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 become isomorphic over k sthe 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 which is closely related to quadratic extensions of the field k.
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