Journal of Mathematical Sciences

, Volume 146, Issue 2, pp 5697–5716 | Cite as

On arithmetic properties of values of theta-constants

  • Yu. V. Nesterenko


This article describes results about the transcendence and algebraic independence of values of theta constants (Nullthetawerte) and direct methods for proving these results. Values of other functions related to theta constants are discussed. We also present some conjectures and open questions.


Elliptic Function Theta Function Homogeneous Polynomial Algebraic Number Modular Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. I. Ahiezer, Elements of the Theory of Elliptic Functions, Translations of Mathematical Monographs, Vol. 79, American Mathematical Society, Providence (1990).Google Scholar
  2. 2.
    Y. Andre, Quelques Conjectures de Transcendance Issues de la Géométrie Algébrique, preprint de l’Inst. Math. de Jussieu, No. 121 (1997).Google Scholar
  3. 3.
    K. Barré-Sirieix, G. Diaz, F. Gramain, and G. Philibert, “Une preuve de la conjecture de Mahler-Manin,” Invent. Math., 124, 1–9 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    D. Bertrand, “Theta functions and transcendence,” Ramanujan J., 1, No. 4, 339–350 (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    W. D. Brownawell and D. W. Masser, “Multiplicity estimates for analytic functions. I,” J. Reine Angew. Math., 313, 200–216 (1980).MathSciNetGoogle Scholar
  6. 6.
    G. Chudnovsky, Contributions to the Theory of Transcendental Numbers, AMS, Providence (1984).zbMATHGoogle Scholar
  7. 7.
    P. Cohen, “On the coefficients of the transformation polynomials for the elliptic modular function,” Math. Proc. Cambridge Philos. Soc., 95, 389–402 (1984).zbMATHMathSciNetGoogle Scholar
  8. 8.
    N. I. Feldman, Hilbert’s Seventh Problem [in Russian], Izd. Mosk. Univ., Moscow (1982).Google Scholar
  9. 9.
    A. O. Gelfond, Transcendental and Algebraic Numbers, Dover (1960).Google Scholar
  10. 10.
    G. Halphen, “Sur une systéme d’equations différentielles,” C. R. Acad. Sci. Paris, 92, 1101–1103 (1881).Google Scholar
  11. 11.
    S. Lang, Algebra, Addison-Wesley, Reading (1965).zbMATHGoogle Scholar
  12. 12.
    S. Lang, Elliptic Functions, Addison-Wesley, Reading (1973).zbMATHGoogle Scholar
  13. 13.
    D. F. Lawden, Elliptic Functions and Applications, Springer, Berlin (1989).zbMATHGoogle Scholar
  14. 14.
    K. Mahler, “On algebraic differential equations satisfied by automorphic functions,” J. Austral. Math. Soc., 10, 445–450 (1969).zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    K. Mahler, “Remarks on a paper by W. Schwarz,” J. Number Theory, 1, 512–521 (1969).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    K. Mahler, Lectures on Transcendental Numbers, Lect. Notes Math., Vol. 546, Springer, Berlin (1976).Google Scholar
  17. 17.
    K. Mahler and J. Popken, “Ein neues Prinzip für Transzendenzbeweize,” Proc. Akad. Amsterdam, 38, 864–871 (1935).zbMATHGoogle Scholar
  18. 18.
    Yu. I. Manin, “Cyclotomic fields and modular curves,” Russian Math. Surveys, 26, No. 6, 7–78 (1971).zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Yu. V. Nesterenko, “Multiplicity estimates for theta-constants,” Fund. Prikl. Mat., 5, No. 2, 557–562 (1999).zbMATHMathSciNetGoogle Scholar
  20. 20.
    A. B. Schidlovskii, Transcendental Numbers, Walter de Gruyter, Berlin (1989).Google Scholar
  21. 21.
    Th. Schneider, “Arithmetische Untersuchungen elliptischer Integrale,” Math. Ann., 113, 1–13 (1937).CrossRefMathSciNetGoogle Scholar
  22. 22.
    Th. Schneider, “Zur Theorie der Abelschen Functionen und Integrale,” J. Reine Angew. Math., 183, 110–128 (1941).zbMATHMathSciNetGoogle Scholar
  23. 23.
    Th. Schneider, Einführung in die Transzendenten Zahlen, Springer, Berlin (1957).zbMATHGoogle Scholar
  24. 24.
    P. K. Suetin, Classical Orthogonal Polynomials [in Russian], Nauka, Moscow (1976).Google Scholar
  25. 25.
    E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Vol. 2, Cambridge (1927).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Mechanics and Mathematics FacultyMoscow State UniversityRussia

Personalised recommendations