Unconditional construction of K3 surfaces over finite fields with given L-function in large characteristic

TL;DR

This paper provides an unconditional method to construct K3 surfaces over finite fields with specified L-functions, extending previous results that relied on assumptions like semistable reduction.

Contribution

It removes the need for semistable reduction assumptions by using complex K3 surfaces with CM and advanced reduction techniques to construct the desired surfaces over finite fields.

Findings

Constructs K3 surfaces with prescribed L-functions unconditionally.

Extends previous results by removing semistable reduction assumptions.

Uses advanced reduction techniques to achieve the construction.

Abstract

We give an unconditional construction of K3 surfaces over finite fields with given L-function, up to finite extensions of the base fields, under some mild restrictions on the characteristic. Previously, such results were obtained by Taelman assuming semistable reduction. The main contribution of this paper is to make Taelman's proof unconditional. We use some results of Nikulin and Bayer-Fluckiger to construct an appropriate complex projective K3 surface with CM which admits an elliptic fibration with a section, or an ample line bundle of low degree. Then using Saito's construction of strictly semistable models and applying a slight refinement of Matsumoto's good reduction criterion for K3 surfaces, we obtain a desired K3 surface over a finite field.