Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 003, 23 pages      arXiv:1405.2002
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

Galois Groups of Difference Equations of Order Two on Elliptic Curves

Thomas Dreyfus a and Julien Roques b
a) Université Paul Sabatier - Institut de Mathématiques de Toulouse, 18 route de Narbonne, 31062 Toulouse, France
b) Institut Fourier, Université Grenoble 1, CNRS UMR 5582, 100 rue des Maths, BP 74, 38402 St Martin d'Hères, France

Received August 06, 2014, in final form January 08, 2015; Published online January 13, 2015

This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups. For instance, our results combined with a result from transcendence theory due to Schneider allow us to identify a large class of discrete Lamé equations with difference Galois group $\operatorname{GL}_{2}(\mathbb C)$.

Key words: linear difference equations; difference Galois theory; elliptic curves.

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  1. André Y., Différentielles non commutatives et théorie de Galois différentielle ou aux différences, Ann. Sci. École Norm. Sup. (4) 34 (2001), 685-739, math.GM/0203274.
  2. Baker A., Transcendental number theory, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990.
  3. Bertrand D., Masser D., Linear forms in elliptic integrals, Invent. Math. 58 (1980), 283-288.
  4. Bugeaud V., Groupe de Galois local des équations aux $q$-différences irrégulières, Ph.D. Thesis, Institut de Mathématiques de Toulouse, 2012.
  5. Casale G., Roques J., Dynamics of rational symplectic mappings and difference Galois theory, Int. Math. Res. Not. 2008 (2008), 103, 23 pages, arXiv:0803.3951.
  6. Casale G., Roques J., Non-integrability by discrete quadratures, J. Reine Angew. Math. 687 (2014), 87-112.
  7. 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, Cambridge University Press, Cambridge, 2008, 73-109, arXiv:0705.2975.
  8. 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, math.NT/0104178.
  9. Di Vizio L., Hardouin C., Courbures, groupes de Galois génériques et $D$-groupoï de de Galois d'un système aux $q$-différences, C. R. Math. Acad. Sci. Paris 348 (2010), 951-954.
  10. Etingof P.I., Galois groups and connection matrices of $q$-difference equations, Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 1-9.
  11. Franke C.H., Picard-Vessiot theory of linear homogeneous difference equations, Trans. Amer. Math. Soc. 108 (1963), 491-515.
  12. Franke C.H., Solvability of linear homogeneous difference equations by elementary operations, Proc. Amer. Math. Soc. 17 (1966), 240-246.
  13. Franke C.H., A note on the Galois theory of linear homogeneous difference equations, Proc. Amer. Math. Soc. 18 (1967), 548-551.
  14. Franke C.H., A characterization of linear difference equations which are solvable by elementary operations, Aequationes Math. 10 (1974), 97-104.
  15. Hardouin C., Singer M.F., Differential Galois theory of linear difference equations, Math. Ann. 342 (2008), 333-377, arXiv:0801.1493.
  16. Hartshorne R., Algebraic geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York - Heidelberg, 1977.
  17. Helmer O., Divisibility properties of integral functions, Duke Math. J. 6 (1940), 345-356.
  18. Hendriks P.A., An algorithm for computing a standard form for second-order linear $q$-difference equations, J. Pure Appl. Algebra 117/118 (1997), 331-352.
  19. Hendriks P.A., An algorithm determining the difference Galois group of second order linear difference equations, J. Symbolic Comput. 26 (1998), 445-461.
  20. Lang S., On quasi algebraic closure, Ann. of Math. 55 (1952), 373-390.
  21. Masser D., Elliptic functions and transcendence, Lecture Notes in Math., Vol. 437, Springer-Verlag, Berlin - New York, 1975.
  22. Mumford D., Tata lectures on theta. I, Progress in Mathematics, Vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983.
  23. Nguyen K.A., van der Put M., Top J., Algebraic subgroups of ${\text{GL}}_2({\mathbb{C}})$, Indag. Math. (N.S.) 19 (2008), 287-297.
  24. Nguyen P., Hypertranscedance de fonctions de Mahler du premier ordre, C. R. Math. Acad. Sci. Paris 349 (2011), 943-946.
  25. Ovchinnikov A., Wibmer M., $\sigma$-Galois theory of linear difference equations, Int. Math. Res. Not., to appear, arXiv:1304.2649.
  26. 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, B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007, 167-193, math.QA/0611521.
  27. Ramis J.-P., Sauloy J., The $q$-analogue of the wild fundamental group. II, Astérisque (2009), 301-324, arXiv:0711.4034.
  28. Ramis J.-P., Sauloy J., The $q$-analogue of the wild fundamental group and the inverse problem of the galois theory of $q$-difference equations, Ann. Sci. École Norm. Sup., to appear, arXiv:1207.0107.
  29. Roques J., Galois groups of the basic hypergeometric equations, Pacific J. Math. 235 (2008), 303-322.
  30. Sauloy J., Galois theory of Fuchsian $q$-difference equations, Ann. Sci. École Norm. Sup. 36 (2003), 925-968, math.QA/0210221.
  31. Schneider T., Arithmetische Untersuchungen elliptischer Integrale, Math. Ann. 113 (1936), 1-13.
  32. Serre J.-P., Corps locaux, Hermann, Paris, 1968.
  33. Silverman J.H., The arithmetic of elliptic curves, Graduate Texts in Mathematics, Vol. 106, 2nd ed., Springer, Dordrecht, 2009.
  34. van der Put M., Reversat M., Galois theory of $q$-difference equations, Ann. Fac. Sci. Toulouse Math. 16 (2007), 665-718, math.QA/0507098.
  35. van der Put M., Singer M.F., Galois theory of difference equations, Lecture Notes in Math., Vol. 1666, Springer-Verlag, Berlin, 1997.

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