The Brauer group of Kummer surfaces and torsion of elliptic curves

TL;DR

This paper investigates the Brauer group of Kummer surfaces derived from elliptic curves, establishing conditions under which this group is trivial, thereby linking geometric properties to elliptic curve torsion structures.

Contribution

It provides new results connecting the Brauer group of Kummer surfaces to the properties of elliptic curves over the rationals, especially regarding triviality conditions.

Findings

Many Kummer surfaces from elliptic curve products have trivial Brauer group.

The Brauer group of the Kummer surface relates to that of the associated abelian surface.

Results apply to elliptic curves over the rational numbers.

Abstract

We relate the Brauer group of a Kummer surface to the Brauer group of the corresponding abelian surface. For many pairs of elliptic curves over the rational numbers we prove that the Kummer surface attached to their product has trivial Brauer group.