Transcendental Brauer-Manin obstructions on singular K3 surfaces

Abstract

Let E and $E^{\text{'}}$ be elliptic curves over $Q$ with complex multiplication by the ring of integers of an imaginary quadratic field K and let $Y=\text{Kum}(E\times E^{\text{'}})$ be the minimal desingularisation of the quotient of $E\times E^{\text{'}}$ by the action of $-1$ . We study the Brauer groups of such surfaces Y and use them to furnish new examples of transcendental Brauer-Manin obstructions to weak approximation.

Keywords: 11G05; 14J28; Primary: 14G05; Secondary: 14F22.