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Regulator of modular units and Mahler measures

Published online by Cambridge University Press:  02 January 2014

WADIM ZUDILIN
WADIM ZUDILIN*
Affiliation:
School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia. e-mail: [email protected]
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Abstract

We present a proof of the formula, due to Mellit and Brunault, which evaluates an integral of the regulator of two modular units to the value of the L-series of a modular form of weight 2 at s=2. Applications of the formula to computing Mahler measures are discussed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 2 , March 2014 , pp. 313 - 326
Copyright
Copyright © Cambridge Philosophical Society 2014 

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References

REFERENCES

[1]Baker, M. H., González–Jiménez, E., González, J. and Poonen, B.Finiteness results for modular curves of genus at least 2. Amer. J. Math. 127 (2005), 1325–1387.Google Scholar
[2]Bertin, M. J.Mesure de Mahler dune famille de polynômes. J. Reine Angew. Math. 569 (2004), 175–188.Google Scholar
[3]Boyd, D.Mahler's measure and special values of L-functions. Experiment. Math. 7 (1998), 37–82.Google Scholar
[4]Boyd, D. Mahler's measure and L-functions of elliptic curves evaluated at s=3, slides from a lecture at the SFU/UBC number theory seminar (7 December 2006), http://www.math.ubc.ca~boyd/sfu06.ed.pdf.Google Scholar
[5]Brunault, F.Beilinson–Kato elements in K 2 of modular curves. Acta Arith. 134 (2008), 283–298.Google Scholar
[6]Condon, J. D. Mahler measure evaluations in terms of polylogarithms. Dissertation. The University of Texas at Austin (2004).Google Scholar
[7]Deninger, C.Deligne periods of mixed motives, K-theory and the entropy of certain $\mathbb Z^n$-actions. J. Amer. Math. Soc. 10 (1997), 259–281.Google Scholar
[8]Katz, N. M.p-adic interpolation of real analytic Eisenstein series. Ann. of Math., Ser. (2) 104 (1976), 459–571.CrossRefGoogle Scholar
[9]Lalín, M. N.On a conjecture by Boyd. Intern. J. Number Theory 6 (2010), 705–711.Google Scholar
[10]Lalín, M. N. and Rogers, M. D.Functional equations for Mahler measures of genus-one curves. Algebra and Number Theory 1 (2007), 87–117.CrossRefGoogle Scholar
[11]Mellit, A.Mahler measures and q-series. In Explicit Methods in Number Theory (MFO, Oberwolfach, Germany, 17–23 July 2011), Oberwolfach Reports 8 (2011), no. 3, 1990–1991.Google Scholar
[12]Mellit, A. Regulator of two modular units formula, unpublished note (12 June 2012).Google Scholar
[13]Rodriguez Villegas, F.Modular Mahler measures. I. In Topics in Number Theory (University Park, PA, 1997), Math. Appl. 467 (Kluwer Academic Publisher, Dordrecht, 1999), 17–48.Google Scholar
[14]Rogers, M. and Zudilin, W.From L-series of elliptic curves to Mahler measures. Compositio Math. 148 (2012), 385–414.Google Scholar
[15]Rogers, M. and Zudilin, W.On the Mahler measure of 1+X+1/X+Y+1/Y. Internat. Math. Res. Not., to appear; doi: 10.1093/imrn/rns285.Google Scholar
[16]Schoeneberg, B.Elliptic modular functions: an introduction, translated from the German by Smart, J. R. and Schwandt, E. A. Die Grundlehren der mathematischen Wissenschaften 203 (Springer-Verlag, 1974).Google Scholar
[17]Yang, Y.Transformation formulas for generalized Dedekind eta functions. Bull. London Math. Soc. 36 (2004), 671–682.Google Scholar
[18]Zudilin, W.Period(d)ness of L-values. In Number Theory and Related Fields, in memory of Alf van der Poorten, Borwein, J. M.et al. (eds.). Springer Proceedings in Math. Stat. 43 (Springer, New York, 2013), 381–395.Google Scholar