Arithmetic of singular Enriques surfaces

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 $H\left(d\right)$. Our main theorem is that the same holds true for any Enriques quotient of $X$. It is based on a study of Néron–Severi groups of singular K3 surfaces. We also comment on Galois actions on divisors of Enriques surfaces.