Explicit bounds on the transcendental Brauer group of K3 surfaces with principal complex multiplication

Abstract

Let $X$ be a K3 surface defined over a number field $k$, with principal complex multiplication by a CM field $E$. We find explicit bounds, in terms of $k$ and $E$, on the size of the transcendental Brauer group $\operatorname{Br}(X)/\operatorname{Br}_1(X)$ of $X$. Bounding the size of this group is important for computing the Brauer--Manin obstruction, which is conjectured by Skorobogatov to be the only obstruction to the Hasse principle for K3 surfaces. Our methods are built on top of earlier work by Valloni, who related the group $\operatorname{Br}(X)/\operatorname{Br}_1(X)$ to the arithmetic structure of the CM field $E$. It is from this arithmetic structure that we deduce our bounds.