Complex multiplication and Brauer groups of K3 surfaces

TL;DR

This paper explores the arithmetic and classification of Brauer groups of K3 surfaces with complex multiplication, providing algorithms to compute their Brauer groups and fields of moduli, especially for quadratic imaginary CM fields.

Contribution

It introduces an algorithm to determine Brauer groups of CM K3 surfaces over number fields, extending previous computational methods to new cases.

Findings

Algorithm computes Brauer groups for CM K3 surfaces.

Classifies fields of moduli for CM K3 surfaces.

Extends computational techniques to quadratic imaginary CM fields.

Abstract

We study K3 surfaces with complex multiplication following the classical work of Shimura on CM abelian varieties. After we translate the problem in terms of the arithmetic of the CM field and its id\`{e}les, we proceed to study some abelian extensions that arise naturally in this context. We then make use of our computations to determine the fields of moduli of K3 surfaces with CM and to classify their Brauer groups. More specifically, we provide an algorithm that given a number field $K$ and a CM number field $E$, returns a finite lists of groups which contains $\mathrm{Br}(\overline{X})^{G_K}$ for any K3 surface $X/K$ that has CM by the ring of integers of $E$. We run our algorithm when $E$ is a quadratic imaginary field (a condition that translates into $X$ having maximal Picard rank) generalizing similar computations already appearing in the literature.