K3 surfaces over finite fields with given L-function
Algebraic Geometry
2016-08-03 v2 Number Theory
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 semi-stable 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.
Cite
@article{arxiv.1507.08547,
title = {K3 surfaces over finite fields with given L-function},
author = {Lenny Taelman},
journal= {arXiv preprint arXiv:1507.08547},
year = {2016}
}
Comments
(v2: minor corrections, added numerical evidence by Kedlaya and Sutherland)