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arXiv · 2209.12864

Quadratic twists of genus one curves and Diophantine quintuples

Matija Kazalicki
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Abstract

Motivated by the theory of Diophantine $m$-tuples, we study rational points on quadratic twists $H^d:d y^2=(x^2+6x-18)(-x^2+2x+2)$, where $|d|$ is a prime. If we denote by $S(X)=\{ d \in \mathbb{Z}: H^d(\mathbb{Q})\ne \emptyset, |d| \textrm{ is a prime}\textrm{ and } |d| < X\},$ then, by assuming some standard conjectures about the ranks of elliptic curves in the family of quadratic twists, we prove that as $X \rightarrow \infty$ $$\frac{43}{256}+o(1)\le \frac{\#S(X)}{2\pi(X)}\le \frac{46}{256}+o(1).$$

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Matija Kazalicki. 2022-09-26. Quadratic twists of genus one curves and Diophantine quintuples. https://arxiv.org/abs/2209.12864

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