# Fields of definition of K3 surfaces with complex multiplication

## TL;DR

This paper investigates the fields over which K3 surfaces with complex multiplication can be defined, explicitly constructing models over certain abelian extensions and analyzing their Galois representations.

## Contribution

It provides explicit methods to define CM K3 surfaces over abelian extensions and studies their Galois properties and minimal fields of definition.

## Key findings

- K3 surfaces with CM can be defined over explicit abelian extensions
- Constructed models of K3 surfaces over these fields using Galois descent
- Derived bounds for minimal fields of definition based on class number and discriminant

## Abstract

Let $X/ \mathbb{C}$ be a K3 surface with complex multiplication by the ring of integers of a CM field $E$. We show that $X$ can always be defined over an Abelian extension $K/E$ explicitly determined by the discriminant form of the lattice $\mathrm{NS}(X)$. We then construct a model of $X$ over $K$ via Galois-descent and we study some of its basic properties, in particular we determine its Galois representation explicitly. Finally, we apply our results to give upper and lower bounds for a minimal field of definition for $X$ in terms of the class number of $E$ and the discriminant of $\mathrm{NS}(X)$.