Rank Growth of Elliptic Curves in Non-Abelian Extensions

Abstract

Abstract Given an elliptic curve $E/\mathbb{Q}$, it is a conjecture of Goldfeld that asymptotically half of its quadratic twists will have rank zero and half will have rank one. Nevertheless, higher rank twists do occur: subject to the parity conjecture, Gouvêa and Mazur constructed $X^{1/2-\epsilon }$ twists by discriminants up to $X$ with rank at least two. For any $d\geq 3$, we build on their work to consider the rank growth of $E$ in degree $d$ $S_d$-extensions of $\mathbb{Q}$ with discriminant up to $X$. We prove that there are at least $X^{c_d-\epsilon }$ such fields where the rank grows, where $c_d$ is a positive constant that tends to $1/4$ as $d\to \infty $. Moreover, subject to a suitable parity conjecture, we obtain the same result for fields for which the rank grows by at least two.