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arXiv · 1810.04018

Rank growth of elliptic curves in nonabelian extensions

Robert J. Lemke Oliver·Frank Thorne
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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.

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Robert J. Lemke Oliver, Frank Thorne. 2018-10-09. Rank growth of elliptic curves in nonabelian extensions. https://arxiv.org/abs/1810.04018

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