Page 166 -

7

Galois Cohomology and Isomorphism Classification of

Elliptic Curves over Arbitrary Fields

In Chapter 3 we saw that j(E) is an isomorphism invariant for elliptic curves defined
over algebraically closed fields. In this chapter we describe all elliptic curves over a
given field k which becomes isomorphic over k s the separable algebraic closure of k,
up to k isomorphism. This is done using the Galois group of k s over k and its action
on the automorphism group of the elliptic curve over k s . The answer is given in terms
of a certain first Galois cohomology set closely related to quadratic extensions of the
field k.
We introduce basic Galois cohomology which is used to analyse how a group
G
0
G acts on a group E. The zeroth cohomology group H (G, E) is equal to E ,the
subgroup of E consisting of all elements of E left fixed by the action of G.The
1
1
first cohomology set H (G, E) is a set with base point, i.e., H (G, E) is a pointed
set, and it has an abelian group structure when E is an abelian group. For an exact
sequence of G-groups there is a six-term exact sequence involving the three zeroth
cohomology groups and the three first cohomology sets. This six-term sequence is
used for most of the elementary calculations of Galois cohomology.

§1. Galois Theory: Theorems of Dedekind and Artin

We give a short resume of the basic properties of Galois groups and extensions which
are used in the study of elliptic curves. For details the reader should consult the book
of E. Artin, Galois Theory, Notre Dame Mathematical Lectures.
If K is an extension field of a field F, then K is a vector space over F from
the multiplication on K, and we denote the dimension dim F K of K over F by
[K : F]. For a group G of automorphisms of a field K,wedenotebyFix(G) the
fixed elements of K under G, i.e., the set of a ∈ K with s(a) = a for all s ∈ G.
Then Fix(G) is a subfield of K.
(1.1) Definition. A field extension K over F is Galois provided there is a group of
automorphisms G of K such that F = Fix(G). The group G is denoted by Gal(K/F)
and is called the Galois group of the extension K over F.