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arXiv · 2102.07918

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

Bo-Hae Im·Joachim König
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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.

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Bo-Hae Im, Joachim König. 2021-02-16. Rank gain of Jacobians over number field extensions with prescribed Galois groups. https://arxiv.org/abs/2102.07918

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