Complex multiplication and Brauer groups of K3 surfaces

Abstract

Inspired by the classical theory of CM abelian varieties, in this paper we discuss the theory of complex multiplication for K3 surfaces. Let $X$ be a complex K3 surface with complex multiplication by the maximal order $\mathcal{O}_E$ of a CM field $E$. We compute the field of moduli of triples $(T(X), B, \iota)$, where $T(X)$ denotes the transcendental lattice of $X$, $B \subset \text{Br}(X)$ a finite, $\mathcal{O}_E$-invariant subgroup and $\iota \colon E \rightarrow \text{End}_{\text{Hdg}}(T(X)_{\mathbb{Q}})$ an isomorphism. If $X$ is defined over a number field $K$, we show how our results can be efficiently implemented to study the Galois-invariant part of the geometric Brauer group of $X$. As an application, we list all the possible groups that can appear as $\text{Br}(X)^{\Gamma_K}$ when $X$ has (geometric) maximal Picard rank, $K$ is the field of moduli of $(T(X)_{\mathbb{C}}, \iota)$ and $\Gamma_K$ denotes its absolute Galois group