K3 surfaces over finite fields with given L-function

TL;DR

This paper investigates the inverse problem of whether a given zeta function satisfying certain constraints can be realized by a K3 surface over a finite field, showing existence under semi-stable reduction and field extension.

Contribution

It proves the existence of K3 surfaces with prescribed zeta functions satisfying all constraints, assuming semi-stable reduction and allowing finite field extensions.

Findings

Existence of K3 surfaces with given zeta functions under certain conditions.

Construction of complex projective K3 surfaces with specified complex multiplication.

Validation of the converse Honda-Tate style question for K3 surfaces.

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.