The growth of the rank of Abelian varieties upon extensions

Abstract

We study the growth of the rank of elliptic curves and, more generally, Abelian varieties with respect to finite extensions of number fields. First, we show that if $A$ is an Abelian variety over a number field $K$ and $L/K$ is a finite Galois extension such that ${{\mathrm{Gal}}}(L/K)$ does not have an index 2 subgroup, then ${{\mathrm{rk}}}A(L)-{{\mathrm{rk}}}A(K)$ can never be 1. We show that ${{\mathrm{rk}}}A(L)-{{\mathrm{rk}}}A(K)$ is either 0 or $\ge p-1$, where $p$ is the smallest prime divisor of $\# {{\mathrm{Gal}}}(L/K)$, and we obtain more precise results when ${{\mathrm{Gal}}}(L/K)$ is alternating, ${{\mathrm{SL}}}_2(\mathbb {F}_p)$ or ${{\mathrm{PSL}}}_2(\mathbb {F}_p)$ for $p>2$. This implies a restriction on ${{\mathrm{rk}}}E(K(E[p]))-{{\mathrm{rk}}}E(K(\zeta _p))$ when $E/K$ is an elliptic curve whose mod $p$ Galois representation is surjective. We obtain similar results for the growth of the rank over certain non-Galois extensions. Second, we show that for every $n\ge 2$ there exists an elliptic curve $E_n$ over a number field $K_n$ such that $\mathbb {Q}\otimes {{\mathrm{End}}}_\mathbb {Q}{{\mathrm{Res}}}_{K_n/\mathbb {Q}} E_n$ contains a number field of degree $2^n$. We ask whether every elliptic curve $E/K$ has infinite rank over $K\mathbb {Q}(2)$, where $\mathbb {Q}(2)$ is the compositum of all quadratic extensions of $\mathbb {Q}$. We show that if the answer is yes, then for any $n\ge 2$, there exists an elliptic curve $E_n$ over a number field $K_n$ admitting infinitely many quadratic twists whose rank is a positive multiple of $2^n$.