Weil-étale cohomology and zeta-values of arithmetic schemes at negative integers
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866911426081193984 |
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| author | Beshenov, Alexey |
| author_facet | Beshenov, Alexey |
| contents | Following the ideas of Flach and Morin (Doc. Math. 23 (2018), 1425--1560), we state a conjecture in terms of Weil-étale cohomology for the vanishing order and special value of the zeta function $ζ(X, s)$ at $s = n < 0$, where $X$ is a separated scheme of finite type over $\operatorname{Spec} \mathbb{Z}$. We prove that the conjecture is compatible with closed-open decompositions of schemes and with affine bundles, and consequently, that it holds for cellular schemes over certain one-dimensional bases.
This is a continuation of arXiv:2012.11034, which gives a construction of Weil-étale cohomology for $n < 0$ under the mentioned assumptions on $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_12114 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Weil-étale cohomology and zeta-values of arithmetic schemes at negative integers Beshenov, Alexey Algebraic Geometry Number Theory Following the ideas of Flach and Morin (Doc. Math. 23 (2018), 1425--1560), we state a conjecture in terms of Weil-étale cohomology for the vanishing order and special value of the zeta function $ζ(X, s)$ at $s = n < 0$, where $X$ is a separated scheme of finite type over $\operatorname{Spec} \mathbb{Z}$. We prove that the conjecture is compatible with closed-open decompositions of schemes and with affine bundles, and consequently, that it holds for cellular schemes over certain one-dimensional bases. This is a continuation of arXiv:2012.11034, which gives a construction of Weil-étale cohomology for $n < 0$ under the mentioned assumptions on $X$. |
| title | Weil-étale cohomology and zeta-values of arithmetic schemes at negative integers |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2102.12114 |