Galois theory of q-difference equations

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Choose q∈ 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 Eq:=C*/qᶻ. 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 Fₚ. 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 Eq, corresponding to a difference module M over K, with a connection ∇ₘ. If M is regular singular, then ∇ₘ is essentially determined by the absense of singularities and `unit circle monodromy'. More precisely, the monodromy of the connection (v(M),∇ₘ) coincides with the action of two topological generators of the universal regular singular difference Galois group. For irregular difference modules, ∇ₘ will have singularities and there are various Tannakian choices for M (v(M),∇ₘ). Explicit computations are difficult, especially for the case of non integer slopes.