Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tate's conjecture and the Tate-Shafarevich group over global function
  fields   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.

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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: cs
24
The categories of this paper are: dg-ga
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The categories of this paper are: hep-th
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The categories of this paper are: comp-gas
33
The categories of this paper are: plasm-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: solv-int