Families of twists of tuples of hyperelliptic curves
Beyza Mevl\"ude Amir, Mohammad Sadek, Nermine El-Sissi

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 be a square-free polynomial of degree at least , , , odd positive integers, and , , non-zero rational numbers. We show the existence of a rational function such that the Jacobian of the quadratic twist of and the Jacobian of the -twist, respectively -twist, of , , by are all of positive Mordell-Weil ranks. As an application, we present families of hyperelliptic curves with large Mordell-Weil rank.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Differential Equations and Dynamical Systems · Cryptography and Residue Arithmetic
