The number of automorphic representations of $\mathrm{GL}_{2}$ with exceptional eigenvalues

Dohoon Choi , Min Lee , Youngmin Lee and Subong Lim Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, Republic of Korea [email protected] School of Mathematics, University of Bristol, Bristol BS8 1TW, U.K. [email protected] School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea [email protected] Department of Mathematics Education, Sungkyunkwan University, Jongno-gu, Seoul 03063, Republic of Korea [email protected]

Abstract.

We obtain an upper bound for the dimension of the cuspidal automorphic forms for $\mathrm{GL}_{2}$ over a number field, whose archimedean local representations are not tempered. More precisely, we prove the following result.

Let $F$ be a number field and $\mathbb{A}_{F}$ be the ring of adeles of $F$ . Let $\mathcal{O}_{F}$ be the ring of integers of $F$ . Let $\mathfrak{X}_{F,\mathrm{ex}}$ be the set of irreducible cuspidal automorphic representations $\pi$ of $\mathrm{GL}_{2}(\mathbb{A}_{F})$ with the trivial central character such that for each archimedean place $v$ of $F$ , the local representation of $\pi$ at $v$ is an unramified principal series and is not tempered. For an ideal $J$ of $\mathcal{O}_{F}$ , let $\mathrm{K}_{0}(J)$ be the subgroup of $\mathrm{GL}_{2}(\mathbb{A}_{F})$ corresponding to $\Gamma_{0}(J)\subset\mathrm{SL}_{2}(\mathcal{O}_{F})$ . Let $r_{1}$ be the number of real embeddings of $F$ and $r_{2}$ be the number of conjugate pairs of complex embeddings of $F$ . Using the Arthur-Selberg trace formula, we have

\sum_{\pi\in\mathfrak{X}_{F,\mathrm{ex}}}\dim\pi^{\mathrm{K}_{0}(J)}\ll_{F}\frac{[\mathrm{SL}_{2}(\mathcal{O}_{F}):\Gamma_{0}(J)]}{(\log(N_{F/\mathbb{Q}}(J)))^{2r_{1}+3r_{2}}}\quad\text{ as }\quad|N_{F/\mathbb{Q}}(J)|\to\infty.

From this result, we obtain the result on an upper bound for the number of Hecke-Maass cusp forms of weight $0$ on $\Gamma_{0}(N)$ which do not satisfy the Selberg eigenvalue conjecture.

2020 Mathematics Subject Classification:

11F72 (Primary) ; 11F12 (Secondary)

Keywords : Selberg eigenvalue conjecture, Arthur-Selberg trace formula

1. Introduction

Let $\Gamma$ be a congruence subgroup of $\mathrm{SL}_{2}(\mathbb{Z})$ and $f$ be a Maass cusp form of weight $0$ on $\Gamma$ . Let $\Delta:=-y^{2}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right)$ be the hyperbolic Laplace operator on the upper half plane $\mathbb{H}$ , and $\lambda_{f}$ be the eigenvalue of $\Delta$ on $f$ . In [19], Selberg proved that $\lambda_{f}\geq\frac{3}{16}$ , and asserted that $\lambda_{f}\geq\frac{1}{4}$ . It is called the Selberg eigenvalue conjecture. The best known lower bound for $\lambda_{f}$ is that

\lambda_{f}\geq\frac{1}{4}-\left(\frac{7}{64}\right)^{2},

which was proved by Kim and Sarnak [15]. This conjecture has been proven for some congruence subgroups (see [2, 3, 11]).

For $\lambda\in\mathbb{R}$ , the space $V_{\lambda}$ of Maass cusp forms of weight $0$ on $\Gamma$ with the Laplacian eigenvalue $\lambda$ is a finite-dimensional vector space over $\mathbb{C}$ . If $\lambda_{f}<\frac{1}{4}$ , then we call $\lambda_{f}$ an exceptional eigenvalue, and its multiplicity is equal to the dimension of $V_{\lambda_{f}}$ over $\mathbb{C}$ . The number of exceptional eigenvalues with multiplicities also has been studied in various aspects, as shown in [10, 11, 12, 14]. For $\Gamma=\Gamma_{0}(N)$ with a positive integer $N$ , Iwaniec and Szmidt [14] and Huxley [11] independently proved that if $e=2$ , then the number of exceptional eigenvalues $\lambda_{f}$ with multiplicities satisfying $0<\lambda_{f}<\frac{1}{4}-\delta^{2}$ is less than $C_{\epsilon}\cdot N^{1-e\delta+\epsilon}$ for some constant $C_{\epsilon}$ depending on $\epsilon$ with $\epsilon>0$ . Later, Iwaniec [12] proved that it holds for $e=4$ .

Let $F$ be a number field and $\mathbb{A}_{F}$ be the ring of adeles of $F$ . In this paper, we consider the number of Maass cusp forms of $\mathrm{GL}_{2}(\mathbb{A}_{F})$ with exceptional eigenvalues. The Selberg eigenvalue conjecture for Maass cusp forms of $\mathrm{GL}_{2}(\mathbb{A}_{F})$ can be stated via automorphic representations as follows. Let $\mathfrak{X}_{F}$ denote the set of irreducible cuspidal automorphic representations $\pi\cong\otimes_{v}\pi_{v}$ of $\mathrm{GL}_{2}(\mathbb{A}_{F})$ with the trivial central character such that the local representation $\pi_{v}$ of $\pi$ at each archimedean place $v$ of $F$ is an unramified principal series. When $F=\mathbb{Q}$ and $\Gamma$ is a congruence subgroup of $\mathrm{SL}_{2}(\mathbb{Z})$ , there is a correspondence between Maass cusp forms of weight $0$ on $\Gamma$ and the fixed vectors under $\Gamma_{\mathbb{A}_{\mathbb{Q}}}$ of automorphic representations $\pi\in\mathfrak{X}_{\mathbb{Q}}$ . Here, $\Gamma_{\mathbb{A}_{\mathbb{Q}}}$ is the subgroup of $\mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$ corresponding to $\Gamma$ . For a Maass cusp form $f$ of weight $0$ on $\Gamma$ , let $\pi_{f}\in\mathfrak{X}_{\mathbb{Q}}$ be the irreducible cuspidal automorphic representation of $\mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$ corresponding to $f$ . By the $(\mathfrak{g},K)$ -module theory due to Harish-Chandra, $\lambda_{f}\geq\frac{1}{4}$ if and only if $(\pi_{f})_{\infty}$ is tempered. Thus, the Selberg eigenvalue conjecture is equivalent to that $(\pi_{f})_{\infty}$ is tempered for every Maass cusp form $f$ of weight $0$ on $\Gamma$ . Moreover, the number of exceptional eigenvalues with multiplicities is equal to the sum of $\dim\pi_{f}^{\Gamma_{\mathbb{A}_{\mathbb{Q}}}}$ with respect to $f$ , where $f$ is a Maass form of weight $0$ on $\Gamma$ such that $(\pi_{f})_{\infty}$ is not tempered. Here, for $\pi\in\mathfrak{X}_{F}$ and $H\leq\mathrm{GL}_{2}(\mathbb{A}_{F})$ , $\pi^{H}$ denotes the space of fixed vectors under $H$ of $\pi$ , i.e.,

\pi^{H}:=\left\{v\in V_{\pi}:\pi(g)v=v\text{ for }g\in H\right\},

where $V_{\pi}$ is the underlying vector space of $\pi$ . Let $\mathfrak{X}_{F,\mathrm{ex}}$ be the subset of $\mathfrak{X}_{F}$ consisting of $\pi$ such that for every archimedean place $v$ of $F$ , $\pi_{v}$ is not tempered.

Let $\mathcal{O}_{F}$ be the ring of integers of $F$ . For each place $v$ of $F$ , let $F_{v}$ be the completion of $F$ at $v$ and $\mathcal{O}_{F_{v}}$ be the ring of integers of $F_{v}$ . Let $S_{F,\mathrm{fin}}$ (resp. $S_{F,\infty}$ ) be the set of non-archimedean (resp. archimedean) places of $F$ . For $v\in S_{F,\mathrm{fin}}$ , let $\mathfrak{p}_{v}$ be the prime ideal of $\mathcal{O}_{F}$ corresponding to $v$ . For a non-negative integer $e$ , we define $\mathrm{K}_{v,e}$ by

\mathrm{K}_{v,e}:=\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\mathrm{GL}_{2}(\mathcal{O}_{F_{v}}):c\in\mathfrak{p}_{v}^{e}\right\}.

For each $v\in S_{F,\infty}$ , let $\mathrm{K}_{v}^{0}$ be the maximal connected compact subgroup of $\mathrm{GL}_{2}(F_{v})$ defined by

\mathrm{K}_{v}^{0}:=\begin{cases}\mathrm{SO}(2)\quad&\text{if }F_{v}=\mathbb{R},\\ \mathrm{U}(2)\quad&\text{if }F_{v}=\mathbb{C}.\end{cases}

Let $J$ be an ideal of $\mathcal{O}_{F}$ , and $N_{F/\mathbb{Q}}(J):=[\mathcal{O}_{F}:J]$ be the absolute norm of $J$ . Since $\mathcal{O}_{F}$ is a Dedekind domain, there is a unique non-negative integer $\mathrm{val}_{v}(J)$ for each $v\in S_{F,\mathrm{fin}}$ such that

J=\prod_{v\in S_{F,\mathrm{fin}}}\mathfrak{p}_{v}^{\mathrm{val}_{v}(J)}.

Then, we let

\mathrm{K}(J):=\prod_{v\in S_{F,\mathrm{fin}}}\mathrm{K}_{v,\mathrm{val}_{v}(J)}\cdot\prod_{v\in S_{F,\infty}}\mathrm{K}_{v}^{0}

and

\mathcal{N}(J):=\sum_{\pi\in\mathfrak{X}_{F,\mathrm{ex}}}\dim\pi^{\mathrm{K}(J)}.

In the following theorem, we obtain an upper bound for $\mathcal{N}(J)$ .

Theorem 1.1.

Assume that $F$ is a number field. Assume that $r_{1}$ is the number of real embeddings of $F$ and that $r_{2}$ is the number of conjugate pairs of complex embeddings of $F$ . Then, for non-trivial ideals $J$ in $\mathcal{O}_{F}$ ,

\mathcal{N}(J)\ll_{F}\frac{[\mathrm{SL}_{2}(\mathcal{O}_{F}):\Gamma_{0}(J)]}{(\log(N_{F/\mathbb{Q}}(J)))^{2r_{1}+3r_{2}}},\quad|N_{F/\mathbb{Q}}(J)|\to\infty.

Let us consider the case when the number of archimedean places of $F$ is $1$ . In this case, $F$ is the field of rational numbers or an imaginary quadratic field, and $\mathcal{N}(J)$ means the number of exceptional eigenvalues with multiplicities for a congruence subgroup corresponding to $J$ .

When $F$ is an imaginary quadratic field, an automorphic form on $\mathbb{H}^{3}$ corresponding to a fixed vector of $\pi\in\mathfrak{X}_{F}$ is called a Maass cusp form of weight $0$ over $F$ (for details, see [4]). Note that the Laplace–Beltrami operator $\Delta_{\mathbb{H}^{3}}$ on $\mathbb{H}^{3}:=\{x+iy+jr:x,y,r\in\mathbb{R}\text{ and }r>0\}$ is defined by

\Delta_{\mathbb{H}^{3}}:=r^{2}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial r^{2}}\right)-r\frac{\partial}{\partial r}.

Assume that $f$ is a Maass cusp form of weight $0$ over an imaginary quadratic field $F$ and $\pi_{f}\in\mathfrak{X}_{F}$ is an automorphic representation corresponding to $f$ . Let $\lambda_{f}$ be the eigenvalue of $\Delta_{\mathbb{H}^{3}}$ on $f$ . Then, the Selberg eigenvalue conjecture is equivalent to that $\lambda_{f}\geq 1$ .

Let $T_{v}$ be the Hecke operator for $v\in S_{F,\mathrm{fin}}$ . For an ideal $J$ of $\mathcal{O}_{F}$ , a Hecke-Maass cusp form of weight $0$ on $\Gamma_{0}(J)$ over $F$ is a Maass cusp form of weight $0$ on $\Gamma_{0}(J)$ over $F$ that is an eigenform for all $T_{v}$ with $\mathrm{val}_{v}(J)=0$ . A Hecke-Maass cusp form $f$ is called a normalized Hecke-Maass cusp form if the $L^{2}$ -norm of $f$ is $1$ . Then, the vector space $V_{\lambda}$ of Maass cusp forms of weight $0$ on $\Gamma_{0}(J)$ over $F$ with the Laplacian eigenvalue $\lambda$ has a basis consisting of normalized Hecke-Maass cusp forms. The following corollary is immediately implied by Theorem 1.1.

Corollary 1.2.

Assume that the number of archimedean places of $F$ is $1$ . Let $J$ be a non-trivial ideal of $\mathcal{O}_{F}$ and $\mathcal{N}(\Gamma_{0}(J))$ be the number of normalized Hecke-Maass cusp forms of weight $0$ on $\Gamma_{0}(J)$ over $F$ that do not satisfy the Selberg eigenvalue conjecture. Then, the followings are true.

(1)

Assume that $F=\mathbb{Q}$ and $N$ is a positive integer with $N>1$ . For convenience, we write $\Gamma_{0}(N)$ for $\Gamma_{0}((N))$ . Then, we have

$\mathcal{N}(\Gamma_{0}(N))\ll\frac{[\mathrm{SL}_{2}(\mathbb{Z}):\Gamma_{0}(N)]}{(\log N)^{2}},\quad N\to\infty.$

(2)

Assume that $F$ is an imaginary quadratic field. Then, we have

\mathcal{N}(\Gamma_{0}(J))\ll_{F}\frac{[\mathrm{SL}_{2}(\mathcal{O}_{F}):\Gamma_{0}(J)]}{(\log(N_{F/\mathbb{Q}}(J)))^{3}},\quad|N_{F/\mathbb{Q}}(J)|\to\infty.

Remark 1.3.

Recalling the result of Iwaniec [12], the number of normalized Hecke-Maass cusp forms $f$ of weight $0$ on $\Gamma_{0}(N)$ satisfying $0<\lambda_{f}<\frac{1}{4}-\delta^{2}$ is less than $C_{\epsilon}\cdot N^{1-4\delta+\epsilon}$ , where $C_{\epsilon}$ is a constant depending only on $\epsilon>0$ . If we take $\delta=0$ , then we have

(1.1)

\mathcal{N}(\Gamma_{0}(N))\ll_{\epsilon}N^{1+\epsilon},\quad N\to\infty.

Since $[\mathrm{SL}_{2}(\mathbb{Z}):\Gamma_{0}(N)]\ll N\log\log N$ , Corollary 1.2 implies

\mathcal{N}(\Gamma_{0}(N))\ll\frac{N\log\log N}{(\log N)^{2}},\quad N\to\infty.

The rest of this paper is organized as follows. In Section 2, we review the Arthur-Selberg trace formula which is mainly used to prove Theorem 1.1 and describe how to obtain an upper bound for $\mathcal{N}(J)$ by using the Arthur-Selberg trace formula. In Section 3, we compute the geometric side of the Arthur-Selberg trace formula. In Section 4, we prove Theorem 1.1.

Acknowledgments.

The authors appreciate Andrew Knightly for his kind and helpful comments. The second author is supported by a Royal Society University Research Fellowship.

2. Preliminaries

We introduce some notions as follows. Let $\mathrm{G}:=\mathrm{GL}_{2}$ and $\overline{\mathrm{G}}:=\mathrm{G}/\mathrm{Z}$ , where $\mathrm{Z}$ is defined by the center of $\mathrm{G}$ . Let $F$ be a number field and $S_{F,\mathrm{fin}}$ (resp. $S_{F,\infty}$ ) be the set of non-archimedean (resp. archimedean) places of $F$ , and $S_{F}:=S_{F,\mathrm{fin}}\cup S_{F,\infty}$ be the set of places of $F$ . Let $\mathbb{A}_{F}$ be the ring of adeles of $F$ and $\mathbb{A}_{F,\infty}$ be the ring of infinite adeles of $F$ . For each $v\in S_{F}$ , let $F_{v}$ be the completion of $F$ at $v$ and $\mathcal{O}_{F_{v}}$ be the ring of integers of $F_{v}$ . Let $\mathrm{K}_{v}$ be the maximal compact subgroup of $\mathrm{G}(F_{v})$ defined by

\mathrm{K}_{v}:=\begin{cases}\mathrm{G}(\mathcal{O}_{F_{v}})\quad&\text{if $v\in S_{F,\mathrm{fin}}$,}\\ \mathrm{O}(2)\quad&\text{if $v\in S_{F,\infty}$ and $F_{v}=\mathbb{R}$,}\\ \mathrm{U}(2)\quad&\text{if $v\in S_{F,\infty}$ and $F_{v}=\mathbb{C}$.}\end{cases}

For each $v\in S_{F,\mathrm{fin}}$ , let $\varpi_{v}$ be the uniformizer of $\mathcal{O}_{F_{v}}$ and $q_{v}:=[\mathcal{O}_{F_{v}}:(\varpi_{v})]$ . Let $\mathfrak{p}_{v}:=(\varpi_{v})$ be a unique prime ideal of $\mathcal{O}_{F_{v}}$ . For $g=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\mathrm{G}(F_{v})$ with $v\in S_{F,\mathrm{fin}}$ , a Haar measure $dg$ on $\mathrm{G}(F_{v})$ is defined by

(2.1)

dg:=\frac{q_{v}^{3}}{(q_{v}-1)^{2}(q_{v}+1)}\cdot\frac{d_{v}a\,d_{v}b\,d_{v}c\,d_{v}d}{|ad-bc|_{v}^{2}},

where $d_{v}x$ is a Haar measure on $F_{v}$ satisfying $\mathrm{Vol}(\mathcal{O}_{F_{v}})=1$ , and $|\cdot|_{v}$ denotes the $v$ -adic absolute value on $F_{v}$ . From (2.1), we have $\mathrm{Vol}(\mathrm{K}_{v})=1$ . For $v\in S_{F,\mathrm{fin}}$ , a multiplicative Haar measure $d_{v}^{\times}x$ on $F_{v}^{\times}$ is defined by

d_{v}^{\times}x:=\frac{q_{v}}{q_{v}-1}\cdot\frac{d_{v}x}{|x|_{v}}.

For $v\in S_{F,\infty}$ , a multiplicative Haar measure $d_{v}^{\times}x$ on $F_{v}^{\times}$ is defined by

d_{v}^{\times}x:=\begin{cases}\frac{d_{\mathbb{R}}x}{2|x|_{v}}\quad&\text{if }F_{v}=\mathbb{R},\\ \frac{d_{\mathbb{C}}x}{\pi|x|_{v}}\quad&\text{if }F_{v}=\mathbb{C},\end{cases}

and $|y|_{v}:=|y|^{[F_{v}:\mathbb{R}]}$ . If $g\in\mathrm{G}(F_{v})$ with $v\in S_{F,\infty}$ , then by the Iwasawa decomposition, there are $y,z\in F_{v}^{\times}$ , $x\in F_{v}$ and $\kappa\in\mathrm{K}_{v}$ such that $g=\left(\begin{smallmatrix}z&0\\ 0&z\end{smallmatrix}\right)\left(\begin{smallmatrix}y&0\\ 0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right)\kappa$ . Then, a Haar measure $dg$ on $\mathrm{G}(F_{v})$ is defined by

dg:=d_{v}^{\times}z\,d_{v}^{\times}y\,d_{v}x\,d_{v}\kappa,

where $d_{v}\kappa$ is a Haar measure on $\mathrm{K}_{v}$ . For any $v\in S_{F}$ , we set a Haar measure $d_{v}\overline{\kappa}$ on $\mathrm{Z}(F_{v})\backslash\mathrm{Z}(F_{v})\mathrm{K}_{v}$ such that the volume of $\mathrm{Z}(F_{v})\backslash\mathrm{Z}(F_{v})\mathrm{K}_{v}$ is equal to $1$ . Then, a Haar measure $dg$ on $\overline{\mathrm{G}}(F_{v})$ is defined by

dg:=d_{v}^{\times}y\,d_{v}x\,d_{v}\overline{\kappa},

where $g=\left(\begin{smallmatrix}y&0\\ 0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right)\overline{\kappa}$ . Thus, a Haar measure on $\overline{\mathrm{G}}(\mathbb{A}_{F})$ is defined by the product of Haar measures on $\overline{\mathrm{G}}(F_{v})$ for all $v\in S_{F}$ .

For a non-archimedean place $v$ of $F$ and a non-negative integer $e$ , let

\mathrm{K}_{v,e}:=\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\mathrm{G}(\mathcal{O}_{F_{v}}):\varpi_{v}^{e}\mid c\right\}.

Assume that $J$ is an ideal of $\mathcal{O}_{F}$ . Then, for each $v\in S_{F,\mathrm{fin}}$ , there is a unique non-negative integer $\mathrm{val}_{v}(J)$ such that

J=\prod_{v\in S_{F,\mathrm{fin}}}\mathfrak{p}_{v}^{\mathrm{val}_{v}(J)}.

For convenience, let $\mathrm{val}_{v}(m):=\mathrm{val}_{v}((m))$ for $m\in\mathcal{O}_{F}$ , where $(m)=m\mathcal{O}_{F}$ denotes the principal ideal generated by $m$ in $\mathcal{O}_{F}$ . Let $N_{F/\mathbb{Q}}(J):=[\mathcal{O}_{F}:J]$ be the absolute norm of $J$ . Let $\mathrm{K}_{\mathrm{fin}}(J):=\prod_{v\in S_{F,\mathrm{fin}}}\mathrm{K}_{v,\mathrm{val}_{v}(J)}$ and $\mathrm{K}(J):=\mathrm{K}_{\mathrm{fin}}(J)\cdot\prod_{v\in S_{F,\infty}}\mathrm{K}_{v}^{0}$ , where

(2.2)

\mathrm{K}_{v}^{0}:=\begin{cases}\mathrm{SO}(2)\quad&\text{if }F_{v}=\mathbb{R},\\ \mathrm{U}(2)\quad&\text{if }F_{v}=\mathbb{C}\end{cases}

is the maximal connected compact subgroup of $\mathrm{G}(F_{v})$ .

For each $v\in S_{F,\infty}$ and $s\in\mathbb{C}$ , let

\mathbb{H}_{v}\left(|\cdot|_{v}^{s},|\cdot|_{v}^{-s}\right):=\left\{f:\mathrm{G}(F_{v})\to\mathbb{C}\quad\bigg{|}\int_{\mathrm{K}_{v}}|f(\kappa_{v})|^{2}d_{v}\kappa_{v}<\infty\text{ and }f\left(\left(\begin{smallmatrix}a&x\\ 0&b\end{smallmatrix}\right)g\right)=\left|\frac{a}{b}\right|_{v}^{s+\frac{1}{2}}f(g)\right\}.

Recall that $\mathfrak{X}_{F}$ is defined by the set of irreducible cuspidal automorphic representations $\pi$ of $\mathrm{G}(\mathbb{A}_{F})$ with the trivial central character such that the local representation $\pi_{v}$ of $\pi$ at each archimedean place $v$ of $F$ is an unramified principal series, and that $\mathfrak{X}_{F,\mathrm{ex}}$ is the subset of $\mathfrak{X}_{F}$ consisting of $\pi$ such that $\pi_{v}$ is not tempered for every $v\in S_{F,\infty}$ . In other words, if $\pi\in\mathfrak{X}_{F}$ , then for each $v\in S_{F,\infty}$ , there is a unique spectral parameter $\nu(\pi_{v})\in\mathbb{C}$ such that $\pi_{v}$ is a right regular representation of $\mathrm{G}(F_{v})$ on $\mathbb{H}_{v}\left(|\cdot|_{v}^{\nu(\pi_{v})},|\cdot|_{v}^{-\nu(\pi_{v})}\right)$ . Note that for $\pi\in\mathfrak{X}_{F}$ , the spectral parameter $\nu(\pi_{v})$ is either a purely imaginary number or a non-zero real number with $|\nu(\pi_{v})|\leq\frac{1}{2}$ . Since $\pi_{v}$ is tempered if and only if $\nu(\pi_{v})$ is a purely imaginary number, it follows that $\mathfrak{X}_{F,\mathrm{ex}}$ is the subset of $\mathfrak{X}_{F}$ consisting of $\pi$ such that $\nu(\pi_{v})\in\mathbb{R}^{\times}$ for all $v\in S_{F,\infty}$ . For $\pi\in\mathfrak{X}_{F}$ , let $V_{\pi}$ be the underlying vector space of $\pi$ and

\pi^{\mathrm{K}(J)}:=\left\{v\in V_{\pi}:\pi(g)v=v\text{ for }g\in\mathrm{K}(J)\right\}.

As in Section 1, we define

\mathcal{N}(J):=\sum_{\pi\in\mathfrak{X}_{F,\mathrm{ex}}}\dim\pi^{\mathrm{K}(J)}.

The goal of this section is to derive the formula for an upper bound for $\mathcal{N}(J)$ by using the Arthur-Selberg trace formula.

For each $v\in S_{F,\infty}$ , let $\phi_{v}$ be a smooth function on $\mathrm{G}(F_{v})$ satisfying the following conditions :

(a)

$\phi_{v}(z\kappa_{1}g\kappa_{2})=\phi_{v}(g)$ for every $z\in\mathrm{Z}(F_{v})$ , $\kappa_{1},\kappa_{2}\in\mathrm{K}_{v}^{0}$ and $g\in\mathrm{G}(F_{v})$ ,
(b)

$\phi_{v}$ is compactly supported modulo $\mathrm{Z}(F_{v})$ ,
(c)

If $F_{v}=\mathbb{R}$ , then the support of $\phi_{v}$ is contained in $\mathrm{GL}_{2}^{+}(\mathbb{R})$ .

Since $\phi_{v}$ is a $\mathrm{Z}(F_{v})$ -invariant function, throughout this paper we see that $\phi_{v}$ is a function on $\overline{\mathrm{G}}(F_{v})$ by abusing the notation. Let $\widehat{h_{\phi_{v}}}:\mathbb{R}\to\mathbb{C}$ and $h_{\phi_{v}}:\mathbb{C}\to\mathbb{C}$ be defined by

(2.3)

\widehat{h_{\phi_{v}}}(t):=2\pi^{1+\epsilon_{v}}\int_{\frac{2\pi|t|}{1+\epsilon_{v}}}^{\infty}\frac{\phi_{v}\left(\left(\begin{smallmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{smallmatrix}\right)\right)}{(\sinh^{2}(\frac{r}{2})-\sinh^{2}(\pi t))^{\frac{1-\epsilon_{v}}{2}}}\sinh(r)\,dr,

where $\epsilon_{v}:=[F_{v}:\mathbb{R}]-1$ , and

(2.4)

h_{\phi_{v}}(z):=\int_{\mathbb{R}}\widehat{h_{\phi_{v}}}(x)e^{-2\pi izx}dx.

Assume that $\widehat{h}:\mathbb{R}\to\mathbb{R}_{\geq 0}$ and $h(z):=\int_{\mathbb{R}}\widehat{h}(x)e^{-2\pi izx}dx$ satisfy the following conditions :

(1)

$\widehat{h}$ is smooth and compactly supported,
(2)

$\widehat{h}$ is even,
(3)

$\widehat{h}(0)=\int_{\mathbb{R}}h(x)dx=1$ ,
(4)

$h$ is entire,
(5)

$h$ is rapidly decreasing on horizontal strips,
(6)

$h(x)\geq 0$ on $x\in\mathbb{R}$ ,
(7)

$h(x)>0$ and $h(ix)>0$ on $x\in[-\frac{1}{2},\frac{1}{2}]$ .

To show the existence of $h$ and $\widehat{h}$ , let us take $g_{0}:\mathbb{R}\to\mathbb{R}$ such that $g_{0}$ is smooth, even, non-negative on $\mathbb{R}$ and supported on $[-1,1]$ . Let $g_{1}:=g_{0}*g_{0}$ . Then, $g_{1}$ is supported on $[-2,2]$ and

g_{1}(0)=\int_{-\infty}^{\infty}g_{0}(x)^{2}dx>0.

Let $\widehat{h_{1}}:=g_{1}/g_{1}(0)$ . Since $h_{1}=\widehat{\widehat{h_{1}}}$ , we have for $x\in\mathbb{R}$ ,

(2.5)

h_{1}(x)=\frac{1}{g_{1}(0)}\widehat{g_{0}}(x)^{2}\geq 0,

and $h_{1}(0)>0$ . Note that if $x\in\mathbb{R}$ , then $h_{1}(x)\in\mathbb{R}$ and $h_{1}(ix)\in\mathbb{R}$ . Thus, there is $\delta>0$ such that $h_{1}(x)$ and $h_{1}(ix)$ are positive on $x\in[-\frac{\delta}{2},\frac{\delta}{2}]$ . Let $h(z):=h_{1}(\delta z)$ . Then, $\widehat{h}$ and $h$ satisfy the conditions (1) $\sim$ (7). Using the Abel inversion formula, we obtain the following lemma.

Lemma 2.1.

Let $h:\mathbb{C}\to\mathbb{C}$ be a function satisfying the conditions $(1)\sim(7)$ . Assume that a function $\phi_{v}:\mathrm{G}(F_{v})\to\mathbb{C}$ satisfies the conditions $(a)\sim(c)$ such that

h_{\phi_{v}}=h.

Then, when $F_{v}=\mathbb{R}$ , we have for $r\in\mathbb{R}$ ,

(2.6)		$\displaystyle\phi_{v}\left(\begin{pmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{pmatrix}\right)$	$\displaystyle=-\frac{1}{4\pi^{2}}\int_{\frac{r}{2\pi}}^{\infty}\frac{\widehat{h_{\phi_{v}}}^{\prime}(t)}{\sqrt{\sinh^{2}(\pi t)-\sinh^{2}(r/2)}}dt$
(2.7)			$\displaystyle=\frac{1}{2\pi}\int_{\frac{r}{2\pi}}^{\infty}\int_{-\infty}^{\infty}xh_{\phi_{v}}(x)\frac{\sin(2\pi xt)}{\sqrt{\sinh^{2}(\pi t)-\sinh^{2}(r/2)}}dxdt.$

When $F_{v}=\mathbb{C}$ , we have for $r\in\mathbb{R}$ ,

(2.8)

\phi_{v}\left(\begin{pmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{pmatrix}\right)=-\frac{\widehat{h_{\phi_{v}}}^{\prime}(r/\pi)}{2\pi^{3}\sinh(r)}=\frac{1}{\pi^{2}}\int_{-\infty}^{\infty}xh_{\phi_{v}}(x)\frac{\sin(2xr)}{\sinh(r)}dx.

Proof.

When $F_{v}=\mathbb{R}$ , see [8, Proposition 4.1, pp 15-16] and when $F_{v}=\mathbb{C}$ , see [4, Lemma 3.5.5, pp 121]. ∎

Conversely, for a function $h$ satisfying the conditions $(1)\sim(7)$ , we define a function $\phi_{v}$ on $\mathrm{G}(F_{v})$ satisfying the conditions $(a)\sim(c)$ as follows. If $F_{v}=\mathbb{R}$ , then

\phi_{v}\left(z\kappa_{1}\begin{pmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{pmatrix}\kappa_{2}\right)=\phi_{v}\left(\begin{pmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{pmatrix}\right):=-\frac{1}{4\pi^{2}}\int_{\frac{r}{2\pi}}^{\infty}\frac{\widehat{h}^{\prime}(t)}{\sqrt{\sinh^{2}(\pi t)-\sinh^{2}(r/2)}}dt

and if $F_{v}=\mathbb{C}$ , then

\phi_{v}\left(z\kappa_{1}\begin{pmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{pmatrix}\kappa_{2}\right)=\phi_{v}\left(\begin{pmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{pmatrix}\right):=-\frac{\widehat{h}^{\prime}(r/\pi)}{2\pi^{3}\sinh(r)}.

Here, $z\in\mathrm{Z}(F_{v})$ , $\kappa_{1},\kappa_{2}\in\mathrm{K}_{v}^{0}$ and $r\geq 0$ . By (2.3), we see that $h_{\phi_{v}}$ is an even function. When $F_{v}=\mathbb{R}$ , we have for $t\geq 0$ ,

(2.9)	$\displaystyle\widehat{h_{\phi_{v}}}(t)$	$\displaystyle=2\pi\int_{2\pi t}^{\infty}\frac{\phi_{v}\left(\left(\begin{smallmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{smallmatrix}\right)\right)}{\sqrt{\sinh^{2}(\frac{r}{2})-\sinh^{2}(\pi t))}}\sinh(r)dr$
		$\displaystyle=-\frac{1}{2\pi}\int_{2\pi t}^{\infty}\int_{\frac{r}{2\pi}}^{\infty}\frac{\sinh(r)}{\sqrt{\sinh^{2}(\frac{r}{2})-\sinh^{2}(\pi t)}}\cdot\frac{\widehat{h}^{\prime}(x)}{\sqrt{\sinh^{2}(\pi x)-\sinh^{2}(\frac{r}{2})}}dxdr$
		$\displaystyle=-\frac{1}{2\pi}\int_{t}^{\infty}\widehat{h}^{\prime}(x)\int_{2\pi t}^{2\pi x}\frac{\sinh(r)}{\sqrt{\sinh^{2}(\frac{r}{2})-\sinh^{2}(\pi t)}\sqrt{\sinh^{2}(\pi x)-\sinh^{2}(\frac{r}{2})}}drdx.$

The last equality holds by Fubini’s theorem. By changing the variable $y=\sinh^{2}(\frac{r}{2})$ , we have

\int_{2\pi t}^{2\pi x}\frac{\sinh(r)}{\sqrt{\sinh^{2}(\frac{r}{2})-\sinh^{2}(\pi t)}\sqrt{\sinh^{2}(\pi x)-\sinh^{2}(\frac{r}{2})}}dr=2\pi.

Thus, (2.9) becomes

\widehat{h_{\phi_{v}}}(t)=-\int_{t}^{\infty}\widehat{h}^{\prime}(x)dx=\widehat{h}(t).

When $F_{v}=\mathbb{C}$ , for $t\geq 0$ , we have

	$\displaystyle\widehat{h_{\phi_{v}}}(t)$	$\displaystyle=2\pi^{2}\int_{\pi t}^{\infty}\phi_{v}\left(\begin{pmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{pmatrix}\right)\sinh(r)dr$
		$\displaystyle=-\frac{1}{\pi}\int_{\pi t}^{\infty}\widehat{h}^{\prime}\left(\frac{r}{\pi}\right)dr$
		$\displaystyle=\widehat{h}(t).$

Since $\widehat{h}$ is smooth and compactly supported, it follows that $\phi_{v}$ is also smooth and compactly supported modulo $\mathrm{Z}(F_{v})$ . Hence, $\phi_{v}$ satisfies the conditions $(a)\sim(c)$ and

h_{\phi_{v}}=h.

Let $(\pi,V_{\pi})$ be an irreducible cuspidal automorphic representation of $\mathrm{G}(\mathbb{A}_{F})$ with the trivial central character. Assume that a smooth function $\phi$ on $\mathrm{G}(\mathbb{A}_{F})$ is $\mathrm{Z}(\mathbb{A}_{F})$ -invariant and compactly supported modulo $\mathrm{Z}(\mathbb{A}_{F})$ . We define an operator $\pi(\phi)$ on $V_{\pi}$ by

\pi(\phi)v:=\int_{\overline{\mathrm{G}}(\mathbb{A}_{F})}\phi(g)\pi(g)vdg.

Then, the operator $\pi(\phi)$ is a trace class.

For $v\in S_{F,\mathrm{fin}}$ , let $\phi_{v}$ be a function on $\mathrm{G}(F_{v})$ defined by the characteristic function of $\mathrm{Z}(F_{v})\mathrm{K}_{v,\mathrm{val}_{v}(J)}$ . From the definition of $\phi_{v}$ , we see that

\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(g)dg=\frac{1}{[\mathrm{K}_{v}:\mathrm{K}_{v,\mathrm{val}_{v}(J)}]}.

We define a constant $A_{J}$ by

(2.10)

\displaystyle A_{J}:

\displaystyle=\prod_{v\in S_{F,\mathrm{fin}}}\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(g)dg=\prod_{v\in S_{F,\mathrm{fin}}}\frac{1}{[\mathrm{K}_{v}:\mathrm{K}_{v,\mathrm{val}_{v}(J)}]}=\frac{1}{[\mathrm{SL}_{2}(\mathcal{O}_{F}):\Gamma_{0}(J)]}.

The following lemma provides the expression of $\mathrm{tr}\pi(\phi)$ in terms of $h_{\phi_{v}}$ for $v\in S_{F,\infty}$ .

Lemma 2.2.

Let $F$ be a number field and $\pi=\otimes_{v\in S_{F}}\pi_{v}\in\mathfrak{X}_{F}$ . Let $r_{1}$ be the number of archimedean places $v$ of $F$ with $F_{v}=\mathbb{R}$ . Let $J$ be an ideal of $\mathcal{O}_{F}$ and for each $v\in S_{F,\mathrm{fin}}$ , $\phi_{v}$ is defined as the characteristic function of $\mathrm{Z}(F_{v})\mathrm{K}_{v,\mathrm{val}_{v}(J)}$ . Assume that for each $v\in S_{F,\infty}$ , $\phi_{v}$ satisfies the conditions $(a)\sim(c)$ . Let $\phi$ be a function on $\mathrm{G}(\mathbb{A}_{F})$ defined by $\phi:=\prod_{v\in S_{F}}\phi_{v}$ . Then, we have

\mathrm{tr}\pi(\phi)=\frac{A_{J}}{2^{r_{1}}}\cdot\prod_{v\in S_{F,\infty}}h_{\phi_{v}}(\nu(\pi_{v})/i)\cdot\dim\pi^{\mathrm{K}(J)}.

Proof.

For any $\mathbf{v}\in V_{\pi}$ and $\kappa\in\mathrm{K}(J)$ , we have

	$\displaystyle\pi(\kappa)\left(\pi(\phi)\mathbf{v}\right)$	$\displaystyle=\pi(\kappa)\left(\int_{\bar{\mathrm{G}}(\mathbb{A}_{F})}\phi(g)\pi(g)\mathbf{v}dg\right)$
		$\displaystyle=\int_{\bar{\mathrm{G}}(\mathbb{A}_{F})}\phi(g)\pi(\kappa g)\mathbf{v}dg$
		$\displaystyle=\int_{\bar{\mathrm{G}}(\mathbb{A}_{F})}\phi(\kappa^{-1}g)\pi(g)\mathbf{v}dg.$

Since $\phi(\kappa^{-1}g)=\phi(g)$ for any $\kappa\in\mathrm{K}(J)$ and $g\in\mathrm{G}(\mathbb{A}_{F})$ , it follows that $\pi(\phi)\mathbf{v}\in\pi^{\mathrm{K}(J)}$ . Thus, the trace of $\pi(\phi)$ is the same as the trace of the restriction of $\pi(\phi)$ to the space $\pi^{\mathrm{K}(J)}$ . To complete the proof of Lemma 2.2, we prove that for any $\mathbf{v}\in\pi^{\mathrm{K}(J)}$ ,

(2.11)

\pi(\phi)(\mathbf{v})=\left(\frac{A_{J}}{2^{r_{1}}}\cdot\prod_{v\in S_{F,\infty}}h_{\phi_{v}}(\nu(\pi_{v})/i)\right)\cdot\mathbf{v}.

For each place $v$ of $F$ , let $(\pi_{v},V_{\pi_{v}})$ be the local representation of $\pi$ at $v$ . Assume $\mathbf{v}=\otimes_{v\in S_{F}}\mathbf{v}_{v}\in\pi^{\mathrm{K}(J)}$ . Then, for each $v\in S_{F,\mathrm{fin}}$ , we have

(2.12)	$\displaystyle\pi_{v}(\phi_{v})\mathbf{v}_{v}$	$\displaystyle=\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(g)\pi_{v}(g)\mathbf{v}_{v}dg$
		$\displaystyle=\int_{\overline{\mathrm{K}_{v,\mathrm{val}_{v}(J)}}}\mathbf{v}_{v}dg$
		$\displaystyle=\mathrm{Vol}\left(\overline{\mathrm{K}_{v,\mathrm{val}_{v}(J)}}\right)\cdot\mathbf{v}_{v},$

where $\overline{\mathrm{K}_{v,\mathrm{val}_{v}(J)}}$ is the image of $\mathrm{K}_{v,\mathrm{val}_{v}(J)}$ under the canonical projection modulo $\mathrm{Z}(F_{v})$ . Following the proof of [18, Theorems 4.1 and 5.1], if $v\in S_{F,\infty}$ , then we have

(2.13)

\pi_{v}(\phi_{v})\mathbf{v}_{v}=\begin{cases}\frac{1}{2}h_{\phi_{v}}(\nu(\pi_{v})/i)\mathbf{v}_{v}\quad&\text{if }F_{v}=\mathbb{R},\\ h_{\phi_{v}}(\nu(\pi_{v})/i)\mathbf{v}_{v}\quad&\text{if }F_{v}=\mathbb{C}.\end{cases}

Combining (2.12) and (2.13), we complete the proof of Lemma 2.2. ∎

Note that for $v\in S_{F,\infty}$ and $z\in\mathbb{R}$ , we have

(2.14)

h_{\phi_{v}}(iz)=\int_{\mathbb{R}}\widehat{h_{\phi_{v}}}(x)e^{2\pi zx}dx=2\int_{0}^{\infty}\widehat{h_{\phi_{v}}}(x)\cosh(2\pi zx)dx\geq 2\int_{0}^{\infty}\widehat{h_{\phi_{v}}}(x)dx=h_{\phi_{v}}(0).

Thus, we obtain the following lemma which provides the relation between $\sum_{\pi\in\mathfrak{X}_{F}}\mathrm{tr}\pi(\phi)$ and $\mathcal{N}(J)$ .

Lemma 2.3.

Let $J$ be an ideal of $\mathcal{O}_{F}$ . Let $\phi$ be a function on $\mathrm{G}(\mathbb{A}_{F})$ defined as in Lemma 2.2. Then, we have

\sum_{\pi\in\mathfrak{X}_{F}}\mathrm{tr}\pi(\phi)\geq\frac{A_{J}\cdot\mathcal{N}(J)}{2^{r_{1}}}\left(\prod_{v\in S_{F,\infty}}h_{\phi_{v}}(0)\right).

Proof.

By Lemma 2.2, we have

(2.15)

\sum_{\pi\in\mathfrak{X}_{F}}\mathrm{tr}\pi(\phi)=\frac{A_{J}}{2^{r_{1}}}\cdot\sum_{\pi\in\mathfrak{X}_{F}}\left(\prod_{v\in S_{F,\infty}}h_{\phi_{v}}(\nu(\pi_{v})/i)\cdot\dim\pi^{\mathrm{K}(J)}\right).

Assume that $\pi\in\mathfrak{X}_{F}$ and $v\in S_{F,\infty}$ . Note that $\nu(\pi_{v})\in[-\frac{1}{2},\frac{1}{2}]$ or $\nu(\pi_{v})/i\in\mathbb{R}^{\times}$ . By the assumption of $h_{\phi_{v}}$ and (2.14), if $x\in\mathbb{R}$ or $x/i\in[-\frac{1}{2},\frac{1}{2}]$ , then $h_{\phi_{v}}(x)\geq 0$ . It follows that

(2.16)		$\displaystyle\sum_{\pi\in\mathfrak{X}_{F}}\left(\prod_{v\in S_{F,\infty}}h_{\phi_{v}}(\nu(\pi_{v})/i)\cdot\dim\pi^{\mathrm{K}(J)}\right)$	$\displaystyle\geq\sum_{\pi\in\mathfrak{X}_{F,\mathrm{ex}}}\left(\prod_{v\in S_{F,\infty}}h_{\phi_{v}}(\nu(\pi_{v})/i)\cdot\dim\pi^{\mathrm{K}(J)}\right)$
(2.16)			$\displaystyle\geq\left(\prod_{v\in S_{F,\infty}}h_{\phi_{v}}(0)\right)\cdot\mathcal{N}(J).$

∎

By the Arthur-Selberg trace formula, we have

(2.17)

\sum_{\pi\in\mathfrak{X}_{F}}\mathrm{tr}\pi(\phi)=-S_{\mathrm{one}}(\phi)+S_{\mathrm{id}}(\phi)+S_{\rm ell}(\phi)+S_{\rm hyp}(\phi)+S_{\rm par}(\phi)+S_{\rm Eis}(\phi)+S_{\rm Res}(\phi).

Here, the definitions of $S_{\mathrm{one}}(\phi),S_{\mathrm{id}}(\phi),S_{\rm ell}(\phi),S_{\rm hyp}(\phi),S_{\rm par}(\phi),S_{\rm Eis}(\phi)$ , and $S_{\rm Res}(\phi)$ are in Section 3. Then, Lemma 2.3 implies that

	$\displaystyle\mathcal{N}(J)$	$\displaystyle\leq\frac{2^{r_{1}}}{A_{J}\cdot\prod_{v\in S_{F,\infty}}h_{\phi_{v}}(0)}$
		$\displaystyle\times\left(\|S_{\mathrm{one}}(\phi)\|+\|S_{\mathrm{id}}(\phi)\|+\|S_{\rm ell}(\phi)\|+\|S_{\rm hyp}(\phi)\|+\|S_{\rm par}(\phi)\|+\|S_{\rm Eis}(\phi)\|+\|S_{\rm Res}(\phi)\|\right).$

Therefore, we have to compute an upper bound for

\frac{|S_{\mathrm{one}}(\phi)|+|S_{\mathrm{id}}(\phi)|+|S_{\rm ell}(\phi)|+|S_{\rm hyp}(\phi)|+|S_{\rm par}(\phi)|+|S_{\rm Eis}(\phi)|+|S_{\rm Res}(\phi)|}{\prod_{v\in S_{F,\infty}}h_{\phi_{v}}(0)}.

3. Geometric side

In this section, we compute the geometric side of the Arthur-Selberg trace formula. For two non-negative functions $X$ and $Y$ of $n$ variables $t_{1},\dots,t_{n}$ , we write $X\ll_{t_{1},\dots,t_{m}}Y$ if there exists a constant $C(t_{1},\dots,t_{m})$ , dependent on $t_{1},\dots,t_{m}$ such that

C(t_{1},\dots,t_{m})\cdot X\leq Y.

Here, $n$ and $m$ are positive integers with $m\leq n$ . For simplicity, if there is a constant $C$ which is independent on $t_{1},\dots,t_{n}$ such that

C\cdot X\leq Y,

then we write $X\ll Y$ . We follow the notation in Section 2. Throughout this section, we fix an ideal $J$ of $\mathcal{O}_{F}$ and $e_{v}:=\mathrm{val}_{v}(J)$ for each $v\in S_{F,\mathrm{fin}}$ . Moreover, $\phi$ is a function on $\mathrm{G}(\mathbb{A}_{F})$ defined as in Lemma 2.2.

3.1. One-dimensional representations

In [6], $S_{\mathrm{one}}(\phi)$ is defined by

S_{\mathrm{one}}(\phi):=\sum_{\begin{subarray}{c}\chi^{2}=1\\ \chi\text{ idele class character }\end{subarray}}\int_{\bar{\rm G}(\mathbb{A})}\phi(g)\chi(\mathrm{det}g)dg.

Then, we compute $S_{\mathrm{one}}(\phi)$ in the following lemma.

Lemma 3.1.

Let $J$ be an ideal of $\mathcal{O}_{F}$ . Then, we have

S_{\mathrm{one}}(\phi)=A_{J}\cdot\prod_{v\in S_{F,\infty}}h_{\phi_{v}}\left(\frac{i}{2}\right).

Proof.

Assume that $\chi:F^{\times}\backslash\mathbb{A}_{F}^{\times}\to\mathbb{C}^{\times}$ is an idele class character with $\chi^{2}=1$ such that

(3.1)

\int_{\bar{\mathrm{G}}(\mathbb{A}_{F})}\phi(g)\chi(\mathrm{det}g)dg\neq 0.

Let $\chi=\otimes_{v\in S_{F}}\chi_{v}$ , where $\chi_{v}:F_{v}^{\times}\to\mathbb{C}^{\times}$ is a character. Then, we have

\int_{\bar{\mathrm{G}}(\mathbb{A}_{F})}\phi(g)\chi(\mathrm{det}g)dg=\left(\prod_{v\in S_{F,\mathrm{fin}}}\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(g)\chi_{v}(\mathrm{det}g)dg\right)\cdot\left(\prod_{v\in S_{F,\infty}}\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(g)\chi_{v}(\mathrm{det}g)dg\right).

For each $v\in S_{F,\mathrm{fin}}$ , we assume $g_{v}\in\mathrm{Z}(F_{v})\mathrm{K}_{v,\mathrm{val}_{v}(J)}$ . Since $\phi_{v}$ is the characteristic function of $\mathrm{Z}(F_{v})\mathrm{K}_{v,\mathrm{val}_{v}(J)}$ , it follows that $\phi_{v}(gg_{v})=\phi_{v}(g)$ for any $g\in\mathrm{G}(F_{v})$ . Then, we have

	$\displaystyle\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(g)\chi_{v}(\mathrm{det}g)dg$	$\displaystyle=\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(gg_{v})\chi_{v}(\mathrm{det}(gg_{v}))d(gg_{v})$
		$\displaystyle=\chi_{v}(\mathrm{det}g_{v})\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(g)\chi_{v}(\mathrm{det}g)dg.$

Hence, we obtain that $\chi_{v}(\mathrm{det}g_{v})=1$ for $g_{v}\in\mathrm{Z}(F_{v})\mathrm{K}_{v,\mathrm{val}_{v}(J)}$ . Since the set of determinants of all elements in $\mathrm{Z}(F_{v})\mathrm{K}_{v,\mathrm{val}_{v}(J)}$ contains $F_{v}^{\times}$ , it follows that $\chi_{v}=1$ .

For each $v\in S_{F,\infty}$ , if $F_{v}=\mathbb{C}$ and $\chi_{v}^{2}=1$ , then $\chi_{v}$ is the trivial character since the image of $\chi_{v}$ is connected. If $F_{v}=\mathbb{R}$ , then the characters $\chi_{v}:\mathbb{R}^{\times}\to\mathbb{C}^{\times}$ satisfying $\chi_{v}^{2}=1$ are only the trivial character and the sign character. Since the support of $\phi_{v}$ is contained in $\mathrm{GL}_{2}^{+}(\mathbb{R})$ when $F_{v}=\mathbb{R}$ , we obtain that

	$\displaystyle S_{\mathrm{one}}(\phi)$	$\displaystyle=\left(\prod_{v\in S_{F,\mathrm{fin}}}\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(g)dg\right)$
		$\displaystyle\times\prod_{F_{v}=\mathbb{R}}\left(\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(g)dg+\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(g)\mathop{\rm sgn}(\mathrm{det}g)d_{v}g\right)\cdot\prod_{F_{v}=\mathbb{C}}\left(\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(g)dg\right)$
		$\displaystyle=2^{r_{1}}A_{J}\cdot\prod_{F_{v}=\mathbb{R}}\left(\int_{\overline{\mathrm{GL}_{2}^{+}}(\mathbb{R})}\phi_{v}(g)dg\right)\cdot\prod_{F_{v}=\mathbb{C}}\left(\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(g)dg\right),$

where $r_{1}$ denotes the number of archimedean places $v$ of $F$ satisfying $F_{v}=\mathbb{R}$ , and $\overline{\mathrm{GL}_{2}^{+}}(\mathbb{R})$ denotes the image of $\mathrm{GL}_{2}^{+}(\mathbb{R})$ modulo $\mathrm{Z}(\mathbb{R})$ . Following the proof of Lemma 2.2, we obtain that if $F_{v}=\mathbb{R}$ , then

\int_{\overline{\mathrm{GL}_{2}^{+}}(\mathbb{R})}\phi_{v}(g)dg=\frac{1}{2}h_{\phi_{v}}\left(\frac{i}{2}\right),

and if $F_{v}=\mathbb{C}$ , then

\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(g)dg=h_{\phi_{v}}\left(\frac{i}{2}\right).

∎

3.2. Identity contribution

Let $I_{2}$ be the identity matrix of size $2$ . In [6], $S_{\mathrm{id}}(\phi)$ is defined by

(3.2)

S_{\mathrm{id}}(\phi):=\mathrm{Vol}(\bar{\mathrm{G}}(F)\backslash\bar{\mathrm{G}}(\mathbb{A}_{F}))\phi(I_{2}).

Note that $\phi_{v}(I_{2})=1$ for each $v\in S_{F,\mathrm{fin}}$ . Then, we have the following lemma.

Lemma 3.2.

For each $v\in S_{F,\infty}$ , let $\epsilon_{v}:=[F_{v}:\mathbb{R}]-1$ . Then, we have

\phi(I_{2})=\prod_{v\in S_{F,\infty}}\left(\frac{1+\epsilon_{v}}{(2-\epsilon_{v})^{2(1-\epsilon_{v})}\pi^{1+\epsilon_{v}}}\int_{-\infty}^{\infty}x^{1+\epsilon_{v}}h_{v}(x)\tanh^{1-\epsilon_{v}}(\pi x)\,dx\right).

Proof.

By (2.7), when $F_{v}=\mathbb{R}$ , we get

\phi_{v}(I_{2})=\frac{1}{2\pi}\int_{-\infty}^{\infty}xh_{\phi_{v}}(x)\left(\int_{0}^{\infty}\frac{\sin(2\pi xt)}{\sinh(\pi t)}\,dt\right)\,dx=\frac{1}{4\pi}\int_{-\infty}^{\infty}xh_{\phi_{v}}(x)\tanh(\pi x)dx

since by [7, 3.986.2],

\int_{0}^{\infty}\frac{\sin(2\pi xt)}{\sinh(\pi t)}dt=\frac{1}{2}\tanh(\pi x).

When $F_{v}=\mathbb{C}$ , we use (2.8). Then, we get

\phi_{v}(I_{2})=\frac{2}{\pi^{2}}\int_{-\infty}^{\infty}x^{2}h_{\phi_{v}}(x)dx.

∎

3.3. Elliptic contribution

For $\gamma\in\bar{\mathrm{G}}(F)$ , let $\bar{\mathrm{G}}_{\gamma}(F)$ (resp. $\bar{\mathrm{G}}_{\gamma}(\mathbb{A}_{F})$ ) be the centralizer of $\gamma$ in $\bar{\mathrm{G}}(F)$ (resp. $\bar{\mathrm{G}}(\mathbb{A}_{F})$ ). For an elliptic matrix $\gamma\in\bar{\mathrm{G}}(F)$ , the conjugacy class $[\gamma]$ of $\gamma$ is defined by

[\gamma]:=\left\{\delta^{-1}\gamma\delta\;:\;\delta\in\bar{\mathrm{G}}_{\gamma}(F)\backslash\bar{\mathrm{G}}(F)\right\}.

Let $\mathrm{G}_{\gamma}(F)$ (resp. $\mathrm{G}_{\gamma}(\mathbb{A}_{F})$ ) be the centralizer of $\tilde{\gamma}$ in $\mathrm{G}(F)$ (resp. $\mathrm{G}(\mathbb{A}_{F})$ ), where $\tilde{\gamma}\in\mathrm{G}(F)$ is a representative of $[\gamma]$ . Let $\overline{\mathrm{G}_{\gamma}(F)}:=\mathrm{Z}(F)\backslash\mathrm{G}_{\gamma}(F)$ and $\overline{\mathrm{G}_{\gamma}(\mathbb{A}_{F})}:=\mathrm{Z}(\mathbb{A}_{F})\backslash\mathrm{G}_{\gamma}(\mathbb{A}_{F})$ . Then, by [16, Lemma 3.4], we see that

[\bar{\mathrm{G}}_{\gamma}(F):\overline{\mathrm{G}_{\gamma}(F)}]=\begin{cases}1&\text{if }\mathrm{tr}\gamma\neq 0,\\ 2&\text{otherwise.}\end{cases}

In [6, pp. 244], $S_{\mathrm{ell}}(\phi)$ is defined by

S_{\mathrm{ell}}(\phi):=\sum_{\text{elliptic }\gamma\in\bar{\mathrm{G}}(F)}\int_{\bar{\mathrm{G}}(F)\backslash\bar{\mathrm{G}}(\mathbb{A}_{F})}\phi(g^{-1}\gamma g)dg.

Since $g^{-1}\delta^{-1}\gamma\delta g=g^{-1}\gamma g$ for $\delta\in\bar{\mathrm{G}}_{\gamma}(\mathbb{A}_{F})$ and $\phi=\prod_{v\in S_{F}}\phi_{v}$ is factorizable, we obtain that

	$\displaystyle S_{\mathrm{ell}}(\phi)$	$\displaystyle=\sum_{[\gamma]\text{ elliptic }}\int_{\bar{\mathrm{G}}(F)\backslash\bar{\mathrm{G}}(\mathbb{A}_{F})}\sum_{\delta\in\bar{\mathrm{G}}_{\gamma}(F)\backslash\bar{\mathrm{G}}(F)}\phi(g^{-1}\delta^{-1}\gamma\delta g)dg$
		$\displaystyle=\sum_{[\gamma]\text{ elliptic }}\int_{\bar{\mathrm{G}}_{\gamma}(F)\backslash\bar{\mathrm{G}}(\mathbb{A}_{F})}\phi(g^{-1}\gamma g)dg$
		$\displaystyle=\sum_{[\gamma]\text{ elliptic }}C_{\gamma}\int_{\overline{\mathrm{G}_{\gamma}(F)}\backslash\bar{\mathrm{G}}(\mathbb{A}_{F})}\phi(g^{-1}\gamma g)dg$
		$\displaystyle=\sum_{[\gamma]\text{ elliptic }}\mathrm{Vol}(\overline{\mathrm{G}_{\gamma}(F)}\backslash\overline{\mathrm{G}_{\gamma}(\mathbb{A}_{F})})\cdot S_{[\gamma]}(\phi),$

where for each conjugacy class $[\gamma]$ ,

(3.3)

S_{[\gamma]}(\phi):=\prod_{v\in S_{F,\mathrm{fin}}}\int_{\overline{\mathrm{G}_{\gamma}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\gamma g)dg\cdot\prod_{v\in S_{F,\infty}}\int_{\overline{\mathrm{G}_{\sigma_{v}(\gamma)}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\sigma_{v}(\gamma)g)dg.

Here, $C_{\gamma}:=[\bar{\mathrm{G}}_{\gamma}(F):\overline{\mathrm{G}_{\gamma}(F)}]^{-1}$ and $\sigma_{v}:F\to F_{v}$ is an embedding corresponding to $v\in S_{F,\infty}$ .

Assume that $S_{[\gamma]}(\phi)\neq 0$ . Then, for $v\in S_{F,\mathrm{fin}}$ , we have

(3.4)

\int_{\overline{\mathrm{G}_{\gamma}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\gamma g)dg\neq 0.

Since the centralizer $\mathrm{G}_{\tilde{\gamma}}(F_{v})$ of $\tilde{\gamma}$ in $\mathrm{G}(F_{v})$ contains $\mathrm{Z}(F_{v})$ , we see that

\int_{\overline{\mathrm{G}_{\gamma}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\gamma g)dg=\int_{\mathrm{G}_{\tilde{\gamma}}(F_{v})\backslash\mathrm{G}(F_{v})}\phi_{v}(g^{-1}\tilde{\gamma}g)dg.

Note that the integral $\int_{\mathrm{G}_{\tilde{\gamma}}(F_{v})\backslash\mathrm{G}(F_{v})}\phi_{v}(g^{-1}\tilde{\gamma}g)dg$ is independent of the choice of the representative $\tilde{\gamma}$ of $[\gamma]$ since a Haar measure is invariant under the right multiplication. By (3.4), there are $g_{v}\in\mathrm{G}(F_{v})$ and $\alpha_{v}\in\mathrm{Z}(F_{v})$ such that $\alpha_{v}(g_{v}^{-1}\tilde{\gamma}g_{v})\in\mathrm{G}(\mathcal{O}_{F_{v}})$ . It follows that $\mathrm{det}(\tilde{\gamma})\in\mathcal{O}_{F_{v}}^{\times}F_{v}^{\times^{2}}$ for each $v\in S_{F,\mathrm{fin}}$ . Since the entries of $\tilde{\gamma}$ are in $F$ , we have $\mathrm{det}(\tilde{\gamma})\in\mathcal{O}_{F}^{\times}F^{\times^{2}}$ . By multiplying a non-zero element in $F$ , we can assume that the determinant of $\tilde{\gamma}$ is in $\mathcal{O}_{F}^{\times}$ . Since $\alpha_{v}(g_{v}^{-1}\tilde{\gamma}g_{v})\in\mathrm{G}(\mathcal{O}_{F_{v}})$ , it follows that $\alpha_{v}\in\mathrm{Z}(\mathcal{O}_{F_{v}})$ for every $v\in S_{F,\mathrm{fin}}$ . From this, we obtain that $g_{v}^{-1}\tilde{\gamma}g_{v}\in\mathrm{G}(\mathcal{O}_{F_{v}})$ . Thus, $\mathrm{tr}(\tilde{\gamma})\in\mathcal{O}_{F_{v}}$ for each $v\in S_{F,\mathrm{fin}}$ , and then $\mathrm{tr}(\tilde{\gamma})\in\mathcal{O}_{F}$ . Note that $\tilde{\gamma}$ is similar to $\left(\begin{smallmatrix}0&1\\ -\mathrm{det}\tilde{\gamma}&\mathrm{tr}\tilde{\gamma}\end{smallmatrix}\right)$ over $F$ . Therefore, there is a unique pair $(m,u)\in\mathcal{O}_{F}\times\left(\mathcal{O}_{F}^{\times}/\mathcal{O}_{F}^{\times^{2}}\right)$ such that $\left(\begin{smallmatrix}0&1\\ -u&m\end{smallmatrix}\right)$ is in $[\gamma]$ .

Let $U_{F}$ be the set of representatives of $\mathcal{O}_{F}^{\times}/\mathcal{O}_{F}^{\times^{2}}$ . Dirichlet’s unit theorem implies that $U_{F}$ is a finite set. Let $\gamma_{m,u}:=\left(\begin{smallmatrix}0&1\\ -u&m\end{smallmatrix}\right)$ with $(m,u)\in\mathcal{O}_{F}\times U_{F}$ . For a ring $R$ containing $\mathcal{O}_{F}$ , the centralizer $\mathrm{G}_{\gamma_{m,u}}(R)$ of $\gamma_{m,u}$ in $\mathrm{G}(R)$ is equal to

\mathrm{G}_{\gamma_{m,u}}(R)=\left\{\begin{pmatrix}a&-cu^{-1}\\ c&a-cmu^{-1}\end{pmatrix}:a,c\in R\text{ and }a^{2}-acmu^{-1}+c^{2}u^{-1}\in R^{\times}\right\}.

Here, $R^{\times}$ denotes the set of units in $R$ . Let $E:=F(\sqrt{m^{2}-4u})$ . Since $\gamma_{m,u}$ is an elliptic matrix over $F$ , it follows that $[E:F]=2$ . We define an isomorphism $\psi:\mathrm{G}_{\gamma_{m,u}}(F)\to E^{\times}$ by

(3.5)

\psi(aI_{2}-cu^{-1}\gamma_{m,u}):=a-cu^{-1}\left(\frac{m+\sqrt{m^{2}-4u}}{2}\right).

The isomorphism (3.5) deduces that

(3.6)

\mathrm{G}_{\gamma_{m,u}}(\mathbb{A}_{F})\cong\mathbb{A}_{E}^{\times}.

Note that a Haar measure on $\mathbb{A}_{E}^{\times}$ is given by an isomorphism (3.6). From this isomorphism (3.6), we compute the volume of $\overline{\mathrm{G}_{\gamma_{m,u}}(F)}\backslash\overline{\mathrm{G}_{\gamma_{m,u}}(\mathbb{A}_{F})}$ in the following lemma.

For a number field $K$ , let $\mathbb{A}_{K}^{1}$ (resp. $\mathbb{A}_{K,\infty}^{1}$ ) be the subset of $\mathbb{A}_{K}$ (resp. $\mathbb{A}_{K,\infty}$ ) consisting of $(k_{v})_{v\in S_{K}}$ (resp. $(k_{v})_{v\in S_{K,\infty}}$ ) such that $\prod_{v\in S_{K}}|k_{v}|_{v}=1$ (resp. $\prod_{v\in S_{K,\infty}}|k_{v}|_{v}=1$ ).

Lemma 3.3.

Let $(m,u)\in\mathcal{O}_{F}\times\mathcal{O}_{F}^{\times}$ and $E:=F(\sqrt{m^{2}-4u})$ . Let $\mathrm{cl}(E)$ be the class number of $E$ . Let $i_{\infty}:E\to\mathbb{A}_{E,\infty}$ be an inclusion map defined by

i_{\infty}(e):=(\sigma_{v}(e))_{v\in S_{E,\infty}}.

Assume that $[E:F]=2$ . Then, we have

\mathrm{Vol}\left(\overline{\mathrm{G}_{\gamma_{m,u}}(F)}\backslash\overline{\mathrm{G}_{\gamma_{m,u}}(\mathbb{A}_{F})}\right)=\frac{2\cdot\mathrm{cl}(E)\cdot\mathrm{Vol}\left(i_{\infty}(\mathcal{O}_{E}^{\times})\backslash\mathbb{A}^{1}_{E,\infty}\right)\cdot\prod_{w\in S_{E,\mathrm{fin}}}\mathrm{Vol}(\mathcal{O}_{E_{w}}^{\times})}{\mathrm{Vol}\left(F^{\times}\backslash\mathbb{A}_{F}^{1}\right)}.

Remark 3.4.

The exact volume of $\overline{\mathrm{G}_{\gamma_{m,u}}(F)}\backslash\overline{\mathrm{G}_{\gamma_{m,u}}(\mathbb{A}_{F})}$ is obtained by [16, Theorem 4.16]. For the calculation of the orbital integral $\int_{\overline{\mathrm{G}_{\gamma}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\gamma g)dg$ , we construct a Haar measure of each set in Section 3.3. Note that the difference between these constructed Haar measures and the Haar measures in [16] is a constant multiple.

Proof of Lemma 3.3.

By an isomorphism (3.6), $\overline{\mathrm{G}_{\gamma_{m,u}}(F)}\backslash\overline{\mathrm{G}_{\gamma_{m,u}}(\mathbb{A}_{F})}=\mathrm{Z}(\mathbb{A}_{F})\mathrm{G}_{\gamma_{m,u}}(F)\backslash\mathrm{G}_{\gamma_{m,u}}(\mathbb{A}_{F})$ can be identified $\left(\mathbb{A}_{F}^{\times}\cdot E^{\times}\right)\backslash\mathbb{A}_{E}^{\times}$ . By (4.31) in [16], we have

\mathrm{Vol}\left(\left(\mathbb{A}_{F}^{\times}\cdot E^{\times}\right)\backslash\mathbb{A}_{E}^{\times}\right)=\frac{\mathrm{Vol}\left(E^{\times}\backslash\mathbb{A}_{E}^{1}\right)}{\frac{1}{2}\cdot\mathrm{Vol}\left(F^{\times}\backslash\mathbb{A}_{F}^{1}\right)}=\frac{2\cdot\mathrm{Vol}\left(E^{\times}\backslash\mathbb{A}_{E}^{1}\right)}{\mathrm{Vol}\left(F^{\times}\backslash\mathbb{A}_{F}^{1}\right)}.

Following the proof of [20, Theorem 4.3.2], we conclude that

\mathrm{Vol}\left(\overline{\mathrm{G}_{\gamma_{m,u}}(F)}\backslash\overline{\mathrm{G}_{\gamma_{m,u}}(\mathbb{A}_{F})}\right)=\frac{2\cdot\mathrm{cl}(E)\cdot\mathrm{Vol}\left(i_{\infty}(\mathcal{O}_{E}^{\times})\backslash\mathbb{A}^{1}_{E,\infty}\right)\cdot\prod_{w\in S_{E,\mathrm{fin}}}\mathrm{Vol}(\mathcal{O}_{E_{w}}^{\times})}{\mathrm{Vol}\left(F^{\times}\backslash\mathbb{A}_{F}^{1}\right)}.

∎

To compute the volume of $\mathcal{O}^{\times}_{E_{w}}$ for $w\in S_{E,\mathrm{fin}}$ , we define a Haar measure on $E_{w}^{\times}$ as follows. For $v\in S_{F,\mathrm{fin}}$ , a Haar measure $dg$ on $\mathrm{G}_{\gamma_{m,u}}(F_{v})$ is defined by

(3.7)

dg:=\frac{q_{v}^{3}}{(q_{v}-1)^{2}(q_{v}+1)}\cdot\frac{d_{v}a\,d_{v}c}{|a^{2}-acmu^{-1}+c^{2}u^{-1}|_{v}},

where $g=\left(\begin{smallmatrix}a&-cu^{-1}\\ c&a-cmu^{-1}\end{smallmatrix}\right)\in\mathrm{G}_{\gamma_{m,u}}(F_{v})$ . Here, $d_{v}a$ is an additive measure on $F_{v}$ satisfying $\mathrm{Vol}(\mathcal{O}_{F_{v}})=1$ . If there exists an element $x\in F_{v}$ such that $x^{2}=m^{2}-4u$ , then the prime ideal $\mathfrak{p}_{v}$ corresponding to $v$ splits completely in $E$ . Thus, there are exactly two places $w\in S_{E,\mathrm{fin}}$ lying over $v$ , and then there is an isomorphism $\psi_{v}:\mathrm{G}_{\gamma_{m,u}}(F_{v})\to\prod_{w\mid v}E_{w}^{\times}$ defined by

(3.8)

\psi_{v}(aI_{2}-cu^{-1}\gamma_{m,u}):=\left(a-cu^{-1}\left(\frac{m+x_{1}}{2}\right),a-cu^{-1}\left(\frac{m+x_{2}}{2}\right)\right),

where $x_{1}$ and $x_{2}$ are roots of $x^{2}=m^{2}-4u$ in $F_{v}$ . If there is no $x\in F_{v}$ satisfying $x^{2}=m^{2}-4u$ , then there is a unique $w\in S_{F,\mathrm{fin}}$ lying over $v$ . Thus, there is an isomorphism $\psi_{v}:\mathrm{G}_{\gamma_{m,u}}(F_{v})\to E_{w}^{\times}$ defined by

(3.9)

\psi_{v}(aI_{2}-cu^{-1}\gamma_{m,u}):=a-cu^{-1}\left(\frac{m+\sqrt{m^{2}-4u}}{2}\right).

Hence, a Haar measure on $\prod_{w\mid v}E_{w}^{\times}$ is given by the isomorphism $\psi_{v}$ for each $v\in S_{F,\mathrm{fin}}$ . Thus, the volume of $\prod_{w\mid v}\mathcal{O}_{E_{w}}^{\times}$ is equal to the volume of the preimage of it under $\psi_{v}$ , i.e.,

\mathrm{Vol}\left(\prod_{w\mid v}\mathcal{O}_{E_{w}}^{\times}\right)=\mathrm{Vol}\left(\psi_{v}^{-1}\left(\prod_{w\mid v}\mathcal{O}_{E_{w}}^{\times}\right)\right).

The following lemma provides an upper bound for $\mathrm{Vol}(\prod_{w\mid v}\mathcal{O}_{E_{w}}^{\times})$ in terms of $q_{v}$ when $v\in S_{F,\mathrm{fin}}$ .

Lemma 3.5.

Assume that $v\in S_{F,\mathrm{fin}}$ and that $(m,u)\in\mathcal{O}_{F_{v}}\times\mathcal{O}_{F_{v}}^{\times}$ . Let $N_{v}$ be defined by the number of $t\in\mathcal{O}_{F_{v}}/\varpi_{v}\mathcal{O}_{F_{v}}$ satisfying $t^{2}-mt+u=0$ . Let $f_{v}:=\mathrm{val}_{v}(m^{2}-4u)$ . If $f_{v}>0$ , then

\mathrm{Vol}\left(\prod_{w\mid v}\mathcal{O}_{E_{w}}^{\times}\right)\leq\frac{q_{v}^{3+2\cdot[\frac{f_{v}+1}{2}]}}{(q_{v}-1)^{2}(q_{v}+1)},

and if $f_{v}=0$ , then

\mathrm{Vol}\left(\prod_{w\mid v}\mathcal{O}_{E_{w}}^{\times}\right)\leq\frac{q_{v}(q_{v}+1-N_{v})}{q_{v}^{2}-1}.

Proof.

By the definition of $\psi_{v}$ , we have

	$\displaystyle\psi_{v}^{-1}$	$\displaystyle\left(\prod_{w\mid v}\mathcal{O}_{E_{w}}^{\times}\right)$
		$\displaystyle\subset\left\{\begin{pmatrix}a&-cu^{-1}\\ c&a-cmu^{-1}\end{pmatrix}:\min\left(\mathrm{val}_{v}(a),\mathrm{val}_{v}(c)\right)\geq-\frac{f_{v}}{2}\text{ and }\mathrm{val}_{v}(a^{2}-acmu^{-1}+c^{2}u^{-1})=0\right\}.$

Recall that the Haar measure on $\mathrm{G}_{\gamma}(F_{v})$ is given by (3.7). If $f_{v}>0$ , then we have

	$\displaystyle\mathrm{Vol}\left(\psi_{v}^{-1}\left(\prod_{w\mid v}\mathcal{O}_{E_{w}}^{\times}\right)\right)$	$\displaystyle\leq\frac{q_{v}^{3}}{(q_{v}-1)^{2}(q_{v}+1)}\int_{\varpi_{v}^{-[\frac{f_{v}+1}{2}]}\mathcal{O}_{F_{v}}}\int_{\varpi_{v}^{-[\frac{f_{v}+1}{2}]}\mathcal{O}_{F_{v}}}d_{v}a\,d_{v}c$
		$\displaystyle=\frac{q_{v}^{3}}{(q_{v}-1)^{2}(q_{v}+1)}\cdot q_{v}^{2\cdot[\frac{f_{v}+1}{2}]}.$

Assume that $f_{v}=0$ . Since $a^{2}-acmu^{-1}+c^{2}u^{-1}=c^{2}u^{-2}((au/c)^{2}-m(au/c)+u)$ , it follows that

		$\displaystyle\{(a,c)\in\mathcal{O}_{F_{v}}\times\mathcal{O}_{F_{v}}:\mathrm{val}_{v}(a^{2}-acmu^{-1}+c^{2}u^{-1})=0\}$
		$\displaystyle=\mathcal{O}_{F_{v}}\times\mathcal{O}_{F_{v}}-\{(ctu^{-1}+a_{1},c):c\in\mathcal{O}_{F_{v}}^{\times},a_{1}\in\varpi_{v}\mathcal{O}_{F_{v}}\text{ and }\varpi_{v}\mid t^{2}-mt+u\}-\varpi_{v}\mathcal{O}_{F_{v}}\times\varpi_{v}\mathcal{O}_{F_{v}}.$

Therefore, we conclude that

	$\displaystyle\mathrm{Vol}\left(\prod_{w\mid v}\mathcal{O}_{E_{w}}^{\times}\right)$	$\displaystyle\leq\frac{q_{v}^{3}}{(q_{v}-1)^{2}(q_{v}+1)}\left(1-\frac{N_{v}(q_{v}-1)}{q_{v}^{2}}-\frac{1}{q_{v}^{2}}\right)$
		$\displaystyle=\frac{q_{v}(q_{v}+1-N_{v})}{q_{v}^{2}-1}.$

∎

Since $\mathrm{G}_{\gamma_{m,u}}(F_{v})$ contains $\mathrm{Z}(F_{v})$ , it follows that $\overline{\mathrm{G}_{\gamma_{m,u}}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})$ is isomorphic to $\mathrm{G}_{\gamma_{m,u}}(F_{v})\backslash\mathrm{G}(F_{v})$ . In the following lemma, we compute the set of representatives of $\mathrm{G}_{\gamma_{m,u}}(F_{v})\backslash\mathrm{G}(F_{v})$ .

Lemma 3.6.

Assume that $v\in S_{F,\mathrm{fin}}$ and that $(m,u)\in\mathcal{O}_{F}\times\mathcal{O}_{F}^{\times}$ satisfies $m^{2}-4u\not\in F^{2}$ . If there is $\alpha\in F_{v}$ such that $\alpha^{2}-m\alpha+u=0$ , then

\left\{\begin{pmatrix}1&u^{-1}\alpha_{i}\\ \alpha_{i}&1\end{pmatrix}:i\in\{1,2\}\right\}\cup\left\{\begin{pmatrix}1&x\\ \alpha_{i}&0\end{pmatrix}:x\in F_{v}^{\times}\text{ and }i\in\{1,2\}\right\}\cup\left\{\begin{pmatrix}1&x\\ 0&y\end{pmatrix}:x\in F_{v}\text{ and }y\in F_{v}^{\times}\right\}

is a set of representatives of $\mathrm{G}_{\gamma_{m,u}}(F_{v})\backslash\mathrm{G}(F_{v})$ . Here, $\alpha_{1}$ and $\alpha_{2}$ are roots of $x^{2}-mx+u=0$ in $F_{v}$ . If there is no $\alpha\in F_{v}$ satisfying $\alpha^{2}-m\alpha+u=0$ , then

\left\{\begin{pmatrix}1&x\\ 0&y\end{pmatrix}:x\in F_{v}\text{ and }y\in F_{v}^{\times}\right\}

is a set of representatives of $\mathrm{G}_{\gamma_{m,u}}(F_{v})\backslash\mathrm{G}(F_{v})$ .

Proof.

For $\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\mathrm{G}(F_{v})$ with $a^{2}-macu^{-1}+c^{2}u^{-1}\neq 0$ , we have

(3.10)

\begin{pmatrix}a&b\\ c&d\end{pmatrix}=\begin{pmatrix}a&-cu^{-1}\\ c&a-cmu^{-1}\end{pmatrix}\begin{pmatrix}1&\frac{ab+cdu^{-1}-bcmu^{-1}}{a^{2}-acmu^{-1}+c^{2}u^{-1}}\\ 0&\frac{ad-bc}{a^{2}-acmu^{-1}+c^{2}u^{-1}}\end{pmatrix}.

Moreover, if $(x_{1},y_{1})\neq(x_{2},y_{2})$ , then

\begin{pmatrix}1&x_{1}\\ 0&y_{1}\end{pmatrix}\begin{pmatrix}1&x_{2}\\ 0&y_{2}\end{pmatrix}^{-1}=\frac{1}{y_{2}}\begin{pmatrix}y_{2}&x_{1}-x_{2}\\ 0&y_{1}\end{pmatrix}\not\in\mathrm{G}_{\gamma_{m,u}}(F_{v}).

Thus, if there is no root of $x^{2}-mx+u$ in $F_{v}$ , then

\left\{\begin{pmatrix}1&x\\ 0&y\end{pmatrix}:x\in F_{v}\text{ and }y\in F_{v}^{\times}\right\}

is a set of representatives of $\mathrm{G}_{\gamma_{m,u}}(F_{v})\backslash\mathrm{G}(F_{v})$ .

Assume that there is $\alpha\in F_{v}$ satisfying $\alpha^{2}-m\alpha+u=0$ . Let $\alpha_{1}$ and $\alpha_{2}$ be roots of $x^{2}-mx+u=0$ in $F_{v}$ . Note that if $a^{2}-acmu^{-1}+c^{2}u^{-1}=0$ , then $c=a\alpha_{i}$ for $i\in\{1,2\}$ . Thus, we assume that $c=a\alpha$ with $\alpha\in\{\alpha_{1},\alpha_{2}\}$ . If $b\neq du^{-1}\alpha$ , then

(3.11)		$\displaystyle\begin{pmatrix}a&b\\ a\alpha&d\end{pmatrix}$	$\displaystyle=\begin{pmatrix}ab(b-du^{-1}\alpha)^{-1}&-adu^{-1}(b-du^{-1}\alpha)^{-1}\\ ad(b-du^{-1}\alpha)^{-1}&a(b-mu^{-1}d)(b-du^{-1}\alpha)^{-1}\end{pmatrix}\begin{pmatrix}1&a^{-1}(b-du^{-1}\alpha)\\ \alpha&0\end{pmatrix}$
(3.11)			$\displaystyle\in\mathrm{G}_{\gamma_{m,u}}(F_{v})\begin{pmatrix}1&a^{-1}(b-du^{-1}\alpha)\\ \alpha&0\end{pmatrix}.$

Note that for any $a$ and $d$ in $F_{v}^{\times}$ , we get

	$\displaystyle a-du^{-1}\alpha^{2}-m(a-d)u^{-1}\alpha$	$\displaystyle=a-du^{-1}\alpha^{2}-m(a-d)u^{-1}\alpha+(d-a)u^{-1}(\alpha^{2}-m\alpha+u)$
		$\displaystyle=d-au^{-1}\alpha^{2}$

and

(3.12)

\begin{pmatrix}a&du^{-1}\alpha\\ a\alpha&d\end{pmatrix}=\frac{1}{1-u^{-1}\alpha^{2}}\begin{pmatrix}a-du^{-1}\alpha^{2}&-(a-d)u^{-1}\alpha\\ (a-d)\alpha&-au^{-1}\alpha^{2}+d\end{pmatrix}\begin{pmatrix}1&u^{-1}\alpha\\ \alpha&1\end{pmatrix}\in\mathrm{G}_{\gamma}(F_{v})\begin{pmatrix}1&u^{-1}\alpha\\ \alpha&1\end{pmatrix}.

Let $S$ be the collection of $\left(\begin{smallmatrix}1&u^{-1}\alpha_{i}\\ \alpha_{i}&1\end{smallmatrix}\right)$ , $\left(\begin{smallmatrix}1&y\\ \alpha_{i}&0\end{smallmatrix}\right)$ , and $\left(\begin{smallmatrix}1&x\\ 0&y\end{smallmatrix}\right)$ , where $i\in\{1,2\}$ , $x\in F_{v}$ and $y\in F_{v}^{\times}$ . By (3.10), (3.11) and (3.12), we see that for any $g\in\mathrm{G}(F_{v})$ , there is $g_{0}\in S$ such that

(3.13)

g\in\mathrm{G}_{\gamma_{m,u}}(F_{v})\cdot g_{0}.

Moreover, by a calculation, we deduce that there is a unique $g_{0}$ in $S$ satisfying (3.13). Therefore, the set $S$ is a set of representatives of $\mathrm{G}_{\gamma_{m,u}}(F_{v})\backslash\mathrm{G}(F_{v})$ . ∎

Since $\mathrm{G}_{\gamma_{m,u}}(F_{v})$ is an abelian group and $\mathrm{G}(F_{v})$ is a unimodular group, there is the unique right $\mathrm{G}(F_{v})$ -invariant measure on the homogeneous space $\mathrm{G}_{\gamma_{m,u}}(F_{v})\backslash\mathrm{G}(F_{v})$ up to constant multiplication. Thus, Haar measures on $\mathrm{G}(F_{v})$ and $\mathrm{G}_{\gamma_{m,u}}(F_{v})$ naturally induce a right $\mathrm{G}(F_{v})$ -invariant Haar measure on $\mathrm{G}_{\gamma_{m,u}}(F_{v})\backslash\mathrm{G}(F_{v})$ as follows. By using Lemma 3.6, a measure $dg$ on $\mathrm{G}_{\gamma_{m,u}}(F_{v})\backslash\mathrm{G}(F_{v})$ is defined by

(3.14)

dg:=\begin{cases}\frac{d_{v}xd_{v}y}{|y|_{v}^{2}}\quad&\text{if }g=\mathrm{G}_{\gamma_{m,u}}(F_{v})\left(\begin{smallmatrix}1&x\\ 0&y\end{smallmatrix}\right),\\ 0\quad&\text{otherwise,}\end{cases}

where $x\in F_{v}$ and $y\in F_{v}^{\times}$ . We have to check whether Haar measures on $\mathrm{G}(F_{v})$ , $\mathrm{G}_{\gamma_{m,u}}(F_{v})$ , and $\mathrm{G}_{\gamma_{m,u}}(F_{v})\backslash\mathrm{G}(F_{v})$ are compatible. Let $h_{1}:=aI_{2}-cu^{-1}\gamma_{m,u}\in\mathrm{G}_{\gamma_{m,u}}(F_{v})$ , and let $h_{2}$ be an element in the set of representatives of $\mathrm{G}_{\gamma_{m,u}}(F_{v})\backslash\mathrm{G}(F_{v})$ as given in Lemma 3.6. Let $g:=h_{1}h_{2}\in\mathrm{G}(F_{v})$ . Assume that $h_{2}=\left(\begin{smallmatrix}1&x\\ 0&y\end{smallmatrix}\right)$ with $x\in F_{v}$ and $y\in F_{v}^{\times}$ . By the definitions of Haar measures on $\mathrm{G}_{\gamma_{m,u}}(F_{v})$ and $\mathrm{G}_{\gamma_{m,u}}(F_{v})\backslash\mathrm{G}(F_{v})$ , we have

dh_{1}\,dh_{2}=\frac{q_{v}^{3}}{(q_{v}-1)^{2}(q_{v}+1)}\cdot\frac{d_{v}a\,d_{v}c}{|a^{2}-acmu^{-1}+c^{2}u^{-1}|_{v}}\cdot\frac{d_{v}x\,d_{v}y}{|y|_{v}^{2}}.

In another way, since $g=\left(\begin{smallmatrix}a&ax-cu^{-1}y\\ c&ay+cx-cmu^{-1}y\end{smallmatrix}\right)$ , it follows that

\displaystyle dg=\frac{q_{v}^{3}}{(q_{v}-1)^{2}(q_{v}+1)}\cdot\frac{d_{v}a\,d_{v}c\,d_{v}(ax-cu^{-1}y)\,d_{v}(ay+cx-cmu^{-1}y)}{|y(a^{2}-acmu^{-1}+c^{2}u^{-1})|_{v}^{2}}=dh_{1}\,dh_{2}.

Similarly, when $h_{2}$ does not have the form $\left(\begin{smallmatrix}1&x\\ 0&y\end{smallmatrix}\right)$ , we can obtain $dg=dh_{1}\,dh_{2}$ .

In the following lemma, we obtain an upper bound for the integral

\int_{\overline{\mathrm{G}_{\gamma_{m,u}}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\gamma_{m,u}g)dg.

Lemma 3.7.

Assume that $v\in S_{F,\mathrm{fin}}$ . Let $m$ , $u$ , $f_{v}$ and $N_{v}$ be defined as in Lemma 3.5. Then, we have

\left|\int_{\overline{\mathrm{G}_{\gamma_{m,u}}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\gamma_{m,u}g)dg\right|\leq\begin{cases}\frac{1}{q_{v}}(q_{v}-1+N_{v})\quad&\text{if }e_{v}=f_{v}=0,\\ 4\cdot q_{v}^{-e_{v}+\frac{f_{v}}{2}}\quad&\text{otherwise}.\end{cases}

Here, the measure on $\overline{\mathrm{G}_{\gamma_{m,u}}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})$ is given by (3.14).

Proof.

Assume that $g=\mathrm{G}_{\gamma_{m,u}}(F_{v})\left(\begin{smallmatrix}1&x\\ 0&y\end{smallmatrix}\right)\in\mathrm{G}_{\gamma_{m,u}}(F_{v})\backslash\mathrm{G}(F_{v})$ with $x\in F_{v}$ and $y\in F_{v}^{\times}$ satisfies $\phi_{v}(g^{-1}\gamma_{m,u}g)\neq 0$ . Then, we get

(3.15)

\displaystyle g^{-1}\gamma_{m,u}g=\begin{pmatrix}1&x\\ 0&y\end{pmatrix}^{-1}\begin{pmatrix}0&1\\ -u&m\end{pmatrix}\begin{pmatrix}1&x\\ 0&y\end{pmatrix}=\frac{1}{y}\begin{pmatrix}xu&x^{2}u-mxy+y^{2}\\ -u&-ux+my\end{pmatrix}\in\mathrm{Z}(F_{v})\cdot\mathrm{K}_{v,e_{v}}.

Since $\mathrm{det}(g^{-1}\gamma_{m,u}g)\in\mathcal{O}_{F_{v}}^{\times}$ , it follows that $g^{-1}\gamma_{m,u}g\in\mathrm{K}_{v,e_{v}}$ . By (3.15), if $\phi_{v}(g^{-1}\gamma_{m,u}g)\neq 0$ , then we obtain that $(x,y)$ is in

(3.16)

\left\{(x,y)\in F\times F^{\times}:y^{-1}\in\varpi_{v}^{e_{v}}\mathcal{O}_{F_{v}},\quad xy^{-1}\in\mathcal{O}_{F_{v}},\text{ and }(x^{2}u-mxy+y^{2})y^{-1}\in\mathcal{O}_{F_{v}}\right\}.

If $e_{v}=0$ and $y\in\mathcal{O}_{F_{v}}^{\times}$ , then any $x\in\mathcal{O}_{F_{v}}$ satisfies the condition (3.16). If $\mathrm{val}_{v}(y)<0$ and $\mathrm{val}_{v}(x)>\mathrm{val}_{v}(y)$ , then we have

\mathrm{val}_{v}\left((x^{2}u-mxy+y^{2})y^{-1}\right)=\mathrm{val}_{v}(y)<0.

Thus, if $(x,y)$ is in the set (3.16) with $\mathrm{val}_{v}(y)<0$ , then $\mathrm{val}_{v}(x)=\mathrm{val}_{v}(y)$ . From this, we obtain that if $e_{v}=0$ , then the set (3.16) is equal to

\{(x,y)\in\mathcal{O}_{F_{v}}\times\mathcal{O}_{F_{v}}^{\times}\}\cup\{(\varpi_{v}^{-\alpha}x_{0},\varpi_{v}^{-\alpha}y_{0}):\alpha>0,\quad x_{0},y_{0}\in\mathcal{O}_{F_{v}}^{\times}\text{ and }x_{0}^{2}u-mx_{0}y_{0}+y_{0}^{2}\in\varpi_{v}^{\alpha}\mathcal{O}_{F_{v}}\},

and if $e_{v}>0$ , then the set (3.16) is equal to

\{(\varpi_{v}^{-\alpha}x_{0},\varpi_{v}^{-\alpha}y_{0}):\alpha\geq e_{v},\quad x_{0},y_{0}\in\mathcal{O}_{F_{v}}^{\times}\text{ and }x_{0}^{2}u-mx_{0}y_{0}+y_{0}^{2}\in\varpi_{v}^{\alpha}\mathcal{O}_{F_{v}}\}.

For a positive integer $\alpha$ , let

S_{v,\alpha,m,u}:=\{t\in\mathcal{O}_{F_{v}}/\varpi_{v}^{\alpha}\mathcal{O}_{F_{v}}:t^{2}-mt+u=0\}.

Then, we have

		$\displaystyle\{(\varpi_{v}^{-\alpha}x_{0},\varpi_{v}^{-\alpha}y_{0}):x_{0},y_{0}\in\mathcal{O}_{F_{v}}^{\times}\text{ and }x_{0}^{2}u-mx_{0}y_{0}+y_{0}^{2}\in\varpi_{v}^{\alpha}\mathcal{O}_{F_{v}}\}$
		$\displaystyle=\{(\varpi_{v}^{-\alpha}x_{0},\varpi_{v}^{-\alpha}x_{0}t_{\alpha}+y):x_{0}\in\mathcal{O}_{F_{v}}^{\times},y\in\mathcal{O}_{F_{v}}\text{ and }t_{\alpha}\in S_{v,\alpha,m,u}\}$
		$\displaystyle=\{(\varpi_{v}^{-\alpha}t_{\alpha}^{-1}(x_{0}t_{\alpha}+\varpi_{v}^{\alpha}y)-t_{\alpha}^{-1}y,\varpi_{v}^{-\alpha}(x_{0}t_{\alpha}+\varpi_{v}^{\alpha}y)):x_{0}\in\mathcal{O}_{F_{v}}^{\times},y\in\mathcal{O}_{F_{v}}\text{ and }t_{\alpha}\in S_{v,\alpha,m,u}\}$
		$\displaystyle=\{(\varpi_{v}^{-\alpha}t_{\alpha}^{-1}y_{1}+x_{1},\varpi_{v}^{-\alpha}y_{1}):x_{1}\in\mathcal{O}_{F_{v}},y_{1}\in\mathcal{O}_{F_{v}}^{\times}\text{ and }t_{\alpha}\in S_{v,\alpha,m,u}\}.$

Hence, the set of $g=\left(\begin{smallmatrix}1&x\\ 0&y\end{smallmatrix}\right)$ satisfying $\phi_{v}(g^{-1}\gamma_{m,u}g)\neq 0$ is equal to

(3.17)			$\displaystyle\left\{\begin{pmatrix}1&\varpi_{v}^{-\alpha}t_{\alpha}^{-1}y_{1}+x_{1}\\ 0&\varpi_{v}^{-\alpha}y_{1}\end{pmatrix}:\alpha\geq 1,x_{1}\in\mathcal{O}_{F_{v}},y_{1}\in\mathcal{O}_{F_{v}}^{\times}\text{ and }t_{\alpha}\in S_{v,\alpha,m,u}\right\}$
(3.17)			$\displaystyle\cup\left\{\begin{pmatrix}1&x\\ 0&y\end{pmatrix}:x\in\mathcal{O}_{F_{v}}\text{ and }y\in\mathcal{O}_{F_{v}}^{\times}\right\}$

if $e_{v}=0$ , and

(3.18)

\left\{\begin{pmatrix}1&\varpi_{v}^{-\alpha}t_{\alpha}^{-1}y_{1}+x_{1}\\ 0&\varpi_{v}^{-\alpha}y_{1}\end{pmatrix}:\alpha\geq e_{v},x_{1}\in\mathcal{O}_{F_{v}},y_{1}\in\mathcal{O}_{F_{v}}^{\times}\text{ and }t_{\alpha}\in S_{v,\alpha,m,u}\right\}

if $e_{v}>0$ .

For $t_{\alpha}\in S_{v,\alpha,m,u}$ , the volume of

\left\{\begin{pmatrix}1&\varpi_{v}^{-\alpha}t_{\alpha}^{-1}y_{1}+x_{1}\\ 0&\varpi_{v}^{-\alpha}y_{1}\end{pmatrix}:x_{1}\in\mathcal{O}_{F_{v}}\text{ and }y_{1}\in\mathcal{O}_{F_{v}}^{\times}\right\}

is computed by

(3.19)

\displaystyle\int_{\varpi_{v}^{-\alpha}\mathcal{O}_{F_{v}}^{\times}}\int_{\mathcal{O}_{F_{v}}}\frac{d_{v}xd_{v}y}{|y|^{2}_{v}}=q_{v}^{-\alpha}\int_{\mathcal{O}_{F_{v}}^{\times}}\int_{\mathcal{O}_{F_{v}}}d_{v}xd_{v}y=q_{v}^{-\alpha-1}(q_{v}-1).

Similarly, we get

\mathrm{Vol}\left(\left\{\begin{pmatrix}1&x\\ 0&y\end{pmatrix}:x\in\mathcal{O}_{F_{v}}\text{ and }y\in\mathcal{O}_{F_{v}}^{\times}\right\}\right)=q_{v}^{-1}(q_{v}-1).

To complete the proof of Lemma 3.7, we compute the number of elements in $S_{v,\alpha,m,u}$ . Assume that $f_{v}=0$ and $\alpha\geq 1$ . For each $t\in S_{v,\alpha,m,u}$ , there is $t_{0}\in\mathcal{O}_{F_{v}}$ such that the reduction of $t_{0}$ modulo $\varpi_{v}^{\alpha}$ is equal to $t$ . Since $t\in S_{v,\alpha,m,u}$ and $f_{v}=0$ , there is $s\in\mathcal{O}_{F_{v}}$ such that

t_{0}^{2}-mt_{0}+u=\varpi_{v}^{\alpha}s,

and then $2t_{0}-m\in\mathcal{O}_{F_{v}}^{\times}$ . Note that for $a\in\mathcal{O}_{F_{v}}$ , we have

(t_{0}+a\varpi_{v}^{\alpha})^{2}-m(t_{0}+a\varpi_{v}^{\alpha})+u=\varpi_{v}^{\alpha}(s+a(2t_{0}-m))+\varpi_{v}^{2\alpha}a^{2}.

Since $2t_{0}-m$ is a unit in $\mathcal{O}_{F_{v}}$ , the reduction of $a$ modulo $\varpi_{v}$ is equal to the reduction of $-(2t_{0}-m)^{-1}s$ modulo $\varpi_{v}$ if and only if the reduction of $t_{0}+a\varpi_{v}^{\alpha}$ modulo $\varpi_{v}^{\alpha+1}$ is in $S_{v,\alpha+1,m,u}$ . Thus, if $f_{v}=0$ and $\alpha\geq 1$ , then $\#S_{v,\alpha,m,u}=\#S_{v,\alpha+1,m,u}$ .

If $e_{v}=f_{v}=0$ , then $\#S_{v,\alpha,m,u}=\#S_{v,1,m,u}=N_{v}$ for $\alpha\geq 1$ . Thus, we have

	$\displaystyle\left\|\int_{\overline{\mathrm{G}_{\gamma_{m,u}}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\gamma_{m,u}g)dg\right\|$	$\displaystyle\leq q_{v}^{-1}(q_{v}-1)+\sum_{\alpha\geq 1}N_{v}q_{v}^{-\alpha-1}(q_{v}-1)$
		$\displaystyle=\frac{q_{v}-1+N_{v}}{q_{v}}.$

For any $v\in S_{F,\mathrm{fin}}$ and $\alpha\geq 0$ , if $t\in S_{v,\alpha,m,u}$ , then

\left(t-\frac{m}{2}\right)^{2}-\frac{m^{2}-4u}{4}\in\varpi_{v}^{\alpha}\mathcal{O}_{F_{v}}.

This implies that

\#S_{v,\alpha,m,u}\leq 4q_{v}^{\frac{f_{v}}{2}}.

Therefore, if $e_{v}+f_{v}>0$ , then we conclude that

	$\displaystyle\left\|\int_{\overline{\mathrm{G}_{\gamma_{m,u}}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\gamma_{m,u}g)dg\right\|$	$\displaystyle\leq\sum_{\alpha\geq e_{v}}\#S_{v,\alpha,m,u}\cdot q_{v}^{-\alpha-1}(q_{v}-1)$
		$\displaystyle\leq 4q_{v}^{-e_{v}+\frac{f_{v}}{2}}.$

∎

Now, we assume that $v\in S_{F,\infty}$ . Suppose that there are two roots $\alpha_{1}$ and $\alpha_{2}$ of $x^{2}-\sigma_{v}(m)x+\sigma_{v}(u)=0$ in $F_{v}$ . Since $\left(\begin{smallmatrix}\alpha_{1}&0\\ 0&\alpha_{2}\end{smallmatrix}\right)$ is similar to $\left(\begin{smallmatrix}0&1\\ -\sigma_{v}(u)&\sigma_{v}(m)\end{smallmatrix}\right)$ over $F_{v}$ , we take $\gamma=\left(\begin{smallmatrix}\alpha_{1}&0\\ 0&\alpha_{2}\end{smallmatrix}\right)$ as a representative of the conjugacy class $[\gamma_{m,u}]$ . Then, we have

\mathrm{G}_{\gamma}(F_{v})=\left\{\begin{pmatrix}z_{1}&0\\ 0&z_{2}\end{pmatrix}:z_{1},z_{2}\in F_{v}^{\times}\right\}

and

\mathrm{G}_{\gamma}(F_{v})\backslash\mathrm{G}(F_{v})=\left\{\begin{pmatrix}1&x\\ 0&1\end{pmatrix}\overline{\kappa}:x\in F_{v}\text{ and }\overline{\kappa}\in\left(\mathrm{Z}(F_{v})\cap\mathrm{K}_{v}\right)\backslash\mathrm{K}_{v}\right\}.

Thus, there is an isomorphism $\psi_{v}:\mathrm{G}_{\gamma}(F_{v})\to\prod_{w\mid v}E_{w}^{\times}$ given by

(3.20)

\psi_{v}\left(\begin{pmatrix}z_{1}&0\\ 0&z_{2}\end{pmatrix}\right):=(z_{1},z_{2}).

An isomorphism (3.20) induces a Haar measure $dg$ on $\prod_{w\mid v}E_{w}^{\times}$ defined by

(3.21)

dg:=d_{w}^{\times}z_{1}\,d_{w}^{\times}z_{2},

where $g=(z_{1},z_{2})\in\prod_{w\mid v}E_{w}^{\times}$ . In this case, $F_{v}$ is isomorphic to $E_{w}$ for $w\mid v$ . Also, a Haar measure $dg$ on $\mathrm{G}_{\gamma}(F_{v})\backslash\mathrm{G}(F_{v})$ is defined by

(3.22)

dg:=d_{v}x\,d_{v}\overline{\kappa},

where $g=\left(\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right)\overline{\kappa}\in\mathrm{G}_{\gamma}(F_{v})\backslash\mathrm{G}(F_{v})$ . Here, $d_{v}\overline{\kappa}$ is a Haar measure on $(\mathrm{Z}(F_{v})\cap\mathrm{K}_{v})\backslash\mathrm{K}_{v}$ .

Assume that there is no root of $x^{2}-\sigma_{v}(m)x+\sigma_{v}(u)=0$ in $F_{v}$ . This only occurs when $F_{v}=\mathbb{R}$ and $\sigma_{v}(m^{2}-4u)<0$ . Then, $\gamma=\left(\begin{smallmatrix}r&0\\ 0&r\end{smallmatrix}\right)\left(\begin{smallmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{smallmatrix}\right)$ is similar to $\left(\begin{smallmatrix}0&1\\ -\sigma_{v}(u)&\sigma_{v}(m)\end{smallmatrix}\right)$ over $\mathbb{R}$ , where

(3.23)

r:=\sqrt{\sigma_{v}(u)},\text{ and }\theta:=\tan^{-1}\left(\frac{\sqrt{\sigma_{v}(4u-m^{2})}}{\sigma_{v}(m)}\right).

Thus, we obtain that

\mathrm{G}_{\gamma}(F_{v})=\left\{\begin{pmatrix}r&0\\ 0&r\end{pmatrix}\cdot\kappa:r>0\text{ and }\kappa\in\mathrm{SO}(2)\right\}

and that

\mathrm{G}_{\gamma}(F_{v})\backslash\mathrm{G}(F_{v})=\left\{\begin{pmatrix}y&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&x\\ 0&1\end{pmatrix}:y\in\mathbb{R}^{\times}\text{ and }x\in\mathbb{R}\right\}.

By the Iwasawa decomposition, for $g\in\mathrm{G}(F_{v})$ , we have

g=\begin{pmatrix}r_{1}&0\\ 0&r_{1}\end{pmatrix}\begin{pmatrix}y_{1}&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&x_{1}\\ 0&1\end{pmatrix}\kappa_{1}=\begin{pmatrix}r_{2}&0\\ 0&r_{2}\end{pmatrix}\kappa_{2}\begin{pmatrix}y_{2}&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&x_{2}\\ 0&1\end{pmatrix}.

Since a Haar measure on $\mathrm{G}(\mathbb{R})$ is a unique up to constant multiple, there is a constant $C>0$ such that

(3.24)

dg=d^{\times}_{v}r_{1}\,d_{v}^{\times}y_{1}\,d_{v}x_{1}\,d_{v}\kappa_{1}=C\cdot d^{\times}_{v}r_{2}\,d_{v}\kappa_{2}\,d_{v}^{\times}y_{2}\,d_{v}x_{2}.

Note that there is an isomorphism $\psi_{v}:\mathrm{G}_{\gamma}(F_{v})\to\prod_{w\mid v}E_{w}^{\times}$ given by

(3.25)

\psi_{v}\left(\begin{pmatrix}r&0\\ 0&r\end{pmatrix}\cdot\kappa_{\theta}\right):=r\cdot e^{i\theta}.

An isomorphism (3.25) induces a Haar measure $dg$ on $E_{w}^{\times}$ defined by

(3.26)

dg:=C\cdot d_{v}^{\times}r\,d\kappa_{\theta}=C\cdot\frac{dr\,d\theta}{2\pi r}=C\cdot\frac{d^{\times}_{\mathbb{C}}g}{2},

where $C$ is given as in (3.24) and $g=re^{i\theta}$ . Also a Haar measure $dg$ on $\mathrm{G}_{\gamma}(F_{v})\backslash\mathrm{G}(F_{v})$ is defined by

(3.27)

dg:=d_{v}^{\times}y\,d_{v}x,

where $g=\left(\begin{smallmatrix}y&0\\ 0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right)\in\mathrm{G}_{\gamma}(F_{v})\backslash\mathrm{G}(F_{v})$ .

By (3.21), (3.26) and the definition of the regulator of a number field, we have

\mathrm{Vol}\left(i_{\infty}(\mathcal{O}_{E}^{\times})\backslash\mathbb{A}^{1}_{E,\infty}\right)=\left(\frac{C}{2}\right)^{a}\cdot\frac{\mathrm{reg}(E)}{w_{E}}.

Here, $w_{E}$ denotes the number of roots of unity in $E$ , $\mathrm{reg}(E)$ denotes the regulator of $E$ , and $a$ is the number of $v\in S_{F,\infty}$ such that $F_{v}=\mathbb{R}$ and $\sigma_{v}(m^{2}-4u)<0$ . It follows that

(3.28)

\mathrm{Vol}\left(i_{\infty}(\mathcal{O}_{E}^{\times})\backslash\mathbb{A}^{1}_{E,\infty}\right)\ll_{F}\mathrm{reg}(E).

Combining Lemmas 3.3, 3.5, 3.7 and (3.28), we obtain

		$\displaystyle\mathrm{Vol}\left(\overline{\mathrm{G}_{\gamma_{m,u}}(F)}\backslash\overline{\mathrm{G}_{\gamma_{m,u}}(\mathbb{A}_{F})}\right)\prod_{v\in S_{F,\mathrm{fin}}}\left\|\int_{\overline{\mathrm{G}_{\gamma_{m,u}}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\gamma_{m,u}g)dg\right\|$
		$\displaystyle\ll_{F}\mathrm{cl}(E)\cdot\mathrm{reg}(E)\cdot\prod_{\begin{subarray}{c}v\in S_{F,\mathrm{fin}}\\ f_{v}>0\end{subarray}}\left(\frac{q_{v}^{2[\frac{f_{v}+1}{2}]+3}}{(q_{v}-1)^{2}(q_{v}+1)}\right)\cdot\prod_{\begin{subarray}{c}v\in S_{F,\mathrm{fin}}\\ f_{v}=0\end{subarray}}\left(\frac{q_{v}(q_{v}+1-N_{v})}{q_{v}^{2}-1}\right)$
		$\displaystyle\times\prod_{\begin{subarray}{c}v\in S_{F,\mathrm{fin}}\\ e_{v}+f_{v}=0\end{subarray}}\left(\frac{q_{v}-1+N_{v}}{q_{v}}\right)\cdot\prod_{\begin{subarray}{c}v\in S_{F,\mathrm{fin}}\\ e_{v}+f_{v}>0\end{subarray}}\left(4q_{v}^{-e_{v}+\frac{f_{v}}{2}}\right).$

Note that we have

\prod_{\begin{subarray}{c}v\in S_{F,\mathrm{fin}},\\ f_{v}>0\end{subarray}}\frac{q_{v}^{2[\frac{f_{v}+1}{2}]+3}}{(q_{v}-1)^{2}(q_{v}+1)}\leq\prod_{\begin{subarray}{c}v\in S_{F,\mathrm{fin}},\\ f_{v}>0\end{subarray}}q_{v}^{5f_{v}}=\left|N_{F/\mathbb{Q}}(m^{2}-4u)\right|^{5}.

Let $n:=[F:\mathbb{Q}]$ and $\omega(m)$ be the number of prime divisors of an integer $m$ . Since there are at most $n$ prime ideals in $\mathcal{O}_{F}$ lying over a prime $p\in\mathbb{Z}$ , the number of $v\in S_{F,\mathrm{fin}}$ such that $e_{v}>0$ is less than or equal to $n\cdot\omega(|N_{F/\mathbb{Q}}(J)|)$ . Thus, we obtain

		$\displaystyle\prod_{\begin{subarray}{c}v\in S_{F,\mathrm{fin}}\\ f_{v}=0\end{subarray}}\left(\frac{q_{v}(q_{v}+1-N_{v})}{q_{v}^{2}-1}\right)\cdot\prod_{\begin{subarray}{c}v\in S_{F,\mathrm{fin}}\\ e_{v}+f_{v}=0\end{subarray}}\left(\frac{q_{v}-1+N_{v}}{q_{v}}\right)$
		$\displaystyle=\prod_{\begin{subarray}{c}v\in S_{F,\mathrm{fin}}\\ f_{v}=0\end{subarray}}\left(\frac{q_{v}(q_{v}+1-N_{v})}{q_{v}^{2}-1}\cdot\frac{q_{v}-1+N_{v}}{q_{v}}\right)\cdot\prod_{\begin{subarray}{c}v\in S_{F,\mathrm{fin}}\\ f_{v}=0,\,e_{v}>0\end{subarray}}\left(\frac{q_{v}-1+N_{v}}{q_{v}}\right)^{-1}$
		$\displaystyle\leq 2^{n\cdot\omega(\|N_{F/\mathbb{Q}}(J)\|)}.$

The last inequality holds since $\frac{q_{v}}{q_{v}-1+N_{v}}\leq 2$ . Similarly, we get

\prod_{\begin{subarray}{c}v\in S_{F,\mathrm{fin}}\\ e_{v}+f_{v}>0\end{subarray}}4q_{v}^{-e_{v}+\frac{f_{v}}{2}}\leq 4^{n\cdot\omega(|N_{F/\mathbb{Q}}(J)|\cdot|N_{F/\mathbb{Q}}(m^{2}-4u)|)}\cdot|N_{F/\mathbb{Q}}(J)|^{-1}\cdot|N_{F/\mathbb{Q}}(m^{2}-4u)|^{\frac{1}{2}}.

By [17, Theorem 1], we also have

(3.29)

\mathrm{cl}(E)\cdot\mathrm{reg}(E)\leq\frac{w_{E}}{2}\cdot\left(\frac{2}{\pi}\right)^{r_{2,E}}\cdot\left(\frac{e\log\Delta_{E}}{4(2n-1)}\right)^{2n-1}\sqrt{\Delta_{E}},

where $e$ denotes the Euler number, $r_{2,E}$ denotes the number of conjugate pairs of complex embeddings of $E$ , and $\Delta_{E}$ denotes the discriminant of $E$ . Note that the number of roots of unity of a number field $K$ is bounded if the extension degree $[K:\mathbb{Q}]$ is fixed. Since $E:=F(\sqrt{m^{2}-4u})$ , it follows that $\Delta_{E}=\Delta_{F}^{2}\cdot N_{F/\mathbb{Q}}(m^{2}-4u)$ . Note that for a fixed positive integer $M$ and for any $\epsilon>0$ , we have

M^{\omega(N)}\ll_{\epsilon}N^{\epsilon},\quad N\to\infty.

Thus, we conclude that

(3.30)		$\displaystyle\mathrm{Vol}\left(\overline{\mathrm{G}_{\gamma_{m,u}}(F)}\backslash\overline{\mathrm{G}_{\gamma_{m,u}}(\mathbb{A}_{F})}\right)$	$\displaystyle\prod_{v\in S_{F,\mathrm{fin}}}\left\|\int_{\overline{\mathrm{G}_{\gamma_{m,u}}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\gamma_{m,u}g)dg\right\|$
(3.30)			$\displaystyle\ll_{F}\|N_{F/\mathbb{Q}}(J)\|^{-\frac{1}{2}}\cdot\|N_{F/\mathbb{Q}}(m^{2}-4u)\|^{7}.$

To complete the calculation of $S_{\mathrm{ell}}(\phi)$ , we compute the integral

(3.31)

\int_{\overline{\mathrm{G}_{\sigma_{v}(\gamma)}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\sigma_{v}(\gamma)g)dg

for $v\in S_{F,\infty}$ . Since the Haar measure on $\overline{\mathrm{G}_{\sigma_{v}(\gamma)}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})$ is defined differently in the two cases, we compute (3.31) in the following two lemmas.

Lemma 3.8.

Assume that $v\in S_{F,\infty}$ and that there are two roots of $x^{2}-\sigma_{v}(m)x+\sigma_{v}(u)=0$ in $F_{v}$ . Let

\sigma_{v}(\alpha_{\gamma}):=\frac{\sigma_{v}(m)+\sqrt{\sigma_{v}(m^{2}-4u)}}{\sigma_{v}(m)-\sqrt{\sigma_{v}(m^{2}-4u)}}.

Then, we have

(3.32)

\left|\int_{\overline{\mathrm{G}_{\sigma_{v}(\gamma_{m,u})}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\sigma_{v}(\gamma_{m,u})g)dg\right|\leq\frac{(1+\epsilon_{v})}{2\pi}\left|\frac{\sigma_{v}(u)}{\sigma_{v}(m^{2}-4u)}\right|^{\frac{1+\epsilon_{v}}{2}}.

Proof.

Since $\sigma_{v}(\alpha_{\gamma})$ is the ratio of two roots of $x^{2}-\sigma_{v}(m)x+\sigma_{v}(u)=0$ in $F_{v}$ , there are $g_{0}\in\mathrm{G}(F_{v})$ and $z_{0}\in\mathrm{Z}(F_{v})$ such that

\gamma_{v}:=\begin{pmatrix}\sigma_{v}(\alpha_{\gamma})&0\\ 0&1\end{pmatrix}=z_{0}g_{0}\gamma_{m,u}g_{0}^{-1}.

Thus, we have

\int_{\overline{\mathrm{G}_{\sigma_{v}(\gamma_{m,u})}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\sigma_{v}(\gamma_{m,u})g)dg=\int_{\overline{\mathrm{G}_{\gamma_{v}}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\gamma_{v}g)dg.

By the Iwasawa decomposition,

\overline{\rm G_{\gamma_{v}}(F_{v})}\backslash\bar{\rm G}(F_{v})\cong\left\{\begin{pmatrix}1&x\\ 0&1\end{pmatrix}:x\in F_{v}\right\}\overline{{\rm K}_{v}}.

Since $\phi_{v}$ is bi- $\mathrm{K}_{v}$ -invariant and $\mathrm{Vol}(\overline{\mathrm{K}_{v}})=1$ , we have

(3.33)

\int_{\overline{\rm G_{\gamma_{v}}(F_{v})}\backslash\bar{\rm G}(F_{v})}\phi_{v}(g^{-1}\gamma_{v}g)dg=\int_{F_{v}}\phi_{v}\left(\begin{pmatrix}1&-x\\ 0&1\end{pmatrix}\begin{pmatrix}\sigma_{v}(\alpha_{\gamma})&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&x\\ 0&1\end{pmatrix}\right)d_{v}x.

Assume that $F_{v}=\mathbb{R}$ . Since the support of $\phi_{v}$ is contained in $\mathrm{GL}_{2}^{+}(\mathbb{R})$ , it follows that

\phi_{v}\left(\begin{pmatrix}1&-x\\ 0&1\end{pmatrix}\begin{pmatrix}\sigma_{v}(\alpha_{\gamma})&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&x\\ 0&1\end{pmatrix}\right)=\phi_{v}\left(\begin{pmatrix}\sigma_{v}(\alpha_{\gamma})&x(\sigma_{v}(\alpha_{\gamma})-1)\\ 0&1\end{pmatrix}\right)=0

unless $\sigma_{v}(\alpha_{\gamma})>0$ . If $\sigma_{v}(\alpha_{\gamma})>0$ , then by the Cartan decomposition, there exist $r>0$ and $\kappa_{1},\kappa_{2}\in\mathrm{K}_{v}$ such that

(3.34)

g:=\begin{pmatrix}\sigma_{v}(\alpha_{\gamma})&x(\sigma_{v}(\alpha_{\gamma})-1)\\ 0&1\end{pmatrix}=\sigma_{v}(\alpha_{\gamma})^{\frac{1}{2}}\kappa_{1}\begin{pmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{pmatrix}\kappa_{2}.

Then, we have the following relation between $x$ and $r$ :

{\rm tr}(gg^{*})=\sigma_{v}(\alpha_{\gamma})^{2}+x^{2}(\sigma_{v}(\alpha_{\gamma})-1)^{2}+1=2\sigma_{v}(\alpha_{\gamma})\cosh(r),

where $g^{*}$ denotes the conjugate transpose of $g$ . Since $\phi_{v}$ is $\mathrm{Z}(F_{v})$ -invariant and bi- $\mathrm{K}_{v}$ -invariant, it follows that

\phi_{v}(g)=\phi_{v}\left(\begin{pmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{pmatrix}\right).

Let $g_{1}:=\left(\begin{smallmatrix}\sigma_{v}(\alpha_{\gamma})&|x(\sigma_{v}(\alpha_{\gamma})-1)|\\ 0&1\end{smallmatrix}\right)$ . Since the trace of $gg^{*}$ is equal to the trace of $g_{1}g_{1}^{*}$ , we have

(3.35)

\phi_{v}\left(\begin{pmatrix}\sigma_{v}(\alpha_{\gamma})&x(\sigma_{v}(\alpha_{\gamma})-1)\\ 0&1\end{pmatrix}\right)=\phi_{v}\left(\begin{pmatrix}\sigma_{v}(\alpha_{\gamma})&|x(\sigma_{v}(\alpha_{\gamma})-1)|\\ 0&1\end{pmatrix}\right).

Hence, by changing the variable $x^{2}(\sigma_{v}(\alpha_{\gamma})-1)^{2}=2\sigma_{v}(\alpha_{\gamma})\cosh(r)-\sigma_{v}(\alpha_{\gamma})^{2}-1$ , we have

(3.36)	$\displaystyle\int_{F_{v}}$	$\displaystyle\phi_{v}\left(\begin{pmatrix}\sigma_{v}(\alpha_{\gamma})&x(\sigma_{v}(\alpha_{\gamma})-1)\\ 0&1\end{pmatrix}\right)dx$
		$\displaystyle=\frac{\sigma_{v}(\alpha_{\gamma})^{\frac{1}{2}}}{\|\sigma_{v}(\alpha_{\gamma})-1\|}\int_{\log\sigma_{v}(\alpha_{\gamma})}^{\infty}\frac{\phi_{v}\left(\left(\begin{smallmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{smallmatrix}\right)\right)\sinh(r)}{\sqrt{\sinh^{2}(r/2)-\sinh^{2}\left(\frac{\log\sigma_{v}(\alpha_{\gamma})}{2}\right)}}dr$
		$\displaystyle=\frac{\sigma_{v}(\alpha_{\gamma})^{\frac{1}{2}}}{2\pi\|\sigma_{v}(\alpha_{\gamma})-1\|}\widehat{h_{\phi_{v}}}\left(\frac{\log\sigma_{v}(\alpha_{\gamma})}{2\pi}\right).$

The last equality holds by (2.3).

Assume that $F_{v}=\mathbb{C}$ . Similarly, for $x\in\mathbb{C}$ , we have

\phi_{v}\left(\begin{pmatrix}\sigma_{v}(\alpha_{\gamma})&x(\sigma_{v}(\alpha_{\gamma})-1)\\ 0&1\end{pmatrix}\right)=\phi_{v}\left(\begin{pmatrix}|\sigma_{v}(\alpha_{\gamma})|&|x||\sigma_{v}(\alpha_{\gamma})-1|\\ 0&1\end{pmatrix}\right).

Then, we have

(3.37)		$\displaystyle\int_{F_{v}}\phi_{v}\left(\begin{pmatrix}\sigma_{v}(\alpha_{\gamma})&x(\sigma_{v}(\alpha_{\gamma})-1)\\ 0&1\end{pmatrix}\right)dx$	$\displaystyle=\int_{F_{v}}\phi_{v}\left(\begin{pmatrix}\|\sigma_{v}(\alpha_{\gamma})\|&\|x\|\|\sigma_{v}(\alpha_{\gamma})-1\|\\ 0&1\end{pmatrix}\right)dx$
(3.37)			$\displaystyle=2\pi\int_{0}^{\infty}\phi_{v}\left(\begin{pmatrix}\|\sigma_{v}(\alpha_{\gamma})\|&y\|\sigma_{v}(\alpha_{\gamma})-1\|\\ 0&1\end{pmatrix}\right)ydy.$

Similarly, by changing the variable $y^{2}|\sigma_{v}(\alpha_{\gamma})-1|^{2}=2|\sigma_{v}(\alpha_{\gamma})|\cosh(r)-|\sigma_{v}(\alpha_{\gamma})|^{2}-1$ , we have

(3.38)		$\displaystyle 2\pi\int_{0}^{\infty}\phi_{v}\left(\begin{pmatrix}\|\sigma_{v}(\alpha_{\gamma})\|&y\|\sigma_{v}(\alpha_{\gamma})-1\|\\ 0&1\end{pmatrix}\right)ydy$	$\displaystyle=\frac{2\pi\|\sigma_{v}(\alpha_{\gamma})\|}{\|\sigma_{v}(\alpha_{\gamma})-1\|^{2}}\int_{\log\|\sigma_{v}(\alpha_{\gamma})\|}^{\infty}\phi_{v}\left(\begin{pmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{pmatrix}\right)\sinh(r)dr$
(3.38)			$\displaystyle=\frac{\|\sigma_{v}(\alpha_{\gamma})\|}{\pi\|\sigma_{v}(\alpha_{\gamma})-1\|^{2}}\widehat{h_{\phi_{v}}}\left(\frac{\log\|\sigma_{v}(\alpha_{\gamma})\|}{\pi}\right).$

Since $h_{v}$ is a non-negative even function on $\mathbb{R}$ , for any $x\in\mathbb{R}$ , we have

\displaystyle\left|\widehat{h_{v}}(x)\right|=\left|\int_{\mathbb{R}}h_{v}(z)e^{-2\pi ixz}dz\right|=\left|\int_{\mathbb{R}}h_{v}(z)\cos(2\pi xz)dz\right|\leq\left|\int_{\mathbb{R}}h_{v}(z)dz\right|=1.

Thus, we complete the proof of Lemma 3.8. ∎

Lemma 3.9.

Assume that $v\in S_{F,\infty}$ and that there is no root of $x^{2}-\sigma_{v}(m)x+\sigma_{v}(u)=0$ in $F_{v}$ . Then, we have

\left|\int_{\overline{\mathrm{G}_{\sigma_{v}(\gamma_{m,u})}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\sigma_{v}(\gamma_{m,u})g)dg\right|\leq\frac{1}{\sqrt{2}}\left|\frac{\sigma_{v}(u)}{\sigma_{v}(4u-m^{2})}\right|\cdot h_{\phi_{v}}(0).

Proof.

Following the proof of the case for $D<0$ of [1, Lemma 2.1], we have

(3.39)

\int_{\overline{\mathrm{G}_{\sigma_{v}(\gamma_{m,u})}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\sigma_{v}(\gamma_{m,u})g)dg=\frac{1}{4\sqrt{2}}\int_{-\infty}^{\infty}\widehat{h_{\phi_{v}}}(x)\frac{\cosh(\pi x)}{\sinh^{2}(\pi x)+\sigma_{v}\left(\frac{4u-m^{2}}{4u}\right)}dx.

Since

\displaystyle\left|\sinh^{2}(\pi x)+\sigma_{v}\left(\frac{4u-m^{2}}{4u}\right)\right|\geq\left|\sigma_{v}\left(\frac{4u-m^{2}}{4u}\right)\cdot\cosh(\pi x)\right|,

we conclude that

	$\displaystyle\left\|\frac{1}{4\sqrt{2}}\int_{-\infty}^{\infty}\widehat{h_{\phi_{v}}}(x)\frac{\cosh(\pi x)}{\sinh^{2}(\pi x)+\sigma_{v}\left(\frac{4u-m^{2}}{4u}\right)}dx\right\|$	$\displaystyle\leq\frac{1}{4\sqrt{2}}\left\|\frac{\sigma_{v}(4u)}{\sigma_{v}(4u-m^{2})}\right\|\cdot\left\|\int_{-\infty}^{\infty}\widehat{h_{\phi_{v}}}(x)dx\right\|$
		$\displaystyle=\frac{1}{\sqrt{2}}\left\|\frac{\sigma_{v}(u)}{\sigma_{v}(4u-m^{2})}\right\|\cdot h_{\phi_{v}}(0).$

∎

3.4. Hyperbolic contribution

For each place $v$ of $F$ , we define a function $H_{v}$ on $\mathrm{G}(F_{v})$ by

(3.40)

H_{v}\left(\begin{pmatrix}a&x\\ 0&b\end{pmatrix}\kappa\right):=\left|\frac{a}{b}\right|_{v},

where $a,b,x\in F_{v}$ and $\kappa_{v}\in\mathrm{K}_{v}$ . Let $H$ be a function on $\mathrm{G}(\mathbb{A}_{F})$ defined by

H:=\prod_{v\in S_{F}}H_{v}.

In [6, (6.35)], $S_{\mathrm{hyp}}(\phi)$ is defined by

(3.41)

S_{\mathrm{hyp}}(\phi):=-\frac{{\mathrm{Vol}}(F^{\times}\backslash\mathbb{A}_{F}^{1})}{2}\sum_{\begin{subarray}{c}\alpha\in F^{\times}\\ \alpha\neq 1\end{subarray}}J_{\alpha}(\phi),

where for each $\alpha\in F^{\times}$ with $\alpha\neq 1$ ,

(3.42)

J_{\alpha}(\phi):=\int_{\mathrm{K}}\int_{\mathbb{A}_{F}}\phi\left(\kappa^{-1}\begin{pmatrix}1&-x\\ 0&1\end{pmatrix}\begin{pmatrix}\alpha&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&x\\ 0&1\end{pmatrix}\kappa\right)\sum_{v\in S_{F}}\log H_{v}\left(\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\begin{pmatrix}1&x\\ 0&1\end{pmatrix}\kappa\right)dx\,d\kappa.

If $J_{\alpha}(\phi)\neq 0$ , then there are $\kappa_{v}\in\mathrm{K}_{v}$ and $x_{v}\in F_{v}$ such that $\phi_{v}\left(\kappa_{v}^{-1}\left(\begin{smallmatrix}1&-x_{v}\\ 0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}\alpha&0\\ 0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&x_{v}\\ 0&1\end{smallmatrix}\right)\kappa_{v}\right)$ is non-zero for each $v\in S_{F,\mathrm{fin}}$ . Since $\phi_{v}$ is a characteristic function of $\mathrm{Z}(F_{v})\mathrm{K}_{v,e_{v}}$ , we have

\kappa_{v}^{-1}\begin{pmatrix}1&-x_{v}\\ 0&1\end{pmatrix}\begin{pmatrix}\alpha&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&x_{v}\\ 0&1\end{pmatrix}\kappa_{v}\in\mathrm{Z}(F_{v})\mathrm{K}_{v,e_{v}}\subset\mathrm{Z}(F_{v})\mathrm{K}_{v}.

This implies that

\begin{pmatrix}1&-x_{v}\\ 0&1\end{pmatrix}\begin{pmatrix}\alpha&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&x_{v}\\ 0&1\end{pmatrix}=\begin{pmatrix}\alpha&x_{v}(\alpha-1)\\ 0&1\end{pmatrix}\in\mathrm{Z}(F_{v})\mathrm{K}_{v}.

Hence, if $J_{\alpha}(\phi)\neq 0$ , then $\alpha\in\mathcal{O}_{F_{v}}^{\times}$ for each $v\in S_{F,\mathrm{fin}}$ , and so $\alpha\in\mathcal{O}_{F}^{\times}$ .

For $v,w\in S_{F}$ , if $v\neq w$ , then let

J_{\alpha,v}^{w}(\phi_{v}):=\int_{\mathrm{K}_{v}}\int_{F_{v}}\phi_{v}\left(\kappa^{-1}\begin{pmatrix}1&-x\\ 0&1\end{pmatrix}\begin{pmatrix}\alpha&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&x\\ 0&1\end{pmatrix}\kappa\right)d_{v}x\,d_{v}\kappa,

and if $v=w$ , then

J_{\alpha,v}^{v}(\phi_{v}):=\int_{\mathrm{K}_{v}}\int_{F_{v}}\phi_{v}\left(\kappa^{-1}\begin{pmatrix}1&-x\\ 0&1\end{pmatrix}\begin{pmatrix}\alpha&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&x\\ 0&1\end{pmatrix}\kappa\right)\log H_{v}\left(\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\begin{pmatrix}1&x\\ 0&1\end{pmatrix}\kappa\right)d_{v}x\,d_{v}\kappa.

Let $\sigma_{v}:F\to F_{v}$ be an embedding corresponding to an archimedean place $v$ of $F$ . By (3.42), we have

(3.43)

J_{\alpha}(\phi)=\sum_{w\in S_{F}}\prod_{v\in S_{F,\mathrm{fin}}}J_{\alpha,v}^{w}(\phi_{v})\prod_{v\in S_{F,\infty}}J_{\sigma_{v}(\alpha),v}^{w}(\phi_{v}).

Lemma 3.10.

Let $\alpha\in\mathcal{O}_{F}^{\times}$ and $v\in S_{F,\mathrm{fin}}$ . If $\mathrm{val}_{v}(\alpha-1)=0$ , then $J_{\alpha,v}^{v}(\phi_{v})=0$ .

Proof.

Assume that $\kappa_{v}\in\mathrm{K}_{v}$ and $x_{v}\in F_{v}$ satisfying

\phi_{v}\left(\kappa_{v}^{-1}\begin{pmatrix}1&-x_{v}\\ 0&1\end{pmatrix}\begin{pmatrix}\alpha&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&x_{v}\\ 0&1\end{pmatrix}\kappa_{v}\right)\neq 0.

Since the determinant of $\kappa_{v}^{-1}\left(\begin{smallmatrix}1&-x_{v}\\ 0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}\alpha&0\\ 0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&x_{v}\\ 0&1\end{smallmatrix}\right)\kappa_{v}$ is in $\mathcal{O}_{F}^{\times}$ , we have

\kappa_{v}^{-1}\begin{pmatrix}1&-x_{v}\\ 0&1\end{pmatrix}\begin{pmatrix}\alpha&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&x_{v}\\ 0&1\end{pmatrix}\kappa_{v}\in\mathrm{K}_{v,e_{v}}.

Let $\kappa_{v}=\left(\begin{smallmatrix}a_{v}&b_{v}\\ c_{v}&d_{v}\end{smallmatrix}\right)\in\mathrm{K}_{v}$ . By a computation, we have

		$\displaystyle\kappa_{v}^{-1}\begin{pmatrix}1&-x_{v}\\ 0&1\end{pmatrix}\begin{pmatrix}\alpha&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&x_{v}\\ 0&1\end{pmatrix}\kappa_{v}$
		$\displaystyle=(\mathrm{det}\kappa_{v})^{-1}\begin{pmatrix}\alpha a_{v}d_{v}+(\alpha-1)x_{v}c_{v}d_{v}-b_{v}c_{v}&\alpha b_{v}d_{v}+(\alpha-1)x_{v}d_{v}^{2}-b_{v}d_{v}\\ \ -(\alpha-1)c_{v}(a_{v}+x_{v}c_{v})&-\alpha b_{v}c_{v}-(\alpha-1)x_{v}c_{v}d_{v}+a_{v}d_{v}\end{pmatrix}.$

Thus, we obtain that

(3.44)

(\alpha-1)x_{v}c_{v}d_{v}\in\mathcal{O}_{F_{v}},\quad(\alpha-1)x_{v}d_{v}^{2}\in\mathcal{O}_{F_{v}},\text{ and }(\alpha-1)c_{v}(a_{v}+x_{v}c_{v})\in\varpi_{v}^{e_{v}}\mathcal{O}_{F_{v}}.

From the assumption that $\alpha-1\in\mathcal{O}_{F_{v}}^{\times}$ , the conditions (3.44) are equivalent to

x_{v}c_{v}d_{v}\in\mathcal{O}_{F_{v}},\quad x_{v}d_{v}^{2}\in\mathcal{O}_{F_{v}},\text{ and }c_{v}(a_{v}+x_{v}c_{v})\in\varpi_{v}^{e_{v}}\mathcal{O}_{F_{v}}.

Since $\mathrm{det}(\kappa_{v})=a_{v}d_{v}-b_{v}c_{v}\in\mathcal{O}_{F_{v}}^{\times}$ , it follows that at least one of $c_{v}$ and $d_{v}$ is in $\mathcal{O}_{F_{v}}^{\times}$ . If $c_{v}\in\mathcal{O}_{F_{v}}^{\times}$ , then $x_{v}\in\mathcal{O}_{F_{v}}$ because $c_{v}(x_{v}c_{v}+a_{v})\in\mathcal{O}_{F_{v}}$ . Otherwise, if $d_{v}\in\mathcal{O}_{F_{v}}^{\times}$ , then we also have $x_{v}\in\mathcal{O}_{F_{v}}$ by $x_{v}d_{v}^{2}\in\mathcal{O}_{F_{v}}$ . Thus, we obtain that $x_{v}\in\mathcal{O}_{F_{v}}$ . This implies that

(3.45)

\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\begin{pmatrix}1&x_{v}\\ 0&1\end{pmatrix}\kappa_{v}\in\mathrm{K}_{v}.

Therefore, we have

\log H_{v}\left(\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\begin{pmatrix}1&x_{v}\\ 0&1\end{pmatrix}\kappa_{v}\right)=0.

∎

By (3.43) and Lemma 3.10, we can rewrite $J_{\alpha}(\phi)$ as

(3.46)

J_{\alpha}(\phi)=\sum_{\begin{subarray}{c}w\in S_{F,\mathrm{fin}},\\ \mathrm{val}_{w}(\alpha-1)>0\end{subarray}}\prod_{v\in S_{F,\mathrm{fin}}}J_{\alpha,v}^{w}(\phi_{v})\prod_{v\in S_{F,\infty}}J_{\sigma_{v}(\alpha),v}^{w}(\phi_{v})+\sum_{w\in S_{F,\infty}}\prod_{v\in S_{F,\mathrm{fin}}}J_{\alpha,v}^{w}(\phi_{v})\prod_{v\in S_{F,\infty}}J_{\sigma_{v}(\alpha),v}^{w}(\phi_{v}).

The following lemma provides an upper bound for $|J^{w}_{\alpha,v}(\phi_{v})|$ when $v\in S_{F,\mathrm{fin}}$ .

Lemma 3.11.

Assume that $v\in S_{F,\mathrm{fin}}$ and $\alpha\in\mathcal{O}_{F}^{\times}$ . Let $g_{v}:=\mathrm{val}_{v}(\alpha-1)$ . Then,

\left|J_{\alpha,v}^{w}(\phi_{v})\right|\leq\begin{cases}q_{v}^{g_{v}-\frac{e_{v}}{2}}\quad&\text{if }w\neq v,\\ 2f_{v}\cdot\log q_{v}\cdot q_{v}^{g_{v}-\frac{e_{v}}{2}}\quad&\text{if }w=v.\end{cases}

Proof.

First, we assume that $w\neq v$ . Let $R_{v}$ be the collection of all $(\kappa_{v},x_{v})\in\mathrm{K}_{v}\times F_{v}$ satisfying the condition $\phi_{v}\left(\kappa_{v}^{-1}\left(\begin{smallmatrix}1&-x_{v}\\ 0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}\alpha&0\\ 0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&x_{v}\\ 0&1\end{smallmatrix}\right)\kappa_{v}\right)\neq 0$ . Since $\phi_{v}$ is a characteristic function of $\mathrm{Z}(F_{v})\mathrm{K}_{v,e_{v}}$ , it follows that $|J_{\alpha,v}^{w}(\phi_{v})|$ is equal to the volume of $R_{v}$ . By (3.44), we obtain that $\left(\left(\begin{smallmatrix}a_{v}&b_{v}\\ c_{v}&d_{v}\end{smallmatrix}\right),x_{v}\right)\in R_{v}$ if and only if

(3.47)

x_{v}c_{v}d_{v}\in\varpi_{v}^{-g_{v}}\mathcal{O}_{F_{v}},\quad x_{v}d_{v}^{2}\in\varpi_{v}^{-g_{v}}\mathcal{O}_{F_{v}},\text{ and }c_{v}(a_{v}+x_{v}c_{v})\in\varpi_{v}^{e_{v}-g_{v}}\mathcal{O}_{F_{v}}.

By dividing the set $R_{v}$ according to $\mathrm{val}_{v}(c_{v})$ , we have

(3.48)

R_{v}=\left(R_{v}\cap\left(\mathrm{K}_{v,[\frac{e_{v}}{2}]+1}\times F_{v}\right)\right)\bigcup\left(\bigcup_{i=0}^{[\frac{e_{v}}{2}]}R_{v}\cap\left(\left(\mathrm{K}_{v,i}\backslash\mathrm{K}_{v,i+1}\right)\times F_{v}\right)\right).

If $\mathrm{val}_{v}(c_{v})\geq[\frac{e_{v}}{2}]+1>0$ , then $d_{v}\in\mathcal{O}_{F_{v}}^{\times}$ . Then, we have $x_{v}\in\varpi_{v}^{-g_{v}}\mathcal{O}_{F_{v}}$ by (3.47). Thus, we get

(3.49)		$\displaystyle\mathrm{Vol}\left(R_{v}\cap\left(\mathrm{K}_{v,[\frac{e_{v}}{2}]+1}\times F_{v}\right)\right)$	$\displaystyle\leq\mathrm{Vol}\left(\left\{\left(\left(\begin{smallmatrix}a_{v}&b_{v}\\ c_{v}&d_{v}\end{smallmatrix}\right),x_{v}\right):\mathrm{val}_{v}(c_{v})\geq[\frac{e_{v}}{2}]+1\text{ and }x_{v}\in\varpi_{v}^{-g_{v}}\mathcal{O}_{F_{v}}\right\}\right)$
(3.49)			$\displaystyle=\mathrm{Vol}\left(\mathrm{K}_{v,[\frac{e_{v}}{2}]+1}\right)\times\mathrm{Vol}\left(\varpi_{v}^{-g_{v}}\mathcal{O}_{F_{v}}\right).$

Now, we assume that $\mathrm{val}_{v}(c_{v})\leq[\frac{e_{v}}{2}]$ . Since $c_{v}(a_{v}+c_{v}x_{v})\in\varpi_{v}^{e_{v}-g_{v}}\mathcal{O}_{F_{v}}$ , it follows that $x_{v}=-a_{v}c_{v}^{-1}+y_{v}$ , where $y_{v}\in\varpi_{v}^{e_{v}-g_{v}-2\mathrm{val}_{v}(c_{v})}\mathcal{O}_{F_{v}}$ . Thus, for each integer $i\in\{0,\dots[\frac{e_{v}}{2}]\}$ , we have

(3.50)			$\displaystyle\mathrm{Vol}\left(R_{v}\cap\left(\left(\mathrm{K}_{v,i}\backslash\mathrm{K}_{v,i+1}\right)\times F_{v}\right)\right)$
			$\displaystyle\leq\mathrm{Vol}\left(\left\{\left(\left(\begin{smallmatrix}a_{v}&b_{v}\\ c_{v}&d_{v}\end{smallmatrix}\right),x_{v}\right):\mathrm{val}_{v}(c_{v})=i\text{ and }x_{v}=-a_{v}c_{v}^{-1}+y_{v}\text{ for some }y_{v}\in\varpi_{v}^{e_{v}-g_{v}-2i}\mathcal{O}_{F_{v}}\right\}\right)$
			$\displaystyle=\mathrm{Vol}\left(\mathrm{K}_{v,i}\backslash\mathrm{K}_{v,i+1}\right)\times\mathrm{Vol}\left(\varpi_{v}^{e_{v}-g_{v}-2i}\mathcal{O}_{F_{v}}\right).$

Combining (3.48) $\sim$ (3.50), we obtain that

\mathrm{Vol}(R_{v})\leq\mathrm{Vol}\left(\mathrm{K}_{v,[\frac{e_{v}}{2}]+1}\right)\times\mathrm{Vol}\left(\varpi_{v}^{-g_{v}}\mathcal{O}_{F_{v}}\right)+\sum_{i=0}^{[\frac{e_{v}}{2}]}\mathrm{Vol}\left(\mathrm{K}_{v,i}\backslash\mathrm{K}_{v,i+1}\right)\times\mathrm{Vol}\left(\varpi_{v}^{e_{v}-g_{v}-2i}\mathcal{O}_{F_{v}}\right).

Similar to deduce $[\mathrm{SL}_{2}(\mathbb{Z}):\Gamma_{0}(N)]=N\prod_{p\mid N}(1+\frac{1}{p})$ , we have

(3.51)

\mathrm{Vol}(\mathrm{K}_{v,i})=\frac{1}{[\mathrm{G}(\mathcal{O}_{F_{v}}):\mathrm{K}_{v,i}]}=\begin{cases}1\quad&\text{if }i=0,\\ q_{v}^{1-i}(q_{v}+1)^{-1}\quad&\text{if }i>0,\end{cases}

and

\mathrm{Vol}(\varpi_{v}^{i}\mathcal{O}_{F_{v}})=q_{v}^{-i}.

Then, we have

(3.52)			$\displaystyle\mathrm{Vol}\left(\mathrm{K}_{v,[\frac{e_{v}}{2}]+1}\right)\times\mathrm{Vol}\left(\varpi_{v}^{-g_{v}}\mathcal{O}_{F_{v}}\right)+\sum_{i=0}^{[\frac{e_{v}}{2}]}\mathrm{Vol}\left(\mathrm{K}_{v,i}\backslash\mathrm{K}_{v,i+1}\right)\times\mathrm{Vol}\left(\varpi_{v}^{e_{v}-g_{v}-2i}\mathcal{O}_{F_{v}}\right)$
			$\displaystyle=q_{v}^{g_{v}}(q_{v}+1)^{-1}\left(q_{v}^{1-e_{v}+[\frac{e_{v}}{2}]}+q_{v}^{-[\frac{e_{v}}{2}]}\right)$
			$\displaystyle\leq q_{v}^{g_{v}-\frac{e_{v}}{2}}.$

Thus, we complete the proof of Lemma 3.11 for the case when $w\neq v$ .

For the case when $w=v$ , it is enough to show that for $(\kappa_{v},x_{v})\in R_{v}$ , $|\log H_{v}(\left(\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}\right)\left(\begin{smallmatrix}1&x_{v}\\ 0&1\end{smallmatrix}\right)\kappa_{v})|$ is less than or equal to $2g_{v}\log q_{v}$ . If $x_{v}\in\mathcal{O}_{F_{v}}$ , then by the definition of $H_{v}$ in (3.40) we have

\log H_{v}\left(\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\begin{pmatrix}1&x_{v}\\ 0&1\end{pmatrix}\kappa_{v}\right)=0.

If $x_{v}=\varpi_{v}^{-m}u$ for some $m>0$ and $u\in\mathcal{O}_{F_{v}}^{\times}$ , then we have

(3.53)

\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\begin{pmatrix}1&\varpi_{v}^{-m}u\\ 0&1\end{pmatrix}=\begin{pmatrix}\varpi_{v}^{m}&-u^{-1}\\ 0&\varpi_{v}^{-m}\end{pmatrix}\begin{pmatrix}-u^{-1}&0\\ -\varpi_{v}^{m}&-u\end{pmatrix}.

From (3.53), we obtain

(3.54)

\log H_{v}\left(\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\begin{pmatrix}1&x_{v}\\ 0&1\end{pmatrix}\kappa_{v}\right)=-2m\log q_{v}.

Assume that $(\kappa_{v},x_{v}):=\left(\left(\begin{smallmatrix}a_{v}&b_{v}\\ c_{v}&d_{v}\end{smallmatrix}\right),x_{v}\right)\in R_{v}$ . If $\mathrm{val}_{v}(c_{v})=0$ , then by (3.47) we have $a_{v}+x_{v}c_{v}\in\varpi_{v}^{e_{v}-g_{v}}\mathcal{O}_{F_{v}}$ . It follows that $\mathrm{val}_{v}(x_{v})\geq\min(0,e_{v}-g_{v})$ . If $\mathrm{val}_{v}(c_{v})>0$ , then $d_{v}\in\mathcal{O}_{F_{v}}^{\times}$ . Again, by (3.47), we obtain that $\mathrm{val}_{v}(x_{v})\geq-g_{v}$ . Thus, we conclude that if $(\kappa_{v},x_{v})\in R_{v}$ , then $\mathrm{val}_{v}(x_{v})\geq-g_{v}$ . Combining (3.54), we obtain for $(\kappa_{v},x_{v})\in R_{v}$ ,

|\log H_{v}(\left(\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}\right)\left(\begin{smallmatrix}1&x_{v}\\ 0&1\end{smallmatrix}\right)\kappa_{v})|\leq 2g_{v}\log q_{v}.

∎

Note that

\prod_{v\in S_{F,\mathrm{fin}}}q_{v}^{g_{v}-\frac{e_{v}}{2}}=\left|N_{F\backslash\mathbb{Q}}(\alpha-1)\right|\cdot\left|N_{F\backslash\mathbb{Q}}(J)\right|^{-\frac{1}{2}}.

Combining (3.46) and Lemma 3.11, we have

(3.55)		$\displaystyle\|J_{\alpha}(\phi)\|$	$\displaystyle\leq\left\|N_{F\backslash\mathbb{Q}}(\alpha-1)\right\|\cdot\left\|N_{F\backslash\mathbb{Q}}(J)\right\|^{-\frac{1}{2}}$
(3.55)			$\displaystyle\times\left(2\log\|N_{F\backslash\mathbb{Q}}(\alpha-1)\|\cdot\left\|\prod_{v\in S_{F,\infty}}J_{\sigma_{v}(\alpha),v}^{w_{0}}(\phi_{v})\right\|+\sum_{w\in S_{F,\infty}}\left\|\prod_{v\in S_{F,\infty}}J_{\sigma_{v}(\alpha),v}^{w}(\phi_{v})\right\|\right),$

where $w_{0}$ is an arbitrary non-archimedean place of $F$ . In the following lemma, we compute $J_{\alpha,v}^{w}(\phi_{v})$ when $v$ is an archimedean place of $F$ .

Lemma 3.12.

Assume that $v\in S_{F,\infty}$ . Fix $\alpha\in F_{v}^{\times}$ and $\alpha\neq 1$ . When $w\neq v$ ,

(3.56)

\left|J_{\alpha,v}^{w}(\phi_{v})\right|\leq\frac{(1+\epsilon_{v})\cdot|\alpha|^{\frac{1+\epsilon_{v}}{2}}}{2\pi|\alpha-1|^{1+\epsilon_{v}}}.

When $w=v$ and $F_{v}=\mathbb{R}$ ,

(3.57)

\left|J^{v}_{\alpha,v}(\phi_{v})\right|\leq\frac{2|\alpha|}{|\alpha-1|^{2}}\int_{\frac{\log|\alpha|}{2\pi}}^{\infty}\widehat{h_{\phi_{v}}}(t)\cosh(\pi t)dt.

When $w=v$ and $F_{v}=\mathbb{C}$ ,

(3.58)

\left|J_{\alpha,v}^{v}(\phi_{v})\right|\leq-\frac{2|\alpha|}{|\alpha-1|^{2}}\int_{\frac{\log|\alpha|}{\pi}}^{\infty}\widehat{h_{\phi_{v}}}(t)\frac{\sinh(\pi t)}{\cosh(\pi t)-\frac{\Re(\alpha)}{|\alpha|}}\,dt.

Proof.

Following the proof of Lemma 3.8, we obtain (3.56). By the Iwasawa decomposition, we have

\begin{pmatrix}0&1\\ -1&-x\end{pmatrix}\in\begin{pmatrix}\frac{1}{|x|^{2}+1}&-\frac{\bar{x}}{|x|^{2}+1}\\ 0&1\end{pmatrix}\mathrm{Z}(F_{v}){\rm K}_{v}.

Then, we get

\log H_{v}\left(\begin{pmatrix}0&1\\ -1&-x\end{pmatrix}\right)=-(1+\epsilon_{v})\log(|x|^{2}+1).

By (3.35) and (3.37), we have

	$\displaystyle J_{\alpha,v}^{v}(\phi_{v})$	$\displaystyle=-(1+\epsilon_{v})\int_{F_{v}}\phi_{v}\left(\begin{pmatrix}\alpha&x(\alpha-1)\\ 0&1\end{pmatrix}\right)\log(\|x\|^{2}+1)dx$
		$\displaystyle=-2\pi^{\epsilon_{v}}(1+\epsilon_{v})\int_{0}^{\infty}\phi_{v}\left(\begin{pmatrix}\|\alpha\|&x\|\alpha-1\|\\ 0&1\end{pmatrix}\right)\log(x^{2}+1)x^{\epsilon_{v}}dx$
		$\displaystyle=-\frac{2\pi^{\epsilon_{v}}(1+\epsilon_{v})}{\|\alpha-1\|^{1+\epsilon_{v}}}\int_{0}^{\infty}\phi_{v}\left(\begin{pmatrix}\|\alpha\|&x\\ 0&1\end{pmatrix}\right)\log(x^{2}\|\alpha-1\|^{-2}+1)x^{\epsilon_{v}}dx.$

As in the proof of Lemma 3.8, there are $r>0$ , $\kappa_{1},\kappa_{2}\in\mathrm{K}_{v}$ such that

\begin{pmatrix}|\alpha|&x\\ 0&1\end{pmatrix}=|\alpha|^{\frac{1}{2}}\kappa_{1}\begin{pmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{pmatrix}\kappa_{2},

and that

(3.59)

|\alpha|^{2}+x^{2}+1=2|\alpha|\cosh(r).

Assume that $F_{v}=\mathbb{R}$ and $\alpha>0$ . Since $\phi_{v}$ is $\mathrm{Z}(F_{v})$ -invariant and bi- $\mathrm{K}_{v}$ -invariant, we have by Lemma 2.1

\displaystyle\phi_{v}\left(\begin{pmatrix}\alpha&x\\ 0&1\end{pmatrix}\right)=\phi_{v}\left(\begin{pmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{pmatrix}\right)=-\frac{1}{4\pi^{2}}\int_{\frac{r}{2\pi}}^{\infty}\frac{\widehat{h_{\phi_{v}}}^{\prime}(t)}{\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{r}{2}\right)}}dt.

Then, we have

J^{v}_{\alpha,v}(\phi_{v})=\frac{1}{2\pi^{2}|\alpha-1|}\int_{0}^{\infty}\int_{\frac{r}{2\pi}}^{\infty}\frac{\widehat{h_{\phi_{v}}}^{\prime}(t)}{\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{r}{2}\right)}}\log(x^{2}|\alpha-1|^{-2}+1)dt\,dx.

Note that $\frac{r}{2\pi}\leq t$ implies

x^{2}=2|\alpha|\cosh(r)-|\alpha|^{2}-1\leq 2|\alpha|\cosh(2\pi t)-|\alpha|^{2}-1=4|\alpha|\left(\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{\log|\alpha|}{2}\right)\right).

Using the relation $\sinh^{2}(\frac{r}{2})=\sinh^{2}(\frac{\log|\alpha|}{2})+\frac{x^{2}}{4|\alpha|}$ and Fubini’s theorem, we have

		$\displaystyle J^{v}_{\alpha,v}(\phi_{v})$
		$\displaystyle=\frac{1}{2\pi^{2}\|\alpha-1\|}\int_{\frac{\log\|\alpha\|}{2\pi}}^{\infty}\widehat{h_{\phi_{v}}}^{\prime}(t)\int_{0}^{2\sqrt{\|\alpha\|}\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{\log\|\alpha\|}{2}\right)}}\frac{\log\bigg{(}\frac{x^{2}}{\|\alpha-1\|^{2}}+1\bigg{)}}{\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{r}{2}\right)}}dx\,dt$
		$\displaystyle=\frac{1}{2\pi^{2}\|\alpha-1\|}\int_{\frac{\log\|\alpha\|}{2\pi}}^{\infty}\widehat{h_{\phi_{v}}}^{\prime}(t)\int_{0}^{2\sqrt{\|\alpha\|}\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{\log\|\alpha\|}{2}\right)}}\frac{\log\bigg{(}\frac{x^{2}}{\|\alpha-1\|^{2}}+1\bigg{)}}{\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{\log\|\alpha\|}{2}\right)-\frac{x^{2}}{4\|\alpha\|}}}dx\,dt$
		$\displaystyle=\frac{\sqrt{\alpha\|}}{\pi^{2}\|\alpha-1\|}\int_{\frac{\log\|\alpha\|}{2\pi}}^{\infty}\widehat{h_{\phi_{v}}}^{\prime}(t)\int_{0}^{\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{\log\|\alpha\|}{2}\right)}}\frac{\log\bigg{(}\frac{4\|\alpha\|x^{2}}{\|\alpha-1\|^{2}}+1\bigg{)}}{\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{\log\|\alpha\|}{2}\right)-x^{2}}}dx\,dt.$

By [7, 4.295(38)], and then applying the integration by parts, we have

	$\displaystyle J_{\alpha,v}^{v}(\phi_{v})$	$\displaystyle=\frac{\sqrt{\|\alpha\|}}{\pi^{2}\|\alpha-1\|}\int_{\frac{\log\|\alpha\|}{2\pi}}^{\infty}\widehat{h_{\phi_{v}}}^{\prime}(t)\pi\log\bigg{(}\frac{1+\sqrt{1+\frac{4\|\alpha\|}{\|\alpha-1\|^{2}}\left(\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{\log\|\alpha\|}{2}\right)\right)}}{2}\bigg{)}dt$
		$\displaystyle=-\frac{\sqrt{\|\alpha\|}}{\pi\|\alpha-1\|}\int_{\frac{\log\|\alpha\|}{2\pi}}^{\infty}\widehat{h_{\phi_{v}}}(t)\frac{\pi\frac{4\|\alpha\|}{\|\alpha-1\|^{2}}\sinh(\pi t)\cosh(\pi t)}{\left(1+\sqrt{1+\frac{4\|\alpha\|}{\|\alpha-1\|^{2}}f_{\alpha}(t)}\right)\sqrt{1+\frac{4\|\alpha\|}{\|\alpha-1\|^{2}}f_{\alpha}(t)}}dt,$

where

f_{\alpha}(t):=\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{\log|\alpha|}{2}\right).

Since $|\alpha-1|^{2}\geq(|\alpha|-1)^{2}$ , it follows that

1+\frac{4|\alpha|}{|\alpha-1|^{2}}f_{\alpha}(t)\geq\frac{4|\alpha|}{|\alpha-1|^{2}}\sinh^{2}(\pi t)

and that

\left|\left(1+\sqrt{1+\frac{4|\alpha|}{|\alpha-1|^{2}}f_{\alpha}(t)}\right)\sqrt{1+\frac{4|\alpha|}{|\alpha-1|^{2}}f_{\alpha}(t)}\right|>\frac{2\sqrt{|\alpha|}}{|\alpha-1|}\sinh(\pi t).

Hence, we get (3.57).

Assume that $F_{v}=\mathbb{C}$ . Then, we have by Lemma 2.1

\phi_{v}\left(\begin{pmatrix}|\alpha|&x\\ 0&1\end{pmatrix}\right)=\phi_{v}\left(\begin{pmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{pmatrix}\right)=-\frac{1}{2\pi^{3}\sinh(r)}\widehat{h_{\phi_{v}}}^{\prime}\left(\frac{r}{\pi}\right),

where $|\alpha|^{2}+x^{2}+1=2|\alpha|\cosh(r)$ . By changing the variable $x^{2}=2|\alpha|\cosh(r)-|\alpha|^{2}-1$ , we obtain

	$\displaystyle J^{v}_{\alpha,v}(\phi_{v})$	$\displaystyle=\frac{2\|\alpha\|}{\pi^{2}\|\alpha-1\|^{2}}\int_{\log\|\alpha\|}^{\infty}\widehat{h_{\phi_{v}}}^{\prime}\left(\frac{r}{\pi}\right)\log\left(\frac{2\|\alpha\|\cosh(r)-\|\alpha\|^{2}-1+\|\alpha-1\|^{2}}{\|\alpha-1\|^{2}}\right)dr$
		$\displaystyle=\frac{2\|\alpha\|}{\pi\|\alpha-1\|^{2}}\int_{\frac{\log\|\alpha\|}{\pi}}^{\infty}\widehat{h_{\phi_{v}}}^{\prime}(t)\log\left(\frac{2\|\alpha\|\cosh(\pi t)-\|\alpha\|^{2}-1+\|\alpha-1\|^{2}}{\|\alpha-1\|^{2}}\right)dt$

Applying the integration by parts, we get (3.58). ∎

3.5. Parabolic contribution

For $s\in\mathbb{C}$ with $\mathrm{Re}(s)>1$ , $Z(s,\phi)$ is defined by

(3.60)

Z(s,\phi):=\prod_{v\in S_{F}}Z_{v}(s,\phi_{v}),

where for each $v\in S_{F}$ ,

Z_{v}(s,\phi_{v}):=\int_{\mathrm{K}_{v}}\int_{F_{v}^{\times}}\phi_{v}\left(\kappa^{-1}\left(\begin{smallmatrix}1&y\\ 0&1\end{smallmatrix}\right)\kappa\right)|y|_{v}^{s}d^{\times}_{v}y\,d_{v}\kappa.

In [6, (6.34)], $S_{\mathrm{par}}(\phi)$ is defined by

S_{\mathrm{par}}(\phi):=\lim_{s\to 1}\bigg{(}Z(s,\phi)-\text{ principal part of $Z(s,\phi)$ at $s=1$}\bigg{)}.

In the following lemma, we compute $Z_{v}(s,\phi_{v})$ for $v\in S_{F,\mathrm{fin}}$ .

Lemma 3.13.

For each $v\in S_{F,\mathrm{fin}}$ , let

g_{v}(s):=\frac{1}{q_{v}+1}\left(q_{v}^{1-e_{v}s}+(q_{v}-1)q_{v}^{(2-e_{v})s-1}\frac{q_{v}^{(2s-1)[\frac{e_{v}}{2}]}-1}{q_{v}^{2s-1}-1}+q_{v}^{-[\frac{e_{v}}{2}]}\right).

Then, we have

Z_{v}(s,\phi_{v})=\frac{g_{v}(s)}{1-q_{v}^{-s}}.

Proof.

Suppose that $e_{v}=0$ . If $\phi_{v}\left(\kappa^{-1}\left(\begin{smallmatrix}1&y\\ 0&1\end{smallmatrix}\right)\kappa\right)$ is non-zero for some $\kappa\in\mathrm{G}(\mathcal{O}_{F_{v}})$ and $y\in F_{v}^{\times}$ , then $\kappa^{-1}\left(\begin{smallmatrix}1&y\\ 0&1\end{smallmatrix}\right)\kappa\in\mathrm{Z}(F_{v})\cdot\mathrm{G}(\mathcal{O}_{F_{v}})$ . This implies that $y\in\mathcal{O}_{F_{v}}-\{0\}$ . Conversely, if $y\in\mathcal{O}_{F_{v}}-\{0\}$ and $\kappa\in\mathrm{K}_{v}$ , then we have $\phi_{v}\left(\kappa^{-1}\left(\begin{smallmatrix}1&y\\ 0&1\end{smallmatrix}\right)\kappa\right)$ is $1$ . Thus, we have

	$\displaystyle Z_{v}(s,\phi_{v})$	$\displaystyle=\mathrm{Vol}(\mathrm{K}_{v})\int_{\mathcal{O}_{F_{v}}-\{0\}}\|y\|^{s}_{v}d_{v}^{\times}y$
		$\displaystyle=\sum_{m=0}^{\infty}\int_{\varpi_{v}^{m}\mathcal{O}_{v}^{\times}}\|y\|_{v}^{s}d_{v}^{\times}y$
		$\displaystyle=\frac{1}{1-q_{v}^{-s}}.$

Now, we assume that $e_{v}$ is a positive integer. Let $\kappa=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\mathrm{K}_{v}$ . Since $\phi_{v}$ is the characteristic function of $\mathrm{Z}(F_{v})\mathrm{K}_{v,e_{v}}$ , it follows that $\phi_{v}\left(\kappa^{-1}\left(\begin{smallmatrix}1&y\\ 0&1\end{smallmatrix}\right)\kappa\right)$ is non-zero if and only if

(3.61)

\kappa^{-1}\begin{pmatrix}1&y\\ 0&1\end{pmatrix}\kappa=\frac{1}{ad-bc}\begin{pmatrix}cdy+ad-bc&d^{2}y\\ -c^{2}y&-cdy+ad-bc\end{pmatrix}\in\mathrm{Z}(F_{v})\cdot\mathrm{K}_{v,e_{v}}.

Since the determinant of $\kappa^{-1}\begin{pmatrix}1&y\\ 0&1\end{pmatrix}\kappa$ is $1$ , the condition (3.61) is equivalent to

(3.62)

cdy\in\mathcal{O}_{F_{v}},\quad d^{2}y\in\mathcal{O}_{F_{v}},\text{ and }c^{2}y\in\varpi_{v}^{e_{v}}\mathcal{O}_{F_{v}}.

Note that at least one of $c$ and $d$ is in $\mathcal{O}_{F_{v}}^{\times}$ since $\kappa\in\mathrm{K}_{v}$ . Thus, (3.62) is equivalent to

(3.63)

y\in\mathcal{O}_{F_{v}}\text{ and }c^{2}y\in\varpi_{v}^{e_{v}}\mathcal{O}_{F_{v}}.

As in the proof of Lemma 3.11, we consider the condition (3.63) according to $\mathrm{val}_{v}(c)$ . If $\mathrm{val}_{v}(c)\leq[\frac{e_{v}}{2}]$ , then $y\in\varpi_{v}^{e_{v}-2\mathrm{val}_{v}(c)}\mathcal{O}_{F_{v}}$ , and if $\mathrm{val}_{v}(c)>[\frac{e_{v}}{2}]$ , then $y\in\mathcal{O}_{F_{v}}$ . Thus, we have

(3.64)

\displaystyle Z_{v}(s,\phi_{v})

\displaystyle=\sum_{i=0}^{[\frac{e_{v}}{2}]}\mathrm{Vol}(\mathrm{K}_{v,i}\backslash\mathrm{K}_{v,i+1})\cdot\int_{\varpi^{e_{v}-2i}\mathcal{O}_{F_{v}}-\{0\}}|y|_{v}^{s}d_{v}^{\times}y+\mathrm{Vol}(\mathrm{K}_{v,[\frac{e_{v}}{2}]+1})\cdot\int_{\mathcal{O}_{F_{v}}-\{0\}}|y|_{v}^{s}d_{v}^{\times}y.

For each integer $i$ , we have

(3.65)

\int_{\varpi_{v}^{e_{v}-2i}\mathcal{O}_{F_{v}}-\{0\}}|y|_{v}^{s}d_{v}^{\times}y=\frac{q_{v}^{-(e_{v}-2i)s}}{1-q_{v}^{-s}}.

Combining (3.51), (3.64) and (3.65), we complete the proof of Lemma 3.13. ∎

By Lemma 3.13, we have

Z(s,\phi)=\zeta_{F}(s)\cdot\prod_{v\in S_{F,\mathrm{fin}}}g_{v}(s)\cdot\prod_{v\in S_{F,\infty}}Z_{v}(s,\phi_{v}),

where $\zeta_{F}$ denotes the Dedekind zeta function of $F$ . Let

g(s):=\prod_{v\in S_{F,\mathrm{fin}}}g_{v}(s)

and

Z_{\infty}(s,\phi_{\infty}):=\prod_{v\in S_{F,\infty}}Z_{v}(s,\phi_{v}).

Note that the Laurent series of $\zeta_{F}$ at $s=1$ has the form

\zeta_{F}(s)=\frac{\lambda_{-1}}{s-1}+\lambda_{0}+\sum_{n=1}^{\infty}\lambda_{n}(s-1)^{n}.

By the definition of $g_{v}$ , we see that $g_{v}(s)$ is holomorphic at $s=1$ for each $v\in S_{F,\mathrm{fin}}$ . Also, $Z_{\infty}(s,\phi_{\infty})$ is holomorphic at $s=1$ (see [6, pp. 242]). Thus, we have

	$\displaystyle Z(s,\phi)$	$\displaystyle=\left(\frac{\lambda_{-1}}{s-1}+\lambda_{0}+\cdots\right)\left(g(1)+g^{\prime}(1)(s-1)+\cdots\right)\left(Z_{\infty}(1,\phi_{\infty})+Z_{\infty}^{\prime}(1,\phi_{\infty})(s-1)+\cdots\right)$
		$\displaystyle=\frac{\lambda_{-1}\cdot g(1)\cdot Z_{\infty}(1,\phi_{\infty})}{s-1}+\left(\lambda_{0}\cdot g(1)\cdot Z_{\infty}(1,\phi_{\infty})+\lambda_{-1}\left(g(1)\cdot Z^{\prime}_{\infty}(1,\phi_{\infty})+g^{\prime}(1)\cdot Z_{\infty}(1,\phi_{\infty})\right)\right)+\cdots.$

Then, we get

(3.66)

S_{\mathrm{par}}(\phi)=\lambda_{0}\cdot g(1)\cdot Z_{\infty}(1,\phi_{\infty})+\lambda_{-1}\left(g(1)\cdot Z^{\prime}_{\infty}(1,\phi_{\infty})+g^{\prime}(1)\cdot Z_{\infty}(1,\phi_{\infty})\right).

From Lemma 3.13, we have for each $v\in S_{F,\mathrm{fin}}$ ,

|g_{v}(1)|=\left|(q_{v}+1)^{-1}\left(q_{v}^{[\frac{e_{v}}{2}]+1-e_{v}}+q_{v}^{-[\frac{e_{v}}{2}]}\right)\right|\leq q_{v}^{-\frac{e_{v}}{2}}

and

	$\displaystyle\|g_{v}^{\prime}(1)\|$	$\displaystyle=\left\|\log q_{v}\cdot(q_{v}+1)^{-1}\cdot\left(\left(2\left[\frac{e_{v}}{2}\right]-e_{v}\right)\cdot q_{v}^{[\frac{e_{v}}{2}]-e_{v}+1}-2(q_{v}-1)^{-1}\left(q_{v}^{[\frac{e_{v}}{2}]-e_{v}+1}-q_{v}^{1-e_{v}}\right)\right)\right\|$
		$\displaystyle\leq 6\log q_{v}\cdot q_{v}^{-\frac{e_{v}}{2}}.$

Then, we get

|g(1)|=\left|\prod_{v\in S_{F,\mathrm{fin}}}g_{v}(1)\right|\leq\left|N_{F\backslash\mathbb{Q}}(J)\right|^{-\frac{1}{2}}

and

|g^{\prime}(1)|=\left|\sum_{w\in S_{F,\mathrm{fin}}}g_{w}^{\prime}(1)\prod_{\begin{subarray}{c}v\in S_{F,\mathrm{fin}}\\ v\neq w\end{subarray}}g_{v}(1)\right|\leq 6\left|N_{F\backslash\mathbb{Q}}(J)\right|^{-\frac{1}{2}}\log\left|N_{F\backslash\mathbb{Q}}(J)\right|.

Again by (3.66), we conclude that

(3.67)

|S_{\rm par}(\phi)|\leq\left|N_{F\backslash\mathbb{Q}}(J)\right|^{-\frac{1}{2}}\left(\lambda_{0}Z_{\infty}(1,\phi_{\infty})+\lambda_{-1}\left(Z^{\prime}_{\infty}(1,\phi_{\infty})+6\cdot Z_{\infty}(1,\phi_{\infty})\log\left|N_{F\backslash\mathbb{Q}}(J)\right|\right)\right).

In the following lemma, we compute $Z_{v}(1,\phi_{v})$ and $Z_{v}^{\prime}(1,\phi_{v})$ for $v\in S_{F,\infty}$ .

Lemma 3.14.

Assume that $v\in S_{F,\infty}$ . Let $\epsilon_{v}:=[F_{v}:\mathbb{R}]-1$ . Then, we have

Z_{v}(1,\phi_{v})=\frac{(1+\epsilon_{v})^{2}}{4\pi^{1+\epsilon_{v}}}

and

(3.68)

Z_{v}^{\prime}(1,\phi_{v})=C_{v,1}+C_{v,2}\cdot h_{\phi_{v}}(0)+C_{v,3}\cdot\int_{-\infty}^{\infty}h_{\phi_{v}}(t)\frac{\Gamma^{\prime}}{\Gamma}(1+(1+\epsilon_{v})it)dt,

where $C_{v,1},C_{v,2}$ and $C_{v,3}$ are constants depending on $F_{v}$ .

Proof.

See [9, Theorem 6.2] or [4, Proposition 6.5.3(1)]. ∎

3.6. Eisenstein series and residual contribution

Let $\chi=\otimes_{v\in S_{F}}\chi_{v}$ be an idele class character of $F^{\times}\backslash\mathbb{A}_{F}^{\times}$ and $\chi_{v}$ be a character of $F_{v}^{\times}$ . For $a:=(a_{v})_{v\in S_{F}}\in\mathbb{A}_{F}$ , let $\|a\|:=\prod_{v\in S_{F}}|a_{v}|_{v}$ . For a complex number $s$ , we define $V_{\chi,s}$ by the space of functions $\varphi$ on $\mathrm{G}(\mathbb{A}_{F})$ satisfying

(1)

$\varphi\left(\left(\begin{smallmatrix}a&x\\ 0&b\end{smallmatrix}\right)g\right)=\chi(ab^{-1})\left\|\frac{a}{b}\right\|^{s+\frac{1}{2}}\varphi(g)$ ,
(2)

$\int_{\mathrm{K}}|\varphi(k)|^{2}dk<\infty$ .

Similarly, for each place $v$ of $F$ , we define $V_{\chi_{v},s}$ by the space of functions $\varphi_{v}$ on $\mathrm{G}(F_{v})$ satisfying

(1)

$\varphi_{v}\left(\left(\begin{smallmatrix}a&x\\ 0&b\end{smallmatrix}\right)g\right)=\chi_{v}(ab^{-1})\left|\frac{a}{b}\right|_{v}^{s+\frac{1}{2}}\varphi_{v}(g)$ ,
(2)

$\int_{\mathrm{K}_{v}}|\varphi(k)|^{2}dk<\infty$ .

For $\varphi\in V_{\chi,s}$ and $\varphi_{v}\in V_{\chi_{v},s}$ , $M\left(\eta_{\chi,s}\right)(\varphi)$ and $M\left(\eta_{\chi_{v},s}\right)(\varphi_{v})$ are defined by

\left(M\left(\eta_{\chi,s}\right)(\varphi)\right)(g):=\int_{\mathbb{A}_{F}}\varphi\left(w\begin{pmatrix}1&x\\ 0&1\end{pmatrix}g\right)dx,

and

\left(M\left(\eta_{\chi_{v},s}\right)(\varphi_{v})\right)(g):=\int_{F_{v}}\varphi_{v}\left(w\begin{pmatrix}1&x\\ 0&1\end{pmatrix}g\right)d_{v}x,

where $w:=\left(\begin{smallmatrix}0&-1\\ 1&0\end{smallmatrix}\right)$ . Let $\pi_{\chi,s}(\phi):V_{\chi,s}\to V_{\chi,s}$ and $\pi_{\chi_{v},s}(\phi_{v}):V_{\chi_{v},s}\to V_{\chi_{v},s}$ be defined by

\left(\pi_{\chi,s}(\phi)\varphi\right)(g):=\int_{\bar{\mathrm{G}}(\mathbb{A}_{F})}\phi(x)\varphi(gx)dx

and

\left(\pi_{\chi_{v},s}(\phi_{v})\varphi_{v}\right)(g):=\int_{\bar{\mathrm{G}}(F_{v})}\phi_{v}(x)\varphi_{v}(gx)d_{v}x.

Then, we have by [6, pp. 243]

S_{\rm Eis}(\phi)=\frac{1}{4\pi}\sum_{\chi}\int_{-i\infty}^{i\infty}\mathrm{tr}\left(M(\eta_{\chi,s})^{-1}M^{\prime}(\eta_{\chi,s})\pi_{\chi,s}(\phi)\right)ds,

where the sum is over all $\chi$ whose restriction to $F_{\infty}^{+}$ is trivial. Here, $F_{\infty}^{+}$ is the subset of $\mathbb{A}_{F}$ consisting of $a:=(a_{v})_{v\in S_{F}}$ such that $a_{v}=1$ for all $v\in S_{F,\mathrm{fin}}$ , and $a_{v}>0$ for all $v\in S_{F,\infty}$ .

Note that $M(\eta_{\chi_{v},s})$ acts by a scalar multiplication. Thus, there is a constant $(a_{\chi_{v},s})_{v}\in F_{v}$ such that for all $\varphi_{v}\in V_{\chi_{v},s}$ ,

\left(M(\eta_{\chi_{v},s})\varphi_{v}\right)=(a_{\chi_{v},s})_{v}\cdot\varphi_{v}.

Then, for $\varphi\in V_{\chi,s}$ , we obtain that

(3.69)

M(\eta_{\chi,s})\varphi=\left(\prod_{v\in S_{F}}(a_{\chi_{v},s})_{v}\right)\varphi

and

M(\eta_{\chi,s})^{-1}M(\eta_{\chi,s})^{\prime}\varphi=\left(\sum_{v\in S_{F}}\frac{(a_{\chi_{v},s})^{\prime}_{v}}{(a_{\chi_{v},s})_{v}}\right)\varphi.

It implies that

(3.70)

\mathrm{tr}\left(M(\eta_{\chi,s})^{-1}M^{\prime}(\eta_{\chi,s})\pi_{\chi,s}(\phi)\right)=\left(\sum_{v\in S_{F}}\frac{(a_{\chi_{v},s})^{\prime}_{v}}{(a_{\chi_{v},s})_{v}}\right)\mathrm{tr}\pi_{\chi,s}(\phi).

Assume that there is $v_{0}\in S_{F,\infty}$ such that $F_{v_{0}}=\mathbb{C}$ and $\chi_{v_{0}}$ is non-trivial on the unit circle $S^{1}$ . Then, there is $\theta_{0}\in[0,2\pi)$ such that $\chi_{v_{0}}(e^{i\theta_{0}})\neq 1$ . Let $g_{0}:=(g_{0,v})_{v\in S_{F}}\in\mathrm{G}(\mathbb{A}_{F})$ be defined by

g_{0,v}:=\begin{cases}\left(\begin{smallmatrix}e^{i\theta_{0}}&0\\ 0&1\end{smallmatrix}\right)\quad&\text{if }v=v_{0},\\ I_{2}\quad&\text{otherwise}.\end{cases}

Then, we have

	$\displaystyle\left(\pi_{\chi,s}(\phi)\varphi\right)(I_{2})$	$\displaystyle=\int_{\bar{\mathrm{G}}(\mathbb{A}_{F})}\phi(x)\varphi(x)dx$
		$\displaystyle=\int_{\bar{\mathrm{G}}(\mathbb{A}_{F})}\phi(g_{0}x)\varphi(g_{0}x)dx$
		$\displaystyle=\chi_{v_{0}}(e^{i\theta_{0}})\cdot\left(\pi_{\chi,s}(\phi)\varphi\right)(I_{2}).$

This implies that $\left(\pi_{\chi,s}(\phi)\varphi\right)(I_{2})=0$ and that $\left(\pi_{\chi,s}(\phi)\varphi\right)(\kappa)=0$ for all $\kappa\in\prod_{v\in S_{F,\infty}}\mathrm{K}_{v}^{0}$ since

	$\displaystyle\left(\pi_{\chi,s}(\phi)\varphi\right)(\kappa)$	$\displaystyle=\int_{\bar{\mathrm{G}}(\mathbb{A}_{F})}\phi(x)\varphi(\kappa x)dx$
		$\displaystyle=\int_{\bar{\mathrm{G}}(\mathbb{A}_{F})}\phi(\kappa^{-1}x)\varphi(x)dx$
		$\displaystyle=\int_{\bar{\mathrm{G}}(\mathbb{A}_{F})}\phi(x)\varphi(x)dx$
		$\displaystyle=\left(\pi_{\chi,s}(\phi)\varphi\right)(I_{2}).$

By the Iwasawa decomposition, we conclude that $\pi_{\chi,s}(\phi)=0$ if there is $v_{0}\in S_{F,\infty}$ such that $F_{v_{0}}=\mathbb{C}$ and $\chi_{v_{0}}$ is non-trivial on $S^{1}$ .

Thus, if $\mathrm{tr}\pi_{\chi,s}(\phi)\neq 0$ and the restriction of $\chi$ to $F^{+}_{\infty}$ is trivial, then we have

(3.71)

\chi_{v}=\begin{cases}\chi_{0,v}\text{ or }\mathop{\rm sgn}\quad&\text{if }F_{v}=\mathbb{R},\\ \chi_{0,v}\quad&\text{otherwise.}\end{cases}

Here, $\chi_{0,v}$ is the trivial character of $F_{v}^{\times}$ for each $v\in S_{F}$ and $\mathop{\rm sgn}$ denotes the sign function. Assume that $\chi=\otimes_{v\in S_{F}}\chi_{v}$ is an idele class character of $F^{\times}\backslash\mathbb{A}_{F}^{\times}$ such that $\chi_{v}$ satisfies (3.71) for each $v\in S_{F}$ . If $\chi$ is a non-trivial idele class character, then there is $v_{0}\in S_{F,\infty}$ such that $\chi_{v_{0}}$ is a non-trivial character of $F_{v_{0}}^{\times}$ . By the strong approximation theorem, there is $a\in F^{\times}$ such that $\sigma_{v_{0}}(a)<0$ and $\sigma_{v}(a)>0$ for $v\in S_{F,\infty}$ with $F_{v}=\mathbb{R}$ and $v\neq v_{0}$ . This contradicts our assumption that $\chi$ is trivial on $F^{\times}$ . Hence, we get

(3.72)

S_{\rm Eis}(\phi)=\frac{1}{4\pi}\int_{-i\infty}^{i\infty}\mathrm{tr}\left(M(\eta_{\chi_{0},s})^{-1}M^{\prime}(\eta_{\chi_{0},s})\pi_{\chi_{0},s}(\phi)\right)ds.

Here, $\chi_{0}$ is the trivial idele class character of $F^{\times}\backslash\mathbb{A}_{F}^{\times}$ . Let $V_{\chi_{0},s}^{\mathrm{K}(J)}$ denote the subspace of $V_{\chi_{0},s}$ consisting of $v$ such that $v=\pi_{\chi_{0},s}(k)v$ for all $k\in\mathrm{K}(J)$ . From (3.72), we compute $S_{\mathrm{Eis}}(\phi)$ in the following lemma.

Lemma 3.15.

Let $r_{1}$ be the number of real embeddings of $F$ and $r_{2}$ be the number of conjugate pairs of complex embeddings of $F$ . Let $\zeta_{F}$ be the Dedekind zeta function of $F$ and $\Delta_{F}$ be the discriminant of $F$ . Let $\Lambda_{F}(s)$ be the completed zeta function defined by

\Lambda_{F}(s):=|\Delta_{F}|^{s/2}(\pi^{-s/2}\Gamma(s/2))^{r_{1}}((2(2\pi)^{-s}\Gamma(s))^{r_{2}}\zeta_{F}(s).

Let $A_{J}$ be defined as in (2.10). Then, we have

S_{\rm Eis}(\phi)=\frac{A_{J}}{2^{r_{1}+2}\pi}\int_{\mathbb{R}}\bigg{(}\frac{\Lambda_{F}^{\prime}(2it)}{\Lambda_{F}(2it)}-\frac{\Lambda_{F}^{\prime}(2it+1)}{\Lambda_{F}(2it+1)}\bigg{)}\prod_{v\in S_{F,\infty}}h_{\phi_{v}}(t)\cdot\dim V_{it}^{\mathrm{K}(J)}dt

Proof.

By (3.72), it is enough to show that

\mathrm{tr}\left(M(\eta_{\chi_{0},it})^{-1}M^{\prime}(\eta_{\chi_{0},it})\pi_{\chi_{0},it}(\phi)\right)=2^{-r_{1}}A_{J}\bigg{(}\frac{\Lambda_{F}^{\prime}(2it)}{\Lambda_{F}(2it)}-\frac{\Lambda_{F}^{\prime}(2it+1)}{\Lambda_{F}(2it+1)}\bigg{)}\prod_{v\in S_{F,\infty}}h_{\phi_{v}}(t)\cdot\dim V_{it}^{\mathrm{K}(J)}.

Assume that $v\in S_{F,\mathrm{fin}}$ . We take $\varphi_{v}\in V_{\chi_{0,v},s}$ such that the $\varphi_{v}|_{\mathrm{K}_{v}}=1$ . Then, we have

(3.73)	$\displaystyle\left(M(\eta_{\chi_{0,v},s})\varphi_{v}\right)(w)$	$\displaystyle=\int_{F_{v}}\varphi_{v}\left(w\begin{pmatrix}1&x\\ 0&1\end{pmatrix}w\right)d_{v}x$
		$\displaystyle=\int_{F_{v}}\varphi_{v}\left(\begin{pmatrix}1&0\\ -x&1\end{pmatrix}\right)d_{v}x$
		$\displaystyle=\sum_{n\in\mathbb{Z}}q_{v}^{-n}(1-q_{v}^{-1})\int_{\mathcal{O}_{F_{v}}^{\times}}\varphi_{v}\left(\begin{pmatrix}1&0\\ -u\varpi_{v}^{n}&1\end{pmatrix}\right)d_{v}^{\times}u.$

When $n\geq 0$ , we have $\left(\begin{smallmatrix}1&0\\ -u\varpi_{v}^{n}&1\end{smallmatrix}\right)\in\mathrm{K}_{v}$ for all $u\in\mathcal{O}_{F_{v}}^{\times}$ . Thus, we have by (3.73)

(3.74)

\left(M(\eta_{\chi_{0,v},s})\varphi_{v}\right)(w)=\sum_{n\geq 0}q_{v}^{-n}(1-q_{v}^{-1})+\sum_{n<0}q_{v}^{-n}(1-q_{v}^{-1})\int_{\mathcal{O}_{F_{v}}^{\times}}\varphi_{v}\left(\begin{pmatrix}1&0\\ -u\varpi_{v}^{n}&1\end{pmatrix}\right)d_{v}^{\times}u.

For $n<0$ , we have

\begin{pmatrix}1&0\\ -u\varpi_{v}^{n}&1\end{pmatrix}=\begin{pmatrix}u^{-1}\varpi_{v}^{-n}&1\\ 0&-u\varpi_{v}^{n}\end{pmatrix}\begin{pmatrix}0&1\\ 1&-u^{-1}\varpi_{v}^{-n}\end{pmatrix}.

Since $\left(\begin{smallmatrix}0&1\\ 1&-u^{-1}\varpi_{v}^{-n}\end{smallmatrix}\right)\in\mathrm{K}_{v}$ , it follows that

(3.75)

\varphi_{v}\left(\begin{pmatrix}1&0\\ -u\varpi_{v}^{n}&1\end{pmatrix}\right)=q_{v}^{2n(s+\frac{1}{2})}=q_{v}^{2ns+n}.

Combining (3.74) and (3.75), we deduce that

(3.76)

(a_{\chi_{0,v},s})_{v}=\left(M(\eta_{\chi_{0,v},s})\varphi_{v}\right)(w)=\frac{1-q_{v}^{-2s-1}}{1-q_{v}^{-2s}}.

Now, we consider an archimedean place $v$ of $F$ . When $F_{v}=\mathbb{R}$ , since $\varphi_{v}$ is a bi- $\mathrm{SO}(2)$ -invariant function, we have

	$\displaystyle\left(M(\eta_{\chi_{0,v},s})\varphi_{v}\right)(I_{2})$	$\displaystyle=\int_{\mathbb{R}}\varphi_{v}\left(w\begin{pmatrix}1&x\\ 0&1\end{pmatrix}\right)dx$
		$\displaystyle=\int_{\mathbb{R}}\varphi_{v}\left(w\begin{pmatrix}1&x\\ 0&1\end{pmatrix}w^{-1}\right)dx.$

Note that there is $\kappa\in\mathrm{SO}(2)$ such that

w\begin{pmatrix}1&x\\ 0&1\end{pmatrix}w^{-1}=\begin{pmatrix}\frac{1}{\sqrt{x^{2}+1}}&-\frac{x}{\sqrt{x^{2}+1}}\\ 0&\sqrt{x^{2}+1}\end{pmatrix}\kappa.

Thus, for $v\in S_{F,\infty}$ with $F_{v}=\mathbb{R}$ , we have by [7, 3.251(2)]

(3.77)

(a_{\chi_{0,v},s})_{v}=\left(M(\eta_{\chi_{0,v},s})\varphi_{v}\right)(I_{2})=\int_{\mathbb{R}}\varphi_{v}\left(\left(\begin{smallmatrix}\frac{1}{\sqrt{x^{2}+1}}&-\frac{x}{\sqrt{x^{2}+1}}\\ 0&\sqrt{x^{2}+1}\end{smallmatrix}\right)\right)dx=\int_{\mathbb{R}}(x^{2}+1)^{-s-\frac{1}{2}}dx=\frac{\sqrt{\pi}\Gamma\left(s\right)}{\Gamma\left(s+\frac{1}{2}\right)}.

When $F_{v}=\mathbb{C}$ , by taking $z=re^{i\theta}$ , we have

(M((\eta_{\chi_{0,v},s})_{v})\varphi_{v})(I_{2})=\int_{\mathbb{C}}\varphi_{v}\left(w\left(\begin{smallmatrix}1&z\\ 0&1\end{smallmatrix}\right)\right)dz=\int_{0}^{2\pi}\int_{0}^{\infty}\varphi_{v}\left(\left(\begin{smallmatrix}w\left(\begin{smallmatrix}1&re^{i\theta}\\ 0&1\end{smallmatrix}\right)w^{-1}\end{smallmatrix}\right)\right)rdr\,d\theta.

Similarly, we have

w\left(\begin{smallmatrix}1&re^{i\theta}\\ 0&1\end{smallmatrix}\right)w^{-1}=\left(\begin{smallmatrix}e^{-i\theta}&\\ &1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&0\\ -r&1\end{smallmatrix}\right)\left(\begin{smallmatrix}e^{i\theta}&\\ &1\end{smallmatrix}\right)\in\left(\begin{smallmatrix}e^{-i\theta}&\\ &1\end{smallmatrix}\right)\left(\begin{smallmatrix}\frac{1}{\sqrt{r^{2}+1}}&\frac{-r}{\sqrt{r^{2}+1}}\\ 0&\sqrt{r^{2}+1}\end{smallmatrix}\right)\mathrm{U}(2).

Then, we get by [7, 3.251(2)]

	$\displaystyle(M((\eta_{\chi_{0,v},s})_{v})f_{v})(I_{2})$	$\displaystyle=\int_{0}^{2\pi}\int_{0}^{\infty}\varphi_{v}\left(\left(\begin{smallmatrix}\frac{e^{-i\theta}}{\sqrt{r^{2}+1}}&\frac{-re^{-i\theta}}{\sqrt{r^{2}+1}}\\ 0&\sqrt{r^{2}+1}\end{smallmatrix}\right)\right)rdr\,d\theta$
		$\displaystyle=2\pi\int_{0}^{\infty}(r^{2}+1)^{-2s-1}rdr$
		$\displaystyle=\frac{\pi\Gamma\left(2s\right)}{\Gamma\left(2s+1\right)}$

for $\Re(s)>-\frac{3}{4}$ . This implies that

(3.78)

a((\eta_{\chi_{0,v},s})_{v})=\frac{\pi\Gamma\left(2s\right)}{\Gamma\left(2s+1\right)}.

Combining (3.70), (3.76), (3.77), and (3.78), we obtain that

\mathrm{tr}\left(M(\eta_{\chi_{0},it})^{-1}M^{\prime}(\eta_{\chi_{0},it})\pi_{\chi_{0},it}(\phi)\right)=\bigg{(}\frac{\Lambda_{F}^{\prime}(2it)}{\Lambda_{F}(2it)}-\frac{\Lambda_{F}^{\prime}(2it+1)}{\Lambda_{F}(2it+1)}\bigg{)}\mathrm{tr}\pi_{\chi_{0},it}(\phi).

By Lemma 2.2, we complete the proof of Proposition 3.15. ∎

In [6, pp. 244], $S_{\mathrm{Res}}(\phi)$ is given by

S_{\mathrm{Res}}(\phi):=-\frac{1}{4}\sum_{\chi^{2}=\chi_{0}}\mathrm{tr}\left(M(\eta_{\chi,0})\pi_{\chi,0}(\phi)\right).

By a similar argument to the Eisenstein series contribution, if an idele class character $\chi$ of $F^{\times}\backslash\mathbb{A}_{F}^{\times}$ satisfies $\chi^{2}=\chi_{0}$ and $\mathrm{tr}(M(\eta_{\chi,0})\pi_{\chi,0}(\phi))\neq 0$ , then $\chi=\chi_{0}$ . This implies that

S_{\mathrm{Res}}(\phi)=-\frac{1}{4}\mathrm{tr}\left(M(\eta_{\chi_{0},0})\pi_{\chi_{0},0}(\phi)\right).

Combining (3.76), (3.77), and (3.78), we obtain the following lemma which provides the formula for $S_{\mathrm{Res}}(\phi)$ .

Lemma 3.16.

With the above notation, we have

S_{\rm Res}(\phi)=-2^{-r_{1}-r_{2}-2}\cdot\sqrt{\Delta_{F}}\cdot\lim_{s\to 0}\frac{\Lambda_{F}(2s)}{\Lambda_{F}(2s+1)}\cdot A_{J}\cdot\prod_{v\in S_{F,\infty}}h_{\phi_{v}}(0)\cdot\dim V_{0}^{\mathrm{K}(J)}.

4. Proof of Theorem 1.1

In this section, we prove Theorem 1.1. We follow the notation in Sections 2 and 3. In the definitions of $S_{\mathrm{ell}}(\phi)$ and $S_{\mathrm{hyp}}(\phi)$ , the sums seem to be infinite sums. However, the following lemma guarantees that these sums are finite.

Lemma 4.1.

Let $v\in S_{F,\infty}$ and $\epsilon_{v}:=[F_{v}:\mathbb{R}]-1$ . Assume that the support of $\widehat{h_{\phi_{v}}}$ is contained in $[-a,a]$ for some positive real number $a$ . If $\gamma\in\mathrm{SL}_{2}(F_{v})$ satisfies

\mathrm{tr}(\gamma\gamma^{*})\geq e^{-\frac{2\pi a}{1+\epsilon_{v}}}+e^{\frac{2\pi a}{1+\epsilon_{v}}},

then $\phi_{v}(\gamma)=0$ .

Proof.

By the Cartan decomposition, there are $\kappa_{1},\kappa_{2}\in\mathrm{K}_{v}^{0}$ and a non-negative real number $a(\gamma)$ such that

\gamma=\kappa_{1}\begin{pmatrix}e^{-\frac{a(\gamma)}{2}}&0\\ 0&e^{\frac{a(\gamma)}{2}}\end{pmatrix}\kappa_{2}.

Then, the trace of $\gamma\gamma^{*}$ is equal to $e^{-a(\gamma)}+e^{a(\gamma)}$ . Since $\phi_{v}$ is a bi- $\mathrm{K}_{v}^{0}$ -invariant function, we have

(4.1)

\phi_{v}(\gamma)=\phi_{v}\left(\begin{pmatrix}e^{-\frac{a(\gamma)}{2}}&0\\ 0&e^{\frac{a(\gamma)}{2}}\end{pmatrix}\right).

By Lemma 2.1, we have if $F_{v}=\mathbb{R}$ , then

(4.2)

\phi_{v}\left(\begin{pmatrix}e^{-\frac{a(\gamma)}{2}}&0\\ 0&e^{\frac{a(\gamma)}{2}}\end{pmatrix}\right)=-\frac{1}{4\pi^{2}}\int_{\frac{a(\gamma)}{2\pi}}^{\infty}\frac{\widehat{h_{\phi_{v}}}^{\prime}(t)}{\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{a(\gamma)}{2}\right)}}dt,

and if $F_{v}=\mathbb{C}$ , then

\phi_{v}\left(\begin{pmatrix}e^{-\frac{a(\gamma)}{2}}&0\\ 0&e^{\frac{a(\gamma)}{2}}\end{pmatrix}\right)=-\frac{\widehat{h_{\phi_{v}}}^{\prime}(a(\gamma)/\pi)}{2\pi^{3}\sinh(a(\gamma))}.

Thus, we obtain that if $a(\gamma)\geq\frac{2\pi a}{1+\epsilon_{v}}$ , then $\phi_{v}(\gamma)=0$ . ∎

For each $v\in S_{F,\infty}$ , assume that $\widehat{h_{v}}:\mathbb{R}\to\mathbb{R}_{\geq 0}$ and $h_{v}:\mathbb{C}\to\mathbb{C}$ satisfy the conditions $(1)\sim(7)$ in Section 2. For a real positive number $t$ , let

\widehat{h_{v,t}}(x):=\widehat{h_{v}}(tx).

Then, $\widehat{h_{v,t}}$ and $h_{v,t}$ also satisfy the conditions $(1)\sim(7)$ in Section 2. Thus, we may assume that the support of $\widehat{h_{v}}$ is contained in $[-1,1]$ for all $v\in S_{F,\infty}$ . For each $v\in S_{F,\infty}$ , we fix $h_{v}$ . Let $a_{v}$ be a positive real number. Then, the support of $\widehat{h_{v,a_{v}}}$ is contained in $[-\frac{1}{a_{v}},\frac{1}{a_{v}}]$ and

(4.3)

h_{v,a_{v}}(z)=\frac{1}{a_{v}}h_{v}\left(\frac{z}{a_{v}}\right).

Note that there is a function $\phi_{v,a_{v}}$ on $\mathrm{G}(F_{v})$ satisfying the conditions $(a)\sim(c)$ in Section 2 such that

h_{\phi_{v,a_{v}}}=h_{v,a_{v}}.

For $\mathbf{a}:=(a_{v})_{v\in S_{F,\infty}}\in\mathbb{R}_{>0}^{\#S_{F,\infty}}$ , let $\phi^{(\mathbf{a})}:=\prod_{v\in S_{F,\mathrm{fin}}}\phi_{v}\cdot\prod_{v\in S_{F,\infty}}\phi_{v,a_{v}}$ be a function on $\mathrm{G}(\mathbb{A}_{F})$ . Recalling that for each $v\in S_{F,\mathrm{fin}}$ , $\phi_{v}$ is the characteristic function of $\mathrm{Z}(F_{v})\mathrm{K}_{v,\mathrm{val}_{v}(J)}$ .

In the following lemmas, we compute upper bounds for $S_{\mathrm{one}}(\phi^{(\mathbf{a})})$ , $S_{id}(\phi^{(\mathbf{a})})$ , $S_{\rm ell}(\phi^{(\mathbf{a})})$ , $S_{\rm hyp}(\phi^{(\mathbf{a})})$ , $S_{\rm par}(\phi^{(\mathbf{a})})$ and $S_{\rm Eis}(\phi^{(\mathbf{a})})$ .

Lemma 4.2.

With the above notation, we have

\displaystyle\left|S_{\mathrm{one}}(\phi^{(\mathbf{a})})\right|\leq A_{J}\cdot\prod_{v\in S_{F,\infty}}\left(e^{\frac{\pi}{a_{v}}}\cdot h_{v,a_{v}}(0)\right)

Proof.

By Lemma 3.1, we have

S_{\mathrm{one}}(\phi^{(\mathbf{a})})=A_{J}\cdot\prod_{v\in S_{F,\infty}}h_{v,a_{v}}\left(\frac{i}{2}\right).

Note that we get

	$\displaystyle\left\|h_{v,a_{v}}\left(\frac{i}{2}\right)\right\|$	$\displaystyle=\frac{1}{a_{v}}\left\|h_{v}\left(\frac{i}{2a_{v}}\right)\right\|$
		$\displaystyle=\frac{1}{a_{v}}\left\|\int_{\mathbb{R}}\widehat{h_{v}}(x)e^{\frac{\pi x}{a_{v}}}dx\right\|$
		$\displaystyle=\frac{2}{a_{v}}\left\|\int_{0}^{1}\widehat{h_{v}}(x)\cosh\left(\frac{\pi x}{a_{v}}\right)dx\right\|$
		$\displaystyle\leq\cosh\left(\frac{\pi}{a_{v}}\right)\cdot h_{v,a_{v}}(0)$
		$\displaystyle\leq e^{\frac{\pi}{a_{v}}}\cdot h_{v,a_{v}}(0).$

Therefore, we conclude that

\displaystyle\left|S_{\mathrm{one}}(\phi^{(\mathbf{a})})\right|\leq A_{J}\cdot\prod_{v\in S_{F,\infty}}\left(e^{\frac{\pi}{a_{v}}}\cdot h_{v,a_{v}}(0)\right).

∎

Lemma 4.3.

With the above notation, we have

\left|S_{\rm id}(\phi^{(\mathbf{a})})\right|\ll_{F}\prod_{v\in S_{F,\infty}}\left(a_{v}^{2+\epsilon_{v}}\cdot h_{v,a_{v}}(0)\right).

Proof.

By Lemma 3.2, we have

	$\displaystyle S_{\rm id}(\phi^{(\mathbf{a})})$	$\displaystyle=\mathrm{Vol}(\bar{\mathrm{G}}(F)\backslash\bar{\mathrm{G}}(\mathbb{A}_{F}))\cdot\prod_{v\in S_{F,\infty}}\left(\frac{1+\epsilon_{v}}{(2-\epsilon_{v})^{2(1-\epsilon_{v})}\pi^{1+\epsilon_{v}}}\int_{-\infty}^{\infty}x^{1+\epsilon_{v}}h_{v,a_{v}}(x)\tanh^{1-\epsilon_{v}}(\pi x)\,dx\right)$
		$\displaystyle=\mathrm{Vol}(\bar{\mathrm{G}}(F)\backslash\bar{\mathrm{G}}(\mathbb{A}_{F}))\cdot\prod_{v\in S_{F,\infty}}\left(\frac{2(1+\epsilon_{v})}{(2-\epsilon_{v})^{2(1-\epsilon_{v})}\pi^{1+\epsilon_{v}}}\int_{0}^{\infty}x^{1+\epsilon_{v}}h_{v,a_{v}}(x)\tanh^{1-\epsilon_{v}}(\pi x)\,dx\right)$

Since $|\tanh(x)|\leq 1$ , for each $v\in S_{F,\infty}$ , we have

	$\displaystyle\left\|\int_{0}^{\infty}x^{1+\epsilon_{v}}h_{v,a_{v}}(x)\tanh^{1-\epsilon_{v}}(\pi x)\,dx\right\|$	$\displaystyle\leq\left\|\int_{0}^{\infty}x^{1+\epsilon_{v}}h_{v,a_{v}}(x)\,dx\right\|$
		$\displaystyle=\left\|\int_{0}^{\infty}\frac{x^{1+\epsilon_{v}}}{a_{v}}\cdot h_{v}\left(\frac{x}{a_{v}}\right)\,dx\right\|$
		$\displaystyle=a_{v}^{1+\epsilon_{v}}\left\|\int_{0}^{\infty}x^{1+\epsilon_{v}}h_{v}(x)dx\right\|$
		$\displaystyle=\left(\frac{1}{h_{v}(0)}\left\|\int_{0}^{\infty}x^{1+\epsilon_{v}}h_{v}(x)dx\right\|\right)a_{v}^{2+\epsilon}\cdot h_{v,a_{v}}(0).$

∎

Now, we introduce the following lemma which is useful to compute upper bounds for $S_{\rm ell}(\phi^{(\mathbf{a})})$ and $S_{\rm hyp}(\phi^{(\mathbf{a})})$ .

Lemma 4.4.

For each $v\in S_{F,\infty}$ , let $M_{v}$ be a positive real number. Let $n:=[F:\mathbb{Q}]$ . Then, we have

\#\left\{a\in\mathcal{O}_{F}:|\sigma_{v}(a)|\leq M_{v}\text{ for all }v\in S_{F,\infty}\right\}\leq n\cdot\left(2\cdot\prod_{v\in S_{F,\infty}}(|M_{v}|+1)^{1+\epsilon_{v}}\right)^{n}.

Proof.

Assume that $a\in\mathcal{O}_{F}$ satisfies $|\sigma_{v}(a)|\leq M_{v}$ for all $v\in S_{F,\infty}$ . Let $f_{a}(x)\in\mathbb{Z}[x]$ be a monic irreducible polynomial of $a$ . Note that the coefficients of $f_{a}$ can be expressed as a symmetric function of a subset of $\{\sigma_{v}(a)\}_{v\in S_{\infty}}$ and that the absolute value of each coefficient of $f_{a}$ is less than or equal to

\prod_{v\in S_{F,\infty}}(|M_{v}|+1)^{1+\epsilon_{v}}.

Since the degree of $f_{a}$ is less than or equal to $n$ and $f_{a}$ has at most $n$ roots, we complete the proof of Lemma 4.4. ∎

For convenience, let $A:=|N_{F/\mathbb{Q}}(J)|$ and $B_{m,u}:=|N_{F/\mathbb{Q}}(m^{2}-4u)|$ .

Lemma 4.5.

With the above notation, assume that $a_{v}\leq 1$ for all $v\in S_{F,\infty}$ . Then, we have

\left|S_{\rm ell}(\phi^{(\mathbf{a})})\right|\ll_{F}A^{-\frac{1}{2}}\cdot\prod_{v\in S_{F,\infty}}\left(e^{\frac{2\pi(n+13)}{a_{v}}}\cdot h_{v,a_{v}}(0)\right).

Proof.

Recalling (3.3), $S_{[\gamma]}(\phi)$ is defined by

S_{[\gamma]}(\phi):=\prod_{v\in S_{F,\mathrm{fin}}}\int_{\overline{\mathrm{G}_{\gamma}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\gamma g)dg\cdot\prod_{v\in S_{F,\infty}}\int_{\overline{\mathrm{G}_{\sigma_{v}(\gamma)}(F_{v})}\backslash\bar{\mathrm{G}}(F_{v})}\phi_{v}(g^{-1}\sigma_{v}(\gamma)g)dg.

First, we find the condition of $(m,u)\in\mathcal{O}_{F}\times\mathcal{O}_{F}^{\times}$ such that $S_{[\gamma_{m,u}]}(\phi^{(\mathbf{a})})\neq 0$ . Suppose that $S_{[\gamma_{m,u}]}(\phi^{(\mathbf{a})})\neq 0$ . For each $(m,u)\in\mathcal{O}_{F}\times\mathcal{O}_{F}^{\times}$ , let $S_{m,u}^{1}$ be the subset of $S_{F,\infty}$ consisting of $v$ such that there are two roots of $x^{2}-\sigma_{v}(m)x+\sigma_{v}(u)=0$ in $F_{v}$ and $S_{m,u}^{2}:=S_{F,\infty}\backslash S_{m,u}^{1}$ . Note that $v\in S_{m,u}^{2}$ if and only if $F_{v}=\mathbb{R}$ and $\sigma_{v}(m^{2}-4u)<0$ . Thus, if $v\in S_{m,u}^{2}$ , then we have

|\sigma_{v}(m)|<2\sqrt{|\sigma_{v}(u)|}\leq 2\cdot e^{\frac{2\pi}{a_{v}(1+\epsilon_{v})}}\sqrt{|\sigma_{v}(u)|}

and

|\sigma_{v}(m^{2}-4u)|\leq|\sigma_{v}(m^{2})|+4|\sigma_{v}(u)|\leq 8|\sigma_{v}(u)|\leq 8\cdot e^{\frac{4\pi}{a_{v}(1+\epsilon_{v})}}\cdot|\sigma_{v}(u)|.

If $v\in S_{m,u}^{1}$ , then we have by (3.36) and (3.38)

e^{-\frac{4\pi}{a_{v}(1+\epsilon_{v})}}<\left|\sigma_{v}(\alpha_{\gamma})\right|<e^{\frac{4\pi}{a_{v}(1+\epsilon_{v})}}.

This implies that

(4.4)

\left|\sigma_{v}(m)\pm\sqrt{\sigma_{v}(m^{2}-4u)}\right|^{2}<4\cdot e^{\frac{4\pi}{a_{v}(1+\epsilon_{v})}}\cdot|\sigma_{v}(u)|.

Note that we have

(4.5)

\left|\sigma_{v}(m)+\sqrt{\sigma_{v}(m^{2}-4u)}\right|^{2}+\left|\sigma_{v}(m)-\sqrt{\sigma_{v}(m^{2}-4u)}\right|^{2}=2(|\sigma_{v}(m)|^{2}+|\sigma_{v}(m^{2}-4u)|).

Combining (4.4) and (4.5), we get

|\sigma_{v}(m)|^{2}+|\sigma_{v}(m^{2}-4u)|<4\cdot e^{\frac{4\pi}{a_{v}(1+\epsilon_{v})}}\cdot|\sigma_{v}(u)|.

It immediately implies that

(4.6)

|\sigma_{v}(m)|<2\cdot e^{\frac{2\pi}{a_{v}(1+\epsilon_{v})}}\sqrt{|\sigma_{v}(u)|}\text{ and }|\sigma_{v}(m^{2}-4u)|<8\cdot e^{\frac{4\pi}{a_{v}(1+\epsilon_{v})}}\cdot|\sigma_{v}(u)|.

Hence, we conclude that if $S_{[\gamma]}(\phi^{(\mathbf{a})})$ is non-zero, then $(m,u)$ satisfies (4.6) for all $v\in S_{F,\infty}$ .

Recalling Section 3, let $U_{F}$ be a set of representatives of $\mathcal{O}_{F}^{\times}/\mathcal{O}_{F}^{\times^{2}}$ . Then, Dirichlet’s unit theorem implies that $U_{F}$ is a finite set. In Section 3.3, we see that if $S_{[\gamma]}(\phi^{(\mathbf{a})})$ is non-zero, then there is a unique pair $(m,u)\in\mathcal{O}_{F}\times U_{F}$ corresponding to $\gamma$ such that $\gamma_{m,u}:=\left(\begin{smallmatrix}0&1\\ -u&m\end{smallmatrix}\right)\in[\gamma]$ . For $\mathbf{a}:=(a_{v})_{v\in S_{F,\infty}}$ , let $X_{\mathbf{a}}$ be the subset of $\mathcal{O}_{F}\times U_{F}$ consisting of $(m,u)$ satisfying (4.6) for all $v\in S_{F,\infty}$ . Then, we have

(4.7)

\left|S_{\rm ell}(\phi^{(\mathbf{a})})\right|\leq\sum_{(m,u)\in X_{\mathbf{a}}}\mathrm{Vol}(\overline{\mathrm{G}_{\gamma_{m,u}}(F)}\backslash\overline{\mathrm{G}_{\gamma_{m,u}}(\mathbb{A}_{F})})\left|S_{[\gamma_{m,u}]}(\phi^{(\mathbf{a})})\right|.

Note that we have

(4.8)

B_{m,u}:=N_{F/\mathbb{Q}}(m^{2}-4u)=\prod_{v\in S_{F,\infty}}|\sigma_{v}(m^{2}-4u)|^{1+\epsilon_{v}}.

Combining (3.30), (4.8), Lemmas 3.8 and 3.9, for each $(m,u)\in X_{\mathbf{a}}$ , we have

		$\displaystyle\mathrm{Vol}(\overline{\mathrm{G}_{\gamma_{m,u}}(F)}\backslash\overline{\mathrm{G}_{\gamma_{m,u}}(\mathbb{A}_{F})})\left\|S_{[\gamma_{m,u}]}(\phi^{(\mathbf{a})})\right\|$
		$\displaystyle\ll_{F,u}A^{-\frac{1}{2}}\cdot\prod_{v\in S_{m,u}^{1}}\left\|\sigma_{v}(m^{2}-4u)\right\|^{\frac{13(1+\epsilon_{v})}{2}}\prod_{v\in S_{m,u}^{2}}\left(\left\|\sigma_{v}(m^{2}-4u)\right\|^{6}\cdot h_{v,a_{v}}(0)\right)$
		$\displaystyle\ll_{F,u}A^{-\frac{1}{2}}\cdot\prod_{v\in S_{F,\infty}}e^{\frac{26\pi}{a_{v}}}\cdot\prod_{v\in S_{m,u}^{2}}h_{v,a_{v}}(0)$
		$\displaystyle\ll_{F,u}A^{-\frac{1}{2}}\cdot\prod_{v\in S_{F,\infty}}e^{\frac{26\pi}{a_{v}}}\cdot\prod_{v\in S_{m,u}^{1}}h_{v,a_{v}}^{-1}(0)\cdot\prod_{v\in S_{F,\infty}}h_{v,a_{v}}(0)$
		$\displaystyle\ll_{F,u}A^{-\frac{1}{2}}\cdot\prod_{v\in S_{F,\infty}}\left(e^{\frac{26\pi}{a_{v}}}\cdot h_{v,a_{v}}(0)\right).$

Here, the last inequality holds since a positive real number $a_{v}$ is less than $1$ for each $v\in S_{F,\infty}$ . Since $U_{F}$ is a finite set, it follows that

(4.9)

\mathrm{Vol}(\overline{\mathrm{G}_{\gamma_{m,u}}(F)}\backslash\overline{\mathrm{G}_{\gamma_{m,u}}(\mathbb{A}_{F})})\left|S_{[\gamma_{m,u}]}(\phi^{(\mathbf{a})})\right|\ll_{F}A^{-\frac{1}{2}}\cdot\prod_{v\in S_{F,\infty}}\left(e^{\frac{26\pi}{a_{v}}}\cdot h_{v,a_{v}}(0)\right).

By the finiteness of $U_{F}$ , Lemma 4.4 implies

(4.10)

\#X_{\mathbf{a}}\ll_{F}\prod_{v\in S_{F,\infty}}e^{\frac{2\pi n}{a_{v}}}.

Therefore, we complete the proof of Lemma 4.5 by combining (4.7), (4.9) and (4.10). ∎

Lemma 4.6.

With the above notation, assume that $a_{v}\leq 1$ for all $v\in S_{F,\infty}$ . Then, we have

\left|S_{\rm hyp}(\phi^{(\mathbf{a})})\right|\ll_{F}A^{-\frac{1}{2}}\cdot\prod_{v\in S_{F,\infty}}\left(e^{\frac{5\pi}{a_{v}}}\cdot h_{v,a_{v}}(0)\right).

Proof.

By (3.43), if $J_{\alpha}(\phi^{(\mathbf{a})})\neq 0$ , then there is $w\in S_{F}$ such that

\prod_{v\in S_{F,\infty}}J_{\sigma_{v}(\alpha),v}^{w}(\phi_{v,a_{v}})\neq 0.

Lemma 3.12 implies that for all $v\in S_{F,\infty}$ , we have

(4.11)

e^{-\frac{4\pi}{a_{v}(1+\epsilon_{v})}}<|\sigma_{v}(\alpha)|<e^{\frac{4\pi}{a_{v}(1+\epsilon_{v})}}.

Let $Y_{\mathbf{a}}$ be the subset of $\mathcal{O}_{F}^{\times}$ consisting of $\alpha$ satisfying (4.11) for all $v\in S_{F,\infty}$ . Then, we get

\left|S_{\rm hyp}(\phi^{(\mathbf{a})})\right|\leq\frac{\mathrm{Vol}(F^{\times}\backslash\mathbb{A}_{F}^{1})}{2}\sum_{\alpha\in Y_{\mathbf{a}}}\left|J_{\alpha}(\phi^{(\mathbf{a})})\right|.

Assume that $\alpha\in Y_{\mathbf{a}}$ . Since $\alpha-1\in\mathcal{O}_{F}$ , it follows that

(4.12)

1\leq\left|N_{F/\mathbb{Q}}(\alpha-1)\right|=\prod_{v\in S_{F,\infty}}\left|\sigma_{v}(\alpha-1)\right|^{1+\epsilon_{v}}.

Then, for $w\in S_{F,\infty}$ , we have

(4.13)

|\sigma_{w}(\alpha-1)|=|\sigma_{w}(\alpha)-1|\leq|\sigma_{w}(\alpha)|+1<e^{\frac{4\pi}{a_{w}(1+\epsilon_{w})}}+1\leq 2\cdot e^{\frac{4\pi}{a_{w}(1+\epsilon_{w})}}.

Combining (4.12) and (4.13), for any $v\in S_{F,\infty}$ , we get

(4.14)

\frac{1}{|\sigma_{v}(\alpha-1)|^{1+\epsilon_{v}}}<2^{n-1}\prod_{w\neq v}e^{\frac{4\pi}{a_{w}}}.

Since $\alpha\in Y_{\mathbf{a}}$ , it follows that $|\sigma_{v}(\alpha)|<e^{\frac{4\pi}{a_{v}(1+\epsilon_{v})}}$ and that

(4.15)

\left(\frac{|\sigma_{v}(\alpha)|^{\frac{1}{2}}}{|\sigma_{v}(\alpha)-1|}\right)^{1+\epsilon_{v}}<2^{n-1}e^{\frac{2\pi}{a_{v}}}\prod_{w\neq v}e^{\frac{4\pi}{a_{w}}}.

Note that $\frac{\Re(\alpha)}{|\alpha|}=\frac{\Re(\alpha^{-1})}{|\alpha^{-1}|}$ , $\frac{|\alpha|}{|\alpha-1|^{2}}=\frac{|\alpha^{-1}|}{|\alpha^{-1}-1|^{2}}$ and $\widehat{h_{\phi_{v}}}$ is an even function. By Lemma 3.12, we may assume that $|\sigma_{v}(\alpha)|\geq 1$ without loss of generality. If $v\neq w$ , then Lemma 3.12 says that

(4.16)

\left|J_{\sigma_{v}(\alpha),v}^{w}(\phi_{v,a_{v}})\right|\leq\frac{1+\epsilon_{v}}{2\pi}\cdot\frac{|\sigma_{v}(\alpha)|^{\frac{1+\epsilon_{v}}{2}}}{|\sigma_{v}(\alpha)-1|^{1+\epsilon_{v}}}.

Assume that $F_{v}=\mathbb{R}$ . By Lemma 3.12 and (4.15), we have

(4.17)	$\displaystyle\left\|J_{\sigma_{v}(\alpha),v}^{v}(\phi_{v,a_{v}})\right\|$	$\displaystyle\leq\frac{2\|\sigma_{v}(\alpha)\|}{\|\sigma_{v}(\alpha)-1\|^{2}}\int_{\frac{\log\|\sigma_{v}(\alpha)\|}{2\pi}}^{\frac{1}{a_{v}}}\widehat{h_{v,a_{v}}}(x)\cosh(\pi x)dx$
		$\displaystyle\leq\frac{\|\sigma_{v}(\alpha)\|}{\|\sigma_{v}(\alpha)-1\|^{2}}\cdot\cosh\left(\frac{\pi}{a_{v}}\right)\cdot h_{v,a_{v}}(0)$
		$\displaystyle<\frac{\|\sigma_{v}(\alpha)\|^{\frac{1}{2}}}{\|\sigma_{v}(\alpha)-1\|}\left(2^{n-1}e^{\frac{2\pi}{a_{v}}}\prod_{w\neq v}e^{\frac{4\pi}{a_{w}}}\right)\cdot\cosh\left(\frac{\pi}{a_{v}}\right)\cdot h_{v,a_{v}}(0)$
		$\displaystyle<\frac{2^{n-1}\|\sigma_{v}(\alpha)\|^{\frac{1}{2}}}{\|\sigma_{v}(\alpha)-1\|}\prod_{w\neq v}\frac{1}{h_{w}(0)}\prod_{w\in S_{F,\infty}}e^{\frac{4\pi}{a_{w}}}\cdot h_{w,a_{w}}(0).$

The last inequality holds since $h_{w,a_{w}}(0)=\frac{1}{a_{w}}h_{w}(0)\geq h_{w}(0)>0$ .

Assume that $F_{v}=\mathbb{C}$ . Since $\widehat{h_{v}}$ is a smooth function with compact support, there is a positive real number $M_{v}$ such that

\left|\widehat{h_{v,a_{v}}}(x)\right|\leq M_{v}

for all $x\in\mathbb{R}$ . By Lemma 3.12, we have

(4.18)		$\displaystyle\left\|J_{\sigma_{v}(\alpha),v}^{v}(\phi_{v,a_{v}})\right\|$	$\displaystyle\leq\frac{2\|\sigma_{v}(\alpha)\|}{\|\sigma_{v}(\alpha)-1\|^{2}}\cdot M_{v}\cdot\int_{\frac{\log(\|\sigma_{v}(\alpha)\|)}{\pi}}^{\frac{1}{a_{v}}}\frac{\sinh(\pi x)}{\cosh(\pi x)-\frac{\Re(\sigma_{v}(\alpha))}{\|\sigma_{v}(\alpha)\|}}dx$
(4.18)			$\displaystyle=\frac{2\|\sigma_{v}(\alpha)\|}{\|\sigma_{v}(\alpha)-1\|^{2}}\cdot M_{v}\cdot\frac{\left(\log s_{2}-\log s_{1}\right)}{\pi},$

where

s_{1}:=\cosh(\log|\sigma_{v}(\alpha)|)-\frac{\Re(\sigma_{v}(\alpha))}{|\sigma_{v}(\alpha)|}=\frac{|\sigma_{v}(\alpha)-1|^{2}}{2|\sigma_{v}(\alpha)|}\text{ and }s_{2}:=\cosh\left(\frac{\pi}{a_{v}}\right)-\frac{\Re(\sigma_{v}(\alpha))}{|\sigma_{v}(\alpha)|}.

By (4.15), we obtain

(4.19)	$\displaystyle\log s_{2}-\log s_{1}$	$\displaystyle\leq\log\left(\cosh\left(\frac{\pi}{a_{v}}\right)+1\right)+\log\left(\frac{2\|\sigma_{v}(\alpha)\|}{\|\sigma_{v}(\alpha)-1\|^{2}}\right)$
		$\displaystyle\leq\frac{\pi}{a_{v}}+\log\left(2^{n}e^{\frac{2\pi}{a_{v}}}\prod_{w\neq v}e^{\frac{4\pi}{a_{w}}}\right)$
		$\displaystyle\leq n\log 2+4\pi\sum_{w\in S_{F,\infty}}\frac{1}{a_{w}}$
		$\displaystyle\leq(4\pi+2\log 2)\sum_{w\in S_{F,\infty}}\frac{1}{a_{w}}.$

The last inequality holds since $\sum_{w\in S_{F,\infty}}\frac{1}{a_{w}}\geq\frac{n}{2}$ . Thus, combining (4.18) and (4.19), we have

(4.20)	$\displaystyle\left\|J_{\sigma_{v}(\alpha),v}^{v}(\phi_{v,a_{v}})\right\|$	$\displaystyle\leq\frac{2(4\pi+2\log 2)M_{v}\|\sigma_{v}(\alpha)\|}{\pi\|\sigma_{v}(\alpha)-1\|^{2}}\sum_{w\in S_{F,\infty}}\frac{1}{a_{w}}$
		$\displaystyle=\frac{2(4\pi+2\log 2)M_{v}\|\sigma_{v}(\alpha)\|}{\pi\|\sigma_{v}(\alpha)-1\|^{2}}\cdot\frac{\left(\prod_{w\in S_{F,\infty}}a_{w}\right)\left(\sum_{w\in S_{F,\infty}}\frac{1}{a_{w}}\right)}{\prod_{w\in S_{F,\infty}}h_{w}(0)}\prod_{w\in S_{F,\infty}}h_{w,a_{w}}(0)$
		$\displaystyle\leq\frac{2(4\pi+2\log 2)M_{v}\|\sigma_{v}(\alpha)\|}{\pi\|\sigma_{v}(\alpha)-1\|^{2}}\cdot\frac{n}{\prod_{w\in S_{F,\infty}}h_{w}(0)}\prod_{w\in S_{F,\infty}}h_{w,a_{w}}(0)$
		$\displaystyle\leq\frac{2(4\pi+2\log 2)M_{v}\|\sigma_{v}(\alpha)\|}{\pi\|\sigma_{v}(\alpha)-1\|^{2}}\cdot\frac{n}{\prod_{w\in S_{F,\infty}}h_{w}(0)}\prod_{w\in S_{F,\infty}}e^{\frac{4\pi}{a_{w}}}\cdot h_{w,a_{w}}(0).$

Recalling (3.55), we have

(4.21)		$\displaystyle\left\|J_{\alpha}(\phi^{(\mathbf{a})})\right\|$	$\displaystyle\leq A^{-\frac{1}{2}}\cdot\left\|N_{F\backslash\mathbb{Q}}(\alpha-1)\right\|$
(4.21)			$\displaystyle\times\left(2\log\|N_{F\backslash\mathbb{Q}}(\alpha-1)\|\cdot\left\|\prod_{v\in S_{F,\infty}}J_{\sigma_{v}(\alpha),v}^{w_{0}}(\phi_{v,a_{v}})\right\|+\sum_{w\in S_{F,\infty}}\left\|\prod_{v\in S_{F,\infty}}J_{\sigma_{v}(\alpha),v}^{w}(\phi_{v,a_{v}})\right\|\right)$

for any $w_{0}\in S_{F,\mathrm{fin}}$ . Combining (4.16), (4.17) and (4.20), we deduce that (4.21) becomes

	$\displaystyle\left\|J_{\alpha}(\phi^{(\mathbf{a})})\right\|\leq$	$\displaystyle A^{-\frac{1}{2}}\cdot\left\|N_{F/\mathbb{Q}}(\alpha-1)\right\|\cdot\prod_{v\in S_{F,\infty}}\frac{\|\sigma_{v}(\alpha)\|^{\frac{1+\epsilon_{v}}{2}}}{\|\sigma_{v}(\alpha-1)\|^{1+\epsilon_{v}}}$
	$\displaystyle\times$	$\displaystyle\bigg{(}2\cdot C_{1}\cdot\log\left\|N_{F/\mathbb{Q}}(\alpha-1)\right\|+\sum_{w\in S_{F,\infty}}C_{w,1}\prod_{v\in S_{F,\infty}}\left(e^{\frac{4\pi}{a_{v}}}\cdot h_{v,a_{v}}(0)\right)\bigg{)}$
	$\displaystyle=$	$\displaystyle A^{-\frac{1}{2}}\cdot\left\|N_{F/\mathbb{Q}}(\alpha)\right\|^{\frac{1}{2}}\bigg{(}2\cdot C_{1}\cdot\log\left\|N_{F/\mathbb{Q}}(\alpha-1)\right\|+\sum_{w\in S_{F,\infty}}C_{w,1}\prod_{v\in S_{F,\infty}}\left(e^{\frac{4\pi}{a_{v}}}\cdot h_{v,a_{v}}(0)\right)\bigg{)}$

for some constants $C_{1}$ and $C_{w,1}$ with $w\in S_{F,\infty}$ . Since $\alpha\in\mathcal{O}_{F}^{\times}$ , the absolute value of the norm $N_{F/\mathbb{Q}}(\alpha)$ of $\alpha$ is $1$ . By (4.13), we have

	$\displaystyle\log\left\|N_{F/\mathbb{Q}}(\alpha-1)\right\|$	$\displaystyle=\log\left(\prod_{w\in S_{F,\infty}}\|\sigma_{w}(\alpha-1)\|^{1+\epsilon_{w}}\right)$
		$\displaystyle\leq n\log 2+4\pi\sum_{w\in S_{F,\infty}}\frac{1}{a_{w}}$
		$\displaystyle\leq(4\pi+2\log 2)\sum_{w\in S_{F,\infty}}\frac{1}{a_{w}}$
		$\displaystyle\leq\frac{n(2\log 2+4\pi)}{\prod_{w\in S_{F,\infty}}h_{w}(0)}\prod_{w\in S_{F,\infty}}h_{w,a_{w}}(0).$

Therefore, we complete the proof of Lemma 4.6. ∎

Lemma 4.7.

With the above notation, assume that $a_{v}\leq 1$ for all $v\in S_{F,\infty}$ . Then, we have

(4.22)

\left|S_{\rm par}(\phi^{(\mathbf{a})})\right|\ll_{F}A^{-\frac{1}{2}}\log A\cdot\prod_{v\in S_{F,\infty}}h_{v,a_{v}}(0).

Proof.

Assume that $v\in S_{F,\infty}$ . Since $h_{v,a_{v}}$ is an even function, it follows that

\int_{\mathbb{R}}h_{v,a_{v}}(t)\cdot\frac{\Gamma^{\prime}}{\Gamma}(1+(1+\epsilon_{v})it)dt=2\int_{0}^{\infty}h_{v,a_{v}}(t)\cdot\Re\left(\frac{\Gamma^{\prime}}{\Gamma}(1+(1+\epsilon_{v})it)\right)dt.

By the asymptotic behavior of $\frac{\Gamma^{\prime}}{\Gamma}(1+it)$ , there are non-negative real numbers $A$ and $B$ such that

\left|\Re\left(\frac{\Gamma^{\prime}}{\Gamma}(1+(1+\epsilon_{v})it)\right)\right|\leq A+Bt

for all non-negative real numbers $t$ . Thus,

	$\displaystyle\left\|\int_{0}^{\infty}h_{v,a_{v}}(t)\cdot\Re\left(\frac{\Gamma^{\prime}}{\Gamma}(1+(1+\epsilon_{v})it)\right)dt\right\|$	$\displaystyle\leq\int_{0}^{\infty}\frac{1}{a_{v}}\cdot h_{v}\left(\frac{t}{a_{v}}\right)(A+Bt)dt$
		$\displaystyle=A\int_{0}^{\infty}\frac{1}{a_{v}}\cdot h_{v}\left(\frac{t}{a_{v}}\right)dt+B\int_{0}^{\infty}\frac{t}{a_{v}}\cdot h_{v}\left(\frac{t}{a_{v}}\right)dt$
		$\displaystyle=A\int_{0}^{\infty}h_{v}(t)dt+a_{v}\cdot B\int_{0}^{\infty}th_{v}(t)dt.$

Since $a_{v}\leq 1$ , we have

\left|\int_{\mathbb{R}}h_{v,a_{v}}(t)\cdot\frac{\Gamma^{\prime}}{\Gamma}(1+it)dt\right|\ll_{v}1.

Then, Lemma 3.14 implies that there are positive constants $C_{v}^{(1)}$ and $C_{v}^{(2)}$ satisfying

\left|Z_{v}^{\prime}(1,\phi_{v,a_{v}})\right|\leq C_{v}^{(1)}+C_{v}^{(2)}\cdot h_{v,a_{v}}(0).

Again, since $a_{v}\leq 1$ for all $v\in S_{F,\infty}$ , we obtain (4.22) by (3.67). ∎

Lemma 4.8.

With the above notation, assume that $a_{v}\leq 1$ for all $v\in S_{F,\infty}$ . Then, we have

\left|S_{\rm Eis}(\phi^{(\mathbf{a})})\right|\ll_{F}A_{J}\cdot A^{\frac{1}{2}}\cdot\prod_{v\in S_{F,\infty}}h_{v,a_{v}}(0).

Proof.

Recalling Lemma 3.15, we have

S_{\rm Eis}(\phi^{(\mathbf{a})})=\frac{A_{J}}{2^{r_{1}+2}\pi}\cdot\int_{\mathbb{R}}\bigg{(}\frac{\Lambda_{F}^{\prime}(2it)}{\Lambda_{F}(2it)}-\frac{\Lambda_{F}^{\prime}(2it+1)}{\Lambda_{F}(2it+1)}\bigg{)}\cdot\left(\prod_{v\in S_{F,\infty}}h_{v,a_{v}}(t)\right)\cdot\dim V_{it}^{\mathrm{K}(J)}dt.

There are only finitely many $v\in S_{F,\mathrm{fin}}$ such that $\mathrm{val}_{v}(J)>0$ . By [5, pp. 73], we have

(4.23)

\dim V_{it}^{\mathrm{K}(J)}\leq 2^{r_{1}}\cdot\prod_{v\in S_{F,\infty}}(1+\mathrm{val}_{v}(J)).

For each $v\in S_{F,\mathrm{fin}}$ , let $r_{v}:=[\mathcal{O}_{F}/\mathfrak{p}_{v}:\mathbb{F}_{p}]$ , where $\mathfrak{p}_{v}$ denotes a prime ideal of $\mathcal{O}_{F}$ corresponding to $v$ and $p$ denotes a prime satisfying $(p):=\mathfrak{p}_{v}\cap\mathbb{Z}$ . Then, we have

(4.24)

N_{F/\mathbb{Q}}(J)=\prod_{p}\left(\prod_{\mathfrak{p}_{v}\mid p}N_{F/\mathbb{Q}}(\mathfrak{p}_{v})^{\mathrm{val}_{v}(J)}\right)=\prod_{p}p^{\sum_{\mathfrak{p}_{v}\mid p}r_{v}\cdot\mathrm{val}_{v}(J)}.

Let $d(N)$ be the number of divisors of an integer $N$ . Let $n:=[F:\mathbb{Q}]$ . Since the number of prime ideals $\mathfrak{p}_{v}$ of $\mathcal{O}_{F}$ lying over $p$ is less than or equal $n$ , it follows that

\left(d\left(N_{F/\mathbb{Q}}(J)\right)\right)^{n}=\prod_{p}\left(1+\sum_{\mathfrak{p}_{v}\mid p}r_{v}\cdot\mathrm{val}_{v}(J)\right)^{n}\geq\prod_{p}\left(\prod_{\mathfrak{p}_{v}\mid p}(1+\mathrm{val}_{v}(J))\right)=\prod_{v\in S_{F,\mathrm{fin}}}(1+\mathrm{val}_{v}(J)).

Combining (4.23) and (4.24), we get

\frac{\dim V_{it}^{\mathrm{K}_{0}(J)}}{2^{r_{1}}}\leq\left(d\left(N_{F/\mathbb{Q}}(J)\right)\right)^{n}=d(A)^{n}.

Since $d(A)\ll_{\epsilon}A^{\epsilon}$ for $\epsilon>0$ , it follows that

\displaystyle\left|S_{\rm Eis}(\phi^{(\mathbf{a})})\right|

\displaystyle\ll_{F}A_{J}\cdot A^{\frac{1}{2}}\cdot\int_{\mathbb{R}}\left|\frac{\Lambda_{F}^{\prime}(2it)}{\Lambda_{F}(2it)}-\frac{\Lambda_{F}^{\prime}(2it+1)}{\Lambda_{F}(2it+1)}\right|\prod_{v\in S_{F,\infty}}h_{v,a_{v}}(t)dt.

By [13, Proposition 5.7], there are positive real numbers $C_{1}$ and $C_{2}$ such that for any $t\in\mathbb{R}$ ,

\left|\frac{\Lambda_{F}^{\prime}(2it)}{\Lambda_{F}(2it)}-\frac{\Lambda_{F}^{\prime}(2it+1)}{\Lambda_{F}(2it+1)}\right|\leq C_{1}+C_{2}|t|.

Thus, we obtain that

\displaystyle\left|S_{\rm Eis}(\phi^{(\mathbf{a})})\right|

\displaystyle\ll_{F}A_{J}\cdot A^{\frac{1}{2}}\cdot\int_{0}^{\infty}(C_{1}+C_{2}t)\prod_{v\in S_{F,\infty}}h_{v,a_{v}}(t)dt.

Since $\widehat{h_{v,a_{v}}}$ is a non-negative function, $h_{v,a_{v}}(t)$ is less than or equal to $h_{v,a_{v}}(0)$ . Then, we have

	$\displaystyle\left\|\int_{0}^{\infty}(C_{1}+C_{2}t)\prod_{v\in S_{F,\infty}}h_{v,a_{v}}(t)dt\right\|$	$\displaystyle\leq\prod_{v\neq w}h_{v,a_{v}}(0)\cdot\left\|\int_{0}^{\infty}(C_{1}+C_{2}t)h_{w,a_{w}}(t)dt\right\|$
		$\displaystyle=\prod_{v\neq w}h_{v,a_{v}}(0)\cdot\left\|C_{1}\cdot\int_{0}^{\infty}h_{w}(t)dt+C_{2}\cdot a_{w}\int_{0}^{\infty}th_{w}(t)dt\right\|$
		$\displaystyle\ll_{F}\prod_{v\in S_{F,\infty}}h_{v,a_{v}}(0).$

Here, the last inequality holds because $a_{w}$ is a positive real number with $a_{w}\leq 1$ . ∎

Using the argument in the proof of Lemma 4.8, we have

\dim V_{0}^{\mathrm{K}(J)}\ll_{F}A^{\frac{1}{2}}.

Thus, Lemma 3.16 implies the following lemma.

Lemma 4.9.

With the above notation, we have

\left|S_{\mathrm{Res}}(\phi^{(\mathbf{a})})\right|\ll_{F}A_{J}\cdot A^{\frac{1}{2}}\cdot\prod_{v\in S_{F,\infty}}h_{v,a_{v}}(0).

Now, we are ready to prove Theorem 1.1.

Proof of Theorem 1.1.

Let $(\mathbf{a}):=(a_{v})_{v\in S_{F,\infty}}\in\mathbb{R}_{>0}^{\#S_{F,\infty}}$ . For each $v\in S_{F,\infty}$ , assume that $a_{v}\leq 1$ . By Lemma 2.3 and (2.17), we have

(4.25)	$\displaystyle\mathcal{N}(J)$	$\displaystyle\leq\frac{2^{r_{1}}}{A_{J}\cdot\prod_{v\in S_{F,\infty}}h_{v,a_{v}}(0)}\sum_{\pi\in\mathfrak{X}_{F}}\mathrm{tr}\pi(\phi^{(\mathbf{a})})$
		$\displaystyle\leq\frac{2^{r_{1}}}{A_{J}\cdot\prod_{v\in S_{F,\infty}}h_{v,a_{v}}(0)}\bigg{(}\left\|S_{\mathrm{one}}(\phi^{(\mathbf{a})})\right\|+\left\|S_{\rm id}(\phi^{(\mathbf{a})})\right\|+\left\|S_{\rm ell}(\phi^{(\mathbf{a})})\right\|+\left\|S_{\rm hyp}(\phi^{(\mathbf{a})})\right\|$
		$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\qquad+\left\|S_{\rm par}(\phi^{(\mathbf{a})})\right\|+\left\|S_{\rm Eis}(\phi^{(\mathbf{a})})\right\|+\left\|S_{\mathrm{Res}}(\phi^{(\mathbf{a})})\right\|\bigg{)}.$

By Lemmas 4.2, 4.3, 4.5, 4.6, 4.7, 4.8, and 4.9, we have

(4.26)		$\displaystyle\mathcal{N}(J)\ll_{F}\frac{1}{A_{J}}\bigg{(}$	$\displaystyle A_{J}\cdot\prod_{v\in S_{F,\infty}}e^{\frac{\pi}{a_{v}}}+\prod_{v\in S_{F,\infty}}a_{v}^{2+\epsilon_{v}}+A^{-\frac{1}{2}}\cdot\prod_{v\in S_{F,\infty}}e^{\frac{2\pi(n+13)}{a_{v}}}$
(4.26)			$\displaystyle+A^{-\frac{1}{2}}\cdot\prod_{v\in S_{F,\infty}}e^{\frac{5\pi}{a_{v}}}+A^{-\frac{1}{2}}\log A+A_{J}\cdot A^{\frac{1}{2}}\bigg{)}.$

For each $v\in S_{F,\infty}$ , let

a_{v}:=\frac{8\pi n(n+13)}{\log A}.

We see that $a_{v}\leq 1$ as $A\to\infty$ . Since $A=|N_{F/\mathbb{Q}}(J)|\leq A_{J}^{-1}$ , the main term of the right hand side of $\eqref{eq 86}$ comes from $A_{J}^{-1}\prod_{v\in S_{F,\infty}}a_{v}^{2+\epsilon_{v}}$ . Therefore, we conclude that

\mathcal{N}(J)\ll_{F}\frac{1}{A_{J}(\log A)^{2r_{1}+3r_{2}}}=\frac{[\mathrm{SL}_{2}(\mathcal{O}_{F}):\Gamma_{0}(J)]}{(\log(N_{F/\mathbb{Q}}(J)))^{2r_{1}+3r_{2}}},\quad\left|N_{F/\mathbb{Q}}(J)\right|\to\infty.

∎

References

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[3] A. R. Booker and A. Strömbergsson. Numerical computations with the trace formula and the Selberg eigenvalue conjecture. J. Reine Angew. Math., 607:113–161, 2007.
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[5] S. Gelbart. Automorphic forms on adèle groups. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. Annals of Mathematics Studies, No. 83.
[6] S. Gelbart and H. Jacquet. Forms of ${\rm GL}(2)$ from the analytic point of view. In Automorphic forms, representations and $L$ -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, pages 213–251. Amer. Math. Soc., Providence, R.I., 1979.
[7] I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products. Elsevier/Academic Press, Amsterdam, seventh edition, 2007. Translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX).
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[15] H. H. Kim. Functoriality for the exterior square of ${\rm GL}_{4}$ and the symmetric fourth of ${\rm GL}_{2}$ . J. Amer. Math. Soc., 16(1):139–183, 2003. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak.
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[18] M. Palm. The character of $GL(2)$ automorphic forms. J. Number Theory, 160:679–699, 2016.
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(3.38)		$\displaystyle 2\pi\int_{0}^{\infty}\phi_{v}\left(\begin{pmatrix}\|\sigma_{v}(\alpha_{\gamma})\|&y\|\sigma_{v}(\alpha_{\gamma})-1\|\\ 0&1\end{pmatrix}\right)ydy$	$\displaystyle=\frac{2\pi\|\sigma_{v}(\alpha_{\gamma})\|}{\|\sigma_{v}(\alpha_{\gamma})-1\|^{2}}\int_{\log\|\sigma_{v}(\alpha_{\gamma})\|}^{\infty}\phi_{v}\left(\begin{pmatrix}e^{-\frac{r}{2}}&0\\ 0&e^{\frac{r}{2}}\end{pmatrix}\right)\sinh(r)dr$
(3.38)			$\displaystyle=\frac{\|\sigma_{v}(\alpha_{\gamma})\|}{\pi\|\sigma_{v}(\alpha_{\gamma})-1\|^{2}}\widehat{h_{\phi_{v}}}\left(\frac{\log\|\sigma_{v}(\alpha_{\gamma})\|}{\pi}\right).$

(3.55)		$\displaystyle\|J_{\alpha}(\phi)\|$	$\displaystyle\leq\left\|N_{F\backslash\mathbb{Q}}(\alpha-1)\right\|\cdot\left\|N_{F\backslash\mathbb{Q}}(J)\right\|^{-\frac{1}{2}}$
(3.55)			$\displaystyle\times\left(2\log\|N_{F\backslash\mathbb{Q}}(\alpha-1)\|\cdot\left\|\prod_{v\in S_{F,\infty}}J_{\sigma_{v}(\alpha),v}^{w_{0}}(\phi_{v})\right\|+\sum_{w\in S_{F,\infty}}\left\|\prod_{v\in S_{F,\infty}}J_{\sigma_{v}(\alpha),v}^{w}(\phi_{v})\right\|\right),$

	$\displaystyle J_{\alpha,v}^{v}(\phi_{v})$	$\displaystyle=-(1+\epsilon_{v})\int_{F_{v}}\phi_{v}\left(\begin{pmatrix}\alpha&x(\alpha-1)\\ 0&1\end{pmatrix}\right)\log(\|x\|^{2}+1)dx$
		$\displaystyle=-2\pi^{\epsilon_{v}}(1+\epsilon_{v})\int_{0}^{\infty}\phi_{v}\left(\begin{pmatrix}\|\alpha\|&x\|\alpha-1\|\\ 0&1\end{pmatrix}\right)\log(x^{2}+1)x^{\epsilon_{v}}dx$
		$\displaystyle=-\frac{2\pi^{\epsilon_{v}}(1+\epsilon_{v})}{\|\alpha-1\|^{1+\epsilon_{v}}}\int_{0}^{\infty}\phi_{v}\left(\begin{pmatrix}\|\alpha\|&x\\ 0&1\end{pmatrix}\right)\log(x^{2}\|\alpha-1\|^{-2}+1)x^{\epsilon_{v}}dx.$

		$\displaystyle J^{v}_{\alpha,v}(\phi_{v})$
		$\displaystyle=\frac{1}{2\pi^{2}\|\alpha-1\|}\int_{\frac{\log\|\alpha\|}{2\pi}}^{\infty}\widehat{h_{\phi_{v}}}^{\prime}(t)\int_{0}^{2\sqrt{\|\alpha\|}\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{\log\|\alpha\|}{2}\right)}}\frac{\log\bigg{(}\frac{x^{2}}{\|\alpha-1\|^{2}}+1\bigg{)}}{\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{r}{2}\right)}}dx\,dt$
		$\displaystyle=\frac{1}{2\pi^{2}\|\alpha-1\|}\int_{\frac{\log\|\alpha\|}{2\pi}}^{\infty}\widehat{h_{\phi_{v}}}^{\prime}(t)\int_{0}^{2\sqrt{\|\alpha\|}\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{\log\|\alpha\|}{2}\right)}}\frac{\log\bigg{(}\frac{x^{2}}{\|\alpha-1\|^{2}}+1\bigg{)}}{\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{\log\|\alpha\|}{2}\right)-\frac{x^{2}}{4\|\alpha\|}}}dx\,dt$
		$\displaystyle=\frac{\sqrt{\alpha\|}}{\pi^{2}\|\alpha-1\|}\int_{\frac{\log\|\alpha\|}{2\pi}}^{\infty}\widehat{h_{\phi_{v}}}^{\prime}(t)\int_{0}^{\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{\log\|\alpha\|}{2}\right)}}\frac{\log\bigg{(}\frac{4\|\alpha\|x^{2}}{\|\alpha-1\|^{2}}+1\bigg{)}}{\sqrt{\sinh^{2}(\pi t)-\sinh^{2}\left(\frac{\log\|\alpha\|}{2}\right)-x^{2}}}dx\,dt.$

	$\displaystyle\left\|h_{v,a_{v}}\left(\frac{i}{2}\right)\right\|$	$\displaystyle=\frac{1}{a_{v}}\left\|h_{v}\left(\frac{i}{2a_{v}}\right)\right\|$
		$\displaystyle=\frac{1}{a_{v}}\left\|\int_{\mathbb{R}}\widehat{h_{v}}(x)e^{\frac{\pi x}{a_{v}}}dx\right\|$
		$\displaystyle=\frac{2}{a_{v}}\left\|\int_{0}^{1}\widehat{h_{v}}(x)\cosh\left(\frac{\pi x}{a_{v}}\right)dx\right\|$
		$\displaystyle\leq\cosh\left(\frac{\pi}{a_{v}}\right)\cdot h_{v,a_{v}}(0)$
		$\displaystyle\leq e^{\frac{\pi}{a_{v}}}\cdot h_{v,a_{v}}(0).$

The number of automorphic representations of GL2\mathrm{GL}_{2} with exceptional eigenvalues

Abstract.

2020 Mathematics Subject Classification:

1. Introduction

Theorem 1.1.

Corollary 1.2.

Remark 1.3.

Acknowledgments.

2. Preliminaries

Lemma 2.1.

Proof.

Lemma 2.2.

Proof.

Lemma 2.3.

Proof.

3. Geometric side

3.1. One-dimensional representations

Lemma 3.1.

Proof.

3.2. Identity contribution

Lemma 3.2.

Proof.

3.3. Elliptic contribution

Lemma 3.3.

Remark 3.4.

Proof of Lemma 3.3.

Lemma 3.5.

Proof.

Lemma 3.6.

Proof.

Lemma 3.7.

Proof.

Lemma 3.8.

Proof.

Lemma 3.9.

Proof.

3.4. Hyperbolic contribution

Lemma 3.10.

Proof.

Lemma 3.11.

Proof.

Lemma 3.12.

Proof.

3.5. Parabolic contribution

Lemma 3.13.

Proof.

Lemma 3.14.

Proof.

3.6. Eisenstein series and residual contribution

Lemma 3.15.

Proof.

Lemma 3.16.

4. Proof of Theorem 1.1

Lemma 4.1.

Proof.

Lemma 4.2.

Proof.

Lemma 4.3.

Proof.

Lemma 4.4.

Proof.

Lemma 4.5.

Proof.

Lemma 4.6.

Proof.

Lemma 4.7.

Proof.

Lemma 4.8.

Proof.

Lemma 4.9.

Proof of Theorem 1.1.

References

The number of automorphic representations of $\mathrm{GL}_{2}$ with exceptional eigenvalues