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Equidistribution and generalized Mahler measures

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Number Theory, Analysis and Geometry

Abstract

If K is a number field and \(\varphi: {\mathcal{P}}_{K}^{1}\rightarrow {\mathcal{P}}_{K}^{1}\) is a rational map of degree d > 1, then at each place v of K, one can associate to φ a generalized Mahler measure for polynomials F ∈ K[t]. These Mahler measures give rise to a formula for the canonical height h φ(β) of an element \(\beta\in \overline{K}\); this formula generalizes Mahler’s formula for the usual Weil height h(β). In this paper, we use Diophantine approximation to show that the generalized Mahler measure of a polynomial F at a place v can be computed by averaging log | F | v over the periodic points of φ.

Mathematics Subject Classification (2010): Primary 37P30; Secondary 11J68

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Acknowledgements

We would like to thank M. Baker, A. Chambert-Loir, L. DeMarco, C. Petsche, R. Rumely, and S. Zhang for many helpful conversations. In particular, we thank M. Baker, L. DeMarco, and R. Rumely for suggesting some of the applications mentioned in Section 7. The first author was partially supported by NSF Grant 0071921. The second author was partially supported by NSF Grant 0101636.

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Authors and Affiliations

  1. Program in Mathematics, Graduate Center of CUNY, 365 Fifth Avenue, New York, NY, 10016-4309, USA

    L. Szpiro

  2. Department of Mathematics, Hylan Building, University of Rochester, Rochester, NY, 14627, USA

    T. J. Tucker

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  1. L. Szpiro
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  2. T. J. Tucker
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Correspondence to L. Szpiro .

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  1. , Department of Mathematics, Columbia University, Broadway 2990, New York, 10027, New York, USA

    Dorian Goldfeld

  2. , Department of Mathematics, City College of New York, New York, 10031, New York, USA

    Jay Jorgenson

  3. , Department of Mathematics, Yale University, Hillhouse Ave. 10, New Haven, 06520, Connecticut, USA

    Peter Jones

  4. , Department of Mathematics, California Institute of Technology, E. California Blvd. 1200, Pasadena, 91125, California, USA

    Dinakar Ramakrishnan

  5. , Department of Mathematics, University of California at Berkeley, Evans Hall 925, Berkeley, 94720-3840, California, USA

    Kenneth Ribet

  6. , Department of Mathematics, Harvard University, Science Center/One Oxford Street, Cambridge, 02138, Massachusetts, USA

    John Tate

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Dedicated to the memory of Serge Lang, who taught the world number theory for more than fifty years, through his research, lectures, and books

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Szpiro, L., Tucker, T.J. (2012). Equidistribution and generalized Mahler measures. In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds) Number Theory, Analysis and Geometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1260-1_26

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