$p$-adic hypergeometric $\mathscr{D}^{\dagger}(\infty)$-module and exponential sums on reductive groups

TL;DR

This paper introduces $p$-adic hypergeometric $ abla$-modules on reductive groups, exploring their properties as overholonomic arithmetic $ abla$-modules with Frobenius structures, and applies this to analyze hypergeometric exponential sums.

Contribution

It develops the theory of $p$-adic hypergeometric $ abla$-modules on reductive groups and connects them with $L$-functions of exponential sums, advancing the understanding of $p$-adic arithmetic $ abla$-modules.

Findings

Hypergeometric $ abla$-modules are overholonomic with Frobenius structures.

They define $F$-isocrystals over non-degenerate loci.

Application to hypergeometric exponential sums on reductive groups.

Abstract

We study the $p$-adic analogue of the $\ell$-adic hypergeometric sheaves for reductive groups, called the hypergeometric $\mathscr{D}^{\dagger}(\infty)$-modules. They are overholonomic objects in the derived category of arithmetic $\mathscr{D}$-modules with Frobenius structures. Over the non-degenerate locus, the hypergeometric $\mathscr{D}^{\dagger}(\infty)$-modules define $F$-isocrystals overconvergent along the complement of the non-degenerate locus. As an application, we use the theory of $L$-functions of overholonomic arithmetic $\mathscr{D}$-modules to study hypergeometric exponential sums on reductive groups.