Vol. 1, No. 1, 2007

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
K3 surfaces with Picard number one and infinitely many rational points

Ronald van Luijk

Vol. 1 (2007), No. 1, 1–15

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.

K3 surface, Néron–Severi group, Picard group, rational points, arithmetic geometry
Mathematical Subject Classification 2000
Primary: 14J28, 14C22, 14G05
Received: 20 January 2007
Revised: 13 April 2007
Accepted: 25 April 2007
Published: 1 February 2007
Ronald van Luijk
Department of Mathematics
Simon Fraser University
Burnaby, BC V5A 1S6