Galois Groups of Difference Equations of Order Two on Elliptic Curves

TL;DR

This paper investigates the Galois groups of second-order difference equations on elliptic curves, providing general properties and applications that identify large classes of such equations with the full general linear group.

Contribution

It establishes new properties of difference Galois groups for order two equations on elliptic curves and applies transcendence theory to identify their structure.

Findings

Large class of discrete Lamé equations have Galois group GL_2(C)

General properties of difference Galois groups on elliptic curves

Applications to calculating specific difference Galois groups

Abstract

This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups. For instance, our results combined with a result from transcendence theory due to Schneider allow us to identify a large class of discrete Lam\'e equations with difference Galois group $\operatorname{GL}_{2}(\mathbb C)$.