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

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$\mathbb {Q}$ 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.