# Generalized hypergeometric arithmetic D-modules under a p-adic non-Liouvilleness condition

## TL;DR

This paper demonstrates that under a p-adic non-Liouvilleness condition, hypergeometric arithmetic D-modules can be decomposed into convolutions of rank-one modules, establishing their overholonomicity.

## Contribution

It introduces a novel decomposition of hypergeometric arithmetic D-modules into rank-one convolutions under a p-adic non-Liouvilleness condition.

## Key findings

- Hypergeometric arithmetic D-modules are described as convolutions of rank-one modules.
- Overholonomicity of hypergeometric arithmetic D-modules is established.
- The results depend on a p-adic non-Liouvilleness condition.

## Abstract

We prove that the arithmetic $\mathscr{D}$-modules associated with the $p$-adic generalized hypergeometric differential operators, under a $p$-adic non-Liouvilleness condition on parameters, are described as an iterative multiplicative convolution of (hypergeometric arithmetic) $\mathscr{D}$-modules of rank one. As a corollary, we prove the overholonomicity of hypergeometric arithmetic $\mathscr{D}$-modules under a $p$-adic non-Liouvilleness condition.