On geometric Brauer groups and Tate-Shafarevich 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 prime-to-$p$ 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 Tate-Shafarevich group of $A$, generalizing a theorem of Schneider for abelian varieties over global function fields.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2020
- DOI:
- 10.48550/arXiv.2012.01681
- arXiv:
- arXiv:2012.01681
- Bibcode:
- 2020arXiv201201681Q
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Number Theory;
- 14F22
- E-Print:
- Theorem 1.2 was proved by Cadoret-Hui-Tamagawa. The preprint will not be submitted for publication