Sur la conjecture de Tate pour les diviseurs

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.

in French language. The proof of Proposition 3 was too imprecise; I clarified it. To appear in Ess. Numb. Theory

References in corpus (1)

• A note on Tate's conjectures for abelian varieties