The theory of complex multiplication for K3 surfaces

Domenico Valloni

Published 2018-04-23Version 1

Inspired by the classical theory of CM abelian varieties, in this paper we discuss the theory of complex multiplication for K3 surfaces. In particular, we compute the field of moduli of data of the form $(T(X), B, \iota)$, where $T(X)$ is the transcendental lattice of a principal K3 surface $X/ \mathbb{C}$ with CM by $E$, $B \subset \text{Br}(X)$ an admissible subgroup and $\iota \colon E \xrightarrow{\sim} \text{End}(T(X)_\mathbb{Q})$ an isomorphism, and show how these computations enable us to understand which groups can appear as $\text{Br}(\overline{X})^{\Gamma_K}$, where $X$ is a K3 surface with CM over a number field $K$ and $\Gamma_K$ the absolute Galois group of $K$.