Transcendental Brauer groups of products of CM elliptic curves

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 $\text{Kum}(E\times E)$. 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 $\text{Kum}(E\times E)$, while there is no obstruction coming from the algebraic part of the Brauer group.

arXiv commentCorrigendum added as an appendix

References in corpus (7)

• Transcendental obstructions to weak approximation on general K3 surfaces
• On the Brauer group of diagonal quartic surfaces
• Transcendental Brauer-Manin obstruction on a pencil of elliptic curves
• Point counting on reductions of CM elliptic curves
• On good reduction of some K3 surfaces related to abelian surfaces
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Cited by in corpus (7)

• Finiteness theorems for K3 surfaces and abelian varieties of CM type
• Effective bounds for Brauer groups of Kummer surfaces over number fields
• Arithmetic of rational points and zero-cycles on products of Kummer varieties and K3 surfaces
• A uniform bound on the Brauer groups of certain log K3 surfaces
• An example of a Brauer-Manin obstruction to weak approximation at a prime with good reduction
• Arithmetic of rational points and zero-cycles on Kummer varieties
• Transcendental Brauer-Manin obstructions on singular K3 surfaces