Skip to main content
SpringerLink
Log in

On the Mahler measure of a family of genus 2 curves

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berndt, B.C.: Ramanujan’s Notebooks, Part V. Springer, New York (1998)

    Book  MATH  Google Scholar 

  2. Bertin, M.J.: Une mesure de Mahler explicite. C. R. Acad. Sci. Paris Sér. I Math. 333(1), 1–3 (2001)

    Article  MathSciNet  Google Scholar 

  3. Bertin, M.J., Zudilin, W.: On the Mahler measure of hyperelliptic families, Preprint (2016); arXiv:1601.07583 [math.NT]

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bosman, J.: Boyd’s conjecture for a family of genus \(2\) curves, Thesis (2004). http://www.uni-due.de/~ada649b/papers/scriptie.pdf

  6. Boyd, D.: Mahler’s measure and special values of \(L\)-functions. Exp. Math. 7(1), 37–82 (1998)

    Article  MATH  Google Scholar 

  7. Goursat, É.: Sur la réduction des intégrales hyperelliptiques. Bull. Soc. Math. Fr. 13, 143–162 (1885)

    MathSciNet  Google Scholar 

  8. Mellit, A.: Elliptic dilogarithms and parallel lines, Preprint (2009, 2011); arXiv:1207.4722 [math.NT]

  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. 17–48

  10. Rogers, M., Zudilin, W.: From \(L\)-series of elliptic curves to Mahler measures. Compos. Math. 148(2), 385–414 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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. 253–268

  12. Zudilin, W.: Regulator of modular units and Mahler measures. Math. Proc. Camb. Philos. Soc. 156(2), 313–326 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Mathématiques, Université Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75252, Paris, France

    Marie José Bertin

  2. School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW, 2308, Australia

    Wadim Zudilin

Authors
  1. Marie José Bertin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wadim Zudilin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wadim Zudilin.

Additional information

W. Zudilin is supported by Australian Research Council Grant DP140101186.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bertin, M.J., Zudilin, W. On the Mahler measure of a family of genus 2 curves. Math. Z. 283, 1185–1193 (2016). https://doi.org/10.1007/s00209-016-1637-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-016-1637-6

Keywords

Mathematics Subject Classification

Search

Navigation