The Tate Conjecture for Motivic Endomorphisms of K3 Surfaces over Finite Fields

TL;DR

This paper provides a geometric proof of the Tate conjecture for K3 surfaces over finite fields with characteristic at least 5, utilizing twisted derived equivalences to connect to finiteness results.

Contribution

It introduces a new geometric approach to prove the Tate conjecture for K3 surfaces, extending previous results with a novel method involving derived equivalences.

Findings

Proof of the Tate conjecture for K3 surfaces in characteristic ≥ 5

Use of twisted derived equivalences to establish finiteness results

Connection between geometric methods and Tate's conjecture

Abstract

The Tate conjecture for squares of K3 surfaces over finite fields was recently proved by Ito-Ito-Koshikawa. We give a more geometric proof when the characteristic is at least 5. The main idea is to use twisted derived equivalences between K3 surfaces to link the Tate conjecture to finiteness results over finite fields, in the spirit of Tate.