Arithmetic hypergeometric $\mathcal{D}$-modules and exponential sums on reductive groups

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. 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.