Arithmetic of singular Enriques Surfaces

TL;DR

This paper investigates the arithmetic properties of Enriques surfaces derived from singular K3 surfaces, demonstrating that their models are defined over specific number fields and analyzing Galois actions on their divisors.

Contribution

It extends known results about singular K3 surfaces to their Enriques quotients, showing they share similar arithmetic models over ring class fields.

Findings

Enriques quotients of singular K3 surfaces have models over the same ring class fields.

The study of Neron-Severi groups is key to understanding the arithmetic of these surfaces.

Galois actions on divisors of Enriques surfaces are characterized.

Abstract

We study the arithmetic of Enriques surfaces whose universal covers are singular K3 surfaces. If a singular K3 surface X has discriminant d, then it has a model over the ring class field d. Our main theorem is that the same holds true for any Enriques quotient of X. It is based on a study on Neron-Severi groups of singular K3 surfaces. We also comment on Galois actions on divisors of Enriques surfaces.