The growth of the rank of Abelian varieties upon extensions

TL;DR

This paper investigates how the rank of Abelian varieties, especially elliptic curves, changes over field extensions, establishing restrictions based on Galois group properties and exploring the existence of curves with large rank growth over composita of quadratic fields.

Contribution

It provides new restrictions on rank growth under certain Galois extensions and constructs examples of elliptic curves with arbitrarily large rank over specific field extensions.

Findings

Rank difference cannot be 1 for certain Galois extensions.

Existence of elliptic curves with arbitrarily large rank over composita of quadratic fields.

Restrictions on rank growth related to Galois group structure.

Abstract

We study the growth of the rank of elliptic curves and, more generally, Abelian varieties upon extensions of number fields. First, we show that if $L/K$ is a finite Galois extension of number fields such that $\Gal(L/K)$ does not have an index 2 subgroup and $A/K$ is an Abelian variety, then $\rk A(L)-\rk A(K)$ can never be 1. We obtain more precise results when $\Gal(L/K)$ is of odd order, alternating, $\SL_2(\F_p)$ or $\PSL_2(\F_p)$. This implies a restriction on $\rk E(K(E[p]))-\rk E(K(\zeta_p))$ when $E/K$ is an elliptic curve whose mod $p$ Galois representation is surjective. Similar results are obtained for the growth of the rank in certain non-Galois extensions. Second, we show that for every $n\ge2$ there exists an elliptic curve $E$ over a number field $K$ such that $\Q\otimes_\Q\Res_{K/\Q} E$ contains a number field of degree $2^n$. We ask whether every elliptic curve $E/K$ has infinite rank over $K\Q(2)$, where $\Q(2)$ is the compositum of all quadratic extensions of $\Q$. We show that if the answer is yes, then for any $n\ge2$, there exists an elliptic curve $E/K$ admitting infinitely many quadratic twists whose rank is a positive multiple of $2^n$.