K3 surfaces with Picard rank 20

authored by

Matthias Schütt

Abstract

We determine all complex K3 surfaces with Picard rank 20 over ℚ. Here the Néron-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 apply our methods to general singular K3 surfaces, that is, those with Néron-Severi group of rank 20, but not necessarily generated by divisors over ℚ.

Organisation(s)

Institute of Algebraic Geometry

Type

Article

Journal

Algebra and Number Theory

Volume

4

Pages

335-356

No. of pages

22

ISSN

1937-0652

Publication date

2010

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Algebra and Number Theory

Electronic version(s)

https://doi.org/10.2140/ant.2010.4.335 (Access: Open)