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8

Descent and Galois Cohomology

Central to the proof of the Mordell theorem is the idea of descent which was present
in the criterion for a group to be finitely generated, see 6(1.4). This criterion was
based on the existence of a norm, which came out of the theory of heights, and the
finiteness of the index (E(Q) :2E(Q)), or more generally (E(k) : nE(k)). In this
chapter we will study the finiteness of these indices from the point of view of Galois
cohomology with the hope of obtaining a better hold on the rank of E(Q), see 6(3.3).
These indices are orders of the cokernel of multiplication by n, and along the same
ϕ
2
lines, we consider the cokernel of the isogeny E[a, b] → E[−2a, a − 4b].
There is a new version of the descent procedure when the index is studied for
i
larger and larger n,thatis, n equals m , powers of a fixed number m which is usually
a prime. We are missing an important result at this stage of the book, namely that
multiplication by n is surjective on E(k) for k separable algebraically closed and n
prime to the characteristic of k.Inthe caseof m = 2 we know by 1(4.1) that multi-
plication by 2 is surjective for certain fields k, in particular, separable algebraically
closed fields in characteristic different from 2.
We begin by considering some examples of homogeneous curves over an elliptic
curve E.

§1. Homogeneous Spaces over Elliptic Curves

Let E be an elliptic curve over k, and let k s be a separable algebraic closure of k with
Galois group Gal(k s /k). By 7(3.9) we have a natural bijection
1
Prin (Gal(k s /k), E(k s )) → H (Gal(k s /k), E(k s )) .
In the context of elliptic curves E over k, a homogeneous space X is a curve over
k together with a map X × E → X over k defining a principal action. Over the
separable algebraic closure k s there is an isomorphism E → X, and this means that
we are considering certain curves X over k which become isomophic to E over k s .