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Bounds on canonical Green's functions

Published online by Cambridge University Press:  17 May 2006

J. Jorgenson  and
J. Kramer
J. Jorgenson
Affiliation:
Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, NY 10031, USAjjorgenson@mindspring.com
J. Kramer
Affiliation:
Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germanykramer@math.hu-berlin.de
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Abstract

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A fundamental object in the theory of arithmetic surfaces is the Green's function associated to the canonical metric. Previous expressions for the canonical Green's function have relied on general functional analysis or, when using specific properties of the canonical metric, the classical Riemann theta function. In this article, we derive a new identity for the canonical Green's function involving the hyperbolic heat kernel. As an application of our results, we obtain bounds for the canonical Green's function through covers and for families of modular curves.

Keywords

Green's functionsArakelov theorymodular curveshyperbolic heat kernels14G4030F1011F7230C40
Type
Research Article
Information
Compositio Mathematica , Volume 142 , Issue 3 , May 2006 , pp. 679 - 700
Copyright
Foundation Compositio Mathematica 2006