The local information of difference equations
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.
Cite
@article{arxiv.1803.08611,
title = {The local information of difference equations},
author = {Moisés Herradón Cueto},
journal= {arXiv preprint arXiv:1803.08611},
year = {2019}
}
Comments
29 pages. v2: Improved the exposition, added explicit computations and explicit link to the monodromy matrix. v3: fixed an error in Proposition 3.12 and Corollary 3.13