Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-26T06:57:58.886Z Has data issue: false hasContentIssue false
  • English
  • Français

A Mahler Measure of a K3 Surface Expressed as a Dirichlet L-Series

Published online by Cambridge University Press:  20 November 2018

Marie José Bertin
Marie José Bertin*
Affiliation:
Université Pierre et Marie Curie (Paris 6), Institut de Mathématiques, 175 rue du Chevaleret, 75013 Paris e-mail: bertin@math.jussieu.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present another example of a 3-variable polynomial defining a $K3$-hypersurface and having a logarithmic Mahler measure expressed in terms of a Dirichlet $L$-series.

Keywords

1114D14Jmodular Mahler measureEisenstein–Kronecker seriesL-series of K3-surfacesl-adic representationsLivné criterionRankin–Cohen brackets
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 1 , 01 March 2012 , pp. 26 - 37
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Beilinson, A., Higher regulators of modular curves. In: Application of Algebraic K-theory to Algebraic Geometry and Number Theory. Contemp. Math. 55, American Mathematical Society, Providence, RI, 1986, pp. 1–34.Google Scholar
[2] Bertin, M. J., Une mesure de Mahler explicite. C. R. Acad. Sci. Paris Sér. I Math. 333(2001), no. 1, 1–3.Google Scholar
[3] Bertin, M. J., Mesure de Mahler d’hypersurfaces K3 . J. Number Theory 128(2008), no. 11, 2890–2913. doi:10.1016/j.jnt.2007.12.012Google Scholar
[4] Bertin, M. J., Mahler's measure and L-series of K3 hypersurfaces. In: Mirror Symmetry V. AMS/IP Studies in Advanced Mathematics 38, American Mathematical Society, Providence, RI, 2006, pp. 3–18.Google Scholar
[5] Bertin, M. J., The Mahler measure and the L-series of a singular K3-surface. arXiv:0803.0413v1math.NT(math.AG).Google Scholar
[6] Boyd, D. W., Mahler's measure and special values of L-functions. Experiment. Math. 7(1998), no. 1, 37–82.Google Scholar
[7] Elkies, N. and Schütt, M., Modular forms and K3 surfaces. arXiv:math.AG/0809.0830v1.Google Scholar
[8] Hulek, K., Kloosterman, R., and Schütt, M., Modularity of Calabi-Yau varieties. In: Global Aspects of Complex Geometry. Springer, Berlin, 2006, pp. 271–309.Google Scholar
[9] Lalin, M. and Rogers, M., Functional equations for Mahler measures of genus-one curves. Algebra Number Theory 1(2007), no. 1, 87–117. doi:10.2140/ant.2007.1.87Google Scholar
[10] Livné, R., Cubic exponential sums and Galois representations. In: Current Trends in Arithmetical Algebraic Geometry. Contemp. Math. 67, American Mathematical Society, Providence, RI, 1987, pp. 247–261.Google Scholar
[11] Livné, R., Motivic orthogonal two-dimensional representations of . Israel J. of Math 92(1995), no. 1-3, 149–156. doi:10.1007/BF02762074Google Scholar
[12] Peters, C., Top, J., and van der Vlugt, M., The Hasse zeta function of a K3 surface related to the number of words of weight5 in the Melas codes. J. Reine Angew. Math. 432(1992), 151–176.Google Scholar
[13] Peters, C., Top, J., and van der Vlugt, M., Modular Mahler measures. I. In: Topics in Number Theory. Math. Appl. 467, Kluwer, Dordrecht, 1999, pp. 17–48.Google Scholar
[14] Shioda, T., On elliptic modular surfaces. J. Math. Soc. Japan 24(1972), 20–59. doi:10.2969/jmsj/02410020Google Scholar
[15] Schütt, M., CM newforms with rational coefficients. arXiv:math.NT/0511228v5.Google Scholar
[16] Williams, K., Some Lambert series expansions of products of theta functions. Ramanujan Journal 3(1999), no. 4, 367–384. doi:10.1023/A:1009853106329Google Scholar
[17] Yui, N., Arithmetic of certain Calabi-Yau varieties and mirror symmetry. In: Arithmetic Algebraic Geometry. IAS/Park City Math. Ser. 9, American Mathematical Society, Providence, RI, 2001, pp. 507–569.Google Scholar
[18] Zagier, D. and Gangl, H., Classical and elliptic polylogarithms and special values of L-series. In: The Arithmetic and Geometry of Algebraic Cycles. Nato Sciences Series C 548, NATO Sci. Ser. C Math. Phys. Sci. 548, Kluwer, Dordrecht, 2000, pp. 561–615.Google Scholar
[19] Zucker, I. and McPhedran, R., Dirichlet L-series with real and complex characters and their application to solving double sums. arXiv:0708.1224v1[math-ph]. doi:10.1098/rspa.2007.0162Google Scholar
[20] Zucker, I. and Robertson, M., A systematic approach to the evaluation of Σ (m,n≠0,0)(am 2 + bmn + cn 2 )–s . J. Phys. A: 9(1976), no. 8, 1215–1225. doi:10.1088/0305-4470/9/8/007Google Scholar