Constructible tori over Dedekind schemes
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910929717821440 |
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| author | Morin, Adrien Suzuki, Takashi |
| author_facet | Morin, Adrien Suzuki, Takashi |
| contents | We introduce an exact category of torsion-free constructible tori and an abelian category of constructible tori over a Dedekind scheme with perfect residue fields. The first one has an explicit description as $2$-term complexes of smooth commutative group algebraic spaces. Using the second-named author's duality results arXiv:1806.07641, we prove that they are equivalent to the opposite of the categories of torsion-free $\mathbb{Z}$-constructible sheaves and all $\mathbb{Z}$-constructible sheaves, respectively. We then define $L$-functions for constructible tori over a Dedekind scheme proper over $\mathrm{Spec}(\mathbb{Z})$ in terms of their étale realizations and prove a special value formula at $s=0$ using the Weil-étale formalism developed by the first-named author in arXiv:2210.09102. This extends the results of the first-named author by removing the tame ramification hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03634 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructible tori over Dedekind schemes Morin, Adrien Suzuki, Takashi Algebraic Geometry Number Theory We introduce an exact category of torsion-free constructible tori and an abelian category of constructible tori over a Dedekind scheme with perfect residue fields. The first one has an explicit description as $2$-term complexes of smooth commutative group algebraic spaces. Using the second-named author's duality results arXiv:1806.07641, we prove that they are equivalent to the opposite of the categories of torsion-free $\mathbb{Z}$-constructible sheaves and all $\mathbb{Z}$-constructible sheaves, respectively. We then define $L$-functions for constructible tori over a Dedekind scheme proper over $\mathrm{Spec}(\mathbb{Z})$ in terms of their étale realizations and prove a special value formula at $s=0$ using the Weil-étale formalism developed by the first-named author in arXiv:2210.09102. This extends the results of the first-named author by removing the tame ramification hypothesis. |
| title | Constructible tori over Dedekind schemes |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2505.03634 |