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...