Andreas-Stephan Elsenhans, Jörg Jahnel
Published 2010-06-09Version 1
We construct examples of $K3$ surfaces of geometric Picard rank $1$. Our method is a refinement of that of R. van Luijk. It is based on an analysis of the Galois module structure on \'etale cohomology. This allows 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 which require significantly less computation time.
On the distribution of the Picard ranks of the reductions of a $K3$ surface
Kummer surfaces and the computation of the Picard group
Determinantal quartics and the computation of the Picard group
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