Twists of hyperelliptic curves by integers in progressions modulo $p$

TL;DR

This paper investigates the distribution of squarefree integers that produce hyperelliptic curves with rational or integral points, focusing on their behavior modulo primes.

Contribution

It introduces new results on the distribution patterns of such integers in progressions modulo primes for hyperelliptic curves.

Findings

Distribution patterns of squarefree integers with rational points analyzed

Results on the density of such integers in residue classes modulo primes

Insights into the arithmetic of hyperelliptic curves and their twists

Abstract

Let $f(x)$ be a nonconstant polynomial with integer coefficients and nonzero discriminant. We study the distribution modulo primes of the set of squarefree integers $d$ such that the curve $dy^2=f(x)$ has a nontrivial rational or integral point.