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| Formato: | Preprint |
| Publicado: |
2021
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| Acceso en línea: | https://arxiv.org/abs/2108.00811 |
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| _version_ | 1866913575952449536 |
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| author | Morin, Adrien |
| author_facet | Morin, Adrien |
| contents | We construct a well-behaved Weil-étale complex for a large class of $\mathbb{Z}$-constructible sheaves on a regular irreducible scheme $U$ of finite type over $\mathbb{Z}$ and of dimension $1$. We then give a formula for the special value at $s=0$ of the $L$-function associated to any $\mathbb{Z}$-constructible sheaf on $U$ in terms of Euler characteristics of Weil-étale cohomology; for smooth proper curves, we obtain the formula of arXiv:2009.14504. We deduce a special value formula for Artin $L$-functions twisted by a singular irreducible scheme $X$ of finite type over $\mathbb{Z}$ and of dimension $1$. This generalizes and improves all results in arXiv:1611.01720; as a special case, we obtain a special value formula for the arithmetic zeta function of $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_00811 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Special values of $L$-functions on regular arithmetic schemes of dimension $1$ Morin, Adrien Number Theory We construct a well-behaved Weil-étale complex for a large class of $\mathbb{Z}$-constructible sheaves on a regular irreducible scheme $U$ of finite type over $\mathbb{Z}$ and of dimension $1$. We then give a formula for the special value at $s=0$ of the $L$-function associated to any $\mathbb{Z}$-constructible sheaf on $U$ in terms of Euler characteristics of Weil-étale cohomology; for smooth proper curves, we obtain the formula of arXiv:2009.14504. We deduce a special value formula for Artin $L$-functions twisted by a singular irreducible scheme $X$ of finite type over $\mathbb{Z}$ and of dimension $1$. This generalizes and improves all results in arXiv:1611.01720; as a special case, we obtain a special value formula for the arithmetic zeta function of $X$. |
| title | Special values of $L$-functions on regular arithmetic schemes of dimension $1$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2108.00811 |