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Examples of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K3$ surfaces with real multiplication

Part of: Surfaces and higher-dimensional varieties Arithmetic algebraic geometry Cycles and subschemes

Published online by Cambridge University Press:  01 August 2014

Andreas-Stephan Elsenhans  and
Jörg Jahnel
Andreas-Stephan Elsenhans
Affiliation:
School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia email stephan@maths.usyd.edu.au
Jörg Jahnel
Affiliation:
Département Mathematik, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany email jahnel@mathematik.uni-siegen.de

Abstract

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We construct explicit $K3$ surfaces over $\mathbb{Q}$ having real multiplication. Our examples are of geometric Picard rank 16. The standard method for the computation of the Picard rank provably fails for the surfaces constructed.

MSC classification

Secondary: 14J28: $K3$ surfaces and Enriques surfaces 14C30: Transcendental methods, Hodge theory , Hodge conjecture 11G15: Complex multiplication and moduli of abelian varieties 14C22: Picard groups

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