Algebra & Number Theory

Arithmetic of singular Enriques surfaces

Klaus Hulek and Matthias Schütt

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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.

Article information

Algebra Number Theory, Volume 6, Number 2 (2012), 195-230.

Received: 13 March 2010
Revised: 28 November 2010
Accepted: 29 December 2010
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 11E16: General binary quadratic forms 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22] 11G35: Varieties over global fields [See also 14G25] 14J27: Elliptic surfaces

Enriques surface singular K3 surface elliptic fibration Néron–Severi group Mordell–Weil group complex multiplication


Hulek, Klaus; Schütt, Matthias. Arithmetic of singular Enriques surfaces. Algebra Number Theory 6 (2012), no. 2, 195--230. doi:10.2140/ant.2012.6.195.

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