Vol. 5, No. 8, 2011
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The Picard group of a $K3$ surface and its reduction modulo $p$

Andreas-Stephan Elsenhans and Jörg Jahnel
Vol. 5 (2011), No. 8, 1027–1040
DOI: 10.2140/ant.2011.5.1027
Abstract

We present a method to compute the geometric Picard rank of a K3 surface over ℚ. Contrary to a widely held belief, we show that it is possible to verify Picard rank 1 using reduction at a single prime.

Keywords
$K3$ surface, Picard group, Picard scheme, deformation, Artin approximation, Van Luijk's method
Mathematical Subject Classification 2010
Primary: 14C22
Secondary: 14D15, 14J28, 14Q10
Milestones
Received: 31 March 2010
Revised: 1 March 2011
Accepted: 1 April 2011
Published: 5 June 2012
Authors
Andreas-Stephan Elsenhans
Mathematisches Institut
Universität Bayreuth
Universitätsstraße 30
D-95440 Bayreuth
Germany
http://www.staff.uni-bayreuth.de/~btm216
Jörg Jahnel
Fachbereich 6 Mathematik
Universität Siegen
Walter-Flex-Straße 3
D-57068 Siegen
Germany
http://www.uni-math.gwdg.de/jahnel