The local information of difference equations

TL;DR

This paper develops a local theory for difference modules on the affine line, including restrictions, vanishing cycles, and a Mellin transform, paralleling concepts from D-module theory.

Contribution

It introduces a new local framework for difference modules, defining restrictions, vanishing cycles, and a Mellin transform, bridging difference equations and D-module concepts.

Findings

Equivalent description of difference modules via restrictions and isomorphisms.

Definition of vanishing cycles for difference modules.

Establishment of a local Mellin transform as an equivalence.

Abstract

We give a definition for the restriction of a difference module on the affine line to a formal neighborhood of an orbit, trying to mimic the analogous definition and properties for a D-module. We show that this definition is reasonable in two ways. First, we show that specifying a difference module on the affine line is equivalent to giving its restriction to the complement of an orbit, together with its restriction to a neighborhood of an orbit and an isomorphism between the restriction of both to the intersection. We also give a definition for vanishing cycles of a difference module and define a local Mellin transform, which is an equivalence between vanishing cycles of a difference module and nearby cycles of its Mellin transform, a D-module.