Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 003, 23 pages      arXiv:1405.2002      https://doi.org/10.3842/SIGMA.2015.003
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

Galois Groups of Difference Equations of Order Two on Elliptic Curves

Thomas Dreyfus a and Julien Roques b
a) Université Paul Sabatier - Institut de Mathématiques de Toulouse, 18 route de Narbonne, 31062 Toulouse, France
b) Institut Fourier, Université Grenoble 1, CNRS UMR 5582, 100 rue des Maths, BP 74, 38402 St Martin d'Hères, France

Received August 06, 2014, in final form January 08, 2015; Published online January 13, 2015

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 Lamé equations with difference Galois group $\operatorname{GL}_{2}(\mathbb C)$.

Key words: linear difference equations; difference Galois theory; elliptic curves.

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