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Krichever modules for difference and differential equations. (English) Zbl 1086.12001
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, 207-225 (2004).
The construction of Drinfeld modules involves function fields with a Frobenius endomorphism. If the latter is replaced with a differential operator, one gets Krichever modules, which are a step towards Geometric Langlands Theory for differential equations. In this paper, the authors develop a similar theory for difference and \(q\)-difference equations and consequences for the classification of differential, difference and \(q\)-difference abelian extensions, including an analogue of the Kronecker-Weber theorem.
For the entire collection see [Zbl 1061.00006].

MSC:
12H05 Differential algebra
12H10 Difference algebra
39A05 General theory of difference equations
39A13 Difference equations, scaling (\(q\)-differences)
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