Elliptic (p,q)-difference modules

TL;DR

This paper explores elliptic difference modules over elliptic functions, establishing a structure theorem linking them to vector bundles on elliptic curves and proving an elliptic analogue of a number theory conjecture.

Contribution

It introduces a structure theorem for elliptic (p,q)-difference modules and connects them to the classification of vector bundles on elliptic curves, extending previous conjectures.

Findings

Established a structure theorem for elliptic (p,q)-difference modules

Connected these modules to Atiyah's classification of vector bundles

Proved an elliptic analogue of Loxton and van der Poorten's conjecture

Abstract

We study finite dimensional vector spaces over fields of elliptic functions equipped with two commuting aotomorphisms \sigma and \tau induced by isogenies of relatively prime orders. We give a structure theorem for such objects, that reveals a connection to the classification of vector bundles on elliptic curves by Atiyah. As an application we prove an elliptic analogue of a conjecture of Loxton and van der Poorten which has been recently proved by Adamczewski and Bell.