Galois Cohomology - Elliptic Curves

Let

G

be a group and

A

be a

G

-module (an abelian group with an action of

G

via group homomorphisms).

G

-module means exactly the same thing as

ℤ [G]

-module.

Theorem 15.1. Assuming that:

we have a short exact sequence of $G$ -modules

$0 \to A \overset{ϕ}{\to} B \overset{ψ}{\to} C \to 0 .$

Then it gives rise to a long exact sequence of abelian groups:

0 \to A^{G} \overset{ϕ}{\to} B^{G} \overset{ψ}{\to} C^{G} \overset{δ}{\to} H^{1} (G, A) \overset{ϕ^{*}}{\to} H^{1} (G, B) \overset{ψ^{*}}{\to} H^{1} (G, C) .

Definition ( $d e l t a$ ). Let $c \in C^{G}$ . Then $b \in B$ such that $ψ (b) = c$ . Then

ψ (σ b - b) = σ c - c = 0 \forall σ \in G

so $σ b - b = ϕ (a_{σ})$ for some $a_{σ} \in A$ .

Can check ${(a_{σ})}_{σ \in G} \in Z^{1} (G, A)$ .

Then $δ (c) =$ class of ${(a_{σ})}_{σ \in G}$ in $H^{1} (G, A)$ .

Theorem 15.2. Assuming that:

$A$ is a $G$ -module
$H ⊴ G$ a normal subgroup

Then there is an inflation restriction exact sequence

0 \to H^{1} (G ∕ H, A^{H}) \overset{\inf}{\to} H^{1} (G, A) \overset{res}{\to} H^{1} (H, A)

Let

K

be a perfect field. Then

Gal (\bar{K} ∕ K)

is a topological group with basis of open subgroups being the

Gal (\bar{K} ∕ L)

for

[L : K] < \infty

G = Gal (\bar{K} ∕ K)

then we modify the definition of

H^{1} (G, A)

by insisting:

Proof. Let $G = Gal (L ∕ K)$ . Let ${(a_{σ})}_{σ \in G} \in Z^{1} (G, L^{*})$ . Distinct automorphisms are linearly independent, so there exists $y \in L$ such that

\underset{= x}{\underset{⏟}{\sum_{τ \in G} a_{τ}^{- 1} τ (y)}} \neq 0 .

Then

\begin{array}{l} σ (x) & = \sum_{τ \in G} σ {(a_{τ})}^{- 1} σ τ (y) \\ = a_{σ} \underset{= x}{\underset{⏟}{\sum_{τ \in G} a_{σ τ}^{- 1} σ τ (y)}} \end{array}

Then $a_{σ} = \frac{σ (x)}{x}$ for all $σ \in G$ . ( $a_{σ} τ = σ (a_{τ}) a_{σ}$ $⟹$ $σ {(a_{τ})}^{- 1} = a_{σ} a_{σ τ}^{- 1}$ ).

So ${(a_{σ})}_{σ \in G} \in B^{1} (G, L^{*})$ . Therefore $H^{1} (G, L^{*}) = 0$ . □

Application: Assume

char K ∤ n

. There is a short exact sequence of

Gal (\bar{K} ∕ K)

-modules

Finite subgroups of

{Hom}_{cts} (Gal (\bar{K} ∕ K), μ_{n})

are of the form

Hom (Gal (L ∕ K), μ_{n})

for

L ∕ K

a finite abelian extension of

K

of exponent dividing

n

Let

ϕ : E \to E^{'}

be an isogeny of elliptic curves over

K

. Short exact sequence of

Gal (\bar{K} ∕ K)

-modules

For each place

v

, fix an embedding

\bar{K} \subset {\bar{K}}_{v}

. Then

Gal ({\bar{K}}_{v} ∕ K_{v}) \subset Gal (\bar{K} ∕ K)

Definition (Selmer-group). We define the $ϕ$ -Selmer group

\begin{array}{l} S^{(ϕ)} (E ∕ K) & = \ker (H^{1} (K, E [ϕ]) \to \prod_{v} H^{1} (K_{v}, E)) \\ = {α \in H^{1} (K, E [ϕ]) | {res}_{v} (α) \in Im (δ_{v}) \forall v} \end{array}

(the map $H^{1} (K, E [ϕ]) \to \prod_{v} H^{1} (K_{v}, E)$ is as in the commutative diagram above).

Proof. For $L ∕ K$ a finite Galois extension,

◜-------fin◞it◟e-------◝
0 H1 (Gal(L∕K ),E(L)[n ]) H1 (K,E [n]) H1(L,E [n])

inr⊃⊃fes S (n)(E ∕K ) S(n)(E∕L )

Therefore by extending our field, we may assume

E [n] \subset E (K)

and hence by the Weil pairing

μ_{n} \subset K

Therefore $E [n] ≅ μ_{n} \times μ_{n}$ as a $Gal (\bar{K} ∕ K)$ -module. Then

\begin{array}{l} H^{1} (K, E [n]) & ≅ H^{1} (K, μ_{n}) \times H^{1} (K, μ_{n}) \\ ≅ K^{*} ∕ {(K^{*})}^{n} \times K^{*} ∕ {(K^{*})}^{n} \end{array}

Let

S = {primes of bad reduction for E} \cup {v | n \infty}

(a finite set of places).

Define the subgroup of $H^{1} (K, A)$ unramified outside of $S$ as

H^{1} (K, A; S) = \ker (H^{1} (K, A) \to \prod_{v \notin S} H^{1} (K_{v}^{nr}, A)) .

There is a commutative diagram with exact rows:

E (Kv ) E(Kv ) H1(Kv,E [n])

×δr×0nvesn E (Knr ) E(Kvnr ) H1(Knr,E [n])
v v

The bottom map

\times n

is surjective

\forall v \notin S

(see Theorem 9.8).

Therefore

\begin{array}{l} S^{(n)} (E ∕ K) & \subset H^{1} (K, E [n]; S) \\ ≅ H^{1} (K, μ_{n}; S) \times H^{1} (K, μ_{n}; S) \end{array}

But

\begin{array}{l} H^{1} (K, μ_{n}; S) & = \ker (\frac{K^{*}}{{(K^{*})}^{n}} \to \prod_{v \notin S} \frac{{(K_{v}^{nr})}^{*}}{{(K_{v}^{nr})}^{* n}}) \\ \subset K (S, n) \end{array}

which is finite by Lemma 11.4. □

15 Galois Cohomology