An approach to the Tate conjecture for surfaces over a finite field

TL;DR

This paper reformulates the Tate conjecture for surfaces over finite fields using affine open subsets and explores three approaches to prove this reformulation, linking some methods to Gersten's conjecture.

Contribution

It introduces a new reformulation of the Tate conjecture for surfaces over finite fields and investigates multiple proof strategies, connecting them to existing conjectures.

Findings

Three proof attempts were made, each falling short.

Two approaches relate to techniques used in Gersten's conjecture.

The reformulation offers a new perspective on the Tate conjecture.

Abstract

We give a reformuation of the Tate conjecture for a surface over a finite field in terms of suitable affine open subsets. We then present three attempts to prove this reformulation, each of them falling short. Interestingly, the last two are related to techniques used in proofs of Gersten's conjecture.