Relative Brauer groups of genus 1 curves

TL;DR

This paper develops methods to compute the relative Brauer group of genus 1 curves, revealing cases where it can be infinite and providing explicit descriptions for certain rational curves, connecting to elliptic curve theory.

Contribution

It introduces techniques for calculating the relative Brauer group of genus 1 curves, including a family of 'cyclic type' curves with explicit Brauer group descriptions.

Findings

The relative Brauer group can be infinite over some fields.

Explicit computation of the Brauer group for certain rational genus 1 curves.

Connection established between the pairing on elliptic curves and the period-index problem.

Abstract

In this paper we develop techniques for computing the relative Brauer group of curves, focusing particularly on the case where the genus is 1. We use these techniques to show that the relative Brauer group may be infinite (for certain ground fields) as well as to determine this group explicitly for certain curves defined over the rational numbers. To connect to previous descriptions of relative Brauer groups in the literature, we describe a family of genus 1 curves, which we call "cyclic type" for which the relative Brauer group can be shown to have a particularly nice description. In order to do this, we discuss a number of formulations of the pairing between the points on an elliptic curve and its Weil-Chat\^elet group into the Brauer group of the ground field, and draw connections to the period-index problem for genus 1 curves.