Rank growth of elliptic curves in nonabelian extensions

TL;DR

This paper investigates the growth of ranks of elliptic curves in nonabelian extensions, establishing lower bounds on the number of twists with positive rank and extending previous quadratic twist results to degree d extensions.

Contribution

It generalizes the construction of high-rank twists from quadratic to degree d nonabelian extensions, providing new lower bounds on their abundance.

Findings

At least X^{c_d - ε} twists with positive rank for degree d extensions.

As d increases, the constant c_d approaches 1/4.

Under a parity conjecture, similar results hold for rank at least two.

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, Gouv\^ea 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 twists by 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 twists with positive rank, 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 twists with rank at least two.