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

2021/02/16 by Bo-Hae Im, Im, Bo-Hae, Joachim König +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.2102.07918

arxiv created 2021/08/02 · arxiv updated 2021/08/03

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.) 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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