Torsors for finite group schemes of bounded height
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866916136277245952 |
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| author | Darda, Ratko Yasuda, Takehiko |
| author_facet | Darda, Ratko Yasuda, Takehiko |
| contents | Let $F$ be a global field. Let $G$ be a non trivial finite étale tame $F$-group scheme. We define height functions on the set of $G$-torsors over $F,$ which generalize the usual heights such as discriminant. As an analogue of the Malle conjecture for group schemes, we formulate a conjecture on the asymptotic behavior of the number of $G$-torsors over $F$ of bounded height. This is a special case of our more general Stacky Batyrev-Manin conjecture from arXiv:2207.03645. The conjectured asymptotic is proven for the case $G$ is commutative. When $F$ is a number field, the leading constant is expressed as a product of certain arithmetic invariants of $G$ and a volume of a space attached to $G$. Moreover, an equidistribution property of $G$-torsors in the space is established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_03642 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Torsors for finite group schemes of bounded height Darda, Ratko Yasuda, Takehiko Number Theory 11R45, 11R32, 11R34, 11G50, 11R56, 14L15 Let $F$ be a global field. Let $G$ be a non trivial finite étale tame $F$-group scheme. We define height functions on the set of $G$-torsors over $F,$ which generalize the usual heights such as discriminant. As an analogue of the Malle conjecture for group schemes, we formulate a conjecture on the asymptotic behavior of the number of $G$-torsors over $F$ of bounded height. This is a special case of our more general Stacky Batyrev-Manin conjecture from arXiv:2207.03645. The conjectured asymptotic is proven for the case $G$ is commutative. When $F$ is a number field, the leading constant is expressed as a product of certain arithmetic invariants of $G$ and a volume of a space attached to $G$. Moreover, an equidistribution property of $G$-torsors in the space is established. |
| title | Torsors for finite group schemes of bounded height |
| topic | Number Theory 11R45, 11R32, 11R34, 11G50, 11R56, 14L15 |
| url | https://arxiv.org/abs/2207.03642 |