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 group permutation‐isomorphic to prescribed G (in short, “ ‐extensions”). In particular, for alternating groups (an infinite family of) projective linear , we show that most curves (for example) infinitely many ‐extensions, conditional only on parity conjecture. More provide theoretical criterion, which allows deduce “many” conjecture existence geometric realizations with certain local properties.