K3 surfaces over finite fields with given L-function

Taelman, Lenny

doi:10.2140/ant.2016.10.1133

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K3 surfaces over finite fields with given $L$-function

Lenny Taelman

Vol. 10 (2016), No. 5, 1133–1146

DOI: 10.2140/ant.2016.10.1133

Abstract

The zeta function of a K3 surface over a finite field satisfies a number of obvious (archimedean and $ℓ$ -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.

Keywords

K3 surfaces, zeta functions, finite fields

Mathematical Subject Classification 2010

Primary: 14J28

Secondary: 14G15, 14K22, 11G25

Milestones

Received: 17 August 2015

Revised: 27 November 2015

Accepted: 27 December 2015

Published: 28 July 2016

Authors

Lenny Taelman
Korteweg-de Vries Instituut Universiteit van Amsterdam P.O. Box 94248 1090 GE Amsterdam Netherlands

Vol. 10, No. 5, 2016