Weil-étale cohomology and the equivariant Tamagawa number conjecture for constructible sheaves in characteristic $p$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910705858379776 |
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| author | Morin, Adrien |
| author_facet | Morin, Adrien |
| contents | Let $X$ be a variety over a finite field. Given an order $R$ in a semi-simple algebra over the rationals and a constructible étale sheaf $F$ of $R$-modules over $X$, one can consider a natural non-commutative $L$-function associated with $F$. We prove a special value formula at negative integers for this $L$-function, expressed in terms of Weil-étale cohomology; this is a geometric analogue of, and implies, the equivariant Tamagawa number conjecture for an Artin motive and its negative twists over a global function field. It also generalizes the results of Lichtenbaum and Geisser on special values at negative integers for zeta functions of varieties, and the work of Burns--Kakde in the case of non-commutative L-functions coming from a Galois cover of varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07896 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weil-étale cohomology and the equivariant Tamagawa number conjecture for constructible sheaves in characteristic $p$ Morin, Adrien Algebraic Geometry Number Theory Let $X$ be a variety over a finite field. Given an order $R$ in a semi-simple algebra over the rationals and a constructible étale sheaf $F$ of $R$-modules over $X$, one can consider a natural non-commutative $L$-function associated with $F$. We prove a special value formula at negative integers for this $L$-function, expressed in terms of Weil-étale cohomology; this is a geometric analogue of, and implies, the equivariant Tamagawa number conjecture for an Artin motive and its negative twists over a global function field. It also generalizes the results of Lichtenbaum and Geisser on special values at negative integers for zeta functions of varieties, and the work of Burns--Kakde in the case of non-commutative L-functions coming from a Galois cover of varieties. |
| title | Weil-étale cohomology and the equivariant Tamagawa number conjecture for constructible sheaves in characteristic $p$ |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2411.07896 |