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

Abstract

We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non-Galois extensions whose Galois closure has a 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 (for example) 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.