K3 surfaces with Picard rank 20

Matthias Schuett

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arXiv · 0804.1558

K3 surfaces with Picard rank 20

Matthias Schuett

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Abstract

We determine all complex K3 surfaces with Picard rank 20 over Q. Here the Neron-Severi group has rank 20 and is generated by divisors which are defined over Q. Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over Q is impossible for an elliptic K3 surface. We then apply our methods to general singular K3 surfaces, i.e. with Neron-Severi group of rank 20, but not necessarily generated by divisors over Q.

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Matthias Schuett. 2010-01-01. K3 surfaces with Picard rank 20. https://arxiv.org/abs/0804.1558

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