p-adic Generalized Hypergeometric Equations from the Viewpoint of Arithmetic D-modules

TL;DR

This paper explores p-adic hypergeometric equations through arithmetic D-modules, demonstrating that certain hypergeometric isocrystals possess overconvergent F-isocrystal structures, advancing understanding in p-adic arithmetic geometry.

Contribution

It introduces a novel approach using multiplicative convolution of arithmetic D-modules to analyze p-adic hypergeometric equations and establishes the overconvergence of specific hypergeometric isocrystals.

Findings

Hypergeometric isocrystals with rational parameters are overconvergent F-isocrystals.

The use of multiplicative convolution provides new insights into p-adic hypergeometric equations.

The results connect hypergeometric equations with arithmetic D-module theory.

Abstract

We study the $p$-adic (generalized) hypergeometric equations by using the theory of multiplicative convolution of arithmetic $\mathscr{D}$-modules. As a result, we prove that the hypergeometric isocrystals with suitable rational parameters have a structure of overconvergent $F$-isocrystals.