K3 surfaces with Picard rank 20

Details K3 surfaces with Picard rank 20 K3 surfaces with Picard rank 20

2008-04-09

arxiv.org/abs/0804.1558v3

Mathematics

Number Theory

19 pages, 1 table; v2: section 10 on Q-isomorphism classes added; v3: final version with minor corrections and additions

Scientific paper

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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