Families of twists of tuples of hyperelliptic curves

TL;DR

This paper constructs families of hyperelliptic curves with large Mordell-Weil ranks by demonstrating the existence of a rational function that simultaneously twists multiple curves, ensuring their Jacobians have positive rank.

Contribution

It introduces a method to produce families of hyperelliptic curves with positive Mordell-Weil rank through specific rational function twists, expanding the understanding of rational points on these curves.

Findings

Existence of a rational function D ensuring positive rank for twisted Jacobians.

Construction of families of hyperelliptic curves with large Mordell-Weil rank.

Application to explicit examples of hyperelliptic curves with high rank.

Abstract

Let $f \in \mathbb Q[x]$ be a square-free polynomial of degree at least $3$, $m_i$, $i=1,2,3$, odd positive integers, and $a_i$, $i=1,2,3$, non-zero rational numbers. We show the existence of a rational function $D\in\mathbb{Q}(v_1,v_2,v_3,v_4)$ such that the Jacobian of the quadratic twist of $y^2=f(x)$ and the Jacobian of the $m_i$-twist, respectively $2m_i$-twist, of $y^2=x^{m_i}+a_i^2$, $i=1,2,3$, by $D$ are all of positive Mordell-Weil ranks. As an application, we present families of hyperelliptic curves with large Mordell-Weil rank.