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Galois theory of $$q$$-difference equations. (English) Zbl 1213.39011
Summary: 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 $$0$$. A systematic treatment of classification and moduli is developed. It turns out that a difference module $$M$$ over $$K$$ induces in a functorial way a vector bundle $$v(M)$$ on the Tate curve $$E_q:=\mathbb C^*/q^{\mathbb Z}$$ that was known for modules with “integer slopes” [J. Sauloy, in: Complex analysis, dynamical systems, summability of divergent series and Galois theories. I. Astérisque 296, 227–251 (2004; Zbl 1075.39020)]. 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 $$p$$ in order to derive Atiyah’s results for elliptic curves for which the $$j$$-invariant is not algebraic over $$\mathbb F_p$$.

##### MSC:
 12H10 Difference algebra 39A13 Difference equations, scaling ($$q$$-differences) 14H60 Vector bundles on curves and their moduli
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##### References:
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