Mahler measures of a family of non-tempered polynomials and Boyd’s conjectures

  • Yotsanan Meemark
  • Detchat SamartEmail author


We prove an identity relating Mahler measures of a certain family of non-tempered polynomials to those of tempered polynomials. Evaluations of Mahler measures of some polynomials in the first family are also given in terms of special values of L-functions and logarithms.Finally, we prove Boyd’s conjectures for conductor 30 elliptic curves using our new identity, Brunault–Mellit–Zudilin’s formula and additional functional identities for Mahler measures.



This research is supported financially by the Thailand Research Fund (TRF) Grant from the TRF and the Office of the Higher Education Commission, under the Contract No. MRG6280045. Part of this work was done during the second author’s visit to the Vietnam Institute for Advanced Study in Mathematics (VIASM) in the Summer semester of year 2018, partially supported under the institute’s research Contract No. NC/2018/VNCCCT. The second author is grateful to the VIASM for their support and hospitality. The authors would like to thank Bruce Berndt and Wadim Zudilin for their valuable comments on an earlier version of this paper. Lastly, the authors thank the anonymous referee for insightful remarks which greatly help to improve the exposition of this paper and for a suggestion about possible approach to completing the proof of Boyd’s conductor 30 conjectures.

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Berndt, B.C.: Ramanujan’s Notebooks. Part V. Springer, New York (1998)CrossRefGoogle Scholar
  2. 2.
    Bertin, M.J., Zudilin, W.: On the Mahler measure of a family of genus 2 curves. Math. Z. 283(3–4), 1185–1193 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beĭlinson, A.A.: Higher regulators and values of \(L\)-functions of curves. Funktsional. Anal. i Prilozhen. 14(2), 46–47 (1980)MathSciNetGoogle Scholar
  4. 4.
    Bloch, S.J.: Higher Regulators, Algebraic \(K\)-Theory, and Zeta Functions of Elliptic Curves. CRM Monograph Series, vol. 11. American Mathematical Society, Providence, RI (2000)Google Scholar
  5. 5.
    Boyd, D.W.: Mahler’s measure and special values of \(L\)-functions. Exp. Math. 7(1), 37–82 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brunault, F.: Parametrizing elliptic curves by modular units. J. Aust. Math. Soc. 100(1), 33–41 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brunault, F.: Regulators of Siegel units and applications. J. Number Theory 163, 542–569 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chan, H.H., Lang, M.L.: Ramanujan’s modular equations and Atkin–Lehner involutions. Isr. J. Math. 103, 1–16 (1998)MathSciNetCrossRefGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Garvan, F.: A q-product tutorial for a q-series MAPLE package. ArXiv Mathematics e-prints (1998). arXiv: math/9812092
  11. 11.
    Giard, A.: Mahler measure of a non-tempered Weierstrass form. J Number Theory (2019). CrossRefGoogle Scholar
  12. 12.
    Jia, Y.: One Special Identity between the complete elliptic integrals of the first and the third kind. ArXiv e-prints. arXiv:0802.3977 (2008)
  13. 13.
    Lalín, M., Mittal, T.: The Mahler measure for arbitrary tori. Res. Number Theory 4(2), 23 (2018). Art. 16MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lalín, M., Samart, D., Zudilin, W.: Further explorations of Boyd’s conjectures and a conductor 21 elliptic curve. J. Lond. Math. Soc. (2) 93(2), 341–360 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lalín, M.N.: Equations for Mahler measure and isogenies. J. Théor. Nombres Bordx. 25(2), 387–399 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lalín, M.N., Ramamonjisoa, F.: The Mahler measure of a Weierstrass form. Int. J. Number Theory 13(8), 2195–2214 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mellit, A.: Elliptic dilogarithms and parallel lines. J. Number Theory 204, 1–24 (2019)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V. (eds.): NIST Digital Library of Mathematical Functions. Release 1.0.23 of 2019-06-15. Accessed 16 June 2019
  19. 19.
    Ono, K.: The web of modularity: arithmetic of the coefficients of modular forms and q-series. CBMS Regional Conference Series in Mathematics. In: Published for the Conference Board of the Mathematical Sciences, vol. 102, Washington, DC, American Mathematical Society, Providence, RI (2004)Google Scholar
  20. 20.
    Rogers, M., Yuttanan, B.: Modular equations and lattice sums. In: Bailey, D.H., Bauschke, H.H., Borwein, P., Garvan, F., Théra, M., Vanderwerff, J., et al. (eds.) Computational and Analytical Mathematics. Springer Proceedings in Mathematics and Statistics, vol. 50, pp. 667–680. Springer, New York (2013)zbMATHGoogle Scholar
  21. 21.
    Rogers, M., Zudilin, W.: From \(L\)-series of elliptic curves to Mahler measures. Compos. Math. 148(2), 385–414 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    SageMath Inc.: CoCalc Collaborative Computation Online (2019). Accessed 29 May 2019
  23. 23.
    Smyth, C.J.: An explicit formula for the Mahler measure of a family of 3-variable polynomials. J. Théor. Nombres Bordx. 14(2), 683–700 (2002)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sturm, J.: On the congruence of modular forms. In: Chudnovsky, D.V., Chudnovsky, G.V., Cohn, H., Nathanson, M.B. (eds.) Number Theory (New York, 1984–1985), Lecture Notes in Mathematics, vol. 1240, pp. 275–280. Springer, Berlin (1987)Google Scholar
  25. 25.
    Villegas, F.R.: Modular Mahler measures. I. In: Ahlgren, S.D., Andrews, G.E., Ono, K. (eds.) Topics in Number Theory (University Park, PA, 1997), Mathematics and its Applications, vol. 467, pp. 17–48. Kluwer Academic Publishers, Dordrecht (1999)CrossRefGoogle Scholar
  26. 26.
    Zudilin, W.: Regulator of modular units and Mahler measures. Math. Proc. Camb. Philos. Soc. 156(2), 313–326 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Faculty of ScienceChulalongkorn UniversityBangkokThailand
  2. 2.Department of Mathematics, Faculty of ScienceBurapha UniversityChonburiThailand
  3. 3.Center of Excellence in Mathematics, CHEBangkokThailand

Personalised recommendations