Boundedness of the \(p\)-primary torsion of the Brauer group of an abelian variety. (English) Zbl 1541.14034
Summary: We prove that the \(p^\infty \)-torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic \(p>0\) is bounded. This answers a (variant of a) question asked by A. N. Skorobogatov and Y. G. Zarhin [J. Algebr. Geom. 17, No. 3, 481–502 (2008; Zbl 1157.14008)] for abelian varieties. To do this, we prove a ‘flat Tate conjecture’ for divisors. We also study other geometric Galois-invariant \(p^\infty \)-torsion classes of the Brauer group which are not in the transcendental Brauer group. These classes, in contrast with our main theorem, can be infinitely \(p\)-divisible. We explain how the existence of these \(p\)-divisible towers is naturally related to the failure of surjectivity of specialisation morphisms of Néron-Severi groups in characteristic \(p\).
MSC:
| 14F22 | Brauer groups of schemes |
| 19E15 | Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) |
| 14F42 | Motivic cohomology; motivic homotopy theory |
| 14C25 | Algebraic cycles |
| 14F30 | \(p\)-adic cohomology, crystalline cohomology |
Citations:
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