Pure and Applied Mathematics Quarterly

Volume 14 (2018)

Number 3-4

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

Pages: 515 – 527

DOI: https://dx.doi.org/10.4310/PAMQ.2018.v14.n3.a4


Emiliano Ambrosi (Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, Palaiseau, France)


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 codimension are given.

Received 15 October 2018

Published 5 November 2019