Stokes structure of wild difference modules

TL;DR

This paper establishes a Riemann--Hilbert correspondence linking wild difference modules with Stokes-filtered modules, extending classical and mild difference module correspondences.

Contribution

It formulates and proves a new Riemann--Hilbert correspondence for wild difference modules and Stokes-filtered modules, generalizing previous results.

Findings

Established a correspondence between wild difference modules and Stokes-filtered modules

Generalized the Riemann--Hilbert correspondence for mild difference modules

Extended classical Riemann--Hilbert results to wild difference modules

Abstract

We formulate and prove a Riemann--Hilbert correspondence between two categories: wild difference modules and wild Stokes-filtered $\mathscr{A}_{\rm{per}}$-modules. This correspondence is motivated by the Riemann--Hilbert correspondence for germs of meromorphic connections in one variable due to Deligne--Malgrange. It also generalizes the Riemann--Hilbert correspondence for mild difference modules.