Selmer groups of elliptic curves in degree $p$ extensions

TL;DR

This paper investigates how the $p$-Selmer group of an elliptic curve changes in degree $p$ Galois extensions, linking growth to local cohomology and establishing criteria for when growth occurs.

Contribution

It provides a detailed analysis of the growth of Selmer groups in degree $p$ extensions, identifying local cohomology conditions that govern this behavior.

Findings

Growth of Selmer groups is determined by local cohomology groups.

Necessary and sufficient conditions for triviality of these groups are established.

Criteria for growth of the full $p$-Selmer group in degree $p$ extensions are provided.

Abstract

We study the growth of the Galois invariants of the $p$-Selmer group of an elliptic curve in a degree $p$ Galois extension. We show that this growth is determined by certain local cohomology groups and determine necessary and sufficient conditions for these groups to be trivial. Under certain hypotheses this allows us to give necessary and sufficient conditions for there to be growth in the full $p$-Selmer group in a degree $p$ Galois extension.