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arXiv · 1006.1724

On the computation of the Picard group for $K3$ surfaces

Andreas-Stephan Elsenhans·Jörg Jahnel
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Abstract

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 étale 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.

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Andreas-Stephan Elsenhans, Jörg Jahnel. 2010-06-09. On the computation of the Picard group for $K3$ surfaces. https://doi.org/10.1017/s0305004111000326

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