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164 8. Descent and Galois Cohomology

(3.1) Notations. Let E be an elliptic curve over a number field k. For each place v
of k choose an embedding k → k v , and, hence, the decomposition subgroup G v =
Gal(k v /k v ) inside G = Gal(k/k). This induces a morphism ofthe short exact se-
quence associated with cohomology
E(k) f
1
1
¯
¯
0 → → H G, m E(k) −−−−→ m H G, E(k) → 0
mE(k)
 
f
 

E (k v ) 1 1
¯
¯
0 → → H G v , m E(k) −−−−→ m H G v , E(k) → 0.
mE (k v )
v v v
(3.2) Definition. The Selmer group for m-descent S (m) = S (m) (E) isthekernelof
1 1 ˇ
f defined H (G, m E(k)) → v m H (G v , E(k)), and the Tate–Sarafeviˇ c group
1
for m-descent X m = X m (E) is thekernelof f defined m H (G, E(k)) →
1
v m H (G v , E(k)).

The group E(k)/mE(k), which is our main interest, is the kernel of f ,and
the following proposition is the exact sequence for the three kernels of the three

morphisms in a composite f = f f .
(3.3) Proposition. The mapping between the cohomology short exact sequences in
(3.1) yields the short exact sequence
E(K) (m)
0 → → S (E) → X m (E) → 0
mE(K)
The finiteness of E(k)/mE(K) will follow from the finiteness of S (m) (E) and
the group S (m) (E) can be computed in some important cases. Consider the situation
where K is large enough so that m E(K) = m E(K).Thenwehave
¯

Z

2
1
H G, m E K ¯ = Hom G, m E K ¯ = Hom G, ,
mZ
and each element corresponds to an abelian extension L x /K.
1
(3.4) Assertion. Let x in H (G, m E(K)) have as corresponding abelian extension
L x /K.If x is in the Selmer group S (m) (E), then L x is unramified outside the set S
ofplaces v where either E has bad reduction or v(m)> 0.
1
Proof. Let x project as in (3.1) to (x v ) in H (G v , m E(K v )). Since f (x) = 0,
v
¯
there exists P v mod m E(K) mapping to x v .Let P v = mQ v in E(K) and form the
cocycle s → s(Q v ) − Q v to obtain a representative in the cohomology class of
x v . The field L x is generated by the coordinates of Q v and the inertia group I v acts
trivially on L x for v outside S. Hence L x is unramified outside of S, and this proves
the assertion.
Now we appeal to a basic finiteness result in algebraic number theory.