Bounds for canonical Green’s function at cusps

Abstract

In this article, we derive bounds for the canonical Green’s function defined on a noncompact hyperbolic Riemann surface, when evaluated at two inequivalent cusps.

1 Introduction

Notation and main results Let $X$ be a noncompact hyperbolic Riemann surface of finite volume ${{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)$ with genus $g\ge 1$. We assume that $X$ is devoid of elliptic fixed points. Then, from the uniformization theorem from complex analysis, $X$ can be realized as the quotient space $\Gamma \backslash \mathbb {H}$, where $\Gamma \subset \mathrm {PSL}_{2}(\mathbb {R})$ is a Fuchsian subgroup of the first kind acting on the hyperbolic upper half-plane $\mathbb {H}$, via fractional linear transformations. Let $\mathcal {P}$ denote the set of cusps of $\Gamma$. Put $\overline{X}=X\cup \mathcal {P}$.

The compact Riemann surface $\overline{X}$ is embedded in its Jacobian variety $\mathrm {Jac}(\overline{X})$ via the Abel-Jacobi map. Then, the pull back of the flat Euclidean metric by the Abel-Jacobi map is called the canonical metric, and the (1,1)-form associated to it is denoted by ${{\mathrm{\widehat{\mu }_{can}}}}(z)$. We denote its restriction to $X$ by ${{\mathrm{\mu _{can}}}}(z)$. Put

$$\begin{aligned} d_{X}=\sup _{z\in X}\frac{{{\mathrm{\mu _{can}}}}(z)}{{{\mathrm{\mu _{shyp}}}}(z)}. \end{aligned}$$

The canonical Green’s function is defined as the unique solution of the differential equation (which is to be interpreted in terms of currents)

$$\begin{aligned} d_{z}d^{c}_{z}{{\mathrm{{\textit{g}_{{\mathrm {can}}}}}}}(z,w)+ \delta _{w}(z)={{\mathrm{\mu _{can}}}}(z), \end{aligned}$$

with the normalization condition

$$\begin{aligned} \int _{X}{{\mathrm{{\textit{g}_{{\mathrm {can}}}}}}}(z,w){{\mathrm{\mu _{can}}}}(z)=0. \end{aligned}$$

Let $c_{X}$ denote a certain constant related to the Selberg zeta function (see Eq. (11) for definition). Let $\lambda _{1}$ denote the first non-zero eigenvalue of the hyperbolic Laplacian ${{\mathrm{\Delta _{hyp}}}}$ acting on smooth functions defined on $X$. Let $\kappa _{p}(z)$ denote the Kronecker’s limit function associated to a cusp $p\in \mathcal {P}$, which is the constant term in the Laurent expansion of the Eisenstein series associated to the cusp $p\in \mathcal {P}$ at $s=1$. Let $k_{p,q}(0)$ $\in$ $\mathbb {C}$ denote the zeroth Fourier coefficient of $\kappa _{p}(z)$ at the cusp $q\in \mathcal {P}$.

Main theorem

With notation as above, we have the following upper bound

$$\begin{aligned}&\big | g_{{\mathrm {can}}}(p,q)\big |\le 4\pi \big |k_{p,q}(0)\big |+ \frac{2\pi }{g}\bigg (\sum _{\begin{array}{c} s\in \mathcal {P}\\ s\not = p \end{array}}|k_{s,p}(0)\big | \sum _{\begin{array}{c} s\in \mathcal {P}\\ s\not = q \end{array}}\big |k_{s,q}(0)\big |\bigg )+\nonumber \\&\quad +\frac{1}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg (\frac{4\pi (d_{X}+1)^{2}}{\lambda _{1}}+\frac{\big |4\pi c_{X}\big |}{g}+ \frac{ 43|\mathcal {P}|}{g}+4\pi \bigg )+\frac{2\log (4\pi )}{g}. \end{aligned}$$

Arithmetic significance Bounds for the canonical Green’s function are very essential for calculating various arithmetic invariants like the Faltings height function and the Faltings delta function. Especially bounds for the canonical Green’s function evaluated at two inequivalent cusps are essential for calculating the arithmetic self-intersection number of the dualizing sheaf defined on an arithmetic surface.

In [1], while bounding the arithmetic self-intersection number of the dualizing sheaf defined on the modular curve $X_{0}(N)$, Abbes and Ullmo computed bounds for the canonical Green’s function evaluated at the cusps $0$ and $\infty$. In [11], Mayer has done the same for the modular curve $X_{1}(N)$.

Furthermore, in [12] Kühn has also derived bounds for the arithmetic self-intersection number of the dualizing sheaf defined on any curve defined over a number field. However, his bounds for the analytic contribution of the arithmetic self-intersection number of the dualizing sheaf are not explicit. That is, he bounds the analytic contribution by a certain constant, which is not effective. As our bounds hold true for any noncompact hyperbolic Riemann surface of genus $g\ge 1$, they can be directly used in [12]. We hope that this leads to more explicit bounds for Kühn.

As our bounds depend on the zeroth Fourier coefficients of the Kronecker’s limit function, they depend on the uniformization of the Riemann surface. However, in [7], Jorgenson and Kramer have shown that for modular curves, the remaining terms involved in our bounds are independent of the uniformization. If one can relate the zeroth Fourier coefficients of the Kronecker’s limit functions of two Riemann surfaces, where one is a finite degree cover of the other, then one can apply techniques from [7] to show that our bounds are also independent of the uniformization, for any noncompact modular curve of genus $g\ge 1$.

Lastly, using results from [2] and [3], our bounds can be easily extended to the case when $X$ admits elliptic fixed points.

Organization of the paper In the first section, we set up our notation, introduce basic notions and recall some results. In the second section, we compute bounds for the canonical Green’s functions evaluated at two inequivalent cusps.

2 Background material

Let $\Gamma \subset \mathrm {PSL}_{2}(\mathbb {R})$ be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane $\mathbb {H}$. Let $X$ be the quotient space $\Gamma \backslash \mathbb {H}$, and let $g$ denote the genus of $X$. The quotient space $X$ admits the structure of a Riemann surface.

Let $\mathcal {P}(\Gamma )$ and $H(\Gamma )$ denote the set of parabolic and hyperbolic elements of $\Gamma$, respectively. We assume that $\Gamma$ does not admit elliptic fixed points. Let $\mathcal {P}$ be the finite set of cusps of $X$, respectively. Let $\overline{X}$ denote $\overline{X}=X\cup \mathcal {P}$. Locally, away from the cusps, we identity $\overline{X}$ with its universal cover $\mathbb {H}$, and hence, denote the points on $\overline{X}\backslash \mathcal {P}$ by the same letter as the points on $\mathbb {H}$.

Structure of $\overline{X}$ as a Riemann surface The quotient space $\overline{X}$ admits the structure of a compact Riemann surface. We refer the reader to section 1.8 in [13], for the details regarding the structure of $\overline{X}$ as a compact Riemann surface. For the convenience of the reader, we recall the coordinate functions for the neighborhoods of cusps.

Let $p\in \mathcal {P}$ be a cusp and let $w\in U_{r}(p)$, where $U_{r}(p)$ denotes an open coordinate disk of radius $r$ around the cusp $p$. Then $\vartheta _{p}(w)$ is given by

$$\begin{aligned} \vartheta _{p}(w)= e^{2\pi i \sigma _{p}^{-1}w}, \end{aligned}$$

where $\sigma _{p}$ is a scaling matrix of the cusp $p$ satisfying the following relations

$$\begin{aligned} \sigma _{p}i\infty = p \quad {\mathrm {and}} \quad \sigma _{p}^{-1}\Gamma _{p}\sigma _{p} = \langle \gamma _{\infty }\rangle ,\quad {\mathrm {where}}\,\,\, \gamma _{\infty }=\left( \begin{array}{ccc} 1 &{}\quad 1\\ 0 &{}\quad 1 \end{array}\right) \quad&{\mathrm {and}}\quad \Gamma _{p}=\langle \gamma _{p}\rangle \end{aligned}$$

denotes the stabilizer of the cusp $p$ with generator $\gamma _{p}$.

Hyperbolic metric We denote the (1,1)-form corresponding to the hyperbolic metric of $X$, which is compatible with the complex structure on $X$ and has constant negative curvature equal to minus one, by ${{\mathrm{\mu _{hyp}}}}(z)$. Locally, for $z\in X$, it is given by

$$\begin{aligned} {{\mathrm{\mu _{hyp}}}}(z)= \frac{i}{2}\cdot \frac{dz\wedge d\overline{z}}{{{{\mathrm{Im}}}(z)}^{2}}. \end{aligned}$$

Let ${{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)$ be the volume of $X$ with respect to the hyperbolic metric ${{\mathrm{\mu _{hyp}}}}$. It is given by the formula

$$\begin{aligned} {{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X) = 2\pi \big (2g -2 + |\mathcal {P}|\big ). \end{aligned}$$

The hyperbolic metric ${{\mathrm{\mu _{hyp}}}}(z)$ is singular at the cusps, and the rescaled hyperbolic metric

$$\begin{aligned} {{\mathrm{\mu _{shyp}}}}(z)= \frac{{{\mathrm{\mu _{hyp}}}}(z)}{ {{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)} \end{aligned}$$

measures the volume of $X$ to be one.

For $z=x+iy \in \mathbb {H}$, the hyperbolic Laplacian ${{\mathrm{\Delta _{hyp}}}}$ is given by

$$\begin{aligned} \Delta _{\mathrm {hyp}} = -y^{2}\bigg (\frac{\partial ^{2}}{\partial x^{2}} + \frac{\partial ^{2}}{\partial y^{2}}\bigg ) = -4y^{2}\bigg (\frac{\partial ^{2}}{\partial z \partial \overline{z}} \bigg ). \end{aligned}$$

Recall that $d=\left( \partial + \overline{\partial } \right)$, $d^{c}=\dfrac{1}{4\pi i}\left( \partial - \overline{\partial }\right)$, and $dd^{c}= -\dfrac{\partial \overline{\partial }}{2\pi i}$. Furthermore, we have

$$\begin{aligned} dd^{c}={{\mathrm{\mu _{hyp}}}}{{\mathrm{\Delta _{hyp}}}}. \end{aligned}$$

(1)

Canonical metric Let $S_{2}(\Gamma )$ denote the $\mathbb {C}$-vector space of cusp forms of weight 2 with respect to $\Gamma$ equipped with the Petersson inner product. Let $\lbrace f_{1},\ldots ,f_{g}\rbrace$ denote an orthonormal basis of $S_{2}(\Gamma )$ with respect to the Petersson inner product. Then, the (1,1)-form ${{\mathrm{\mu _{can}}}}(z)$ corresponding to the canonical metric of $X$ is given by

$$\begin{aligned} {{\mathrm{\mu _{can}}}}(z)=\frac{i}{2g} \sum _{j=1}^{g}\left| f_{j}(z)\right| ^{2}dz\wedge d\overline{z}. \end{aligned}$$

The canonical metric ${{\mathrm{\mu _{can}}}}(z)$ remains smooth at the cusps, and measures the volume of $X$ to be one. We denote the smooth (1,1)-form defined by ${{\mathrm{\mu _{can}}}}(z)$ on $\overline{X}$ by ${{\mathrm{\widehat{\mu }_{can}}}}(z)$.

For $z\in X$, we put,

$$\begin{aligned} d_{X}=\sup _{z\in X}\frac{{{\mathrm{\mu _{can}}}}(z)}{{{\mathrm{\mu _{shyp}}}}(z)}. \end{aligned}$$

(2)

As the canonical metric ${{\mathrm{\mu _{can}}}}(z)$ remains smooth at the cusps and at the elliptic fixed points, and the hyperbolic metric is singular at these points, the quantity $d_{X}$ is well-defined.

Canonical Green’s function For $z, w \in X$, the canonical Green’s function ${{\mathrm{{\textit{g}_{{\mathrm {can}}}}}}}(z,w)$ is defined as the solution of the differential equation (which is to be interpreted in terms of currents)

$$\begin{aligned} d_{z}d^{c}_{z}{{\mathrm{{\textit{g}_{{\mathrm {can}}}}}}}(z,w)+ \delta _{w}(z)={{\mathrm{\mu _{can}}}}(z), \end{aligned}$$

(3)

with the normalization condition

$$\begin{aligned} \int _{X}{{\mathrm{{\textit{g}_{{\mathrm {can}}}}}}}(z,w){{\mathrm{\mu _{can}}}}(z)=0. \end{aligned}$$

From Eq. (3), it follows that ${{\mathrm{{\textit{g}_{{\mathrm {can}}}}}}}(z,w)$ admits a $\log$-singularity at $z=w$, i.e., for $z, w\in X$, it satisfies

$$\begin{aligned} \lim _{w\rightarrow z}\big ({{\mathrm{{\textit{g}_{{\mathrm {can}}}}}}}(z,w)+ \log |\vartheta _{z}(w)|^{2}\big )= O_{z}(1). \end{aligned}$$

(4)

Parabolic Eisenstein series For $z\in X$ and $s\in \mathbb {C}$ with ${{\mathrm{Re}}}(s)> 1$, the parabolic Eisenstein series $\mathcal {E}_{\mathrm {par},p}(z,s)$ corresponding to a cusp $p\in \mathcal {P}$ is defined by the series

$$\begin{aligned} \mathcal {E}_{\mathrm {par},p}(z,s) = \sum _{\eta \in \Gamma _{p}\backslash \Gamma } {{\mathrm{Im}}}(\sigma _{p}^{-1}\eta z)^{s}. \end{aligned}$$

The series converges absolutely and uniformly for ${{\mathrm{Re}}}(s) >1$. It admits a meromorphic continuation to all $s\in \mathbb {C}$ with a simple pole at $s = 1$, and the Laurent expansion at $s=1$ is of the form

$$\begin{aligned} \mathcal {E}_{\mathrm {par},p}(z,s) = \frac{1}{(s-1){{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)} + \kappa _{p}(z) + O_{z}(s-1), \end{aligned}$$

(5)

where $\kappa _{p}(z)$ the constant term of $\mathcal {E}_{{\mathrm {par}},p}(z,s)$ at $s=1$ is called Kronecker’s limit function (see Chapter 6 of [5]). For $z\in X$, and $p,q \in \mathcal {P}$, the Kronecker’s limit function $\kappa _{p}(\sigma _{q}z)$ satisfies the following equation (see Theorem 1.1 of [10] for the proof)

$$\begin{aligned} \kappa _{p}(\sigma _{q}z)= \sum _{n < 0} k_{p,q}(n)e^{2 \pi in\overline{z}}+ \delta _{p,q}{{\mathrm{Im}}}(z)+ k_{p,q}(0)- \frac{\log \big ({{\mathrm{Im}}}(z)\big )}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)} + \sum _{n > 0}k_{p,q}(n)e^{2\pi i nz}, \end{aligned}$$

(6)

with Fourier coefficients $k_{p,q}(n)$ $\in$ $\mathbb {C}$. Let $p,q \in \mathcal {P}$, then for $z\in X$ and $s\in \mathbb {C}$ with ${{\mathrm{Re}}}(s)> 1$, the parabolic Eisenstein series $\mathcal {E}_{\mathrm {par,}p}(\sigma _{q}z,s)$ associated to $p\in \mathcal {P}$, admits a Fourier expansion of the form

$$\begin{aligned} \mathcal {E}_{\mathrm {par},p}(\sigma _{q}z,s)= \delta _{p,q}y^{s} + \alpha _{p,q}(s)y^{1-s} + \sum _{n\not =0}\alpha _{p,q}(n,s)W_{s}(nz), \end{aligned}$$

(7)

where $\alpha _{p,q}(s)$, $\alpha _{p,q}(n,s)$, and $W_{s}(nz)$ the Whittaker function are given by equations (3.21), (3.22), and (1.37), respectively in [5].

Heat Kernels For $t \in \mathbb {R}_{> 0}$ and $z, w \in \mathbb {H}$, the hyperbolic heat kernel $K_{\mathbb {H}}(t;z,w)$ on $\mathbb {R}_{> 0}\times \mathbb {H} \times \mathbb {H}$ is given by the formula

$$\begin{aligned} K_{\mathbb {H}}(t;z,w)= \frac{\sqrt{2}e^{- t\slash 4}}{(4\pi t)^{3\slash 2}} \int _{d_{\mathbb {H}}(z,w)}^{\infty }\frac{re^{-r^{2}\slash 4t}}{\sqrt{\cosh (r)-\cosh (d_{\mathbb {H}}(z,w))}}dr, \end{aligned}$$

where $d_{\mathbb {H}}(z,w)$ is the hyperbolic distance between $z$ and $w$.

For $t \in \mathbb {R}_{> 0}$ and $z, w \in X$, the hyperbolic heat kernel ${{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z,w)$ on $\mathbb {R}_{> 0}\times X\times X$ is defined as

$$\begin{aligned} {{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z,w)=\sum _{\gamma \in \Gamma }K_{\mathbb {H}}(t;z,\gamma w). \end{aligned}$$

For $z,w\in X,$ the hyperbolic heat kernel ${{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z,w)$ satisfies the differential equation

$$\begin{aligned} \bigg (\Delta _{\text {hyp},z} + \frac{\partial }{\partial t}\bigg ){{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z,w)&=0, \end{aligned}$$

(8)

Furthermore for a fixed $w\in X$ and any smooth function $f$ on $X$, the hyperbolic heat kernel ${{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z,w)$ satisfies the equation

$$\begin{aligned} \lim _{t\rightarrow 0}\int _{X}{{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z,w)f(z){{\mathrm{\mu _{hyp}}}}(z)&= f(w). \end{aligned}$$

(9)

To simplify notation, we write ${{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z)$ instead of ${{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z,z)$, when $z=w$.

For $t\in \mathbb {R}_{> 0}$ and $z\in X$, put

$$\begin{aligned} {{\mathrm{{\textit{HK}_{{\mathrm {hyp}}}}}}}(t;z)=\sum _{\gamma \in \mathcal {P}(\Gamma )}{{\mathrm{{{\textit{K}_{\mathbb {H}}}}}}}(t;z,\gamma z). \end{aligned}$$

The convergence of the above series follows from the convergence of the hyperbolic heat kernel ${{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z)$ and the fact that ${{\mathrm{{{\textit{K}_{\mathbb {H}}}}}}}(t;z,\gamma z)$ is positive for all $t\in \mathbb {R}_{> 0}$, $z\in \mathbb {H}$, and $\gamma \in \Gamma$.

Selberg constant The hyperbolic length of the closed geodesic determined by a primitive non-conjugate hyperbolic element $\gamma \in \mathcal {H}(\Gamma )$ on $X$ is given by

$$\begin{aligned} \ell _{\gamma }=\inf \lbrace {d_{\mathbb {H}}(z,\gamma z)|\,z\in \mathbb {H}\rbrace }. \end{aligned}$$

For $s\in \mathbb {C}$ with ${{\mathrm{Re}}}(s)>1$, the Selberg zeta function associated to $X$ is defined as

$$\begin{aligned} Z_{X}(s)= \prod _{\gamma \in \mathcal {H}(\Gamma )}Z_{\gamma }(s), \,\,\,\,\,\,\mathrm {where}\,\,\,Z_{\gamma }(s)= \prod _{n=0}^{\infty }\big (1-e^{-(s+n)\ell _{\gamma }}\big ). \end{aligned}$$

The Selberg zeta function $Z_{X}(s)$ admits a meromorphic continuation to all $s\in \mathbb {C}$, with zeros and poles characterized by the spectral theory of the hyperbolic Laplacian. Furthermore, $Z_{X}(s)$ has a simple zero at $s=1$, and the following constant is well-defined

$$\begin{aligned} c_{X}= \lim _{s\rightarrow 1}\bigg (\frac{Z^{'}_{X}(s)}{Z_{X}(s)}- \frac{1}{s-1}\bigg ). \end{aligned}$$

(10)

For $t\in \mathbb {R}_{> 0}$, the hyperbolic heat trace is given by the integral

$$\begin{aligned} H{\mathrm {Tr}}{{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t)=\int _{X}{{\mathrm{{\textit{HK}_{{\mathrm {hyp}}}}}}}(t;z){{\mathrm{\mu _{hyp}}}}(z). \end{aligned}$$

The convergence of the integral follows from the celebrated Selberg trace formula. Furthermore, from Lemma 4.2 in [8], we have the following relation

$$\begin{aligned} \int _{0}^{\infty } \big (H\mathrm {Tr}{{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t)-1\big )dt=c_{X}-1. \end{aligned}$$

(11)

Automorphic Green’s function For $z, w \in \mathbb {H}$ with $z \not = w$, and $s$ $\in$ $\mathbb {C}$ with ${{\mathrm{Re}}}(s)> 0$, the free-space Green’s function $g_{\mathbb {H},s}(z,w)$ is defined as

$$\begin{aligned} g_{\mathbb {H},s}(z,w) = g_{\mathbb {H},s}(u(z,w))= \dfrac{\Gamma (s)^{2}}{\Gamma (2s)}u^{-s} F(s,s;2s,-1\slash u), \end{aligned}$$

where $u=u(z,w)=|z-w|^{2}\slash ( 4{{\mathrm{Im}}}(z){{\mathrm{Im}}}(w))$ and $F(s,s;2s,-1\slash u)$ is the hypergeometric function.

There is a sign error in the formula defining the free-space Green’s function given by equation (1.46) in [5], i.e., the last argument $-1\slash u$ in the hypergeometric function has been incorrectly stated as $1\slash u$, which we have corrected in our definition. We have also normalized the free-space Green’s function defined in [5] by multiplying it by $4\pi .$

For $z, w \in X$ with $z\not = w$, and $s\in \mathbb {C}$ with ${{\mathrm{Re}}}(s) > 1$, the automorphic Green’s function $g_{\mathrm {hyp},s}(z,w)$ is defined as

$$\begin{aligned} g_{\mathrm {hyp},s}(z,w) = \sum _{\gamma \in \Gamma }g_{\mathbb {H},s}(z,\gamma w). \end{aligned}$$

The series converges absolutely uniformly for $z\not = w$ and ${{\mathrm{Re}}}(s) > 1$ (see Chapter 5 in [5]).

For $z, w \in X$ with $z \not = w$, and $s\in \mathbb {C}$ with ${{\mathrm{Re}}}(s) > 1$, the automorphic Green’s function satisfies the following properties (see Chapters 5 and 6 in [5]):

• (1) For ${{\mathrm{Re}}}(s(s-1)) > 1$, we have

  $$\begin{aligned} g_{\mathrm {hyp},s}(z,w) = 4\pi \int _{0}^{\infty }{{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z,w)e^{-s(s-1)t}dt. \end{aligned}$$

• (2) It admits a logarithmic singularity along the diagonal, i.e.,

  $$\begin{aligned} \lim _{w\rightarrow z}\big (g_{\mathrm {hyp},s}(z,w) + \log {|\vartheta _{z}(w)|^{2}}\big )= O_{s,z}(1). \end{aligned}$$

• (3) The automorphic Green’s function $g_{\mathrm {hyp},s}(z,w)$ admits a meromorphic continuation to all $s\in \mathbb {C}$ with a simple pole at $s=1$ with residue $4\pi \slash {{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)$, and the Laurent expansion at $s=1$ is of the form

  $$\begin{aligned} g_{\mathrm {hyp},s}(z,w)= \frac{4\pi }{s(s-1){{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)} + g^{(1)}_{\mathrm {hyp}}(z,w) + O_{z,w}(s-1), \end{aligned}$$

  where $g_{\mathrm {hyp}}^{(1)}(z,w)$ is the constant term of $g_{\mathrm {hyp},s}(z,w)$ at $s=1$.

• (4) Let $p,q\in \mathcal {P}$ be two cusps. Put

  $$\begin{aligned} C_{p,q} = \min \bigg \lbrace c > 0\,\bigg | \bigg (\begin{array}{ccc} a &{}b\\ c &{} d \end{array}\bigg ) \in \sigma _{p}^{-1}\Gamma \sigma _{q} \bigg \rbrace , \end{aligned}$$

  and $C_{p,p}=C_{p}$. Then, for $z,w\in X$ with ${{\mathrm{Im}}}(w) > {{\mathrm{Im}}}(z)$ and ${{\mathrm{Im}}}(w){{\mathrm{Im}}}(z) > C_{p,q}^{-2}$, and $s\in \mathbb {C}$ with ${{\mathrm{Re}}}(s) > 1$, the automorphic Green’s function admits the Fourier expansion

  $$\begin{aligned}&g_{\mathrm {hyp},s}(\sigma _{p}z,\sigma _{q}w)= \frac{4\pi {{\mathrm{Im}}}(w)^{1-s}}{2s-1} \mathcal {E}_{\mathrm {par},q}(\sigma _{p}z,s) -\delta _{p,q} \log \big |1- e^{2\pi i (w-z)}\big |^{2}\\&\quad +O\big (e^{-2\pi ({{\mathrm{Im}}}( w)-{{\mathrm{Im}}}(z))}\big ). \end{aligned}$$

  This equation has been proved as Lemma 5.4 in [5], and one of the terms was wrongly estimated in the proof of the lemma. We have corrected this error, and stated the corrected equation.

The space $C_{\ell ,\ell \ell }(X)$ Let $C_{\ell ,\ell \ell }(X)$ denote the set of complex-valued functions $f:X\rightarrow \mathbb {P}^{1}( \mathbb {C})$, which admit the following type of singularities at finitely many points $\mathrm {Sing}(f)\subset X$, and are smooth away from $\mathrm {Sing}(f)$:

• (1) If $s\in \mathrm {Sing}(f)$, then as $z$ approaches $s$, the function $f$ satisfies

  $$\begin{aligned} f(z)= c_{f,s}\log |\vartheta _{s}(z)| + O_{z}(1), \end{aligned}$$

  (12)

  for some $c_{f,s}\in \mathbb {C}$.

• (2) As $z$ approaches a cusp $p\in \mathcal {P}$, the function $f$ satisfies

  $$\begin{aligned} f(z)=c_{f,p}\log \big (-\log |\vartheta _{p}(z)|\big ) + O_{z}(1), \end{aligned}$$

  (13)

  for some $c_{f,p}\in \mathbb {C}$.

Hyperbolic Green’s function For $z, w \in X$ and $z\not = w$, the hyperbolic Green’s function is defined as

$$\begin{aligned} {{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,w) = 4\pi \int _{0}^{\infty }\bigg ({{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z,w)-\frac{1}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg )dt. \end{aligned}$$

For $z, w \in X$ with $z \not = w$, the hyperbolic Green’s function satisfies the following properties:

• (1) For $z, w \in X$, we have

  $$\begin{aligned} \lim _{w\rightarrow z}\big ( {{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,w) + \log {|\vartheta _{z}(w)|^{2}}\big )= O_{z}(1). \end{aligned}$$

  (14)

• (2) For $z, w \in X$, the hyperbolic Green’s function satisfies the differential equation (which is to be interpreted in terms of currents)

  $$\begin{aligned} d_{z}d_{z}^{c}{{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,w) +\delta _{w}(z)&= {{\mathrm{\mu _{shyp}}}}(z), \end{aligned}$$

  (15)

  with the normalization condition

  $$\begin{aligned} \int _{X}{{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,w){{\mathrm{\mu _{hyp}}}}(z)&= 0. \end{aligned}$$

  (16)

• (3) For $z,w\in X$ and $z\not =w$, we have

  $$\begin{aligned} {{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,w)= g^{(1)}_{\mathrm {hyp}}(z,w)= \lim _{s\rightarrow 1}\bigg (g_{\text {hyp},s}(z,w) - \frac{4\pi }{s(s-1){{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg ). \end{aligned}$$

  (17)

  The above properties follow from the properties of the heat kernel ${{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z,w)$ [Eqs. (8) and (9)] or from that of the automorphic Green’s function $g_{\mathrm {hyp},s}(z,w)$.

Lemma 1.1

For a fixed $w\in X$, and $z\in X$ with ${{\mathrm{Im}}}(\sigma _{p}^{-1}z)>{{\mathrm{Im}}}(\sigma _{p}^{-1} w)$, and ${{\mathrm{Im}}}(\sigma _{p}^{-1}z){{\mathrm{Im}}}(\sigma _{p}^{-1}w)$

$>C_{p}^{-2}$, we have

$$\begin{aligned}&{{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,w) = 4\pi \kappa _{p}(w) - \frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-\frac{4\pi \log \big ({{\mathrm{Im}}}(\sigma _{p}^{-1}z)\big )}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-\nonumber \\&\log \big |1-e^{2\pi i(\sigma _{p}^{-1}z - \sigma _{p}^{-1}w)}\big |^{2}+ O\big (e^{-2\pi ({{\mathrm{Im}}}(\sigma _{p}^{-1}z)-{{\mathrm{Im}}}(\sigma _{p}^{-1}w))}\big ), \end{aligned}$$

(18)

i.e., for a fixed $w\in X$, as $z\in X$ approaches a cusp $p\in \mathcal {P}$, we have

$$\begin{aligned} {{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,w)&= -\frac{4\pi \log \big ({{\mathrm{Im}}}(\sigma _{p}^{-1}z)\big )}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+ O_{z,w}(1) = -\frac{4\pi \log \big (-\log |\vartheta _{p}(z)|\big )}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+ O_{z,w}(1). \end{aligned}$$

(19)

Proof

We refer the reader to Proposition 2.4.1 in [4] (or Proposition 2.1 in [2]) for the proof. $\square$

Lemma 1.2

For any $f\in C_{\ell ,\ell \ell }(X)$ and for any fixed $w\in X\backslash \mathrm {Sing}(f)$, we have the equality of integrals

$$\begin{aligned} \int _{X}{{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,w)d_{z}d_{z}^{c}f(z) + f(w)+ \sum _{s\in \mathrm {Sing}(f)} \frac{c_{f,s}}{2}{{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(s,w)= \int _{X}f(z){{\mathrm{\mu _{shyp}}}}(z). \end{aligned}$$

(20)

Proof

We refer the reader to Corollary 3.1.8 in [4] (or Corollary 2.5 in [2]) for the proof. $\square$

Certain Convergence results For $z\in \mathbb {H}$, put

$$\begin{aligned} P(z)= \sum _{\gamma \in \mathcal {P}(\Gamma )}{{\mathrm{{\textit{g}_{\mathbb {H}}}}}}(z,\gamma z). \end{aligned}$$

The above series is invariant under the action $\Gamma$, and hence, defines a function on $X$. Furthermore, from Proposition 4.2.4 in [4] (or from 2.2 in [3]), the above series converges absolutely and uniformly for all $z\in X$, and satisfies the following equation

$$\begin{aligned} P(z)=\sum _{p\in \mathcal {P}}\sum _{\eta \in \Gamma _{p}\backslash \Gamma }P_{\mathrm {gen},p}(\eta z), \end{aligned}$$

(21)

where $P_{\mathrm {gen},p}(z)=\displaystyle \sum _{ n\not = 0}g_{\mathbb {H}}(z,\gamma _{p}^{n}z)$.

Furthermore, from the absolute and uniform convergence of $P(z)$, and from that of the following series from Lemma 5.2 in [6]

$$\begin{aligned} \sum _{\gamma \in \mathcal {P}(\Gamma )}{{\mathrm{\Delta _{hyp}}}}g_{\mathbb {H}}(z,\gamma z), \end{aligned}$$

we get

$$\begin{aligned}&\sum _{\gamma \in \mathcal {P}(\Gamma )}{{\mathrm{\Delta _{hyp}}}}g_{\mathbb {H}}(z,\gamma z) ={{\mathrm{\Delta _{hyp}}}}P(z)= \sum _{p\in \mathcal {P}}\sum _{\eta \in \Gamma _{p}\backslash \Gamma }{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},p}( \eta z),\end{aligned}$$

(22)

$$\begin{aligned}&{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},p}(z)=\sum _{n\not = 0}{{\mathrm{\Delta _{hyp}}}}g_{\mathbb {H}}(\sigma _{p}^{-1} z,\gamma _{\infty }^{n} \sigma _{p}^{-1}z)= 2\bigg (\frac{2\pi {{\mathrm{Im}}}(\sigma _{p}^{-1}z)}{\sinh (2\pi {{\mathrm{Im}}}(\sigma _{p}^{-1}z))}\bigg )^{2}-2. \end{aligned}$$

(23)

For $z\in X$, put

$$\begin{aligned} H(z)=4\pi \int _{0}^{\infty }\bigg ({{\mathrm{{\textit{HK}_{{\mathrm {hyp}}}}}}}(t;z)-\frac{1}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg )dt. \end{aligned}$$

(24)

The function $H(z)$ is invariant under the action of $\Gamma$, and hence, defines a function on $X$. Furthermore, from Proposition 4.3.2 (or from Proposition 2.9), it follows that $H(z)$ is well-defined on $X$, and for $z,w\in X$, we have

$$\begin{aligned} H(z)= \lim _{w\rightarrow z}\big ({{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,w)-g_{\mathbb {H}}(z,w)\big )-P(z). \end{aligned}$$

From the above equation, for $z\in X$, we find

$$\begin{aligned} {{\mathrm{\Delta _{hyp}}}}P(z)+ {{\mathrm{\Delta _{hyp}}}}H(z)= {{\mathrm{\Delta _{hyp}}}}\lim _{w\rightarrow z}\big (g_{\mathrm {hyp}}(z,w)-g_{\mathbb {H}}(z,w)\big ). \end{aligned}$$

For $z\in X$, since the integral

$$\begin{aligned} 4\pi \int _{0}^{\infty }\bigg ({{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z,z)-K_{\mathbb {H}}(t;0)-\frac{1}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg )dt, \end{aligned}$$

as well as the integral of the derivatives of the integrand are absolutely convergent, we can take the Laplace operator ${{\mathrm{\Delta _{hyp}}}}$ inside the integral. So for $z\in X$, we find

$$\begin{aligned} {{\mathrm{\Delta _{hyp}}}}P(z)+{{\mathrm{\Delta _{hyp}}}}H(z)= 4\pi \int _{0}^{\infty }{{\mathrm{\Delta _{hyp}}}}K_{\mathrm {hyp}}(t;z)dt. \end{aligned}$$

(25)

From Lemma 5.2 and Proposition 7.3 in [9], for $z\in X$, we have the following relation

$$\begin{aligned} 4\pi \int _{0}^{\infty }{{\mathrm{\Delta _{hyp}}}}{{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z)dt=\sum _{\gamma \in \Gamma \backslash \lbrace {{\mathrm{\mathrm {id}}}}\rbrace }{{\mathrm{{\textit{g}_{\mathbb {H}}}}}}(z,\gamma z) \end{aligned}$$

and the right-hand side of above equation remains bounded at the cusps. So we deduce that the left-hand side also remains bounded at the cusps.

Lemma 1.3

For $z\in X$ approaching a cusp $p\in \mathcal {P}$, we have

$$\begin{aligned} H(z)&\,= -\frac{8\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\log \big ({{\mathrm{Im}}}(\sigma _{p}^{-1}z)\big )-\frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+ 4\pi k_{p,p}(0)+O\big ({{\mathrm{Im}}}(\sigma _{p}^{-1}z)^{-1}\big )\nonumber \\&\,=-\frac{8\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\log \big (-\log |\vartheta _{p}(z)|\big )+O_{z}(1). \end{aligned}$$

(26)

Proof

We refer the reader to Proposition 4.3.3 in [4] (or Proposition 2.10 in [3]) for the proof. $\square$

Hence, we can conclude that the function $H(z)\in C_{\ell ,\ell \ell }(X)$ with $\mathrm {Sing}(f)=\emptyset$. Lastly, from Eq. (11), we have

$$\begin{aligned} \int _{X}H(z){{\mathrm{\mu _{hyp}}}}(z)=4\pi (c_{X}-1). \end{aligned}$$

(27)

An auxiliary identity For notational brevity, put

$$\begin{aligned} C_{\mathrm {hyp}}&= \int _{X}\int _{X}{{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(\zeta ,\xi )\bigg (\int _{0}^{\infty }{{\mathrm{\Delta _{hyp}}}}{{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;\zeta )dt\bigg )\bigg (\int _{0}^{\infty }{{\mathrm{\Delta _{hyp}}}}{{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;\xi )dt\bigg )\\&\quad \times {{\mathrm{\mu _{hyp}}}}(\xi ){{\mathrm{\mu _{hyp}}}}(\zeta ). \end{aligned}$$

From Proposition 2.6.4 in [4] (or from Proposition 2.8 in [2]), for $z,w\in X$, we have

$$\begin{aligned} {{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,w)-{{\mathrm{{\textit{g}_{{\mathrm {can}}}}}}}(z,w)= \phi (z) + \phi (w), \end{aligned}$$

(28)

where from Corollary 3.2.7 in [4] (or from Remark 2.16 in [2]), the function $\phi (z)$ is given by the formula

$$\begin{aligned}&\phi (z)= \frac{1}{2g}\int _{X}{{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,\zeta ) \left( \int _{0}^{\infty }{{\mathrm{\Delta _{hyp}}}}{{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;\zeta )dt\right) {{\mathrm{\mu _{hyp}}}}(\zeta )-\frac{C_{\mathrm {hyp}}}{8g^{2}}. \end{aligned}$$

(29)

Lemma 1.4

For $z\in X$, we have

$$\begin{aligned} \phi (z)= \frac{H(z)}{2g}+\frac{1}{8\pi g}\int _{X}g_{\mathrm {hyp}}(z,\zeta ){{\mathrm{\Delta _{hyp}}}}P(\zeta ) {{\mathrm{\mu _{hyp}}}}(\zeta ) - \frac{C_{\mathrm {hyp}}}{8g^{2}}-\frac{2\pi (c_{X}-1)}{g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}. \end{aligned}$$

(30)

Proof

The lemma has been proved as Theorem 4.3.8 in [4] (and Corollary 2.12 in [3]). However, for the convenience of the reader, we recall the proof here.

Combining Eqs. (25) and (29), we have

$$\begin{aligned} \phi (z)= \frac{1}{8\pi g}\int _{X}{{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,\zeta )\big ({{\mathrm{\Delta _{hyp}}}}P(\zeta ){{\mathrm{\mu _{hyp}}}}(\zeta )+{{\mathrm{\Delta _{hyp}}}}H(\zeta )\big ){{\mathrm{\mu _{hyp}}}}(\zeta ) -\frac{C_{\mathrm {hyp}}}{8g^{2}}. \end{aligned}$$

(31)

As $H(z)\in C_{\ell ,\ell \ell }(X)$, from Lemma 1.2, and Eqs. (1) and (27), we derive

$$\begin{aligned}&\frac{1}{8\pi g}\int _{X}{{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,\zeta ){{\mathrm{\Delta _{hyp}}}}H(\zeta ){{\mathrm{\mu _{hyp}}}}(\zeta ) = -\frac{1}{2 g}\int _{X}{{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,\zeta )d_{\zeta }d_{\zeta }^{c} H(\zeta ){{\mathrm{\mu _{hyp}}}}(\zeta )\nonumber \\ {}&= \frac{H(z)}{2g}-\frac{1}{2g}\int _{X}H(\zeta ){{\mathrm{\mu _{shyp}}}}(\zeta )=\frac{H(z)}{2g}-\frac{2\pi (c_{X}-1)}{g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}. \end{aligned}$$

(32)

Combining Eqs. (31) and (32) completes the proof of the lemma. $\square$

Key identity For $z \in X$, we have the relation of differential forms

$$\begin{aligned}&g{{\mathrm{\mu _{can}}}}(z) =\bigg (\frac{1}{4\pi }+ \frac{1}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg ){{\mathrm{\mu _{hyp}}}}(z)+ \frac{1}{2}\bigg (\int _{0}^{\infty }{{\mathrm{\Delta _{hyp}}}}{{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z) dt\bigg ){{\mathrm{\mu _{hyp}}}}(z). \end{aligned}$$

This relation has been established as Theorem 3.4 in [6], when $X$ is compact, which easily extends to our case.

Lemma 1.5

For any $f\in C_{\ell ,\ell \ell }(X)$, we have the following equality of integrals

$$\begin{aligned}&g\int _{X}f(z){{\mathrm{\mu _{can}}}}(z) \nonumber \\&=\bigg (\frac{1}{4\pi }+\frac{1}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)} \bigg ) \int _{X}f(z){{\mathrm{\mu _{hyp}}}}(z) + \frac{1}{2}\int _{X}f(z)\bigg (\int _{0}^{\infty } {{\mathrm{\Delta _{hyp}}}}{{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;z)dt \bigg ){{\mathrm{\mu _{hyp}}}}(z). \end{aligned}$$

(33)

Proof

We refer the reader to Corollary 3.2.5 in [4] (or Corollary 2.15 in [2]) for the proof.

$\square$

3 Bounds for canonical Green’s functions at cusps

Let $p,q \in \mathcal {P}$ be two cusps with $p\not =q$. Then, from equation (28), we find

$$\begin{aligned} g_{\mathrm {can}}(p,q)= \lim _{z\rightarrow p}\lim _{w\rightarrow q} \big (g_{\mathrm {hyp}}(z,w)-\phi (z)-\phi (w)\big ). \end{aligned}$$

(34)

From Lemmas (1.1) and (1.3), we know the asymptotics of the functions ${{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(z,w)$ and $H(z)$ at the cusps, respectively. So if we can compute the asymptotics of the integral

$$\begin{aligned} \int _{X}g_{\mathrm {hyp}}(z,\zeta ){{\mathrm{\Delta _{hyp}}}}P(\zeta ){{\mathrm{\mu _{hyp}}}}(\zeta ) \end{aligned}$$

at the cusps, we will be able to compute an upper bound for the canonical Green’s function when evaluated at two different cusps.

For the remaining part of the article, for $p\in \mathcal {P}$ a cusp and $z\in \mathbb {H}$, we denote ${{\mathrm{Im}}}(\sigma _{p}^{-1}z)$ by $y_{p}.$

In the following two lemmas, we compute the zeroth Fourier coefficient of the automorphic Green’s function and the hyperbolic Green’s function.

Lemma 2.1

Let $p,q\in \mathcal {P}$ be two cusps. Then, for $z\in X$ and $w=u+iv\in \mathbb {H}$ with $y_{p}> v$ and $v y_{p}> 1$, and $s\in \mathbb {C}$ with ${{\mathrm{Re}}}(s)> 1$, we have

$$\begin{aligned} \int _{0}^{1}g_{\mathrm {hyp},s}(z,\sigma _{q}w)du=\frac{4\pi v^{1-s}}{2s-1}\mathcal {E}_{\mathrm {par},q}(z,s)+ \frac{ 4\pi \delta _{p,q}}{2s-1}\big (v^{s}y_{p}^{1-s}-v^{1-s}y_{p}^{s}\big ). \end{aligned}$$

(35)

Furthermore, for $v > y_{p}$ and $v y_{p}> 1$, and $s\in \mathbb {C}$ with ${{\mathrm{Re}}}(s)> 1$, we have

$$\begin{aligned} \int _{0}^{1}g_{\mathrm {hyp},s}(z,\sigma _{q}w)du=\frac{4\pi v^{1-s}}{2s-1}\mathcal {E}_{\mathrm {par},q}(z,s). \end{aligned}$$

(36)

Proof

For $z\in X$ and $w=u+iv\in \mathbb {H}$ with $y_{p}> v$ and $v y_{p}> 1$, and $s\in \mathbb {C}$ with ${{\mathrm{Re}}}(s)> 1$, combining Lemmas 5.1 and 5.2 of [5], we have

$$\begin{aligned}&\int _{0}^{1}g_{\mathrm {hyp},s}(z,\sigma _{q}w)du= \frac{4\pi y_{p}^{1-s}}{2s-1}\left( \delta _{p,q} v^{s} + \alpha _{p,q}(s)v^{1-s}\right) \\&\quad +\frac{4\pi v^{1-s}}{2s-1}\sum _{n\not = 0} \alpha _{p,q}(n,s)W_{s}(n\sigma _{p}^{-1}z). \end{aligned}$$

The expression on the right-hand side of the above equation can be rewritten as

$$\begin{aligned}&\frac{4\pi v^{1-s}}{2s-1}\bigg (\delta _{p,q}y_{p}^{s}+\alpha _{p,q}(s)y_{p}^{1-s}+ \sum _{n\not = 0}\alpha _{p,q}(n,s)W_{s}(n\sigma _{p}^{-1}z)\bigg )\nonumber \\&\quad +\frac{ 4\pi \delta _{p,q}}{2s-1}\big (v^{s}y_{p}^{1-s}-v^{1-s}y_{p}^{s}\big ). \end{aligned}$$

(37)

For $s\in \mathbb {C}$ and ${{\mathrm{Re}}}(s)> 1$, from the Fourier expansion of the parabolic Eisenstein series $\mathcal {E}_{\mathrm {par},q}(z,s)$ described in Eq. (7), we get

$$\begin{aligned} \frac{4\pi v^{1-s}}{2s-1}\bigg (\delta _{p,q}y_{p}^{s}+\alpha _{p,q}(s)y_{p}^{1-s}+\sum _{n\not = 0}\alpha _{p,q} (n,s)W_{s}(n\sigma _{p}^{-1}z)\bigg )=\frac{4\pi v^{1-s}}{2s-1}\mathcal {E}_{\mathrm {par},q}(z,s). \end{aligned}$$

(38)

Combining Eqs. (37) and (38) proves Eq. (35).

For $v > y_{p}$ and $v y_{p}> 1$, and $s\in \mathbb {C}$ with ${{\mathrm{Re}}}(s)> 1$, combining Lemmas 5.1 and 5.2 of [5], we have

$$\begin{aligned}&\int _{0}^{1}g_{\mathrm {hyp},s}(z,\sigma _{q}w)du=\frac{4\pi v^{1-s}}{2s-1}\bigg (\delta _{p,q}y_{p}^{s}+\alpha _{p,q}(s)y_{p}^{1-s} +\sum _{n\not = 0}\alpha _{p,q}(n,s)W_{s}(n\sigma _{p}^{-1}z)\bigg ). \end{aligned}$$

From Eq. (38), we derive that

$$\begin{aligned} \int _{0}^{1}g_{\mathrm {hyp},s}(z,\sigma _{q}w)du=\frac{4\pi v^{1-s}}{2s-1}\mathcal {E}_{\mathrm {par},q}(z,s), \end{aligned}$$

which proves Eq. (36), and completes the proof of the lemma. $\square$

Lemma 2.2

Let $p,q\in \mathcal {P}$ be two cusps. Then, for $z\in X$ and $w=u+iv\in \mathbb {H}$ with $y_{p}> v$ and $v y_{p}> 1$, we have

$$\begin{aligned} \int _{0}^{1}g_{\mathrm {hyp}}(z,\sigma _{q}w)du= 4\pi \kappa _{q}(z)-\frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-\frac{4\pi \log v}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+ 4\pi \delta _{p,q}(v-y_{p}). \end{aligned}$$

(39)

Furthermore, for $v > y_{p}$ and $v y_{p}> 1$, we have

$$\begin{aligned} \int _{0}^{1}g_{\mathrm {hyp}}(z,\sigma _{q}w)du= 4\pi \kappa _{q}(z)-\frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}- \frac{4\pi \log v}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}. \end{aligned}$$

(40)

Proof

Observe that

$$\begin{aligned}&\int _{0}^{1}g_{\mathrm {hyp}}(z,\sigma _{q}w)du=\int _{0}^{1}\lim _{s\rightarrow 1}\bigg (g_{\mathrm {hyp},s} (z,\sigma _{q}w)-\frac{4\pi }{s(s-1){{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg )du\nonumber \\&=\lim _{s\rightarrow 1}\bigg (\int _{0}^{1} g_{\mathrm {hyp},s}(z,\sigma _{q}w)du-\frac{4\pi }{(s-1){{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg )+\frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}. \end{aligned}$$

(41)

For $z\in X$ and $w=u+iv\in \mathbb {H}$ with $y_{p}> v$ and $v y_{p}> 1$, combining Eqs. (35) and (41), we find that the right-hand side of the above equation decomposes into the following expression

$$\begin{aligned} \lim _{s\rightarrow 1}\bigg (\frac{4\pi v^{1-s}}{2s-1} \mathcal {E}_{\mathrm {par},q}(z,s)-\frac{4\pi }{(s-1){{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg ) +4\pi \delta _{p,q}(v-y_{p})+ \frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}. \end{aligned}$$

To evaluate the above limit, we compute the Laurent expansions of $\mathcal {E}_{\mathrm {par},p}(w,s)$, ${{\mathrm{Im}}}(\sigma _{p}^{-1}z)^{1-s}$, and $(2s-1)^{-1}$ at $s=1$. The Laurent expansions of ${{\mathrm{Im}}}{(\sigma _{p}^{-1}z)^{1-s}}$ and $(2s-1)^{-1}$ at $s=1$ are easy to compute, and are of the form

$$\begin{aligned}&{{\mathrm{Im}}}{(\sigma _{p}^{-1}z)}^{1-s}=1 - (s-1)\log \big ({{\mathrm{Im}}}{(\sigma _{p}^{-1}z)}\big ) + O\big ((s-1)^{2}\big );\\&\quad \frac{1}{2s-1} = 1- 2(s-1) + O\big ((s-1)^{2}\big ). \end{aligned}$$

Combining the above two equations with Eq. (5), we find

$$\begin{aligned}&4\pi \lim _{s\rightarrow 1}\bigg (\frac{{{\mathrm{Im}}}(\sigma _{p}^{-1}z)^{1-s}}{2s-1} \mathcal {E}_{\mathrm {par},p}(w,s)-\frac{1}{(s-1){{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg )\nonumber \\&= 4\pi \kappa _{p}(w) -\frac{8\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}- \frac{4\pi \log \big ({{\mathrm{Im}}}(\sigma _{p}^{-1}z)\big )}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}, \end{aligned}$$

(42)

Combining the above computation with Eq. (41), we arrive at

$$\begin{aligned}&\int _{0}^{1}g_{\mathrm {hyp}}(z,\sigma _{q}w)du= 4\pi \kappa _{q}(z)- \frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-\frac{4\pi \log v}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+4\pi \delta _{p,q}(v-y_{p}), \end{aligned}$$

which proves Eq. (39).

We now prove Eq. (40). For $v > y_{p}$ and $v y_{p}> 1$, combining Eqs. (36) and (41), we find

$$\begin{aligned}&\int _{0}^{1}g_{\mathrm {hyp}}(z,\sigma _{q}w)du=\lim _{s\rightarrow 1}\bigg (\frac{4\pi v^{1-s}}{2s-1} \mathcal {E}_{\mathrm {par},q}(z,s)-\frac{4\pi }{(s-1){{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg ) +\frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}. \end{aligned}$$

(43)

Combining Eqs. (43) and (42), we find

$$\begin{aligned}&\int _{0}^{1}g_{\mathrm {hyp}}(z,\sigma _{q}w)du=4\pi \kappa _{q}(z)-\frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-\frac{4\pi \log v}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}, \end{aligned}$$

which proves Eq. (40), and hence, completes the proof of the lemma. $\square$

Proposition 2.3

Let $p\in \mathcal {P}$ be a cusp. For $z\in X$ and $w=u+iv\in \mathbb {H}$ with $y_{p}>1$, we have the formal decomposition

$$\begin{aligned}&\int _{X}g_{\mathrm {hyp}}(z,w){{\mathrm{\Delta _{hyp}}}}P(w){{\mathrm{\mu _{hyp}}}}(w)\!=\!\sum _{q\in \mathcal {P}}\int _{0}^{1\slash y_{p}} \int _{0}^{1}g_{\mathrm {hyp}}(z,\sigma _{q}w){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\frac{dudv}{v^{2}}\nonumber \\&\quad +\sum _{q\in \mathcal {P}}\int _{1\slash y_{p}}^{\infty }\bigg (4\pi \kappa _{q}(z)-\frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}- \frac{4\pi \log v}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg ){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w) \frac{dv}{v^{2}}\nonumber \\&\quad + 4\pi \int _{1\slash y_{p}}^{y_{p}} (v-y_{p}){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},p} (\sigma _{p}w)\frac{dv}{v^{2}}. \end{aligned}$$

(44)

The formal unfolding of the above integral translates into an equality of integrals, only if each of the three integrals on the right-hand side of the above equation converges absolutely.

Proof

As the series ${{\mathrm{\Delta _{hyp}}}}P(w)$ is absolutely and uniformly convergent, we have

$$\begin{aligned}&\int _{X}g_{\mathrm {hyp}}(z,w){{\mathrm{\Delta _{hyp}}}}P(w){{\mathrm{\mu _{hyp}}}}(w)= \sum _{q\in \mathcal {P}}\sum _{\eta \in \Gamma _{q}\backslash \Gamma } \int _{X}g_{\mathrm {hyp}}(z,w){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\eta w){{\mathrm{\mu _{hyp}}}}(w), \end{aligned}$$

(45)

After making the substitution $w\mapsto \eta ^{-1}\sigma _{q}w$, from the $\Gamma$-invariance of $g_{\mathrm {hyp}}(z,w)$, and from the $\mathrm {PSL}_{2}(\mathbb {R})$-invariance of ${{\mathrm{\mu _{hyp}}}}(z)$, formally for $w=u+iv\in \mathbb {H}$, we find

$$\begin{aligned}&\sum _{q\in \mathcal {P}}\sum _{\eta \in \Gamma _{q}\backslash \Gamma }\int _{X}g_{\mathrm {hyp}}(z,w){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}( \eta w){{\mathrm{\mu _{hyp}}}}(w)\nonumber \\&=\sum _{q\in \mathcal {P}}\int _{0}^{\infty }\int _{0}^{1}g_{\mathrm {hyp}} (z,\sigma _{q}w){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\frac{dudv}{v^{2}}. \end{aligned}$$

(46)

Recall from Eq. (23), that for any $w=u+iv\in \mathbb {H}$, the function $P_{\mathrm {gen},q}(\sigma _{q}w)$ does not depend on $u$. So the right-hand side of Eq. (46) further decomposes to give

$$\begin{aligned}&\sum _{q\in \mathcal {P}}\int _{0}^{1\slash y_{p}}\int _{0}^{1}g_{\mathrm {hyp}}(z,\sigma _{q}w) {{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\frac{dudv}{v^{2}}+\sum _{q\in \mathcal {P}}\int _{1\slash y_{p}}^{y_{p}}\bigg (\int _{0}^{1} g_{\mathrm {hyp}}(z,\sigma _{q}w)du\bigg )\nonumber \\&\times {{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\frac{dv}{v^{2}}+\sum _{q\in \mathcal {P}} \int _{y_{p}}^{\infty }\bigg (\int _{0}^{1} g_{\mathrm {hyp}}(z,\sigma _{q}w)du\bigg ){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\frac{dv}{v^{2}}. \end{aligned}$$

(47)

Since in the second line of formula (47) we have $1\slash y_{p}<v<y_{p}$, we can apply Eq. (39), and rewrite the second line of formula (47) as

$$\begin{aligned}&\sum _{q\in \mathcal {P}}\int _{1\slash y_{p}}^{y_{p}}\bigg (4\pi \kappa _{q}(z)-\frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}- \frac{4\pi \log v}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg ){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\frac{dv}{v^{2}}\nonumber \\&+ 4\pi \int _{1\slash y_{p}} ^{y_{p}}( v-y_{p}){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},p}(\sigma _{p}w)\frac{dv}{v^{2}}. \end{aligned}$$

(48)

Since in the third line of formula (47) we have $v >y_{p}>1\slash y_{p}$, we can apply Eq.  (40), and rewrite the third line of formula (47) as

$$\begin{aligned} \sum _{q\in \mathcal {P} }\int _{y_{p}}^{\infty }\bigg (4\pi \kappa _{q}(z)- \frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-\frac{4\pi \log v}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg ){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\frac{dv}{v^{2}}. \end{aligned}$$

(49)

The proof of the proposition follows from combining Eqs.  (48) and (49). $\square$

In the following lemmas, we prove the absolute convergence of each of the three integrals on the right-hand side of Eq.  (44).

Lemma 2.4

Let $p,q\in \mathcal {P}$ be two cusps. For $z\in X$ and $w=u+iv\in \mathbb {H}$, the integral

$$\begin{aligned} \int _{0}^{1\slash y_{p}}\int _{0}^{1}g_{\mathrm {hyp}}(z,\sigma _{q}w){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q} (\sigma _{q}w)\frac{dudv}{v^{2}} \end{aligned}$$

converges absolutely. Furthermore as $z\in X$ approaches a cusp $p\in \mathcal {P}$, we have

$$\begin{aligned} \sum _{q\in \mathcal {P}}\int _{0}^{1\slash y_{p}}\int _{0}^{1}g_{\mathrm {hyp}}(z,\sigma _{q}w){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w) \frac{dudv}{v^{2}} =o_{z}(1), \end{aligned}$$

(50)

where the contribution from the term $o_{z}(1)$ is a smooth function in $z$, which approaches zero, as $z\in X$ approaches the cusp $p\in \mathcal {P}$.

Proof

For $v\in \mathbb {R}_{> 0}$, from the formula for the function ${{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)$ from Eq. 23, we derive that

$$\begin{aligned} \frac{{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)}{v^{2}}=\frac{8\pi ^{2}}{\sinh ^{2}(2\pi v)}-\frac{2}{v^{2}} \end{aligned}$$

remains bounded. So it suffices to show that the integral

$$\begin{aligned} \int _{0}^{1\slash y_{p}}\int _{0}^{1}g_{\mathrm {hyp}}(z,\sigma _{q}w)dudv \end{aligned}$$

converges absolutely. Let $\mathcal {I}$ denote the set $[0,1]\times [0,1\slash y_{p}]$. We view the above integral as a real-integral on the compact subset $\mathcal {I}\subset \mathbb {R}^{2}$. The hyperbolic Green’s function $g_{\mathrm {hyp}}(z,\sigma _{q}w)$ is at most $\log$-singular on a measure zero subset of the interior points of $\mathcal {I}$. Furthermore from Eq. (19), the hyperbolic Green’s function $g_{\mathrm {hyp}}(z,\sigma _{q}w)$ is at most $\log \log$-singular on a measure zero subset of the boundary points of $\mathcal {I}$. Hence, it is absolutely integrable on $\mathcal {I}$. This implies that the integral

$$\begin{aligned} \int _{0}^{1\slash y_{p}}\int _{0}^{1}g_{\mathrm {hyp}}(z,\sigma _{q}w){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w) \frac{dudv}{v^{2}} \end{aligned}$$

converges absolutely, and also proves the asymptotic relation asserted in Eq. (50). $\square$

Lemma 2.5

Let $p,q\in \mathcal {P}$ be two cusps. For $z\in X$ and $w=u+iv\in \mathbb {H}$, the integral

$$\begin{aligned} \int _{1\slash y_{p}}^{\infty }\bigg (4\pi \kappa _{q}(z)-\frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg ){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w) \frac{dv}{v^{2}} \end{aligned}$$

(51)

converges absolutely. Furthermore, as $z\in X$ approaches the cusp $p\in \mathcal {P}$, we have

$$\begin{aligned}&\sum _{q\in \mathcal {P}}\int _{1\slash y_{p}}^{\infty }\bigg (4\pi \kappa _{q}(z)- \frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg ){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\frac{dv}{v^{2}}\nonumber \\&\quad =16\pi ^{2}\bigg (-y_{p}+\frac{|\mathcal {P}|\big (\log y_{p}+1\big )}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)} -\sum _{q\in \mathcal {P}}k_{q,p}(0)+ \frac{2\pi }{3}\bigg )+O\bigg (\frac{\log y_{p}}{y_{p}}\bigg ). \end{aligned}$$

Proof

Substituting the formula for the function ${{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)$ from Eq. (22), we have

$$\begin{aligned}&\int _{1\slash y_{p}}^{\infty }\bigg (4\pi \kappa _{q}(z)-\frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg ){{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w) \frac{dv}{v^{2}} \\&\quad = \bigg (8\pi \kappa _{q}(z)-\frac{8\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg )\int _{1\slash y_{p}}^{\infty }\bigg (\bigg (\frac{2\pi v }{\sinh (2\pi v)}\bigg )^{2}-1\bigg )\frac{dv}{v^{2}}. \end{aligned}$$

The integral on the right-hand side of the above equation further simplifies to give

$$\begin{aligned}&\bigg (8\pi \kappa _{q}(z)-\frac{8\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg )\bigg [ \frac{1}{v}-2\pi \coth (2\pi v)\bigg ]_{1\slash y_{p}}^{\infty }\nonumber \\&\quad = \bigg (8\pi \kappa _{q}(z)-\frac{8\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg )\bigg (-2\pi -y_{p}+2\pi \coth \bigg (\frac{2\pi }{y_{p}}\bigg )\bigg ). \end{aligned}$$

(52)

Hence, from Eq. (52), we can conclude that the integral (51) converges absolutely.

We now compute the asymptotics of the expression obtained on the right-hand side of Eq. (52), as $z\in X$ approaches the cusp $p\in \mathcal {P}.$ We first compute the asymptotics for the expression in the second bracket on the right-hand side of Eq. (52).

For $t\in \mathbb {R}_{>0}$, recall that the Taylor series expansion of the function $\coth (t)$ as $t$ approaches zero is of the form

$$\begin{aligned} \coth (t)=\frac{1}{t}+\frac{t}{3}+O(t^3). \end{aligned}$$

As $z\in X$ approaches $p\in \mathcal {P}$, the quantity $1\slash y_{p}$ approaches zero. So as $z\in X$ approaches $p\in \mathcal {P}$, using the Taylor expansion of $\coth (2\pi \slash y_{p})$, we have the asymptotic relation

$$\begin{aligned} -2\pi -y_{p}+ 2\pi \coth \bigg (\frac{2\pi }{y_{p}}\bigg )&= -2\pi -y_{p}+2\pi \bigg (\frac{y_{p}}{2\pi }+\frac{2\pi }{3y_{p}}+ O\bigg (\frac{1}{y_{p}^{3}}\bigg )\bigg )\nonumber \\&\quad =-2\pi +\frac{4\pi ^{2}}{3y_{p}}+ O\left( \frac{1}{y_{p}^{3}}\right) . \end{aligned}$$

(53)

As $z\in X$ approaches $p\in \mathcal {P}$, from the Fourier expansion of Kronecker’s limit function $\kappa _{q}(z)$ described in Eq. (6), we have the following asymptotic relation

$$\begin{aligned}&8\pi \kappa _{q}(z)-\frac{8\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}= 8\pi \delta _{p,q} y_{p}-\frac{8\pi \log y_{p}}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+8\pi k_{q,p}(0)-\frac{8\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)} +O\big (e^{-2\pi y_{p}}\big ). \end{aligned}$$

(54)

Combining Eqs. (53) and (54), as $z\in X$ approaches $p\in \mathcal {P}$, we have the asymptotic relation for the right-hand side of Eq. (52)

$$\begin{aligned}&\bigg (8\pi \delta _{p,q} y_{p}-\frac{8\pi \log y_{p}}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+8\pi k_{q,p}(0)-\frac{8\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+ O\big (e^{-2\pi y_{p}}\big )\bigg )\\&\quad \bigg (-2\pi +\frac{4\pi ^{2}}{3y_{p}} +O\bigg (\frac{1}{y_{p}^{3}}\bigg )\bigg )= 16\pi ^{2}\bigg (-\delta _{p,q}y_{p}+ \frac{(\log y_{p}+1)}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-k_{q,p}(0)\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad +\frac{2\pi }{3}\delta _{p,q}+O\bigg (\frac{\log y_{p}}{y_{p}}\bigg )\bigg ). \end{aligned}$$

Hence, taking the summation over all $q\in \mathcal {P}$ completes the proof of the lemma. $\square$

Lemma 2.6

Let $p,q\in \mathcal {P}$ be two cusps. For $z\in X$ and $w=u+iv\in \mathbb {H}$, the integral

$$\begin{aligned} \int _{1\slash y_{p}}^{\infty }\frac{4\pi \log v}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\frac{dv}{v^{2}} \end{aligned}$$

(55)

converges absolutely. Furthermore, we have the upper bound

$$\begin{aligned} \sum _{q\in \mathcal {P}}\int _{1\slash y_{p}}^{\infty } \Bigg |\frac{4\pi \log v}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\Bigg |\frac{dv}{v^{2}}\le \frac{8\pi |\mathcal {P}|}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)} \bigg (1+ \frac{4\pi ^{2}}{3}\bigg ). \end{aligned}$$

(56)

Proof

We prove the upper bound asserted in (56), which also proves the absolute convergence of the integral in (55). Observing the elementary estimate

$$\begin{aligned}&\int _{1\slash y_{p}}^{\infty }\bigg |\log v{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q} (\sigma _{q}w)\bigg |\frac{dv}{v^{2}}\nonumber \\&\le \int _{0}^{1}\bigg |\log v{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\bigg |\frac{dv}{v^{2}}+ \int _{1}^{\infty }\bigg |\log v{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\bigg |\frac{dv}{v^{2}}, \end{aligned}$$

(57)

we proceed to bound the two integrals on the right-hand side of the above inequality. For $v\in \mathbb {R}_{> 0}$, from the Eq.  (22), we find that the function

$$\begin{aligned} -\frac{{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)}{v^{2}}=\frac{2}{v^{2}}-\frac{8\pi ^{2} }{\sinh ^{2}(2\pi v)} \end{aligned}$$

is a positive monotone decreasing function, and hence, attains its maximum value at $v= 0$. So we compute the limit

$$\begin{aligned} -\lim _{v\rightarrow 0}\frac{{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)}{v^{2}} =\lim _{v\rightarrow 0}\bigg (\frac{2}{v^{2}}-\frac{8\pi ^{2} }{\sinh ^{2}(2\pi v)}\bigg )=\frac{8\pi ^{2}}{3}. \end{aligned}$$

So using the fact that, for $v\in (0,1]$, $|\log v|=-\log v$, we have the following upper bound for the first integral on the right-hand side of inequality (57)

$$\begin{aligned}&\int _{0}^{1}\bigg |\log v{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\bigg |\frac{dv}{v^{2}}\le -\frac{8\pi ^{2}}{3}\int _{0}^{1}\log v dv=\frac{8\pi ^{2}}{3}. \end{aligned}$$

(58)

Again using formula (22), we derive

$$\begin{aligned} \sup _{v\in \mathbb {R}_{>0}}\big |{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\big |= \sup _{v\in \mathbb {R}_{>0}}\bigg (2-\frac{8\pi ^{2}v^{2} }{\sinh ^{2}(2\pi v)} \bigg )=2. \end{aligned}$$

Using the above bound, we derive the following upper bound for the second integral on the right-hand side of inequality (57)

$$\begin{aligned}&\int _{1}^{\infty }\bigg |\log v{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\bigg |\frac{dv}{v^{2}}\le 2\int _{1}^{\infty }\frac{\log v }{v^{2}}dv=2\bigg (\bigg [-\frac{\log v}{v}\bigg ]_{1}^{\infty }+ \bigg [-\frac{1}{v}\bigg ]_{1}^{\infty }\bigg )=2. \end{aligned}$$

(59)

Hence, combining the upper bounds derived in Eqs.  (58) and (59) proves the lemma. $\square$

Lemma 2.7

Let $p\in \mathcal {P}$ be a cusp. For $z\in X$ and $w=u+iv\in \mathbb {H}$ with $y_{p}> 1$, the integral

$$\begin{aligned} -4\pi y_{p}\int _{1\slash y_{p}}^{y_{p}}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},p}(\sigma _{p}w)\frac{dv}{v^{2}} \end{aligned}$$

(60)

converges absolutely. Furthermore, as $z\in X$ approaches the cusp $p\in \mathcal {P}$, we have

$$\begin{aligned} -&4\pi y_{p}\int _{1\slash y_{p}}^{y_{p}}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},p}(\sigma _{p}w)\frac{dv}{v^{2}}= 4\pi \bigg (4\pi y_{p}\coth (2\pi y_{p})-2-\frac{8\pi ^{2}}{3}\bigg ) + O\bigg (\frac{1}{y_{p}^{2}} \bigg ). \end{aligned}$$

(61)

Proof

From Eq. (22), for a cusp $p\in \mathcal {P}$, we find

$$\begin{aligned} -&4\pi y_{p}\int _{1\slash y_{p}}^{y_{p}}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},p}(\sigma _{p}w) \frac{dv}{v^{2}}= -8\pi y_{p}\int _{1\slash y_{p}}^{y_{p}}\bigg (\frac{4\pi ^2}{\sinh ^{2}(2\pi v)}-\frac{1}{v^2}\bigg )dv\nonumber \\&\quad = -8\pi y_{p}\bigg [\frac{1}{v}-2\pi \coth (2\pi v)\bigg ]_{1\slash y_{p}}^{y_{p}}=- 8\pi y_{p}\bigg (\frac{1}{y_{p}}-2\pi \coth (2\pi y_{p})-y_{p}\nonumber \\&\quad +2\pi \coth \bigg (\frac{2\pi }{y_{p}}\bigg )\bigg )= -8\pi +16\pi ^{2}y_{p}\coth (2\pi y_{p})-8\pi y_{p}\bigg (-y_{p}+ 2\pi \coth \bigg (\frac{2\pi }{y_{p}}\bigg ) \bigg ). \end{aligned}$$

(62)

This implies that the integral (60) converges absolutely.

As $z\in X$ approaches the cusp $p\in \mathcal {P}$, from the Taylor expansion of $\coth (2\pi \slash y_{p})$ already used in Eq.  (53), we get

$$\begin{aligned} -8\pi y_{p}\bigg (-y_{p}+ 2\pi \coth \bigg (\frac{2\pi }{y_{p}}\bigg )\bigg )=-8\pi y_{p}\bigg ( \frac{4\pi ^{2}}{3 y_{p}}+O\bigg (\frac{1}{y_{p}^{3}}\bigg )\bigg ) -\frac{32\pi ^{3}}{3 }+O\bigg (\frac{1}{y_{p}^{2}}\bigg ), \end{aligned}$$

which together with Eq. (62) completes the proof of the lemma. $\square$

Lemma 2.8

Let $p\in \mathcal {P}$ be a cusp. For $z\in X$ and $w=u+iv\in \mathbb {H}$ with $y_{p}> 1$, the integral

$$\begin{aligned} 4\pi \int _{1\slash y_{p}}^{y_{p}} {{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},p}(\sigma _{p}w)\frac{dv}{v}. \end{aligned}$$

(63)

converges absolutely. Furthermore, as $z\in X$ approaches the cusp $p\in \mathcal {P}$, we have

$$\begin{aligned} 4\pi \int _{1\slash y_{p}}^{y_{p}} {{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},p}(\sigma _{p}w)\frac{dv}{v}= -8\pi \log y_{p}+8\pi (1-\log (4\pi ))+O\bigg (\frac{1}{y_{p}}\bigg ). \end{aligned}$$

(64)

Proof

Using Eq. (22), for a cusp $p\in \mathcal {P}$, we find

$$\begin{aligned}&\int _{1\slash y_{p}}^{y_{p}}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},p}(\sigma _{p}w)\frac{dv}{v}= 2\int _{1\slash y_{p}}^{y_{p}}\bigg (-\frac{1}{v}+\frac{4\pi ^{2}v}{\sinh ^{2}(2\pi v)}\bigg )dv\\&= 2\bigg [-\log v\!-\!2\pi v\coth (2\pi v)+\log (\sinh (2\pi v)) \bigg ]_{1\slash y_{p}}^{y_{p}}. \end{aligned}$$

Substituting the formulae for $\coth (2\pi v)$ and $\sinh (2\pi v)$, the right-hand side of the above equation can be further simplified to

$$\begin{aligned} 2\bigg [-\log v-4\pi v-\frac{4\pi v}{e^{4\pi v}-1}+\log \bigg (\frac{e^{4\pi v}-1}{2}\bigg )\bigg ]_ {1\slash y_{p}}^{y_{p}}. \end{aligned}$$

Observe that

$$\begin{aligned}&\left[ -\log v-4\pi v-\frac{4\pi v}{e^{4\pi v}-1}+\log \bigg (\frac{e^{4\pi v}-1}{2}\bigg )\right] _{1\slash y_{p}}^{y_{p}}=-\log y_{p}-4\pi y_{p}-\frac{4\pi y_{p}}{e^{4\pi y_{p}}-1}\nonumber \\&+ \log \big (e^{4\pi y_{p}}-1\big )+\log \bigg (\frac{1}{y_{p}}\bigg )+ \frac{4\pi }{y_{p}}+\frac{4\pi }{y_{p}\big (e^{4\pi \slash y_{p}}-1\big )}-\log \big (e^{4\pi \slash y_{p}}-1\big )\nonumber \\&\quad =-\log y_{p}-\log \bigg (\frac{e^{4\pi y_{p}}}{e^{4\pi y_{p}}\!-\!1}\bigg )\!-\!\frac{4\pi y_{p}}{e^{4\pi y_{p}}-1}\!+\! \frac{4\pi }{y_{p}}\!+\!\frac{4\pi }{y_{p}\big (e^{4\pi \slash y_{p}}\!-\!1\big )}\!-\!\log \big (y_{p}\big (e^{4\pi \slash y_{p} }\!-\! 1\big )\big ) , \end{aligned}$$

(65)

which proves that the integral (63) converges absolutely.

We now compute the asymptotic expansion of each of the terms in the above expression, as $z\in X$ approaches the cusp $p\in \mathcal {P}$. As $z\in X$ approaches the cusp $p\in \mathcal {P}$, we have the asymptotic relation for the first and second terms of (65)

$$\begin{aligned}&-\log y_{p}-\log \bigg (\frac{e^{4\pi y_{p}}}{e^{4\pi y_{p}}-1}\bigg )=-\log y_{p}+O\left( e^{-4\pi y_{p}}\right) ; \end{aligned}$$

(66)

the third and fourth terms of (65) satisfy the asymptotic relation

$$\begin{aligned} -\frac{4\pi y_{p}}{e^{4\pi y_{p}}-1}+\frac{4\pi }{y_{p}}=O\bigg (\frac{1}{ y_{p}}\bigg ); \end{aligned}$$

(67)

the fifth term satisfies the asymptotic relation

$$\begin{aligned}&\frac{4\pi }{y_{p} \big (e^{4\pi \slash y_{p}}-1\big )}=\frac{4\pi }{y_{p}\bigg (\displaystyle \sum _{n=1}^{\infty } \frac{(4\pi )^{n}}{n!\,y_{p}^{n}}\bigg )}=1+O\bigg (\frac{1}{y_{p}}\bigg ); \end{aligned}$$

(68)

and the sixth term satisfies the asymptotic relation

$$\begin{aligned}&-\log \big (y_{p}\big (e^{4\pi \slash y_{p}} -1\big )\big )=-\log \bigg ( \sum _{n=1}^{\infty }\frac{(4\pi )^{n}}{n!\,y_{p}^{n-1}}\bigg )\nonumber \\&\quad = -\log \bigg (4\pi +\sum _{n=1}^{\infty }\frac{(4\pi )^{n+1}}{(n+1)!\,y_{p}^{n}}\bigg )= -\log ( 4\pi ) + O\bigg (\frac{1}{y_{p}}\bigg ). \end{aligned}$$

(69)

Substituting the asymptotic relations obtained in Eqs.  (66), (67), (68), and (69) into (65), we derive the asymptotic relation

$$\begin{aligned} 4\pi \int _{1\slash y_{p}}^{y_{p}} {{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},p}(\sigma _{p}w)\frac{dv}{v}= -8\pi \log y_{p}+8\pi (1-\log (4\pi ))+O\bigg (\frac{1}{y_{p}}\bigg ), \end{aligned}$$

as $z\in X$ approaches the cusp $p\in \mathcal {P}$, which completes the proof of the lemma. $\square$

In the following proposition, combining all the asymptotics established in this section, we compute the asymptotics of the integral

$$\begin{aligned} \int _{X}g_{\mathrm {hyp}}(z,w){{\mathrm{\Delta _{hyp}}}}P(w){{\mathrm{\mu _{hyp}}}}(w), \end{aligned}$$

as $z\in X$ approaches a cusp $p\in \mathcal { P}.$

Proposition 2.9

Let $p\in \mathcal {P}$ be a cusp. Then, as $z\in X$ approaches the cusp $p\in \mathcal {P}$, we have

$$\begin{aligned}&\int _{X}g_{\mathrm {hyp}}(z,w){{\mathrm{\Delta _{hyp}}}}P(w){{\mathrm{\mu _{hyp}}}}(w)\nonumber \\&\quad =-\frac{32\pi ^{2}(g-1)\log y_{p}}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}- \sum _{q\in \mathcal {P}}\int _{1\slash y_{p}}^{\infty }\frac{4\pi \log v}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q} (\sigma _{q}w)\frac{dv}{v^{2}}+\alpha _{p}+o_{z}(1),\nonumber \\&\mathrm {where}\,\,\alpha _{p}=\frac{16\pi ^{2}|\mathcal {P}|}{ {{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-16\pi ^{2}\sum _{q\in \mathcal {P}}k_{q,p}(0)-8\pi \log (4\pi ), \end{aligned}$$

(70)

and the contribution from the term $o_{z}(1)$ is a smooth function in $z$, which approaches zero, as $z\in X$ approaches the cusp $p\in \mathcal {P}$.

Proof

From Lemmas 2.4, 2.5, 2.6, 2.7, and 2.8, it follows that each of the integrals on the right-hand side of the Eq. (44) is absolutely convergent. This implies that the equality of integrals described in Eq. (44) indeed holds true for all $z\in X$ provided that $y_{p}>1$.

As $z\in X$ approaches the cusp $p\in \mathcal {P}$, combining Lemmas 2.4 and 2.5, we find that the first two integrals on the right-hand side of Eq. (44) yield

$$\begin{aligned}&16\pi ^{2}\bigg (-y_{p}+\frac{|\mathcal {P}|\big (\log y_{p}+1\big )}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)} -\sum _{q\in \mathcal {P}}k_{q,p}(0)+ \frac{2\pi }{3}\bigg )\nonumber \\&\quad -\sum _{q\in \mathcal {P}} \int _{1\slash y_{p}}^{\infty }\frac{4\pi \log v}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w) \frac{dv}{v^{2}}+ o_{z}(1), \end{aligned}$$

(71)

where the contribution from the term $o_{z}(1)$ is a smooth function in $z$, which approaches zero, as $z\in X$ approaches $p\in \mathcal {P}$. As $z\in X$ approaches the cusp $p\in \mathcal {P}$, combining Lemmas 2.7 and 2.8, we find that the third integral on the right-hand side of Eq.  (44) yields

$$\begin{aligned} 16\pi ^{2}y_{p}\coth (2\pi y_{p})-8\pi \log y_{p}-\frac{32\pi ^{3}}{3}-8\pi \log (4\pi )+ O\bigg (\frac{1}{y_{p}}\bigg ). \end{aligned}$$

(72)

Combining (71) and (72), as $z\in X$ approaches the cusp $p\in \mathcal {P}$, the right-hand side of Eq. (44) simplifies to

$$\begin{aligned} -&16\pi ^{2}y_{p}+16\pi ^{2}y_{p}\coth (2\pi y_{p})+\frac{16\pi ^{2}|\mathcal {P}| \log y_{p}}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-8\pi \log y_{p}+\frac{16\pi ^{2}|\mathcal {P}|}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-\nonumber \\&16\pi ^{2}\sum _{q\in \mathcal {P}}k_{q,p}(0)-\sum _{q\in \mathcal {P}} \int _{1\slash y_{p}}^{\infty }\frac{4\pi \log v}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w) \frac{dv}{v^{2}}-8\pi \log (4\pi )+ o_{z}(1). \end{aligned}$$

(73)

As $z\in X$ approaches the cusp $p\in \mathcal {P}$, we have the following asymptotic relation for the first two terms in the above expression

$$\begin{aligned}&16\pi ^{2}y_{p}\big (\coth (2\pi y_{p})-1\big )=16\pi ^{2}y_{p} \bigg (\frac{\cosh (2\pi y_{p})-\sinh (2\pi y_{p})}{\sinh (2\pi y_{p})}\bigg )= O\big (e^{-y_{p}}\big ). \end{aligned}$$

Furthermore, as $z\in X$ approaches the cusp $p\in \mathcal {P}$ the third and fourth terms in expression (73) give

$$\begin{aligned} \frac{16\pi ^{2}|\mathcal {P}|\log y_{p}}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-8\pi \log y_{p}= -\frac{32\pi ^{2}(g-1)\log y_{p}}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}. \end{aligned}$$

Hence, as $z\in X$ approaches the cusp $p\in \mathcal {P}$, the expression in (73) further reduces to give

$$\begin{aligned} -&\frac{32\pi ^{2}(g-1)\log y_{p}}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)} -\sum _{q\in \mathcal {P}} \int _{1\slash y_{p}}^{\infty }\frac{4\pi \log v}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w) \frac{dv}{v^{2}}\nonumber \\&\quad +\frac{16\pi ^{2}|\mathcal {P}|}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-16\pi ^{2}\sum _{q\in \mathcal {P}}k_{q,p}(0)- 8\pi \log (4\pi )+ o_{z}(1), \end{aligned}$$

which completes the proof of the proposition. $\square$

Corollary 2.10

Let $p\in \mathcal {P}$ be a cusp. Then, as $z\in X$ approaches the cusp $p\in \mathcal {P}$, we have

$$\begin{aligned}&\phi (z)=-\frac{4\pi \log y_{p}}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-\sum _{q\in \mathcal { P}} \int _{1\slash y_{p}}^{\infty }\frac{\log v}{2g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}{{\mathrm{\Delta _{hyp}}}}P_{{\mathrm {gen}},q}(\sigma _{q}w) \frac{dv}{v^{2}}\nonumber \\&\quad +\frac{\alpha _{p}}{8\pi g}+\frac{2\pi k_{p,p}(0)}{g}- \frac{C_{\mathrm {hyp}}}{8g^{2}}-\frac{2\pi c_{X}}{g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+o_{z}(1), \end{aligned}$$

where the constant $\alpha _{p}$ is as defined in (70), and the contribution from the term $o_{z}(1)$ is a smooth function in $z$, which approaches zero, as $z\in X$ approaches the cusp $p\in \mathcal {P}$.

Proof

As $z\in X$ approaches the cusp $p\in \mathcal {P}$, from Eq.  (26), we have

$$\begin{aligned}&\frac{H(z)}{2g}= -\frac{4\pi \log y_{p}}{g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-\frac{2\pi }{g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+\frac{2\pi k_{p,p}(0)}{g}+O\bigg (\frac{1}{y_{p}}\bigg ). \end{aligned}$$

(74)

Furthermore, from Proposition 2.9, we find that

$$\begin{aligned}&\frac{1}{8\pi g}\int _{X}g_{\mathrm {hyp}}(z,\zeta ){{\mathrm{\Delta _{hyp}}}}P(\zeta ){{\mathrm{\mu _{hyp}}}}(\zeta )= -\frac{4\pi \log y_{p}}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+\frac{4\pi \log y_{p}}{g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\nonumber \\&\quad -\sum _{q\in \mathcal {P}}\int _{1\slash y_{p}}^{\infty } \frac{\log v}{2g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},q}(\sigma _{q}w)\frac{dv}{v^{2}}+ \frac{\alpha _{p}}{8\pi g} +o_{z}(1), \end{aligned}$$

(75)

where the contribution from the term $o_{z}(1)$ is a smooth function in $z$, which approaches zero, as $z\in X$ approaches the cusp $p\in \mathcal {P}$. The proof of the corollary follows from combining Eqs. (30), (74), and (75). $\square$

The following proposition has been proved as Proposition 6.1.9 in [4] (or Proposition 4.10 in [3]). However, for the convenience of the reader, we reproduce the proof here.

Proposition 2.11

We have the following upper bound

$$\begin{aligned} \frac{\big |C_{\mathrm {hyp}}\big |}{8g^{2}} \le \frac{2\pi \left( d_{X}+1\right) ^{2}}{\lambda _{1} {{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}, \end{aligned}$$

where $\lambda _{1}$ denotes the first non-zero eigenvalue of the hyperbolic Laplacian acting on smooth functions defined on $X$.

Proof

Recall that $C_{\mathrm {hyp}}$ is defined as

$$\begin{aligned}&C_{\mathrm {hyp}}= \int _{X}\int _{X}{{\mathrm{{\textit{g}_{{\mathrm {hyp}}}}}}}(\zeta ,\xi )\bigg (\int _{0}^{\infty }{{\mathrm{\Delta _{hyp}}}}{{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;\zeta )dt\bigg )\bigg (\int _{0}^{\infty }{{\mathrm{\Delta _{hyp}}}}{{\mathrm{{{\textit{K}_{{\mathrm {hyp}}}}}}}}(t;\xi )dt\bigg )\\&\quad {{\mathrm{\mu _{hyp}}}}(\xi ){{\mathrm{\mu _{hyp}}}}(\zeta ). \end{aligned}$$

From formulae (28), (30), we have

$$\begin{aligned}&{{\mathrm{\Delta _{hyp}}}}\phi (z)=\frac{4\pi {{\mathrm{\mu _{can}}}}(z)}{{{\mathrm{\mu _{hyp}}}}(z)}-\frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\Longrightarrow \int _{X}{{\mathrm{\Delta _{hyp}}}}\phi (z) {{\mathrm{\mu _{hyp}}}}(z)=0,\\&\phi (z)= \frac{1}{2g}\int _{X}g_{\mathrm {hyp}}(z,\zeta )\bigg ( \int _{0}^{\infty }{{\mathrm{\Delta _{hyp}}}}K_{\mathrm {hyp}}(t;\zeta )dt\bigg ){{\mathrm{\mu _{hyp}}}}(\zeta )-\frac{C_{\mathrm {hyp}}}{8g^{2}},\nonumber \end{aligned}$$

(76)

respectively. So combining the above two equations, we get

$$\begin{aligned} -\frac{1}{4\pi }\int _{X}\phi (z){{\mathrm{\Delta _{hyp}}}}\phi (z){{\mathrm{\mu _{hyp}}}}(z)&=\frac{1}{2g}\int _{X}\int _{X} g_{\mathrm {hyp}}(z,\zeta )\bigg (\int _{0}^{\infty }{{\mathrm{\Delta _{hyp}}}}K_{\mathrm {hyp}}(t;\zeta )dt\bigg )\nonumber \\&\quad \times {{\mathrm{\mu _{hyp}}}}(\zeta ) {{\mathrm{\mu _{can}}}}(z). \end{aligned}$$

(77)

Observe that

$$\begin{aligned} \int _{X}g_{\mathrm {hyp}}(z,\zeta )\bigg (\int _{0}^{\infty }{{\mathrm{\Delta _{hyp}}}}K_{\mathrm {hyp}}(t;\zeta )dt\bigg ) {{\mathrm{\mu _{hyp}}}}(\zeta )=2g\phi (z)+\frac{C_{\mathrm {hyp}}}{4g}\in C_{\ell ,\ell \ell }(X). \end{aligned}$$

So using Eqs. (33) and (77), we derive

$$\begin{aligned} \int _{X}\phi (z){{\mathrm{\Delta _{hyp}}}}\phi (z){{\mathrm{\mu _{hyp}}}}(z)&=\frac{\pi }{ g^{2}}\int _{X}\int _{X} g_{\mathrm {hyp}}(z,\zeta )\left( \int _{0}^{\infty }{{\mathrm{\Delta _{hyp}}}}K_{\mathrm {hyp}}(t;\zeta )dt\right) \nonumber \\&\quad \times \left( \int _{0}^{\infty }{{\mathrm{\Delta _{hyp}}}}K_{\mathrm {hyp}}(t;z)dt\right) {{\mathrm{\mu _{hyp}}}}(\zeta ){{\mathrm{\mu _{hyp}}}}(z)=\frac{\pi C_{\mathrm {hyp}}}{ g^{2}}. \end{aligned}$$

(78)

From Eq. (76), we have

$$\begin{aligned}&\sup _{z\in X}|{{\mathrm{\Delta _{hyp}}}}\phi (z)|\le \sup _{z\in X}\bigg |\frac{4\pi {{\mathrm{\mu _{can}}}}(z)}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X){{\mathrm{\mu _{shyp}}}}(z)}\bigg | + \frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}=\frac{4\pi \left( d_{X}+1\right) }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}, \end{aligned}$$

(79)

where $d_{X}$ is as defined in (2). As the function $\phi (z)\in L^{2}(X)$, it admits a spectral expansion in terms of the eigenfucntions of the hyperbolic Laplacian ${{\mathrm{\Delta _{hyp}}}}$. So from the arguments used to prove Proposition 4.1 in [7], we have

$$\begin{aligned}&\bigg |\int _{X}\phi (z){{\mathrm{\Delta _{hyp}}}}\phi (z){{\mathrm{\mu _{hyp}}}}(z) \bigg | \le \sup _{z\in X} \frac{|{{\mathrm{\Delta _{hyp}}}}\phi (z)|^{2}}{\lambda _{1}}\int _{X}{{\mathrm{\mu _{hyp}}}}(z), \end{aligned}$$

(80)

where $\lambda _{1}$ denotes the first non-zero eigenvalue of the hyperbolic Laplacian ${{\mathrm{\Delta _{hyp}}}}$. Hence, from Eq.  (78), and combining estimates (79) and (80), we arrive at the estimate

$$\begin{aligned} \big |C_{\mathrm {hyp}}\big | =\frac{ g^{2}}{\pi }\bigg | \int _{X}\phi (z){{\mathrm{\Delta _{hyp}}}}\phi (z) {{\mathrm{\mu _{hyp}}}}(z) \bigg |&\le \frac{g^{2}}{\pi \lambda _{1}}\int _{X}|{{\mathrm{\Delta _{hyp}}}}\phi (z)|^{2}{{\mathrm{\mu _{hyp}}}}(z)\\&\le \frac{16\pi g^{2}\left( d_{X}+1\right) ^{2}}{\lambda _{1}{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}, \end{aligned}$$

which completes the proof of the proposition. $\square$

Theorem 2.12

Let $p,q \in \mathcal { P}$ be two cusps with $p\not = q$. Then, we have the upper bound

$$\begin{aligned}&\big |g_{\mathrm {can}}(p,q)\big |\le 4\pi \big |k_{p,q}(0)\big |+ \frac{2\pi }{g}\bigg (\sum _{\begin{array}{c} s\in \mathcal {P}\\ s\not = p \end{array}}\big |k_{s,p}(0)\big |+ \sum _{\begin{array}{c} s\in \mathcal {P}\\ s\not = q \end{array}}\big |k_{s,q}(0)\big |\bigg )\nonumber \\&\quad + \frac{1}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg (\frac{4\pi (d_{X}+1)^{2}}{\lambda _{1}}+\frac{\big |4\pi c_{X}\big |}{g}+ \frac{ 43|\mathcal {P}|}{g}+4\pi \bigg )+\frac{2\log (4\pi )}{g}. \end{aligned}$$

Proof

For $z,w\in X$, from Eq. (28), we have

$$\begin{aligned} g_{\mathrm {can}}(p,q)=\lim _{z\rightarrow p}\lim _{w\rightarrow q}\big ( g_{\mathrm {hyp}}(z,w)-\phi (z)-\phi (w)\big ). \end{aligned}$$

Combining Eq. (18) with Corollary 2.10, for a fixed $w\in X$ with $z\in X$ approaching the cusp $p\in \mathcal {P}$, we have

$$\begin{aligned}&\lim _{z\rightarrow p}\big (g_{\mathrm {hyp}}(z,w)-\phi (z)\big )=4\pi \kappa _{p}(w)- \frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-\frac{\alpha _{p}}{8\pi g}-\frac{2\pi k_{p,p}(0)}{g}\nonumber \\&\quad + \frac{C_{\mathrm {hyp}}}{8g^{2}}+\frac{2\pi c_{X}}{g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+ \lim _{y_{p}\rightarrow \infty }\sum _{s\in \mathcal {P}}\int _{1\slash y_{p}}^{\infty } \frac{\log \zeta }{2g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},s}(\sigma _{s}\xi )\frac{d\zeta }{\zeta ^{2}}, \end{aligned}$$

(81)

where $\zeta ={{\mathrm{Im}}}(\xi ).$ As $w\in X$ approaches the cusp $q\in \mathcal { P}$ with $q\not = p$, from the Fourier expansion of the Kronecker’s limit function $\kappa _{p}(w)$, stated in Eq.  6, we have

$$\begin{aligned} 4\pi \kappa _{p}(w)=4\pi k_{p,q}(0) -\frac{4\pi \log v_{q}}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+O\big (e^{-2\pi v_{q}} \big ). \end{aligned}$$

So using Corollary 2.10 one more time, and substituting the above asymptotic relation into Eq. (81), we compute the limit

$$\begin{aligned}&\lim _{z\rightarrow p}\lim _{w\rightarrow q}\big (g_{\mathrm {hyp}}(z,w)-\phi (z)-\phi (w) \big )= 4\pi k_{p,q}(0)-\frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-\frac{\alpha _{p}}{8\pi g}-\frac{2\pi k_{p,p}(0)}{g} \nonumber \\&\quad -\frac{\alpha _{q}}{8\pi g}\!-\!\frac{2\pi k_{q,q}(0)}{g}\!+\!\frac{C_{\mathrm {hyp}}}{4g^{2}} \!+\!\frac{4\pi c_{X}}{g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\!+\!\lim _{y_{p}\rightarrow \infty }\sum _{s\in \mathcal {P}}\int _{1\slash y_{p}}^{\infty } \frac{\log \zeta }{2g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},s}\nonumber \\&\quad (\sigma _{s}\xi )\frac{d\zeta }{\zeta ^{2}}+\lim _{v_{q}\rightarrow \infty }\sum _{s\in \mathcal { P}} \int _{1\slash v_{q}}^{\infty }\frac{\log \zeta }{2g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)} {{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},s}(\sigma _{s}\xi )\frac{d\zeta }{\zeta ^{2}}. \end{aligned}$$

(82)

Using the definition of the constant $\alpha _{p}$ from (70), we find that the first six terms on the right-hand side of the above equation give

$$\begin{aligned}&4\pi k_{p,q}(0)-\frac{1}{ g}\bigg (\frac{2\pi |\mathcal {P}|}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-2\pi \sum _{s\in \mathcal {P}} k_{s,p}(0)-\log (4\pi )\bigg )-\frac{2\pi k_{p,p}(0)}{g}\\&\quad -\frac{4\pi }{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-\frac{1}{g}\bigg (\frac{2\pi |\mathcal {P}|}{{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}-2\pi \sum _{s\in \mathcal {P}}k_{s,q}(0)-\log (4\pi )\bigg )-\frac{2\pi k_{q,q}(0)}{g}\\&\quad =4\pi k_{p,q}(0)+\frac{2\pi }{g}\bigg (\sum _{\begin{array}{c} s\in \mathcal {P}\\ s\not = p \end{array}} k_{s,p}(0)+\sum _{\begin{array}{c} s\in \mathcal {P}\\ s\not = q \end{array}}k_{s,q}(0)\bigg ) -\frac{4\pi (|\mathcal {P}|+g)}{g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+\frac{2\log (4\pi )}{g}. \end{aligned}$$

Furthermore, the expression on the right-hand side of the above equation can be bounded by

$$\begin{aligned} 4\pi \big |k_{p,q}(0)\big |+\frac{2\pi }{g}\left( \sum _{\begin{array}{c} s\in \mathcal {P} \\ s\not = p \end{array}}\big |k_{s,p}(0)\big |+\sum _{\begin{array}{c} s\in \mathcal {P}\\ s\not = q \end{array}}\big | k_{s,q}(0)\big |\right) +\frac{13|\mathcal {P}|+4\pi g}{g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+ \frac{2\log (4\pi )}{g}. \end{aligned}$$

(83)

Using Proposition 2.11, we derive the upper bound for the next two terms on the right-hand side of Eq.  (82)

$$\begin{aligned} \frac{C_{\mathrm {hyp}}}{4g^{2}}+\frac{4\pi c_{X}}{g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)} \le \frac{4\pi \left( d_{X}+1\right) ^{2}}{\lambda _{1}{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}+\frac{\big |4\pi c_{X}\big |}{g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)} . \end{aligned}$$

(84)

From Lemma 2.6, we have the upper bound for the absolute value of the last two terms on the right-hand side of Eq.  (82)

$$\begin{aligned}&\lim _{y_{p}\rightarrow \infty }\sum _{s\in \mathcal {P}}\int _{1\slash y_{p}}^{\infty } \Bigg |\frac{\log \zeta }{2g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},s}(\sigma _{s}\xi )\Bigg |\frac{d\zeta }{\zeta ^{2}}\nonumber \\&+\lim _{v_{q}\rightarrow \infty }\sum _{s\in \mathcal {P}}\int _{1\slash v_{q}}^{\infty } \Bigg |\frac{\log \zeta }{2g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}{{\mathrm{\Delta _{hyp}}}}P_{\mathrm {gen},s}(\sigma _{s}\xi ) \Bigg |\frac{d\zeta }{\zeta ^{2}}\le \frac{2|\mathcal {P}|}{g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}\bigg (1+ \frac{4\pi ^{2}}{3}\bigg )\nonumber \\&\quad \le \frac{30|\mathcal {P}|}{g{{\mathrm{{\mathrm {vol}}_{{\mathrm {hyp}}}}}}(X)}. \end{aligned}$$

(85)

The proof of the theorem follows from combining the estimates obtained in Eqs. (83), (84), and (85). $\square$