K3 surfaces with Picard number one and infinitely many rational points

van Luijk, Ronald

doi:10.2140/ant.2007.1.1

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K3 surfaces with Picard number one and infinitely many rational points

Ronald van Luijk

Vol. 1 (2007), No. 1, 1–15

DOI: 10.2140/ant.2007.1.1

Abstract

In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Néron–Severi group over an algebraic closure of the base field, is high enough, more structure is known and more can be said. However, until recently not a single explicit K3 surface was known to have geometric Picard number one. We give explicit examples of such surfaces over the rational numbers. This solves an old problem that has been attributed to Mumford. The examples we give also contain infinitely many rational points, thereby answering a question of Swinnerton-Dyer and Poonen.

Keywords

K3 surface, Néron–Severi group, Picard group, rational points, arithmetic geometry

Mathematical Subject Classification 2000

Primary: 14J28, 14C22, 14G05

Milestones

Received: 20 January 2007

Revised: 13 April 2007

Accepted: 25 April 2007

Published: 1 February 2007

Authors

Ronald van Luijk
Department of Mathematics Simon Fraser University Burnaby, BC V5A 1S6 Canada
http://www.cecm.sfu.ca/~rluijk

Vol. 1, No. 1, 2007