Zeta-values of one-dimensional arithmetic schemes at strictly negative integers
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866917142832611328 |
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| author | Beshenov, Alexey |
| author_facet | Beshenov, Alexey |
| contents | Let $X$ be an arithmetic scheme (i.e., separated, of finite type over $\operatorname{Spec} \mathbb{Z}$) of Krull dimension $1$. For the associated zeta function $ζ(X,s)$, we write down a formula for the special value at $s = n < 0$ in terms of the étale motivic cohomology of $X$ and a regulator. We prove it in the case when for each generic point $η\in X$ with $\operatorname{char} κ(η) = 0$, the extension $κ(η)/\mathbb{Q}$ is abelian. We conjecture that the formula holds for any one-dimensional arithmetic scheme.
This is a consequence of the Weil-étale formalism developed by the author in [arXiv:2012.11034] and [arXiv:2102.12114], following the work of Flach and Morin (Doc. Math. 23 (2018), 1425--1560). We also calculate the Weil-étale cohomology of one-dimensional arithmetic schemes and show that our special value formula is a particular case of the main conjecture from [arXiv:2102.12114]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_13398 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Zeta-values of one-dimensional arithmetic schemes at strictly negative integers Beshenov, Alexey Algebraic Geometry Number Theory Let $X$ be an arithmetic scheme (i.e., separated, of finite type over $\operatorname{Spec} \mathbb{Z}$) of Krull dimension $1$. For the associated zeta function $ζ(X,s)$, we write down a formula for the special value at $s = n < 0$ in terms of the étale motivic cohomology of $X$ and a regulator. We prove it in the case when for each generic point $η\in X$ with $\operatorname{char} κ(η) = 0$, the extension $κ(η)/\mathbb{Q}$ is abelian. We conjecture that the formula holds for any one-dimensional arithmetic scheme. This is a consequence of the Weil-étale formalism developed by the author in [arXiv:2012.11034] and [arXiv:2102.12114], following the work of Flach and Morin (Doc. Math. 23 (2018), 1425--1560). We also calculate the Weil-étale cohomology of one-dimensional arithmetic schemes and show that our special value formula is a particular case of the main conjecture from [arXiv:2102.12114]. |
| title | Zeta-values of one-dimensional arithmetic schemes at strictly negative integers |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2111.13398 |