Year: 2024
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Fu, Lei, Li, Xuanyou
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4404570631