Transcendental Brauer groups of products of CM elliptic curves

TL;DR

This paper computes the transcendental Brauer group of products of CM elliptic curves and explores its implications for rational points and obstructions on associated K3 surfaces.

Contribution

It explicitly determines the transcendental Brauer group of E×E for CM elliptic curves and links these results to Brauer-Manin obstructions on K3 surfaces.

Findings

Explicit description of transcendental Brauer groups for CM elliptic curves

Identification of Brauer-Manin obstructions to weak approximation

Application to K3 surfaces derived from elliptic curve products

Abstract

Let $L$ be a number field and let $E/L$ be an elliptic curve with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$. We use class field theory and results of Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the abelian surface $E\times E$. The results for the odd order torsion also apply to the Brauer group of the K3 surface $\textrm{Kum}(E\times E)$. We describe explicitly the elliptic curves $E/\mathbb{Q}$ with complex multiplication by $\mathcal{O}_K$ such that the Brauer group of $E\times E$ contains a transcendental element of odd order. We show that such an element gives rise to a Brauer-Manin obstruction to weak approximation on $\textrm{Kum}(E\times E)$, while there is no obstruction coming from the algebraic part of the Brauer group.