f(hg) = hf(g) for every hE V, and similarly for Indv(Wft.) C Wit. Lemma 4.6.7. If L is a finite extension of K, unramified at q, then with notation as above we have a natural commutative diagram with exact columns

0

0

0.

4.6. LOCAL BEHAVIOR AT PRIMES NOT DIVIDING pt

95

Proof. The three columns come from GL-cohomology of the short exact sequences

0 --+ WM --+ WM --+ WM /WM --+ 0 0--+ Indv(WM) --+ WM --+ WM/lndv(WM) --+ 0 0 --+ Indv(Wk) --+ W~ --+ W~/Indv(Wk) --+ 0 respectively (the left-hand column is Proposition 4.4.5(i)), using Corollaries B.4.4 and B.5.2. The horizontal arrows are the natural ones, and the commutativity follows from the functoriality of all the maps involved. 0 We now need a local analogue of Proposition 4.4.8. If KerF Q E SF{r), and .sIt, then by Proposition 4.6.1, (cF{s))Q E H~r(F(.s)Q,T) = H 1 (F(.s)'Q/F(.s)Q,Tio.)

eK

00 ,

e H 1 (F(.s)Q,Tio.)

(4.7)

Proposition 4.6.8. Suppose c is an Euler system and two families of O[GK]-module maps

t

{dF,q: XF{r) --t (WM/Indv(WM))°F: KerF

{d~,q: XF(r) --t (W~/Indv(Wk))°F: KerF lifting c, i.e., such that if KerFerF'

eK

00

E R. There are

e Koo} e Koo}

then

(i) the maps dF,q (resp d~,q) are compatible with respect to the norm maps XF'(r)-+ XF(r)> (WM/Indv(WM))°F'-+ (WM/Indv(WM))°F,

(W~ /Indv(Wk ))°F' --t (W~ /Indv(Wk ))°F,

(ii) for every KerF

eK

00

and every .s dividing t, the compositions

dFq OF( n + (ISEp (K, W~)l + l)nw + indo{c) {if indo{c) is infinite then there is nothing to prove). Apply Lemma 5.2.3 with this group C, let ~ be a set of primes of K produced by that lemma, and apply Lemma 5.2.5 with this set ~Combining the inequality of Lemma 5.2.5 with Theorem 1.7.3{iii) shows that Therefore fo(tm(SEp(K, W~))) :::; fo(tm(SEuEp(K, W~))) + nw +indo{ c).

Lemma 5.2.3{iv) shows that tm{SEuEp{K, W;)) C H 1 (fl/K, W*) nSEp(K, W*),

nw = fo(H 1 {fl/ K, W*) n SEp (K, W*)), so we see that lo(tm(SEp (K, W;))) :::; indo{c) + nw + nw.

and by definition

Since this holds for every m, and SEp(K, W*) orem 2.2.2 follows.

= lli!1 tm(SEp (K, W~)), The"'

D

5.3. Bounding the Exponent of the Selmer Group

The proof of Theorem 2.2.3 is similar to that of Theorem 2.2.2; it is easier in the sense that one can work with a single prime q instead of a finite set of primes, but more difficult in the sense that one must keep track of extra "error terms". The idea is as follows. Given 1J E SEp(K, W_M), we use Lemma 5.3.1 below to choose a prime q such that • H}(Kq, WM-) and H!{Kq, WM) are "almost" isomorphic to 0/MO, • order{{K[q,MJ)~,H!{Kq, WM)) is approximately ordpM- indo( c), • order({7J)q,H}{Kq, WM-)) is approximately order{7J,H 1 (K, WM-)).

5.3. BOUNDING THE EXPONENT OF THE SELMER GROUP

Since the Kolyvagin derivative class ll:[q,M] belongs to duality Theorem 1. 7.3 shows that order((K[q,MJ);,Hi(Kq, WM))

sEpq

115

(K, WM), the

+ order((17)q,H}(Kq, WM-))

is "approximately" bounded by ordpM, and so order(17,H 1 (K, WM-)) is "approximately" bounded by indo(c). Since 11 E SEp(K, WM-) is arbitrary, if we can bound all the error terms independently of M, this will prove Theorem 2.2.3. In the remainder of this section we sketch the details of this argument. Keep the notation of §5.1 and §5.2. Suppose the Euler system c satisfies the hypotheses Hyp(K, V), and fixaTE GK as in hypothesis Hyp(K, V)(i). We now allow p = 2. Let a be the smallest positive integer such that pa annihilates the maximal GK-stable 0-submodule of (r -1)W and of (r -1)W*. Hypothesis Hyp(K, V)(ii) ensures that a is finite, since any divisible GK-stable subgroup of (r - 1) W would be the image of a GK-stable subgroup of (r - 1) V, which must be zero. We have the following variant of Lemma 5.2.1. Lemma 5.3.1. Fix a power M of p. Suppose L is a Galois extension of K such that G L acts trivially on WM and on WM-. If 11:

E H 1 (K, WM),

11 E H 1 (K, Wk)

then there is an element 'Y E G L satisfying (i) order(~~:('Yr), WM/(r -1)WM)) ~ order((t~:)L,H 1 (L, WM))- a -1, (ii) order(7J('Yr), WM-/(r -1)WM-) ~ order((7J)L,H 1 (L, WM-))- a -1. Proof. The proof is similar to that of Lemma 5.2.1, once we note that every GK-submodule of WM which projects to zero in WM/(r -1)WM is killed by pa, and similarly for WM-. The extra '1' takes care of the case p = 2. D Let 0

= K(l)K(W,JLpoo, (0~.)1/Poo) as in §5.2.

Lemma 5.3.2. If T =/; 0 and T =/; 0(1) then both H 1 (0JK, W) and H 1 (0JK, W*) are finite.

Proof. This is Corollary C.2.2 applied with F = K.

D

Proof of Theorem 2.2.3. If T = 0(1) then Proposition 1.6.1 shows that

SEp (K, W*) C Hom(AK, D), where AK is the ideal class group of K, so SEp(K, W*) is finite. The theorem assumes that T =/; 0, so by Lemma 5.3.2 we may assume from

5. BOUNDING THE SELMER GROUP

116

now on that H 1 (f!IK, W) and H 1 (f!IK, W*) are finite. Let

n

= max.{lo(H 1 (0.IK, W)),lo(H 1 (f!IK, W*))}.

Suppose M is a power of p and TJ E SEp (K, W,V ). Apply Lemma 5.3.1 with L = K(l)K(WM,#LM,(O~.)IIM) C f!, with this TJ, and with "'[t,M] E H 1 (K, WM), and let 'Y E GL be an element satisfying the conclusions of that lemma. Then since H 1 (f!IK, W) is the kernel of the restriction map H 1 (K, W) -t H 1 (f!, W),

"'=

order("'[t,M]('Yr), WMI(r -1)WM)) ~ order(tM("'[t,MJ)o,H 1 (f!, W))- a -1 ~ order(tM("'[t,M]),H 1 (K, W))- a -1- n

= ordpM- indo( c)- a -1- n

(5.17)

by Lemma 5.1.1. Similarly order(TJ('Yr), W,VI(r -1)W,V) ~ order(TJ,H 1 (K, W,V))- a -1- n. (5.18) Let L' denote the fixed field of ker(("'[l,Mj)L)

n ker((TJ)L)

and, using the Tchebotarev theorem, choose a prime q of K, not dividing N, whose Frobenius in L' I K, for some choice of prime above q, is "'fT. By Lemma 4.1.3, we have q E RM. As in the proof of Lemma 5.2.3, we conclude from (5.17) and (5.18) that

and order((TJ)q,H}(Kq, W,V)) ~ order(TJ,H 1 (K, W,V))- a -1- n.

(5.19)

Let b = lo (wr=l I (wr=l )div), where (wr=l )div is the maximal divisible submodule of wr=l. By Theorem 4.5.4 and Corollary A.2.6, order(("'[q,MJ)~,H;(Kq, WM)) ~ order(("'[t,MJ)q,H}(Kq, WM))- 2b ~

ordpM- indo( c)- a- 1- n- 2b.

By Lemma 1.4.7(i),

lo(H;(Kq, WM))

= lo((WMY= 1 ) = lo((wr=l)M)

Thus, applying Theorem 1.7.3(iii) with

:E = :Epq, :Eo= :Ep, TJ E SEp(K, W,V),

:::; ordpM +b.

5.3. BOUNDING THE EXPONENT OF THE SELMER GROUP

117

we conclude that order((ry)q,H}(Kq, WM-)) ~ lo(coker(loc~pq,Ep)) ~ lo(H:(Kq, WM))- order((~[q,MJ)~, H:(Kq, WM)) ~

indo (c)

+ a + 1 + n + 3b

since ~[q,M] E sEpq (K, WM ). Combining this with (5.19) shows that order(ry, H 1 (K, WM-)) ~ 2 + 2a + 3b + 2n + indoc. This inequality holds for every M and every 77 E SEp (K, WM-). SEp (K, W*) is the direct limit of the SEp (K, WM- ), if we set

Since

m = p2+2a+3b+2n+indoc

then we conclude that mSEp (K, W*) = 0. As is well-known, this implies that SEp (K, W*) is finite: Lemma 1.5.4 shows that

SEp(K, W*)

= SEp(K, W*)m

c S(K, W*)m

and the latter is finite by Lemma 1.5.7(i).

= Lm(S(K, w;)) 0

CHAPTER 6

Twisting In this chapter we extend the methods of §2.4 to twist Euler systems by characters of infinite order. This will be used in Chapter 7 when we prove Theorems 2.3.2, 2.3.3, and 2.3.4. If pis a character of Gal(K00 /K), then • Theorem 6.3.5 says that an Euler system c for (T, Koo) gives rise to an Euler system cP for (T ® p, K 00 ), • Theorem 6.4.1 shows that the theorems of §2.3 hold forT and c if and only if they hold for T ® p and cP, and • Lemma 6.1.3 allows us to choose a particular p which avoids certain complications. We keep the setting of Chapter 2, so K is a number field, T is a p-adic representation of GK ramified at only finitely many primes, and K 00 is an abelian extension of K satisfying Gal(Koo/ K) !:!:! Let r = Gal(Koo/ K), and recall that A is the Iwasawa algebra

z:.

A

= O[[r]] =

¥!!!

O[Gal(F/ K)],

KCrFCKoo

a complete local noetherian unique factorization domain. The characteristic ideal char(B) of a finitely generated A-module B was defined in §2.3. 6.1. Twisting Representations Definition 6.1.1. Suppose p: GK --+ox is a continuous character, possibly of infinite order. As in Example 1.1.2 we will write Op for a free rank-one 0-module with GK acting via p, and if B is a GK-module we will abbreviate B®p = B® 0 0p. Then B ®pis isomorphic to B as an 0-module but not (in general) as a GK-module. If p: r--+ ox define Twp: A~ A to be the 0-linear isomorphism induced by 'Y '"""* p('Y)'Y for 'Y E r. 119

120

6. TWISTING

Lemma 6.1.2. Suppose B is a finitely genemted torsion A-module and p : r -t is a chamcter. Then B ® p is a finitely genemted torsion A -module and

ox

(i) Twp(char(B ® p)) = char(B), (ii) iff E A then f · (B ® p) = 0 (V* f(r -l)V*)

= dim(vT= 1 ) = 1.

Definition 7.1.1. Fix an isomorphism ()* :

W* /(r -l)W* ~ D.

Recall that n = K(l)K(W,P.poo,(0~) 1 1P 00 ). Define noo nt? be the fixed field of T in {loo. 129

= Koon and let

7. IWASAWA THEORY

130

There is a natural evaluation homomorphism Ev* : G0 tl -----+ Hom(S:r:p (K00 , W*), D)

=X

00 ,

defined as follows. For every u E Gn{-r) Uoo and class c E S:r:p (K00 , W*), we set Ev*(u)(c) = (}*(c(u)) where c(u) means any cocycle in the class c, evaluated at u. Then c(u) is well-defined modulo (u- 1)W*, and u acts on W* through Gal(floo/Ot?} which is (topologically) generated by T, so

(u- 1)W* C (T- 1)W*

= ker(9*).

Thus Ev*(u) is well-defined, and the cocycle relation shows that Ev* is a homomorphism. Definition 7.1.2. Define a positive integer aT by aT = [wT= 1 : (WT= 1)div] · max{IZI, IZ*I}

z

where (WT= 1 )div is the maximal divisible submodule of wT= 1 , and (resp., Z*) is the maximal GKoo-stable submodule of (T -1)W (resp., (T- 1)W*). Lemma 7.1.3. (i) aT is finite. (ii) If T and T satisfy hypotheses Hyp(N00 , T) then aT

= 1.

Proof. Let Z be as in Definition 7.1.2. The maximal divisible submodule of Z gives rise to a GK -stable subspace Vo C (T -1)V ~ V. Hypothesis Hyp(K00 , V)(ii) asserts that V is irreducible, so we must have Vo = 0. Hence Z is finite, and similarly Z~ is finite. The index [WT= 1 : (WT= 1)div] is finite simply because W has finite Zp-corank. This proves (i). Now suppose hypotheses Hyp(K00 , T) hold. Then Wp is an irreducible GKoo-module (where p is the maximal ideal of 0), and Wp rt. (T -1)W because WM f (T - 1) WM is free of rank one over 0/ M 0 for every nonzero M, so it follows that Z = 0. Similarly Z* = 0, and Proposition A.2.5 shows that wT= 1 = (WT= 1)div· This proves (ii). D 00

Recall that .N is the ideal of Definition 2.1.1 corresponding to c. Consider the following two extra assumptions. Assumption 7.1.4. For every KctF C K 00 , both Apfchar(X00 )AF and X 00 ® Ap are finite.

Assumption 7.1.5. For every prime A of K dividing .N, the decomposition group of A in G K contains an element 'Y>. with the property that P"-

T"'f>. - 1

= (T*)'Y>-P"-- 1 = 0

for every n ~ 0.

7.1. OVERVIEW

131

Remark 7.1.6. Suppose that X 00 is a torsion A-module. If p: r -t ox is a character, and we replace T by its twist T®p, then Corollary 6.2.2 shows that Xoo is replaced by X 00 ® p. Thus Lemma 6.1.3 applied to T EEl T* (in part (i)) and to Xoo EEl A/char(X00 ) (in part (ii)) shows that there is a p such that after twisting by p, Assumptions 7.1.4 and 7.1.5 are satisfied. Theorem 6.4.1 shows that Theorems 2.3.2, 2.3.3, and 2.3.4 forT and c are equivalent to Theorems 2.3.2, 2.3.3, and 2.3.4, respectively, for T ® p and the twisted Euler system cP of §6.3. Suppose now that Theorem 2.3.2 holds (the proof will be given in §7.3), so that Xoo is a torsion A-module. Then the discussion above shows that, without loss of generality, to prove Theorems 2.3.3 and 2.3.4 we may assume that 7.1.4 and 7.1.5 are satisfied. We will assume this for the rest of this section. As discussed in §2.3, since Xoo is a torsion A-module we can fix an injective pseudo-isomorphism (7.1) i=1

where JI, ... , fr E A satisfy fi+1 I h for 1 :::; i :::; r - 1. The sequence of principal ideals (elementary divisors) JIA, ... , frA is uniquely determined by these conditions, and the characteristic ideal of Xoo is r

char(Xoo)

= IJ fiA.

(7.2)

i=1

Since Xoo is a torsion A-module, all the h are nonzero. Assume for the rest of this section that, in addition to hypotheses Hyp(K00 , V), hypothesis Hyp(Koo/K) is satisfied as well.

Proposition 7.1.7. With r as above, there are elements Z1, ••. ,zr E Xoo and ideals 91, ... , Qr C A such that for 1 :::; k :::; r we have (i) Zk E Ev"(rGn.J, (ii) aTgk C fkA, and if k < r then Qk C Qk+l, (iii) there is a split exact sequence

0

-t

k-1

k

i=1

i=1

L Azi - t L Azi - t A/gk - t 0,

(iv) aT(Xoo/ E~= 1 Azi) is pseudo-null. The proof of Proposition 7.1.7 will be given in §7.6. Using (7.1) it is easy to find {zi}, with Qi = /iA, satisfying (ii), (iii), and (iv), but condition (i) will be essential for our purposes.

132

7. IWASAWA THEORY

Definition 7.1.8. Fix a sequence z1, ... , Zr E X 00 as in Proposition 7.1.7 and define r

Zoo

= L:Azi C X

00 •

i=l

Let M denote the maximal ideal of A. If 0 ::; k ::; r, a Selmer sequence u of length k is a k-tuple (a 1 , ... , O'k) of elements of rGnoo satisfying Ev*(ai)- Zi E MZoo

=

for 1 ::; i ::; k. (When k 0, the empty sequence is a Selmer sequence.) Note that by Proposition 7.1.7{i), Selmer sequences exist, for example with all the above differences equal to zero. Suppose M is a power of p. Let OM= K(1)K(WM,I£M,(0~) 1 1M), and if KerF C K 00 let LF,M ::::> FOM be the fixed field of the subgroup

n

ker((c)FnM) c GFnM'

cESEp(F,WA:t)

The restriction of S'Ep (F, W.M) to FOM is a finite {Lemma 1.5. 7) subgroup of Hom(GFnM, W_M), so LF,M is a finite abelian extension of FOM. It is straightforward to check that LF,M I K is Galois and unramified outside primes above p, oo, and primes where T is ramified. For 0::; k::; r we call a k-tuple (q1, ... , qk) of primes of K a Kolyvagin sequence {for F and M) if there is a Selmer sequence u of length k such that for 1 ::; i ::; k, • qi does not divide the ideal N of Definition 2.1.1, and • Frq, is (a conjugate of) O'i on LF,M (all primes not dividing N are unramified in L F,M I K). If 1r is a Kolyvagin sequence of length k we define

i=l

By Lemma 4.1.3, t{7r) belongs to the set nF,M defined in Definition 4.1.1. Let II{k, F, M) be the set of all Kolyvagin sequences of length k for F and M. When k = 0, we make the convention that II{k, F, M) has a single element, the empty sequence {independent ofF and M). Define an ideal in AF,M

'IJ!(k, F, M)

=

L

11'EII(k,F,M)

L .,P(K[F.~(11'),Mj)

c AF,M

1/J

where K[F,r(11'),M] is the Euler system derivative class constructed in §4.4, (K[F.~(11'),MJ) is the AF,M-submodule of H 1 (F, WM) generated by K[F.~(11'),M]• and the inner sum is over '1/J E HomA{(K[F.~(11'),Mj),AF,M)· In other words,

7.1. OVERVIEW

133

'l!(k, F, M) is the ideal of AF,M generated by all homomorphic images of modules (K:[F,t(.,..),MJ) as 1r runs through II(k, F, M). Proposition 7.1.9. There is an element hE A such that

(i) h is relatively prime to char(Xoo), (ii) for every K CrF C Koo there is a power Np of p such that if M is a power of p and 0::; k < r, then a~h'l!(k, F, M N F )AF,M C fk+1 'l!(k

+ 1, F, M).

Proposition 7.1.9 is the key to the proofs of Theorems 2.3.3 and 2.3.4; it will be proved in §7.7. We now show how to use Proposition 7.1.9 to complete the proof of Theorems 2.3.3 and 2.3.4. Recall that if ~ is a set of places of K, then KE denotes the maximal extension of Kin k which is unramified outside ~. Corollary 7.1.10. Suppose KerF C K 00 and hE A satisfies Proposition 7.1.9. Let L: be a set of places of K containing all primes above p, all primes where T is ramified, and all infinite places, If¢ E HomA(H 1(KE/F,T),Ap), then a~rhr'lj;(cp) E char(Xoo)Ap.

Proof. Note that cp E H 1(KE/F,T) by Corollary B.3.6. Let N F be as in Proposition 7.1.9(ii). Suppose 0 ::; k < r and M is a power of p. Proposition 7.1.9(ii) shows that a~h'l!(k, F, M N[..-k)AF,M C fk+l 'l!(k

so by induction, writing M'

+ 1, F, M Np-k- 1)AF,M,

= M N[;. and using (7.2), we conclude that

a~r hr\1!(0, F, M')AF,M c

r

(II fi) 'l!(r, F, M) i=1

r

C

(II /i) AF,M = char(Xoo)AF,M.

(7.3)

i=1

By Lemma 4.4.13(i), K:[F,l,M'] is the image of cp under the injection

H 1(KE/F,T)/M'H 1(KE/F,T) 1, { Av = Anno[[Dv]]{W°Koo,w /(W°Koo,w )div) if v·l p and Dv = Zp, Anno[[Dv]](H 1 (K~,w/Koo,w, Wiv /(Wiv)div)) if V f p,

AN= ITAvA. viN We define W*.

A;Iob•

A:, and Ai,r in exactly the same way with W replaced by

Lemma 7.4.2. The ideals height at least two in A.

Aglob,

AN,

A;lob•

and Ai,r defined above have

Proof. This is clear from the definitions of these ideals.

0

Lemma 7 .4.3. Suppose K Cr F C Koo and i ~ 1. (i) Hi(Koo/F, W°Koo) is finite and annihilated by Aglob· (ii) Hi(K00 / F, (W*)°Koo) is finite and annihilated by A;lob· (iii) If v is a prime of K above p and w is a prime of Koo above v, then Hi(Koo,w/ Fw, (W*)°Koo,w) is finite and annihilated by A:. Proof. Let W' = W°Koo' (W*)°Koo' or (W*)°Koo,w' and G = Gal(Koo/ F), Gal{Koo/ F), or Gal(Koo,w/ Fw), respectively. By Assumption 7.1.5, we can .choose a 'Y E GF.. c G F such that TF 1 = (T*p=l = 0. Let i' E r denote the restriction of 'Y to K 00 •

146

7. IWASAWA THEORY

Since r is abelian, the annihilator of W' annihilates Hi(G, W') for every i. H f(x) = det(1 - 1xiT EB T*) E O[x], then the Cayley-Hamilton theorem shows that /(i- 1 ) annihilates W', so in particular /(i- 1 ) annihilates Hi(G, W'). Since G acts trivially on Hi(G, W'), it follows that /(1) annihilates Hi(G, W'). Our hypothesis on 'Y ensures that /(1) =F 0, so it follows without difficulty (since G is finitely generated and W' is co-finitely generated) that Hi(G, W') is finite. This proves the finiteness in all three cases, and the annihilation when rankzp(G) > 1. Suppose now that G ~ Zp, and use the exact sequences Hi(G, W~iv)---+ Hi(G, W')---+ Hi(G, W'/W~iv)---+ Hi+ 1 (G, W~iv)· If i > 1 then Hi(G, W~iv) = 0 because G has cohomological dimension 1, and if a is a topological generator of G then (Lemma B.2.8)

H 1 (G, W~iv) ~ W~ivf(a- l)W~iv

=0

because W~iv/(a-1)W~iv is a quotient of W~iv/(i-l)W~iv· Thus for every i > 0 we have Hi(G, W') ~ Hi(G, W' /W~iv), so we see that the annihilator of W' /W~iv annihilates Hi(G, W').

0

Proposition 7 .4.4. Suppose K Cr F C K 00 and M is a power of p. (i) The kernel of the restriction map

H 1 (F, W) ---+ H 1 (K00 , W)°F is finite and is annihilated by

Aglob.

(ii) The kernel of the natuml map

H 1 (F, WM) ---+ H 1 (F, W)M is finite with order bounded independently of M, and is annihilated by AnnA(W°Koo ). (iii) The cokernel of the restriction map SEp(F, W*) ---+ SEP(K001 W*)°F is finite and is annihilated by A;IobA.N· (iv) If SEp (K00 , W*)°F is finite, then there is a power MF of p such that if M 2:: MF is a power of p, then A;IobA.N annihilates the cokernel of the natuml map SEP(F, W_M) ---+ SEP(K00 , W*)°F.

7.5. GALOIS EQUIVARIANCE OF THE EVALUATION MAPS

147

(v) The cokernel of the natural map S:r;p (F, WM-) ---+ S:r;p (F, W*) M is finite and bounded independently of M. Proof. The inflation-restriction exact sequence shows that the kernel of the restriction map in (i) is H 1 (K00 /F, W°Koo ), so (i) follows from Lemma 7.4.3(i). By Lemma 1.2.2(i), the kernel (ii) is W°F /MW°F' which in turn is a quotient of W°F /(W°F)div, and (ii) follows. Assertion (iii) is immediate from Lemmas 7.3.1, 7.4.3, and 1.3.5(iii). Suppose further that S:r;p (K00 , W*)°F is finite. Since S:r;p(F, W*)

= ~S:r;p(F, WM-),

we can choose Mp so that the image of S:r;P(F, WM-F) in H 1 (K00 , W*) contains the image of resK"" (S:r;P (F, W*)). With this choice (iv) follows from (iii). By Lemma 1.5.4, the map SEp(F, WM-) -t SEp(F, W*)M is surjective. Thus the cokernel in (v) is isomorphic to a subquotient of E9wlvker (H 1 (Fw, WM-) ---+ H 1 (Fw, W*)). For each w dividing p, Lemma 1.2.2(i) shows that the above kernel is

(W*)GF,. /M(W*)GF,.' which is a quotient of the finite group (W*)°F,. /((W*)°F,. )div and hence is bounded independently of M. This proves (v). 0 7.5. Galois Equivariance of the Evaluation Maps

For the proofs of Propositions 7.1.7 and 7.1.9 in the following sections, it would be convenient if Go"" were a A-module and Ev and Ev* were Amodule homomorphisms. Unfortunately this makes no sense, since Go"" is not a A-module. We will get around this by defining an action of a subring of A on a quotient of Go"" , and Ev and Ev* will behave well with respect to this action. Proposition 7.5.1. There are a subgroup ro of finite index in r, charan abelian extension L of 0 00 , and an action of acters x, x* : ro -+ Zp[[r0 ]] on Gal(L/0 00 ) such that

ox,

(i) Ev and Ev* on Go"" factor through Gal(L/0 00 ), (ii) if TJ E ro and 'Y E Gal(L/0 00 ) then Ev('YTI)

= X(TJ)TJ(Ev('Y)),

Ev*('YTI)

= x*(TJ)TJ(Ev*('Y)).

7. IWASAWA THEORY

148

Proof. Let L be the maximal abelian p-extension of Koo(P.voo, W)

= Koo(P.voo, W*) = Koo(W, W*).

Then 0 00 C L, and every cocycle in H 1 {K00 , W) or in H 1 {K00 , W*) vanishes on GL, so (i) is satisfied. Consider the diagram of fields in Figure 2. By Proposition C.1.7,

L

K(W)

Ko

I K FIGURE

2

there is a finite extension Ko of Kin K(W) n K 00 such that the center of Gal(K(W)JK) maps onto Gal{(K(W) nKoo)/K0 ). Define ro

= Gal(Koo/Ko).

Fix once and for all a set of independent topological generators {rl , . . . , 'Yd} of ro, and for 1 :$ i :$ d fix a lift .:Yi E Gal{Koo(P.poo• W)/Ko) of 'Yi such that the restriction of .::Yi to K(W) belongs to the center of Gal(K(W)/ K). Since Koo(P.poo, W) is the compositum of K(W) with an abelian extension of K, each .:Yi belongs to the center of Gal{Koo(P.poo, W)/K). Therefore these choices extend by multiplicativity to define a homomorphism ro ~ Gal{Koo(P.poo, W)/Ko),

7.5. GALOIS EQUIVARIANCE OF THE EVALUATION MAPS

149

whose image lies in the center of Gal(Koo(ILpoo, W)f K), which is a section for the projection map Gal(Koo(ILpoo• W)/Ko) -t ro. We will denote this map by 'Y t-+ .:Y, and we will use this map to define an action of r 0 on Gal(Lfnoo): for 'Y E Gal(Lfnoo) and 'TJ E ro, define

'Y 11

= ii'Yii- 1 •

This definition extends to give an action of Zp[[r0 ]] on Gal(Ljn 00 ). It is not canonical, since it depends on our choice of the i'i· By Lemma C.l.6, since Vis assumed to be irreducible, every element of the center of Gal(K(W)/K) acts on W by a scalar in ox. Thus the choice above defines a character

X:

ro --+Ox,

X(TJ)

= ij E Aut(W).

Similarly, if 'TJ E r 0 then ij belongs to the center of Gal(K(W*)/K) so we get a second character

x* :

ro --+ox,

x*(TJ) = ij E Aut(W*).

Suppose that c E H 1 (Koo, W), that 'Y E Gal(Lfnoo), and that TJ E ro. Since Ev('Y) E Hom(H 1 (K00 , W),D), we have (TJEv('Y))(c)

= Ev('Y)(TJ- 1c) = O((TJ- 1c)('Y)) = O(ij-1(c('Y.,.,))) = x(TJ-1)Ev('Y11)(c).

In other words,

Ev('Y11 ) = X(TJ)TJ(Ev('Y)), and similarly with Ev* and x*. This proves (ii). Recall the involution

'TJ t-+ 'TJ•

0

of A given by Definition 7.2.3.

Proposition 7.5.2. Suppose X' is a A-submodule of Xoo and Xoo/X' is pseudo-null. Then there is an ideal .Ao of height at least two in A such that for every KerF C Koo,

AoarAnnA(W°Koo )• AnnA(H1(n 00 jK00 , W)).Hom(H 1 (F, WM), OJMO) C 0EvF,M({Ev*)- 1(X') n Gnoo). In other words, if 1/J belongs to

AoarAnnA(W°Koo )• AnnA(H1(n 00 jK00 , W)).Hom(H 1 (F, WM), OJMO) then there are 'Y1, ... , 'Yk E Gnoo and c1, ... , Ck E 0 such that Ev* ('Yi) E X' for every i and k

L CiEVF,Mbi) = 1/J. i=1

7. IWASAWA THEORY

150

Proof. Let ro, L,

x, and x* be as in Proposition 7.5.1. We define

Twx : O[[ro]] ----+ O[[ro]] by "Y t-t

xbh

and similarly for Twx•, and then Proposition 7.5.1 shows that for every 17 E Zp[[ro]] and "Y E Gal(L/floo),

Ev(7 71 )

= Twx(TJ)(Ev("()),

Ev*("Y 71 )

= Twx•(TJ)(Ev*("f)).

(7.6)

Note that a pseudo-null A-module is also pseudo-null as a Zp[[ro]]module, and conversely if A is an ideal of Zp[[ro]] of height at least two then AA is an ideal of A of height at least two. Define A

= Tw;!(Anno([ron(Xoo/ X')) n Zp[[ro]].

Since Xoo/ X' is assumed to be a pseudo-null A-module, A is an ideal of height at least two in Zp[[ro]]. By (7.6), Ev*(AGal(L/0 00 ))

= Twx• (A)Ev*(Gnoo)

c X',

and by (7.6) and Lemma 7.2.4(ii), for every K CrF C K 00 , OEv(AGal(L/floo))

= OTwx(A)Ev(Gnoo)

:J Twx(A)aTAnnA(H 1 (floo/Koo, W))-.Hom(H 1 (Koo, W), D).

By Proposition 7.4.4(i) and (ii), the image of the composition Hom(H 1 (K00 , W),D) ----+ Hom(H 1 (F, W),D)

----+ Hom(H 1 (F, WM),O/MO) contains A:lobAnnA(W°Koo )•Hom(H 1 (F, WM), 0/MO). Combining these inclusions, we see that the proposition holds with

Ao = A:lobTwx(A), which has height at least two by Lemma 7.4.2.

0

Remark 7.5.3. When r e:! Zp there is a simpler proof of Proposition 7.5.2, which does not rely on the noncanonical construction of Proposition 7.5.1. In that case X 00 / X' is finite, so (Ev*)- 1 (X')nGnoo has finite index in Gnoo, so by Lemma 7.2.4(ii), 0Ev((Ev*)- 1 (X') n Gnoo) contains a subgroup of finite index (not a priori a A-submodule) of aTAnnA(H 1 (floo/K00 , W)).Hom(H 1 (K00 , W),D).

7.6. PROOF OF PROPOSITION 7.1.7

151

But every subgroup of finite index contains a submodule of finite index, and hence there is a j ~ 0 such that .

1

•

1

M 3 aTAnnA(H (f>.oo/K00 , W)) Hom(H (K00 , W),D)

c 0Ev((Ev*)- 1 (X') n Gn.,J where we recall that M is the maximal ideal of A. By Proposition 7.4.4{i) and {ii), A;IohAnnA(W°Kcc )• annihilates the cokernel of the map Hom{H 1 {K00 , W),D)

--t

Hom{H 1 {F, WM),D),

so the proposition is satisfied with Ao = Mi A;lob

7.6. Proof of Proposition 7.1.7 Proposition 7.1.7 is very easy to prove in the following (fairly common, see the examples of Chapter 3) special case. Suppose that hypotheses Hyp(K00 , T) are satisfied {so aT = 1 by Lemma 7.1.3{ii)), 0 = Zp, and H 1 {0. 00 / K 00 , W*) = 0. Use {7.1) to choose a sequence z1, ... , Zr E Xoo such that E9Azi ~ E9A/ fiA. By Lemma 7.2.4{iii), under our assumptions we have Ev*(rGn"") = Ev*{r) +Ev*(Gn"") = Ev*(r) +Xoo = X 00 , so Proposition 7.1.7 holds with these Zi and with tli = fiA. The rest of this section is devoted to the proof of Proposition 7.1.7 in the general case, which unfortunately is more complicated. We say that two ideals A and B of A are relatively prime if A + B has height at least two.

Lemma 7.6.1. The ideal char{Xoo) is relatively prime to each of the ideals AnnA{W°Kcc ), AnnA{H 1 {f200 /K00 , W)), AnnA{H 1 {f>.oo/Koo, W*))•. Proof. The proofs for all three ideals are similar. If W°Kcc is finite or if rankzp(r) > 1 then AnnA(W°Kcc) has height at least two and the first assertion holds trivially. We have assumed that V is irreducible over G K"", so if W°Kcc is infinite then GKcc acts trivially on T. Thus (using hypothesis Hyp(Koo/ K)) the first assertion follows from Lemma 7.3.4{i). The other two assertions follow similarly, using Lemma 7.3.4 and Corollary C.2.2. We sketch briefly the proof for the third ideal. Corollary C.2.2 applied toT*, with F = Koo and n = 0. 00 , gives three cases. In case {i), H 1 (0. 00 /K00 , W*) is finite, so AnnA{H 1 {0. 00 /K00 , W*)) has height at least two, and hence is relatively prime to everything. In case {ii) (resp. {iii)), GK acts on T* via a character p of r (resp. ccycP), and H 1 (0. 00 / KXJ, W*) has a subgroup Coffinite index on which GK acts via p.

152

7. IWASAWA THEORY

Then GK acts on T via ccycP- 1 (resp. p- 1 ), so AnnA(Ct :::> AnnA{T{-1)) (resp. AnnA(Ct :::> AnnA{T)). Since AnnA{H 1 {0oo/Koo, W*)) :::> AnnA{C)AnnA{H 1 {0 00 /K00 , W*)/C) and H 1 (0 00 /K00 , W*)/C is finite, the lemma in this case follows from Lemma 7.3.4. 0 Lemma 7.6.2. Suppose B is a torsion A-module and x,y E B. Suppose further that gz,gy E A are such that AnnA{x) C gzA and AnnA(Y) C gyA. Then there is an n E Z such that

AnnA{x + ny) C [gz,gy]A where [gz, gy] denotes the least common multiple of gz and gy. Proof. Suppose s.p is a {height-one) prime divisor of [gz,gy], and define s'lt

= {n E z: AnnA{X + ny) ct. s.pord'P[9zo9y]}.

Recall that p is the maximal ideal of 0. We will show that S'lt has at most one element if s.p =f. pA, and S'lt is contained in a congruence class modulo p if s.p = pA. Then it will follow that Z- U'ltS'lt is nonempty, and every n in this set satisfies the conclusion of the lemma. Suppose n, mE s'lt, and let A= AnnA(X + ny)nAnnA(X +my). Then A¢. s_pk, where k = ord'lt[gz,gy]· But (n- m)A annihilates both y and x, so (n - m)A C s.pk and we conclude that n - m E s.p. If s.p =f. pA it follows that n = m, and if s.p = pA then n m (mod p). This completes the proof. 0

=

Lemma 7.6.3. Suppose B is a finitely generated torsion A-module, and B is pseudo-isomorphic to $~= 1 A/ hiA, where hi+l I hi for 1 ~ i < k. Suppose we are given a subring Ao of A such that A is finitely generated as a Ao-module, a Ao-submodule B 0 C B, and an element t E B such that t and Bo generate B over A. Then there are elements x1 E t + Bo and x2, ... ,xk E Bo such that

(i) Ax1 ~ A/~1 where ~1 C h1A and h1A/~1 is pseudo-null, (ii) if 2 ~ j ~ k there is a split exact sequence 0 ---+ If t

j-1

j

i=1

i=1

L Axi ---+ L Axi ---+ A/hi A ---+ 0.

= 0 then we can replace (i)

(i') Ax1

~

by

A/h1A, i.e., (ii) holds for j

= 1 as well.

7.6. PROOF OF PROPOSITION 7.1.7

153

Proof. We will prove the lemma by induction on k. If A is an ideal of A then char( A/ A) is the unique principal ideal containing A with pseudo-null quotient. For every x E B 0 write A~:

= char(A/AnnA(x)).

By Lemma 7.6.2 (applied successively with x = t and y running through a sequence of elements of Bo) we can choose x1 E t + Bo such that Ax 1 C Ax for every x E t + Bo. Since t and Bo generate B over A, we must have Ax 1 = h1A, so (i) is satisfied. This proves the lemma when k = 1 and t =I= 0. If t = 0 then choose g E A0 , prime (in A) to h 1, which annihilates the pseudo-null A-module h 1A/ AnnA(xi), and replace x 1 by gx 1 • This element has annihilator exactly h 1A, so this completes the proof when k 1. If k > 1, choose x1 as above. Let B' = B / Ax1, let Bb be the image of Bo in B', and let t' = 0. Then B' is pseudo-isomorphic to $f=2 A/ hi A, so by the induction hypothesis (in the "t = 0" case) we can choose x2, ... ,xk E Bb leading to split exact sequences

=

j

j-1

0 ---+ LAxi ---+ LAxi ---+ Ajh1 A ---+ 0 i=2

i=2

if 2 ::; j ::; k. Now fori~ 2 choose Xi to be any lift of Xi to Bo. We claim the lemma is satisfied with this choice of x 1, ... , Xk. It will suffice to check that the exact sequences j

0

j

---+ Ax1 ---+ L Axi ---+ i=1

L Axi ---+ 0

(7.7)

i=2

split for 2 ::; j ::; k. Let ~ = AnnA(B). Then ~ C h 1A and h1 1 ~ is pseudo-null. By our induction hypothesis we can choose elements fh, ... , fik E 2 Axi such that Ayi ~ A/hiA for each i and Ayi = Axi. Let Yi be a lift of 2 2 'iii to 1 Axi. For each i we have hiYi E Ax1, say hiYi = CiX1. Then hi 1 ~ annihilates cix1, i.e., Ci~ C hi~ 1 , and we conclude that hi divides Ci. Now the map Yi 1-t Yi- (cdhi)x1 gives a splitting of (7.7). D

L::=

L::=

L::=

L::=

Proof of Proposition 7.1.7. Recall that we have a pseudo-isomorphism

fiJ'i=1A/ fiA ---+ Xoo. Define a A-submodule

Xo

= AEv*(r) + AEv*(Gnoo)

C Xoo.

154

7. IWASAWA THEORY

Then Xoo ::) Xo ::) Xo n arX00 , and Lemmas 7.2.4{iii) and 7.6.1 show that (arXoo)/(Xo n arXoo) is pseudo-null. Thus we can find a new injective pseudo-isomorphism ffir=1A/giA - t Xo with elements 9i E A satisfying, for every i, 9i+1

I 9i,

fi I argi,

9i

Ik

Apply Lemma 7.6.3 with B = Xo, with hi= 9i, with Bo = Ev*(Gn.,J, and with t = Ev*(r) to produce a sequence Xt, ... ,xn E Xo. (Note that B 0 satisfies the hypotheses of Lemma 7.6.3 with A0 = Twx• (Zp[[r 0 ]]), where ro and x* are as in Proposition 7.5.1, and Twx• is as in the proof of Proposition 7.5.2.) Define z1 = x1 E Ev*(rGnoo) and g1 = ~1· For 2 ~ i ~ r define Zi

= X1 + Xi

E Ev* (rGnoo), gi

= giA.

Lemma 7.6.3 shows that char(:E Azi) = 11 9i = char(Xo), so Xo/ :E Azi is pseudo-null. The other conclusions of Proposition 7.1.7 for these Zi and gi also follow immediately from Lemma 7.6.3. D Corollary 7 .6.4. Suppose z1, ... , Zk and g1, ... , gk are as in Proposition 7.1.7. If 1 ~ k ~ r then :E~= 1 Azi ~ ffif= 1A/gi and :E~= 1 Azi is a direct summand of :E~= 1 Azi. Proof. This follows easily by induction on k from Proposition 7.1.7{iii).

D

7. 7. Proof of Proposition 7.1.9 In this section we will prove Proposition 7.1.9, and thereby complete the proof of Theorems 2.3.3 and 2.3.4 begun in §7.1. Keep the notation of §7.1. In particular recall that r

Zoo =

r

L Azi ~ (B A/gi C Xoo

i=1 i=1 where the Zi and gi are given by Proposition 7.1.7. If u is a Selmer sequence of length k, as in Definition 7.1.8, define k

Zcr

=L

AEv*{ai) C Z 00 • i=1 Lemma 7.7.1. If u is a Selmer sequence of length k, then

Zcr ~ ffi~=1A/giA and Zcr is a direct summand of Zoo. If k < r and u' is a Selmer sequence of length k + 1 extending u, then Zcr' / Zcr ~ A/ gk+l·

7.7. PROOF OF PROPOSITION 7.1.9

155

Proof. Let Yk = L:~= 1 Azi· By Corollary 7.6.4, Yk e! EB~= 1 A/gi and there is a complementary submodule Y; C Zoo such that Yk EB Y; = Z 00 • The image of Za + Y; in Z 00 /MZ00 contains the image ofYk + Y; = Z 00 , so by Nakayama's Lemma Za + Y; = Z 00 • We will show that Zan Yk = 0, and thus Zoo = Za EB Y; and Za ~ Zoo/Y;

e!

Yk

e!

EB~=1A/giA.

If k < r and u' extends u, we can repeat the argument above with k replaced by k + 1. We can choose Y;+l to be contained in Y;, and then YUY;+l ~ A/gk+1 and

Za' EB Y£+1

= Zoo = Za EB Y;,

so

Za' ~ Za EB Y;/Y;+l = Za EB A/gk+l· It remains to show that Za

n Y; = 0. For 1 ~ i

Ev*(ai) where

vi

E MYk and

Wi

~

k write

= Zi +Vi+ Wi

E MY{ Suppose k

:~::::aiEv*(ai) i=1

E Y£

with ai E A; we need to show that L:~= 1 aiEv*(ai) = 0. Projecting into Yk we see that k

L ai(Zi +Vi) = 0.

(7.8)

i=1

Using Proposition 7.1.7(iii) (see also Corollary 7.6.4), fix YI. ... ,yk E Yk so that for 1 ~ i ~ k we have Yi

and Ayi tors, as

~

i

i

j=l

j=l

= L:Az; = ffiAy;

A/.gi. We can rewrite (7.8) in matrix form, using these genera(at, ... ,ak)A E (gtA, ... ,gkA)

where A is a k x k matrix with entries in A. Modulo M, we see that A is lower-triangular with invertible diagonal entries (since Zi E Yi, and the projection of Zi generates Yi/Yi-1 = Ayi, and the Vi vanish modulo M). Therefore A is invertible, and, since gi C gk for every i ~ k, we conclude

7. IWASAWA THEORY

156

that ai E gk for every i. But gk annihilates Yk because gk C gi fori so we deduce that k

LaiEv*(ai) i=l

~

k,

k

= LaiWi = 0. i=l

D

This completes the proof of the lemma.

Lemma 7.7.2. For every Selmer sequence u, every power M of p, and every K CrF C K 00 , the ideal AnnA(Xoo/ Zoo) annihilates the kernel of the map

Zcr ® AF,M --t Xoo ® AF,M · Proof. By Lemma 7.7.1, Zcr is a direct summand of Z 00 , so Zcr ® AF,M injects into Z 00 ® AF,M. If .:TF,M is the kernel of the map A --t AF,M, then the kernel of Zoo® AF,M --t Xoo ® AF,M is (Zoo n JF,MXoo)/.:TF,MZoo, which is annihilated by AnnA(Xoo/Z00 ). This proves the lemma. D For the rest of this section fix a field F such that K Cr F C K 00 • By Assumption 7.1.4, AF/ ftAF is finite. Fix a power of NF of p such that NF ~ IAF/!IAFI and such that NF is at least as large as the integer MF of Proposition 7.4.4(iv). Let Bo = (A;Job)•(Ai,.r )• AnnA(Xoo/ Zoo)· Corollary 7. 7 .3. If u is a Selmer sequence and M

~

N F is a power of p,

then B0 annihilates the kernel of the natural map Zcr®AF,M --t Hom(SEp(F,W_M),OfMO). Proof. The map in question is the composition Zcr®AF,M --t Xoo®AF,M --t Hom(SEp(F,W_M),O/MO). It follows from Assumption 7.1.4 that SEp (K00 , W*)°F is finite, so the corollary follows from Proposition 7.4.4(iv) and Lemma 7.7.2. D If t E n, recall that Ep~ denotes the set of primes of K dividing pt. Recall also from Definition 7.1.8 that II(k, F, M) is the set of Kolyvagin sequences of length k for F and M, and if1r = (q 1, ... ,qk) E IT(k,F,M) then t(1r) = ll~=l qi.

Lemma 7.7.4. Suppose that M is a power of p, that u is a Selmer sequence of length k, and that 1r is a Kolyvagin sequence corresponding to u. Then the map of Corollary 7.7.3 factors through a surjective map

Zcr ® AF,M

--*

Hom(SEp(F, W_M)/SEP•(F, W_M),O/MO).

7.7. PROOF OF PROPOSITION 7.1.9

157

Proof. Write u = {al,··· ,ak) and 1r = (ql,··· ,qk)· The image of Zcr in Hom(SEp(F, WM),O/MO) is Hom(SEp{F, WM)fB,OfMO), where B

=

n

ker(Evj;.,M(ai)'Y)

=

n

ker(Evj;.,M{FrQ)).

Q ofF

l = 0, Lemma 4.2.5(i) shows that T°KC•> = 0 for every t. Thus if we replace WM by 1l' = Maps(GK, T) in Proposition 4.4.5 we get a short exact sequence

0 ----t

'Jl'°F(•}

----t (11' /T)°F(•)

lifting c. Projecting this map to (WM /WM)°FC•> we can proceed exactly as in Definition 4.4.10 to define II:[F,~,MJ· Analogue of Theorem 4.5.1. We will use Corollary 4.6.5 instead of Theorem 4.5.1. Corollary 4.6.5 follows directly from Proposition 4.6.1, which is included as part of (ii')(b) and (ii')(c). (In §4.6 we used assumption (ii')(a) and Corollary B.3.5 to prove Proposition 4.6.1.) Analogue of Theorem 4.5.4. Theorem 4.5.4 follows directly from Lemma 4. 7.3, so it will suffice to prove a form of that lemma. Suppose first that (ii') (b) holds with an element 'Y E GK. Fix tq E 'R, a power M of p,

9.1. RIGIDITY

177

and a power M' of p divisible by MP('yiT; I). By the definition of,, we have P(riT; I) =F 0. Let n = IP.voo n Kl. Choose a prime l of K such that (a) Fr, ='Yon K(l)K(WM•,IJ.nM'• (0~.)1/(nM')), (b) Fr 1 =I on K(tq), (c) Fr, =F I on K(,VI(np)) where >-.OK = qh with h equal to the order of q in the ideal class group of K. (Exercise: show that these conditions can be satisfied simultaneously.) One can imitate the proof of Lemma 4.7.3 by using the extensions K(l)/K in place of the finite extensions of Kin K 00 • Condition (a) and the definition ' of 'Y ensure that nM' I [K(l) : K(I)]. Condition (c) ensures that the decomposition group of q has index dividing n in Gal(K(l)/ K), and therefore has order at least M'. The key point is that although cK{t) and cK{tq) are not "universal norms" from K(d) and K(tql} (as they would be from K 00 (t) and K 00 (tq)), the Euler system distribution relation shows that P(Fr[ 1 1T*;Fr[1 )cK{t) is a norm from K(d) and similarly with t replaced by tq. Conditions (a) and (b) imply that in O[Gal(K(tq)/K)], P(Fr[ 1 IT*;Fr( 1 )

= P(Fr[ 1 IT*; I)

:= P(r- 1 IT*; I)

(mod M)

= P(riT; I).

Now imitating the proof of Lemma 4. 7.3 one can show that, with notation as in the statement of that lemma, if tq E nK,M' then P('yiT; I)(Nqrd(xF{tq))- P(Fr; 1 IT*; Fr; 1 )'Yd(xF(t)))

=0

E WM'

This suffices to prove that II:[F,t,M] and II:[F,tq,M] satisfy the equality of Theorem 4.5.4. Now suppose (ii')(c) holds. In §4.8 we used Lemma 4.7.3 to prove the congruence of Corollary 4.8.1. Under the assumptions (ii')(c) we can just reverse the argument to prove Lemma 4.7.3, and then Theorem 4.5.4. Example 9.1.1 (cyclotomic units revisited). With this expanded definition, we can redefine the cyclotomic unit Euler system of §3.2. Namely, for every m > I prime to p define Cm = ((m -I)((,; 1 -I) E (Q(p.m)+)x C H 1 (Q(p.m)+,zp(I))

and set c1 = 1. This collection is not an Euler system, even under our expanded Definition 2.1.1. (If it were, then for every prime l =F p, the class CQ(~£l) would belong to sEp(Q(p.t), Zp(I)), but this is not the case because CQ(~£l) ¢ Hj(Q(p.t)t, Zp(I)).) However, suppose x : GQ -t is a nontrivial character of finite order, and its conductor f is prime top. Then we can twist c by x- 1 as in Definition 2.4.I, and with the modified definition above, the collection

ox

178

9. VARIANTS

c = cx- 1 is an Euler system for (Zp(1)®x- 1 ,Qab,P,fp), where Qab,p is the maximal abelian extension of Q unramified outside p. Namely, although condition (ii')(a) does not hold, (ii')(b) (with 'Y E GQ(I-'poo) and X('Y) =F 1) and (ii')(c) (see Example 4.8.2) both do hold. With this Euler system we can remove one of the hypotheses from Theorem 3.2.3 and Corollary 3.2.4. With notation as in §3.2 (so L is the field cut out by x), we have the following theorem. Theorem 9.1.2. Suppose p > 2 and conductor prime to p. Then

x is

a nontrivial even character of

IA11 = [£f: CL,x]· Sketch of proof. If x(p) =F 1 this is Corollary 3.2.4. So we may assume that the conductor of x is prime to p and use the Euler system constructed above. For this Euler system, c1 generates CL,x• so exactly as in the proof of Theorem 3.2.3 we deduce from Theorem 2.2.2 that

ISEp(Q, (QpfZp) ® x)l divides [£f: CL,x1· Unfortunately, Proposition 1.6.1 shows that SEp(Q, (QpfZp) ® x) = Hom(A1JP,D) where P is the subgroup of A! generated by the classes of primes of L above p. This is not quite what we need. To complete the proof, we observe that the derivative classes K[K,r,M] attached to our Euler system all lie in sE~(Q,JLM ® x- 1 ), not just in sE~p(Q,JLM ® x- 1 ) as Theorem 4.5.1 shows in the general case. (This follows from the fact that CQ(p~) E S(Q(JLr), Zp(l) ® x- 1 ) for every r. See for example [Ru3] Proposition 2.4.) Therefore we can repeat the proof of Theorem 2.2.2, but using Eo = 0 and E =Erin Theorem 1.7.3 instead of Eo= {p} and E = ErP• to conclude that IA11 = IS(Q, (QpfZp) ® x)l divides [£f : CL,x1· Now the equality of the theorem follows from the analytic class number formula exactly as in Corollary 3.2.4. 0 9.2. Finite Primes Splitting Completely in K 00 / K Definition 2.1.1 of an Euler system requires a Z~-extension K 00 / K, with Koo C K., such that no finite prime splits completely in Koo/K. In fact, the assumption that no prime splits completely is unnecessarily strong. We can remove this hypothesis if we assume instead that

9.3. EULER SYSTEMS OF FINITE DEPTH

179

{*) for every prime q of K which splits completely in Koo/K, and for every finite extension F of KinK, we have (cF)q E H~r{Fq,T). The set of primes which split completely in K 00 / K has density zero, so such primes do not interfere with our Tchebotarev arguments. Using this fact and (*), the proofs in Chapters 4 through 7 work without significant modification in this more general setting. 9.3. Euler Systems of Finite Depth

Definition 9.3.1. Fix a nonzero ME 0. An Euler system for WM {or an Euler system of depth M) is a collection of cohomology classes satisfying all the properties of Definition 2.1.1 except that instead of c F E H 1 ( F, T) we require CF E H 1 {F, WM)· An Euler system in the sense of Definition 2.1.1 can be viewed as an Euler system of infinite depth. Remark 9.3.2. For this definition we could replace WM by a free 0/MOmodule of finite rank with an action of GK; it is not necessary that it can be written as T f MT for some T. The construction of the derivative classes II:.[F,r,MJ in Chapter 4 only used the images of the classes CL (for various L) in H 1 {£, WM)· Thus if c is an Euler system for WM then we can define the classes II:.[F,r,M) exactly as in Chapter 4. The proof of Theorem 4.5.4 also only used the images of the Euler system classes in H 1 {£, WM), so that theorem still holds for the derivative classes of an Euler system for WM. However, the proof of Theorem 4.5.1 used the images of the Euler system classes in H 1 (L,T), so that proof breaks down in this setting. However, as discussed in §9.1 above {and see Remark 4.6.4), we can still prove a weaker version of Theorem 4.5.1, and this will suffice for some applications. For example, the proofs in Chapters 4 and 5 will prove the following theorem. Keep the setting and notation of Chapter 2 (so in particular, for simplicity, WM = TfMT). Theorem 9.3.3. Suppose ME 0 is nonzero and c is an Euler system for W M. Suppose that Hypotheses Hyp(K, T) hold, that the error terms nw and of Theorem 2.2.2 are both zero, and that = 0. Let

nw

a1

=

n

wzx

Anno(Wiq/(Wiq)div)

primes q of K qfp

and let a2 C 0 be the annihilator of a1 cK in H 1 (K, WM)· Then

(M/a2)a1SEp(K, WM-)

= 0.

9. VARIANTS

180

In particular if «1 CK

# 0 then SEp (K, W*)

is finite.

Remark 9.3.4. The ideal a1 of Theorem 9.3.3 is finite, since wz I (WI)div is finite for all q and is zero if T is unramified at q. See the proof of Corollary 4.6.5. One could reformulate Theorem 9.3.3 for a general G K-module W which is free of finite rank over OIMO, i.e., one which does not come from a "T", but one would have to redefine the Selmer group since our definition depends on T, not just on WM. 9.4. Anticyclotomic Euler Systems The "Euler system of Heegner points", one of Kolyvagin's original Euler systems, is not an Euler system under our Definition 2.1.1. If one tries to make the definition fit with K = Q, the problem is that the cohomology classes (Heegner points) are not defined over abelian extensions of Q, but rather over abelian extensions of an imaginary quadratic field which are "anticyclotomic" (and hence not abelian) over Q. On the other hand, if one tries to make the definition fit by taking K to be an appropriate imaginary quadratic field, then the problem is that the Heegner points are not defined over large enough abelian extensions of K, but only over those which are anticyclotomic over Q. We will not discuss Heegner points in any detail (see instead [Ko2], [Ru2], or [Gro2]), but in this section we propose an expanded definition of Euler systems that will include "anticyclotomic" Euler systems such as Heegner points as examples. Fix a number field K and a p-adic representation T of G K as in §2.1. Suppose d is a positive integer dividing p - 1, and X : G K -t is a character of order d. Let K' = .i(ker(x) be the cyclic extension of degree d of K cut out by X· For every prime q of K not dividing p let K' (q)x denote the maximal p-extension of K' inside the ray class field of K' modulo q, such that Gal(K' I K) acts on Gal(K'(q)xl K') via the character X· Similarly, let K'(l)x denote the x-part of the maximal unramified p-extension of K'. Now suppose IC' is an (infinite) abelian p-extension of K' and N is an ideal of K divisible by p, the conductor of x, and all primes where T is ramified, and such that IC' contains K'(q)x for every prime q of Knot dividing N.

z;

Definition 9.4.1. A collection of cohomology classes c

= {cp E H 1 (F,T): K'ctF c IC'}

9.4. ANTICYCLOTOMIC EULER SYSTEMS

181

is a x-anticyclotomic Euler system for (T, JC', .N) (or simply for T) if (i) whenever K' CrFCrF' C JC', then CorF'fF(cp•)

=(

IJ

P(Fr~ 1 IT*;Fr~ 1 ))cp

qEE(F'/F)

where YJ(F' I F) is the set of primes of K not dividing N which ramify in F' but not in F, as always Fr q is a Frobenius of q in G K, and P(Fr~ 1 IT*; x) = det(1- Fr~ 1 xiT*) E O[x], (ii) at least one of the following analogues of the hypotheses (ii') of §9.1 holds: (a) JC' contains a z;-extension K:X, of K' such that no finite prime splits completely in K:X,I K', and Gal(K' I K) acts on Gal(K:X,I K') via x, (b) CK·(~)x E sEp(K'(t)x,T) for every t, and there is a 'Y E GK such that ccyc('y) = x('y), and T'Y= 1 = 0, and 'Yd = 1 on K'(1)x(P.p"", (0~, ) 1/P"" ), (c) cK(~) E sEp (K(t), T) for every t, the classes {cp} satisfy the appropriate analogue of the congruence of Corollary 4.8.1, and for every q not dividing N, and every power n of p, Fr~ -1 is injective on T. Remark 9.4.2. If d = 1, then X is trivial, K' = K~ and thus a x-anticyclotomic Euler system for T is the same as an Euler system for T in the sense of Definition 2.1.1 (or §9.1). Suppose K = Q and xis an odd quadratic character, sod= 2. Then K' is an imaginary quadratic field and JC' is an anticyclotomic p-extension of K'. If T is the Tate module of a modular elliptic curve, and we make the additional assumption that x(q) = 1 for every prime q dividing the conductor of x, then the Heegner points in anticyclotomic extensions of K' give a x-anticyclotomic Euler system forT. (One must modify the Heegner points slightly, as in §9.6 below, to obtain the correct distribution relation.) Note that in this situation we could take JC' to contain the anticyclotomic Zv-extension K:X, of K'. However, all rational primes which are inert in K' split completely \n K:X,I K' so condition (ii)(a) of Definition 9.4.1 fails. However, both (ii)(b) and (ii)(c) hold.

Let For every i E Z let indo(c,xi)

= sup{n: c~,

E pnH 1 (K',T)

+ H 1 (K',T)tors} ~

oo,

182

9. VARIANTS

where c~, denotes the projection of CK• into the subgroup H 1 (K', T)X; of H 1 (K',T) on which Gal(K'/K) acts via xi. In this setting of anticyclotomic Euler systems one can prove the following theorem. Theorem 9.4.3. Suppose c is a x-anticyclotomic Euler system for T. Suppose further that H 1 (0'/K';W) = H 1 (0'/K', W*) = 0, that T ® k is an irreducible k(GK·]-module, and that there is aTE GK such that

• ccyc{T) = X(T), • Td is the identity on K' {1) x (J.Lpoo , {0~,) 1I p"" ) , • if ( E J.Ld Cox then Tj(r- ()Tis free of rank one over 0. Then for every i, pindo{c,x;)SEP(K', W*)Xl-i

= 0.

Remark 9.4.4. In the setting of the Heegner point Euler system mentioned in Remark 9.4.2, K' is an imaginary quadratic field, and we can take T to be a complex conjugation. Theorem 9.4.3 then says that the "minus part" of the Heegner point in E(K') controls the "plus part" of the Selmer group of E over K', and vice versa.

Sketch of proof. Given a x-anticyclotomic Euler system and a power M of p, one can proceed exactly as in §4.4 to define derivative classes

II:[K' .~.M] E H 1 (K', W M) for every t E 'R,K' ,M,r. where 'RK' ,M,r is the set of squarefree ideals of K divisible only by primes q such that q f .N and such that the Frobenius of q in Gal(K'(1)x(J.LM, (0~,) 1 /M, WM)/K is (conjugate to) r. These classes satisfy analogues of Theorems 4.5.1 and 4.5.4, and can be used along with global duality {Theorem 1.7.3) to bound the appropriate Selmer group. The main difference between the case of trivial x (i.e., Theorem 2.2.2) and nontrivial x is the way the powers of x appear in the statement of Theorem 9.4.3. This is due to the "anticyclotomic" version of Theorem 4.5.4, which states that for tq E 'RK' ,M,r. we have

loc:{II:[K',~q,MJ) = cf>£ 8 (11:[K',~,MJ) where cf>£ 8

:

Hj(K~, WM)

-t

H!(K~, WM)·

As usual we write Hj(K~, WM) = EflvlqH}(K~, WM) and similarly for Hi(K~, WM), so that both are Gal{K'/K)-modules. However, cf>£ 8 is not Gal(K' / K)-equivariant; for q E 'R,K' ,M,r. one can show that

A.!B(Hf1(K'q• wM )Xi) c H 81(K'q• wM )Xi-l ·

'f'q

9.5. ADDITIONAL LOCAL CONDITIONS

183

Thus, taking t = 1 and letting q vary, we obtain a large collection of classes in H 1 (K', W M p:i-l, ramified at only one prime of Knot dividing p, whose ramification is expressed in terms of c~,, and these classes can be used to annihilate classes in Sr;p(K', W*)X 1 - i . This is how Theorem 9.4.3 is proved. D Remark 9.4.5. To prove an analogue of Theorem 2.2.2 and bound the order of the various components of Sr;P (K', W*), we would need to proceed by induction as in Chapter 5. Unfortunately this is not at all straightforward, because at each step of the induction we move to a different component. We will not attempt to formulate, much less prove, such a statement here. In the case of Kolyvagin's Euler system of Heegner points, the induction succeeds by using the fact that T* ~ T. When d > 2 there is no obvious property to take the place of this self-duality. Also, if d = 2, then x takes values ±1, so if L is an abelian extension of K' it makes sense to ask if Gal(K' I K) acts on Gal(LI K') via X· When d > 2, this only makes sense when L I K' is a p-extension. This is sufficient to discuss and work with Euler systems, but it raises the question of whether one should expect x-anticyclotomic Euler systems with d > 2 to exist. 9.5. Additional Local Conditions

Inspired both by work on Stark's conjectures (see for example [Grot] or [Ru6]) and by the connection between Euler systems and £-functions (see Chapter 8), we now allow the imposition of additional local conditions on Euler system cohomology classes. Suppose E and E' are disjoint finite sets of places of K. If A is T, W, WM, T*, W* or WM-, define S~,(K,A)

= ker(SE(K,A) -tE9ver;•H1 (Kv,A))

and similarly with K replaced by a finite extension. For example, S~, (K, T) consists of all classes c E H 1 (K, T) satisfying the local conditions • cv E H}(Kv,T) ifv ¢ EUE', • Cv = 0 if V E E', • no restriction for v E E.

Definition 9.5.1. Suppose cis an Euler system for (T,K-,.N), and E is a finite set of primes of K not dividing p. We say c is trivial at E if CF E S~P (F, T) for every F. If an Euler system is trivial at E, we can use it to bound the Selmer groupS~p (K, W*). The proof will be the same as the original case where E is empty, once we have the following strengthening of Theorem 4.5.1.

9. VARIANTS

184

Theorem 9.5.2. Let E be a finite set of primes of K not dividing p. If c is an Euler system forT, trivial at E, then the derivative classes ~~;[F,~,MJ constructed in §4.4 satisfy

1\;[F.~.MJ

E

s~P•(F, wM)·

Proof. By Theorem 4.5.1, we only need to show that (~~;[F,~,MJ)q = 0 if q E E. The proof is similar to that of Theorem 4.5.1 in §4.6. We use the

notation of that proof. Fix a lift d: XF(~) -t WM /WM of cas in Proposition 4.4.8 and write dq for the image of din Hom(XF(~)• WM/IndgK(WM)) in the diagram of Lemma 4.6.7. Then dq is a lift of c in the sense of Proposition 4.6.8, but so is the zero map, since (cF(~))q = 0. Therefore the uniqueness portion of Proposition 4.6.8 shows that dq E image(Hom(XF(~)• W:.:!•>)), and from this it follows without difficulty, as in the proof of Theorem 4.5.1, that (~~;[F.~,MJ)q = 0. D The following analogue of Theorem 2.2.2 (using the same notation) is an example of the kind of bound that comes from using an Euler system which is trivial at E. Theorem 9.5.3. Suppose that p > 2 and that T satisfies Hyp(K, T). Let E be a finite set of primes of K not dividing p. If c is an Euler system for T, trivial atE, then lo(S~p (K, W*)) :5 indo(c)

+ nw + nw

where

nw nw

= lo(H 1 (0fK,W)nS~P(K,W)), = lo(H 1 (0/K, W*) nSEp(K, W*)).

Proof. The proof is identical to that of Theorem 2.2.2, using Theorem 9.5.2 instead of Theorem 4.5.1. D

Remarks 9.5.4. There are similar analogues of the other theorems of Chapter 2, bounding S~P (K, W*) and S~ (Koo, W*). up By taking E to be large, we can ensure that the error term nw in Theorem 9.5.3 is small. In the spirit of Chapter 8, if we think of Euler systems as corresponding to p-adic £-functions, then an Euler system which is trivial at E corresponds to a p-adic £-function with modified Euler factors at primes in E. As in [Grol] §1 (where our E is denoted T), these Euler factors can be used to remove denominators from the original p-adic £-function (see Remark 8.2.5 and Conjecture 8.2.6).

9.6. VARYING THE EULER FACTORS

185

9.6. Varying the Euler Factors It may happen that one has a collection of cohomology classes satisfying distribution relations different from the ones in Definition 2.1.1. Under certain conditions one can modify the given classes to obtain an Euler system. Return again to the setting of §2.1. We fix a number field K and a p-adic representation T of GK. Suppose X:. is an abelian extension of K and N is an ideal of K divisible by p and all primes where T is ramified. If KCrFCrF' C X:., let E(F'IF) denote the set of primes of Knot dividing N which ramify in F' I K but not in F I K.

Lemma 9.6.1. Suppose {!q E O[x] : q f N} and {gq E O[x] : q f N} are two collections of polynomials such that fq(x) gq(x) (mod N(q) -1) for every q, and suppose {cF E H 1 ( F, T) : K Cr F C K} is a collection of cohomology classes such that if KcrFCrF' C X:., then

=

CorF'fF(cF') = (

II

Jq(Fr; 1))cF. qEE(F'/F)

Then there is a collection of classes {CF E H 1(F, T) : K Cr F C K} satisfying the following properties.

(i) If K Cr F Cr F' C X:., then CorF'fF(cF•)

II

=(

gq(Fr; 1))cF.

qEE(F' /F)

(ii) If KerF C X:. and FIKis unramified outside N, then CF

= CF·

(iii) Suppose K Cr F C X:. and X is a character of Gal(FI K) of conductor f. If every prime which ramifies in F I K divides Nf, then

L

X('Y)'YCF

-yEGaJ(F/K)

=

L

X('Y)'YCF.

-yEGal(F/K)

Proof. If K CrF C X:. let E(F) = E(FIK), and if Sis a finite set of primes of K let Fs be the largest extension of Kin F which is unramified outside Sand N. If q f N let dq = gq(Fr; 1)- Jq(Fr; 1). For every F define

_ CF -

II

"" TiqeE(F)-S dq ( f q (Fr-1))LJ [F . F ] q CF5 • SCE(F) . S qES-E(Fs)

(Let Iq(FIK) denote the inertia group of q in Gal(FI K). Then Gal(FI Fs) is generated by {Iq : q E E(F)- S}, and IIql divides (N(q)- 1) in 0, so

9. VARIANTS

186

[F: Fs] divides Tiqei:(F)-s(N(q) -1). Since dq E (N(q) -l)O[Gal(F/K)], the fractions above belong to O[Gal(F/K)].) With this definition, (ii) is clear. Assertion (iii) (of which (ii) is a special case) also holds, because if every prime which ramifies in FfK divides .Nf, and if S is a proper subset of E(F), then X does not factor through Gal(Fs/K) and so E-reGal(F/K) xhhcFs = 0. For (i), observe that for every S, we have F8 n F = Fs. Thus, using the diagram

we see that CorF'/F(cF}) = CorF}F/FCorF'/F}F(cF}) = [F': F~F]CorF}F/F(cF}) [F': F] c (- ) [F': F] ( II f q (Fr-1))= [F' : F] orF}/Fs CF} = [F' : F] q CF s S s 8 qei:(F}/Fs)

5 ,

and so CorF' /F(cF') as

= Esci:(F') asCF

5

= Tiqei:(F')-S dq ( [F' · F'] . S

IT

II

f, (Fr-1)) '[F' : F] ( f, (Fr-1)) q q [F' · F ] q q qES-I:(F}) S . S qEI:(F}/Fs)

II

_ Tiqei:(F')-S dq

Jq(Fr~1). qES-I:(Fs)

[F:Fs]

-

where

Since Fs = Fsni:(F)• we can group together those sets S which have the same intersection with E(F), and we get a new expression CorF'/F(CF•) =

L

SCI:( F)

bsCF5

where bs

= =

S'CI:(F'/F)

Tiqei:(F')-S-8' dq [F: Fs]

TiqeE(F)-S dq [F: Fs]

---'~~-:--

II

II

qESUS' -I:(Fs)

Jq(Fr~1) qES-E(Fs)

9.6. VARYING THE EULER FACTORS

Since

L ( II

II fq(Fr;1) = II

dq)

S' CE(F' I F) qEE(F' I F)-S'

(dq qEE(F' I F)

qES'

187

+ fq(Fr;1n

II

9q(Fr;1), qEE(F' /F)

= nqEE(F' /F) 9q(Fr; 1)cF as desired.

we conclude that CorF' fF(CFt)

0

Example 9.6.2. Suppose that K = Q, that fq(x) = 1 - x, and that gq(x) = 1-q- 1x. Then fq(x) gq(x) (mod (q-1)Zp) for every q # p. By applying Lemma 9.6.1 with these data to the collection {cp E H 1(F, Zp)} constructed in Definition 3.4.2, we obtain an Euler system for Zp.

=

Lemma 9.6.3. Suppose {!q(x) E O[x, x- 1] : q f .N} is a collection of polynomials, {uq E q f .N} is a collection of units, dE Z, and

ox :

{cF E H 1 (F,T): KerF eX::} is a collection of cohomology classes such that if KerF er F' e X:: then

For each q define gq(x) collection of classes

II

=(

CorF' fF(cF')

fq(Frq) )cF. qEE(F'/F)

= uqxdfq(x- 1)

E O[x,x- 1). Then there is a

{cF E H 1 (F,T): KerF eX::} such that

(i) for all F and F' as above, CorF' fF(cF')

=(

II

gq(Fr; 1) )cF,

qEE(F' /F)

(ii) for every finite extension F of K unramified outside .N, CF

= CF.

Proof. For every F define cF

=(

II

uqFr;d)cF qEE{F/K)

where we fix some Frobenius Frq E Gal(Kab / K) (previously we always had Frq acting through an extension unramified at q). Then it is easy to check that this collection has the desired properties. 0

9. VARIANTS

188

Corollary 9.6.4. Suppose {cF E H 1 (F,T): KerF C ,q is a collection of cohomology classes such that if K Cr F Cr F' C JC, then

CorF' JF(cF')

IT

=(

P(Fr~ 1 IT; Frq) )cF.

qEE{F'/F)

Then there is an Euler system {CF} for (T, JC, N) such that for every finite extension F of K unramified outside N, CF

=

CF.

Proof. This will follow directly from the previous two lemmas. For every q we have P(Fr~ 1 IT;x- 1 ) = det(l- Fr~ 1 x- 1 IT) = det(l- N(q)- 1 Frqx- 1 IT*)

= (-N(q))-d det(FrqiT*)x-d det(l- N(q)Fr~ 1 xiT*) where d = rankoT. Thus if we first apply Lemma 9.6.3 with fq

= P(Fr~ 1 IT;x),

uq

= (-N(q))ddet(FrqiT*)- 1 ,

and then apply Lemma 9.6.1 with fq

= P(Fr~ 1 IT*;N(q)x),

we obtain the desired Euler system.

gq

= P(Fr~ 1 IT*;x), 0

Remark 9.6.5. Hone has a collection of cohomology classes satisfying the "wrong" distribution relation, one can either modify the classes as we did above, to get an Euler system, or else one can keep the given cohomology classes and modify the proofs in Chapters 4 through 7 instead.

APPENDIX A

Linear Algebra Suppose for this appendix that 0 is a discrete valuation ring. Let fo(B) denote the length of an 0-module B. A.l. Herbrand Quotients Suppose a, /3 E O[x] are nonzero. Definition A.l.l. If S is an O[x]-module and af3S aS C sf3=0,

= 0, then

{3S C sa=O,

and we define the {additive) Herbrand quotient h(S)

= fo(s 13=0 laS) -L0 (sa=olf3S)

if both lengths are finite. Example A.1.2. If S sf3=0

so h(S)

= O[x]laf30[x] then

= aS = a0[x]laf30[x],

sa=o

= {3S = {30[x]laf30[x],

= 0.

Proposition A.1.3. {i) If S is an O[x]laf30[x]-module and fo(S) is finite, then h(S) = 0. (ii) If 0 --+ S' --+ S --+ S" --+ 0 is an exact sequence of O[x]laf30[x]modules and two of the three Herbrand quotients exist, then the third exists and h(S)

= h(S') + h(S").

Proof. This is a standard fact about Herbrand quotients, see for example [Se3] §VIII.4. If a = (xn - 1)l(x- 1) and {3 = x- 1, and if G is a cyclic group of order n with a generator which acts on S as multiplication by x, then H0( G' S) = s/3=0 I aS and H1( G, S) = sa=O I {3S.

For completeness we sketch a proof in our more general setting. 189

A. LINEAR ALGEBRA

190

Assertion (i) follows from the exact sequences 0 0

---?

sf3=0 I aS

---?

sa=O

---?

---?

S ~ aS

s I aS ...f!...t so=O

---?

---?

0,

sc.=O I {3S

---?

0.

For (ii), multiplication by {3 induces a snake lemma exact sequence 0 -t stf3=0 -t sf3=0 -t snf3=0 ~ S' I {3S' -t s I {3S -t S" I {3S" -t 0. This gives rise to a commutative diagram 0 - - - t coker(1/J) - - - t SI {3S - - - t S" I {3S" - - - t 0

0 ---t

S'f3=0

---t

sf3=0

ker('lj;)

---t

---t

0.

Applying the snake lemma again gives an exact sequence

o ---?

A

---?

sa=o 1f3S

---? ---?

s"a=o 1f3S" stf3=0 I aS'

---?

sf3=0 I aS

---?

B

---?

0

where A and Bare defined by the exact sequence 0

---?

B

---?

snf3=0 I aS" ~ S'C 1 write

an = Gal(Q(I-'n)/Q) and a;t = Gal(Q(I-'n)+ /Q). > 2,

and let w be the Teichmii.ller character giving the action of GQ on l'p. Suppose 0 is the ring of integers of a finite extension of Qp, and X: GQ -t is a nontrivial even character of finite order, unramified at p. If m is the conductor of x then

Lemma D.2.2. Suppose p

ox

= 2r(k)( -27ri)-k L(x- 1wk, k) x { 1-x_(pP~~:x(p)

if P -If k, ifp -11 k.

Proof. We have Dk1i'Jn((p- 1)

If (

= Dk logh'Jn((p- 1)- pk- 1 Dk logh!':!"p'Y(O).

= (p or ( = 1, then

Dk log h'Jn((- 1)

= =

L

{3k Dk logu'Jn((.B- 1) + {3k Dk logu'Jn((f3- 1) .BE(Z; hors uEGal(Q(I'mp)/Q(I'm)+)

Thus by Lemma D.2.1, writing Lr(x- 1 wk, s) for the Dirichlet £-function with Euler factors for primes dividing r removed,

=

L

x-lwk('y) ( Dk logu'Jn((J- 1)- pk-l Dk logu!':!"P'Y(O)) 'YEamp = (-1)k-l r(k)(27ri)-kpk(1 + (-1)kx-lwk( -1))x(p)Lmp(X- 1 Wk, k)

- Pk-1(-1)k-lr(k)(27ri)-k(1

+ (-1)k)x(p)Lm(x-l,k)

L wk('y).

'YEap

218

D. p-ADIC CALCULATIONS IN CYCLOTOMIC FIELDS

Note that x- 1wk(-1) = (-1)k. If (p-1) I k then wk = 1, and the formula above simplifies to 2f(k)( -211"i)-kx(p)L(x-1' k)( -pk(1- x-1(p)p-k) + (p- 1)pk-1) = 2r(k)( -27Ti)-k L(x- 1, k)(1- pk- 1 x(p)). If (p- 1) f k then "E-reap wk('y) = 0, so in that case

L

-rea;\:.

x- 1 ('Y)Dk1l~((p -1)

= -2r(k)( -211"i)-kpkx(p)L(x- 1 wk, k).

o

Bibliography [AI] [BK] [Bo] [BFH] [CW] [Co] [DR] [dSJ [FI] [FPR]

[Fr] [Gi] [Gr1] [Gr2] [Gr3] [Gro1] [Gro2] [GZ] [Iw1]

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[Ru5) - - - : The "main conjectures" of Iwasawa theory for imaginary quadratic fields, Invent. math. 103 (1991} 25-68. [Ru6) - - - : Stark units and Kolyvagin's "Euler systems", J. fii.r die reine und angew. Math. 425 (1992} 141-154. [Ru7) ___ : p-adic £-functions and rational points on elliptic curves with complex multiplication, Invent. math. 107 (1992} 323-350. [Ru8) - - - : A Stark conjecture "over Z" for abelian £-functions with multiple zeros, Ann. Inst. Fourier {Grenoble) 46 (1996} 33--£2. [Ru9) - - - : Euler systems and modular elliptic curves, in: Galois representations in arithmetic algebraic geometry, A. J. Scholl and R. L. Taylor, eds., London Math. Soc. Lect. Notes 254 Cambridge: Cambridge Univ. Press (1998) 351-367. [RW] Rubin, K., Wiles, A.: Mordell-Weil groups of elliptic curves over cyclotomic fields, in: Number Theory related to Fermat's Last Theorem, Prog. in Math. 26, Boston: Birkhauser (1982) 237-254. [Schn) Schneider, P.: p-adic height pairings, II, Inventmath. 79 (1985) 329-374. [Scho) Scholl, A.: An introduction to Kato's Euler systems, in: Galois representations in arithmetic algebraic geometry, A. J. Scholl and R. L. Taylor, eds., London Math. Soc. Lect. Notes 254 Cambridge: Cambridge Univ. Press (1998} 379-460. [Se1) Serre, J-P.: Classes des corps cyclotomiques (d'apres K. Iwasawa), Seminaire Bourbaki expose 174, December 1958, in: Seminaire Bourbaki vol. 5, Paris: Societe Math. de France (1995) 83-93. [Se2) ___ : Cohomologie Galoisienne, Fifth edition. Lecture Notes in Math. 5, Berlin: Springer-Verlag (1994). [Se3) ___ : Corps Locaux, 2nd edition. Paris: Hermann (1968). [Se4) ___ : Proprietes galoisiennes des points d'ordre fini des courbes elliptiques, Invent. math. 15 (1972) 259-331. [Sh) Shatz, S.: Profinite groups, arithmetic and geometry, Annals of Math. Studies 67, Princeton: Princeton Univ. Press (1972). [Si) Silverman, J.: The arithmetic of elliptic curves, Graduate Texts in Math. 106, New York: Springer-Verlag (1986). [T1) Tate, J.: Duality theorems in Galois cohomology over number fields, in: Proc. Intern. Gong. Math., Stockholm (1962} 234-241. [T2] ___ : Global class field theory, in: Algebraic Number Theory, J. W. S. Cassels and A. Frohlich, eds., London: Academic Press (1967} 162-203. [T3] ___ : Algorithm for determining the type of a singular fiber in an elliptic pencil, in: Modular functions of one variable (IV), Lecture Notes in Math. 476, New York: Springer-Verlag (1975) 33-52. [T4) ___ : Relations between K2 and Galois cohomology, Invent. math. 36 (1976) 257-274. [T5] ___ : Les conjectures de Stark sur les fonctions L d'Artin ens= 0, Prog. in Math. 47, Boston: Birkhauser (1984). [Th) Thaine, F.: On the ideal class groups of real abelian number fields, Annals of Math. 128 (1988} 1-18. [Wa) Washington, L.: Introduction to cyclotomic fields, Graduate Texts in Math. 83, New York: Springer-Verlag (1982}. [Wi) Wiles, A.: Higher explicit reciprocity Jaws, Annals of Math. 107 (1978} 235-254.

Index of Symbols Chapter 1 K p 0

GK

T D

v w

WM

9 9 9 9 9 9 9 9 9 9

Op

10

ecyc

10 10 10 10

0(1)

T*

v• w•

Tp(A) I Kur Fr H~.(K, ·)

H}(K, ·) H!(K, ·) (

)div

( , )K

M

KE SE(K, ·) sE(K, ·) S(K, ·) Iv LM O:x

D:x :x 'Dw B" B:X

27 28 28 28 31

Ep lacE loc~,Eo

locf,Eo loc•Ep

Chapter 2 OK

K(q) Frq P(Fr;J 1 IT*; x) Cr

lC

.N Koo c E(F'/F)

10 10 12 12 12 12 14 14 14 18 20 21 21 21 21 22 22 24 24 24 24 25 25

k(t) 1

K(l) F(t) ICmin

p k

Hyp(K,T) Hyp(K,V) indo

lo()

n

K(W)

nw n•w

AF

I'

A char( ) Hyp(Koo/K) Hyp(Koo,T) 223

33 33 33 33 33 34 34 34 34 34 35 35 35 35 35 36 36 37 37 37 37 37 37 37 37 40 40 40 40 41 41

INDEX OF SYMBOLS

224

Hyp(Koo,V) SEp(Koo, W*) H~(K,T) Xoo

CK,oo indA(c) S(K 00 , W*) H~ •• (Kp,T) ~X eX

F

41 41 41 41 41 41 42 42 44 44

Chapter 3 Q(p.m)+ An En Cn,x Aoo Eoo Coo,x

Uoo J w Cx

Om

Zp[P.mJ ij~)

Am Bl,x-1 XA (c) Tw(e) TJ" 8fp""

u

Tan(E;qn,p) expE WE

Cotan(E;qn,p) El(Qn,p)

Pn

E:

AE expE

lq(q-•) Lm(E, s) Lm(E,x,s) nE rE PE,p LE

Col 00

48 50 50 50 51 51 51 51 53 54 54 56 56 56 57 59 60 60 60 60 61 61 64 64 64 64 64 64 64 64 64 65 65 65 66 66 67 69 70

Chapter 4 rq R

rr F(r) rF(tl Nq 'RF,M

XF(t) XF(t)

X Xoo,t Dq Dr

NF(l)/F Dt,F

WM

lh

6t

dp

K[F,t,M) Qq(x) O,.

wf

M

wf

M

lndv(WM)

0Lq,WM I

75 75 76 76 76 76 76 78 78 79 79 84 84 84 84 85 85 87 87 89 90 91 91 91 94 94 94 94

Chapter 5 order( , ) (TJ)L

105 105

Chapter 6 Twp cP

119 123

Chapter 7 AF,M

(J* noo nt?

Ev*

aT Zoo M

u nM

129 129 129 129 130 130 132 132 132 132

INDEX OF SYMBOLS 11"

t( 7r) IT(k, F, M) 'I!(k,F,M) (K:[F,