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Modular Mahler Measures I

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Topics in Number Theory

Part of the book series: Mathematics and Its Applications ((MAIA,volume 467))

Abstract

We relate Boyd’s numerical examples, linking the Mahler measure m(P k ) of certain polynomials P k to special values of L-series of elliptic curves, to the Bloch-Beilinson conjectures. We study m(P k ) as a function of the parameter k and find a relation to modular forms and certain formal similarities with the expansions of mirror symmetry of physics.

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References

  1. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi—Yau hypersurfaces in tonic varieties, J. Algebraic Geom. 3 (1994), 493–535.

    MathSciNet  MATH  Google Scholar 

  2. A. Beilinson, Higher regulators of modular curves, Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), Contemp. Math., vol. 55, Amer. Math. Soc., Providence, R.I., 1986, pp. 1–34

    MathSciNet  Google Scholar 

  3. D. W. Boyd, Mahler’s measure and special values of L-functions, Experiment. Math. 7 (1998), 37–82.

    MathSciNet  MATH  Google Scholar 

  4. D. W. Boyd, Speculations concerning the range of Mahler’s measure, Canad. Math. Bull. 24 (1981), 453–469.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. W. Boyd, Kronecker’s theorem and Lehmer’s problem for polynomials in several variables, J. Number Theory 13 (1981), 116–121.

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Borcherds,Automorphic forms on O8+2, 2(ℝ) and infinite products, Invent. Math.120 (1995), 161–213.

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Bosch, W. LĂĽtkebohmert and M. Raynaud, NĂ©ron models, Ergebnisse der Mathe-matik und ihrer Grenzgebiete ( 3 ), Springer-Verlag, Berlin, 1990.

    MATH  Google Scholar 

  8. S. Bloch & D. Grayson, K 2 and L—functions of elliptic curves: Computer Calcula-tions, Contemp. Math. 55 (1986), 79–88.

    MathSciNet  Google Scholar 

  9. S. Bloch & K. Kato, L-Functions and Tamagawa Numbers of Motives, The Grothen-dieck Festschrift (P. Cartier et al, eds.), vol. I, Birkhäuser, Boston; Progress in Mathematics, vol. 86, 1990, pp. 333–400.

    Google Scholar 

  10. D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen, Plane curves associated to character varieties of 3-manifolds, Invent. Math. 118 (1994), 47–84.

    Article  MathSciNet  MATH  Google Scholar 

  11. V. I. Danilov and A. G. Khovansky, Newton polyhedra and an algorithm for computing Deligne—Hodge numbers, Math. USSR—Izv 29 (1987), 279–298.

    MATH  Google Scholar 

  12. C. Deninger, Deligne periods of mixed motives, K-theory and the entropy of certain Zn -actions, J. Amer Math. Soc. 10 (1997), 259–281.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Griffiths, On the periods of certain rational integrals: I, II, Ann. of Math., 461–541.

    Google Scholar 

  14. A. B. Goncharov and A. M. Levin, Zagier’s conjecture on L(E,2), preprint, 1997.

    Google Scholar 

  15. D. Hensley, Lattice vertex polytopes with interior lattice points, Pacific J. Math. 105 (1983), 183–191.

    MathSciNet  MATH  Google Scholar 

  16. Hosono, Saito, and Stienstra, On the mirror symmetry conjecture for Schoen’s Calabi-Yau threefolds, preprint 1998.

    Google Scholar 

  17. D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), 461–479.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Lefschetz, Algebraic Geometry, Princeton Univ. Press, Princeton, NJ, 1953.

    MATH  Google Scholar 

  19. D. Lind, K. Schmidt, and T. Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math. 101 (1990), 593–629.

    Article  MathSciNet  MATH  Google Scholar 

  20. K. Mahler, On some inequalities for polynomials in several variables, J. London Math. Soc. (2) 37 (1962), 341–344.

    Article  MathSciNet  MATH  Google Scholar 

  21. K. Mahler, symmétriques de courbes elliptiques sur Q Arithmetic Algebraic Geometry (G. van der Geer, F. Oort, and J. Steenbrink, eds.), Progress in Mathematics 89, Birkhäuser, 1991, pp. 209–245.

    Google Scholar 

  22. J. Milnor, Introduction to algebraic K-theory, Annals of Mathematical Studies, vol. 72, Princeton Univ. Press, Princeton, N.J., 1971.

    MATH  Google Scholar 

  23. P. Philippon, Critères pour l’indépendence algébrique, Publ. IHES 64 (1986), 5–52.

    MathSciNet  MATH  Google Scholar 

  24. T. Pierce,The numerical factors of the arithmetic functions jj 1(1 ± Win), Ann. Of Math. 18 (1916–17).

    Google Scholar 

  25. D. Ramakrishnan, Regulators, Algebraic Cycles, and Values of L-functions, Algebraic K-theory and Algebraic Number Theory (M. R. Stein and R. K. Dennis, eds.), Contemporary Mathematics, vol. 83, Amer. Math. Soc., Providence, R.I., 1989, pp. 183–310.

    Google Scholar 

  26. G. A. Ray, Relations between Mahler’s measure and values of L-series, Canad. J. Math. 39 (1987), 649–732.

    Article  Google Scholar 

  27. F. Rodriguez Villegas, Modular Mahler measures II,in preparation.

    Google Scholar 

  28. R. Ross, Ph. D. thesis, Rutgers University, 1989.

    Google Scholar 

  29. K. Rolhausen, Eléments explicites dans K 2 d’une courbe elliptique, Institut de Recherche Mathématique Avancée, Strasbourg, 1996.

    Google Scholar 

  30. P. Sarnak, Spectral behavior of quasi-periodic potentials, Comm. Math. Phys. 84 (1982), 377–401.

    Article  MathSciNet  MATH  Google Scholar 

  31. N. Schappacher, Les conjectures de Beilinson pour !es courbes elliptiques, Journées Arithmétiques, 1989 (Luminy, 1989 ), Astérisque, No. 198–200, 1992, pp. 305–317.

    Google Scholar 

  32. P. R. Scott, On convex lattice polygons, Bull. Austral. Math. Soc. 15 (1976), 395–399.

    Article  MathSciNet  MATH  Google Scholar 

  33. J. H. Silverman, The arithmetic of elliptic curves, Springer Verlag, Berlin and New York, 1986.

    MATH  Google Scholar 

  34. C. Soulé,Geometrie d’Arakelov et théorie des nombres trascendants, Journées Arith-metiques de Luminy (G. Lachaud, eds.), Astérisque, No. 198–200, 1991 pp. 355–371.

    Google Scholar 

  35. C. Soulé, Regulateurs, Seminar Bourbaki, Vol. 1984/85, Astérisque, No. 133–134, 1986, pp. 237–253.

    Google Scholar 

  36. C. J. Smyth, On measures of polynomials in several variables, Bull. Austral. Math. Soc. 23 (1981), 49–63.

    MathSciNet  MATH  Google Scholar 

  37. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Uni-versity Press, Cambridge, 1984.

    Google Scholar 

  38. D. Zagier, A modular identity arising from mirror symmetry, preprint 1998.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA

    F. Rodriguez Villegas

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  1. F. Rodriguez Villegas
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, University Park, PA, USA

    Scott D. Ahlgren , George E. Andrews  & Ken Ono ,  & 

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© 1999 Kluwer Academic Publishers

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Villegas, F.R. (1999). Modular Mahler Measures I. In: Ahlgren, S.D., Andrews, G.E., Ono, K. (eds) Topics in Number Theory. Mathematics and Its Applications, vol 467. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0305-3_2

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  • DOI: https://doi.org/10.1007/978-1-4613-0305-3_2

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7988-1

  • Online ISBN: 978-1-4613-0305-3

  • eBook Packages: Springer Book Archive

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