Sur la conjecture de Tate pour les diviseurs

TL;DR

This paper proves that the Tate conjecture for divisors in codimension 1 over finitely generated fields can be deduced from its validity for surfaces over the prime subfield, providing a new proof that works in all characteristics.

Contribution

It introduces a new proof of the reduction of the Tate conjecture to surfaces, applicable in both positive characteristic and characteristic zero, simplifying previous approaches.

Findings

Reduction of Tate conjecture to surfaces over prime subfield

New proof applicable in characteristic zero

Simplifies previous methods for proving the conjecture

Abstract

We prove that the Tate conjecture in codimension $1$ over a finitely generated field follows from the same conjecture for surfaces over its prime subfield. In positive characteristic, this is due to de Jong--Morrow over $\mathbf{F}_p$ and to Ambrosi for the reduction to $\mathbf{F}_p$. We give a different proof than Ambrosi's, which also works in characteristic $0$; over $\mathbf{Q}$, the reduction to surfaces follows from a simple argument using Lefschetz's $(1,1)$ theorem.