Elliptic curves with p-Selmer growth for all p

TL;DR

This paper demonstrates that many elliptic curves over Q have their p-Selmer groups grow in all relevant extensions, revealing extensive Selmer rank increases across various Galois extensions.

Contribution

It introduces a criterion for elliptic curves to have Selmer groups grow in all extensions of certain Galois groups, expanding understanding of Selmer rank behavior.

Findings

Existence of elliptic curves with Selmer groups increasing in all bi-quadratic extensions.

p-Selmer groups grow in every D_{2p}-extension for these curves.

Generalizations to other Galois groups are discussed.

Abstract

It is known, that for every elliptic curve over Q there exists a quadratic extension in which the rank does not go up. For a large class of elliptic curves, the same is known with the rank replaced by the 2-Selmer group. We show, however, that there exists a large supply of semistable elliptic curves E/Q whose 2-Selmer group goes up in every bi-quadratic extension and for any odd prime p, the p-Selmer group goes up in every D_{2p}-extension and every elementary abelian p-extension of rank at least 2. We provide a simple criterion for an elliptic curve over an arbitrary number field to exhibit this behaviour. We also discuss generalisations to other Galois groups.