Uniform bounds for rational points on hyperelliptic fibrations

TL;DR

This paper develops a uniform upper bound for the number of rational points of bounded height on hyperelliptic fibrations over the projective line, using a variant of the square-sieve method.

Contribution

It introduces a novel application of the square-sieve to obtain uniform bounds for rational points on hyperelliptic fibrations.

Findings

Established a uniform upper bound for rational points on hyperelliptic fibrations.

Demonstrated the effectiveness of the square-sieve in this geometric context.

Applicable to a broad class of surfaces with hyperelliptic fibrations.

Abstract

We apply a variant of the square-sieve to produce a uniform upper bound for the number of rational points of bounded height on a family of surfaces that admit a fibration over the projective line, whose general fibre is a hyperelliptic curve.