Galois theory of q-difference equations

Choose $q\in \mathbb{C}$ with 0<|q|<1. The main theme of this paper is the study of linear q-difference equations over the field K of germs of meromorphic functions at 0. It turns out that a difference module M over K induces in a functorial way a vector bundle v(M) on the Tate curve $E_q:={\mathbb C}^*/q^{\mathbb Z}$. As a corollary one rediscovers Atiyah’s classification of the indecomposable vector bundles on the complex Tate curve. Linear q-difference equations are also studied in positive characteristic in order to derive Atiyah’s results for elliptic curves for which the j-invariant is not algebraic over ${\mathbb{F}}_p$. A universal difference ring and a universal formal difference Galois group are introduced. Part of the difference Galois group has an interpretation as Stokes matrices', the above moduli space is the algebraic tool to compute it. It is possible to provide the vector bundle v(M) on E_q, corresponding to a difference module M over K, with a connection $\nabla_M$. If M is regular singular, then $\nabla_M$ is essentially determined by the absense of singularities and unit circle monodromy’. More precisely, the monodromy of the connection $(v(M),{\mathrm{\nabla }}_M)$ coincides with the action of two topological generators of the universal regular singular difference Galois group. For irregular difference modules, ${\mathrm{\nabla }}_M$ will have singularities and there are various Tannakian choices for $M\mapsto (v(M),{\mathrm{\nabla }}_M)$. Explicit computations are difficult, especially for the case of non integer slopes.