Published online by Cambridge University Press: 14 May 2014
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
$\mathop{\rm gr}\nolimits ^{\rm can}_X$
.
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