On geometric Brauer groups and TateShafarevich groups
Abstract
Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic $p>0$. We proved that the finiteness of the $\ell$primary part of $\mathrm{Br}(X_{K^s})^{G_K}$ for a single prime $\ell\neq p$ will imply the finiteness of the primeto$p$ part of $\mathrm{Br}(X_{K^s})^{G_K}$, generalizing a theorem of Tate and Lichtenbaum for varieties over finite fields. For an abelian variety $A$ over $K$, we proved a similar result for the TateShafarevich group of $A$, generalizing a theorem of Schneider for abelian varieties over global function fields.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.01681
 Bibcode:
 2020arXiv201201681Q
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory;
 14F22
 EPrint:
 Theorem 1.2 was proved by CadoretHuiTamagawa. The preprint will not be submitted for publication