Families of twists of tuples of hyperelliptic curves

TL;DR

The paper develops a framework to produce families of twists of hyperelliptic curves whose Jacobians have positive Mordell-Weil rank. By converting the twist-positivity condition into a Diophantine system and associating it with auxiliary elliptic curves, the authors construct explicit rational functions $D\in\mathbb{Q}(u,v_1,v_2,v_3,v_4)$ (and variants with fewer variables) that simultaneously force positive ranks for the twists $Dy^2=f(x)$ and $y^2=Dx^{m_i}+a_i^2$ (or their even-degree counterparts). Central techniques include parametric solutions to multi-equation systems, analysis of intersections of quadrics yielding elliptic curves with positive rank, and Silverman’s specialization to guarantee rank positivity for a broad set of specializations. The results yield concrete families of quadruples of twisted hyperelliptic curves with large Mordell-Weil rank, contributing to the understanding of rank behavior in higher-dimensional abelian varieties. Overall, the work extends prior constructions for elliptic and lower-genus cases to more general quadruples and mixed-genus hyperelliptic Jacobians, providing explicit D-functions and practical criteria for positive ranks.

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.

Theorems & Definitions (7)

• Proposition 2.1
• Theorem 2.2
• Example 2.3
• Theorem 2.4
• Theorem 3.1
• Theorem 3.2
• Theorem 4.1