Computational problems in arithmetic of linear differential equations. Some diophantine applications

  • D. V. Chudnovsky
  • G. V. Chudnovsky
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1383)


Elliptic Curve Hypergeometric Function Elliptic Curf Linear Differential Equation Elliptic Function 
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  1. [1]
    P. Griffiths, Periods of integrals on algebraic manifolds: summary of main results and discussion of open problems, Bull. Amer. Math. Soc., 75 (1970), 228–296.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    N. Katz, Nilpotent connections and the monodromy theorem: applications of a result of Turrittin, Publ. Math. I.H.E.S., 32 (1970) 232–355.zbMATHGoogle Scholar
  3. [3]
    C.L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys. Math. Kl., 1, 1929.Google Scholar
  4. [4]
    C.L. Siegel, Transcendental Numbers, Princeton University Press, Princeton, 1949.zbMATHGoogle Scholar
  5. [5]
    A.B. Shidlovsky, The arithmetic properties of the values of analytic functions, Trudy Math. Inst. Steklov, 132 (1973), 169–202.MathSciNetGoogle Scholar
  6. [6]
    A. Baker, Transcendental Number Theory, Cambridge University Press, Cambridge, 1975.CrossRefzbMATHGoogle Scholar
  7. [7]
    G. V. Chudnovsky, On some applications of diophantine approximations, Proc. Nat'l. Acad. Sci. USA, 81 (1984), 1926–1930.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    W. M. Schmidt, Diophantine Approximations, Lecture Notes Math., v. 785, Springer, N.Y. 1980.Google Scholar
  9. [9]
    A.L. Galočhkin, Lower bounds of polynomials in the values of a certain class of analytic functions, Mat. Sb., 95 (1974), 396–417.MathSciNetGoogle Scholar
  10. [10]
    G. V. Chudnovsky, Padé approximations and the Riemann monodromy problem, in Bifurcation Phenomena in Mathematical Physics and Related topics, D. Reidel, Boston, 1980, 448–510.Google Scholar
  11. [11]
    G. V Chudnovsky, Measures of irrationality, transcendence and algebraic independence. Recent progress, in Journees Arithmetiques 1980 (Ed. by J.V. Armitage), Cambridge University Press, 1982, 11–82.Google Scholar
  12. [12]
    E. Bombieri, On G-functions, in Recent Progress in Analytic Number Theory (Ed. by H. Halberstram and C. Hooly), Academic Press, N.Y., v. 2, 1981, 1–67.Google Scholar
  13. [13]
    K. Väänänen, On linear forms of certain class of G-functions and p-adic G-functions, Acta Arith., 36 (1980), 273–295.MathSciNetzbMATHGoogle Scholar
  14. [14]
    G. V. Chudnovsky, On applications of diophantine approximations, Proc. Nat'l. Acad. Sci. USA, 81 (1984), 7261–7265.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    D.V. Chudnovsky, G.V. Chudnovsky, Applications of Padé approximations to diophantine inequalities in values of G-functions, Lecture Notes Math., v. 1135, Springer, N.Y., 1985, 9–51.zbMATHGoogle Scholar
  16. [16]
    N. Katz, Algebraic solutions of differential equations, Invent. Math., 18 (1972), 1–118.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    T. Honda, Algebraic differential equations, Symposia Mathematica, v. 24, Academic Press, N.Y., 1981, 169–204.Google Scholar
  18. [18]
    B. Dwork, Arithmetic theory of differential equations, ibid. 225–243.Google Scholar
  19. [19]
    D. V. Chudnovsky, G.V. Chudnovsky, Applications of Padé approximations to the Grothendieck conjecture on linear differential equations, Lecture Notes Math., v. 1135, Springer, N.Y. 1985, 52–100.zbMATHGoogle Scholar
  20. [20]
    D.V. Chudnovsky, G.V. Chudnovsky, Padé approximations and diophantine geometry, Proc. Nat'l. Acad. Sci. USA, 82 (1985), 2212–2216.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J.-P. Serre, Quelques applications du théoreme de densité de Chebtarev, IHES Pulb. Math., 54 (1981), 323–401.Google Scholar
  22. [22]
    G. Faltings, Eudichkeitssätze für abelsche varietäten über zahlkörpern, Invent. Math., 73 (1983), 349–366.MathSciNetCrossRefGoogle Scholar
  23. [23]
    T. Honda, On the theory of commutative formal groups, J. Math. Soc. Japan, 22 (1970), 213–246.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    D.V. Chudnovsky, G.V. Chudnovsky, p-adic properties of linear differential equations and Abelian integrals, IBM Research Report RC 10645, 7/26/84.Google Scholar
  25. [25]
    D.V. Chudnovsky, G.V. Chudnovsky, The Grothendieck conjecture and Padé approximations, Proc. Japan Acad., 61A (1985), 87–90.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    N. Katz, A conjecture in the arithmetic theory of differential equations, Bull. Soc. Math. France, 110 (1982), 203–239; corr., 347–348.MathSciNetzbMATHGoogle Scholar
  27. [27]
    D. V. Chudnovsky, G.V. Chudnovsky, A random walk in higher arithmetic; Adv. Appl. Math., 7 (1986), 101–122.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    G.V. Chudnovsky, A new method for the investigation of arithmetic properties of analytic functions, Ann. Math., 109 (1979), 353–377.MathSciNetCrossRefGoogle Scholar
  29. [29]
    C. Matthews, Some arithmetic problems on automorphisms of algebraic varieties, in Number Theory Related to Fermat's Last Theorem, Birkhauser, 1982, 309–320.Google Scholar
  30. [30]
    B. Dwork, P. Robba, Effective p-adic bounds for solutions of homogeneous linear differential equations, Trans. Amer. Math. Soc., 259 (1980), 559–577.MathSciNetzbMATHGoogle Scholar
  31. [31]
    E. Whittaker, G. Watson, Modern Analysis, Cambridge, 1927.Google Scholar
  32. [32]
    D.V. Chudnovsky, G.V. Chudnovsky, Computer assisted number theory with applications, Lecture Notes. Math., v. 1240, Springer, N.Y. 1987, 1–68.zbMATHGoogle Scholar
  33. [33]
    D.V. Chudnovsky, G.V. Chudnovsky, Remark on the nature of the spectrum of Lamé equation. Problem from transcendence theory., Lett Nuovo Cimento, 29 (1980), 545–550.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    H. Poincaré, Sur les groupes les équations linéaires, Acta Math., 5 (1884), 240–278.CrossRefGoogle Scholar
  35. [35]
    R. Fricke, F. Klein, Vorlesungen über die theorie der Automorphen Functionen, bd. 1., Teubner, 1925.Google Scholar
  36. [36]
    V. I. Smirunov, Sur les équations differentielles linéaires du second ordre et la théorie des fouctions automorphes, Bull. Soc. Math. 45 (2) (1921), 93–120, 126–135.Google Scholar
  37. [37]
    L. Bers, Quasiconformal mappings, with applications to differential equations, function theory and topology, Bull. Amer. Math. Soc., 83 (1977), 1083–1100.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    L. Keen, H.E. Rauch, A.T. Vasques, Moduli of unctured tori and the accessory parameter of Lamé's equation, Trans. Amer. Math. Soc., 255 (1979), 201–229.MathSciNetzbMATHGoogle Scholar
  39. [39]
    E. Hilb, Lineare Differentialgleichungen im komplexen Gebiet, Enzyklopädie der Math. Wissenschaften II, Band 6, Teubner, 1917, 471–562.Google Scholar
  40. [40]
    L. A. Takhtadjan, P.G. Zograf, The Liouville equation action—the generating function for accessory parameter, Funct. Anal., 19 (1975), 67–68.Google Scholar
  41. [41]
    E. G. C. Poole, Introduction to the Theory of Linear Differential Equations, Oxford, 1936.Google Scholar
  42. [42]
    A. J. Van der Poorten, A proof that Euler missed… Appery's proof of the irrationality ζ(3). Math. Intelligeneer, 1 (1978/79), 195–203.CrossRefGoogle Scholar
  43. [43]
    D.V. Chudnovsky, G.V. Chudnovsky, Padé and rational approximations to systems of functions and their arithmetic applications, Lecture Notes Math., v. 1052, Springer, N.Y., 1984, 37–84.zbMATHGoogle Scholar
  44. [44]
    D.V. Chudnovsky, G.V. Chudnovsky, The use of computer algebra for diophantine and differential equations, in Computer Algebra as a Tool for Research in Mathematics and Physics, Proceedings of the New York Conference 1984, M. Dekker, N.Y.(to appear).Google Scholar
  45. [45]
    B. Dwork, A deformation theory for the zeta function of a hypersurface. Proc. Intern. Congr. Math. Stockholm, 1962, Djursholm, 1963, 247–259.Google Scholar
  46. [46]
    C. Maclachlan, G. Rosenberg, Two-generator arithmetic Fuchsian groups, Math. Proc. Cambridge Phil. Soc., 93 (1983), 383–391.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    K. Takeuchi, Arithmetic Fuchsian groups with signature (1; e) J. Math. Soc. Japan, 35 (1983), 381–407.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    D.V. Chudnovsky, G.V. Chudnovsky, Approximations and complex multiplication according to Ramanujan, in Proceedings of the Ramanujan Centenary Conference, Ed. by R. Berndt, Academic Press, (to appear), 97pp.Google Scholar
  49. [49]
    R. Askey, Orthogonal polynomials and theta functions, Proc. of 1988 AMS Summer School on Theta-Functions (in print).Google Scholar
  50. [50]
    T.J. Stieltjes, Sur la réduction en fraction continue d'une série procédant suivant les puissances descendantes d'une variable, Ann. Fac. Sci. Toulouse, 3 (1889), 1–17. ≡ Oeuvres, t. II, Groningen, 1918, 184–200.MathSciNetCrossRefGoogle Scholar
  51. [51]
    T. J. Stieltjes, Recherches sur lest fractions continue, ibid,, 8 (1894), 1–22; 9 (1895), 1–47 ≡ Oeuvres, t. II, Groningen, 1918, 402–566.MathSciNetCrossRefGoogle Scholar
  52. [52]
    L. J. Rogers, On the representation of certain asymptotic series as convergent continued fractions, Proc. London, Math. Soc., (2), 4 (1907), 72–89.MathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    I.J. Schur, Ueber Potenzreihen die im Invern des Einheitskreises beschränkt sind, J. Reing Angev. Math., 147 (1916), 205–232; 148 (1917), 122–145.MathSciNetGoogle Scholar
  54. [54]
    L. Carlitz, Some orthogonal polynomials related to elliptic functions, Duke Math. J., 27 (1960), 443–460.MathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    G.V. Chudnovsky, The inverse scattering problem and its applications to arithmetic, algebra and transcendental numbers, Lecture Notes Physics, v. 120, Springer, N.Y., 1980, 150–198.MathSciNetCrossRefGoogle Scholar
  56. [56]
    G.V. Chudnovsky, Rational and Padé approximations to solutions of linear differential equations and the monodromy theory, in Proceedings of the Les Houches International Colloquium on Complex Analysis and Relativistic Quantum Field Theory, Lecture Notes Physics, c. 126, Springer, N.Y., 1980, 136–169.MathSciNetCrossRefGoogle Scholar
  57. [57]
    G.V. Chudnovsky, Number theoretical applications of polynomials with rational coefficients defined by extremality conditions, in Arithmetic and Geometry, Progress in Mathematics, v. 35, Birkhauser, Boston, 1983, 67–107.MathSciNetGoogle Scholar
  58. [58]
    S. Ramanujan, Modular equations and approximations to π, Collected Papers, Cambridge, 1927, 23–39.Google Scholar
  59. [59]
    A. Weil, Elliptic Functions According to Eisenstein and Kronecker, Springer, 1976.Google Scholar
  60. [60]
    C.L. Siegel, Zum Beneise des Starkschen Satzes, Invent. Math. 5 (1968), 180–191.MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    G.V. Chudnovsky, Padé approximations to the generalized hypergeometric functions I, J. Math. Pures Appl., Paris, 58 (1979), 445–476.MathSciNetzbMATHGoogle Scholar
  62. [62]
    G. Shimura, Automorphic forms and the periods of Abelian varieties, J. Math Soc. Japan, 31 (1979), 561–579.MathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    P. Deligne, Cycles de Hodge absolus et périodes des intégrales des varietés abéiennes, Bull. Soc. Math. de France, Memoire, No 2, 1980, 23–33.Google Scholar
  64. [64]
    G. Shimura, Introduction to the Arithmetic Theory of Automorphic Forms, Princeton University Press, 1971.Google Scholar
  65. [65]
    R. Morris, On the automorphic functions of the group (0, 3; l 1, l 2, l 3), Trans. Amer. Math. Soc., 7 (1906), 425–448.MathSciNetGoogle Scholar
  66. [66]
    K. Takeuchi, Arithmetic triangle groups J. Math. Soc. Japan, 29 (1977), 91–106.MathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    H.P.F. Swimerton-Dyer, Arithmetic groups, in Discrete Groups and Automorphic Functions, Academic Press, 1977, 377–401.Google Scholar
  68. [68]
    G.V. Chudnovsky, Algebraic independence of values of exponential and elliptic functions, Proceedings of the International Congress of Mathematicians, Helsinki 1979, Acad. Sci. Fennica, Helsinki, 1980, v.1, 339–350.MathSciNetGoogle Scholar
  69. [69]
    G.V. Chudnovsky, Contributions to the Theory of Transcendental Numbers, Mathematical Surveys and Monographs, v. 19, Amer. Math. Soc., Providence, R.I., 1984.zbMATHGoogle Scholar
  70. [70]
    C.H. Clemens, A Scrapbook of Complex Curve Theory, Plenum, 1980.Google Scholar
  71. [71]
    Y.I. Manin, Algebraic curves over fields with differentiation, Izv. Akd. Nauk. SSSR, Ser. Mat. 22 (1958), 737–756.MathSciNetzbMATHGoogle Scholar
  72. [72]
    Y.I. Manin, The Hasse-Witt matrix of an algebraic curve, ibid., 25 (1961), 153–172.MathSciNetGoogle Scholar
  73. [73]
    B. Gross, N. Koblitz, Gauss sums and the p-adic γ-function, Ann. Math., 109 (1979), 569–581.MathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    D.V. Chudnovsky, G.V. Chudnovsky, Sequences of numbers generated by addition in formal groups and new primality and factorization tests, Adv. Meth. 7 (1986), 385–434.MathSciNetzbMATHGoogle Scholar
  75. [75]
    P. Henrici, Applied and Computational Complex Analysis, v. 3, John Wiley, 1986.Google Scholar
  76. [76]
    A. Gerasoulis, M. Grigoriadis, L. Sun, A fast algorithm for Trummer's problem, SIAM J. Ser. Stat. Comput., 8 (1987), s135–s137.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • D. V. Chudnovsky
    • 1
  • G. V. Chudnovsky
    • 1
  1. 1.Department of MathematicsColumbia UniversityNew York

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