Vol. 10, No. 5, 2016

Download this article
Download this article For screen
For printing
Recent Issues

Volume 12, 1 issue

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors' Addresses
Editors' Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Author Index
To Appear
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
K3 surfaces over finite fields with given $L$-function

Lenny Taelman

Vol. 10 (2016), No. 5, 1133–1146

The zeta function of a K3 surface over a finite field satisfies a number of obvious (archimedean and -adic) and a number of less obvious (p-adic) constraints. We consider the converse question, in the style of Honda–Tate: given a function Z satisfying all these constraints, does there exist a K3 surface whose zeta-function equals Z? Assuming semistable reduction, we show that the answer is yes if we allow a finite extension of the finite field. An important ingredient in the proof is the construction of complex projective K3 surfaces with complex multiplication by a given CM field.

K3 surfaces, zeta functions, finite fields
Mathematical Subject Classification 2010
Primary: 14J28
Secondary: 14G15, 14K22, 11G25
Received: 17 August 2015
Revised: 27 November 2015
Accepted: 27 December 2015
Published: 28 July 2016
Lenny Taelman
Korteweg-de Vries Instituut
Universiteit van Amsterdam
P.O. Box 94248
1090 GE Amsterdam