On the Galois structure of Selmer groups

TL;DR

This paper studies the Galois module structure of p-primary Selmer groups of abelian varieties over number fields and uses these insights to establish bounds on Selmer rank growth in extensions.

Contribution

It provides explicit descriptions of the Galois structure of Selmer groups and derives new bounds on their rank growth over field extensions.

Findings

Explicit Galois module structure of Selmer groups obtained

New bounds on Selmer rank growth established

Conditions on A and F specified for results

Abstract

Let A be an abelian variety defined over a number field k and F a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we investigate the explicit Galois structure of the p-primary Selmer group of A over F. We also use the results so obtained to derive new bounds on the growth of the Selmer rank of A over extensions of k.