Year: 2026
Type: preprint
Abstract: For a finite family of representations of a reductive group, we define a Laurent polynomial on the group. The exponential sum associated this Laurent polynomial is called a hypergeometric exponential sum. We introduce an arithmetic hypergeometric $\mathcal D$-module to study the hypergeometric exponential sum. It is an overholonomic arithmetic $\mathcal D$-module with a Frobenius structure so that the trace of the Frobenius at a rational point is the exponential sum. We prove that over the locus where the Laurent polynomial is nondegenerate, the arithmetic hypergeometric $\mathcal {D}$-module defines an $F$-isocrystal overconvergent along the degenerate locus. As an application, we get an estimation of the hypergeometric exponential sum.
Source: arXiv (Cornell University)
Authors: Fu, Lei, Li, Xuanyou, Liu, Chenhan
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7172454397