Structured modules. Let K be an “LF”, and M a finite continuous G_{K}-module. Recall that a finite/singular structure on M is simply a choice of subgroup, which we will denote H_{f}^{1}(G_{K}, M) and call the finite part, of H^{1}(G_{K}, M). We denote the quotient group

H_{s}^{1}(G_{K}, M) := H^{1}(G_{K}, M)/H_{f}^{1}(G_{K}, M),

and refer to H_{s}^{1}(G_{K}, M) as the singular part of H^{1}(G_{K}, M). In particular, to give a finite/singular structure on M is the same as giving an exact sequence,

0 → H_{f}^{1}(G_{K}, M) → H^{1}(G_{K}, M) → H_{s}^{1}(G_{K}, M) → 0.

Recall further our

Definition: If the residue characteristic of K does not divide the order of M and the wildramification subgroup P_{K} ⊂ G_{K} act trivially on M, by the standard finite/singular structure on M we mean the “basic exact sequence”,

0 → H^{1}(G_{k}, M^{I_{K}}) → H^{1}(G_{K}, M) → H^{1}(I_{K}, M)^{G_{k}} → 0.

(So, the standard f/s structure on M has finite part H_{f}^{1}(G_{K}, M) := H^{1}(G_{k}, M^{I_{K}}) and singular part H_{s}^{1}(G_{K}, M) := H^{1}(I_{K}, M)^{G_{k}}. )

Example: If the residue characteristic of K does not divide the order of M and M is an unramified G_{K}-module, meaning that the inertia subgroup I_{K} ⊂ G_{K} acts trivially on M, then the standard finite/singular structure on M is given by

0 → H^{1}(G_{k}, M) → H^{1}(G_{K}, M) → Hom(I_{K}, M)^{G_{k}} → 0.

Now let F be a Number Field (i.e., of finite degree over Q) and if v is a place of F, let F_{v} be the completion of F at v. For simplicity of notation, if nothing else, first fix F̄/F and algebraic closure of the global field F, and then, for each place v of F, choose an extension F̄_{v}/F̄ and imbedding F_{v} ↪ F̄_{v} such that F̄_{v} is the algebraic closure of F_{v}. Letting G := G_{F} = Gal(F̄/F), and for places v, G_{v} := G_{F_{v}} = Gal(F̄_{v}/F_{v}) we get closed imbeddings G_{v} ↪ G, giving us a specific choice of decomposition subgroup for each place v. We will sometimes write H^{j}(F, M) for the cohomology group H^{j}(G, M) = H^{j}(G_{F} , M), and similarly for the local Galois cohomology groups H^{j}(F_{v}, M) = H^{j}(G_{v}, M) = H^{j}(G_{F_{v}}, M).

Definition. A structured (finite) Galois module over F is a (continuous) G-module M, together with a rule σ which assigns to each place v of F a f/s structure on H^{1}(G_{v}, M) such that for almost all places v the f/s structure is “standard”.

Remark. Given a finite Galois module M over F, it may be the underlying Galois module for many possible structures σ. Given two structures σ_{1} and σ_{2} on M let us say that σ_{1} is more stringent than σ_{2} if, for all places v, the finite part of H^{1}(G_{v}, M) for the structure σ_{1} is contained in the finite part of H^{1}(G_{v}, M) for the structure σ_{2}.

Basic Example. Let M be a finite Galois module over F. If S is a finite set of places large enough so that every place v ≠ S has the property that it is non-archimedean, the residue characteristic of v is prime to |M| and M is unramified at v, define the structure σ_{S} on the Galois module M to be given as follows. For all v ≠ S we take H_{f}^{1}(G_{v}, M) to be the standard structure (which, since the action of G on M is unramified at v, is simply H_{f}^{1}(G_{v}, M) = H^{1}(G_{k_{v}}, M), where k_{v} is the residue field of F_{v}). For v ∈ S we take the the least stringent f/s structure; i.e., H_{f}^{1}(G_{v}, M) = H^{1}(G_{v}, M). Denote by M_{S} the Galois module M equipped with the structure σ_{S}.

Definition. The Cartier dual M^{∗} of a structured (finite) Galois module M over F is the Cartier dual of M as G_{F} -module equipped with f/s structure such that for each place v, Tate Local Duality (which establishes a Pontrjagin duality between H^{1}(G_{v}, M) and H^{1}(G_{v}, M^{∗})) puts H_{f}^{1}(G_{v}, M) and H_{s}^{1}(G_{v}, M^{∗}) in Pontrjagin duality, and also puts H_{s}^{1}(G_{v}, M) and H_{f}^{1}(G_{v}, M^{∗}) in Pontrjagin duality.

Remark. By what we have already proved in class, M^{∗} with f/s structures for all v, as just defined, is a structured module. That is, for almost all v its structure is standard. If we fix our Galois module M and have two structures σ_{1} and σ_{2} on M, the first more stringent than the second, the corresponding dual structures σ_{1}^{∗} and σ_{2}^{∗} on M^{∗} will have the property that σ_{1}^{∗} is less stringent than σ_{2}^{∗}.

Let M be a structure Galois module over F, and h ∈ H^{1}(G, M).

Proposition. For almost all places v, the image of h under restriction mapping

res_{v} : H^{1}(G, M) → H^{1}(G_{v}, M)

lies in H_{f}^{1}(G_{v}, M).

The way to prove this. Let L/F be a finite Galois extension that effaces the cohomology class h; i.e., such that h is in the kernel of the restriction homomorphism H^{1}(F, M) → H^{1}(L, M), and therefore in the image of H^{1}(Gal(L/F), M) → H^{1}(F, M). Now just do the right diagram-check to see that res_{v}(h) ∈ H_{f}^{1}(G_{v}, M) for all non-archimedean places v of F which are of residual characteristic prime to |M| and are unramified in the finite extensions L/F.

Define the singular restriction homomorphism, call it res_{s,v}, as the composition

res_{s,v} : H^{1}(G, M) → H^{1}(G_{v}, M) → H_{s}^{1}(G_{v}, M).

It follows from the above Proposition that for any given h ∈ H^{1}(G, M) the image of h under the singular restriction homomorphism res_{s,v} is zero for almost all places v. We get a homomorphism which we can call the singular restriction homomorphism,

⨁_{v} res_{s,v} : H^{1}(G, M) → ⨁_{v} H_{s}^{1}(G_{v}, M).

Definition. Let M be a structured Galois module over F. Define the finite part of H^{1}(G, M), denoted H_{f}^{1}(G, M), to be the kernel of the singular restriction homomorphism above. Define the singular part H_{s}^{1}(G, M) to be the quotient of H^{1}(G, M) by H_{f}^{1}(G, M), giving us an exact sequence for global Galois cohomology:

0 → H_{f}^{1}(G, M) → H^{1}(G, M) → H_{s}^{1}(G, M) → 0,

and an injective homomorphism,

H_{s}^{1}(G, M) ↪ ⨁_{v} H_{s}^{1}(G_{v}, M).

Some notes. Of course, this whole set-up depends upon the choice of structure on the structured Galois module M. For example, if M is the structured module denoted M_{S} as described above, and if the G-action on M is trivial, then

H_{f}^{1}(G, M_{S}) = Hom(G_{F,S}^{ab}, M),

where G_{F,S}^{ab} is the Galois group of the maximal abelian extension of F unramified outside S.

In the literature, you will find definitions of the finite part of cohomology, usually for a specific choice of structure (see the hand-out of last Thursday for some discussion of this). The finite part, H_{f}^{1}(G, M), of global cohomology is also sometimes referred to as a “generalized Selmer group”. We’ll see why, soon. We will sometimes refer to an element h in H_{f}^{1}(G, M) as a “Selmer element”. To keep the balance between nomenclature for H_{f}^{1}(G, M) and H_{s}^{1}(G, M) one might call H_{s}^{1}(G, M) the “Kolyvagin group” and its elements “Kolyvagin elements” (although these would be neologisms). Note that by definition, given any element h ∈ H_{f}^{1}(G, M), and any place v, we have res_{v}(h) ∈ H_{f}^{1}(G_{v}, M). If c = (⋯, c_{v}, ⋯) ∈ ⨁_{v} H_{s}^{1}(G_{v}, M), by the support of c, denoted supp(c), let us mean the set of places v for which c_{v} ≠ 0.

The pairing. Let M be a structured (finite) Galois module over F and let M^{∗} be its Cartier dual. Define a pairing (which we will denote (h, c) ↦ < h, c >):

H_{f}^{1}(G, M) ⊗ ⨁_{v} H_{s}^{1}(G_{v}, M^{∗}) → Q/Z

by the rule:

If h ∈ H_{f}^{1}(G, M) and c = (⋯, c_{v}, ⋯) ∈ ⨁_{v} H_{s}^{1}(G_{v}, M^{∗}), then

< h, c >:= ∑_{v} < res_{v}(h), c_{v} >_{v}= ∑_{v∈supp(c)} < res_{v}(h), c_{v} >_{v},

where < , >_{v} refers to the Pontrjagin pairing between H_{f}^{1}(G_{v}, M) and H_{s}^{1}(G_{v}, M^{∗}).

Given any element c ∈ ⨁_{v} H_{s}^{1}(G_{v}, M^{∗}) we may view c as giving a homomorphism H_{f}^{1}(G, M) → Q/Z ( h ↦< h, c > ) and we sometimes call this homomorphism by the same letter c; i.e., c(h) :=< h, c >. We also write, for any place v, h_{v} := res_{v}(h), and c_{v}(h) :=< res_{v}(h), c_{v} >=< h_{v}, c_{v} >, so that, all in all, we have lots of way of writing the above pairing; e.g.,

c(h) = ∑_{v∈supp(c)} c_{v}(h_{v}).

Proposition. The above pairing vanishes on the subgroup

H_{f}^{1}(G, M) ⊗ H_{s}^{1}(G, M^{∗}) ⊂ H_{f}^{1}(G, M) ⊗ ⨁_{v} H_{s}^{1}(G_{v}, M^{∗}).

That is, given a (Kolyvagin element) c ∈ H_{s}^{1}(G, M^{∗}) and h ∈ H_{f}^{1}(G, M), we have the relation:

∑_{v∈supp(c)} c_{v}(h_{v}) = 0.

The way to prove this. This comes directly from the piece of Global Class Field Theory that we reviewed last Thursday.

The point of view to adopt about all this. Any Kolyvagin element c ∈ H_{s}^{1}(G, M^{∗}) imposes some local condition (at places v in its support) that must be satisfied by all Selmer elements h ∈ H_{f}^{1}(G, M). For example: If the support of c ≠ 0 is concentrated at exactly one place v_{o}, then any Selmer element h must have the property that h_{v_{o}} ∈ Ker(c_{v_{o}}) ⊂ H_{f}^{1}(G_{v_{o}}, M). Here is a way of thinking about the local condition detected by this Kolyvagin element c: It tells us that we may strengthen the f/s structure on the Galois module M, without changing the finite part of its (1-dimensional) cohomology. That is, let us define M′ to be the structured module whose underlying Galois module is M again, whose f/s structure for all places v ≠ v_{o} is equal to the f/s structure on M, but on v_{0} we place a more stringent f/s structure on the G_{v_{o}}-module M by setting:

H_{f}^{1}(G_{v_{o}}, M′) := Ker{c_{v_{o}} : H_{f}^{1}(G_{v_{o}}, M) → Q/Z} ⊂ H_{f}^{1}(G_{v_{o}}, M).

The above discussion gives us that the finite part of the 1-dimensional cohomology of the more stringent structure M′ is equal to the finite part of the 1-dimensional cohomology of M. Now this is already a curiously favorable state of affairs, because if we are trying to to show that H_{f}^{1}(G, M) is small, we have (by constructing the Kolyvagin element c) that

H_{f}^{1}(G, M′) = H_{f}^{1}(G, M),

where M′ is more stringent than M, and there is nothing to stop us from trying to do this again inductively, to force the finite cohomology to satisfy more and more stringent local conditions by finding further Kolyvagin elements for these increasingly stringent structures. In the most extreme cases one then shows that H_{f}^{1}(G, M) satisfies such stringent conditions that it is forced to vanish, or else it is generated by specifically constructed cohomology classes. The efficacy of this method is determined by our ability to manufacture Kolyvagin elements with good control over their local components. This appraoch, combined with the following two further ideas is the underlying philosophy behind Kolyvagin’s method:

1. It is possible, at times, to judiciously enlarge the base field F without losing too much of the Selmer group, but this larger base field gives us an even better chance of constructing Kolyvagin elements.

2. In various different contexts, one can find well-behaved collections of objects (these are the Euler systems of algebraic cycles, Heegner points, circular units, elements in algebraic K-groups, etc. ) which produce the desired Kolyvagin elements in cohomology.

Proposition. Let M be a finite structure Galois module over F. Then H_{f}^{1}(G, M) is finite.

Outline of Proof. Let S be a finite set of places large enough so that every place v ≠ S has the property that the residue characteristic of v is prime to |M|, M is unramified at v, and the f/s structure of M at v is standard. Let M_{S} be the underlying Galois module M over F, equipped with the structure σ_{S} as described under the rubric of “Basic Example” above. Since H_{f}^{1}(G, M) ⊂ H_{f}^{1}(G, M_{S}), it suffices to prove finiteness of H_{f}^{1}(G, M) when M = M_{S}.

For this, let L/F be the “splitting field for the G_{F} -action on M” ;, i.e., L/F is a finite Galois extension such that the action of G_{F} on M factors through a faithful action of Gal(L/F) on M. Now let T be the set of places of L which lie over the places of F in S. Denote by M_{T} the G_{L} Galois module with M as underlying G_{L}-module, with the structure σ_{T}.

Lemma 1. H_{f}^{1}(G_{L}, M_{T}) = Hom(G_{L,T}^{ab} , M) is finite. Remark. There are two ways to prove this: via Class Field Theory, or, using less machinery: Kummer theory and the fact that the group of S-units in O_{F} is finitely generated .

Lemma 2. There is a natural injection of

Ker{H_{f}^{1}(G, M_{S}) → H_{f}^{1}(G_{L}, M_{T})}

into H^{1}(Gal(L/K), M).

Prove both of these as Exercises.

The Proposition follows from Lemmas 1 and 2 because H^{1}(Gal(L/K), M) is (visibly) finite.