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On the computation of the Picard group for K3 surfaces

Published online by Cambridge University Press:  10 June 2011

ANDREAS-STEPHAN ELSENHANS  and
JÖRG JAHNEL
ANDREAS-STEPHAN ELSENHANS
Affiliation:
Mathematisches Institut, Universität Bayreuth, Universitätsstraße 30, D-95440 Bayreuth, Germany. e-mail: Stephan.Elsenhans@uni-bayreuth.de
JÖRG JAHNEL
Affiliation:
Fachbereich 6 Mathematik, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany. e-mail: jahnel@mathematik.uni-siegen.de
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Abstract

We present a method to construct examples of K3 surfaces of geometric Picard rank 1. Our approach is a refinement of that of R. van Luijk. It is based on an analysis of the Galois module structure on étale cohomology. This allows us to abandon the original limitation to cases of Picard rank 2 after reduction modulo p. Furthermore, the use of Galois data enables us to construct examples that require significantly less computation time.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 2 , September 2011 , pp. 263 - 270
Copyright
Copyright © Cambridge Philosophical Society 2011

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