Quadratic twists of genus one curves and Diophantine quintuples

TL;DR

This paper investigates rational points on specific quadratic twists of genus one curves related to Diophantine quintuples, providing asymptotic bounds on the density of such twists with rational points under certain conjectures.

Contribution

It establishes asymptotic density bounds for quadratic twists with rational points, assuming conjectures on elliptic curve ranks, linking Diophantine problems to elliptic curve theory.

Findings

Asymptotic density of twists with rational points is between 43/256 and 46/256.

Results depend on standard conjectures about elliptic curve ranks.

Provides a link between Diophantine m-tuples and elliptic curve rational points.

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 $X \rightarrow \infty$ $$\frac{43}{256}+o(1)\le \frac{\#S(X)}{2\pi(X)}\le \frac{46}{256}+o(1).$$