Two-cover descent on hyperelliptic curves

TL;DR

This paper introduces an algorithm for finding unramified covers of hyperelliptic curves, which helps determine the existence of rational points and enhances methods for solving such curves efficiently.

Contribution

The paper presents a novel algorithm for computing unramified covers of hyperelliptic curves, aiding in rational point detection and curve solvability analysis.

Findings

Algorithm successfully detects rational points on hyperelliptic curves.

Empty set output indicates no rational points exist.

Applications extend to genus 1 curves and those with known rational points.

Abstract

We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the hyperelliptic curve has no rational points. This provides a relatively efficiently tested criterion for solvability of hyperelliptic curves. We also discuss applications of this algorithm to curves of genus 1 and to curves with rational points.