Brauer-Manin obstructions on hyperelliptic curves

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.