×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI EuDML arXiv
References:
[1] Atiyah (M.F.).— Vector bundles on elliptic curves, Proc.London.Math. Soc., 7, 414-452 (1957). · Zbl 0084.17305
[2] Collected mathematical papers of George David Birkhoff, Volume 1. Dover publications, 1968.
[3] Forster (O.), Riemannsche Fläche, Heidelberger Taschenbücher, Springer Verlag (1977). · Zbl 0381.30021
[4] Fresnel (J.), van der Put (M.), Rigid Analytic Geometry and its Applications, Progress in Math., 218, 2004. · Zbl 1096.14014
[5] van der Put (M.), Skew differential fields, differential and difference equations, Astérisque, 296, p. 191-207 (2004). · Zbl 1082.12005
[6] van der Put (M.), Reversat (M.), Krichever modules for difference and differential equations, Astérisque, 296, 197-225 (2004). · Zbl 1086.12001
[7] van der Put (M.), Singer (M.F.), Galois theory of difference equations, Lect. Notes in Math., vol 1666, Springer Verlag, 1997. · Zbl 0930.12006
[8] van der Put (M.), Singer (M.F.), Galois theory of linear differential equations, Grundlehren der mathematische Wissenschaften 328, Springer Verlag, 2003. · Zbl 1036.12008
[9] Ramis (J.-P.), Sauloy (J.), Zhang (C.), La variété des classes analytiques d’équations aux \(q\)-différences dans une classe formelle , C.R.Acad.Sci.Paris, Ser. I 338 (2004). · Zbl 1038.39011
[10] Sauloy (J.), Galois theory of fuchsian \(q\)-difference equations, Ann. Sci. Éc. Norm. Sup.\( 4^e\) série 36, no 6, p. 925-968 (2003). · Zbl 1053.39033
[11] Sauloy (J.), Algebraic construction of the Stokes sheaf for irregular linear \(q\)-difference equations, Astérisque 296, p. 227-251 (2004). · Zbl 1075.39020
[12] Sauloy (J.), La filtration canonique par les pentes d’un module aux \(q\)-différences et le gradué associé, Ann. Inst. Fourier 54, no. 1, p.181-210 (2004). · Zbl 1061.39013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.