The Mahler measure for arbitrary tori
We consider a variation of the Mahler measure where the defining integral is performed over a more general torus. We focus our investigation on two particular polynomials related to certain elliptic curve E and we establish new formulas for this variation of the Mahler measure in terms of \(L'(E,0)\).
KeywordsMahler measure Special values of L-functions Elliptic curve Elliptic regulator
Mathematics Subject ClassificationPrimary 11R06 Secondary 11G05 11F66 19F27 33E05
ML, TM have participated in the whole study and drafted the manuscript, and both authors read and approved the final manuscript.
We are thankful to Marie-José Bertin for providing us a copy of Touafek’s doctoral thesis . We are very grateful to the anonymous referees for their dedicated work and for their several corrections and suggestions that greatly improved the exposition. We would like to specially thank the referee who found a mistake in one of our main formulas. This research was supported by the Natural Sciences and Engineering Research Council of Canada [Discovery Grant 355412-2013 to ML] and Mitacs [Globalink Research Internship to TM].
The authors declare that they have no competing interests.
- 1.Bertin, M.-J.: Mahler measure, regulators, and modular units. Workshop lecture at “The Geometry, Algebra and Analysis of Algebraic Numbers”. Banff International Research Station, Banff, Canada (2015)Google Scholar
- 3.Bloch, S., Grayson, D.: \(K_2\) and \(L\)-functions of elliptic curves: computer calculations, Applications of algebraic \(K\)-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., : Contemp. Math., vol. 55, Amer. Math. Soc. Providence, RI 1986, 79–88 (1983)Google Scholar
- 5.Bloch, S.J.: Higher Regulators, Algebraic \(K\)-Theory, and Zeta Functions of Elliptic Curves, CRM Monograph Series, vol. 11. American Mathematical Society, Providence (2000)Google Scholar
- 10.Maillot, V.: Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables, no. 80 (2000)Google Scholar
- 11.Mellit, A.: Elliptic dilogarithms and parallel lines. ArXiv e-prints (2012)Google Scholar
- 13.Rodriguez-Villegas, F.: Mahler, Modular, Measures. I, Topics in Number Theory (UniversityPark, PA, 1997), Math. Appl., vol. 467, pp. 17–48. Kluwer Academic Publisher, Dordrecht (1997)Google Scholar
- 14.Rodriguez-Villegas, F.: Identities Between Mahler Measures, Number Theory for the Millennium, III (Urbana, IL, 2000), vol. 2002, pp. 223–229. A K Peters, Natick (2000)Google Scholar
- 18.Touafek, N.: Thèse de doctorat. Université de Constantine, Algeria (2008)Google Scholar