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(d). 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
Milestones
Received: 13 March 2010
Revised: 28 November 2010
Accepted: 29 December 2010
Published: 24 June 2012
Authors
Klaus Hulek
Institut für Algebraische Geometrie
Leibniz Universität Hannover
Welfengarten 1
30167 Hannover
Germany
Matthias Schütt
Institut für Algebraische Geometrie
Leibniz Universität Hannover
Welfengarten 1
30167 Hannover
Germany