Boundedness of the $p$-primary torsion of the Brauer group of an abelian variety
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913787719712768 |
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| author | D'Addezio, Marco |
| author_facet | D'Addezio, Marco |
| contents | 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 Skorobogatov and Zarhin for abelian varieties. To do this, we prove a "flat Tate conjecture" for divisors. In the text, 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 $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_07526 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Boundedness of the $p$-primary torsion of the Brauer group of an abelian variety D'Addezio, Marco Algebraic Geometry Number Theory 14F22, 19E15, 14F42, 14C25, 14F30 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 Skorobogatov and Zarhin for abelian varieties. To do this, we prove a "flat Tate conjecture" for divisors. In the text, 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 $p$. |
| title | Boundedness of the $p$-primary torsion of the Brauer group of an abelian variety |
| topic | Algebraic Geometry Number Theory 14F22, 19E15, 14F42, 14C25, 14F30 |
| url | https://arxiv.org/abs/2201.07526 |