Arithmetic of singular Enriques surfaces

Hulek, Klaus; Schütt, Matthias

doi:10.2140/ant.2012.6.195

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Arithmetic of singular Enriques surfaces

Klaus Hulek and Matthias Schütt

Vol. 6 (2012), No. 2, 195–230

DOI: 10.2140/ant.2012.6.195

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

Vol. 6, No. 2, 2012