Brauer-Manin obstructions for homogeneous spaces of commutative affine algebraic groups over global fields
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915531350605824 |
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| author | Đonlagić, Azur |
| author_facet | Đonlagić, Azur |
| contents | Questions related to Brauer-Manin obstructions to the Hasse principle and weak approximation for homogeneous spaces of tori over a number field are well-studied, generally using arithmetic duality theorems, starting with works of Sansuc and of Colliot-Thélène. In this article, we prove the analogous statements (and include obstructions to strong approximation over finite places) in the general case of a commutative affine group scheme $G$ of finite type over a global field in any characteristic. We also study finiteness of different variants of the second Tate-Shafarevich kernel (such as $S$-kernels and $ω$-kernels) of the Cartier dual of $G$. All this is made possible by some recent theoretical advancements in positive characteristic, namely the finiteness theorems of B. Conrad and the generalized Tate duality of Z. Rosengarten. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12127 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Brauer-Manin obstructions for homogeneous spaces of commutative affine algebraic groups over global fields Đonlagić, Azur Number Theory Algebraic Geometry 14G12, 20G30 (primary), 11G35, 14G17, 20G10 (secondary) Questions related to Brauer-Manin obstructions to the Hasse principle and weak approximation for homogeneous spaces of tori over a number field are well-studied, generally using arithmetic duality theorems, starting with works of Sansuc and of Colliot-Thélène. In this article, we prove the analogous statements (and include obstructions to strong approximation over finite places) in the general case of a commutative affine group scheme $G$ of finite type over a global field in any characteristic. We also study finiteness of different variants of the second Tate-Shafarevich kernel (such as $S$-kernels and $ω$-kernels) of the Cartier dual of $G$. All this is made possible by some recent theoretical advancements in positive characteristic, namely the finiteness theorems of B. Conrad and the generalized Tate duality of Z. Rosengarten. |
| title | Brauer-Manin obstructions for homogeneous spaces of commutative affine algebraic groups over global fields |
| topic | Number Theory Algebraic Geometry 14G12, 20G30 (primary), 11G35, 14G17, 20G10 (secondary) |
| url | https://arxiv.org/abs/2410.12127 |