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On classical irregular q-difference equations

Part of: Difference equations Difference and functional equations

Published online by Cambridge University Press:  25 July 2012

Julien Roques
Julien Roques*
Affiliation:
Institut Fourier, Université Grenoble 1, UMR CNRS 5582, 100 rue des Maths, BP 74, 38402 St Martin d’Hères, France (email: Julien.Roques@ujf-grenoble.fr)
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Abstract

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The primary aim of this paper is to (provide tools to) compute Galois groups of classical irregular q-difference equations. We are particularly interested in quantizations of certain differential equations that arise frequently in the mathematical and physical literature, namely confluent generalized q-hypergeometric equations and q-Kloosterman equations.

Keywords

q-difference equationsdifference Galois theory

MSC classification

Secondary: 39A13: Difference equations, scaling ($q$-differences)

Information

Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 5 , September 2012 , pp. 1624 - 1644
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[And01]André, Y., Différentielles non commutatives et théorie de Galois différentielle ou aux différences, Ann. Sci. Éc. Norm. Supér (4) 34 (2001), 685–739.CrossRefGoogle Scholar
[BBH88]Beukers, F., Brownawell, W. D. and Heckman, G., Siegel normality, Ann. of Math. (2) 127 (1988), 279–308.CrossRefGoogle Scholar
[Bou75]Bourbaki, N., Groupes et algèbres de Lie: Chapitres 7 et 8 (Hermann, Paris, 1975).Google Scholar
[CR08]Casale, G. and Roques, J., Dynamics of rational symplectic mappings and difference Galois theory, Int. Math. Res. Not. IMRN (2008), Art. ID rnn 103.Google Scholar
[CR11]Casale, G. and Roques, J., Nonintegrability by discrete quadratures, Preprint (2011), available at http://www-fourier.ujf-grenoble.fr/∼jroques/.Google Scholar
[DM81]Deligne, P. and Milne, J. S., Tannakian categories in Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900 (Springer, Berlin, 1981).Google Scholar
[DiV02]Di Vizio, L., Arithmetic theory of q-difference equations: the q-analogue of Grothendieck-Katz’s conjecture on p-curvatures, Invent. Math. 150 (2002), 517–578.CrossRefGoogle Scholar
[DM89]Duval, A. and Mitschi, C., Matrices de Stokes et groupe de Galois des équations hypergéométriques confluentes généralisées, Pacific J. Math. 138 (1989), 25–56.CrossRefGoogle Scholar
[Kat87]Katz, N. M., On the calculation of some differential Galois groups, Invent. Math. 87 (1987), 13–61.CrossRefGoogle Scholar
[Kat90]Katz, N. M., Exponential sums and differential equations, Annals of Mathematics Studies, vol. 124 (Princeton University Press, Princeton, NJ, 1990).CrossRefGoogle Scholar
[KP87]Katz, N. M. and Pink, R., A note on pseudo-CM representations and differential Galois groups, Duke Math. J. 54 (1987), 57–65.CrossRefGoogle Scholar
[Kos58]Kostant, B., A characterization of the classical groups, Duke Math. J. 25 (1958), 107–123.CrossRefGoogle Scholar
[Mit96]Mitschi, C., Differential Galois groups of confluent generalized hypergeometric equations: an approach using Stokes multipliers, Pacific J. Math. 176 (1996), 365–405.CrossRefGoogle Scholar
[RS07]Ramis, J.-P. and Sauloy, J., The q-analogue of the wild fundamental group. I, in Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies, RIMS Kôkyûroku Bessatsu, vol. B2 (Research Institute for Mathematical Sciences (RIMS), Kyoto, 2007), 167–193.Google Scholar
[RS09]Ramis, J. -P. and Sauloy, J., The q-analogue of the wild fundamental group. II, Astérisque 323 (2009), 301–324.Google Scholar
[Roq11]Roques, J., Generalized basic hypergeometric equations, Invent. Math. 184 (2011), 499–528.CrossRefGoogle Scholar
[Sau00]Sauloy, J., Systèmes aux q-différences singuliers réguliers: classification, matrice de connexion et monodromie, Ann. Inst. Fourier (Grenoble) 50 (2000), 1021–1071.CrossRefGoogle Scholar
[Sau03]Sauloy, J., Galois theory of Fuchsian q-difference equations, Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), 925–968.CrossRefGoogle Scholar
[Sau04]Sauloy, J., La filtration canonique par les pentes d’un module aux q-différences et le gradué associé, Ann. Inst. Fourier (Grenoble) 54 (2004), 181–210.CrossRefGoogle Scholar
[Ser67]Serre, J.-P., Sur les groupes de Galois attachés aux groupes p-divisibles, in Proceedings of a conference on local fields (Driebergen, 1966) (Springer, Berlin, 1967), 118–131.CrossRefGoogle Scholar
[Ser79]Serre, J.-P., Groupes algébriques associés aux modules de Hodge-Tate, Astérisque 65 (1979), 155–188 Journées de Géométrie Algébrique de Rennes (Rennes, 1978), vol. III.Google Scholar
[vdPR07]van der Put, M. and Reversat, M., Galois theory of q-difference equations, Ann. Fac. Sci. Toulouse Math. (6) 16 (2007), 665–718.CrossRefGoogle Scholar
[vdPS97]van der Put, M. and Singer, M. F., Galois theory of difference equations, Lecture Notes in Mathematics, vol. 1666 (Springer, Berlin, 1997).CrossRefGoogle Scholar