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

Paper

Abstract

We study the

-adic analogue of the

-adic hypergeometric sheaves for reductive groups, called the hypergeometric

-modules. They are overholonomic objects in the derived category of arithmetic

-modules with Frobenius structures. Over the non-degenerate locus, the hypergeometric

-modules define

-isocrystals overconvergent along the complement of the non-degenerate locus. As an application, we use the theory of

-functions of overholonomic arithmetic

-modules to study hypergeometric exponential sums on reductive groups.