Euler Systems

by Karl Rubin

Overview

One of the most exciting new subjects in Algebraic Number Theory and Arithmetic Algebraic Geometry is the theory of Euler systems. Euler systems are special collections of cohomology classes attached to p-adic Galois representations. Introduced by Victor Kolyvagin in the late 1980s in order to bound Selmer groups attached to p-adic representations, Euler systems have since been used to solve several key problems. These include certain cases of the Birch and Swinnerton-Dyer Conjecture and the Main Conjecture of Iwasawa Theory. Because Selmer groups play a central role in Arithmetic Algebraic Geometry, Euler systems should be a powerful tool in the future development of the field.


Here, in the first book to appear on the subject, Karl Rubin presents a self-contained development of the theory of Euler systems. Rubin first reviews and develops the necessary facts from Galois cohomology. He then introduces Euler systems, states the main theorems, and develops examples and applications. The remainder of the book is devoted to the proofs of the main theorems as well as some further speculations.


The book assumes a solid background in algebraic Number Theory, and is suitable as an advanced graduate text. As a research monograph it will also prove useful to number theorists and researchers in Arithmetic Algebraic Geometry.

Product Details

ISBN-13:	9780691050768
Publisher:	Princeton University Press
Publication date:	05/21/2000
Series:	Annals of Mathematics Studies , #147
Pages:	240
Product dimensions:	7.75(w) x 10.00(h) x (d)
Age Range:	18 Years

About the Author

Karl Rubin, Professor of Mathematics at Stanford University, was awarded the Cole Prize of the American Mathematical Society in 1992. He has been a Guggenheim Fellow, a Sloan Fellow, and a National Science Foundation Presidential Young Investigator.

Read an Excerpt

Euler Systems

---

By Karl Rubin

PRINCETON UNIVERSITY PRESS

Copyright © 2000 Princeton University Press
All rights reserved.
ISBN: 978-0-691-05076-8

---

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, p is a rational prime, and O is the ring of integers of a finite extension Φ of Qp. A p-adic representation of GK = Gal([bar.K]/K), with coefficients in O, is a free O-module T of finite rank with a continuous O -linear action of GK.

Let D denote the divisible module Φ/O. Attached to a p-adic representation T we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

so WM is the M-torsion in W. Note that T determines V and W, and W determines [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and V, but in general there may be different O-modules T giving rise to the same vector space V.

Example 1.1.2. Suppose ρ: GK ->Ox is a character (continuous, but not necessarily of finite order). Then we can take T = Oρ, where Oρ is a free rank-one O-module on which GK acts via ρ. Clearly every one-dimensional representation arises in this way. When ρ is the trivial character we get T [congruent to] O, and when O = Zp and ρ is the cyclotomic character

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For general O we also write O (1) = O [cross product] Zp(1), write Φ(1) = Φ [cross product] Qp(1), and write D(1) = D [cross product] Zp(1).

Definition 1.1.3. If T is a p-adic representation of GK then so is the dual representation

T * = HomO(T, O(1)).

We will also write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 1.1.4. If ρ: GK ->Ox is a continuous character as in Example 1.1.2, and T = Oρ, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 1.1.5. Suppose A is an abelian variety defined over K, and p is a prime different from the characteristic of K. We can take O to be Zp and T to be the p-adic Tate module of A defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the pn-torsion in A([bar.K]). Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If A and A' are isogenous, their Tate modules T = Tp(A) and T' = Tp(A') need not be isomorphic (as GK-modules), but the corresponding Qp-vector spaces V and V' are isomorphic.

If the endomorphism algebra of A over K contains the ring of integers OF of a number field F, and p is a prime of F above p, we can also take Φ = Fp, the completion of F at p, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which has rank 2 dim(A)/[F : Q] over the ring of integers O of Φ. If A is an elliptic curve with complex multiplication by F [subset] K, this is another important source of one-dimensional representations.

1.2. Galois Cohomology

Suppose K is a field. If B is a commutative topological group with a continuous action of GK, then we have the continuous cohomology groups

Hi(K, B) = Hi(GK, B).

If further the action of GK factors through the Galois group Gal(K'/K) for some extension K' of K, we also write Hi(K'/K, B) = Hi(Gal(K'/K), B). See Appendix B for the basic facts which we will need about continuous cohomology groups.

Example 1.2.1. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By Kummer theory and Proposition B.2.3, respectively,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [??] denotes the (p-adically) completed tensor product.

Suppose T is a p-adic representation of GK with coefficients in O as in §1.1, and M [member of] O is nonzero. Recall that V = T [cross product] and W = V/T. We will frequently make use of the following exact sequences and commutative diagram.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

Lemma 1.2.2.Suppose M [member of] O is nonzero.

(i) The exact sequence (1.1) induces an exact sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii) The bottom row of (1.2) induces an exact sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(iii) The kernel of the map H1(K, T) ->H1(K, W) induced by the composition T ->T/MT [??] WM [??] W is

MH1(K, T) + H1 (K, T)tors.

Proof. Assertion (i) is clear, and so is (ii) once we show that the kernel of the natural map H1(K, T) ->H1(K, V) is H1(K, T)tors. But this is immediate from Proposition B.2.4, which says that the map H1(K, T) ->H1(K, V) induces an isomorphism H1(K, V) [congruent to] H1(K,T) [cross product] Qp.

The diagram (1.2) induces a commutative diagram with exact rows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where φ1 is induced by M-1: T ->V. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which proves (iii).

1.3. Local Cohomology Groups

Unramified local cohomology. Suppose for this section that K is a finite extension of Ql for some l, where we allow l to be either a rational prime or oo (in which case Ql = R). Let I denote the inertia subgroup of GK (so I = GK if K is archimedean), let Kur = bar.KI be the maximal unramified extension of K, and if K is nonarchimedean let Fr [member of] Gal(Kur/K) denote the Frobenius automorphism.

Definition 1.3.1. Suppose B is a GK-module. We say that B is unramified if I acts trivially on B. We define the subgroup of unramified cohomology classes H1ur(K, B) [subset] H1(K, B) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that if T is as in §1.1, then

T is unramified [??] V is unramified [??] W is unramified,

and if K is nonarchimedean of residue characteristic different from p, then this is equivalent to T*, V*, and/or W* being unramified.

Lemma 1.3.2.Suppose B is a GK-module which is either a finitely generatedZp-module, or a finite-dimensionalQp-vector space, or a discrete torsionZp-module.

(i) If K is nonarchimedean then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(ii) If K is nonarchimedean of residue characteristic different from p, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(iii) If K is archimedean then H1ur(K, B) = 0.

Proof. Assertion (iii) and the first isomorphism of (i) follow from the inflation-restriction exact sequence (Proposition B.2.5(i)). The second isomorphism of (i) (induced by the map on cocycles c [??] c(Fr)) is Lemma B.2.8.

The hypotheses on B guarantee (see Propositions B.2.5(ii) and B.2.7) that we have an inflation-restriction exact sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If K is nonarchimedean then Gal(Kur/K) has cohomological dimension one, so H2(Kur/K, BI) = 0. This proves (ii).

Corollary 1.3.3.Suppose K is nonarchimedean of residue characteristic different from p, and V is aQp[GK]-module which has finite dimension as aQp-vector space. Then

(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Using Lemma 1.3.2(i) we have an exact sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which proves (i).

Since the residue characteristic of K is different from p, the inertia group I has a unique maximal p-divisible subgroup I', and I/I' [congruent to] Zp (see [Fr] §8 Corollary 3). Thus both I and Gal(Kur/K) have p-cohomological dimension one. It follows that

Hm (Kur/K, Hn (I, V)) = 0

if m > 1 or n > 1. Therefore the Hochschild-Serre spectral sequence (Propositions B.2.5(ii) and B.2.7) shows that

H1 (Kur/K,H1 (I, V)) = H2(K, V).

On the other hand, Lemma 1.3.2 shows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus there is an exact sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is finite (Proposition B.2.7(iii)), this proves (ii).

Special subgroups. As above, let K be a finite extension of Ql with l ≤ ∞. Let T be a p-adic representation of GK, let V = T [cross product] Φ and let W = V/T as in §1.1. Following many authors (for example Bloch and Kato [BK §3, Fontaine and Perrin-Riou [FPR] §1.3.3, or Greenberg [Gr2]) we define special subgroups H1f(K, x) of certain cohomology groups H1 (K, x). We assume first that l ≠ p and l ≠ ∞, and we discuss the other cases in Remarks 1.3.6 and 1.3.7 below.

Definition 1.3.4. Suppose l ≠ p and l ≠ ∞. Define the finite part of H1(K, V) by

H1f (K, V) = H1ur(K, V).

Define H1f (K, T) [subset] H1(K, T) and H1f (K, W) [subset] H1(K, W) to be the inverse image and image, respectively, of H1f (K, V) under the natural maps

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For every M [member of] O define H1f (K, WM) [subset] H1(K, WM) to be the inverse image of H1f (K, W) under the natural map H1(K, WM) ->H1(K, W).

Finally, for V, T, W, or WM define the singular quotient of H1 (K, x) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

so there are exact sequences

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If A is a Zp-module let Adiv denote its maximal divisible subgroup.

Lemma 1.3.5.Suppose T is as above, l ≠ p, and l ≠ ∞. Then:

(i) H1f (K, W) = H1ur(K, W)div.

(ii) H1ur(K, T) [subset] H1f (K, T) with finite index and H1s (K, T) is torsion-free.

(iii) Writing W = WI/(WI)div, there are natuml isomorphisms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(iv) If T is unramified then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. It is immediate from the definitions that H1f (K, W) is divisible and H1s (K, T) is torsion-free. The commutative diagram with exact rows and columns

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

shows that H1f (K, W) [subset] H1ur(K, W) and H1ur(K, T) [subset] H1f(K, T). The rest of assertions (i) and (ii) will follow once we prove (iii), since WI/(WI)div is finite.

Note that the image of VI in WI is (WI)div. Taking I-cohomology and then Gal(Kur/K)-invariants of the exact sequence 0 ->T ->V ->W -> 0 gives an exact sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore using Lemma 1.3.2 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

This proves (iii). If T is unramified then WI = W is divisible, so (iv) is immediate from (iii).

Remark 1.3.6. When the residue characteristic l is equal to p, the choice of a subspace H1f (K, V) is much more subtle. Fortunately, for the purpose of working with Euler systems it is not essential to make such a choice. However, to understand fully the arithmetic significance of the Selmer groups we will define in §1.5, and to get the most out of the applications of Euler systems in Chapter 3, it is necessary to choose a subspace H1f (K, V) in the more difficult case l = p.

In this case, Bloch and Kato define H1f (K, V) using the ring Bcris defined by Fontaine ([BK] §3). Namely, they define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For our purposes we will allow an arbitrary special subspace of H1(K, V), which we will still denote by H1f (K, V). This notation is not as bad as it may seem: in our applications we will always choose a subspace H1f (K, V) which is the same as the one defined by Bloch and Kato, but we need not (and will not) prove they are the same. One could also choose, for example, H1f (K, V) = 0 or H1f (K, V) = H1(K, V).

(Continues...)

---

Excerpted from Euler Systems by Karl Rubin. Copyright © 2000 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

---