Relative Brauer groups of torsors of period two

TL;DR

This paper investigates the computation of the relative Brauer group for torsors of period two under elliptic curves, linking it to the rational points on the curve and extending previous work to cases with unequal period and index.

Contribution

It introduces a method to compute the relative Brauer group of torsors of period two by reducing the problem to finding generators of the elliptic curve's rational points, extending prior results.

Findings

Reduction of the Brauer group computation to rational point generation

Extension of existing methods to torsors with unequal period and index

Numerical examples demonstrating the approach

Abstract

We consider the problem of computing the relative Brauer group of a torsor of period 2 under an elliptic curve E. We show how this problem can be reduced to finding a set of generators for the group of rational points on E. This extends work of Haile and Han to the case of torsors with unequal period and index. Several numerical examples are given.