#### Vol. 6, No. 2, 2012

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Arithmetic of singular Enriques surfaces

### Klaus Hulek and Matthias Schütt

Vol. 6 (2012), No. 2, 195–230
##### 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.

 Dedicated to the memory of Eckart Viehweg
##### Keywords
Enriques surface, singular K3 surface, elliptic fibration, Néron–Severi group, Mordell–Weil group, complex multiplication
##### Mathematical Subject Classification 2000
Primary: 14J28
Secondary: 11E16, 11G15, 11G35, 14J27