Finiteness of K3 surfaces and the Tate conjecture

TL;DR

This paper establishes a deep connection between the Tate conjecture for K3 surfaces over algebraic closures of finite fields and the finiteness of such surfaces over finite extensions, revealing a criterion for the conjecture's validity.

Contribution

It proves an equivalence linking the Tate conjecture for K3 surfaces over algebraic closures to the finiteness of K3 surfaces over finite extensions of the base field.

Findings

Tate conjecture holds iff finitely many K3 surfaces over each finite extension

Provides a criterion connecting conjecture validity to geometric finiteness

Advances understanding of K3 surfaces over finite fields

Abstract

Given a finite field k of characteristic at least 5, we show that the Tate conjecture holds for K3 surfaces defined over the algebraic closure of k if and only if there are only finitely many K3 surfaces over each finite extension of k.