Rank growth of abelian varieties over certain finite Galois extensions

TL;DR

This paper demonstrates that under certain conditions, the rank of abelian varieties increases over specific Galois extensions, with explicit examples of elliptic curves showing rank growth of 2 and 3.

Contribution

It establishes new conditions under which the rank of abelian varieties grows over Galois extensions and provides explicit families of elliptic curves with controlled rank increases.

Findings

Rank increases by at least p over infinitely many Galois extensions.

Conditions on varieties and groups for rank growth are characterized.

Explicit examples of elliptic curves with rank growth of 2 and 3.

Abstract

We prove that if $f:X \rightarrow A$ is a morphism from a smooth projective variety $X$ to an abelian variety $A$ over a number field $K$, and $G$ is a subgroup of automorphisms of $X$ satisfying certain properties, and if a prime $p$ divides the order of $G$, then the rank of $A$ increases by at least $p$ over infinitely many linearly disjoint $G$-extensions $L_i/K$. We also explore the conditions on such varieties $X$ and groups $G$, with applications to Jacobian varieties, and provide two infinite families of elliptic curves with rank growth of $2$ and $3$, respectively.