Euler systems Karl Rubin - Southwest Center for Arithmetic

187 Pages·2001·1.03 MB·English

Euler systems
Karl Rubin
Author address:
Department of Mathematics
Stanford University
Stanford, CA 94305-2125
USA
E-mail address: [email protected]

Contents
Introduction vii
Chapter I. Galois cohomology of p-adic representations 1
1. p-adic representations 1
2. Galois cohomology 2
3. Local cohomology groups 4
4. Local duality 8
5. Global cohomology groups 11
6. Examples of Selmer groups 13
7. Global duality 17
Chapter II. Euler systems: definition and main results 21
1. Euler systems 21
2. Results over K 23
3. Results over K 26
∞
4. Twisting by characters of finite order 30
Chapter III. Examples and Applications 33
1. Preliminaries 33
2. Cyclotomic units 33
3. Elliptic units 39
4. Stickelberger elements 40
5. Elliptic curves 47
6. Symmetric square of an elliptic curve 54
Chapter IV. Derived cohomology classes 55
1. Setup 55
2. The universal Euler system 58
3. Properties of the universal Euler system 59
4. Kolyvagin’s derivative construction 62
5. Local properties of the derivative classes 67
6. Local behavior at primes not dividing pr 68
7. Local behavior at primes dividing r 74
8. The congruence 77
Chapter V. Bounding the Selmer group 79
1. Preliminaries 79
2. Bounding the order of the Selmer group 80
iii
iv CONTENTS
3. Bounding the exponent of the Selmer group 86
Chapter VI. Twisting 89
1. Twisting representations 89
2. Twisting cohomology groups 91
3. Twisting Euler systems 92
4. Twisting theorems 94
5. Examples and applications 94
Chapter VII. Iwasawa theory 97
1. Outline 97
2. Galois groups and the evaluation map 102
3. The kernel and cokernel of the restriction map 105
4. Proof of Theorem II.3.2 110
5. Galois equivariance of the evaluation maps 111
6. Proof of Proposition 1.4 115
7. Proof of Proposition 1.6 117
Chapter VIII. Euler systems and p-adic L-functions 123
1. The setting 123
2. Perrin-Riou’s p-adic L-function and related conjectures 125
3. Connection with Euler systems when d =1 127
−
4. Example: cyclotomic units 129
5. Connection with Euler systems when d >1 131
−
Chapter IX. Variants 133
1. Rigidity 133
2. Finite primes splitting completely in K /K 136
∞
3. Euler systems of finite depth 136
4. Anticyclotomic Euler systems 137
5. Adding additional local conditions 140
6. Varying the Euler factors 141
Appendix A. Linear algebra 145
1. Herbrand quotients 145
2. p-adic representations 146
Appendix B. Continuous cohomology and inverse limits 151
1. Preliminaries 151
2. Continuous cohomology 151
3. Inverse limits 153
4. Induced modules 155
5. Semilocal Galois cohomology 157
Appendix C. Cohomology of p-adic analytic groups 159
1. Irreducible actions of compact groups 159
2. Application to Galois representations 160
CONTENTS v
Appendix D. p-adic calculations in cyclotomic fields 163
1. Local units in cyclotomic fields 163
2. Cyclotomic units 167
Bibliography 171
Notation Index 175
Subject Index 177
vi CONTENTS
Introduction
History. In 1986, Francisco Thaine [Th] discovered a remarkable method to
bound ideal class groups of real abelian extensions of Q. Namely, if F is such a
field, heusedcyclotomicunitsinfieldsF(µ ), foralargeclassofrationalprimes (cid:96),
(cid:96)
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. Thaine’s results were already known (as a Corollary of the proof by Mazur
and Wiles [MW] of Iwasawa’s “Main Conjecture”) but Thaine’s proof was very
much simpler. The author [Ru1] was able to apply Thaine’s method essentially
unchangedtoboundidealclassgroupsofabelianextensionsofimaginaryquadratic
fields in terms of elliptic units, with important consequences for the arithmetic of
elliptic curves with complex multiplication.
Shortlyafterthis,Kolyvagin[Ko1]discoveredindependentlyasimilarremark-
able method, in his case to bound the Selmer group of an elliptic curve. Suppose
E is a modular elliptic curve over Q, with sign +1 in the functional equation of
its L-function. Kolyvagin’s method used Heegner points on E over anticyclotomic
extensions of prime conductor of an imaginary quadratic field K (in place of cy-
clotomic 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 funda-
mentaladvance. Inhispaper[Ko2]heintroducedwhathecalled“Eulersystems.”
In Thaine’s setting (the Euler system of cyclotomic units) Kolyvagin showed how
to use cyclotomic units in fields F(µ ), for a large class of integers r (no longer
r
just primes), to bound the orders of the different Galois-eigencomponents of the
idealclassgroupofF,ratherthanjusttheirexponents. Similarly,byusingalarger
collectionofHeegnerpointsinthesituationdescribedabove,Kolyvaginwasableto
giveaboundfortheorderoftheSelmergroupofE. ThankstothetheoremofGross
andZagier[GZ], whichlinksHeegnerpointswiththeL-functionofE, 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
GaloisgroupofanumberfieldK. (Thaine’ssituationisthecasewhereT islimµ
←− pn
twisted by an even Dirichlet character, and Kolyvagin’s is the case where T is the
vii
viii INTRODUCTION
Tate module of a modular elliptic curve.) We define an Euler system for T to
be a collection of cohomology classes in c ∈ H1(F,T), for a family of abelian
F
extensions F of K, with properties relating c and c when F ⊂ F(cid:48). Our main
F(cid:48) F
results show how the existence of an Euler system leads to bounds on the orders of
Selmer groups attached to the Galois module Hom(T,µ ), bounds which depend
p∞
only on the given Euler system.
The proofs of these theorems in this 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 Perrin-Riou [PR5].
What we do not do here is construct new Euler systems. This is the deepest
andmostdifficultpartofthetheory. SinceKolyvagin’sintroductionoftheconcept
ofanEulersystemtherehavebeenveryfewnewEulersystemsfound,buteachhas
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 Chapter III §5). Flach [Fl] has used a collection of cohomology classes
(but not a complete Euler system in our sense) 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 L-functions (and thereby to p-adic L-functions).
An important benefit of this connection is that the bounds on Selmer groups that
come out of the theory are then linked to L-values. Such bounds then provide
evidence for the Bloch-Kato conjectures [BK], which predict the orders of these
Selmer groups in terms of L-values.
OurdefinitionofEulersystemsaysnothingaboutL-values. IfthereisanEuler
system for T then there is a whole family of them (for example, the collection of
EulersystemcohomologyclassesisaZ -module,aswellasaGal(K¯/K)-module). If
p
onemultipliesanEulersystembyp,onegetsanewEulersystembutaworsebound
on the associated Selmer groups. The philosophy underlying this book, although
it is explicitly discussed only in Chapter VIII, 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 L-function attached to T.
A remark about generality. It is difficult to formulate the “most general” def-
inition 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 general-
ization is quite small. In the end, we choose a definition which does not cover the
caseofKolyvagin’sHeegnerpoints,becausetouseamoreinclusivedefinitionwould
introduce too many difficulties. (In Chapter IX 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.
INTRODUCTION ix
Organization. In Chapter I we introduce the local and global cohomology
groups, and state the duality theorems, which will be required to state and prove
ourmainresults. ChapterIIcontainsthedefinitionofanEulersystem,followedby
the statements of our main theorems bounding the Selmer group of Hom(T,µ )
p∞
over the base field K (§2) and over Zd-extensions K of K (§3).
p ∞
Chapter III contains sample applications of the theorems of Chapter II. We
apply those theorems to three different Euler systems: the first constructed from
cyclotomic units, to study ideal class groups of real abelian fields (§III.2); the
second constructed from Stickelberger elements, to study the minus part of ideal
class groups of abelian fields (§III.4); and the third constructed by Kato from
Beilinson elements in the K-theory of modular curves, to study the Selmer groups
of modular elliptic curves (§III.5).
The proofs of the theorems of Chapter II are given in Chapters IV through
VII. InChapterIVwegiveKolyvagin’s“derivative”construction,takingtheEuler
system cohomology classes defined over abelian extensions of K and using them to
producecohomologyclassesoverK itself. Wethenanalyzethelocalizationsofthese
derived classes, information which is crucial to the proofs of our main theorems.
In Chapter V we bound the Selmer group over K by using the derived classes of
Chapter IV and global duality. Bounding the Selmer group over K is similar but
∞
moredifficult; thisis accomplishedin Chapter VII aftera digressionin ChapterVI
which is used to reduce the proof to a simpler setting.
In Chapter VIII we discuss the conjectural connection between Euler systems
and p-adic L-functions. This connection relies heavily on conjectures of Perrin-
Riou [PR4]. Assuming a strong version of Perrin-Riou’s conjectures, and subject
tosomehypothesesontherepresentationT, weshowthatthereisanEulersystem
for T which is closely related to the p-adic L-function.
Chapter IX 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.
Notation. Equations are numbered consecutively within each chapter. Theo-
rem 4.2 means the theorem numbered 4.2 in section 4 of the current chapter, while
LemmaIII.2.6meansLemma2.6ofChapterIII(andsimilarlyfordefinitions,etc.).
The chapters are numbered I through IX, and the appendices are A through D.
IfF isafield,F¯ willdenoteafixedseparableclosureofF andG =Gal(F¯/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 Fab will denote the maximal abelian extension of F,
and if F is a local field Fur will denote the maximal unramified extension of F. If
F isaglobalfieldandΣisasetofplacesofF,F willbethemaximalextensionof
Σ
F which is unramified outside Σ. If K ⊂ F is an extension of fields, we will write
K⊂ F to indicate that [F :K] is finite.
f
If F is a field and B is a G -module, F(B) will denote the fixed field of the
F
kernelofthemapG →Aut(B),thesmallestextensionofF whoseabsoluteGalois
F
group acts trivially on B.
x INTRODUCTION
If O is a ring and B is an O-module then Ann (B) ⊂ O will denote the
O
annihilator of B in O. If M ∈O then B will denote the kernel of multiplication
M
by M on B, and similarly if M is an ideal. If B is a free O-module and τ is an
O-linear endomorphism of B, we will write
P(τ|B;x)=det(1−τx|B)∈O[x],
the determinant of 1−τx acting on B.
The Galois module of n-th roots of unity will be denoted by µ .
n
If p is a fixed rational prime and F is a field of characteristic different from p,
the cyclotomic character ε : G → Z× is the character giving the action of G
cyc F p F
on µ , and the Teichmu¨ller character ω : G → (Z×) is the character giving
p∞ F p tors
the action of G on µ (if p is odd) or µ (if p = 2). Hence ω has order at most
F p 4
p−1 or 2, respectively (with equality if F =Q) and (cid:104)ε(cid:105)=ω−1ε takes values in
cyc
1+pZ (resp. 1+4Z ).
p 2
IfB isanabeliangroup,B willdenotethemaximaldivisiblesubgroupofB.
div
Ifpisafixedrationalprime, wedefinethep-adiccompletionofB tobethedouble
dual
Bˆ=Hom(Hom(B,Q /Z ),Q /Z )
p p p p
(where Hom always denotes continuous homomorphisms if the groups involved
comes with topologies). For example, if B is a Z -module then Bˆ = B; if B
p
is a finitely generated abelian group then Bˆ = B ⊗ Z . In general Bˆ is a Z
Z p p
module and there is a canonical map from B to Bˆ. If τ is an endomorphism of B
then we will often write Bτ=0 for the kernel of τ, Bτ=1 for the subgroup fixed by
τ, etc.
Most of these notations will be recalled when they first occur.
Acknowledgments. ThisbookisanoutgrowthoftheHermannWeyllectures
I gave at the Institute for Advanced Study in October, 1995. Some of the work
and writing work 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, especially
Avner Ash, Ralph Greenberg, Barry Mazur, Bernadette Perrin-Riou, Alice Silver-
berg, and Warren Sinnott. I would also like to thank Tom Weston and Christophe
Cornut for their careful reading of the manuscript and their comments, and the
audiences of graduate courses I gave at Ohio State University and Stanford Uni-
versity for their patience as I was developing this material. Finally, special thanks
go to Victor Kolyvagin and Francisco Thaine for their pioneering work.