The Mahler measure for arbitrary tori

Research
  • 37 Downloads

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

Keywords

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

Mathematics Subject Classification

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

Notes

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.

References

  1. 1.
    Bertin, M.-J.: Mahler measure, regulators, and modular units. Workshop lecture at “The Geometry, Algebra and Analysis of Algebraic Numbers”. Banff International Research Station, Banff, Canada (2015)Google Scholar
  2. 2.
    Byrd, P.F., Morris, D.: Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Die Grundlehren der mathematischen Wissenschaften, Band 67, 2nd edn. Springer, New York (1971)CrossRefGoogle Scholar
  3. 3.
    Bloch, S., Grayson, D.: \(K_2\) and \(L\)-functions of elliptic curves: computer calculations, Applications of algebraic \(K\)-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., : Contemp. Math., vol. 55, Amer. Math. Soc. Providence, RI 1986, 79–88 (1983)Google Scholar
  4. 4.
    Beĭ linson, A.A.: Higher regulators and values of \(L\)-functions of curves. Funktsional. Anal. i Prilozhen 14(2), 46–47 (1980)MathSciNetGoogle Scholar
  5. 5.
    Bloch, S.J.: Higher Regulators, Algebraic \(K\)-Theory, and Zeta Functions of Elliptic Curves, CRM Monograph Series, vol. 11. American Mathematical Society, Providence (2000)Google Scholar
  6. 6.
    Boyd, D.W.: Mahler’s measure and special values of \(L\)-functions. Exp. Math. 7(1), 37–82 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Deninger, C.: Deligne periods of mixed motives, \(K\)-theory and the entropy of certain \({ Z}^n\)-actions. J. Am. Math. Soc. 10(2), 259–281 (1997)CrossRefMATHGoogle Scholar
  8. 8.
    Lalín, M.N.: Some examples of Mahler measures as multiple polylogarithms. J. Number Theory 103(1), 85–108 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Lundqvist, J.: An explicit calculation of the Ronkin function. Ann. Fac. Sci. Toulouse Math. (6) 24(2), 227–250 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Maillot, V.: Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables, no. 80 (2000)Google Scholar
  11. 11.
    Mellit, A.: Elliptic dilogarithms and parallel lines. ArXiv e-prints (2012)Google Scholar
  12. 12.
    Myerson, G., Smyth, C.J.: Corrigendum: “On measures of polynomials in several variables”. Bull. Aust. Math. Soc. 26(2), 317–319 (1982)CrossRefGoogle Scholar
  13. 13.
    Rodriguez-Villegas, F.: Mahler, Modular, Measures. I, Topics in Number Theory (UniversityPark, PA, 1997), Math. Appl., vol. 467, pp. 17–48. Kluwer Academic Publisher, Dordrecht (1997)Google Scholar
  14. 14.
    Rodriguez-Villegas, F.: Identities Between Mahler Measures, Number Theory for the Millennium, III (Urbana, IL, 2000), vol. 2002, pp. 223–229. A K Peters, Natick (2000)Google Scholar
  15. 15.
    Rogers, M., Zudilin, W.: From \(L\)-series of elliptic curves to Mahler measures. Compos. Math. 148(2), 385–414 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Smyth, C.J.: On measures of polynomials in several variables. Bull. Aust. Math. Soc. 23(1), 49–63 (1981)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Touafek, N.: From the elliptic regulator to exotic relations. An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 16(2), 117–125 (2008)MathSciNetMATHGoogle Scholar
  18. 18.
    Touafek, N.: Thèse de doctorat. Université de Constantine, Algeria (2008)Google Scholar
  19. 19.
    Vandervelde, S.: A formula for the Mahler measure of \(axy+bx+cy+d\). J. Number Theory 100(1), 184–202 (2003)MathSciNetCrossRefMATHGoogle Scholar

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

Personalised recommendations