K3 surfaces with Picard rank 20
Number Theory
2010-01-01 v3 Algebraic Geometry
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.
Keywords
Cite
@article{arxiv.0804.1558,
title = {K3 surfaces with Picard rank 20},
author = {Matthias Schuett},
journal= {arXiv preprint arXiv:0804.1558},
year = {2010}
}
Comments
19 pages, 1 table; v2: section 10 on Q-isomorphism classes added; v3: final version with minor corrections and additions