Galois theory of fuchsian q-difference equations

Jacques Sauloy

Annales scientifiques de l'École Normale Supérieure (2003)

  • Volume: 36, Issue: 6, page 925-968
  • ISSN: 0012-9593

How to cite


Sauloy, Jacques. "Galois theory of fuchsian q-difference equations." Annales scientifiques de l'École Normale Supérieure 36.6 (2003): 925-968. <>.

author = {Sauloy, Jacques},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {-difference equation; Galois groups; density theorem; Tannaka duality; Riemann-Hilbert correspondence},
language = {eng},
number = {6},
pages = {925-968},
publisher = {Elsevier},
title = {Galois theory of fuchsian q-difference equations},
url = {},
volume = {36},
year = {2003},

AU - Sauloy, Jacques
TI - Galois theory of fuchsian q-difference equations
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2003
PB - Elsevier
VL - 36
IS - 6
SP - 925
EP - 968
LA - eng
KW - -difference equation; Galois groups; density theorem; Tannaka duality; Riemann-Hilbert correspondence
UR -
ER -


  1. [1] André Y., Différentielles non-commutatives et théorie de Galois différentielle ou aux différences, Ann. Scient. Éc. Norm. Sup.34 (2001) 685-739. Zbl1010.12004MR1862024
  2. [2] Arnold V.I., Ordinary Differential Equations, Dynamical Systems, Encyclopaedia of Mathematical Sciences, vol. 1, Springer-Verlag, 1980. Zbl0432.34001MR569932
  3. [3] Baranovsky V., Ginzburg V., Conjugacy Classes in Loop Groups and G-Bundles on Elliptic Curves, International Mathematics Research Notes, vol. 15, 1996. Zbl0992.20034
  4. [4] Bertrand D., Groupes algébriques linéaires et théorie de Galois différentielle, Cours de troisième cycle, Université Paris VI, 1986. 
  5. [5] Birkhoff G.D., The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations, Proc. Amer. Acad.49 (1913) 521-568. Zbl44.0391.03JFM44.0391.03
  6. [6] Borel A., Linear Algebraic Groups, Springer-Verlag, 1991. Zbl0726.20030MR1102012
  7. [7] Cano J., Ramis J.-P., Théorie de Galois différentielle, 1999, in preparation. 
  8. [8] Chevalley C., Introduction to the Theory of Algebraic Functions of One Variable, Mathematical Surveys, vol. 6, American Mathematical Society, Providence, RI, 1963. Zbl0045.32301MR181641
  9. [9] Cohen R., Difference Algebra, Interscience Press, 1965. Zbl0127.26402
  10. [10] Deligne P., Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, vol. 163, Springer-Verlag, 1970. Zbl0244.14004MR417174
  11. [11] Deligne P., Catégories tannakiennes, in: Cartier, (Eds.), Grothendieck Festschrift, vol. II, Birkhäuser, 1990. Zbl0727.14010MR1106898
  12. [12] Deligne P., Milne J., Tannakian categories, in: Deligne, (Eds.), Hodge Cycles, Motives and Shimura Varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, 1989. Zbl0477.14004
  13. [13] Di Vizio L., Arithmetic theory of q-difference equations. The q-analogue of Grothendieck–Katz conjecture on p-curvatures, Prépublication de l'Institut de Mathématiques de Jussieu, no 286, 2000. Invent. Math., submitted for publication. Zbl1023.12004
  14. [14] Etingof P.I., Galois groups and connection matrices of q-difference equations, Electronic Research Announcements of the AMS1 (1) (1995). Zbl0844.12004MR1336694
  15. [15] Gasper G., Rahman M., Basic Hypergeometric Series, Encyclopedia of Mathematics, vol. 35, Cambridge University Press, 1990. Zbl1129.33005MR1052153
  16. [16] Hendriks P.A., Algebraic aspects of linear differential and difference equations, Thesis, University of Groningen, 1996. 
  17. [17] Ince E.L., Ordinary Differential Equations, Dover Publications, 1956. Zbl0063.02971MR10757
  18. [18] Katz N.M., On the calculation of some differential Galois groups, Invent. Math.87 (1987) 13-61. Zbl0609.12025MR862711
  19. [19] Praagman C., The formal classification of linear difference equations, Proc. Kon. Ned. Ac. Wet. Ser. A86 (1983). Zbl0519.39003
  20. [20] van der Put M., Singer M.F., Galois Theory of Difference Equations, Lecture Notes in Mathematics, vol. 1666, Springer-Verlag, 1997. Zbl0930.12006MR1480919
  21. [21] Ramanujan S., Collected Works, Chelsea, 1927. MR2280847
  22. [22] Ramis J.-P., About the growth of entire functions solutions to linear algebraic q-difference equations, Annales de Fac. des Sciences de Toulouse Sér. 6I (1) (1992) 53-94. Zbl0796.39005MR1191729
  23. [23] Ramis J.-P., Fonctions θ et équations aux q-différences, Unpublished notes, Strasbourg, 1990. 
  24. [24] Braaksma B.L.J., Immink G.K., van der Put M., The Stokes Phenomenon and Hilbert's 16th Problem, World Scientific, 1996. Zbl0846.00026MR1443683
  25. [25] Ramis J.-P., Sauloy J., Zhang C., Local analytic classification of irregular q-difference equations, 2001, in preparation. 
  26. [26] Sauloy J., Théorie de Galois des équations aux q-différences fuchsiennes, Thèse, Université Paul Sabatier, Toulouse, 1999. 
  27. [27] Sauloy J., Systèmes aux q-différences singuliers réguliers : classification, matrice de connexion et monodromie, Annales de l'Institut Fourier50 (4) (2000) 1021-1071. Zbl0957.05012MR1799737
  28. [28] Sauloy J., La filtration canonique par les pentes d'un module aux q-différences, C. R. Acad. Sci. Paris (janvier 2002). Zbl0998.39011
  29. [29] Sauloy J., La filtration canonique par les pentes des modules aux q-différences et le gradué associé, 2002, in preparation. Zbl1061.39013
  30. [30] Sauloy J., Local Galois theory of irregular q-difference equations, 2002, in preparation. 
  31. [31] Sauloy J., Galois theory of fuchsian q-difference equations, Long version of the present paper, 2002, url: Zbl1036.12007MR2032530
  32. [32] Sauloy J., La filtration canonique par les pentes des modules aux q-différences et le gradué associé, Rédaction d'exposés au Groupe de Travail “Équations aux q-différences”, 2001, url: 
  33. [33] Serre J.-P., Groupes algébriques et corps de classes, Hermann, 1959. Zbl0718.14001MR103191
  34. [34] Seshadri C.S., Fibrés vectoriels sur les courbes algébriques, Astérisque, vol. 96, Société Mathématique de France, 1982. Zbl0517.14008MR699278
  35. [35] Springer T.A., Linear Algebraic Groups, Birkhäuser, 1998. Zbl0927.20024MR1642713
  36. [36] Wasow W., Asymptotic Expansions for Ordinary Differential Equations, Dover Publications, 1965. Zbl0136.08101MR203188
  37. [37] Weil A., Généralisation des fonctions abéliennes, J. Math. Pures Appl.17 (1938) 47-87. Zbl0018.06302JFM64.0361.02
  38. [38] Zhang C., Une sommation discrète pour des équations aux q-différences linéaires et à coefficients analytiques: théorie générale et exemples, in: Proceedings of the Workshop “Differential Equations and Stokes Phenomenon”, Groningen, 2001, also as a preprint of the Université Paul Sabatier, Toulouse. Zbl1041.39013

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