Kolyvagin systems of rank 0 and the structure of the Selmer group of elliptic curves over abelian extensions

Abstract

With the motivation to study the Selmer group of an elliptic curve, we improve the theory of Kolyvagin systems to describe the Fitting ideals of a Selmer group in the core rank zero situation. By relaxing a Selmer structure of rank zero at certain prime, we can construct an auxiliary Kolyvagin system whose localisation determines most of the Fitting ideals of the Selmer group, and all of them when the Galois representation is not self-dual. With this new theory, one can describe, in terms of the modular symbols, the Galois structure of the Selmer group of an elliptic curve $E/\mathbb{Q}$ over a finite abelian extension whose degree is coprime to $p$.