1. Kummer-theoretic analysis of Hom(G_{F,S}^{ab}, μ_{N}).

Let F/Q be a field of finite degree, F̄/F an algebraic closure, S a finite set of places of F which contain all archimedean places, and O_{F,S} ⊂ F the sub-ring of S-integers of F. So, if S consists of the set of archmidean places, then O_{F,S} = O_{F} is the ring of integers in F. Let G_{F,S} be the Galois group of the maximal sub-extension of F̄/F which is unramified outside S. By the group of S-units of O_{F} we mean O_{F,S}^{∗}. By Pic(O_{F,S}) we mean the abelian group of isomorphism classes of O_{F,S}-modules of finite type which are locally free of rank 1, the group-structure being given by tensor product.

We recall that O_{F,S}^{∗} is a finitely generated abelian group, that Pic(O_{F,S}) is a finite abelian group, and that (fixing F, of course) if S is taken large enough, Pic(O_{F,S}) is trivial.

Fix N a positive integer, and assume that F contains μ_{N}(F̄), and that S contains all places of residual characteristic dividing N.

If M is the Galois module μ_{N} over F, then the G_{F} -action on M is trivial by the above hypothesis, and if M_{S} refers to M with its S-structure (recalling previous handouts) then

H_{f}^{1}(G_{F} , M_{S}) = Hom(G_{F,S}^{ab}, μ_{N}).

Theorem. The natural homomorphisms (which we will define in class) make

0 → O_{F,S}^{∗}/O_{F,S}^{∗N} → Hom(G_{F,S}^{ab}, μ_{N}) → Pic(O_{F,S})[N] → 0

exact.

Describe carefully the twist of an element of Pic by an element ϕ ∈ Hom(G_{F,S}^{ab}, μ_{N}).

2. Cohomology of abelian varieties over finite and local fields.

Recall. If N is a positive integer and A is an abelian variety (or a semi-abelian variety) over any field K, putting M := A(K̄)[N] (which we also write as A[N] for short) we have the Kummer sequence

0 → A(K)/NA(K) → H^{1}(G_{K}, M) → H^{1}(G_{K}, A(K̄))[N] → 0.

Theorem (Lang). Let A be any connected smooth (commutative) algebraic group over the finite field k. Then H^{1}(G_{k}, A(k̄)) = 0.

Corollary. If N is a positive integer and A is an abelian variety (or a semi-abelian variety) over the finite field k, then the natural homomorphism coming from the Kummer sequence induces an isomorphism

A(k)/NA(k) ≅ H^{1}(G_{k}, A(k̄))[N].

Proposition. Let K be an “LF” and k its residue field. Let A be an abelian scheme over O_{K}. Then the natural mapping induces an isomorphism of abelian groups

A(O_{K}) ≅ A(K).

If N is a positive number not divisible by the characteristic of k, the natural mapping

A(K)/NA(K) ≅ A(O_{K})/NA(O_{K}) → A(k)/NA(k)

is an isomorphism.

Putting the above Corollary and Proposition together, we have a natural isomorphism (under the hypotheses of the proposition):

(∗) A(K)/NA(K) ≅ H^{1}(G_{k}, A(k̄)[N]).

Now note that under the above hypotheses the G_{K}-module M := A(K̄)[N] ≅ A(k̄)[N] is unramified, and that there is a nice computation for us to do! For we have two natural exact sequences with the same middle term: the Kummer sequence,

0 → A(K)/NA(K) → H^{1}(G_{K}, M) → H^{1}(G_{K}, A(K̄))[N] → 0,

and the Basic exact sequence,

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

Exercise. Assume the hypotheses of the previous proposition. Show that (*) is compatible with the two injections in the the two exact sequences above, yielding an isomorphism between those two exact sequences (and once I learn more plaintex, I will actually be able to draw the relevant exact diagram).

3. The finite/singular structure on the Galois module of N-torsion points of an abelian variety. Let K be any complete local field (with finite residue field k) and A an abelian variety over K. Let N ≥ 1. Consider the finite G_{K}-module

M := A(K̄)[N].

Definition. By the A.V. finite/singular structure on M we mean the finite/singular structure which has:

H_{f}^{1}(G_{K}, M) := Im{A(K)/NA(K) → H^{1}(G_{K}, M)} = Ker{H^{1}(G_{K}, M) → H^{1}(G_{K}, A(K̄))},

and (as would then follow):

H_{s}^{1}(G_{K}, M) ≅ H^{1}(G_{K}, A(K̄))[N].

Remark. The A.V. finite/singular structure on M depends upon the abelian variety A and the isomorphism between M and the group of N-torsion in A.

Proposition. If N is a positive number not divisible by the characteristic of k, and if A is an abelian variety over K which has good reduction over k, and if M = A(K̄)[N], M is an unramified G_{K}-module, and the A.V. finite/singular structure on M is equal to its standard finite/singular structure.

If F is a number field, A an abelian variety over F, N ≥ 1, and M = A(F̄)[N] is the G_{F} -module of N-torsion in A, by the A.V. finite/singular structure on M we mean the rule that assigns to each place v of F the A.V. finite/singular structure on M when viewed as Galois module over G_{F_{v}}.

Proposition. Let F be a number field, A an abelian variety over F, N ≥ 1, and M = A(F̄)[N]. The A.V. finite/singular structure on M is a structure.

Proof. This follows from the Exercise above, plus the fact that an abelian variety over a number field has good reduction for almost all places v.

4. Selmer and Shafarevich-Tate groups. Define these.

Proposition. Let F be a number field, A an abelian variety over F, N ≥ 1, and M = A(F̄)[N] with its A.V. structure. Then

H_{f}^{1}(G_{F} , M) ≅ S_{N}(A/F),

where S_{N}(A/F) is the N-Selmer group of A over F.

Deduce (as last Thursday)

Corollary. The groups A(F)/NA(F) and Sha(A/F)[N] are finite for any positive N.

State the Shafarevich-Tate Conjecture. Interpretation of Sha as locally trivial torsors. Discuss L/K-forms, the (non-abelian) Galois module of Automorphisms of an algebraic variety, connections with torsors, and with (nonabelian) H^{1}. State the theorem of descent. Examples! Elliptic curves, curves of genus 1 over F with with a rational point in F_{v} for every place v. Consider the smooth plane cubic curve E : x^{3} + y^{3} + 60z^{3} = 0. By work of Kolyvagin and Rubin, one knows that

Sha_{E} ≅ Z/3Z × Z/3Z.

Noting that change of sign of ι corresponds to multiplying by −1 in Sha_{E} we see that there are, in toto, four “nontrivial” Σ’s to find here and after some minor computations (cf [M]) one finds all of them again as smooth plane cubics:

Σ_{1} : 3x^{3} + 4y^{3} + 5z^{3} = 0
Σ_{2} : 12x^{3} + y^{3} + 5z^{3} = 0
Σ_{3} : 15x^{3} + 4y^{3} + z^{3} = 0
Σ_{4} : 3x^{3} + 20y^{3} + z^{3} = 0.