The field of moduli of singular K3 surfaces

TL;DR

This paper investigates the field of moduli for singular K3 and abelian surfaces, analyzing its properties over different fields, addressing non-finiteness issues, and proposing a method for explicit computation.

Contribution

It provides new insights into the field of moduli of singular K3 surfaces, including its behavior over CM fields and rationals, and introduces an explicit computational approach.

Findings

The field of moduli can be non-finite over certain fields.

Explicit methods for computing the field of moduli are developed.

The behavior over CM fields and over re analyzed.

Abstract

We study the field of moduli of singular abelian and K3 surfaces. We discuss both the field of moduli over the CM field and over $\Q$. We also discuss non-finiteness with respect to the degree of the field of moduli. Finally, we provide an explicit approach to the computation of the field of moduli.