Geodesic
A Mahler Measure of a K3 Surface Expressed as a Dirichlet L-Series
Canadian mathematical bulletin, Tome 55 (2012) no. 1, pp. 26-37

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2011-067-0
Mots-clés : 11, 14D, 14J, modular Mahler measure, Eisenstein–Kronecker series, L-series of K3-surfaces, l-adic representations, Livné criterion, Rankin–Cohen brackets 
Bertin, Marie José. A Mahler Measure of a K3 Surface Expressed as a Dirichlet L-Series. Canadian mathematical bulletin, Tome 55 (2012) no. 1, pp. 26-37. doi: 10.4153/CMB-2011-067-0
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[1] [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] [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] [3] Bertin, M. J., Mesure de Mahler d’hypersurfaces K3 . J. Number Theory 128(2008), no. 11, 2890–2913. doi:10.1016/j.jnt.2007.12.012 Google Scholar

[4] [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] [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] [6] Boyd, D. W., Mahler's measure and special values of L-functions. Experiment. Math. 7(1998), no. 1, 37–82. Google Scholar

[7] [7] Elkies, N. and Schütt, M., Modular forms and K3 surfaces. arXiv:math.AG/0809.0830v1. Google Scholar

[8] [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] [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.2007.1.87 Google Scholar

[10] [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] [11] Livné, R., Motivic orthogonal two-dimensional representations of . Israel J. of Math 92(1995), no. 1-3, 149–156. doi:10.1007/BF02762074 Google Scholar

[12] [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] [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] [14] Shioda, T., On elliptic modular surfaces. J. Math. Soc. Japan 24(1972), 20–59. doi:10.2969/jmsj/02410020 Google Scholar

[15] [15] Schütt, M., CM newforms with rational coefficients. arXiv:math.NT/0511228v5. Google Scholar

[16] [16] Williams, K., Some Lambert series expansions of products of theta functions. Ramanujan Journal 3(1999), no. 4, 367–384. doi:10.1023/A:1009853106329 Google Scholar

[17] [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] [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] [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.2007.0162 Google Scholar

[20] [20] Zucker, I. and Robertson, M., A systematic approach to the evaluation of Σ (am 2 + bmn + cn 2 )–s . J. Phys. A: 9(1976), no. 8, 1215–1225. doi:10.1088/0305-4470/9/8/007 Google Scholar

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