Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-08T09:27:54.175Z Has data issue: false hasContentIssue false
  • English
  • Français

From L-series of elliptic curves to Mahler measures

Part of: $K$-theory in number theory Algebraic number theory: global fields Other special functions Discontinuous groups and automorphic forms Curves

Published online by Cambridge University Press:  23 January 2012

Mathew Rogers  and
Wadim Zudilin
Mathew Rogers
Affiliation:
Department of Mathematics and Statistics, Université de Montréal, CP 6128 succ. Centre-ville, Montréal Québec H3C 3J7, Canada (email: mathewrogers@gmail.com)
Wadim Zudilin
Affiliation:
School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan NSW 2308, Australia (email: wadim.zudilin@newcastle.edu.au)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove the conjectural relations between Mahler measures and L-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for L-values of elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family (1+X) (1+Y )(X+Y )−αXY, α∈ℝ.

Keywords

Mahler measureL-value of elliptic curvehypergeometric serieslattice sumelliptic dilogarithm

MSC classification

Secondary: 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure 33C20: Generalized hypergeometric series, $_pF_q$ 11F03: Modular and automorphic functions 14H52: Elliptic curves 19F27: Étale cohomology, higher regulators, zeta and $L$-functions 33C75: Elliptic integrals as hypergeometric functions 33E05: Elliptic functions and integrals
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 2 , March 2012 , pp. 385 - 414
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[AB05]Andrews, G. E. and Berndt, B. C., Ramanujan’s lost notebook, part I (Springer, New York, NY, 2005).CrossRefGoogle Scholar
[BF10]Bacher, R. and Flajolet, P., Pseudo-factorials, elliptic functions, and continued fractions, Ramanujan J. 21 (2010), 71–97.CrossRefGoogle Scholar
[Ber91]Berndt, B. C., Ramanujan’s notebooks, part III (Springer, New York, NY, 1991).Google Scholar
[Ber94]Berndt, B. C., Ramanujan’s notebooks, part IV (Springer, New York, NY, 1994).Google Scholar
[Ber98]Berndt, B. C., Ramanujan’s notebooks, part V (Springer, New York, NY, 1998).CrossRefGoogle Scholar
[BG86]Bloch, S. and Grayson, D., K 2 and L-functions of elliptic curves: computer calculations, in Applications of algebraic K-theory to algebraic geometry and number theory, part I, II (Boulder, Colo., 1983), Contemporary Mathematics, vol. 55 (American Mathematical Society, Providence, RI, 1986), 79–88.CrossRefGoogle Scholar
[BBT85]Borwein, D., Borwein, J. M. and Taylor, K. F., Convergence of lattice sums and Madelung’s constant, J. Math. Phys. 26 (1985), 2999–3009.CrossRefGoogle Scholar
[BB91]Borwein, J. M. and Borwein, P. B., A cubic counterpart of Jacobi’s identity and the AGM, Trans. Amer. Math. Soc. 323 (1991), 691–701.Google Scholar
[Boy98]Boyd, D. W., Mahler’s measure and special values of L-functions, Experiment. Math. 7 (1998), 37–82.Google Scholar
[Bru06]Brunault, F., Version explicite du théorème de Beilinson pour la courbe modulaire X 1(N), C. R. Math. Acad. Sci. Paris 343 (2006), 505–510.Google Scholar
[GR94]Gradshteyn, I. S. and Ryzhik, I. M., Table of integrals, series and products (Academic Press, New York, NY, 1994).Google Scholar
[GR10]Guillera, J. and Rogers, M., Mahler measure and the WZ algorithm, Preprint (2010), arXiv:1006.1654 [math.NT].Google Scholar
[KO05]Kurokawa, N. and Ochiai, H., Mahler measures via crystalization, Comment. Math. Univ. St. Pauli 54 (2005), 121–137.Google Scholar
[LR07]Lalín, M. N. and Rogers, M. D., Functional equations for Mahler measures of genus-one curves, Algebra Number Theory 1 (2007), 87–117.Google Scholar
[MO97]Martin, Y. and Ono, K., Eta-quotients and elliptic curves, Proc. Amer. Math. Soc. 125 (1997), 3169–3176.Google Scholar
[Mel11]Mellit, A., Elliptic dilogarithms and parallel lines, Preprint (2011).Google Scholar
[Rod99]Rodriguez-Villegas, F., Modular Mahler measures I, in Topics in number theory (University Park, PA, 1997), Mathematics and Its Applications, vol. 467 (Kluwer Academic Publishers, Dordrecht, 1999), 17–48.Google Scholar
[Rog11]Rogers, M., Hypergeometric formulas for lattice sums and Mahler measures, Int. Math. Res. Not. IMRN 2011 (2011), 4027–4058.Google Scholar
[RY11]Rogers, M. and Yuttanan, B., Modular equations and lattice sums, Preprint (2011), arXiv:1001.4496 [math.NT].Google Scholar
[Sla66]Slater, L. J., Generalized hypergeometric functions (Cambridge University Press, Cambridge, 1966).Google Scholar
[Som]Somos, M., Dedekind eta function product identities, available at http://eta.math.georgetown.edu/.Google Scholar
[Sti06]Stienstra, J., Mahler measure variations, Eisenstein series and instanton expansions, in Mirror symmetry V, AMS/IP Studies in Advanced Mathematics, vol. 38, eds Yui, N., Yau, S.-T. and Lewis, J. D. (International Press & American Mathematical Society, Providence, RI, 2006), 139–150.Google Scholar
[WL88]Wan, Shi Dong and Li, Ji Bin, Fourier series of rational fractions of Jacobian elliptic functions, Appl. Math. Mech. (English Ed.) 9 (1988), 541–556.Google Scholar
[Wol07] The Wolfram function site, Complete elliptic integral of the third kind: identities (formula 08.03.17.0003) (2007), avalaible at http://functions.wolfram.com/08.03.17.0003.01.Google Scholar