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Mahler Measures as Linear Combinations of L–values of Multiple Modular Forms

Published online by Cambridge University Press:  20 November 2018

Detchat Samart
Detchat Samart*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA. e-mail: detchats@math.tamu.edu
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Abstract

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We study the Mahler measures of certain families of Laurent polynomials in two and three variables. Each of the known Mahler measure formulas for these families involves $L$–values of at most one newform and/or at most one quadratic character. In this paper we show, either rigorously or numerically, that the Mahler measures of some polynomials are related to $L$–values of multiple newforms and quadratic characters simultaneously. The results suggest that the number of modular $L$–values appearing in the formulas significantly depends on the shape of the algebraic value of the parameter chosen for each polynomial. As a consequence, we also obtain new formulas relating special values of hypergeometric series evaluated at algebraic numbers to special values of $L$–functions.

Keywords

11F6733C20Mahler measuresEisenstein–Kronecker seriesL–functionshypergeometric series
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 2 , April 2015 , pp. 424 - 449
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Ahlgren, S., Ono, K., and Penniston, D., Zeta function of an infinite family of K3 surfaces. Amer. J. Math. 124(2002), 353–368. http://dx.doi.org/10.1353/ajm.2002.0007 Google Scholar
[2] Berndt, B. C., Ramanujan's notebooks. Part III. Springer–Verlag, New York, NY, 1991.Google Scholar
[3] Berndt, B. C., Bhargava, S., and Garvan, F. G., Ramanujan's theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347(1995), no. 11, 4163–4244.Google Scholar
[4] Bertin, M. J., Mesure de Mahler d’hypersurfaces K3. J. Number Theory 128(2008), no. 11, 2890–2913. http://dx.doi.org/10.1016/j.jnt.2007.12.012 Google Scholar
[5] Boyd, D.W., Mahler's measure and special values of L–functions. Experiment. Math. 7(1998), no. 1, 37–82. http://dx.doi.org/10.1080/10586458.1998.10504357 Google Scholar
[6] Chen, I. and Yui, N., Singular values of Thompson series. In: Groups, difference sets, and the Monster(Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., 4, de Gruyter, Berlin, 1996, pp. 255–326.Google Scholar
[7] Deninger, C., Deligne periods of mixed motives, K–theory and the entropy of certain Zn–actions. J. Amer. Math. Soc. 10(1997), no. 2, 259–281. http://dx.doi.org/10.1090/S0894-0347-97-00228-2 Google Scholar
[8] Glasser, M. L. and Zucker, I. J., Lattice sums. In: Perspectives in theoretical chemistry: Advances and perspectives, 5, Academic Press, New York, NY, 1980, pp. 67–139.Google Scholar
[9] Guillera, J. and Rogers, M., Mahler measure and the WZ algorithm. Proc. Amer. Math. Soc., to appear.Google Scholar
[10] Hartmann, H., Period– and mirror–maps for the quartic K3. Manuscripta Math. 141 2013, no. 3–4, 391–422. http://dx.doi.org/10.1007/s00229-012-0577-7Google Scholar
[11] Kurokawa, N. and Ochiai, H., Mahler measures via the crystalization. Comment. Math. Univ. St. Pauli 54(2005), no. 2, 121–137.Google Scholar
[12] Laĺn, M. N. and Rogers, M. D., Functional equations for Mahler measures of genus–one curves. Algebra Number Theory 1(2007), no. 1, 87–117. http://dx.doi.org/10.2140/ant.2007.1.87 Google Scholar
[13] Livné, R., otivic orthogonal two–dimensional representations of . Israel J. Math. 92(1995), no. 1–3, 149–156. http://dx.doi.org/10.1007/BF02762074 Google Scholar
[14] Long, L., Modularity of elliptic surfaces. Ph.D. Thesis, The Pennsylvania State University, Proquest LLC, Ann Arbor, MI, 2002.Google Scholar
[15] Long, L., On a Shioda–Inose structure of a family of K3surfaces. In: Calabi–Yau varieties and mirror symmetry, Fields Institute Communications, 38, American Mathematical Society, Providence, RI, 2003, pp. 201–207.Google Scholar
[16] Long, L., On Shioda–Inose structure of one–parameter families of K3surfaces. J. Number Theory, 109(2004), no. 2, 299–318.http://dx.doi.org/10.1016/j.jnt.2004.06.009 Google Scholar
[17] Morrison, D. R., On K3surfaces with large Picard number. Invent. Math. 75(1984), no. 1, 105–121. http://dx.doi.org/10.1007/BF01403093 Google Scholar
[18] Ono, K., The web of modularity: arithmetic of the coefficients of modular forms and q–series. BMS Regional Conference Series in Mathematics, 102, Conference Board of the Mathematical Sciences, Washington, DC; Amererican Mathematical Society, Providence, RI, 2004.Google Scholar
[19] Rodriguez Villegas, F., Modular Mahler measures. I. In: Topics in Number Theory (University Park, PA, 1997), Math. Appl., 467, Kluwer, Dordrecht, 1999, pp. 17–48.Google Scholar
[20] Rogers, M. D., Hypergeometric formulas for lattice sums and Mahler measures. Int. Math. Res. Not. IMRN 2011, no. 17, 4027–4058.Google Scholar
[21] Rogers, M. D., New 5F4hypergeometric transformations, three-variable Mahler measures, and formulas for 1/π Ramanujan J. 18(2009), no. 3, 327–340. http://dx.doi.org/10.1007/s11139-007-9040-x Google Scholar
[22] Rogers, M. and Yuttanan, B., Modular equations and lattice sums. In: Computational and Analytical Mathematics, Springer Proceedings in Mathematics, to appear.Google Scholar
[23] Rogers, M. and Zudilin, W., From L–series of elliptic curves to Mahler measures. Compos. Math. 148(2012), no. 2, 385–414. http://dx.doi.org/10.1112/S0010437X11007342 Google Scholar
[24] Rogers, M. and Zudilin, W., On the Mahler measure of + X + 1=X + Y + 1= Y. Int. Math. Res. Notices to appear.Google Scholar
[25] Samart, D., Three–variable Mahler measures and special values of modular and Dirichlet L–series. Ramanujan J. 32(2013) no. 2, 245–268. http://dx.doi.org/10.1007/s11139-013-9464-4 Google Scholar
[26] Samart, D., The elliptic trilogarithm and Mahler measures of K3surfaces. http://arxiv:1309.7730 Google Scholar
[27] Schütt, M., CM newforms with rational coefficients. Ramanujan J. 19(2009), no. 2, 187–205. http://dx.doi.org/10.1007/s11139-008-9147-8 Google Scholar
[28] Schütt, M., Two lectures on the arithmetic of K3 surfaces. In: Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds, Fields Institute Communications, 67, Springer, New York, 2013, pp. 71–99.Google Scholar
[29] Schütt, M. and Shioda, T., Elliptic surfaces. In: Algebraic geometry in East Asia–Seoul 2008, Adv. Stud. Pure Math., 60, Math. Soc. Japan, Tokyo, 2010, pp. 51–160.Google Scholar
[30] Silverman, J. H., Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics, 151, Springer–Verlag, New York, 1994.Google Scholar
[31] Weber, H., Lehrbuch der Algebra. Bd. III, F. Vieweg & Sohn, Braunschweig, 1908.Google Scholar
[32] Yui, N., Update on the moudularity of Calabi-Yau varieties. In: Calabi–Yau varieties and mirror symmetry, Fields Institute Communications, 38, American Mathematical Society, Providence RI, 2003, pp. 307–362.Google Scholar
[33] Yui, N. and Zagier, D., On the singular values of Weber modular functions. Math. Comp. 66(1997), no. 220, 1645–1662. http://dx.doi.org/10.1090/S0025-5718-97-00854-5 Google Scholar
[34] Zudilin, W., Regulator of modular units and Mahler measures. Math. Proc. Cambridge Philos. Soc. 156(2014), no. 2, 313–326.http://dx.doi.org/10.1017/S0305004113000765 Google Scholar