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

Galois Groups of Difference Equations of Order Two on Elliptic Curves