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Approximations and complex multiplication according to Ramanujan

  • D. V. Chudnovsky
  • G. V. Chudnovsky

Abstract

This talk revolves around two focuses: complex multiplications (for elliptic curves and Abelian varieties) in connection with algebraic period relations, and (diophantine) approximations to numbers related to these periods. Our starting point is Ramanujan’s works [1], [2] on approximations to π via the theory of modular and hypergeometric functions. We describe in chapter 1 Ramanujan’s original quadratic period-quasiperiod relations for elliptic curves with complex multiplication and their applications to representations of fractions of „ and other logarithms in terms of rapidly convergent nearly integral (hypergeometric) series. These representations serve as a basis of our investigation of diophantine approximations to π and other related numbers. In Chapter 2 we look at period relations for arbitrary CM-varieties following Shimura and Deligne. Our main interest lies with modular (Shimura) curves arising from arithmetic Fuchsian groups acting on H. From these we choose arithmetic triangular groups, where period relations can be expressed in the form of hyper-geometric function identities. Particular attention is devoted to two (commensurable) triangle groups, (0,3;2,6,6) and (0,3;2,4,6), arising from the quaternion algebra over Φ with the discriminant D = 2.3.

Keywords

Complex Multiplication Elliptic Curf Abelian Variety Eisenstein Series Monodromy Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    S. Ramanujan, Collected Papers, Cambridge, 1927, 23–39.Google Scholar
  2. [2]
    G.H. Hardy, Ramanujan, Cambridge, 1940.Google Scholar
  3. [3]
    G.N. Watson, Some singular moduli (I);(II);(III); (IV); Quart. J. Math. Oxford, 3 (1932), 81–98; 189–212; Proc. London Math. Soc. 40 (1936), 83–142; Acta Arithmetica, 1 (1936), 284–323.Google Scholar
  4. [4]
    J.M. Borwein, P.B. Borwein, Pi and the AGM, Wiley, 1987.Google Scholar
  5. [5]
    E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, 4 ed., Cambridge, 1927.Google Scholar
  6. [6]
    A. Weil, Elliptic Functions According to Eisenstein and Kronecker, Springer, 1976.Google Scholar
  7. [7]
    C.L. Siegel, Bestimmung der elliptischen Modulfunktionen durch eine Transformations gleichung, Abh. Math. Sem. Univ. Hamburg, 27 (1964), 32–38.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    A. Weil, Sur les périodes des Intégrales Abéliennes, Comm. Pure Appl. Math., 29 (1976), 813–819.MathSciNetzbMATHGoogle Scholar
  9. [9]
    D. Masser, Elliptic Functions and Transcendence, Lecture Notes Math., v. 437, Springer, 1975.Google Scholar
  10. [10]
    S. Lefschetz, On certain numerical invariants of algebraic varieties with application to Abelian varieties, Trans. Amer. Math. Soc., 22 (1921), 327–482.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    E.T. Whittaker, On hyperlemniscate functions, a family of automorphic functions, J. London Math. Soc., 4 (1929), 274–278.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    D.V. Chudnovsky, G.V. Chudnovsky, Computer assisted number theory, Lecture Notes Math., Springer, 1240, 1987, 1–68.MathSciNetGoogle Scholar
  13. [13]
    Ch. Hermite, Sur la Théorie des Equations Modulaires, C.R. Acad. Sci. Paris., 48 (1859), 940–1079–1097; 49 (1859), 16–110–141.Google Scholar
  14. [14]
    H.M. Stark, Class-numbers of complex quadratic fields, Lecture Notes Math., Springer, v. 320, 1973, 153–174.Google Scholar
  15. [15]
    D. Shanks, Dihedral quartic approximation and series for n, J. Number Theory, 14 (1982), 397–423.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    R. Fricke, Die Elliptischen Funktionen und ihre Anwendungen, v. 1, Teubner, 1916.Google Scholar
  17. [17]
    A. Baker, Transcendental Number Theory, Cambridge, 1979.Google Scholar
  18. [18]
    H.M. Stark, A transcendence theorem for class number problems, I; II; Ann. Math. 94 (1971), 153–173; 96 (1972), 174–209.CrossRefGoogle Scholar
  19. [19]
    C.L. Siegel, Zum Beweise des Starkschen Satzes, Invent. Math., 5 (1968), 180–191.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    G. Shimura, Automorphic forms and the periods of Abelian varieties, J. Math. Soc. Japan, 31 (1979), 561–59.zbMATHGoogle Scholar
  21. [21]
    G. Shimura, The arithmetic of certain zeta functions and automorphic forms on orthogonal groups, Ann. of Math., 111 (1980), 313–375.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    P. Deligne, Valeurs de fonctions L et périodes d’integrales, Proc. Symp. Pure Math., v. 33, Part 2, Amer. Math. Soc., Providence, R.I., 313–346.Google Scholar
  23. [23]
    P. Deligne, Cycles de Hodge absolus et périodes des intégrales des variétés abéliennes, Soc. Math. de France, Memoire, N 2, 1980, 23–33.MathSciNetzbMATHGoogle Scholar
  24. [24]
    G. Shimura, Introduction to the Arithmetic Theory of Automorphic Forms, Princeton, University Press, 1971.Google Scholar
  25. [25]
    G. Shimura, Y. Taniyama, Complex Multiplication of Abelian Varieties and Its Applications to Number Theory, Publications of the, Mathematical Society of Japan, N°6, 1961.Google Scholar
  26. [26]
    G.V. Chudnovsky, Algebraic independence of values of exponential and elliptic functions. Proceedings of the International Congress of Mathematicians, Helsinki, 1979, Acad. Sci. Tennice, Helsinki, 1980, v. 1, 339–350.Google Scholar
  27. [27]
    B.H. Gross, N. Koblitz, Gauss sums and the p-adic r-function, Ann. of Math., 109 (1979), 569–581.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    R. Fricke, F. Klein, Vorlesungen über die Theorie der Automorphen Functionen, bd. 2, Tenbner, 1926.Google Scholar
  29. [29]
    R. Morris, On the automorphic functions of the group (0,3;/1,t2,i3), Trans. Amer. Math. Soc., 7 (1906), 425–448.MathSciNetzbMATHGoogle Scholar
  30. [30]
    K. Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan, 29 (1977), 91–106.MathSciNetCrossRefGoogle Scholar
  31. [31]
    H.P.F. Swinnerton-Dyer, Arithmetic groups in Discrete Groups and Automorphic Functions, Academic Press, 1977, 377–401.Google Scholar
  32. [32]
    J.I. Hutchinson, On the automorphic functions of the group (0,3;2,6,6), Trans. Amer. Math. Soc., 5 (1904), 447–460.MathSciNetzbMATHGoogle Scholar
  33. [33]
    D.V. Chudnovsky, G.V. Chudnovsky, Note on Eisenstein’s system of differential equations, in Classical and Quantum Models and Arithmetic Problems, M. Dekker, 1984, 99–116.Google Scholar
  34. [34]
    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, Proceeding of the International Conference, 1984, New York University, M. Dekker, 1987.Google Scholar
  35. [35]
    D.V. Chudnovsky, G.V. Chudnovsky, Applications of Padé approximations to diophantine inequalities in values of G-functions, Lecture Notes Math., v. 1135, Springer, 1985, 9–50.Google Scholar
  36. [36]
    K. Mahler, Perfect systems, Compositio Math., 19 (1968), 95–166.MathSciNetzbMATHGoogle Scholar
  37. [37]
    D.V. Chudnovsky, G.V. Chudnovsky, Applications of Padé approximations to the Grothendieck conjecture on linear differential equations, Lecture Notes Math., v. 1135, Springer, 1985, 51–100.Google Scholar
  38. ]C.L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys. Math. Kl. 1, 1929.Google Scholar
  39. [39]
    G.V. Chudnovsky, Contributions to the Theory of Transcendental Numbers, Mathematical Surveys and Monographs, v. 19, Amer. Math. Soc., Providence, R.I., 1984.Google Scholar
  40. [40]
    G.V. Chudnovsky, Padé approximation and the Riemann monodromy problem, in Bifurcation Phenomena in Mathematical Physics and Related Topics, D. Reidel, Boston, 1980, 448–510.Google Scholar
  41. [41]
    G.V. Chudnovsky, Rational and padé approximation to solutions of linear differential equations and the monodromy theory. Lecture Notes Physics, v. 126, Springer, 1980, 136–169.Google Scholar
  42. [42]
    G.V. Chudnovsky, Pad approximations to the generalized hypergeometric functions. I., J. Math. Pures et Appliques, Paris, 58 (1979), 445–476.MathSciNetzbMATHGoogle Scholar
  43. [43]
    G.V. Chudnovsky, On the method of Thue-Siegel, Ann. of Math., 117 (1983), 325–382.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    C.L. Siegel, Transcendental. Numbers, Princeton University Press, 1949.Google Scholar
  45. [45]
    G.V. Chudnovsky, Rational approximations to linear forms of exponentials and binomials, Proc. Nat’l Acad. Sci. U.S.A., 80 (1983), 3138–3141.MathSciNetzbMATHCrossRefGoogle Scholar
  46. [46]
    D.V. Chudnovsky, G.V. Chudnovsky, A random walk in higher arithmetic, Adv. Appl. Math., 7 (1986), 101–122.MathSciNetzbMATHGoogle Scholar
  47. [47]
    H. Poincare, Sur les groupes des équations lineaires, Acta. Math., 4 (1884) 201–312.MathSciNetCrossRefGoogle Scholar
  48. [48]
    D.A. Hejhel, Monodromy groups and Poincaré series, Bull. Amer. Math. Soc., 84 (1978), 339–376.Google Scholar
  49. [49]
    E.T. Whittaker, On the connexion of algebraic functions with automorphic functions, phil. Trans., 122 A(1898), 1–32.Google Scholar
  50. [50]
    R.A. Rankin, The differential equations associated with the uniformization of certain algebraic curves, proc. Roy. Soc. Edinburgh, 65 (1958) 35–62.MathSciNetzbMATHGoogle Scholar
  51. [51]
    A. Schonhage, V. Strassen, Schnelle Multiplikation grosser Zahlen, Computing, 7 (1971), 281–292.MathSciNetCrossRefGoogle Scholar
  52. [52]
    D.V. Chudnovsky, G.V. Chudnovsky, Algebraic complexities and algebraic curves over finite fields, proc. Natl. Acad. Sci. USA, 84 (1987), 1739–1743.zbMATHCrossRefGoogle Scholar
  53. [53]
    A. Schonhage, Equation solving in terms of computational complexity, proc. International Congress of Mathematicians, Berkeley, 1986.Google Scholar
  54. [54]
    R.P. Brent, Multiple-precision zero-finding methods and the complexity of elementary function evaluation, in Analytic Computational Complexity, J.F. Traub, Ed., Academic Press, 1975, 151–176.Google Scholar
  55. [55]
    R.P. Brent, The complexity of multiple-precision arithmetic, in Complexity of Computational Problem Solving, R.S. Anderson and R.P. Brent, Eds. Univ. of Queensland Press, Brisbane, Australia, 1975, 126–165.Google Scholar
  56. [56]
    D.V. Chudnovsky, G.V. Chudnovsky, On expansions of algebraic functions in power and Puiseux series, I; II; J. of Complexity, 2 (1986), 271–294; 3 (1987), 1–25.CrossRefGoogle Scholar
  57. [57]
    O. Perron, Die Lehre von den Kettenbrüchen, Teubner, 1929.Google Scholar
  58. [58]
    D. Bini, V. Pan, Polynomial division and its computational complexity, J. of Complexity, 2 (1986), 179–203.MathSciNetzbMATHCrossRefGoogle Scholar
  59. [59]
    D. A. Hejhal, A classical approach to a well-known spectral correspondence on quaternion groups, Lecture Notes Math., v. 1135, Springer, 1985, 127–196.Google Scholar

Copyright information

© Springer Science+Business Media New York 1988

Authors and Affiliations

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

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