The Mahler measure for arbitrary tori



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 \(L'(E,0)\).


Mahler measure Special values of L-functions Elliptic curve Elliptic regulator 

Mathematics Subject Classification

Primary 11R06 Secondary 11G05 11F66 19F27 33E05 


Authors' contributions

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


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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Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Départment de Mathématique et de StatistiqueUniversité de Montréal.MontrealCanada
  2. 2.Department of Computer Science and EngineeringIndian Institute of Technology KanpurKanpurIndia

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