#### 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
##### 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