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

TL;DR

This paper establishes explicit bounds on the transcendental Brauer group of K3 surfaces with principal complex multiplication, aiding the understanding of the Brauer--Manin obstruction to the Hasse principle.

Contribution

It provides the first explicit bounds on the transcendental Brauer group for such K3 surfaces, based on the arithmetic of the CM field.

Findings

Bounds depend explicitly on the number field and CM field.

Results improve understanding of the Brauer--Manin obstruction.

Method builds on and extends Valloni's work relating Brauer groups to CM field arithmetic.

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.