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Algebraic construction of the Stokes sheaf for irregular linear $$q$$-difference equations. (English) Zbl 1075.39020
Loday-Richaud, Michèle (ed.), Complex analysis, dynamical systems, summability of divergent series and Galois theories. I. Volume in honor of Jean-Pierre Ramis. Proceedings of the conference, Toulouse, France, September 22–26, 2003 held on the occasion of J.-P. Ramis’ 60th birthday. Paris: Société Mathématique de France (ISBN 2-85629-167-8/pbk). Astérisque 296, 227-251 (2004).
Summary: The local analytic classification of irregular linear$$q$$-difference equations has recently been obtained by J.-P. Ramis, J. Sauloy and C. Zhang [Local analytic classification of irregular $$q$$-difference equations (to appear)]. Their description involves a q-analog of the Stokes sheaf and theorems of Malgrange-Sibuya type and is based on a discrete summation process due to C. Zhang [Une sommation discrète pour des équatins aux $$q$$-differences linéaires et à coefficients analytiques: Théorie générale et exemples. Differential equations and the Stokes phenomenon. Proc. Conf. Groningen. 2001]. We show here another road to some of these results by algebraic means and we describe the $$q$$-Gevrey devissage of the $$q$$-Stokes sheaf by holomorphic vector bundles over an elliptic curve.
For the entire collection see [Zbl 1061.00006].

##### MSC:
 39A13 Difference equations, scaling ($$q$$-differences) 32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation) 34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
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