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

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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.

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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.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX , 77843, USA

    Matthew A. Papanikolas & Detchat Samart

  2. Department of Mathematics and Statistics, Université de Montréal, CP 6128 succ. Centre-ville, Montreal, QC , H3C 3J7, Canada

    Mathew D. Rogers

Authors
  1. Matthew A. Papanikolas
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  2. Mathew D. Rogers
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  3. Detchat Samart
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Correspondence to Matthew A. Papanikolas.

Additional information

Matthew A. Papanikolas and Detchat Samart were partially supported by NSF Grant DMS-1200577.

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Papanikolas, M.A., Rogers, M.D. & Samart, D. The Mahler measure of a Calabi–Yau threefold and special \(L\)-values. Math. Z. 276, 1151–1163 (2014). https://doi.org/10.1007/s00209-013-1238-6

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  • DOI: https://doi.org/10.1007/s00209-013-1238-6

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