Tate's conjecture and the Tate-Shafarevich group over global function fields

Thomas H. Geisser

Published 2018-01-08Version 1

Let $\mathcal X$ be a regular variety, flat and proper over a complete regular curve over a finite field, such that the generic fiber $X$ is smooth. We prove that the Brauer group of $\mathcal X$ is finite if and only Tate's conjecture for divisors on $X$ holds and the Tate-Shafarevich group of the Albanese variety of $X$ is finite, generalizing a theorem of Artin and Grothendieck for surfaces to arbitrary relative dimension.