Remarks on Milnor K-theory and Tate's conjecture for divisors

TL;DR

This paper establishes an equivalence between Tate's conjecture for divisors over finite fields and a specific algebraic problem involving the third Milnor K-group of a function field in three variables.

Contribution

It links Tate's conjecture to an explicit algebraic problem in Milnor K-theory, providing a new perspective on the conjecture.

Findings

Tate's conjecture for divisors is equivalent to a problem in Milnor K-theory.

The equivalence is demonstrated over finite fields.

The work offers a new approach to understanding Tate's conjecture.

Abstract

We show that the Tate conjecture for divisors over a finite field $\mathbb F$ is equivalent to an explicit algebraic problem about the third Milnor K-group of the function field $\bar {\mathbb F}(x,y,z)$ in three variables over $\bar {\mathbb F}$.