Mathematische Zeitschrift

, Volume 283, Issue 3–4, pp 1185–1193 | Cite as

On the Mahler measure of a family of genus 2 curves

  • Marie José Bertin
  • Wadim Zudilin


We establish a general identity between the Mahler measures \(\mathrm {m}(Q_k(x,y))\) and \(\mathrm {m}(P_k(x,y))\) of two polynomial families, where \(Q_k(x,y)=0\) and \(P_k(x,y)=0\) are generically hyperelliptic and elliptic curves, respectively.


Mahler measure L-value Elliptic curve Hyperelliptic curve Elliptic integral 

Mathematics Subject Classification

Primary 11F67 Secondary 11F11 11F20 11G16 11G55 11R06 14H52 19F27 


  1. 1.
    Berndt, B.C.: Ramanujan’s Notebooks, Part V. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bertin, M.J.: Une mesure de Mahler explicite. C. R. Acad. Sci. Paris Sér. I Math. 333(1), 1–3 (2001)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bertin, M.J., Zudilin, W.: On the Mahler measure of hyperelliptic families, Preprint (2016); arXiv:1601.07583 [math.NT]
  4. 4.
    Borwein, J.M., Straub, A., Wan, J., Zudilin, W.: Densities of short uniform random walks. With an appendix by Don Zagier. Can. J. Math. 64(5), 961–990 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bosman, J.: Boyd’s conjecture for a family of genus \(2\) curves, Thesis (2004).
  6. 6.
    Boyd, D.: Mahler’s measure and special values of \(L\)-functions. Exp. Math. 7(1), 37–82 (1998)CrossRefzbMATHGoogle Scholar
  7. 7.
    Goursat, É.: Sur la réduction des intégrales hyperelliptiques. Bull. Soc. Math. Fr. 13, 143–162 (1885)MathSciNetGoogle Scholar
  8. 8.
    Mellit, A.: Elliptic dilogarithms and parallel lines, Preprint (2009, 2011); arXiv:1207.4722 [math.NT]
  9. 9.
    Rodriguez Villegas, F.: Modular Mahler measures. I. In: Topics in Number Theory. University Park, PA, 1997, Math. Appl. 467 (Kluwer Acadamic Publication, Dordrecht, 1999), pp. 17–48Google Scholar
  10. 10.
    Rogers, M., Zudilin, W.: From \(L\)-series of elliptic curves to Mahler measures. Compos. Math. 148(2), 385–414 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Verrill, H.A.: Picard-Fuchs equations of some families of elliptic curves. In: Proceedings on Moonshine and Related Topics (Montréal, QC, 1999), CRM Proc. Lecture Notes 30 (Amer. Math. Soc., Providence, RI, 2001), pp. 253–268Google Scholar
  12. 12.
    Zudilin, W.: Regulator of modular units and Mahler measures. Math. Proc. Camb. Philos. Soc. 156(2), 313–326 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut de MathématiquesUniversité Pierre et Marie Curie (Paris 6)ParisFrance
  2. 2.School of Mathematical and Physical SciencesThe University of NewcastleCallaghanAustralia

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