Mumford curve and its skeleton

For an analog of periodic geodesics in the non-Archimedean case, we need to use the description of the limit set in $\mathbb {P}^{1,an}_k$ . The advantage is that $\mathbb {P}^{1,an}_k$ is a compact real tree, while $\mathbb {P}^1_k$ is discrete.

Let $\Gamma $ be a Schottky group of rank g in $\textrm {PGL}_2(k)$ . As in the special case where $k=\mathbb {C}$ , a point $x\in \mathbb {P}^{1,an}_k$ is a limit point if there exist $x_o\in \mathbb {P}^{1,an}_k$ and a sequence $\{\gamma _n\}_n\subset \Gamma $ of distinct elements such that $\lim _{n\rightarrow \infty }\gamma _nx_0=x$ . Let $L(\Gamma )$ be the limit set of $\Gamma $ on $\mathbb {P}^{1,an}_k$ , that is, the set of limit points. We have that $L(\Gamma )\subset \mathbb {P}^1_k$ [Reference Poineau and Turchetti32, Corollary II.3.14].

Moreover, let $O=\mathbb {P}^{1,an}_k-L(\Gamma )$ . It is shown that O is a domain of discontinuity of $\Gamma $ , that is, it is $\Gamma $ -invariant, and the action of $\Gamma $ on O is free and proper. The quotient space $X=\Gamma \backslash O$ is a Mumford curve. (See [Reference Poineau and Turchetti33, Theorem II.3.18].) Let $\Sigma =O\cap \cup _{x,y\in L}[x,y]$ , where $\mathbb {P}^{1,an}_k$ has the structure of a real tree and $[x,y]$ is the unique injective path from x to y. See [Reference Poineau and Turchetti33, Section 2.5] and [Reference Poineau and Turchetti32, Proposition I.6.12].Footnote ³ Let $\Sigma _X=\Gamma \backslash \Sigma $ , which is called the skeleton of the Mumford curve and is a finite graph of genus g. (See [Reference Poineau and Turchetti33, Notation 4.2.2].) The skeleton $\Sigma _X$ can also be obtained by identifying the Shilov boundary points $p_i,q_i$ of $D_{\gamma _i}^+,D_{\gamma _i^{-1}}^+$ , which is a Schottky figure adapted to $\{\gamma _1,\cdots ,\gamma _g \}$ .

Metric structure

Recall from [Reference Poineau and Turchetti32, Section I.8] and [Reference Poineau and Turchetti33, Section 2.5] the definition of multiplicative length. For $\alpha ,\ \beta \in k$ and $0<r\leq s$ , we define the additive length of the elements $\eta _{\alpha ,r},\eta _{\beta ,s}$ by:

• • if $\max \{r,s\}\geq |\alpha -\beta |_k$ , $d_a(\eta _{\alpha ,r},\eta _{\beta ,s}):=\log (s/r)$ ;

• • if $ \max \{r,s\}< |\alpha -\beta |_k$ , $d_a(\eta _{\alpha ,r},\eta _{\beta ,s}):=\log (|\alpha -\beta |_k/r)+\log (|\alpha -\beta |_k/s)$ .

This length is invariant under the action of Möbius transformations. The tree $\Sigma $ consists of points of $\eta _{a,r}$ with $a\in k$ and $r\in \mathbb {R}_{>0}$ .Footnote ⁴ Using this length, we can define a length on $\Sigma $ . The length of an edge of the graph $\Sigma _X=\Gamma \backslash \Sigma $ comes from the quotient of this additive length.

Definition 2.16. For a loxodromic element $\gamma $ , we define the non-Archimedean length by

$$\begin{align*}\ell^{na}(\gamma)=\log (|\textrm{tr} \tilde\gamma|_k^2/|\det(\tilde\gamma)|_k), \end{align*}$$

where $\tilde \gamma $ is a representative of $\gamma $ in $\mathrm {GL}_2(k)$ .

This definition also coincides with the logarithm of the attraction rate at the attracting fixed point of the loxodromic element. Hence, it serves as a suitable analog of the length of periodic geodesics in the non-Archimedean case.

The following lemma connects the length of loops in the graph with the non-Archimedean length of loxodromic elements in $\Gamma $ .

Lemma 2.17. Let $\gamma $ be a nontrivial loxodromic element in $\Gamma $ and $\alpha ,\alpha '\in \mathbb {P}^1_k$ be its two fixed points. Then for any $x\in [\alpha ,\alpha ']-\{\alpha ,\alpha ' \}$ , we have

$$\begin{align*}d_a(x,\gamma x)=\ell^{na}(\gamma), \end{align*}$$

where $[x,\gamma x]\subset [\alpha ,\alpha ']-\{\alpha ,\alpha ' \}\subset \Sigma $ .

For any $y=\eta _{\beta ,s}\notin [\alpha ,\alpha ']$ with $0<s<\infty $ , we have

$$\begin{align*}d_a(y,\gamma y)>\ell^{na}(\gamma). \end{align*}$$

Ihara zeta function

The non-Archimedean case of the Selberg zeta function is well studied. The analogous zeta function is called the Ihara zeta function for finite graphs. See [Reference Ihara17] and [Reference Horton, Stark and Terras16].

Suppose $G=(V,E)$ is a finite graph. Let $E=\{e_1,\cdots , e_{2J}\}$ be the set of oriented edges of G with $e_j$ and $e_{j+J}$ in opposite directions. Each edge has a positive length. A loop P in G is a finite sequence $P=(e_{i_1},\cdots ,e_{i_n})$ with the endpoint of $e_{i_j}$ equal to the starting point of $e_{i_{j+1}}$ (we use the convention $e_{i_{n+1}}=e_{i_1}$ ). The length $\ell (P)$ of a path P is the sum of the lengths of the $e_{i_j}$ . A path is nonbacktracking if there are no consecutive edges that are inverse to each other; that is, there is no j such that $e_{i_{j+1}}=e_{i_j+J}$ . A loop is primitive if it cannot be written as a multiple of another loop. Two loops that differ only by starting point are defined to be in the same equivalence class of loops, denoted by $[P]$ . Let $\mathcal {P}$ be the set of equivalence classes of primitive nonbacktracking loops.

Theorem 2.18 (Ihara, Hashimoto, Bass).

Let G be a finite graph. Define the Ihara zeta function by

$$\begin{align*}Z_I(G,s)=\prod_{[P]\in \mathcal{P} }(1-e^{-s\ell(P)}). \end{align*}$$

Then the Ihara zeta function extends analytically to the whole complex plane.

See Proposition 5.1 for a proof and a way to compute the Ihara zeta function.

Let $\Gamma $ be a Schottky group in $\textrm {PGL}_2(k)$ . Recall that the skeleton of the Mumford curve associated with $\Gamma $ is the quotient space $\Sigma _X=\Gamma \backslash \Sigma $ . Since $\Sigma $ is a tree, which is simply connected, each nonbacktracking loop P of $\Sigma _X$ corresponds to a conjugacy class $[\gamma _P]$ in $\Gamma $ ; this follows from the standard correspondence between free homotopy classes of loops and conjugacy classes in the fundamental group. The length of each edge in $\Sigma _X$ is given by the length of its lift in $\Sigma $ . Moreover, from Lemma 2.17, we obtain that for the nonbacktracking loop P, we have $\ell (P)=\ell ^{na}(\gamma _P)$ . Therefore, we obtain the following.

Proposition 2.19. Let $Z_I(\Gamma , s)$ be the Ihara zeta function for $\Gamma < \textrm {PGL}_2(k)$ ,Footnote ⁵ that is,

$$\begin{align*}Z_I(\Gamma, s)=\prod_{[\gamma] \in \mathcal{P}}(1-e^{-s\ell^{na}(\gamma)}). \end{align*}$$

Then we have

(2.1) $$ \begin{align} Z_I(\Gamma, s)=Z_I({\Sigma_X},s). \end{align} $$

Remark 2.20. If all the edges of G are of length $1$ , then

$$\begin{align*}Z_I(s)=(1-e^{-2s})^{g-1}\det(1-A_Ge^{-s}+Q_Ge^{-2s}), \end{align*}$$

where A is the adjacent matrix of the graph G, g is the rank of the fundamental group of G, and $Q_G$ is a diagonal matrix with the i-th element equal to the degree minus 1 of the i-th vertex.

Therefore, if $|k^*|=a^{\mathbb {Q}}$ for some $a\in (0,1)$ (in particular, in the case of the algebraic closure of $\mathbb {C}(\!(t)\!)$ ), then the edge lengths of $\Sigma _X$ are rational multiples of $|\log a|$ . By rescaling the metric, we obtain a graph with integer edge lengths. Subdividing these edges with additional vertices yields a graph where every edge has unit length. Consequently, we can apply this formula to compute the Ihara zeta function.

Continuous map $p: \overline {\mathbb {D}}_r\to \mathbb {A}^{3g-3,an}_{A_r}$

Let $p:\mathbb {D}\to \mathbb {C}^{3g-3}$ , $z\mapsto p(z)=(\underline {\alpha }(z),\underline {\beta }(z),\underline {\lambda }(z))$ , be a continuous function in $\overline {\mathbb {D}}_r^{*}$ , holomorphic in $\mathbb {D}_r^*$ , and meromorphic at $0$ . By abusing notation, we construct a map $p: \overline {\mathbb {D}}_r\to \mathbb {A}^{3g-3,an}_{A_r}$ as follows: for any $F=\sum f_n(t) P_n(\underline {\alpha },\underline {\beta },\underline {\lambda })\in A_r[\underline {\alpha },\underline {\beta },\underline {\lambda }]$ , we set

(2.3) $$ \begin{align} |F|_{p(z)}=\begin{cases} \left|\sum f_n(z) P_n(\underline{\alpha}(z),\underline{\beta}(z),\underline{\lambda}(z))\right|{}_{\infty}^{1/\log (1/|z|)},\,\,\,\text{if}\,\,z\neq 0\\ \exp(-{\mathrm{ord}_{z=0}(\sum f_n(z)P_n(\underline{\alpha}(z),\underline{\beta}(z),\underline{\lambda}(z)))}),\,\,\,\text{if}\,\,z=0, \end{cases} \end{align} $$

where $P_n(\underline {\alpha },\underline {\beta },\underline {\lambda })$ is a monomial of total degree n. The key advantage of this construction is that, from [Reference Boucksom and Jonsson4, Proposition A.4], the map $p:\overline {\mathbb {D}}_r\to \mathbb {A}^{3g-3,an}_{A_r}$ is continuous. In the beginning, we deal with meromorphic functions. Using this hybrid construction, we do not have a singularity at $z=0$ and give a sense of the limiting behavior at $z=0$ .

For $z\in \overline {\mathbb {D}}_r^*$ , we have $\mathcal {H}(|\cdot |_{p(z)})=(\mathbb {C},|\cdot |_{\infty }^{1/\log (1/|z|)})$ , and therefore $p(z)$ is an Archimedean point. For $z=0$ , we have $\mathcal {H}(|\cdot |_{p(0)})=(\mathbb {C}(\!(t)\!), |\cdot |_{na})$ , and therefore $p(0)$ is a non-Archimedean point.

Now, we build projective spaces where the Schottky groups attached to the points in the Berkovich affine space act in a natural way. Given $x\in \mathbb {A}^{3g-3,an}_{A_r}$ , let

$$ \begin{align*}\mathbb{P}^{1,an}_{\mathcal{H}(x)} \simeq \mathbb{P}^1(\mathbb{C})\end{align*} $$

if the residue field $\mathcal {H}(x)$ is Archimedean, and $\mathbb {P}^{1,an}_{\mathcal {H}(x)} \simeq $ a tree otherwise. In this context, a point $x\in \mathbb {A}^{3g-3,an}_{A_r}$ belongs to the Schottky space $S^{an}_{g,A_r}$ if $0<|\lambda _j|_x<1$ for all $1\leq j\leq g$ , the points $0,1,2, \alpha _3,\dots ,\alpha _g,\beta _2,\dots ,\beta _g$ are mutually distinct, and the matrices $\Psi (x)\subset \textrm {PGL}_2(\mathcal {H}(x))$ associated with x induce a Schottky group over $\mathbb {P}^{1,an}_{\mathcal {H}(x)}$ . Interestingly enough, Poineau and Turchetti proved an analog of Hejhal’s theorem by establishing that $S^{an}_{g,A_r}$ is an open path-connected subset of $\mathbb {A}^{3g-3,an}_{A_r}$ .

For the family of Schottky groups satisfying condition $(\bigstar )$ , due to Remark 2.14, we know ${p(0)\in S^{an}_{g,A_r}}$ . Due to the openness of $S^{an}_{g,A_r}$ , we recover [Reference Dang and Mehmeti9, Proposition 6.1].

Uniform Schottky basis

For an element $\gamma \in \textrm {PGL}_2(\mathbb {C}(\!(t)\!))$ , whose entries are meromorphic on $\mathbb {D}_r^*$ with a possible pole at $0$ , we define a family of twisted Ford discs:

(2.4) $$ \begin{align} D_{\gamma,\lambda}^+(z)=\begin{cases} &\{x\in\mathbb{C}|\ |c(z)x+d(z)|_\infty^{2/\log(1/|z|)}\leq \lambda |(ad-bc)(z)|_\infty^{1/\log(1/|z|)} \}\text{ if } \ z\neq 0, \\ & \{ x\in \mathbb{A}_{\mathbb{C}(\!(t)\!)}^{1,an}|\ |(cZ+d)(x)|^2\leq \lambda |(ad-bc)(x)|\}\text{ if } \ z=0, \end{cases} \end{align} $$

where $c(z)$ means the complex value of the meromorphic function c at t.

We state a result on uniform Schottky figures for all small parameters. This is a translation of this special case of [Reference Poineau and Turchetti32, Theorem 4.3.2 and Corollary 4.3.3].

Lemma 2.23. Let $\Gamma =\langle M_1,\cdots ,M_g\rangle $ be a family of Schottky groups satisfying condition $(\bigstar )$ . There exist $s\in (0,r)$ and an automorphism $\tau \in Aut(F_g)$ such that the following holds. Let

$$ \begin{align*}N_1=\tau M_1,\cdots,N_g=\tau M_g\in \mathrm{PGL}_2(\mathbb{C}(\!(t)\!)).\end{align*} $$

There exist positive real numbers $\lambda _1,\cdots ,\lambda _g$ such that the family of twisted Ford discs

$$\begin{align*}(D_{N_1,\lambda_1}^+(z),\cdots,D_{N_g,\lambda_g}^+(z), D_{N_1^{-1},\lambda_1^{-1}}^+(z),\cdots, D_{N_g^{-1},\lambda_g^{-1}}^+(z)) \end{align*}$$

is a Schottky figure adapted to the basis $(N_1,\cdots , N_g)$ for each $z\in \mathbb {C}$ with $|z|<s$ .

Proof. We explain the language used in [Reference Poineau and Turchetti33] and how to obtain the lemma.

Recall that $S=S_{g,A_r}^{an}$ is the Schottky space. Let $\mathcal {O}( S)$ be the ring of analytic functions, which means that locally the function can be uniformly approximated by rational functions.Footnote ⁶ Let $\gamma =\begin {pmatrix} a & b \\ c & d \end {pmatrix}\in \textrm {PGL}_2(\mathcal {O}(S))$ be a loxodromic element with $c\neq 0$ and $0<\lambda $ . Let $\mathbb {A}_S^{1,an}$ be the affine line over S with variable Z. In [Reference Poineau and Turchetti33], relative twisted Ford discs are defined by

$$\begin{align*}D_{\gamma,\lambda}^+=\{ x\in \mathbb{A}_S^{1,an}|\ |(cZ+d)(x)|^2\leq \lambda |(ad-bc)(x)|\}. \end{align*}$$

We will prove that this gives the same family of twisted Ford discs as in equation (2.4).

The projection map $\pi $ from $\mathbb {A}_S^{1,an}$ to S is the restriction of the seminorm to the ring $A_r[\underline {\alpha },\underline {\beta },\underline {\lambda }]$ . For any $y\in S$ , the fiber $\pi ^{-1}(y)$ is the affine line $\mathbb {A}_{\mathcal {H}(y)}^{1,an}$ .Footnote ⁷ Relative Ford discs are the union of twisted Ford discs on these fibers.

For the continuous map p, we know $p(0)$ is a non-Archimedean point in S. Then we can apply [Reference Poineau and Turchetti32, Theorem 4.3.2 and Corollary 4.3.3] to find relative twisted Ford discs on an open neighborhood U of $p(0)$ in S, which is a Schottky figure adapted to a chosen basis. The new basis is denoted by $ N_1=\tau M_1,\cdots ,N_g=\tau M_g $ for some $\tau \in Aut(F_g)$ and with parameters $\lambda _1,\cdots ,\lambda _g$ . Moreover, for any $y\in U$ , the restriction of the relative twisted Ford discs to the fiber $\mathbb {A}_{\mathcal {H}(y)}^{1,an}$ gives a Schottky figure on that fiber. Then we only need to prove that this gives the twisted Ford discs in the statement of the lemma.

For $\mathcal {H}(y)$ Archimedean, the norm $(\mathcal {H}(y),|\cdot |_y)$ is always given by $(\mathbb {C},|\cdot |_\infty ^{\epsilon (y)})$ with some $\epsilon (y)\leq 1$ . Due to our choice of $p(z)$ , we know that $\epsilon (p(z))=1/\log (1/|z|)$ for $z\neq 0$ . Over $(\mathbb {C},|\cdot |_\infty ^\epsilon )$ , the analytification gives no new point. Hence, a seminorm $|\cdot |_x\in \mathbb {A}_{\mathcal {H}(p(z))}^{1,an}=\mathbb {A}_{\mathbb {C},|\cdot |_\infty ^\epsilon }^{1,an}$ reads as

$$\begin{align*}|P(Z)(x)|=\left|(\sum a_nZ^n)(x)\right|=\left|\sum a_nZ(x)^n\right|{}_\infty^{1/\log(1/|z|)}, \end{align*}$$

for some unique $Z(x)\in \mathbb {C}$ determined by x and for a polynomial $P\in \mathbb {C}[Z]$ . Due to the openness of U and the continuity of the map p, we can find $0<s<r$ such that $p(\mathbb D_s)$ is contained in U. Therefore, the twisted Ford discs at the fiber over $p(z)$ with $z\neq 0$ and $|z|<s$ read as

$$ \begin{align*} D_{\gamma,\lambda}^+(z)&=\{ x\in \mathbb{A}_{\mathcal{H}(p(z))}^{1,an}|\ |(cZ+d)(x)|^2\leq \lambda |(ad-bc)(x)|\}\\ &=\{x\in\mathbb{C}|\ |c(z)x+d(z)|_\infty^{2/\log(1/|z|)}\leq \lambda |(ad-bc)(z)|_\infty^{1/\log(1/|z|)} \}. \end{align*} $$

The proof is complete.

We define the discs $D_{\mathfrak {a},z}$ to be the Ford discs

(2.5) $$ \begin{align} D_{\mathfrak{a},z}:=D^-_{\mathfrak{a}^{-1},\lambda_{\mathfrak{a}^{-1}}}(z) \end{align} $$

for any generator $\mathfrak {a}$ in the generator set $\{N_1,\cdots , N_g, N_1^{-1},\cdots , N_g^{-1} \}$ , where the inverse is used because we want the attracting fixed point $\mathfrak {a}^+$ of $\mathfrak {a}$ to satisfy $\mathfrak {a}^+\in D_{\mathfrak {a}}$ . For any element $\gamma =\mathfrak {a}_{i_1}\cdots \mathfrak {a}_{i_n}\in \Gamma $ , we define

$$\begin{align*}D_{\gamma,z}=\mathfrak{a}_{i_1,z}\cdots \mathfrak{a}_{i_{n-1},z}D_{\mathfrak{a_{i_n}},z}. \end{align*}$$

We can further suppose the Schottky basis is uniformly bounded for $|z|\ll 1$ . Indeed, we may assume that the limit set and the Schottky figure of $\Gamma $ are in the unit ball $D(0,1)$ of the Berkovich affine line $\mathbb {A}_k^{1,an}$ , which can be realized by suitably conjugating the Schottky group. Then, for small $|z|$ , the Schottky figures of $\Gamma _z$ are in a uniform compact set in $\mathbb {C}$ due to the centers of Schottky discs being meromorphic functions and their radii being functions of the form $\lambda ^{\log (1/|z|)}/|c(z)|$ . Here, $\lambda $ and $c(z)$ are determined by the disc $D^-_{\mathfrak {a}^{-1},\lambda _{\mathfrak {a}^{-1}}}(z)$ corresponding to $\mathfrak a_z=\begin {pmatrix} a(z) & b(z)\\ c(z) & d(z) \end {pmatrix} $ and $\lambda _{\mathfrak a^{-1}}=\lambda $ .

3.2.1 Preliminaries on trace-class operators

We recall some preliminaries on trace-class operators. See [Reference Dyatlov and Zworski11, Appendix B] for more details.

Let $\mathcal {A}:\mathcal {H}_1\to \mathcal {H}_2$ be a compact operator between two Hilbert spaces. The singular values $\mu _{\ell }$ of $\mathcal {A}$ are defined to be the eigenvalues $\lambda _{\ell }$ of $(\mathcal {A}^*\mathcal {A})^{1/2}$ in decreasing order, that is,

$$ \begin{align*} \mu_{\ell}(\mathcal{A}):=\lambda_{\ell}((\mathcal{A}^*\mathcal{A})^{1/2}),\quad \ell=0,1,2,\cdots. \end{align*} $$

We recall the minimax property of singular values:

(3.8) $$ \begin{align} \mu_{\ell}(\mathcal{A})=\min\limits_{\mathrm{{codim}(V, \mathcal{H}_1)=\ell}}\max\limits_{v\in V\setminus \{0\}}\frac{\|\mathcal{A}v\|_{\mathcal{H}_2}}{\|v\|_{\mathcal{H}_1}}. \end{align} $$

This follows directly from the minimax property for self-adjoint operators since $\|\mathcal {A}v\|^2_{\mathcal {H}_2}=(\mathcal {A}^*\mathcal {A}v,v)_{\mathcal {H}_2}$ . As a corollary, for any compact operators $\mathcal {A}:\mathcal {H}_1\to \mathcal {H}_2$ and $\mathcal {B}:\mathcal {H}_1\to \mathcal {H}_2$ , we have the inequality

(3.9) $$ \begin{align} \mu_{\ell}(\mathcal{A}+\mathcal{B})\leq \mu_{\ell_1}(\mathcal{A})+\mu_{\ell_2}(\mathcal{B}),\quad \forall\ \ell=\ell_1+\ell_2. \end{align} $$

Definition 3.8. A compact operator $\mathcal {A}:\mathcal {H}_1\to \mathcal {H}_2$ between two Hilbert spaces is a trace-class operator if

$$ \begin{align*} \|\mathcal{A}\|_1:=\sum\limits_{\ell}\mu_{\ell}(\mathcal{A})<\infty. \end{align*} $$

$\|\cdot \|_1$ is called the trace norm.

For a trace-class operator $\mathcal {A}:\mathcal {H}\to \mathcal {H}$ on a single space, one can define the trace by

(3.10) $$ \begin{align} \mathrm{tr}(\mathcal{A}):=\sum\limits_{j}\langle \mathcal{A} e_j, e_j\rangle, \end{align} $$

where $\{e_j\}$ is an orthonormal basis of $\mathcal {H}$ . One can verify that the definition does not depend on the choice of $\{e_j\}$ . By Lidskii’s theorem [Reference Dyatlov and Zworski11, Proposition B.31], we have

$$ \begin{align*} \mathrm{tr}(\mathcal{A})=\sum_{\ell}\lambda_{\ell}(\mathcal{A}). \end{align*} $$

The Fredholm determinant is defined by

$$ \begin{align*} \det(1-\mathcal{A}):=\prod\limits_{\ell}(1-\lambda_{\ell}(\mathcal{A})). \end{align*} $$

We have the estimates

(3.11) $$ \begin{align} |\mathrm{tr}(\mathcal{A})|\leq \sum\limits_{\ell}|\lambda_{\ell}(\mathcal{A})|\leq \|\mathcal{A}\|_1,\quad |\det(1-\mathcal{A})|\leq \prod\limits_{\ell}(1+\mu_{\ell}(\mathcal{A}))\leq e^{\|\mathcal{A} \|_1}. \end{align} $$

For a trace-class operator $\mathcal {A}:\mathcal {H}_1 \to \mathcal {H}_2$ and a bounded operator $\mathcal {B}:\mathcal {H}_2\to \mathcal {H}_3$ , it follows from the minimax property that $\mu _{\ell }(\mathcal {B}\mathcal {A})\leq \|\mathcal {B}\|_{\mathcal {H}_2\to \mathcal {H}_3} \mu _{\ell }(\mathcal {A})$ and thus

(3.12) $$ \begin{align} \|\mathcal{B}\mathcal{A}\|_{1}\leq \|\mathcal{B}\|_{\mathcal{H}_2\to \mathcal{H}_3} \|\mathcal{A}\|_1. \end{align} $$

Moreover, it is shown in [Reference Dyatlov and Zworski11, Eq (B.4.9)] that for a trace-class operator $\mathcal {A}:\mathcal {H}_1\to \mathcal {H}_2$ and a bounded operator $\mathcal {B}:\mathcal {H}_2\to \mathcal {H}_1$ ,

(3.13) $$ \begin{align} \mathrm{tr}(\mathcal{A}\mathcal{B})=\mathrm{tr}(\mathcal{B}\mathcal{A}). \end{align} $$

3.2.2 Transfer operator: $\mathrm {SL}_2(\mathbb {R})$ case

To estimate the Selberg zeta function in the region where the infinite product expression does not converge, we employ the transfer operator $\mathcal {L}_s$ . The observation is that the transfer operator $\mathcal {L}_s$ is a trace-class operator acting on the space of holomorphic functions with bounded $L^2$ norm, and its Fredholm determinant gives the Selberg zeta function.

We first consider the case of a Schottky group $\Gamma < \mathrm {SL}_2(\mathbb {R})$ of rank $g\geq 2$ . Write $\Gamma =\langle \mathfrak {a}_{1}, \mathfrak {a}_{2},\cdots , \mathfrak {a}_{{2g}} \rangle $ with $\mathfrak {a}_{i}\mathfrak {a}_{i+g}=id$ for $i=1,2,\cdots , g$ . By conjugating $\Gamma $ if necessary, there exist open discs $D_{\mathfrak {a}_1},D_{\mathfrak {a}_2},\cdots , D_{\mathfrak {a}_{2g}}$ in $\mathbb {C}$ centered in $\mathbb {R}$ such that their closures are disjoint, and for each $\mathfrak {a}\in \{\mathfrak {a}_1,\ldots , \mathfrak {a}_{2g}\}$ , we have $\mathfrak {a}(\hat {\mathbb {C}}-\overline {D_{{\mathfrak {a}}^{-1}}})=D_{\mathfrak {a}}$ and $\mathfrak {a}(\hat {\mathbb {R}}- I_{\mathfrak {a}^{-1}}) = I_{\mathfrak {a}}^\circ $ , where $I_{\mathfrak {a}}:=\overline {D_{\mathfrak {a}}}\cap \mathbb {R}$ .

We define the Hilbert space $\mathcal {H}:=\oplus _{i=1}^{2g}\mathcal {H}(D_{\mathfrak {a}_i})$ , where $\mathcal {H}(D_{\mathfrak {a}})=\{u\in L^2(D_{\mathfrak {a}}): u \text { is holomorphic}\}$ is equipped with the norm $\|u\|_{\mathcal {H}(D_{\mathfrak {a}})}^2:=|D_{\mathfrak {a}}|^{-1}\|u\|_{L^2(D_{\mathfrak {a}})}^2$ , where $|D_{\mathfrak {a}}|$ is the Lebesgue measure of the disc $D_{\mathfrak {a}}$ given as in equation (1.7). The transfer operator $\mathcal {L}_s:\mathcal {H}\to \mathcal {H}$ is defined as

(3.14) $$ \begin{align} \mathcal{L}_s u(w):=\sum\limits_{\mathfrak{a}_i\neq \mathfrak{b}^{-1}} \mathfrak{a}_i'(w)^s u(\mathfrak{a}_i(w)) \quad \text{for}\quad w\in D_{\mathfrak{b}}\quad\text{with}\quad\mathfrak{b}\in \{\mathfrak{a}_1,\cdots, \mathfrak{a}_{2g}\}. \end{align} $$

Here, note that the derivative satisfies $\mathfrak {a}_i'(w)>0$ for $w\in D_{\mathfrak {b}}\cap \mathbb {R}$ , and hence we define $\mathfrak {a}_i'(w)^s:=e^{s\log \mathfrak {a}_i'(w)}$ , where we choose the branch of the complex logarithm so that if $s\in \mathbb {R}$ , then $\mathfrak {a}_i'(w)^s$ is the unique holomorphic function on $D_{\mathfrak {b}}$ satisfying $\mathfrak {a}_i'(w)^s>0$ for $w \in D_{\mathfrak {b}}\cap \mathbb {R}$ .

We introduce a few more notations. We use $\mathcal {W}^n$ to denote words of length n with respect to the generators $\mathfrak {a}_1,\cdots ,\mathfrak {a}_{2g}$ , that is, words of the form $\mathfrak {a}_{i_1}\mathfrak {a}_{i_2}\cdots \mathfrak {a}_{i_n}$ with $\mathfrak {a}_{i_{j+1}}\neq \mathfrak {a}_{i_j}^{-1}$ . For $\mathbf {a}=\mathfrak {a}_{i_1}\cdots \mathfrak {a}_{i_n}\in \mathcal {W}^n$ , we denote

$$ \begin{align*} {\mathbf{a}}(w)={\mathfrak{a}_{i_1}}({\mathfrak{a}_{i_2}}(\cdots {\mathfrak{a}_{i_n}}(w))) \end{align*} $$

and $\hat {\mathbf {a}}=\mathfrak {a}_{i_1}\cdots \mathfrak {a}_{i_{n-1}}$ .

For a disc D of radius R, and a real number $\rho>0$ , we use $\rho D$ to denote the disc with the same center and of radius $\rho R$ .

The following lemma shows that $\mathcal {L}_s$ is a trace-class operator (see also [Reference Borthwick5, Lemma 15.7]).

Lemma 3.9. Suppose $\gamma :D_{\mathfrak {b}} \to \rho D_{\mathfrak {a}}$ is a holomorphic function between two discs with $0<\rho <1$ and $\mathfrak {a},\mathfrak {b}\in \{\mathfrak {a}_1,\ldots , \mathfrak {a}_{2g}\}$ . Then the pullback operator

$$ \begin{align*} \gamma^*u(w)=u(\gamma(w)): \mathcal{H}(D_{\mathfrak{a}})\to \mathcal{H}(D_{\mathfrak{b}}) \end{align*} $$

is a trace-class operator satisfying

$$ \begin{align*} \mu_{\ell}(\gamma^*)\leq C(1-\rho)^{-2}\rho^{\ell}\sqrt{\ell+1}. \end{align*} $$

Consequently, for any $s\in \mathbb {C}$ , $\mathcal {L}_s:\mathcal {H}\to \mathcal {H}$ is a trace-class operator.

Proof. We may assume $D_{\mathfrak {a}}$ is centered at $0$ and take an orthonormal basis $\phi _n(w):=\sqrt {n+1}\left (\frac {w}{R}\right )^n$ of $\mathcal {H}(D_{\mathfrak {a}})$ , where R is the radius of $D_{\mathfrak {a}}$ . Then

$$ \begin{align*} \|\gamma^*\phi_n\|_{\mathcal{H}(D_{\mathfrak{b}})}^2=|D_{\mathfrak{b}}|^{-1}\int_{D_{\mathfrak{b}}}(n+1)\frac{|\gamma(w)|^{2n}}{R^{2n}} dm(w)\leq (n+1)\rho^{2n}. \end{align*} $$

By the minimax property equation (3.8), we conclude

$$ \begin{align*} \mu_{\ell}(\gamma^*)\leq\sum\limits_{n\geq \ell}\|\gamma^*\phi_n\|_{\mathcal{H}(D_{\mathfrak{b}})}\leq \sum\limits_{n\geq \ell}\sqrt{n+1}\rho^{n}\leq C(1-\rho)^{-2}\rho^{\ell}\sqrt{\ell+1}. \end{align*} $$

Thus,

$$ \begin{align*} \sum\limits_{\ell}\mu_{\ell}(\gamma^*)\leq C(1-\rho)^{-2}\sum\limits_{\ell} \rho^{\ell}\sqrt{\ell+1}<\infty, \end{align*} $$

which implies that $\gamma ^*$ is a trace-class operator. Since $\mathcal {L}_s$ is a finite direct sum of such pullback operators composed with multiplication by holomorphic functions, we conclude that $\mathcal {L}_s$ is also a trace-class operator using equation (3.9) and equation (3.12).

Since $\mathcal {L}_s$ is a trace-class operator, we can define its Fredholm determinant. We claim $\det (1-\mathcal {L}_s)$ is exactly the Selberg zeta function; see [Reference Borthwick5, Lemma 15.10]. Here, we include a proof for completeness, but we will omit the proof later for similar cases. Note that since $\mathcal {L}_s$ is of trace class and depends holomorphically on $s\in \mathbb {C}$ , $\det (1-\mathcal {L}_s)$ is an entire function. So, this provides an alternative proof of the holomorphic extension of the Selberg zeta function.

Lemma 3.10. For $s\in \mathbb {C}$ , we have

(3.15) $$ \begin{align} \det(1-\mathcal{L}_s)=Z(\Gamma,s). \end{align} $$

Proof. Since both sides are entire functions on $\mathbb {C}$ , it suffices to check the inequality for $\Re s\gg 1$ . For $\Re s\gg 1$ , we have

$$ \begin{align*} \det(1-\mathcal{L}_s)=\exp\left(-\sum\limits_{n=1}^{\infty}\frac{1}{n}\mathrm{tr}(\mathcal{L}_s^n)\right). \end{align*} $$

On the other hand,

$$ \begin{align*} Z(\Gamma,s)=\exp\left(-\sum\limits_{n=1}^{\infty}\frac{1}{n}\sum\limits_{[\gamma],k}e^{-n(s+k)\ell(\gamma)}\right). \end{align*} $$

We again consider the orthonormal basis $\phi _k^{\mathfrak {a}}(w)=\sqrt {k+1}\left (\frac {w-w_{\mathfrak {a}}}{R_{\mathfrak {a}}}\right )^k$ , where $w_{\mathfrak {a}}$ is the center of $D_{\mathfrak {a}}$ and $R_{\mathfrak {a}}$ is the radius of $D_{\mathfrak {a}}$ . We write

$$ \begin{align*} \mathcal{L}_s=\sum\limits_{\mathfrak{a}\neq \mathfrak{b}^{-1}\in \mathcal{W}}\mathcal{L}_{s,\mathfrak{ab}}\quad\text{with}\quad\mathcal{L}_{s, \mathfrak{ab}}:\mathcal{H}(D_{\mathfrak{a}})\to \mathcal{H}({D_{\mathfrak{b}}}) \end{align*} $$

given by

$$ \begin{align*} \quad\mathcal{L}_{s,\mathfrak{ab}}u(w)={\mathfrak{a}}'(w)^su({\mathfrak{a}}(w))\quad \text{for}\,\,w\in {D_{\mathfrak{b}}}. \end{align*} $$

We have

$$ \begin{align*} \mathcal{L}_s^n=\sum\limits_{\mathfrak{a}_{i_1}\mathfrak{a}_{i_2}\cdots \mathfrak{a}_{i_{n+1}}\in \mathcal{W}^{n+1}}\mathcal{L}_{s,\mathfrak{a}_{i_1}\mathfrak{a}_{i_2}\cdots \mathfrak{a}_{i_{n+1}}}\quad \text{with}\quad \mathcal{L}_{s,\mathfrak{a}_{i_1}\mathfrak{a}_{i_2}\cdots \mathfrak{a}_{i_{n+1}}}:=\mathcal{L}_{s,\mathfrak{a}_{i_n}\mathfrak{a}_{i_{n+1}}}\cdots\mathcal{L}_{s,\mathfrak{a}_{i_2}\mathfrak{a}_{i_3}}\mathcal{L}_{s,\mathfrak{a}_{i_1}\mathfrak{a}_{i_2}}. \end{align*} $$

The Schwartz kernel of $\mathcal {L}_{s,\mathfrak {a}_{i_1}\mathfrak {a}_{i_2}\cdots \mathfrak {a}_{i_{n+1}}}$ is given by

$$ \begin{align*} \mathcal{L}_{s,\mathfrak{a}_{i_1}\mathfrak{a}_{i_2}\cdots \mathfrak{a}_{i_{n+1}}}(w,w')=\sum\limits_{k} (\mathfrak{a}_{i_1}\cdots \mathfrak{a}_{i_n})'(w)^s\phi_k^{\mathfrak{a}_{i_1}}(\mathfrak{a}_{i_1}\cdots \mathfrak{a}_{i_n}(w))\overline{\phi_k^{\mathfrak{a}_{i_1}}(w')},\quad w\in D_{\mathfrak{a}_{i_{n+1}}},\, w'\in D_{\mathfrak{a}_{i_1}}. \end{align*} $$

Its restriction to the diagonal is nonzero if and only if $\mathfrak {a}_{i_{n+1}}=\mathfrak {a}_{i_1}$ , since $w=w'$ only when $i_{n+1}=i_1$ . Set $\gamma =\mathfrak {a}_{i_1}\cdots \mathfrak {a}_{i_n}$ and $\mathcal {L}_{s,\gamma }:=\mathcal {L}_{s,\mathfrak {a}_{i_1}\cdots \mathfrak {a}_{i_{n+1}}}$ . So, $\mathcal {L}_{s,\gamma } u(w)=\gamma '(w)^s u(\gamma (w))$ . A change of variables $w=gw'$ gives the operator $(g)^*\circ \mathcal {L}_{s,\gamma }\circ (g^{-1})^*:\mathcal {H}(g^{-1}D_{\mathfrak {a}_{i_1}}) \to \mathcal {H}(g^{-1}D_{\mathfrak {a}_{i_{n+1}}})$ given by

$$ \begin{align*} (g)^*\circ\mathcal{L}_{s,\gamma}\circ (g^{-1})^*u(w')= \gamma'(gw')^s u(g^{-1}\gamma gw'). \end{align*} $$

Let $\gamma _1=g^{-1}\gamma g$ , and $\mathcal {L}_{s,\gamma _1}:\mathcal {H}(g^{-1}D_{\mathfrak {a}_{i_1}}) \to \mathcal {H}(g^{-1}D_{\mathfrak {a}_{i_{n+1}}})$ be given by $\mathcal {L}_{s,\gamma _1}u(w')=\gamma _1'(w)^su(\gamma _1w')$ . Then

$$ \begin{align*} (g)^*\circ\mathcal{L}_{s,\gamma}\circ (g^{-1})^*u(w')= \mathcal{B}^{-1}\mathcal{L}_{s,\gamma_1}\mathcal{B}\quad \text{with}\quad \mathcal{B} u(w')=(g'(w'))^su(w'). \end{align*} $$

Therefore, by equation (3.13), $\mathrm {tr}\mathcal {L}_{s,\gamma }=\mathrm {tr} \mathcal {L}_{s,\gamma _1}$ . So, we may choose the coordinates to make $\mathfrak {a}_{i_1}\cdots \mathfrak {a}_{i_n}$ a diagonal element so that $\mathfrak {a}_{i_1}\cdots \mathfrak {a}_{i_n}(w)=e^{-\ell (\mathfrak {a}_{i_1}\cdots \mathfrak {a}_{i_n})}w$ . Then

$$ \begin{align*} (\mathfrak{a}_{i_1}\cdots \mathfrak{a}_{i_n})'(w)^s=e^{-s\ell(\mathfrak{a}_{i_1}\cdots \mathfrak{a}_{i_n})}. \end{align*} $$

We may assume the center of $D_{\mathfrak {a}_{i_1}}$ is $0$ . Applying the definition of trace (3.10) to the basis $\phi _k^{\mathfrak {a}_{i_1}}(w)$ , we obtain

$$ \begin{align*} \mathrm{tr} \mathcal{L}_{s,\mathfrak{a}_{i_1}\cdots \mathfrak{a}_{i_{n+1}}} = \frac{e^{-s\ell(\mathfrak{a}_{i_1}\cdots \mathfrak{a}_{i_n})}}{\pi R_{\mathfrak{a}_{i_1}}^2} \sum\limits_{k} (k+1) \int_{|w|\leq R_{\mathfrak{a}_{i_1}}} \left(\frac{e^{-\ell(\mathfrak{a}_{i_1}\cdots \mathfrak{a}_{i_n})}w}{R_{\mathfrak{a}_{i_1}}}\right)^k\left(\frac{\bar{w}}{R_{\mathfrak{a}_{i_1}}}\right)^k dm(w)= \sum\limits_{k}e^{-(s+k)\ell(\mathfrak{a}_{i_1}\cdots \mathfrak{a}_{i_n})}. \end{align*} $$

Suppose the closed loop $\mathfrak {a}_{i_1}\cdots \mathfrak {a}_{i_n}=\mathbf {a}^m$ , where $\mathbf {a}$ is primitive. Then

$$ \begin{align*} \sum\limits_{n}\frac{1}{n} \mathrm{tr}(\mathcal{L}_s^n)=\sum\limits_{m}\frac{1}{m}\sum\limits_{[\mathbf{a}]\in\mathcal{P} , k} e^{-m(s+k)\ell({\mathbf{a}})}. \end{align*} $$

The proof is complete.

For further applications, we also introduce modified transfer operators that use only discs with a word length $\geq N$ . For $\mathbf {a}=\mathfrak {a}_{i_1}\cdots \mathfrak {a}_{i_n}\in \mathcal {W}^n$ , let

$$ \begin{align*} D_{\mathbf{a}}:=\mathfrak{a}_{i_1}\cdots \mathfrak{a}_{i_{n-1}}(D_{\mathfrak{a}_{i_n}}). \end{align*} $$

We define the modified transfer operator on $\mathcal {H}=\mathcal {H}_N:=\oplus _{\mathbf {b}\in \mathcal {W}^N} \mathcal {H}(D_{\mathbf {b}})$ :

(3.16) $$ \begin{align} \mathcal{L}_{s,N} u(w) :=\sum\limits_{\mathfrak{a}\neq \mathfrak{a}_{i_1}^{-1}} {\mathfrak{a}}'(w)^s u({\mathfrak{a}}(w)) \quad \text{for}\quad w\in D_{\mathbf{b}}\quad \text{with}\quad\mathbf{b}=\mathfrak{a}_{i_1}\cdots\mathfrak{a}_{i_N}. \end{align} $$

When $N=1$ , it agrees with the previous definition in equation (3.14). When $N\geq 2$ , the modified transfer operator acts on a space of functions with a larger domain. We can still use the proof of Lemma 3.10 to show that for any $N\geq 1$ ,

$$ \begin{align*} \det(1-\mathcal{L}_{s,N})=Z(\Gamma,s). \end{align*} $$

In the computation, we need to use fixed points. By taking larger N, the fixed point of a loxodromic element is still contained in only one disc. The computations are the same.

Now, we can state a much more precise estimate for the transfer operator.

Proposition 3.11. Suppose $\Gamma < \mathrm {SL}_2(\mathbb {R})$ is a Schottky group with the notations above. Suppose for some $A,B>0$ , $N\in \mathbb {Z}_{>0}$ , and $\varphi \in [0,\pi ]$ , we have for any $\mathfrak {a}\in \mathcal {W}$ and $\mathbf {b}=\mathfrak {a}_{i_1}\cdots \mathfrak {a}_{i_N}\in \mathcal {W}^N$ with $\mathfrak {a}_{i_1}\neq \mathfrak {a}^{-1}$ ,

• • $e^{-A}\leq |\mathfrak {a}'(w)|\leq e^{A}$ and $|\arg \mathfrak {a}'(w)|\leq \varphi \leq \pi $ for any $w\in D_{\mathbf {b}}$ ;

• • $\mathfrak {a}(D_{\mathbf {b}})\subset e^{-B}D_{\mathbf {a}}$ with $\mathbf {a}=\mathfrak {a}\hat {\mathbf {b}}$ .

Then the singular values of $\mathcal {L}_{s,N}$ satisfy

(3.17) $$ \begin{align} \mu_{\ell}({\mathcal{L}}_{s,N})\leq C(g,N,B)e^{(A|\Re s|+\varphi |\Im s|-B\ell)/C(g,N)}\sqrt{\ell+1}. \end{align} $$

Proof. The transfer operator $\mathcal {L}_{s,N}$ is the direct sum of operators

$$ \begin{align*}\mathcal{L}_{s,\mathbf{ab}}:\mathcal{H}(D_{\mathbf{a}})\to \mathcal{H}(D_{\mathbf{b}})\quad \text{given by}\quad{\mathcal{L}}_{s,\mathbf{ab}}u(w)={\mathfrak{a}}'(w)^{s}u({\mathfrak{a}}(w)),\end{align*} $$

where $\mathfrak {a}\in \mathcal {W}$ and $\mathbf {b}=\mathfrak {a}_{i_1}\cdots \mathfrak {a}_{i_N}\in \mathcal {W}^N$ with $\mathfrak {a}_{i_1}\neq \mathfrak {a}^{-1}$ and $\mathbf {a}=\mathfrak {a}\hat {\mathbf {b}}$ . We have

• • $|{\mathfrak {a}}'(w)^s|\leq \max (|{\mathfrak {a}}'(w)|^{\Re s},|{\mathfrak {a}}'(w)|^{-\Re s})|\exp ({|\arg {\mathfrak {a}}'(w)||\Im s|})\leq e^{A|\Re s|+\varphi |\Im s|}$ ;

• • By Lemma 3.9, the singular values of the pullback operator $\mathfrak {a}^*:\mathcal {H}(D_{\mathbf {a}})\to \mathcal {H}(D_{\mathbf {b}})$ satisfy

  $$ \begin{align*} \mu_{\ell}({\mathfrak{a}}^*)\leq C(1-e^{-B})^{-2}e^{-B\ell}\sqrt{\ell+1}. \end{align*} $$

Then the proposition follows from equation (3.9) and equation (3.12).

Recall from Lemma 2.23 that, for a family of Schottky groups satisfying condition $(\bigstar )$ , we have a uniform Schottky basis.

Proposition 3.12. Suppose we have a degenerating family of Schottky groups $\Gamma _z<\mathrm {SL}_2(\mathbb {R})$ satisfying $(\bigstar )$ with their Schottky discs uniformly bounded away from $\infty $ . For any $N\geq 1$ , $\mathfrak {a}\in \mathcal {W}$ , $\mathbf {b}=\mathfrak {a}_{i_1}\cdots \mathfrak {a}_{i_N}\in \mathcal {W}^N$ with $\mathfrak {a}\neq \mathfrak {a}_{i_1}^{-1}$ , and sufficiently small $0<|z| \ll _N 1$ , we have for any $w\in D_{\mathbf {b},z}$ , and the disc $D_{\mathbf {a},z}$ with $\mathbf {a}=\mathfrak {a}\hat {\mathbf {b}}$ ,

(3.18) $$ \begin{align} & |\log|{\mathfrak{a}}'(w)||\lesssim_{N} \log(1/|z|),\nonumber\\ & \mathfrak{a}(D_{\mathbf{b},z})\subset \frac{1}{10}D_{\mathbf{a},z}, \end{align} $$

in terms of the absolute norm. Moreover, for any $M>0$ , there exists $N(M)\geq 1$ such that if we require $\mathbf {b}\in \mathcal {W}^{N(M)}$ additionally, then

$$ \begin{align*} \quad |\arg \mathfrak{a}'(w)|\lesssim_M |z|^M. \end{align*} $$

Consequently, given any $C,M>0$ , the family of zeta functions $Z(\Gamma _z,s/\log (1/|z|))$ with $0<|z|<1/e$ is uniformly bounded (depending on C and M) in the region

$$ \begin{align*} |\Re s|\leq C, \quad |\Im s|\leq C|z|^{-M}\log(1/|z|). \end{align*} $$

Proof. By assumption, $\mathfrak {a}\in \mathrm {SL}_2(\mathbb {M}(\mathbb {D}))$ , so we can write $\mathfrak {a}=\begin {pmatrix} a& b\\ c&d\end {pmatrix}$ with $a,b,c,d\in \mathbb {M}(\mathbb {D})$ . Fix any $0<|z|\ll 1$ . The uniform separation $\mathfrak {a}(D_{\mathbf {b},z})\subset \frac {1}{10}D_{\mathbf { a},z}$ follows from Corollary 3.23.

For any $w\in D_{\mathfrak {b},z}$ , we have ${\mathfrak {a}}(w)=\frac {a_zw+b_z}{c_zw+d_z}$ , where the lower subscript means to evaluate the functions at z. Then ${\mathfrak {a}}'(w)=1/(c_zw+d_z)^2$ . The estimate $|z|^A\leq |{\mathfrak {a}}'(w)|$ for some $A>0$ follows from the fact that the Schottky discs are uniformly bounded away from $\infty $ . The other estimate $|{\mathfrak {a}}'(w)|\leq |z|^{-A}$ follows from the definition of uniform Schottky discs (2.4), and the convention (2.5) gives

$$\begin{align*}\inf_{w\in D_{\mathbf{{b},z}}}|w+d_z/c_z|\geq d(\partial D_{\mathfrak{a}^{-1},z}, \frak{a}^{-1}\infty)\geq \mathrm{diam}(D_{\mathfrak{a}^{-1},z})/2.\end{align*}$$

Finally, it follows from Lemma 3.3 that by taking N sufficiently large, we can ensure $\text {diam}(D_{\mathbf {b},z})=O(|z|^{cN})$ . The estimate $|\arg \mathfrak {a}'(w)|\leq |z|^M$ for $w\in D_{\mathbf {b},z}$ follows from the observation that

$$ \begin{align*} \mathfrak{a}'(w)=(c_zw+d_z)^{-2}=c_z^{-2}((\Re w+d_z/c_z)+i\Im w)^{-2} \end{align*} $$

and $D_{\mathbf {b},z}\cap \mathbb {R}\neq \emptyset $ .

For the boundedness of $Z(\Gamma _z,s/\log (1/|z|))$ , it suffices to assume $0<|z|\ll _N 1$ , since otherwise, the statement follows from the continuity of the zeta function. Then the hypothesis of Proposition 3.11 is satisfied with $A=A_0\log (1/|z|)$ , where $A_0$ is fixed and $\varphi =O(|z|^M)$ . We can apply the conclusion of Proposition 3.11 and equation (3.11) to obtain the boundedness of $Z(\Gamma _z,s/\log (1/|z|)$ .

3.2.3 Transfer operator: $\mathrm {SL}_2(\mathbb {C})$ case

Let $\Gamma $ be a classical Schottky group in $\mathrm {SL}_2(\mathbb {C})$ of rank $g\geq 2$ . The discussion is similar but has several complications, which we detail below.

Write $\Gamma =\langle \mathfrak {a}_{1}, \mathfrak {a}_{2},\cdots , \mathfrak {a}_{{2g}} \rangle $ with $\mathfrak {a}_{i}\mathfrak {a}_{i+g}=id$ for $i=1,2,\cdots , g$ . By conjugating $\Gamma $ if necessary, there exist open discs $D_{\mathfrak {a}_1},D_{\mathfrak {a}_2},\cdots , D_{\mathfrak {a}_{2g}}\subset \mathbb {C}$ such that their closures are disjoint, and for each $\mathfrak {a}\in \{\mathfrak {a}_1,\ldots , \mathfrak {a}_{2g}\}$ , we have $\mathfrak {a}(\hat {\mathbb {C}}-\overline {D_{\mathfrak {a}^{-1}}}) = D_{\mathfrak {a}}$ .

To study the Selberg zeta function, we identify $\mathbb {C}$ with $\mathbb {R}^2$ , and given $\gamma =\begin {pmatrix} a& b\\ c& d\end {pmatrix}\in \mathrm {SL}_2(\mathbb {C})$ , complexify the linear fractional transformation $\gamma :\mathbb {R}^2\to \mathbb {R}^2$ :

(3.19) $$ \begin{align} \gamma(x,y)=\left(\Re\frac{a(x+iy)+b}{c(x+iy)+d}, \Im \frac{a(x+iy)+b}{c(x+iy)+d}\right),\quad (x,y)\in \mathbb{R}^2 \end{align} $$

to

(3.20) $$ \begin{align} \widetilde{\gamma}(w_1,w_2)=\frac{1}{2}\left(\frac{a(w_1+iw_2)+b}{c(w_1+iw_2)+d}+\frac{\bar{a}(w_1-iw_2)+\bar{b}}{\bar{c}(w_1-iw_2)+\bar{d}},-i\frac{a(w_1+iw_2)+b}{c(w_1+iw_2)+d}+i\frac{\bar{a}(w_1-iw_2)+\bar{b}}{\bar{c}(w_1-iw_2)+\bar{d}}\right) \end{align} $$

with $(w_1,w_2)\in \mathbb {C}^2$ .

For each generator, suppose $D_{\mathfrak {a}}=\{|w-w_{\mathfrak {a}}|\leq R_{\mathfrak {a}}\}$ . Let $\widetilde {D}_{\mathfrak {a}}$ be the polydisc defined by

$$ \begin{align*} \widetilde{D}_{\mathfrak{a}}=\{(w_1,w_2)\in \mathbb{C}^2 : \max(|w_1-\Re w_{\mathfrak{a}}|,|w_2-\Im w_{\mathfrak{a}}|)\leq R_{\mathfrak{a}}/3\}, \end{align*} $$

and we also view $D_{\mathfrak {a}}\cap \widetilde {D}_{\mathfrak {a}}$ as a subset of $\widetilde {D}_{\mathfrak {a}}$ .

We introduce the Hilbert space $\widetilde {\mathcal {H}}=\oplus _{i=1}^{2g}\mathcal {H}(\widetilde {D}_{\mathfrak {a}_i})$ , where $\mathcal {H}(\widetilde {D}_{\mathfrak {a}_i})=\{u\in L^2(\widetilde {D}_{\mathfrak {a}_i}): u \text { is holomorphic}\}$ with the norm $\|u\|_{\mathcal {H}(\widetilde {D}_{\mathfrak {a}_i})}^2:=|\widetilde {D}_{\mathfrak {a}_i}|^{-1}\|u\|_{L^2(\widetilde {D}_{\mathfrak {a}_i})}^2$ . Suppose $\mathfrak {a}_i$ maps $\widetilde {D}_{\mathfrak {b}}$ into a compact subset of $\widetilde {D}_{\mathfrak {a}_i}$ for any $\mathfrak {b}\neq \mathfrak {a}_i^{-1}$ . The transfer operator $\mathcal {L}_s:\widetilde {\mathcal {H}}\to \widetilde {\mathcal {H}}$ is defined as

(3.21) $$ \begin{align} \mathcal{L}_s u(w_1,w_2):=\sum\limits_{\mathfrak{a}_i\neq \mathfrak{b}^{-1}} [\mathfrak{a}_i'(w_1,w_2)]^s u(\widetilde{\mathfrak{a}}_i(w_1,w_2)) \quad \text{for}\quad (w_1,w_2)\in \widetilde{D}_{\mathfrak{b}}\quad \text{with}\quad \mathfrak{b}\in \{\mathfrak{a}_1,\ldots,\mathfrak{a}_{2g}\}. \end{align} $$

Here, $[\mathfrak {a}^{\prime }_i(w_1,w_2)]$ is a holomorphic function in $\widetilde {D}_{\mathfrak {b}}$ such that $[\mathfrak {a}^{\prime }_i(w_1,w_2)]= |\mathfrak {a}^{\prime }_i(x+iy)|$ for $(w_1,w_2)=(x,y)\in D_{\mathfrak {b}}$ . More precisely, if $\mathfrak {a}_i=\begin {pmatrix} a& b\\ c& d\end {pmatrix}\in \mathrm {SL}_2(\mathbb {C})$ , then

(3.22) $$ \begin{align} [{\mathfrak{a}}^{\prime}_i(w_1,w_2)]=\frac{1}{(c(w_1+iw_2)+d)(\bar{c}(w_1-iw_2)+\bar{d})}. \end{align} $$

Equation (3.21) requires choosing a branch of the complex logarithm such that $\log [\mathfrak {a}^{\prime }_i(w_1,w_2)]$ is a holomorphic function on $\widetilde {D}_{\mathfrak {b}}$ satisfying $\log [\mathfrak {a}^{\prime }_i(w_1,w_2)]=\log |\mathfrak {a}^{\prime }_i(x+iy)|$ for $(w_1,w_2)=(x,y)\in D_{\mathfrak {b}}\cap \widetilde {D}_{\mathfrak {b}}$ . This is achievable because the inequality $|w_{\mathfrak {b}}+d/c|>R_{\mathfrak {b}}$ gives $|w_1+iw_2+d/c|>R_{\mathfrak {b}}/3$ and hence $[\mathfrak {a}^{\prime }_i(w_1,w_2)]\neq 0$ for $(w_1,w_2)\in \widetilde {D}_{\mathfrak {b}}$ . Define $[\mathfrak {a}^{\prime }_i(w_1,w_2)]^s:=\exp ( s \log [\mathfrak {a}^{\prime }_i(w_1,w_2)])$ on $\widetilde {D}_{\mathfrak {b}}$ .

We show that the pullback operator $\widetilde {\gamma }^*$ is again a trace-class operator.

Lemma 3.13. Suppose $\widetilde {\gamma }:\widetilde {D}_{\mathfrak {b}} \to \rho \widetilde {D}_{\mathfrak {a}}$ is a holomorphic function between polydiscs in $\mathbb {C}^2$ with $0<\rho <1$ and $\mathfrak {a}, \mathfrak {b}\in \{\mathfrak {a}_1,\ldots , \mathfrak {a}_{2g}\}$ . Then the pullback operator

$$ \begin{align*} \widetilde{\gamma}^*u(w_1,w_2)=u(\widetilde{\gamma}(w_1,w_2)): \mathcal{H}(\widetilde{D}_{\mathfrak{a}})\to \mathcal{H}(\widetilde{D}_{\mathfrak{b}}) \end{align*} $$

is a trace-class operator satisfying

$$ \begin{align*} \mu_{\ell}(\widetilde{\gamma}^*)\leq C_{\rho}\rho^{\sqrt{\ell}}(\ell+1). \end{align*} $$

In the following, we will use the multiindex notation:

$$ \begin{align*} \alpha=(\alpha_1,\alpha_2)\in \mathbb{Z}^2_{\geq 0}, \quad |\alpha|=\alpha_1+\alpha_2, \quad \text{and}\quad \alpha!=\alpha_1!\alpha_2!. \end{align*} $$

Proof. We may assume $\widetilde {D}_{\mathfrak {a}}:=\{(w_1,w_2)\in \mathbb {C}^2:|w_1|<R_1, |w|_2<R_2\}$ . Consider the orthonormal basis $\phi _{\alpha }(w_1,w_2):=\sqrt {(\alpha _1+1)(\alpha _2+1))}(w_1/R_1)^{\alpha _1}(w_2/R_2)^{\alpha _2}$ of $\mathcal {H}(\widetilde {D}_{\mathfrak {a}})$ . Then

$$ \begin{align*} \|\widetilde{\gamma}^*\phi_\alpha\|_{\mathcal{H}(\widetilde{D}_{\mathfrak{b}})}^2&=|\widetilde{D}_{\mathfrak{b}}|^{-1}\int_{\widetilde{D}_{\mathfrak{b}}}(\alpha_1+1)(\alpha_2+1) \frac{|\pi_1(\gamma (w_1,w_2))|^{2\alpha_1}}{R_1^{2\alpha_1}} \frac{|\pi_2(\gamma(w_1,w_2))|^{2\alpha_2}}{R_2^{2\alpha_2}} dm(w_1,w_2)\\&\leq (|\alpha|+1)^2\rho^{2|\alpha|}, \end{align*} $$

where $\pi _i:\mathbb {C}^2\to \mathbb {C}$ is to project on the i-th coordinate. By the minimax property (3.8), we conclude

$$ \begin{align*} \mu_{\ell}(\widetilde{\gamma}^*)\leq\sum\limits_{|\alpha|^2\geq \ell-10}\|\widetilde{\gamma}^*\phi_\alpha\|_{\mathcal{H}(\widetilde{D}_{\mathfrak{b}})}\leq \sum\limits_{k^2\geq \ell-10}(k+1)^{2}\rho^{k}\leq C_{\rho}\rho^{\sqrt{\ell}}(\ell+1). \end{align*} $$

Thus,

$$ \begin{align*} \sum\limits_{\ell}\mu_{\ell}(\widetilde{\gamma}^*)\leq C_{\rho}\sum\limits_{\ell} \rho^{\sqrt{\ell}}(\ell+1)<\infty, \end{align*} $$

which implies $\widetilde {\gamma }^*$ is a trace-class operator.

Similar to the $\mathrm {SL}_2(\mathbb {R})$ case, we have

$$ \begin{align*} \det(1-\mathcal{L}_s)=Z(\Gamma,s)\quad \text{for any} \quad s\in \mathbb{C}. \end{align*} $$

We refer to [Reference Guillopé, Lin and Zworski14] for more details.

As in the $\mathrm {SL}_2(\mathbb {R})$ case, we also consider the modified transfer operators to obtain further applications. The precise construction is as follows. For $\mathbf {a}=\frak {a}_{i_1}\cdots \frak {a}_{i_n}\in \mathcal {W}^n$ , define the polydisc $\widetilde {D}_{\mathbf {a}}$ in $\mathbb {C}^2$ associated with $D_{\mathbf {a}}=D(w_{\mathbf {a}},R_{\mathbf {a}})\subset \mathbb {C}$ by

$$ \begin{align*} \widetilde{D}_{\mathbf{a}}=\{(w_1,w_2): \max(|w_1-\Re w_{\mathbf{a}}|, |w_2-\Im w_{\mathbf{a}}|)<R_{\mathbf{a}}/3\}. \end{align*} $$

For any $N\in \mathbb {N}$ , the modified transfer operator $\mathcal {L}_{s,N}$ on $\mathcal {H}=\mathcal {H}_N:=\oplus _{b\in \mathcal {W}^N}\mathcal {H}(\widetilde {D}_{\mathbf {b}})$ is defined by

$$ \begin{align*} \mathcal{L}_{s,N} u(w_1,w_2) :=\sum\limits_{\mathfrak{a}\neq \bar{\mathfrak{a}}_{i_1}} [{\mathfrak{a}}'(w_1,w_2)]^s u(\widetilde{{\mathfrak{a}}}(w_1,w_2)) \quad \text{for}\quad (w_1,w_2)\in \widetilde{D}_{\mathbf{b}}\quad\text{with}\quad \mathbf{b}=\mathfrak{a}_{i_1}\cdots\mathfrak{a}_{i_N}. \end{align*} $$

Again, we have for any $N\in \mathbb {N}$ ,

$$ \begin{align*} \det(1-\mathcal{L}_{s,N})=Z(\Gamma,s) \quad \text{for any}\quad s\in \mathbb{C}. \end{align*} $$

We state a proposition similar to Proposition 3.11 for $\mathrm {SL}_2(\mathbb {C})$ .

Proposition 3.14. Suppose $\Gamma <\mathrm {SL}_2(\mathbb {C})$ is a Schottky group with the notations above. Suppose for some $A,B>0$ , $N\in \mathbb {Z}_{>0}$ , and $\varphi \in [0,\pi ]$ , we have for any $\frak {a}\in \mathcal {W}$ and $\mathbf {b}=\frak {a}_{i_1}\cdots \frak {a}_{i_N}\in \mathcal {W}^N$ with $\frak {a}_{i_1}\neq \frak {a}^{-1}$ ,

• • $e^{-A}\leq |[\mathfrak {a}'(w_1,w_2)]|\leq e^{A}$ and $|\arg [\mathfrak {a}'(w_1,w_2)]|\leq \varphi \leq \pi $ for any $(w_1,w_2)\in \widetilde {D}_{\mathbf {b}}$ ;

• • $\widetilde {\mathfrak {a}}(\widetilde {D}_{\mathbf {b}})\subset e^{-B}\widetilde {D}_{\mathbf {a}}$ , where $\mathbf {a}=\mathfrak {a}\hat {\mathbf {b}}$ .

Then the singular values of $\mathcal {L}_s$ satisfy

$$ \begin{align*} \mu_{\ell}(\mathcal{L}_{s,N})\leq C(g,N,B)e^{(A|\Re s|+\varphi |\Im s|-B\ell^{1/2})/C(g,N)}(\ell+1). \end{align*} $$

Consequently, the operator $\mathcal {L}_{s,N}$ is a trace-class operator.

Proof. The proof is similar to that of Proposition 3.11. The operator $\mathcal {L}_{s,N}$ is a direct sum of the following operators: Consider each component of $\mathcal {L}_{s,N}$ :

$$ \begin{align*} \mathcal{L}_{s,\mathbf{ab}}:\mathcal{H}(\widetilde{D}_{\mathbf{a}})\to \mathcal{H}(\widetilde{D}_{\mathbf{b}}),\quad \mathcal{L}_{s, \mathbf{ab}}u(w_1,w_2):=[\mathfrak{a}'(w_1,w_2)]^s u(\widetilde{\mathfrak{a}}(w_1,w_2)),\quad \mathbf{a}=\mathfrak{a}\hat{\mathbf{b}}, \end{align*} $$

where $\frak {a}\in \mathcal {W}$ , $\mathbf {b}=\frak {a}_{i_1}\cdots \frak {a}_{i_N}\in \mathcal {W}^N$ with $\frak {a}_{i_1}\neq \frak {a}^{-1}$ , and $\mathbf {a}=\frak {a}\hat {\mathbf {b}}$ .

We have

• • $|[\mathfrak {a}'(w_1,w_2)]^s|\leq \max \left (|[\mathfrak {a}'(w_1,w_2)]|^{\Re s},|[\mathfrak {a}'(w_1,w_2)]|^{-\Re s}\right )\exp ({|\arg [{\mathfrak {a}}'(w_1,w_2)]||\Im s|})\\ \leq e^{A|\Re s|+\varphi |\Im s|}$ ;

• • By Lemma 3.13, we know

  $$ \begin{align*} \mu_{\ell}(\widetilde{\mathfrak{a}}^*)\leq C(B)e^{-B\ell^{1/2}}(\ell+1). \end{align*} $$

Then the proposition follows from equation (3.9) and equation (3.12).

As in Proposition 3.12, we have the following.

Proposition 3.15. Suppose we have a degenerating family of Schottky groups $\Gamma _z<\mathrm {SL}_2(\mathbb {C})$ satisfying $(\bigstar )$ with their Schottky discs uniformly bounded away from $\infty $ . For $N\geq 1$ , $\frak {a}\in \mathcal {W}$ , $\mathbf {b}=\frak {a}_{i_1}\cdots \frak {a}_{i_N}\in \mathcal {W}^N$ with $\frak {a}\neq \frak {a}_{i_1}^{-1}$ , and sufficiently small $0<|z|\ll _N 1$ , we have for any $(w_1,w_2)\in \widetilde {D}_{\mathbf {b},z}$ , and the disc $\widetilde {D}_{\mathbf {a},z}$ with $\mathbf {a}=\frak {a}\hat {\mathbf {b}}$ ,

(3.23) $$ \begin{align} &\left|\log|[\mathfrak{a}'(w_1,w_2)]|\right|\lesssim \log(1/|z|),\nonumber\\ &\widetilde{\mathfrak{a}}(\widetilde{D}_{\mathbf{b},z})\subset \frac{1}{10}\widetilde{D}_{\mathbf{a},z}, \end{align} $$

in terms of the absolute norm. Moreover, for any $M>0$ , there exists $N(M)\geq 1$ such that if we require $\mathbf {b}\in \mathcal {W}^{N(M)}$ additionally, then

$$ \begin{align*} |\arg [\frak{a}'(w_1,w_2)]|\lesssim |z|^M. \end{align*} $$

Consequently, given any $C,M>0$ , the family of zeta functions $Z(\Gamma _z,s/\log (1/|z|))$ with $0<|z|<1/e$ is uniformly bounded (depending on C and M) in the region

$$ \begin{align*} |\Re s|\leq C,\quad |\Im s|\leq C|z|^{-M}\log(1/|z|). \end{align*} $$

Proof. By Corollary 3.23, there exists $v>0$ such that for any $N\in \mathbb {N}$ , $\frak {a}\in \mathcal {W}$ , $\mathbf {b}=\frak {a}_{i_1}\cdots \frak {a}_{i_N}\in \mathcal {W}^N$ with $\frak {a}\frak {a}_{i_1}\neq id$ , and sufficiently small $0<|z|\ll 1$ , we have

(3.24) $$ \begin{align} \mathfrak{a}(D_{\mathbf{b},z})\subset |z|^{v} D_{\mathbf{a},z}, \end{align} $$

where $\mathbf {a}=\frak {a}\hat {\mathbf {b}}$ .

Fix $0<|z|\ll 1$ . We first show equation (3.23). Suppose $D_{\mathbf {a},z}=D(w_{\mathbf {a},z},R_{\mathbf {a},z})$ and $D_{\mathbf {b},z}=D(w_{\mathbf {b},z},R_{\mathbf {b},z})$ . Then $|\mathfrak {a}(w)-w_{\mathfrak {a},z}|\leq |z|^{v}R_{\mathbf {a},z}$ for $w\in D_{\mathbf {b},z}$ . Using the Cauchy integral formula, we estimate the derivatives of $\frak a$ on $D_{\mathbf {b},z}$ at $w_{\mathbf {b},z}$ : for any $k\in \mathbb {N}$ ,

$$ \begin{align*} |\frac{d^k}{dw^k} \mathfrak{a}(w_{\mathbf{b},z})|=k!\left|\int_{\partial D_{\mathbf{b},z}}\frac{\frak{a}(w)}{(w-w_{\mathbf{b},z})^{k+1}} dw\right|=k!\left|\int_{\partial D_{\mathbf{b},z}}\frac{\frak{a}(w)-w_{\mathbf{a},z}}{(w-w_{\mathbf{b},z})^{k+1}} dw \right| \leq 2\pi\frac{k!}{R_{\mathbf{b},z}^k}|z|^{v}R_{\mathbf{a},z}. \end{align*} $$

We can then form the analytic continuation

$$ \begin{align*} \begin{aligned} &\widetilde{\mathfrak{a}}(w_1,w_2)=\mathfrak{a}(w_{\mathbf{b},z})\\ & \quad + \sum\limits_{\substack{\alpha\in\mathbb{Z}^2_{\geq 0}\\\text{with}\,\,|\alpha|>0}}\frac{1}{\alpha!}(w_1-\Re w_{\mathbf{b},z})^{\alpha_1}(w_2-\Im w_{\mathbf{b},z})^{\alpha_2}\left(\partial_x^{\alpha_1}\partial_y^{\alpha_2}\Re \mathfrak{a}(x+iy), \partial_x^{\alpha_1}\partial_y^{\alpha_2}\Im{\mathfrak{a}}(x+iy)\right)|_{x+iy=w_{\mathbf{b},z}}\in \mathbb{C}^2. \end{aligned} \end{align*} $$

Combining the above two equations, we obtain that for any $(w_1,w_2)\in \widetilde {D}_{\mathbf {b},z}$ ,

$$ \begin{align*} &|\widetilde{\mathfrak{a}}(w_1,w_2)-w_{\mathbf{a},z}|\leq |z|^vR_{\mathbf{a},z}+4\pi\sum\limits_{\alpha}\frac{|\alpha|!}{\alpha!R_{\mathbf{b}}^{|\alpha|}} |z|^{v}R_{\mathbf{a},z}(R_{\mathbf{b},z}/3)^{|\alpha|}\\&\quad =|z|^vR_{\mathbf{a},z}+4\pi\sum\limits_{k} \left(\frac{2}{3}\right)^k |z|^{v}R_{\mathbf{a},z}\leq (8\pi+1)|z|^{v}R_{\mathbf{a},z}. \end{align*} $$

Thus, we have equation (3.23).

Next, we estimate $\left |\log |[\frak {a}'(w_1,w_2)]| \right |$ . We write $\mathfrak {a}=\begin {pmatrix}a &b\\c&d\end {pmatrix}$ with $a,b,c,d\in \mathbb {M}(\mathbb {D})$ . For any $w\in D_{\frak {b},z}$ , we have $\mathfrak {a}(w)=\frac {a_zw+b_z}{c_zw+d_z}$ and $\mathfrak {a}'(w)=1/(c_zw+d_z)^2$ , where the lower subscript means to evaluate the functions at z. By equation (3.22), we have for any $(w_1,w_2)\in \widetilde {D}_{\frak {b},z}$ ,

(3.25) $$ \begin{align} [\mathfrak{a}'(w_1,w_2)]&=|c_z|^{-2} (w_1+iw_2-w_{\frak{b},z}+w_{\frak{b},z}+d_z/c_z)^{-1} (w_1-iw_2-\bar{w}_{\mathbf{b},z}+\bar{w}_{\mathbf{b},z}+\bar{d}_z/\bar{c}_z)^{-1}\nonumber\\ &=|\frak{a}'(w_{\mathbf{b},z})|^2 \left(1+\frac{w_1+iw_2-w_{\mathbf{b},z}}{w_{\mathbf{b},z}+d_z/c_z}\right)^{-1} \left(1+\frac{w_1-iw_2-\bar{w}_{\mathbf{b},z}}{\bar{w}_{\mathbf{b},z}+\bar{d}_z/\bar{c}_z}\right)^{-1}. \end{align} $$

Using $|w_{\mathfrak {b},z}+d_z/c_z|>R_{\mathfrak {b},z}$ , and $\max (|w_1-\Re w_{\mathfrak {b}_z}|,|w_2-\Im w_{\mathfrak {b},z}|)\leq R_{\mathfrak {b},z}/3$ , we have that the absolute values of both the second and third factors are bounded above by $5/3$ and below by $1/3$ . Using the argument of the $\mathrm {SL}_2(\mathbb {R})$ case (proof of Proposition 3.12), we have that there exists $A>0$ , such that $|z|^{A}\leq |\frak {a}'(w_{\mathbf {b},z})|\leq |z|^{-A}$ . Hence, we have established the estimate for $\left |\log |[\frak {a}'(w_1,w_2)]|\right |$ .

We estimate $|\arg [\frak {a}'(w_1,w_2)]|$ . It follows from Corollary 3.23 that $D_{\mathbf {b},z}\subset |z|^vD_{\frak {a}_{i_1},z}$ and ${R_{\mathbf {b},z}\leq |z|^{vN}R_{\frak {a}_{i_1},z}}$ . Then, we have

$$ \begin{align*} \frac{|w_1+iw_2-w_{\mathbf{b},z}|}{|w_{\mathbf{b},z}+d_z/c_z|}\leq \frac{2}{3}|z|^v(1-|z|^v)^{-1}, \end{align*} $$

which yields the estimate $|\arg [\frak {a}'(w_1,w_2)]|\lesssim |z|^{M}$ using equation (3.25).

Finally, we use Proposition 3.14 to obtain the boundedness of the zeta functions.