Brauer-Manin obstructions on hyperelliptic curves

TL;DR

This paper presents a practical algorithm for detecting Brauer-Manin obstructions on hyperelliptic curves over number fields, outperforming descent methods and effectively handling high genus cases.

Contribution

It introduces a new algorithm that computes Brauer-Manin obstructions without relying on complex class group computations, extending descent theory to torsors with restricted ramification.

Findings

Decided existence of rational points for over 99% of sampled genus ≥ 5 curves over Q

Successfully applied to a genus 50 hyperelliptic curve with a Brauer-Manin obstruction

Demonstrated efficiency and effectiveness of the algorithm in high genus cases

Abstract

We describe a practical algorithm for computing Brauer-Manin obstructions to the existence of rational points on hyperelliptic curves defined over number fields. This offers advantages over descent based methods in that its correctness does not rely on rigorous class and unit group computations of large degree number fields. We report on experiments showing it to be a very effective tool for deciding existence of rational points: Among a random samples of curves over the rational numbers of genus at least $5$ we were able to decide existence of rational points for over $99\%$ of curves. We also demonstrate its effectiveness for high genus curves, giving an example of a genus $50$ hyperelliptic curve with a Brauer-Manin obstrution to the Hasse Principle. The main theoretical development allowing for this algorithm is an extension of the descent theory for abelian torsors to a framework of torsors with restricted ramification.