Mathematische Zeitschrift

, Volume 276, Issue 3–4, pp 1151–1163 | Cite as

The Mahler measure of a Calabi–Yau threefold and special \(L\)-values

  • Matthew A. Papanikolas
  • Mathew D. Rogers
  • Detchat Samart
Article

Abstract

The aim of this paper is to prove a Mahler measure formula of a four-variable Laurent polynomial whose zero locus defines a Calabi–Yau threefold. We show that its Mahler measure is a rational linear combination of a special \(L\)-value of the normalized newform in \(S_4(\Gamma _0(8))\) and a Riemann zeta value. This is equivalent to a new formula for a \(_6F_5\)-hypergeometric series evaluated at 1.

Keywords

Mahler measure Hypergeometric series Elliptic integrals Modular form Calabi–Yau threefold 

Mathematics Subject Classification (1991)

Primary 11F67 Secondary 11R06 33C20 33C75 

Notes

Acknowledgments

The authors thank Wadim Zudilin for the useful suggestions which improved the exposition of the paper, and also for bringing Verrill’s paper to our attention. The authors are also grateful to the referee for useful comments, which help improve the exposition of this paper.

References

  1. 1.
    Ahlgren, S., Ono, K.: A Gaussian hypergeometric series evaluation and Apéry number congruences. J. Reine Angew. Math. 518, 187–212 (2000)MATHMathSciNetGoogle Scholar
  2. 2.
    Ahlgren, S., Ono, K.: Modularity of a certain Calabi–Yau threefold. Monatsh. Math. 129(3), 177–190 (2000)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Berndt, B.C.: Ramanujan’s Notebooks Part III. Springer, New York, NY (1991)CrossRefMATHGoogle Scholar
  4. 4.
    Bertin, M.J.: Mahler’s measure and \(L\)-series of \(K3\) hypersurfaces. In: Mirror Symmetry, V, vol. 38, pp. 3–18. AMS/IP Stud. Adv. Math. Amer. Math. Soc., Providence, RI (2006)Google Scholar
  5. 5.
    Bertin, M.J.: Mesure de Mahler d’hypersurfaces \(K3\). J. Number Theory 128(11), 2890–2913 (2008)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bertin, M.J.: Measure de Mahler et série \(L\) d’une surface \(K3\) singulière In: Actes de la Conférence Fonctions \(L\) et Arithmétique, pp. 5–28. Publ. Math. Besançon Algèbre Théorie Nr. (2010)Google Scholar
  7. 7.
    Bertin, M.J., Feaver, A., Fuselier, J., Lalín, M., Manes, M.: Mahler measures of some singular \(K3\)-surfaces, arXiv:1208.6240 (2012)Google Scholar
  8. 8.
    Borwein, J.M.: Ramanujan and Pi, Notices of the AMS in “Srinivasa Ramanujan: Going strong at 125”. 59(11), 534–537 (2012)Google Scholar
  9. 9.
    Borwein, J.M., Borwein, P.B.: Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley, New York, NY (1987)MATHGoogle Scholar
  10. 10.
    Boyd, D.W.: Mahler’s measure and special values of \(L\)-functions. Exp. Math. 7(1), 37–82 (1998)CrossRefMATHGoogle Scholar
  11. 11.
    Chudnovsky, D.B., Chudnovsky, G.V.: Approximations and complex multiplication according to Ramanujan. In: Ramanujan Revisited: Proceedings of the Centenary Conference, pp. 375–472. Urbana-Champaign, IL: Academic Press, Boston, MA (1987)Google Scholar
  12. 12.
    Deninger, C.: Deligne periods of mixed motives, \(K\)-theory and the entropy of certain \(\mathbb{Z}^n\)-actions. J. Am. Math. Soc. 10(2), 259–281 (1997)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Frechette, S., Papanikolas, M.A., Root, J.: Finite Field Hypergeometric Functions and Counting Points on Families of Hypersurfaces, in preparation (2013)Google Scholar
  14. 14.
    Greene, J.: Hypergeometric functions over finite fields. Trans. Am. Math. Soc. 301(1), 77–101 (1987)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Guillera, J.: Hypergeometric identities for 10 extended Ramanujan-type series. Ramanujan J. 15(2), 219–234 (2008)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Guillera, J., Rogers, M.D.: Mahler measure and the WZ algorithm. Proc. Am. Math. Soc. (to appear)Google Scholar
  17. 17.
    Lalín, M.N., Rogers, M.D.: Functional equations for Mahler measures of genus-one curves. Algebra Number Theory 1(1), 87–117 (2007)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    McCarthy, D., Papanikolas, M.A.: A finite field hypergeometric function associated to eigenvalues of a Siegel eigenform, arXiv:1205:1006 (2012)Google Scholar
  19. 19.
    Ramanujan, S.: Modular equations and approximations to \(\pi \), [Q. J. Math. 45, 350–372 (1914)]. In: Collected Papers of Srinivasa Ramanujan, pp. 23–29. AMS Chelsea Publ. Providence, RI (2000)Google Scholar
  20. 20.
    Rodriguez Villegas, F.: Modular Mahler Measures I. Topics in Number Theory (University Park, PA, 1997), pp. 17–48. Kluwer, Dordrecht (1999)Google Scholar
  21. 21.
    Rogers, M.D.: New \(_5F_4\) hypergeometric transformations, three-variable Mahler measures, and formulas for \(1/\pi \). Ramanujan J. 18(3), 327–340 (2009)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Rogers, M., Zudilin, W.: From \(L\)-series of elliptic curves to Mahler measures. Compos. Math. 148(2), 385–414 (2012)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Rogers, M., Zudilin, W.: On the Mahler measure of \(1+X+1/X+Y+1/Y\). Int. Math. Res. Notices (to appear)Google Scholar
  24. 24.
    Samart, D.: Mahler measures as linear combinations of \(L\)-values of multiple modular forms, arXiv:1303.6376 (2013)Google Scholar
  25. 25.
    Samart, D.: Three-variable Mahler measures and special values of modular and Dirichlet \(L\)-series. Ramanujan J. (to appear)Google Scholar
  26. 26.
    van Geemen, B., van Straten, D.: The cusp forms of weight 3 on \(\Gamma _2(2,4,8)\). Math. Comput. 61(204), 849–872 (1993)MATHGoogle Scholar
  27. 27.
    Verrill, H.A., Arithmetic of a certain Calabi-Yau threefold. In: Number theory (Ottawa, ON, 1996): CRM Proceedings of the Lecture Notes, vol. 19, pp. 333–340, Amer. Math. Soc. Providence, RI (1999)Google Scholar
  28. 28.
    Wan, J.G.: Moments of products of elliptic integrals. Adv. Appl. Math. 48(1), 121–141 (2012)CrossRefMATHGoogle Scholar
  29. 29.
    Zudilin, W.: Arithmetic hypergeometric series. Russ. Math. Surv. 66(2), 369–420 (2011)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Matthew A. Papanikolas
    • 1
  • Mathew D. Rogers
    • 2
  • Detchat Samart
    • 1
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Department of Mathematics and StatisticsUniversité de MontréalMontrealCanada

Personalised recommendations