K3 surfaces with Picard rank 20

Schütt, Matthias

doi:10.2140/ant.2010.4.335

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K3 surfaces with Picard rank 20

Matthias Schütt

Vol. 4 (2010), No. 3, 335–356

DOI: 10.2140/ant.2010.4.335

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 $ℚ$ .

Keywords

singular K3 surface, Artin–Tate conjecture, complex multiplication, modular form, class group

Mathematical Subject Classification 2000

Primary: 14J28

Secondary: 11F11, 11G15, 11G25, 11R29

Milestones

Received: 21 July 2009

Revised: 14 November 2009

Accepted: 31 December 2009

Published: 5 February 2010

Authors

Matthias Schütt
Institut für Algebraische Geometrie Leibniz Universität Hannover Welfengarten 1 30167 Hannover Germany
http://www.iag.uni-hannover.de/~schuett/

Vol. 4, No. 3, 2010