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The Mahler measure for arbitrary tori

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Abstract

We consider a variation of the Mahler measure where the defining integral is performed over a more general torus. We focus our investigation on two particular polynomials related to certain elliptic curve E and we establish new formulas for this variation of the Mahler measure in terms of \(L'(E,0)\).

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Notes

  1. Boyd gave precise conjectural conditions for \(\beta \) and k.

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Authors' contributions

ML, TM have participated in the whole study and drafted the manuscript, and both authors read and approved the final manuscript.

Acknowlegements

We are thankful to Marie-José Bertin for providing us a copy of Touafek’s doctoral thesis [18]. We are very grateful to the anonymous referees for their dedicated work and for their several corrections and suggestions that greatly improved the exposition. We would like to specially thank the referee who found a mistake in one of our main formulas. This research was supported by the Natural Sciences and Engineering Research Council of Canada [Discovery Grant 355412-2013 to ML] and Mitacs [Globalink Research Internship to TM].

Competing interests

The authors declare that they have no competing interests.

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Authors and Affiliations

  1. Départment de Mathématique et de Statistique, Université de Montréal., CP 6128, succ. Centre-Ville, Montreal, QC, H3C 3J7, Canada

    Matilde Lalín

  2. Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, Kanpur, India

    Tushant Mittal

Authors
  1. Matilde Lalín
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  2. Tushant Mittal
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Correspondence to Matilde Lalín.

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Lalín, M., Mittal, T. The Mahler measure for arbitrary tori. Res. number theory 4, 16 (2018). https://doi.org/10.1007/s40993-018-0112-3

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