2018/01/08 by Geisser, Thomas H.
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1801.02406
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 and geometrically connected. 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. We also give a formula relating the orders of the group under the assumption that they are finite, generalizing the formula given for a surface.