Explicit bounds on automorphic and canonical Green functions of Fuchsian groups (Q2874610)

In this paper ``the automorphic Green function \(\mathrm{gr}_\Gamma\) on quotients of the hyperbolic plane by cofinite Fuchsian groups, and the canonical Green function \(\mathrm{gr}_X^{\mathrm{can}}\) on the standard compactification \(X\) of such a quotient'' are studied. In more details, ``let \(H = \{ z\in \mathbb{C}\mid \text{Im}(z)>0 \}\) be the hyperbolic plane, and let \(\Gamma\) be a cofinite Fuchsian group. The automorphic Green function \(\mathrm{gr}_\Gamma\) for the Laplace operator \(\Delta_\Gamma\) on \(H/\Gamma\) is an important object in the theory of automorphic forms. The first goal of this paper is to study \(\mathrm{gr}_\Gamma\) quantitatively, and in particular to obtain explicit upper and lower bounds. One result that can be stated without introducing too much notation is the following Theorem.''NEWLINENEWLINETheorem 1.1. Let \(\Gamma_0\) be a cofinite Fuchsian group, let \(Y_0\) be a compact subset of \(H/\Gamma_0\) , and let \(\delta>1\) and \(\eta>0\) be real numbers. There exist real numbers \(A\) and \(B\) such that the following holds. Let \(\Gamma\) be a subgroup of finite index in \(\Gamma_0\) such that all non-zero eigenvalues of \(-\Delta_\Gamma\) are at least \(\eta\). Then for all \(z, w\in H\) whose images in \(H/\Gamma_0\) lie in \(Y_0\), we have NEWLINE\[NEWLINEA\leq \mathrm{gr}_\Gamma(z, w)+\sum_{\substack{\gamma\in\Gamma,\\ u(z,\gamma w)\leq\delta}} L_\delta(z,\gamma w)\leq BNEWLINE\]NEWLINE where the function \(u(z, w)\) is the hyperbolic cosine of the hyperbolic distance, and \(L_\delta(z,w)\), is a real-valued function outside the diagonal on \(H\times H\) with a logarithmic singularity of the form \(-\frac{1}{2\pi} \log |z-w|\) as \(z\to w\).NEWLINENEWLINE``Every compact connected Riemann surface of positive genus has a canonical Green function \(\mathrm{gr}_X^{\mathrm{can}}\). This is a fundamental object in the intersection theory on arithmetic surfaces developed by Arakelov, Faltings and others, where it is used to define local intersection numbers of horizontal divisors at Archimedean places. Let \(X\) be the standard compactification of \(H/\Gamma\) obtained by adding the cusps. We assume that \(X\) has positive genus. The second goal of this paper is to derive explicit bounds on \(\mathrm{gr}_X^{\mathrm{can}}\) from our bounds on \(\mathrm{gr}_\Gamma\). The following theorem illustrates the results. For simplicity, we only give an upper bound; see Theorem (7.1) for more precise results.''NEWLINENEWLINETheorem 1.2. Let \(\Gamma\) be a congruence subgroup of level \(n\) of \(\mathrm{SL}_2(\mathbb{Z})\) such that the compactification \(X\) of \(H/\Gamma\) has positive genus. Then the canonical Green function \(\mathrm{gr}_X^{\mathrm{can}}\) satisfies NEWLINE\[NEWLINE\sup_{X\times X} \mathrm{gr}_X^{\mathrm{can}}\leq 1.6\cdot10^4+7.7n+0.088n^2NEWLINE\]