Separable Pseudo-reductive Bands with Applications to Rational Points
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909847045275648 |
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| author | Đonlagić, Azur |
| author_facet | Đonlagić, Azur |
| contents | We extend the Galois-theoretic Borovoi-Springer interpretation of algebraic bands to a class of étale-locally represented bands on the fppf site of an arbitrary field $k$, which we call separable bands. Next, a band represented étale-locally over $k$ by a pseudo-reductive group is shown to be globally representable when $[k : k^p] = p$, with counterexamples in general.
When $k$ is a global or local field, we deduce a generalization of Borovoi's abelianization theory to separable bands represented by smooth connected algebraic groups. As an application, we prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for a homogeneous space of a pseudo-reductive group (more generally, of a smooth connected affine algebraic group with split unipotent radical) having a smooth connected geometric stabilizer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12973 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Separable Pseudo-reductive Bands with Applications to Rational Points Đonlagić, Azur Number Theory Algebraic Geometry 14G12, 14G17, 20G30 (primary), 11G35, 18G50, 20G10 (secondary) We extend the Galois-theoretic Borovoi-Springer interpretation of algebraic bands to a class of étale-locally represented bands on the fppf site of an arbitrary field $k$, which we call separable bands. Next, a band represented étale-locally over $k$ by a pseudo-reductive group is shown to be globally representable when $[k : k^p] = p$, with counterexamples in general. When $k$ is a global or local field, we deduce a generalization of Borovoi's abelianization theory to separable bands represented by smooth connected algebraic groups. As an application, we prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for a homogeneous space of a pseudo-reductive group (more generally, of a smooth connected affine algebraic group with split unipotent radical) having a smooth connected geometric stabilizer. |
| title | Separable Pseudo-reductive Bands with Applications to Rational Points |
| topic | Number Theory Algebraic Geometry 14G12, 14G17, 20G30 (primary), 11G35, 18G50, 20G10 (secondary) |
| url | https://arxiv.org/abs/2510.12973 |