Rank gain of Jacobians over number field extensions with prescribed Galois groups

TL;DR

This paper studies how elliptic curves and Jacobians increase in rank over specific non-Galois number field extensions with prescribed Galois groups, providing conditions under which rank gain occurs infinitely often.

Contribution

It establishes a theoretical criterion for rank gain over G-extensions, especially for alternating and projective linear groups, conditional on the parity conjecture and geometric realizations.

Findings

Most elliptic curves gain rank over infinitely many G-extensions.

Rank gain is shown for Galois groups like alternating and projective linear groups.

Results depend on the parity conjecture and existence of certain Galois realizations.

Abstract

We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non-Galois extensions whose Galois closure has Galois group permutation-isomorphic to a prescribed group $G$ (in short, "$G$-extensions"). In particular, for alternating groups and (an infinite family of) projective linear groups $G$, we show that most elliptic curves over (e.g.) $\mathbb{Q}$ gain rank over infinitely many $G$-extensions, conditional only on the parity conjecture. More generally, we provide a theoretical criterion which allows to deduce that "many" elliptic curves gain rank over infinitely many $G$-extensions, conditional on the parity conjecture and on the existence of geometric Galois realizations with group $G$ and certain local properties.