Inventiones mathematicae

, Volume 184, Issue 3, pp 499–528 | Cite as

Generalized basic hypergeometric equations

  • Julien Roques


This paper deals with regular singular generalized q-hypergeometric equations with either “large” or “small” Galois groups. In particular, we consider the fundamental problem of finding appropriate Galoisian substitutes for the usual notion of local monodromy.


Galois Group Galois Theory Algebraic Subgroup Connection Component Local Monodromy 
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  1. 1.
    André, Y.: Différentielles non commutatives et théorie de Galois différentielle ou aux différences. Ann. Sci. Éc. Norm. Super. (4) 34(5), 685–739 (2001) zbMATHGoogle Scholar
  2. 2.
    Aomoto, K.: q-analogue of de Rham cohomology associated with Jackson integrals. I. Proc. Jpn. Acad. Ser. A Math. Sci. 66(7), 161–164 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Aomoto, K.: q-analogue of de Rham cohomology associated with Jackson integrals. II. Proc. Jpn. Acad. Ser. A Math. Sci. 66(8), 240–244 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Beukers, F.: Gauss’ hypergeometric function. In: Arithmetic and Geometry Around Hypergeometric Functions. Progr. Math., vol. 260, pp. 23–42. Birkhäuser, Basel (2007) CrossRefGoogle Scholar
  5. 5.
    Beukers, F., Brownawell, W.D., Heckman, G.: Siegel normality. Ann. Math. (2) 127(2), 279–308 (1988) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Beukers, F., Heckman, G.: Monodromy for the hypergeometric function n F n−1. Invent. Math. 95(2), 325–354 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Casale, G., Roques, J.: Integrability by discrete quadratures (submitted) Google Scholar
  8. 8.
    Casale, G., Roques, J.: Dynamics of rational symplectic mappings and difference Galois theory. Int. Math. Res. Not. IMRN, pages Art. ID rnn 103, 23 (2008) Google Scholar
  9. 9.
    Chatzidakis, Z., Hardouin, C., Singer, M.F.: On the definitions of difference Galois groups. In: Model Theory with Applications to Algebra and Analysis, vol. 1. London Math. Soc. Lecture Note Ser., vol. 349, pp. 73–109. Cambridge University Press, Cambridge (2008) CrossRefGoogle Scholar
  10. 10.
    Chatzidakis, Z., Hrushovski, E.: Model theory of difference fields. Trans. Am. Math. Soc. 351(8), 2997–3071 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Deligne, P.: Catégories tannakiennes. In: The Grothendieck Festschrift, vol. II. Progr. Math., vol. 87, pp. 111–195. Birkhäuser Boston, Cambridge (1990) Google Scholar
  12. 12.
    Di Vizio, L.: Arithmetic theory of q-difference equations: the q-analogue of Grothendieck-Katz’s conjecture on p-curvatures. Invent. Math. 150(3), 517–578 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Di Vizio, L., Hardouin, C.: Algebraic and differential Galois groups of q-difference equations (2010, submitted).
  14. 14.
    Drinfel’d, V.G.: Quasi-Hopf algebras. Leningrad J. 1(6), 1419–1457 (1990) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Duval, A., Mitschi, C.: Matrices de Stokes et groupe de Galois des équations hypergéométriques confluentes généralisées. Pac. J. Math. 138(1), 25–56 (1989) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Etingof, P.I.: Galois groups and connection matrices of q-difference equations. Electron. Res. Announc. Am. Math. Soc. 1(1), 1–9 (electronic) (1995) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Franke, C.H.: Picard-Vessiot theory of linear homogeneous difference equations. Trans. Am. Math. Soc. 108, 491–515 (1963) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 96. Cambridge University Press, Cambridge (2004). With a foreword by Richard Askey zbMATHCrossRefGoogle Scholar
  19. 19.
    Goursat, É.: Leñons sur les Síries Hypergíomítriques et sur Quelques Fonctions Qui s’y Rattachent. Hermann, Paris (1936) Google Scholar
  20. 20.
    Hardouin, C., Singer, M.F.: Differential Galois theory of linear difference equations. Math. Ann. 342(2), 333–377 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Katz, N.M.: Exponential Sums and Differential Equations. Annals of Mathematics Studies, vol. 124. Princeton University Press, Princeton (1990) zbMATHGoogle Scholar
  22. 22.
    Klein, F.: Vorlesungen ber die hypergeometrische Funktion. Springer, Berlin, Heidelberg, New York (1933) Google Scholar
  23. 23.
    Kohno, T.: Monodromy representations of braid groups and Yang-Baxter equations. Ann. Inst. Fourier (Grenoble) 37(4), 139–160 (1987) MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kohno, T.: Linear representations of braid groups and classical Yang-Baxter equations. In: Braids, Santa Cruz, CA, 1986. Contemp. Math., vol. 78, pp. 339–363. Am. Math. Soc., Providence (1988) Google Scholar
  25. 25.
    Kostant, B.: A characterization of the classical groups. Duke Math. J. 25, 107–123 (1958) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Krattenthaler, C., Rivoal, T., Zudilin, W.: Séries hypergéométriques basiques, q-analogues des valeurs de la fonction zêta et séries d’Eisenstein. J. Inst. Math. Jussieu 5(1), 53–79 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Mitschi, C.: Differential Galois groups of confluent generalized hypergeometric equations: an approach using Stokes multipliers. Pac. J. Math. 176(2), 365–405 (1996) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Van der Put, M., Reversat, M.: Galois theory of q-difference equations. Preprint: (2005)
  29. 29.
    Van der Put, M., Singer, M.F.: Galois Theory of Difference Equations. Lecture Notes in Mathematics, vol. 1666. Springer, Berlin (1997) zbMATHGoogle Scholar
  30. 30.
    Ramis, J.-P., 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, pp. 167–193. Res. Inst. Math. Sci. (RIMS), Kyoto (2007) Google Scholar
  31. 31.
    Ramis, J.-P., Sauloy, J.: The q-analogue of the wild fundamental group (II). Astérisque 323, 301–324 (2009) MathSciNetGoogle Scholar
  32. 32.
    Ramis, J.-P., Sauloy, J., Zhang, C.: La variété des classes analytiques d’équations aux q-différences dans une classe formelle. C. R. Math. Acad. Sci. Paris 338(4), 277–280 (2004) MathSciNetzbMATHGoogle Scholar
  33. 33.
    Ramis, J.-P., Sauloy, J., Zhang, C.: Développement asymptotique et sommabilité des solutions des équations linéaires aux q-différences. C. R. Math. Acad. Sci. Paris 342(7), 515–518 (2006) MathSciNetzbMATHGoogle Scholar
  34. 34.
    Sabbah, C.: Systèmes holonomes d’équations aux q-différences. In: D-modules and Microlocal Geometry, Lisbon, 1990, pp. 125–147. de Gruyter, Berlin (1993) Google Scholar
  35. 35.
    Sauloy, J.: Systèmes aux q-différences singuliers réguliers: classification, matrice de connexion et monodromie. Ann. Inst. Fourier (Grenoble) 50(4), 1021–1071 (2000) MathSciNetzbMATHGoogle Scholar
  36. 36.
    Sauloy, J.: Galois theory of Fuchsian q-difference equations. Ann. Sci. Éc. Norm. Super. (4) 36(6), 925–968 (2003) MathSciNetzbMATHGoogle Scholar
  37. 37.
    Sauloy, J.: La filtration canonique par les pentes d’un module aux q-différences et le gradué associé. Ann. Inst. Fourier (Grenoble) 54(1), 181–210 (2004) MathSciNetGoogle Scholar
  38. 38.
    Schwarz, H.A.: Über diejenigen fälle in welchen die gaussische hypergeometrische reiheeiner algebraische funktion ihres vierten elementes darstellt. Crelle J 75, 292–335 (1873) CrossRefGoogle Scholar
  39. 39.
    Slater, L.J.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966) zbMATHGoogle Scholar
  40. 40.
    Tarasov, V., Varchenko, A.: Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups. Astérisque (246):vi+135, 1997 Google Scholar
  41. 41.
    Zarkhin, Yu.G.: Weights of simple Lie algebras in the cohomology of algebraic varieties. Izv. Akad. Nauk SSSR Ser. Mat. 48(2), 264–304 (1984) MathSciNetGoogle Scholar

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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Institut Fourier - UMR CNRS 5582Université Grenoble ISt Martin d’Hères cedexFrance

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