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arXiv · 2411.11215

Hypergeometric $\mathcal D$-modules and exponential sums for reductive groups

Lei Fu·Xuanyou Li
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Abstract

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.

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Lei Fu, Xuanyou Li. 2024-11-18. Hypergeometric $\mathcal D$-modules and exponential sums for reductive groups. https://arxiv.org/abs/2411.11215

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