Period(d)ness of L-Values

  • Wadim Zudilin
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 43)


In our recent work with Rogers on resolving some of Boyd’s conjectures on two-variate Mahler measures, a new analytical machinery was introduced to write the values L(E, 2) of L-series of elliptic curves as periods in the sense of Kontsevich and Zagier. Here we outline, in slightly more general settings, the novelty of our method with Rogers and provide two illustrative period evaluations of L(E, 2) and L(E, 3) for a conductor 32 elliptic curve E.

Key words

Modular form L-series Period Arithmetic differential equation Dedekind’s eta function 



I am thankful to Anton Mellit, Mat Rogers, Evgeny Shinder, Masha Vlasenko and James Wan for fruitful conversations on the subject and to Don Zagier for his encouragement to isolate the L-series transformation part from [8, 9]. I thank the anonymous referee for his careful reading of an earlier version and for his suggesting some useful corrections.

The principal part of this work was done during my visit in the Max Planck Institute for Mathematics (Bonn). I would like to thank the staff of the institute for hospitality and enjoyable working conditions.

This work is supported by the Max Planck Institute for Mathematics (Bonn, Germany) and the Australian Research Council (grant DP110104419).


  1. 1.
    W.N. Bailey, Generalized Hypergeometric Series. Cambridge Tracts in Math., vol 32 (Cambridge Univ. Press, Cambridge, 1935); 2nd reprinted edn. (Stechert-Hafner, New York/London, 1964)Google Scholar
  2. 2.
    J.L. Berggren, J.M. Borwein, P. Borwein, Pi: A Source Book, 3rd edn. (Springer, New York, 2004)zbMATHGoogle Scholar
  3. 3.
    J.M. Borwein, R.E. Crandall, Closed forms: what they are and why we care. Notices Am. Math. Soc. 60, 50–65 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    W. Duke, Some entries in Ramanujan’s notebooks. Math. Proc. Camb. Phil. Soc. 144, 255–266 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    M. Kontsevich, D. Zagier, Periods, in Mathematics Unlimited—2001 and Beyond (Springer, Berlin, 2001), pp. 771–808Google Scholar
  6. 6.
    Yu.V. Nesterenko, Integral identities and constructions of approximations to zeta-values. J. Theor. Nombres Bordeaux 15, 535–550 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    M.D. Rogers, A study of inverse trigonometric integrals associated with three-variable Mahler measures, and some related identities. J. Number Theory 121, 265–304 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    M. Rogers, W. Zudilin, From L-series of elliptic curves to Mahler measures. Compos. Math. 148, 385–414 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    M. Rogers, W. Zudilin, On the Mahler measure of \(1 + X + 1/X + Y + 1/Y\). Intern. Math. Research Notices, 22 pp. (2013, in press, doi:10.1093/imrn/rns285)Google Scholar
  10. 10.
    E. Shinder, M. Vlasenko, Linear Mahler measures and double L-values of modular forms (2012), 22 pp. Preprint at
  11. 11.
    A. van der Poorten, A proof that Euler missed…Apéry’s proof of the irrationality of ζ(3). Math. Intell. 1, 195–203 (1978/1979)Google Scholar
  12. 12.
    A. Weil, Remarks on Hecke’s lemma and its use, in Algebraic Number Theory. Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto 1976 (Japan Soc. Promotion Sci., Tokyo, 1977), pp. 267–274Google Scholar
  13. 13.
    Y. Yang, Apéry limits and special values of L-functions. J. Math. Anal. Appl. 343, 492–513 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematical and Physical SciencesThe University of NewcastleCallaghanAustralia

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