Equations for Mahler measure and isogenies
Matilde N. Lalín1
1 Département de mathématiques et de statistique Université de Montréal. CP 6128, succ. Centre-ville. Montreal, QC H3C 3J7, Canada
Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 2, pp. 387-399.

We study some functional equations between Mahler measures of genus-one curves in terms of isogenies between the curves. These equations have the potential to establish relationships between Mahler measure and especial values of L-functions. These notes are based on a talk that the author gave at the “Cuartas Jornadas de Teoría de Números”, Bilbao, 2011.

Nous étudions quelques équations fonctionnelles de la mesure de Mahler de familles de courbes de genre 1 en utilisant des isogénies entre les courbes. Ces équations ont le potentiel d’aider à trouver des relations entre la mesure de Mahler et des valeurs spéciales de fonctions L. Ces notes sont inspirées d’une présentation de l’auteure aux ≪Cuartas Jornadas de Teoría de Números≫, à Bilbao, 2011.

DOI: 10.5802/jtnb.841

Matilde N. Lalín 1

1 Département de mathématiques et de statistique Université de Montréal. CP 6128, succ. Centre-ville. Montreal, QC H3C 3J7, Canada 
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Matilde N. Lalín. Equations for Mahler measure and isogenies. Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 2, pp. 387-399. doi : 10.5802/jtnb.841. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.841/

[1] A. A. Beĭlinson, Higher regulators and values of L-functions of curves. Funktsional. Anal. i Prilozhen. 14 (1980), no. 2, 46–47. | MR | Zbl

[2] M.-J. Bertin, Mesure de Mahler d’une famille de polynômes. J. Reine Angew. Math. 569 (2004), 175–188. | MR | Zbl

[3] M.-J. Bertin, Mesure de Mahler et régulateur elliptique: preuve de deux relations “exotiques”. Number theory 1–12, CRM Proc. Lecture Notes, 36, Amer. Math. Soc., Providence, RI, 2004. | MR | Zbl

[4] S. J.Bloch, Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. CRM Monograph Series, 11. American Mathematical Society, Providence, RI, 2000. x+97 pp. | MR | Zbl

[5] D. W. Boyd, Mahler’s measure and special values of L-functions. Experiment. Math. 7 (1998), 37–82. | MR | Zbl

[6] F. Brunault, Étude de la valeur en s=2 de la fonction L d’une courbe elliptique. Doctoral thesis, Université Paris 7 Denis-Diderot, 2005.

[7] F. Brunault, Version explicite du théorème de Beilinson pour la courbe modulaire X 1 (N). C. R. Math. Acad. Sci. Paris 343 (2006), no. 8, 505–510. | MR | Zbl

[8] J. W. S. Cassels, Lectures on elliptic curves. London Mathematical Society Student Texts, 24. Cambridge University Press, Cambridge, 1991. vi+137 pp. | MR | Zbl

[9] C. Deninger, Deligne periods of mixed motives, K-theory and the entropy of certain Z n -actions. J. Amer. Math. Soc. 10 (1997), no. 2, 259–281. | MR | Zbl

[10] J. Guillera, M. Rogers, Mahler measure and the WZ algorithm. Proc. Amer. Math. Soc., June 2010.

[11] N. Kurokawa and H. Ochiai, Mahler measures via crystalization. Commentarii Mathematici Universitatis Sancti Pauli 54 (2005), 121–137. | MR | Zbl

[12] M. N. Lalín, On a conjecture by Boyd. Int. J. Number Theory 6 (2010), no. 3, 705–711. | MR | Zbl

[13] M. N. Lalín, M. D. Rogers, Functional equations for Mahler measures of genus-one curves. Algebra Number Theory 1 (2007), no. 1, 87 – 117. | MR | Zbl

[14] A. Mellit, Elliptic dilogarithms and parallel lines. Preprint 2012, arXiv:1207.4722 [math.NT].

[15] F. Rodriguez-Villegas, Modular Mahler measures I. Topics in number theory (University Park, PA 1997), 17–48, Math. Appl., 467, Kluwer Acad. Publ. Dordrecht, 1999. | MR | Zbl

[16] F. Rodriguez-Villegas, Identities between Mahler measures. Number theory for the millennium, III (Urbana, IL, 2000), 223–229, A K Peters, Natick, MA, 2002. | MR | Zbl

[17] M. Rogers, Hypergeometric formulas for lattice sums and Mahler measures. Int Math Res Notices 17 (2011), 4027–4058. | MR

[18] M. Rogers, W. Zudilin, From L-series of elliptic curves to Mahler measures. Compositio Math. 148 (2012), no. 2, 385–414. | MR | Zbl

[19] M. Rogers, W. Zudilin, On the Mahler measures of 1+X+1/X+Y+1/Y. Preprint, March 2011. To appear in International Math. Research Notices.

[20] J. H. Silverman, The arithmetic of elliptic curves. Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1992. xii+400 pp. | MR | Zbl

[21] J. H. Silverman, J. Tate, Rational points on elliptic curves. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1992. x+281 pp. | MR | Zbl

[22] J. Top, Descent by 3-isogeny and 3-rank of quadratic fields, Advances in number theory (Kingston, ON, 1991), 303–317, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993. | MR | Zbl

[23] N. Touafek, M. Kerada, Mahler measure and elliptic regulator: some identities. JP J. Algebra Number Theory Appl. 8 (2007), no. 2, 271–285. | MR | Zbl

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