A note on the behaviour of the Tate conjecture under finitely generated field extensions

TL;DR

This paper demonstrates that the Tate conjecture for divisors over finitely generated fields of positive characteristic can be deduced from its validity over finite fields, extending to higher codimension cycles.

Contribution

It establishes a reduction of the Tate conjecture from finitely generated fields to finite fields for divisors and higher codimension cycles.

Findings

Tate conjecture for divisors over finitely generated fields follows from finite field cases.

Results extend to cycles of higher codimension.

Provides a framework for understanding the conjecture's behavior under field extensions.

Abstract

We show that the $\ell$-adic Tate conjecture for divisors on smooth proper varieties over finitely generated fields of positive characteristic follows from the $\ell$-adic Tate conjecture for divisors on smooth projective surfaces over finite fields. Similar results for cycles of higher co-dimension are given.