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

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 $\overline{\mathbb{F}}(x,y,z)$ in three variables over $\overline{\mathbb{F}}$.