Hypergeometric $\mathcal D$-modules and exponential sums for reductive groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913012451901440 |
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| author | Fu, Lei Li, Xuanyou |
| author_facet | Fu, Lei Li, Xuanyou |
| contents | We define the hypergeometric exponential sum associated to a family of representations of a reductive group over a finite field. We introduce the hypergeometric $\ell$-adic sheaf to describe the hypergeometric exponential sum. Motivated by the definition of the hypergeometric sheaf, we introduce the hypergeometric $\mathcal D$-module, prove it is holonomic and estimate its rank. Using the theory of the Fourier transform for vector bundles over a general base developed by Wang, we show how the hypergeometric $\mathcal D$-module controls the general behavior of the hypergeometric sheaf. We apply our results to the estimation of the hypergeometric exponential sum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11215 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypergeometric $\mathcal D$-modules and exponential sums for reductive groups Fu, Lei Li, Xuanyou Algebraic Geometry Primary 14F10, 14F20, Secondary 14M27, 11L07 We define the hypergeometric exponential sum associated to a family of representations of a reductive group over a finite field. We introduce the hypergeometric $\ell$-adic sheaf to describe the hypergeometric exponential sum. Motivated by the definition of the hypergeometric sheaf, we introduce the hypergeometric $\mathcal D$-module, prove it is holonomic and estimate its rank. Using the theory of the Fourier transform for vector bundles over a general base developed by Wang, we show how the hypergeometric $\mathcal D$-module controls the general behavior of the hypergeometric sheaf. We apply our results to the estimation of the hypergeometric exponential sum. |
| title | Hypergeometric $\mathcal D$-modules and exponential sums for reductive groups |
| topic | Algebraic Geometry Primary 14F10, 14F20, Secondary 14M27, 11L07 |
| url | https://arxiv.org/abs/2411.11215 |