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


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.


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

Mathematics Subject Classification (1991)

Primary 11F67 Secondary 11R06 33C20 33C75 



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.


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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

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