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EXPLICIT BOUNDS ON AUTOMORPHIC AND CANONICAL GREEN FUNCTIONS OF FUCHSIAN GROUPS

Part of: Riemann surfaces Arithmetic problems. Diophantine geometry Elliptic equations and systems Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  14 May 2014

Peter Bruin
Peter Bruin*
Affiliation:
Mathematics Institute, Zeeman Building,University of Warwick, Coventry CV4 7AL,U.K. email P.Bruin@warwick.ac.uk
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Abstract

We study the automorphic Green function $\mathop{\rm gr}\nolimits _\Gamma $ on quotients of the hyperbolic plane by cofinite Fuchsian groups $\Gamma $, and the canonical Green function $\mathop{\rm gr}\nolimits ^{\rm can}_X$ on the standard compactification $X$ of such a quotient. We use a limiting procedure, starting from the resolvent kernel, and lattice point estimates for the action of $\Gamma $ on the hyperbolic plane to prove an “approximate spectral representation” for $\mathop{\rm gr}\nolimits _\Gamma $. Combining this with bounds on Maaß forms and Eisenstein series for $\Gamma $, we prove explicit bounds on $\mathop{\rm gr}\nolimits _\Gamma $. From these results on $\mathop{\rm gr}\nolimits _\Gamma $ and new explicit bounds on the canonical $(1,1)$-form of $X$, we deduce explicit bounds on $\mathop{\rm gr}\nolimits ^{\rm can}_X$.

MSC classification

Secondary: 11F72: Spectral theory; Selberg trace formula 30F35: Fuchsian groups and automorphic functions 35J08: Green's functions 14G35: Modular and Shimura varieties 14G40: Arithmetic varieties and schemes; Arakelov theory; heights

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Type
Research Article
Information
Mathematika , Volume 60 , Issue 2 , July 2014 , pp. 257 - 306
Copyright
Copyright © University College London 2014 

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