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160 8. Descent and Galois Cohomology

§2. Primitive Descent Formalism

As indicated in the chapter introduction, the descent procedure will come from the
long exact sequence in Galois cohomology. It is applied to an isogeny ψ : E →
E all defined over a field k such that over a Galois extension k 1 of k the group

homomorphism ψ : E(k 1 ) → E (k 1 ) is surjective leading to the following exact
sequence of Gal(k 1 /k)-modules:
ψ
0 → P → E(k 1 ) → E(k 1 ) → 0.
where P is thekernelof ψ.

(2.1) Remark. An application of the six-term long exact sequence in Galois coho-
mology 7(4.3) yields the exact sequence of pointed sets

0 0 ψ 0 δ 1
0 → H (G, P) → H (G, E(k 1 )) → H (G, E(k 1 )) → H (G, P) →
ψ 1
1
H (G, E(k 1 )) → H (G, E(k 1 )).
where G = Gal(k 1 /k).
1
0
Here the symbol ψ is used for H (ψ) and H (ψ). The middle term is con-
1
tained in a short exact sequence where ψ H (G, E(k 1 )) denotes the kernel of ψ :
1
1
H (G, E(k 1 )) → H (G, E(k 1 )), namely
E(k) δ 1 1
0 → → H (G, P) → ψ H (G, E(k 1 )) → 0.
ψ E(k)
In most applications k 1 = k s the separable algebraic closure of k, and then G =
i
i
Gal(k 1 /k) is the full Galois group and H (G, E(k s )) is denoted by H (k, E(k s ))
1
1
and H (G, P) by H (k, P).
(2.2) Examples. We make use of two examples of ψ. Firstly, we consider ψ equal to
multiplication by n prime to the characteristic of k where P is isomorphic to (Z/nZ) 2
with trivial Galois action exactly when n E(k s ) ⊂ E(k). Secondly, we consider ψ
2
equal to ϕ : E[a, b] → E[−2a, a − 4b] as in 4(5.2) with P = T ={0,(0, 0)}.The
two related short exact sequences are
δ 1 1
0 → E(k)/nE(k) → H (k, n E(k s )) → n H (k, E(k s )) → 0
and
δ 1 1

0 → E (k)/ϕE(k) → H (k, ϕ E(k s )) → ϕ H (k, E(k s )) → 0.
Now we return to the analysis started in 6(3.1) with the curve E[a, b]overthe
rational numbers Q and relate it to the Galois cohomology formalism above. The
2

isogeny appears in exact two sequences for E = E[a, b]and E = E[−2a, a − 4b]