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TATE’S CONJECTURE AND THE TATE–SHAFAREVICH GROUP OVER GLOBAL FUNCTION FIELDS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 20 / Issue 3 / May 2021
- Published online by Cambridge University Press:
- 17 September 2019, pp. 1001-1022
- Print publication:
- May 2021
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- Article
- Export citation
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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 known formula for a surface.
