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EXPLICIT BOUNDS ON AUTOMORPHIC AND CANONICAL GREEN FUNCTIONS OF FUCHSIAN GROUPS

Published online by Cambridge University Press:  14 May 2014

Peter Bruin
Affiliation:
Mathematics Institute, Zeeman Building,University of Warwick, Coventry CV4 7AL,U.K. email P.Bruin@warwick.ac.uk
Corresponding
E-mail address:

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 $\mathop{\rm gr}\nolimits ^{\rm can}_X$.

Type
Research Article
Copyright
Copyright © University College London 2014 

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EXPLICIT BOUNDS ON AUTOMORPHIC AND CANONICAL GREEN FUNCTIONS OF FUCHSIAN GROUPS
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