---
abstract: 'Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a cuspidal automorphic representation $\pi$ of $D_\A^\times$ with trivial central cahracter. We study the zeta functions with period integrals on $\pi$ for the perhomogeneous vector space $(D^\times\times D^\times\times\GL_2, D\oplus D)$. We show their meromorphic continuation and functional equation, determine the location and orders of possible poles and compute the residue. Arguing along the theory of Saito and computing unramified local factors, the explicit formula of the zeta functions is obtained. Counting the order of possible poles of this explicit formula, we show that if $L(1/2, \pi)\neq0$, there are infinitely many quadratic extension $E$ of $F$ which embeds in $D$, such that $\pi$ has nonvanishing toric period with respect to $E$.'
address:
- |
    Miyu Suzuki\
    Faculty of Mathematics and Physics, Institute of Science and Engineering\
    Kanazawa University\
    Kakumamachi, Kanazawa, Ishikawa, 920-1192, JAPAN
- |
    Satoshi Wakatsuki\
    Faculty of Mathematics and Physics, Institute of Science and Engineering\
    Kanazawa University\
    Kakumamachi, Kanazawa, Ishikawa, 920-1192, JAPAN
author:
- Miyu Suzuki and Satoshi Wakatsuki
title: 'Zeta functions and nonvanishing theorems for toric periods on $\GL_2$'
---

Introduction
============

Background and Main result
--------------------------

In a series of papers [@Wal1; @Wal2; @Wal3; @Wal4], Waldspurger studied Shimura correspondence between half integral weight and integral weight modular forms in the framework of automorphic representations of the metaplectic group $\mathrm{Mp}_2$. One of the remarkable results in these papers is a detailed description of the automorphic discrete spectrum of $\mathrm{Mp}_2$. Theta correspondence plays a central role in this representation theoretic treatment. In the course of the study, especially in [@Wal2; @Wal4], Waldspurger found a close relation among special values of $L$-functions, period integrals and nonvanishing of theta correspondence. Our research focuses on nonvanishing of the period integrals with an eye on the work of Waldspurger [@Wal4] which was extended by Friedberg and Hoffstein [@FH]. We briefly review these results.

Let $E$ be a quadratic étale algebra over a number field $F$. Let $\eta$ denote the quadratic character of $\A_F^\times/F^\times$ associated with $E$ via the global class field theory if $E$ is a field and the trivial character otherwise. We write the ring of adèles of $F$ by $\A=\A_F$ and set $\A_E=\A_F\otimes_FE$. We fix an embedding $E\hookrightarrow\M_2(F)$ and regard $\operatorname{Res}_{E/F}\bb{G}_{m, E}$ as an $F$-subgroup of $\GL_2$. Here, $\operatorname{Res}_{E/F}$ denotes the restriction of scalars.

We say that $\pi$ is $E^\times$-distinguished if the integral $$\mP_E(\vp)=\int_{E^\times Z(\A)\bs\A_E^\times}\vp(h)\d h$$ defines a non-zero linear form on $\pi$. Here, $Z$ denotes the center of $\GL_2$.

Fix a finite set $S$ of places of $F$. For $\xi\in F^\times$, let $\chi_\xi$ denote the quadratic character $(\,\cdot, \xi)_\A$ of $\A^\times$, where $(\cdot,\cdot)_\A$ is the product of local Hilbert symbols. The next theorem is [@Wal4 Théorèm 4].

\[thm:W1\] Let $\pi'$ be an irreducible cuspidal automorphic representation of $\GL_2(\A)$ with trivial central character. Take a positive real number $\de_v$ for each $v\in S$. If we have $\ve(1/2, \pi')=1$, there exists $\xi\in F^\times$ such that

-   For any $v\in S$, $|\xi-1|_v<\de_v$.

-   $L(1/2, \pi'\otimes\chi_\xi)\neq0$.

As noted in the introduction of [@GI], this theorem plays an important role in the argument of Waldspurger to complete the full description of the automorphic discrete spectrum of $\mathrm{Mp}_2$.

Friedberg and Hoffstein generalized Theorem \[thm:W1\] to the following result. Fix a quadratic character $\chi_0$ of $\A^\times/F^\times$. Let $\Psi(S; \chi_0)$ denote the set of quadratic characters $\chi$ of $\A^\times/F^\times$ such that $\chi_v=\chi_{0,v}$ for all $v\in S$. For each non-trivial $\chi\in\Psi(S;\chi_0)$, let $E_\chi$ be the corresponding quadratic extension of $F$ via the global class field theory.

\[thm:FH\] Let $\pi'$ be an irreducible cuspidal automorphic representation of $\GL_2(\A)$ which is self-dual. Suppose that we have $\ve(1/2, \pi'\otimes\chi)=1$ for some $\chi\in\Psi(S;\chi_0)$. Then there are infinitely many $\chi\in\Psi(S;\chi_0)$ such that $L(1/2, \pi'\otimes\chi)\neq0$.

In the proof, Waldspurger used in an essential way the result of Flicker on the Shimura lifting from $\GL_2$ to a certain double cover of $\GL_2$. Instead of it, the argument of Friedberg and Hoffstein is based on a technique of analytic number theory. They studied poles of a Dirichlet series with coefficients given by $L$-functions.

These theorems on special values of $L$-functions can be restated in terms of nonvanishing of periods. In order to do that, we recall another theorem of Waldspurger from [@Wal4]. Let $X(E)$ denote the set of quatenion algebras over $F$ which $E$ embeds as an $F$-algebra. For $D\in X(E)$, we set $D_\A=D\otimes_F\A$. For an irreducible cuspidal automorphic representation $\pi$ of $D_\A^\times$, there is a unique irreducible automorphic representation $\pi'=\operatorname{JL}(\pi)$ of $\GL_2(\A)$ such that for almost all places $v$ of $F$ at which $D$ splits, $\pi_v$ is isomorphic to $\pi'_v$.

\[thm:W2\] (1) For $D\in X(E)$, let $\pi$ be an irreducible cuspidal automorphic representation of $D_\A^\times$ with trivial central character such that $\pi'=\operatorname{JL}(\pi)$ is cuspidal. If $\pi$ is $E^\times$-distinguished, then $L(1/2, \pi')L(1/2, \pi'\otimes\eta)\neq0$.

\(2) Let $\pi'$ be an irreducible cuspidal automorphic representation of $\GL_2(\A)$ with trivial central character. If $L(1/2, \pi')L(1/2, \pi'\otimes\eta)\neq0$, then there is a unique $D\in X(E)$ and an irreducible cuspidal automorphic representation $\pi$ of $D_\A^\times$ which is is $E^\times$-distinguished and $\pi'=\operatorname{JL}(\pi)$. Moreover, it was proved by Tunnell [@Tunnell] and Saito [@Saito] that the ramification set of $D$ is $\{v\mid \ve(1/2, \pi'_{E, v})=-1\}$. In other words, $D_v$ is division if and only if $\ve(1/2, \pi'_{E, v})=-1$. Here, $\pi'_E$ is the base change of $\pi'$ to $\GL_2(\A_E)$.

Applying this theorem, we see that Theorem \[thm:FH\] has following corollary

Let $\pi'$ be an irreducible cuspidal automorphic representation of $\GL_2(\A)$ with trivial central character. Suppose that we have $\ve(1/2, \pi'\otimes\chi)=1$ for some $\chi\in\Psi(S;\chi_0)$ and $L(1/2, \pi')\neq0$. Then there are infinitely many $\chi\in\Psi(S;\chi_0)$ such that there exists $D_\chi\in X(E_\chi)$ and an irreducible cuspidal automorphic representation $\pi$ of $D_{\chi, \A}^\times$ which is $E_\chi^\times$-distinguished and $\pi'=\operatorname{JL}(\pi)$.

Now we state our main result. Let $D$ be a quaternion algebra over $F$. The set of quadratic étale algebras over $F$ which $E$ embeds is denoted by $X(D)$.

\[thm:main\] Let $\pi$ be an irreducible cuspidal automorphic representation of $D_\A^\times$ with trivial central character which is not $1$-dimensional. Suppose we have $L(1/2, \pi)\neq0$. Then we can take a quadratic étale algebra $\cE_v$ over $F_v$ for each $v\in S$ so that there are infinitely many $E\in X(D)$ such that $\pi$ is $E^\times$-distinguished and $E_v=\cE_v$ for any $v\in S$.

0Unfortunately, this theorem is not a generalization of Theorem \[thm:FH\] because $S$-components of $E$ are not arbitrary in our theorem. Despite this defect, it has following novelty:

-   Theorem \[thm:main\] is about representations of arbitrary inner forms of $\GL_2$, while Theorem \[thm:W1\] and Theorem \[thm:FH\] only treats representations of $\GL_2$.

-   Theorem \[thm:main\] asserts nonvanishing of infinitely many periods, which is stronger than the assertions of Theorem \[thm:W1\] and Theorem \[thm:FH\] which assert nonvanishing of central values of $L$-functions for infinitely many quadratic twists.

The difference between our theorem and Theorem \[thm:FH\] is reduced to a local problem. In order to see this, let us try to deduce Theorem \[thm:main\] from Theorem \[thm:FH\]. Take a quarternion algebra $D$ and a cuspidal automorphic representation $\pi$ as in Theorem \[thm:main\]. Let $S(\pi)$ denote the finite set of places of $F$ which includes all Archimedean placs and places at which $\pi$ is ramified or $D$ is division. What we have to find is a quadratic extension $E_0$ satisfying two conditions:

-   for any $v\in S(\pi)$, $D_v$ is division if and only if $\ve(1/2, \pi'_{E_0, v})=-1$

-   $\ve(1/2, \pi\otimes\eta_{E_0})=1$,

where $\pi'_{E_0}$ denotes the base change of $\pi'=\operatorname{JL}(\pi)$ to $\GL_2(\A_{E_0})$ and $\eta_{E_0}$ is the qudratic character of $\A^\times$ attached to $E_0$. Suppose we have such an $E_0$. Theorem \[thm:FH\] asserts that there are infinitely many quadratic extension $E$ of $F$ such that $E_v=E_{0, v}$ for any $v\in S(\pi)$ and $L(1/2, \pi\otimes\eta_{E})\neq0$. From Theorem \[thm:W2\] (2), we see that $\pi$ is $E^\times$-distinguished for each such $E$.

Now we assume we have $E_0$ which satisfies (a). For $v\in S(\pi)$, $\pi_v$ is generic unramified and has trivial central character, hence for any quadratic étale algebra $K_v$ over $F_v$, we have $\ve(\pi_{K_v})=1$. Combining this with (a), we get $$1=\ve(1/2, \pi'_{E_0})=\ve(1/2, \pi)\ve(1/2, \pi\otimes\eta_{E_0})\eta_{E_0}(-1)=\ve(1/2, \pi\otimes\eta_{E_0}).$$ Thus (b) follows from (a). The only remaining problem is to find $E_0$ which satisfies (a). This can be done if there is a quadratic étale algebra $\cE_v$ over $F_v$ at each $v\in S(\pi)$ so that we have $\ve(1/2, \pi'_{\cE_v})=-1$ if and only if $D_v$ is division. Once we have such a family $\{\cE_v\}$, we can take a quadratic extension $E_0$ so that $E_{0, v}=K_v$ for each $v\in S(\pi)$ by the strong approximation theorem. However, it seems not easy to prove the existence of such a family $\{\cE_v\}$ in general. Conversely, the existence of such $\{\cE_v\}_v$ follows immediately from Theorem \[thm:main\].

Our method to prove Theorem \[thm:main\] is to use prehomogeneous zeta functions with period integrals. We will show in Theorem \[thm:nonvanishing423\] that if our zeta functions have a pole, then there exists such a quadratic extension $E_0$. The proof of this fact is not so long. Hence, if one combines Theorem \[thm:FH\] with Theorem \[thm:nonvanishing423\], then one obtains a short proof of Theorem \[thm:main\]. On the other hand, we will give an alternative proof in Theorem \[thm:infinite\] by the explicit formula of the zeta functions, that is, Theorem \[thm:main\] is proved by our zeta functions without using Theorem \[thm:FH\]. In the following subsections, we explain the zeta functions, the proofs and possibilities of further applications.

Prehomogeneous zeta functions with period integrals
---------------------------------------------------

Our main tool is the prehomogeneous zeta functions with period integrals which were introduced by F. Sato in [@Sato]. Besides the application to the study of period integrals, this paper has contribution to the theory of prehomogeneous zeta functions with period integrals. In this subsection, we give an overview of prior research on this topic.

After introducing new zeta functions, he focused efforts on proving their global functional equations for several specific prehomogeneous vector spaces. A special case (the case of spherical Maass cusp forms) of the zeta functions we treat in this paper appears in his paper [@Sato3]. In [@Sato3], the global functional equation is obtained in classical setting (not adelic) and from the point of view of [@KS], the zeta functions are shown to be a Dirichlet series of the Fourier coefficients of certain cusp forms of half integral weight. This fact seems to reflect a hidden relation between the prehomogeneous zeta functions with period integrals and the lifting of automorphic forms. In the course of his proof, he obtained the local functional equation of the local prehomogeneous zeta functions with matrix coefficients of real spherical principal series. The local functional equation of the zeta functions attached to spherical representations are shown in general setting under some conditions by Bopp and Rubenthaler [@BR]. In that paper, they aimed to generalize the Godement-Jacquet theory. Recently, the theory of the local functional equations has greatly developed by the works of Wen-Wei Li [@Li1; @Li2; @Li3; @Li4]. According to his results, the local zeta functions for arbitrary admissible representations satisfy certain local functional equations.

On the one hand the local theory is now established in general by Li, but on the other hand nothing is known about the global theory and a lot of open questions are raised in [@Li1 Chapter 8]. We mention that this paper answers those questions for specific prehomogeneous vector spaces. Firstly, he explained in [@Li1 Corollary 8.3.7] that the nonvanishing of the zeta functions implies the automorphic representation is distinguished. In this paper, we will show that the zeta function has a pole provided the central value of the $L$-function is nonzero, and obtain a sufficient condition for distinction. Secondly, the meromorphic continuation and the functional equation of the zeta functions conjectured in [@Li1 Definition 8.4.1] is obtained in Theorem \[thm:global\] and Corollary \[cor:funct\]. Lastly, we will obtain the essential factorization , of our zeta functions, which is an expression as a sum of Euler products of local zeta functions proposed in [@Li1 Definition 8.2.5]. This expression will be obtained from the Saito’s formula and Waldspurger’s formula and plays an important role in the application to the study of the period integrals. In addition, we explicitly compute the local zeta functions and deduce from it the explicit formula of our global zeta functions. As is explained later, the nonvanishing of infinitely many periods immediately follows from this explicit formula.

Methods and details of the results
----------------------------------

The main point of proving Theorem \[thm:main\] is a new way to show nonvanishing of infinitely many period integrals. The setting of our question seems purely algebraic, but it seems that we need the methods and techniques of analytic number theory. A basic idea in analytic number theory to handle such problems is relating them to the existence of poles or zeros of some Dirichlet series or other similar infinite series. One of the most typical examples is the Dirichlet prime number theorem and its generalization called the Chebotarev’s density theorem. These theorems assert the existence of infinite number of prime numbers or places of number fields with certain properties, and are proved by analyzing Dirichlet and Dirichlet-Hecke $L$-functions.

In this paper, we introduce a new method along the same lines as these previous works with a different tool, the prehomogeneous zeta functions with period integrals. This method combines the techniques of analytic number theory with analysis of prehomogeneous zeta functions *à la* H. Saito [@Saito1]. To the best of the authors’ knowledge, this is the first example that the prehomogeneous zeta function with period integrals are used to study nonvanishing of specific period integrals.

Let us switch back to the setting of Theorem \[thm:main\]. Set $G=D^\times\times D^\times\times\GL_2$ and $V=D\oplus D$. The $F$-rational representation $\rho$ of $G$ on $V$ is defined by $$(x, y)\cdot\rho(g_1, g_2, g_3):=(g_1^{-1}xg_2, g_1^{-1}yg_2)g_3, 
\quad (g_1,g_2,g_3)\in G, \quad (x,y)\in V.$$ Then, the triple $(G, \rho, V)$ is an $F$-form of a prehomogeneous vector space. We write the regular locus of $V$ by $V^0$. Let $\pi$ be a cuspidal automorphic representation of $D_\A^\times:=(D\otimes\A)^\times$ with trivial central character. For cusp forms $\phi_1, \phi_2$ in $\pi$ and a Schwartz function $\Phi$ on $V(\A)$, we define the global zeta function with period integrals by $$Z(\Phi, \phi_1, \phi_2, s)=\int_{H(F)\bs H(\A)}|\omega(h)|^s\phi_1(g_1)\phi_2(g_2)\sum_{x\in V^0(F)}\Phi(x\cdot\rho(h))\d h.$$ Here, $H$ is the quotient of $G$ by the kernel of $\rho\,\colon G\rightarrow\GL(V)$, $\omega$ is the character attached to the fundamental relative invariant of $(G, \rho, V)$ and $(g_1, g_2, g_3)$ is a representative in $G$ of $h\in H$.

By the orbit decomposition in Proposition \[prop:orbits\], we obtain $$\label{eq:0418ef}
Z(\Phi,\phi_1,\phi_2,s)=\frac{1}{2}\sum_{E\in X(D)}\int_{H_{x(\delta)}^0(\A)\bsl H(\A)} \cP_E(\phi_1^{g_1})\, \cP_E(\phi_2^{g_2})   \,  |\omega( \tilde{h} )|^s \Phi(x(\delta)\cdot \tilde{h}) \, \d \tilde{h},$$ where $\phi_i^{g_i}=\pi(g_i)\phi_i$, $x(\delta)$ means a representative element of a $H(F)$-orbit in $V^0(F)$ corresponding to $E$, and $\d \tilde{h}$ denotes a quotient measure on $H_{x(\delta)}^0(\A)\bsl H(\A)$. For other unexplained notation, see § \[sec:space\]. From this, we see that $\pi$ is $E^\times$-distinguished for at least one $E\in X(D)$ if the zeta function is not identically zero. By analyzing the principal part of our zeta functions (Theorem \[thm:global\]), we will see that if $L(1/2, \pi)\neq0$, we can take $\Phi$ and $\phi$ so that $Z(s, \Phi, \phi, \bar{\phi}, s)$ has a simple pole at $s=1/2$ under the conditions of Theorem \[thm:main\]. In particular, $\pi$ is $E^\times$-distinguished for at least one $E\in X(D)$.

To show the nonvanishing of infinitely many periods, we need more close analysis on our zeta functions. Applying the technique of essential factorization developed by Saito for ordinary prehomogeneous zeta functions and computing explicitly the unramified components of the Euler factor, we get $$\label{eq:0419ex}
Z(\Phi,\phi,\overline{\phi},s)=\sum_{\cE_S\in X(D_S)}   \prod_{v\in S} Z_{\cE_v}(\phi_v,\Phi_v,s)  \times   \xi(s,D,\phi,\cE_S),$$ where $S$ is a finite set of places of $F$ containing all “bad” places, $\cE_S=\prod_{v\in S}\cE_v$ in $X(D_S):=\prod_{v\in S}X(D_v)$, $Z_{\cE_v}(\phi_v,\Phi_v,s)$ is the local zeta function (cf. ) and $$\begin{aligned}
\xi(s,D,\phi,\cE_S):=& \frac{ 1 }{2 \, |\Delta_F|^4  \, c_F }  \frac{\zeta^S_F(2s-1) \, L^S(2s-1,\pi,\mathrm{Ad}) }{\zeta^S_F(2)^3}  \prod_{v\in S}\alpha_{\cE_v}^\#(\phi_v,\phi_v)^{-1} \\
& \sum_{E\in X(D,\pi,\cE_S)} \frac{ L(1,\eta)^2 \, |\cP_E(\phi)|^2\, \mathcal{D}_E^S(\pi,s)}{N(\mathfrak{f}_\eta^S)^{s-1}}\end{aligned}$$ (Theorem \[thm:20200105t1\]). Here, $\phi=\otimes_v \phi_v$ is a suitably chosen cusp form in $\pi$, $\mathcal{D}_E^S(\pi,s)$ is a function defined as a certain infinite product, and $X(D,\pi,\cE_S)$ denotes the set of $E\in X(D)$ whose $v$-component coincides with $\cE_v$ for $v\in S$ such that $\pi$ is $E^\times$-distinguished. For other unexplained notation, see § \[sec:w0504\], § \[sec:fac\] and § \[sec:exp\]. Together with the assumptions of Theorem \[thm:main\] and an element $\cE_S'=\prod_{v\in S}\cE_v'$ such that $X(D,\pi,\cE_S')$ is not empty, the assertion follows from the following observation and comparison of the order of the pole at $s=1$ of the both sides of the above equation:

-   The global zeta function $Z(\Phi,\phi,\overline{\phi},s)$ is holomorphic at $s=1$.

-   $\zeta_F^S(2s-1)$ has a simple pole at $s=1$.

-   With suitable choices of $\Phi$ and $\phi$, the local zeta function $Z_{\cE_v'}(\phi_v,\Phi_v,s)$ is entire and does not have zero at $s=1$ for each $v\in S$, and $Z_{\cE_v}(\phi_v,\Phi_v,s)=0$ if $\cE_v\neq \cE_v'$.

-   The infinite product defining $\mathcal{D}_E^S(\pi,s)$ converges absolutely for $\mathrm{Re}(s)>1$ and $\mathrm{Re}(\mathcal{D}_E^S(\pi,s))$ tends to $+\infty$ when $s\rightarrow 1+0$.

See the proofs of Theorems \[thm:nonvanishing423\] and \[thm:infinite\] for the detail.

In order to refine our result to get a generalization of Theorem \[thm:FH\], we should study the local zeta functions at “bad places” in more detail. To be more precise, what is needed for such a refinement is nonvanishing of the gamma factors which appear in the local functional equations proved by Wen-Wei Li [@Li4]. In § \[sec:20200424\], we will discuss the prospect of the future works in this direction. A detailed study on analytic properties of local zeta functions provides us quantitative information. For example, from the study of the gamma factors, we might be able to obtain convexity bound of $\xi(s,D,\phi,\cE_S)$, see [@SS Theorem 2 (iii)]. As another application of our global and local zeta functions, probably we can use the Dirichlet series $\xi(s,D,\phi,\cE_S)$ to study the mean value theorem for $\lim_{X\to\inf}X^{-1}\sum_{E\in X(D,\pi,\cE_S),\, N(\mathfrak{f}_\eta^S)^{1/2}<X}L(1,\eta)^2 \, |\cP_E(\phi)|^2$, see [@KY1] and [@Taniguchi2].

The following is the structure of the paper. In the next section, we recall the structure theory of the prehomogeneous vector spaces we consider and Waldspurger’s formula for toric periods. In § \[sec:globalzeta424\], we recall the notion of the prehomogeneous zeta functions with period integrals and show their basic properties such as absolute convergence, meromorphic continuation and the global functional equation. From the analysis of the principal part and the explicit formula of our zeta functions, we prove Theorem \[thm:main\]. Finally in § \[sec:localzeta424\], we will give the the details of the unramified computation and show the explicit formula of the zeta functions.

**Acknowledgments.** The authors thank Wen-Wei Li for helpful discussions and sharing his impeccably written preprints with us. Needless to say, this paper owes its inspiration to a series of his recent pioneering works. The authors also would like to thank Kimball Martin for pointing out that the difference between our result and Theorem \[thm:FH\] is reduced to a local problem. A further thank you to both Atsushi Ichino and Tamotsu Ikeda for useful comments and kindly answering our questions. The first author would like to thank Hang Xue for helpful comments. The second author thanks Akihiko Yukie for helpful discussions.

Preliminaries
=============

The space of pairs of quaternion algebras {#sec:space}
-----------------------------------------

Throughout this subsection, let $F$ be a field of characteristic not two. Let $\bar{F}$ denote an algebraic closure of $F$, and set $\underline{G}:=\GL_2(\bar{F})^3$, and $\underline{V}:=\bar{F}^2\otimes \bar{F}^2\otimes\bar{F}^2$. A rational irreducible representation $\underline{\rho}$ is defined by $v\cdot\underline{\rho}(g)=(v_1g_1)\otimes(v_2g_2)\otimes(v_3g_3)$ where $g=(g_1,g_2,g_3)\in \underline{G}$ and $v=(v_1,v_2,v_3)\in \underline{V}$. Then, the triple $(\underline{G},\underline{\rho},\underline{V})$ is a prehomogeneous vector space, that is, it has an open dense orbit. This space is classified into a special case $(n,m)=(4,2)$ of the irreducible reduced prehomogeneous vector spaces (20) $(\SO_n\times \GL_m,M_{n,m})$ in [@SK Theorem 24 in § 3]. Its split $F$-form $F^2\otimes F^2\otimes F^2$ was introduced in [@WY] as the $D_4$ case from the viewpoint of field extensions, and its $\Z$-structure $\Z^2\otimes \Z^2\otimes \Z^2$ is well-known as Bhargava’s cube [@Bhargava]. Its $F$-forms were classified in \[Saito, Theorem 2.11 (3)\]. The $F$-rational orbits of the non-split $F$-forms $\GL_2(E)\times \GL_2(F)$ and $\GL_2(E')$, where $E$ (resp. $E'$) is a quadratic (resp. cubic) extension of $F$, were studied in [@KY] and [@GS]. In addition, Kable and Yukie studied their zeta functions and density theorems in [@KY1], [@Yukie] and so on. Besides, Taniguchi [@Taniguchi1; @Taniguchi2] studied another type non-split $F$-form, which is a pair of quaternoin algebras. In this section, we review basic properties of the spaces of pairs of quaternion algebras, which include the split case $F^2\otimes F^2\otimes F^2\cong M_2\oplus M_2$.

### Setup

Let $D$ denote a quaternion algebra over $F$. When $D$ is not division, we may identify $D$ with $M_2(F)$, and a standard $F$-involution $\iota$ on $D$ can be chosen as $x^\iota:=\left(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\right)\,{}^t\!x \left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)$. When $D$ is division, there exist a quadratic extension $E$ of $F$ and an element $b\in F^\times$ such that $D$ is regarded as $$D=\left\{  \begin{pmatrix} \xi & \eta \\ b\overline{\eta} & \overline{\xi} \end{pmatrix} \in M_2(E) \mid  \xi,\eta\in E \right\}$$ where $E\ni a\mapsto\overline{a}\in E$ is the non-trivial Galois action on $E/F$, and a standard $F$-involution $\iota$ on $D$ can be chosen as $\left(\begin{smallmatrix} \xi & \eta \\ b\overline{\eta} & \overline{\xi} \end{smallmatrix}\right)^\iota=\left(\begin{smallmatrix} \overline{\xi} & -\eta \\ -b\overline{\eta} & \xi  \end{smallmatrix}\right)$. Set $\det(x):= x\, x^\iota\in F$ and $\Tr(x):=x+x^\iota\in F$ $(x\in D)$.

An algebraic group $G$ over $F$ is defined by $$G:=G_1\times G_2\times G_3, \quad G_1=G_2=\GL_1(D), \quad G_3=\GL_2,$$ and an $8$-dimensional vector space $V$ over $F$ is defined by $V(F)=D\oplus D$. Define a $F$-rational representation $\rho:G\to \GL(V)$ by $$(x,y) \cdot \rho(g_1,g_2,g_3) := (g_1^{-1}xg_2,g_1^{-1}yg_2)g_3, \quad (g_1,g_2,g_3)\in G, \quad (x,y)\in V.$$ Then, the kernel $Z_\rho$ of $\rho$ is $$Z_\rho:=\mathrm{Ker}\rho=\{ (aI_2,bI_2,ab^{-1}I_2)\in G \mid a,b\in\mathbb{G}_m  \}\cong \mathbb{G}_m\times \mathbb{G}_m,$$ where we write $I_n$ for the unit matrix in $M_n$.

Set $H:=Z_\rho\bsl G$, and we identify $H$ with $\rho(G)$. Denote by $x\cdot \rho(h)$ or $x\cdot h$ $(x\in V$, $h\in H)$ the faithful $F$-representation of $H$ obtained from the above representation $\rho$ of $G$. The group $X_F(H)$ of $F$-rational characters of $H$ is generated by $\omega_0$ which is defined by $$\omega_0(h):=\det(g_1)^{-1}\det(g_2)\, \det(g_3) \qquad (h=Z_\rho(g_1,g_2,g_3)\in H).$$ Let $\mathcal{W}$ denote the vector space over $F$ defined by $\mathcal{W}(F)=\{ x\in D \mid  \Tr(x)=0 \}$. A representation $\rho_0$ of $H$ on $\mathcal{W}$ over $F$ is defined by $x\cdot \rho_0(h):=g_1^{-1}x\, g_1$ where $h=Z_\rho(g_1,g_2,g_3)\in H$. Define a $F$-morphism $P_0$ from $V$ to $\mathcal{W}$ by $P_0(x,y):=x\, y^\iota-y \, x^\iota$, $(x,y)\in V$. Then, $P_0$ satisfies $P_0((x,y)\cdot \rho(h))=\omega_0(h)\, P_0(x,y)\cdot\rho_0(h)$.

For each point $x\in V(F)$, we say that $x$ is regular if $x\cdot \rho(G(\bar{F}))$ is open dense in $V(\bar{F})$, and $x$ is singular otherwise. Suppose that $x_0$ is a regular element in $V(F)$. Let $H_x$ denote the stabilizer of $x$ in $H$, and $H_x^0$ the connected component of $1$ in $H$. Set $P(x):=-\det(P_0(x))$ and $\omega(h):=\omega_0(h)^2$. Then, $P(x\cdot\rho(h))=P(x)$ $(\forall h\in H_x(F))$ holds, and $P(x)$ on $V$ is called the fundamental relative invariant. Set $V^0:=\{ x\in V \mid P(x)\neq0 \}$, and then $V^0(F)$ means the set of regular elements in $V(F)$.

### Orbit decompositions {#sec:orbits}

In what follows, we describe $F$-rational orbits and stabilizers for $(G,\rho,V)$. Let $G_x$ denote the stabilizer of $x$ in $G$. We write $O_n$ for the zero matrix in $M_n$, and $E_{ij}$ for the matrix unit of the entry $(i,j)$ in $M_n$. The following propositions can be proved by direct calculations.

\[prop:sor1\] Suppose that $D=M(2,F)$. The following is a list of representative elements of singular $G(F)$-orbits in $V(F)$.

-   $x_0:=(O_2,O_2)$, $G_{x_0}=G$.

-   $x_1:=(O_2,E_{12})$, $G_{x_1}=\{ (  \left(\begin{smallmatrix}a&*  \\ 0 & * \end{smallmatrix}\right)  ,\left(\begin{smallmatrix}*&*  \\ 0 & b \end{smallmatrix}\right)  ,\left(\begin{smallmatrix}*&*  \\ 0 & c \end{smallmatrix}\right)  )\in G \mid a=bc   \}$.

-   $x_2:=(O_2,I_2)$, $G_{x_2}=\{ ( g,h,\left(\begin{smallmatrix}*&*  \\ 0 & c \end{smallmatrix}\right) )\in G \mid g=c h  \}$.

-   $x_3:=(E_{11},E_{12})$, $G_{x_3}\cong G_{x_2}$.

-   $x_4:=(E_{12},E_{22})$, $G_{x_4}\cong G_{x_2}$.

-   $x_5:=(E_{12},I_2)$, $G_{x_5}=\{ ( \left(\begin{smallmatrix}a_1&a_2 \\ 0&a_3 \end{smallmatrix}\right),\left(\begin{smallmatrix}a_1&a_2 \\ 0&a_3 \end{smallmatrix}\right)\left(\begin{smallmatrix}1/c_3 & -c_2/c_1 c_3 \\ 0 & 1/c_3 \end{smallmatrix}\right),\left(\begin{smallmatrix} c_1 & c_2 \\ 0 & c_3 \end{smallmatrix}\right))\in G \mid a_1c_3=a_3c_1 \}$.

\[prop:sor2\] Suppose that $D$ is division. There are only two singular $G(F)$-orbits in $V(F)$. One is the trivial orbit $x_0:=(0,0)$, and the other is $x_2\cdot \rho(G(F))$ where $x_2:=(0,1)$.

Let $E$ be a quadratic étale algebra over $F$. Choose an element $\delta=\delta_E \in E$ such that $E=F(\delta)$ and $\Tr(\delta)=0$. Set $d=d_E=\delta^2$. Define a similitude orthogonal group $\mathrm{GO}_{2,E}$ over $F$ by $$\mathrm{GO}_{2,E}:=\left\{  g\in \GL_2 \mid \exists \mu(g)  \in \mathbb{G}_m \;\; \text{s.t.} \;\; {}^t\!g\begin{pmatrix} 1&0 \\ 0&-d_E \end{pmatrix} g=  \mu(g) \begin{pmatrix} 1&0 \\ 0&-d_E \end{pmatrix}  \right\}.$$ For each algebraic group $\mathscr{U}$ over $F$, let $\mathscr{U}^0$ denote the connected component of $1$ in $\mathscr{U}$. Note that $\mathrm{GO}_{2,E}^0$ is isomorphic to $R_{E/F}(\mathbb{G}_m)$ over $F$.

\[prop:orbits\] Let $X(D)$ denote the set of quadratic étale algebras over $F$, which are embedded in $D$ over $F$. For each $\alpha$ in $D$, we set $x(\alpha):=(1,\alpha)$. A set of representative elements of regular $G(F)$-orbits in $V(F)$ is given by $\{ \, x(\delta_E) \mid E \in X(D) \, \}$. Notice that $P(x(\delta))=4d\in d(F^\times)^2$. In addition, $$G_{x(\delta)}^0= \left\{   (  a_1+ b_1 \, \delta , \, a_2+b_2 \, \delta, \,  g_3)\in G \mid  a_j^2- b_j^2d \neq 0  \; \; (j=1,2), g_3=\left(\begin{smallmatrix} a_1&b_1d \\ b_1&a_1 \end{smallmatrix}\right)\left(\begin{smallmatrix} a_2&b_2d \\ b_2&a_2 \end{smallmatrix}\right)^{-1}   \right\},$$ $$[G_{x(\delta)}:G_{x(\delta)}^0]=2, \quad G_{x(\delta)}^0\cong \mathrm{GO}_{2,E}^0 \times \mathrm{GO}_{2,E}^0.$$ It follows from the Skolem-Noether theorem that there exists an element $\gamma \in D^\times$ such that $(\gamma, \gamma, \diag(1,-1))\in G_{x(\delta)}(F) \setminus G_{x(\delta)}^0(F)$ and $\gamma^{-1} \delta \gamma = -\delta$.

Suppose that $F$ is a number field. Let $\fp_D$ denote the set consisting of places $v$ of $F$ such that $D\otimes F_v$ is division, where $F_v$ denotes the completion of $F$ at $v$. It is known that $E$ belongs to $X(D)$ if and only if $E$ is not split for each $v\in \fp_D$.

Waldspurger’s formula {#sec:w0504}
---------------------

Suppose $E/F$ is a quadratic extension of number fields. Let $D$ be a quaternion algebra over $F$ which $E$ embeds and $\pi=\otimes_v\pi_v$ be a cuspidal automorphic representation of $D_\A^\times$ with trivial central character. We recall the explicit formula relating the central $L$-value $L(1/2, \pi)L(1/2, \pi\otimes\eta)$ and the period $\mP_E$ which was proved by Waldspurger in [@Wal2].

For each place $v$, take a $D_v^\times$-invarian non-degenerate Hermitian pairing $\l\,,\r_v$ on $\pi_v$ so that we have $$\l\,,\r=\prod_v\l\,,\r_v.$$ Here, the left hand side is the Petersson inner prouct on $\pi$. We also take a local Haar measure $\d h_v$ on $F_v^\times\bs E_v^\times$ for each $v$ so that we have $$\d h=\prod_v\d h_v,$$ where the left hand side is the Tamagawa measure on $\A_F^\times\bs\A_E^\times$. Then, the integral $$\al_{E_v}(\vp_v, \vp'_v)=\int_{F_v^\times\bs E_v^\times}\l\pi_v(h_v)\vp_v, \vp'_v\r_v\,\d h_v, \hspace{10pt} \vp_v, \vp'_v\in\pi_v$$ converges absolutely and defines an element of $\operatorname{Hom}_{E_v^\times\times E_v^\times}(\pi_v\boxtimes\bar{\pi}_v, \C)$. We define the normalized $E_v^\times$-invariant Hermitian pairing $\al_{E_v}^\#(\cdot,\cdot)$ by $$\al_{E_v}^\#(\vp_v, \vp'_v)=\frac{L(1, \pi_v, \operatorname{Ad})L(1, \eta_v)}{\zeta_{F_v}(2)L(1/2, \pi_v)L(1/2, \pi_v\otimes\eta_v)}\cdot\al_{E_v}(\vp_v, \vp'_v).$$

\[thm:Wald\] For any $\vp=\otimes_v\vp_v$ and $\vp'=\otimes_v\vp'_v\in\pi$ in $\pi$, one has $$\label{eq:Wald}
\mP_E(\vp)\ol{\mP_E(\vp')}=\frac14\cdot\frac{\zeta_F(2)L(1/2, \pi)L(1/2, \pi\otimes\eta)}{L(1, \pi, \operatorname{Ad})L(1, \eta)}\cdot\prod_v\al_{E_v}^\#(\vp_v, \vp'_v)$$ In particular, we have $\al_{E_v}^\#(\vp_v, \vp'_v)=1$ for almost all $v$.

This explicit formula is used in the next section to obtain the Euler product of the contribution of each orbit to the zeta function.

Global zeta functions with toric periods {#sec:globalzeta424}
========================================

Notations {#sec:notation}
---------

Let $F$ be an algebraic number field, and let $\Sigma$ denote the set of all the places of $F$. For any $v\in\Sigma$, we denote by $F_v$ the completion of $F$ at $v$. If $v<\inf$, we write $\fo_v$ for the ring of integers of $F_v$, and $\varpi_v$ for a prime element of $\fo_v$. We put $q_v=\#(\fo_v/\varpi_v\fo_v)$. Let $\A$ denote the adele ring of $F$. Let $\d x$ denote the Haar measure on $\A$ normalized by $\int_{\A/F} \d x=1$. For each $v<\inf$, we fix a Haar measure $\d x_v$ on $F_v$ normalized by $\int_{\fo_v}\d x_v=1$. Further, we choose Haar measures $\d x_v$ on $F_v$ for $v|\inf$ so that $\d x = \Delta_F^{-1/2} \prod_{v\in\Sigma} \d x_v$ holds, where $\Delta_F$ denotes the absolute discriminant of $F/\Q$.

We denote by $|\; |_v$ the normal valuation of $F_v$. Then, we have $\d (ax_v)=|a|_v \d x_v$ for any $a\in F_v^\times$. Define the idele norm $|\; |=|\; |_\A$ on $\A^\times$ by $|x|=|x|_\A=\prod_{v\in\Sigma} |x_v|_v$ for all $x=(x_v)\in\A^\times$. Choose a non-trivial additive character $\psi_\Q$ on $\A_\Q/\Q$, and set $\psi_F=\psi_\Q \circ \Tr_{F/\Q}$. Then, $\d x$ is the self-dual Haar measure with respect to $\psi_F$. For each $v\in \Sigma$, we set $\psi_{F_v}=\psi_{F}|_{F_v}$. For each $v\in\Sigma$, let $\d^\times x_v$ denote a Haar measure on $F_v^\times$ as ${\displaystyle \d^\times x_v=(1-q_v^{-1})^{-1}\, \tfrac{\d x_v}{|x|_v} }$ if $v<\inf$ and ${\displaystyle \d^\times x_v=\tfrac{\d x_v}{|x|_v} }$ if $v|\inf$. The idele norm $|\;|$ induces an isomorphism $\A^\times/\A^1\to\R_{>0}$. Choose the Haar measure $\d^\times x=\prod_{v\in\Sigma}\d^\times x_v$ on $\A^\times$ and normalize the Haar measure $\d^1 x$ on $\A^1$ in such a way that the quotient measure on $\R_{>0}$ is $\d t/t$, where $\d t$ is the Lebesgue measure on $\R$.

For a finite subset $S$ of $\Sigma$, we set $F_S=\prod_{v\in S}F_v$. Define the norm $|\; |_S$ on $F_S^\times$ by $|x|_S=\prod_{v\in S}|x_v|_v$ for $x=(x_v)\in F_S^\times$. Let $\psi_{F_S}(x)=\prod_{v\in S}\psi_{F_v}(x_v)$ for $x=(x_v)\in F_S$. For any vector space $\mathcal{V}$ over $F$, let $\cS(\mathcal{V}(F_S))$ and $\cS(\mathcal{V}(\A))$ denote the Schwartz spaces of $\mathcal{V}(F_S)$ and $\mathcal{V}(\A)$ respectively.

Global zeta functions and their principal parts
-----------------------------------------------

Let $D$ be a quaternion algebra over $F$, set $D_\A:=D\otimes \A$. Then we have $V(\A)=D_\A\oplus D_\A$. Let $Z$ denote the center of $\GL_1(D)$. Choose an automorphic form $\phi_j\in L^2(Z(\A)D^\times\bsl D_\A^\times)$ $(j=1,2)$. Here, we refer to [@GJ p.145] for the sense of automorphic forms. Suppose that $\phi_1$ and $\phi_2$ are cuspidal.

A bilinear form $\langle \; , \; \rangle$ on $V$ is defined by $$\langle (x_1,x_2) , (y_1,y_2) \rangle:=\Tr(x_1 y_1)+\Tr(x_2  y_2),$$ and the dual space of $V$ is identified with $V$ by $\langle \; , \; \rangle$. Let $\rho^\vee$ denote the contragredient representation of $\rho$ on $V$ with respect to $\langle \;\; , \;\; \rangle$, that is, $\langle x\cdot \rho(g) , y\cdot \rho^\vee(g) \rangle=\langle x , y \rangle$. Then, one has $Z_\rho=\mathrm{Ker}\rho^\vee$ and $$(y_1,y_2)\cdot \rho^\vee(g_1,g_2,g_3) =(g_2^{-1} y_1 g_1,g_2^{-1}y_2g_1)\, ^t{}\!g_3^{-1}.$$ For $s\in\C$ and $\Phi\in\cS(V(\A))$, we define the global zeta functions $Z(\Phi,\phi_1,\phi_2,s)$ and $Z^\vee(\Phi,\phi_1,\phi_2,s)$ as $$Z(\Phi,\phi_1,\phi_2,s):=\int_{H(F)\bsl H(\A)} |\omega(h)|^s \phi_1(g_1)\, \phi_2(g_2)\sum_{x\in V^0(F)} \Phi(x\cdot \rho(h))\, \d h,$$ $$Z^\vee(\Phi,\phi_1,\phi_2,s):=\int_{H(F)\bsl H(\A)} |\omega^\vee(h)|^s \phi_1(g_1)\, \phi_2(g_2)\sum_{x\in V^0(F)} \Phi(x\cdot \rho^\vee(h))\, \d h,$$ where $h=Z_\rho(g_1,g_2,g_3)\in H(\A)$, $\omega^\vee:=\omega^{-1}$ and $\d h$ is a Haar measure on $H(\A)$.

\[lem:abs\] There exists a sufficiently large constant $T>0$ such that $Z(\Phi,\phi_1,\phi_2,s)$ and $Z^\vee(\Phi,\phi_1,\phi_2,s)$ are absolutely and uniformly convergent on any compact set in $\{ s\in \C \mid \Re(s)>T \}$. Hence, they are holomorphic on the domain $\{ s\in \C \mid \Re(s)>T \}$.

See [@Saito2].

We set $$Z_+(\Phi,\phi_1,\phi_2,s):=\int_{H(F)\bsl H(\A) , |\omega(h)|\geq 1} |\omega(h)|^s \phi_1(g_1)\, \phi_2(g_2)\sum_{x\in V^0(F)} \Phi(x\cdot \rho(h))\, \d h ,$$ $$Z_+^\vee(\Phi,\phi_1,\phi_2,s):=\int_{H(F)\bsl H(\A),|\omega^\vee(h)|\geq 1} |\omega^\vee(h)|^s \phi_1(g_1)\, \phi_2(g_2)\sum_{x\in V^0(F)} \Phi(x\cdot \rho^\vee(h))\, \d h .$$ The zeta integrals $Z_+(\Phi,\phi_1,\phi_2,s)$ and $Z_+^\vee(\Phi,\phi_1,\phi_2,s)$ are entire on $\C$ by Lemma \[lem:abs\].

Choose a Haar measure $\d g$ on $D_\A^\times$. Set $$f_{\phi_1,\phi_2}(g'):=\int_{Z(\A) D^\times \bsl D_\A^\times}\phi_1(g)\, \phi_2(gg') \, \d g \qquad (g'\in D_\A^\times), \quad  f^\vee_{\phi_1,\phi_2}=f_{\phi_2,\phi_1}.$$ This function $f_{\phi_1,\phi_2}$ is called a matrix coefficient. A Godement-Jacquet zeta integral $Z^\GJ(\Psi,f_{\phi_1,\phi_2},s)$ is defined by $$Z^\GJ(\Psi,f_{\phi_1,\phi_2},s):= \int_{D_\A^\times} \Psi(g) \, f_{\phi_1,\phi_2}(g) \, |\det(g)|^s \, \d g \qquad (\Psi\in\cS(D_\A))$$ which is absolutely convergent for $\Re(s)>2$, [@GJ Theorem 13.8 in p.179]. In addition, it is analytically continued to the whole $s$-plane. Let $\d x$ denote the Haar measure on $D_\A$ normalized by $\int_{D\bsl D_\A}\d x=1$. Set $\widehat\Psi(y):=\int_{D_\A} \Psi(x) \psi_F(\Tr(xy)) \, \d x$ and then we obtain the functional equation $$Z^\GJ(\Psi,f_{\phi_1,\phi_2},s)=Z^\GJ(\widehat{\Psi},f_{\phi_1,\phi_2}^\vee,2-s).$$

We choose a self-dual Haar measure $\d x$ on $V(\A)$ with respect to $\psi_F(\langle \; ,  \; \rangle)$, that is, we have $\int_{V(F)\bsl V(\A)} \d x=1$. Set $$\widehat\Phi(y):=\int_{V(\A)} \Phi(x) \psi_F(\langle x, y \rangle) \, \d x .$$ Choose a Haar measure $\d g_3$ on $G_3(\A)=\GL_2(\A)$ as $$\d g_3:=\frac{\d t_1}{t_1} \frac{\d t_2}{t_2} \, \d^1a \, \d^1c \, \d b \, \d k$$ for $\diag\left(t_1^{1/2} t_2^{-1/2}, t_1^{1/2}t_2^{1/2}\right) \, \left(\begin{smallmatrix}a&b \\ 0&c\end{smallmatrix}\right) k\in \GL_2(\A)$, $t_1,t_2\in\R_{>0}$, $a,c\in\A^1$, $b\in\A$, $k\in K$, $\operatorname{vol}(F\bsl \A)=\operatorname{vol}(K)=1$, and we set $$c_F:=\operatorname{vol}(F^\times \bsl \A^1).$$ Choose a Haar measure $\d z_j$ $(j=1,2)$ on $Z(\A)$ as $\d z_j:=t^{-1}\d t \, \d^1 a$ for $z=t^{1/2}a I_2 \in Z(\A)$, $a\in\A^1$ and $t\in\R_{>0}$. Let $\d g_j$ $(j=1,2)$ denote the Haar measure on $D^\times_\A$ as above. Then, a Haar measure $\d h$ on $H(\A)$ is chosen by $\d h:=\d g_1 \, \d g_2 \, \d g_3/\d z_1\, \d z_2$ for $h=Z_\rho (g_1,g_2,g_3)\in H(\A)$ and $(z_1I_2,z_2I_2,z_1z_2^{-1}I_2)\in Z_\rho(\A)$. Let $K$ denote the standard maximal compact subgroup in $\GL(2,\A)$. Set $$\Phi_K(x):= \int_K \Phi(x\cdot \rho(1,1,k)) \, \d k.$$ The principal part of $Z(\Phi,\phi_1,\phi_2,s)$ is described as below.

\[thm:global\] Assume that $\phi_1$ and $\phi_2$ are cuspidal, and one of them is orthogonal to the constant functions on $H_j(F)\bsl H_j(\A)$. For any $\Re(s)>T$ and any $\Phi \in \cS(V(\A))$, we obtain $$\begin{aligned}
Z(\Phi,\phi_1,\phi_2,s)=& Z_+(\Phi,\phi_1,\phi_2,s)+Z_+^\vee(\widehat\Phi,\phi_1,\phi_2,2-s) \\
&  +c_F \frac{Z^\GJ(\big(\widehat{\Phi}_K\big)_1,f_{\phi_1,\phi_2}^\vee,1)}{2s-3}-c_F \frac{Z^\GJ(\left(\Phi_K\right)_1,f_{\phi_1,\phi_2},1)}{2s-1},\end{aligned}$$ where $\Phi_1(x):=\Phi(x,0)$. Note that we have $\widehat{(\Phi_K)}=\widehat{\Phi}_K$ and $\Phi_K(x_1,x_2)=\Phi_K(x_2,x_1)$. Hence, $Z(\Phi,\phi_1,\phi_2,s)$ is meromorphically continued to the whole $s$-plane, and might have a simple pole at $s=3/2$ or $s=1/2$.

The proof will be given in § \[sec:principleproof\].

\[cor:funct\] The zeta integrals satisfy the functional equation $$Z(\Phi,\phi_1,\phi_2,s)=Z^\vee(\widehat\Phi,\phi_1,\phi_2,2-s).$$

Comparing $\rho^\vee$ with $\rho$, one can show that $$\begin{aligned}
Z^\vee(\Phi,\phi_1,\phi_2,s)=& Z_+^\vee(\Phi,\phi_1,\phi_2,s)+Z_+(\widehat\Phi,\phi_1,\phi_2,2-s) \\
&  +c_F \frac{Z^\GJ(\big(\widehat{\Phi}_K\big)_1,f_{\phi_1,\phi_2},1)}{2s-3}-c_F\frac{Z^\GJ(\left(\Phi_K\right)_1,f_{\phi_1,\phi_2}^\vee,1)}{2s-1}.\end{aligned}$$ Therefore, the functional equation is proved by this and Theorem \[thm:global\].

Set $\sK:=\{(1,1,k)\in G(\A) \mid k\in K\}$. We say that $\Phi$ is $\sK$-spherical if $\Phi(x\cdot \rho(k))=\Phi(x)$ holds for any $k\in \sK$. We may suppose that $\Phi$ is $\sK$-spherical without loss of generality, because $\phi_1(g_1)$, $\phi_2(g_2)$ and $|\det(g)|^s$ are stable under $\sK$. Actually, $\Phi_K$ is $\sK$-spherical.

\[prop:202001241\] Let $\pi=\otimes_v \pi_v$ be an automoprhic representation of $D_\A^\times$ with trivial central character. When $D$ is division, we suppose that $\pi$ is not one dimensional. When $D=M_2(F)$, we suppose that $\pi$ is cuspidal. If $L(1/2,\pi)\neq 0$, then there exist $\phi=\otimes_v\phi_v\in\pi$ and $\sK$-spherical $\Phi=\otimes_v\Phi_v\in \cS(V(\A))$ such that $Z(\Phi,\phi,\overline{\phi},s)\not\equiv 0$.

Let $K_v$ denote the standard maximal compact subgroup of $\GL_2(F_v)$. Then, we have $K=\prod_{v\in\Sigma}K_v$ and $\sK=\prod_{v\in\Sigma} \sK_v$ where $\sK_v:=\{(1,1,k)\in G(F_v) \mid k\in K_v \}$.

Let $S$ denote a finite set of places which contains all infinite places and finite places $v$ such that $D\otimes F_v$ is division, $\pi_v$ is ramified, $\phi_v$ is non-spherical for $K_v$, or $\Phi_v$ is not the characteristic function of $V(\fo_v)$. Then, we have $$Z^\GJ(\left(\Phi\right)_1,f_{\phi,\overline{\phi}},1) = L(1/2,\pi) \times \prod_{v\in S}  \frac{Z^\GJ((\Phi_v)_1,f_v,1)}{L_v(1/2,\pi_v)}$$ where $f_{\phi,\overline{\phi}}=\otimes_v f_v$ and $f_v(g_v)=\langle \pi_v(g_v)\phi_v,\phi_v\rangle_v$. If this does not vanish, then $Z(\Phi,\phi,\overline{\phi},s)$ has a simple pole at $s=1/2$ by Theorem \[thm:global\], that is, it is not identically zero. It is sufficient to prove that $Z^\GJ((\Phi_v)_1,f_v,1)$ does not vanish for each $v\in S$ and a suitable $\sK_v$-spherical test function $\Phi_v$.

Let $v$ be a finite place in $S$, and denote by $\fO_v$ a maximal order of $D\otimes F_v$. Let $\Psi_{v,m}$ (resp. $\tilde\Psi_{v,m}$) denote the characteristic function of $\varpi_v^{m}\fO_v$ (resp. $\fo_v^\times I_2+\varpi_v^m\fO_v$) where $m\in\Z$. Let $\Phi_{1,v}(x_1,x_2):=\tilde\Psi_{v,m}(x_1)\, \Psi_{v,m}(x_2)$ and $\Phi_v(x):=\int_K \Phi_{1,v}(x\cdot\rho(1,1,k))\, \d k$. Then, we have $(\Phi_v)_1(x_1)=\tilde\Psi_{v,m}(x_1)$ if $m>0$. Therefore, choosing a large $m>0$, there exists a function $\phi_v\in\pi_v$ such that $Z^\GJ((\Phi_v)_1,f_v,1)\neq 0$.

In what follows, we suppose that $v$ is an infinite place of $F$. Consider the case $D\otimes F_v\cong M_2(F_v)$. Set $\Psi_{v,0}(x):=e^{-\pi \Tr(x{}^t\!x)}$ if $F_v\cong \R$, and $\Psi_{v,0}(x):=e^{-2\pi \Tr(x{}^t\!\overline{x})}$ if $F_v\cong \C$. Set $\Phi_v(x_1,x_2):=\Psi_{v,0}(x_1)\, \Psi_{v,0}(x_2)$, which is $\sK_v$-spherical. If $\pi_v$ is spherical, then it is obvious that $Z^\GJ((\Phi_v)_1,f_v,1)\neq 0$ for a spherical vector $\phi_v$ in $\pi_v$, see [@GH]. Hence, by the classification of the unitary representations, we have only to check the four cases: (I) $D\otimes F_v\cong M_2(\R)$ and $\pi_v$ is a discrete series, (II) $D\otimes F_v\cong M_2(\R)$ and $\pi_v$ is a non-spherical unitary principal series, (III) $D\otimes F_v$ is division, (IV) $F_v\cong \C$ and $\pi_v$ is a non-spherical principal series.

Case (I): As a matrix coefficient of $\pi_v$, we can take $\phi_v$ as $f_v(\left(\begin{smallmatrix}a&b \\ c&d\end{smallmatrix}\right))=(a+d+i(b-c))^{-2l}$ for some $l\in\Z_{\geq 1}$. Set $\tilde\Psi_v(x_1):=f_v(x_1)^{-1}\det(x_1)^2\, \Psi_{v,0}(x_1)$, $\Phi_{1,v}(x_1,x_2):=\tilde\Psi_v(x_1)\, \Psi_{v,0}(x_2)$ and $\Phi_v(x):=\int_{K_v} \Phi_{1,v}(x\cdot\rho(1,1,k))\, \d k$. Then, $(\Phi_v)_1=c\times \tilde\Psi_v$ holds for some constant $c>0$. Hence, we obtain $Z^\GJ((\Phi_v)_1,f_v,1)\neq 0$.

Case (II): In this case, there is a vector $\phi_v\in\pi_v$ such that $K_v= \O(2)$ acts on $\phi_v$ by $\det$. Hence, we can do the same argument as in (I) by setting $\tilde\Psi_v(x_1):=\det(x_1)\Psi_{v,0}(x_1)$.

Case (III): In this case, $F_v\cong \R$, $D\otimes \R\cong\mathbb{H}$ ($\mathbb{H}$ denotes the Hamiltonian quaternion) and $\mathbb{H}^\times \cong \R_{>0}\times \mathrm{SU}_2$. In addition, $f_v$ can be chosen as a homogeneous polynomial on $\mathrm{SU(2)}\subset \mathbb{H}/\R_{>0}$ by the Peter-Weyl theorem. Let $x\mapsto x^\iota$ denote the usual involution on $\mathbb{H}$, and $\Psi_{v,0}(x):=e^{-\pi\Tr(xx^\iota)}$. Set $\tilde\Psi_v(x_1):=\overline{f_v(x_1)}\, \det(x_1)\, \Psi_{v,0}(x_1)$, $\Phi_{1,v}(x_1,x_2):=\tilde\Psi_v(x_1)\, \Psi_{v,0}(x_2)$ and $\Phi_v(x):=\int_{K_v} \Phi_{1,v}(x\cdot\rho(1,1,k))\, \d k$. Then, we have $Z^\GJ((\Phi_v)_1,f_v,1)\neq 0$ by the same reason as in (I).

Case (IV): In this case, we consider the induced representation of $\chi\boxtimes \chi^{-1}$, where $\chi$ is a quasi-character on $\C^\times$. If we consider its compact picture as the representation space, then we can do the same argument as in (III).

Proof of Theorem \[thm:global\] {#sec:principleproof}
-------------------------------

For the case that $D$ is division and $\phi_1$, $\phi_2$ belongs to a one dimensional representation of $D_\A^\times$ with trivial central character, Taniguchi determined the principal part in [@Taniguchi1] by using the smooth Eisenstein series, but here we do not use it in order to make the argument simpler. We will prove Theorem \[thm:global\] by using the Poisson summation formula repeatedly and inserting additional terms. Our arguments are similar to some previous works, see, e.g., [@Shintani Proof of Lemma 4] and [@Kogiso], see also [@FHW § 5.2] for the cancellation, which they did not mention. To sum up, the technical problem we need to handle is to find cancellations of divergent terms and suitable additional terms.

For a moment, let us suppose $D= M_2(F)$ and $\phi_1$ and $\phi_2$ are constant functions in the space of the trivial representation. In this case, the principal part of $Z(\Phi,\phi_1,\phi_2,s)$ is not yet determined. We briefly explain the difficulty. There are two points where divergent terms appear in the calculation of the principal part using the Poisson summation formula. One is the point where the series obtained by changing the order of the integral and the sum over singular elements diverges, and the other one is the point where the fundamental domain of the stabilizer has infinite volume. In the case of the zeta functions with cusp forms, the second problem does not occur and we can determine the principal part by the method explained above. On the other hand, when we consider the zeta function for the trivial representation, the second problem occurs for the regular element corresponding to $E=F\oplus F$. As mentioned above, though this regular element is eliminated from the defining sum of the zeta functions, we should take them into account when using the Poisson summation formula. Thus we can not apply the method of this paper to the zeta functions of the trivial representation. The same problem also occurs for the space of binary quadratic forms. They are handled by using the normalization of the multivariable zeta functions [@Shintani], the Eisenstein series [@Yukiebook] and the truncation operators [@HW]. Therefore, also in our spaces, it is expected that we need some additional techniques to deal with this problem and the argument becomes complicated.

### Contributions of singular orbits

Let $H(\A)^1:=\{h\in H(\A) \mid |\omega(h)|=1  \}$, and then we have $H(\A)=\{ (I_2,I_2,t^{1/2}I_2) \mid t\in\R_{>0} \}\, H(\A)^1$. We will use the notation $x_j$ which means the elements given in Propositions \[prop:sor1\] and \[prop:sor2\]. By the Poisson summation formula on $V(F)$, if $\Re(s)$ is sufficiently large, then one obtains $$Z(\Phi,\phi_1,\phi_2,s)=Z_+(\Phi,\phi_1,\phi_2,s)+Z_+^\vee(\widehat\Phi,\phi_1,\phi_2,2-s)+I(\Phi,\phi_1,\phi_2,s),$$ where $$I(\Phi,\phi_1,\phi_2,s):=\int_0^1 t^{2s}  I^1(\Phi_t,\phi_1,\phi_2)  \frac{\d t}{t}, \qquad \Phi_t(x):=\Phi(t^{1/2}x),$$ $$\begin{gathered}
I^1(\Phi,\phi_1,\phi_2):=\\
\int_{H(F)\bsl H(\A)^1} \phi_1(g_1) \, \phi_2(g_2)  \left(  \sum_{x\in V(F)\setminus V^0(F)} \widehat{\Phi}(x\cdot \rho^\vee(h))-  \sum_{x\in V(F)\setminus V^0(F)} \Phi(x\cdot\rho(h))    \right) \, \d^1 h.\end{gathered}$$ For each subset $\mathcal{U}$ in $V(F)\setminus V^0(F)$, the integral $$\int_{H(F)\bsl H(\A)^1} \phi_1(g_1) \, \phi_2(g_2) \left(  \sum_{x\in \mathcal{U}} \widehat{\Phi}(x\cdot \rho^\vee(h))-  \sum_{x\in \mathcal{U}}  \Phi(x\cdot\rho(h))    \right) \, \d^1 h$$ is called the contribution of $\mathcal{U}$ to $I^1(\Phi,\phi_1,\phi_2)$. Notice that the contribution of $x_0=(0,0)$ to $I^1(\Phi,\phi_1,\phi_2)$ vanishes under the assumption of $\phi_1$ and $\phi_2$, because $$\int_{H(F)\bsl H(\A)^1} \left| \phi_1(h_1) \, \phi_2(h_2)    \Phi((0,0))    \right| \, \d^1 h ,\quad \int_{H(F)\bsl H(\A)^1} \left| \phi_1(h_1) \, \phi_2(h_2)    \widehat{\Phi}((0,0))    \right| \, \d^1 h$$ are convergent. Therefore, to prove Theorem \[thm:global\], it is sufficient to determine the contributions of $\sqcup_{j=1}^5 x_j\cdot H(\Q)$ to $I^1(\Phi,\phi_1,\phi_2)$.

By the proof of Theorem \[thm:global\], $$\int_0^\inf t^{2s} \left|  \int_{H(F)\bsl H(\A)^1} |\omega(h)|^{s}\, \phi_1(h_1)\, \phi_2(h_2)    \sum_{x\in V^0(F)} \Phi(t^{1/2}x\cdot \rho(h))    \, \d^1 h\right| \frac{\d t}{t}$$ converges for $\Re(s)>1$.

### The case that $D$ is division

Suppose that $D$ is division. Let $B$ denote the Borel subgroup consisting of upper triangular matrices in $\GL_2$. Set $$H_B\left( \begin{pmatrix}1&u \\ 0&1 \end{pmatrix} \begin{pmatrix}a&0 \\ 0&b \end{pmatrix}k\right):=\log|a/b|  \qquad  (u\in\A, \;\; a,b\in\A^\times , \;\; k\in K).$$

\[lem:absdiv\] The contribution of $x_2\cdot H(\Q)$ to $I^1(\Phi,\phi_1,\phi_2)$ equals $$c_F \{ -Z^\GJ((\Phi_K)_1,f_{\phi_1,\phi_2},1)+Z^\GJ((\widehat{\Phi}_K)_1,f_{\phi_1,\phi_2}^\vee,1)  \}.$$

Let $s\in\C$. First, we will prove that the integrals $$\begin{gathered}
\label{eq:20413e1}
\int_{H(F)\bsl H(\A)^1} \big| \phi_1(g_1) \, \phi_2(g_2) \big| \, \sum_{\gamma \in B(F)\bsl \GL_2(F)} \left|e^{-sH_B(\gamma g_3)} \right| \\
\left|   \sum_{w \in D^\times} \Phi((0,w)\cdot \rho((1,1,\gamma)h))  -  \int_{D_\A} \Phi((0,x)\cdot \rho((1,1,\gamma) h))\, \d x    \right| \, \d^1 h,\end{gathered}$$ $$\begin{gathered}
\label{eq:20413e2}
\int_{H(F)\bsl H(\A)^1}\big| \phi_1(g_1) \, \phi_2(g_2)  \big| \sum_{\gamma \in B(F)\bsl \GL_2(F)}   \left| e^{-sH_B(\gamma g_3)}  \right| \\
\left| \sum_{w \in D^\times} \widehat{\Phi}((w,0)\rho^\vee((1,1,\gamma)h))   -   \int_{D_\A} \widehat{\Phi}((x,0)\rho^\vee((1,1,\gamma)h))\, \d x    \right| \, \d^1 h\end{gathered}$$\[eq:413e1\] are convergent for $-1<\Re(s)<1$. Let $$|\Phi|_B(x):=\int_{B(F)\bsl B(\A)^1 K} |\Phi(  (0,x) \, \rho(1,1,b)) | \,  \d b ,$$ $$|\Phi|_B^{(2)}(x):=  \int_{B(F)\bsl B(\A)^1 K} \left|\int_{D_\A} \Phi((0,z)\, \rho(1,1,b))\, \psi_F(\Tr(xz)) \, \d z \right|  \, \d b .$$ Since $B(F)\bsl B(\A)^1K$ is compact, $|\Phi|_B$ and $|\Phi|_B^{(2)}(x)$ are bounded by certain Schwartz functions on $D_\A$. Using the Poisson summation formula one can show $$\begin{gathered}
\eqref{eq:20413e1}\ll \int_{G_2(F)\bsl G_2(\A)^1}\d^1 g \, \int_1^\inf\frac{\d t}{t} \,  \left| f_{\phi_1,\phi_2}(g) \right| \, t^{s+1} \,  \sum_{w \in D^\times} |\Phi|_B(w g t^{1/2})  \\
+ \int_{G_2(F)\bsl G_2(\A)^1}\d^1 g \, \int_1^\inf\frac{\d t}{t} \,  \left| f_{\phi_1,\phi_2}(g) \right| \, t^{s-1} \, |\Phi|_B^{(2)}(0) \\
+\int_{G_2(F)\bsl G_2(\A)^1}\d^1 g \, \int_0^1 \frac{\d t}{t} \,  \left| f_{\phi_1,\phi_2}(g) \right| \, t^{s-1}   \sum_{w \in D^\times}|\Phi|_B^{(2)}(w {}^t\!g^{-1} t^{-1/2}) \\
+ \int_{G_2(F)\bsl G_2(\A)^1}\d^1 g  \, \int_0^1 \frac{\d t}{t} \,  \left| f_{\phi_1,\phi_2}(g) \right| \, t^{s+1} \, |\Phi|_B(0)  .\end{gathered}$$ Hence, the convergence of is proved. The convergence of can be proved similarly. By using the convergence of and , the integral $$\begin{gathered}
\mathscr{Y}_1(s):=\int_{H(F)\bsl H(\A)^1}  \phi_1(g_1) \, \phi_2(g_2)  \, \sum_{\gamma \in B(F)\bsl \GL_2(F)} e^{-sH_B(\gamma g_3)} \\
\left\{  -\sum_{w \in D^\times} \Phi((0,w)\cdot \rho((1,1,\gamma)h))  + \sum_{w \in D^\times} \widehat{\Phi}((w,0)\rho^\vee((1,1,\gamma)h))   \right\} \, \d^1 h \end{gathered}$$ is convergent in the range $-1<\Re(s)<1$. Note that we have used the cancellation $$\int_{D_\A} \Phi((0,x)\rho(h))\, \d x- \int_{D_\A} \widehat{\Phi}((x,0)\rho^\vee(h))\, \d x =0    \qquad (h\in H(\A)^1).$$ Furthermore, for $-1<\Re(s)<1$ we get $\mathscr{Y}_1(s)=\mathscr{Y}_2(s)$ where $$\begin{gathered}
\mathscr{Y}_2(s):=c_F\int_0^\inf t^{s+1} \,  \int_{\mathrm{diag}(Z(\A)^1)G_1(F)\times G_2(F)\bsl G_1(\A)^1\times G_2(\A)^1}\phi_1(g_1)\phi_2(g_2) \\
\left\{  - \sum_{w\in D^\times} \Phi_K((0,t^{1/2}g_1^{-1}w g_2)) +   \sum_{w\in D^\times} \widehat\Phi_K((t^{1/2}g_2^{-1}w g_1,0)) \right\} \, \d g_1 \, \d g_2 \, \frac{\d t}{t}\end{gathered}$$ $(G_j(\A)^1:=\{g\in G_j(\A)\mid |\det(g)|=1\}$, $Z(\A)^1:=Z(\A)\cap G_j(\A)^1)$. Note that we can exchange $\int_{H(F)\bsl H(\A)^1}$ and $\sum_{\gamma \in B(F)\bsl \GL_2(F)}$ in that range. It is easy to prove that $\mathscr{Y}_2(s)$ is analytically continued to the whole $s$-plane by the Poisson summation formula. In addition, we obtain $$\mathscr{Y}_2(s)=c_F\{-Z^\GJ((\Phi_K)_1,f_{\phi_1,\phi_2},s+1)+Z^\GJ((\widehat{\Phi}_K)_1,f_{\phi_1,\phi_2}^\vee,s+1)\}$$ in the range $\Re(s)>1$ by the bijection $$\begin{gathered}
\{\mathrm{diag}(Z(\A)) G_1(F)\times G_2(F) \bsl G_1(\A)\times G_2(\A) \} \times D^\times \ni (g_1,g_2,w) \\
\mapsto (g_1,g_1^{-1}w g_2) \in (Z(\A)G_1(F)\bsl G_1(\A)) \times G_2(\A).\end{gathered}$$ Thus, the assertion is proved, because the contribution of $x_2\cdot H(\Q)$ to $I^1(\Phi,\phi_1,\phi_2)$ equals $\mathscr{Y}_1(0)=\mathscr{Y}_2(0)=c_F\{-Z^\GJ((\Phi_K)_1,f_{\phi_1,\phi_2},1)+Z^\GJ((\widehat{\Phi}_K)_1,f_{\phi_1,\phi_2}^\vee,1)\}$.

For the case that $D$ is division, we obtain Theorem \[thm:global\] by Lemma \[lem:absdiv\].

### The case $D=M_2(F)$

Assume that $D=M_2(F)$.

\[lem:conv0816\] If $\phi$ is a cuspidal automorphic form on $Z(\A) \GL_2(F)\bsl \GL_2(\A)$, then $$\int_{Z(\A)B(F)\bsl \GL_2(\A)} |\phi(g)| \, e^{(1+\epsilon)\, H_B(g)} \, \d g$$ is convergent for any $\epsilon>0$.

It is well-known that a cuspidal automorphic form $\phi$ is bounded on $\GL_2(\A)^1:=\{g\in\GL_2(\A)\mid |\det(g)|=1\}$ and rapidly decreasing on the Siegel set $\{ pak   \mid p\in \omega$, $a=\diag(e^t,e^{-t})$, $k\in K$, $t>T_0   \}$ for any $T_0\in\R$ and any compact set $\omega$ in $B(\A)$. The assertion follows from these facts.

\[lem:contx2\] The contribution of $x_1\cdot H(\Q)\sqcup x_2\cdot H(\Q)$ to $I^1(\Phi,\phi_1,\phi_2)$ is $$c_F \{ -Z^\GJ((\Phi_K)_1,f_{\phi_1,\phi_2},1)+Z^\GJ((\widehat{\Phi}_K)_1,f_{\phi_1,\phi_2}^\vee,1) \}.$$

One can apply the same argument as in the proof of Lemma \[lem:absdiv\]. Namely, one can prove that $$\begin{gathered}
\int_{H(F)\bsl H(\A)^1} \big| \phi_1(g_1) \, \phi_2(g_2) \big| \, \sum_{\gamma \in B(F)\bsl \GL_2(F)}  \left| e^{-sH_B(\gamma g_3)} \right| \\
\left|   \sum_{w \in M_2(F)\, \mathrm{rank}(w)>0} \Phi((0,w)\cdot \rho((1,1,\gamma)h))  -  \int_{M_2(\A)} \Phi((0,x)\cdot \rho((1,1,\gamma) h))\, \d x    \right| \, \d^1 h ,\end{gathered}$$ $$\begin{gathered}
\int_{H(F)\bsl H(\A)^1}\big| \phi_1(g_1) \, \phi_2(g_2)  \big| \sum_{\gamma \in B(F)\bsl \GL_2(F)}   \left| e^{-sH_B(\gamma g_3)}  \right| \\
\left| \sum_{w \in M_2(F),\mathrm{rank}(w)>0} \widehat{\Phi}((w,0)\cdot\rho^\vee((1,1,\gamma)h))   -   \int_{M_2(\A)} \widehat{\Phi}((x,0)\cdot \rho^\vee((1,1,\gamma)h))\, \d x    \right| \, \d^1 h\end{gathered}$$ are convergent for $-1<\Re(s)<1$ by the cuspidality of $\phi_j$. In the proof, we must divide the integration domain into $H_B(\gamma g_3)>0$ and $H_B(\gamma g_3)<0$ in order to use the Poisson summation formula, and care about the convergence of the sum over $\mathscr{A}_1:= \{w\in M_2(F) \mid \mathrm{rank}(w)=1\}$. We prove, as one example, the convergence of the sum over $\mathscr{A}_1$ of the integral over $H_B(\gamma g_3)<0$. As for the integration over $H_B(\gamma g_3)<0$, we do not apply the Poisson summation formula. In that case, the contribution of $\mathscr{A}_1$ to the first integral is bounded by $$\begin{gathered}
\int_{F^\times\bsl\A^1} \d^1 \alpha \, \int_1^\inf  \frac{\d t}{t} \,  \int_{Z(\A)B(F)\bsl \GL_2(\A)} \d g_1 \,  \int_{Z(\A)B(F)\bsl \GL_2(\A)} \d g_2 \,  \left| \phi_1(g_1) \, \phi_2(g_2) \right| \\
t^{1+\Re(s)} e^{-H_B(g_1)}e^{-H_B(g_2)}   \sum_{u\in F^\times}\left| \Phi((0,\begin{pmatrix} 0& \alpha t^{1/2} e^{-H_B(g_1)/2}e^{-H_B(g_2)/2} u \\ 0&0 \end{pmatrix})\cdot \rho(k_1,k_2,k_3)) \right| \end{gathered}$$ up to constant, where $g_j=b_j k_j$, $b_j\in B(F)\bsl B(\A)$, $k_j\in K$. Thus, from the condition $t>1$, we see that the above integral converges by Lemma \[lem:conv0816\] if we change the variable $t$ by $te^{H_B(g_1)}e^{H_B(g_2)}$ after perturbing the power of $t^{1+\Re(s)}$ by $t^{3+\Re(s)}$. We can prove the convergence of other terms by similar argument.

By the convergence of the above integrals, the contribution of $x_1\cdot H(\Q)\sqcup x_2\cdot H(\Q)$ to $I^1(\Phi,\phi_1,\phi_2)$ is equal to $\mathscr{Y}_3(0)$, where we set $$\begin{gathered}
\mathscr{Y}_3(s):=c_F\int_0^\inf t^{s+1} \,  \int_{\mathrm{diag}(Z(\A)^1)G_1(F)\times G_2(F)\bsl G_1(\A)^1\times G_2(\A)^1}\phi_1(g_1)\phi_2(g_2) \\
\left\{  - \sum_{w \in M_2(F),\mathrm{rank}(w)>0} \Phi_K((0,t^{1/2}g_1^{-1}w g_2)) +  \sum_{w \in M_2(F),\mathrm{rank}(w)>0} \widehat\Phi_K((t^{1/2}g_2^{-1}w g_1,0)) \right\} \, \d g_1 \, \d g_2 \, \frac{\d t}{t},\end{gathered}$$ and $\mathscr{Y}_3(s)$ is convergent in the range $-1<\Re(s)<1$. Furthermore, we can prove that $\mathscr{Y}_3(s)$ is analytically continued to the whole $s$-plane by using the Poisson summation formula. Therefore, by the same argument as in the proof of Lemma \[lem:absdiv\] we find that the contribution of $x_2\cdot H(F)$ is $c_F \{ -Z^\GJ((\Phi_K)_1,f_{\phi_1,\phi_2},1)+Z^\GJ((\widehat{\Phi}_K)_1,f_{\phi_1,\phi_2}^\vee,1) \}$. In addition, the contribution of $x_1\cdot H(\Q)$ vanishes by the cuspidality of $\phi_j$ and Lemma \[lem:conv0816\], because the first term for $\mathrm{rank}(w)=1$ is transformed into $$- 2c_F \prod_{j=1,2}\int_{Z(\A)B(F)\bsl G_j(\A)} \phi_j (g_j)e^{sH_B(g_j)} \d g_j \, \int_{F^\times\bsl \A^\times} |a|^{2+2s} \sum_{x\in F^\times}\Phi_K(0,\begin{pmatrix} 0& ax \\ 0&0 \end{pmatrix}) \, \d^\times a=0$$ if $s>1$, and the second one is computed similarly.

\[lem:contx34\] The contribution of $x_3\cdot H(\Q)\sqcup x_4\cdot H(\Q) $ to $I^1(\Phi,\phi_1,\phi_2)$ is zero.

For $x=\left( \left(\begin{smallmatrix}x_{11}&x_{12} \\ x_{21}&x_{22} \end{smallmatrix}\right),\left(\begin{smallmatrix}y_{11}&y_{12} \\ y_{21}&y_{22} \end{smallmatrix}\right) \right)\in V(F)$, we set $$\sF_3(x):=\left(\begin{pmatrix} x_{11}&x_{12} \\ y_{11}&y_{12} \end{pmatrix},\begin{pmatrix}x_{21}&x_{22} \\ y_{21}&y_{22} \end{pmatrix}\right) ,\quad 
\sF_4(x):=\left(\begin{pmatrix} x_{11}&y_{11} \\ x_{21}&y_{21} \end{pmatrix},\begin{pmatrix}x_{12}&y_{12} \\ x_{22}&y_{22} \end{pmatrix} \right) .$$ Then, we have $\sF_j(x_2\cdot H(F))=x_j\cdot H(F)$ $(j=3,4)$. If $(*)$ is one of the following functions $$\Phi(\sF_3(w,0)\cdot \rho((\gamma,1,1)h))   , \quad   \Phi(\sF_4(0,w)\cdot \rho((1,\gamma,1)h)) ,$$ $$\widehat\Phi( \sF_3(w,0)\cdot \rho^\vee((1,\gamma,1)h)) , \quad \widehat\Phi( \sF_4(0,w)\cdot \rho^\vee((\gamma,1,1)h))  ,$$ then one can prove that $$\int_{H(F)\bsl H(\A)^1} \big| \phi_1(g_1) \, \phi_2(g_2) \big| \, \sum_{\gamma \in B(F)\bsl \GL_2(F)} \left|  \sum_{w \in D^\times} (*)  \right| \, \d^1 h$$ is convergent by dividing the integration domain into $H_B(\gamma g_j)>0$ and $H_B(\gamma g_j)<0$, and using the fact that $\phi_j(g_j) \, e^{r H_B(g_j)}$ is bounded on $\{ g_j\in Z(\A)B(F)\bsl \GL_2(\A)\mid H_B(g_j)>0 \}$ for any $r>0$. Hence, the contribution of $x_3\cdot H(\Q)\sqcup x_4\cdot H(\Q) $ vanishes by the cuspidality of $\phi_1$ and $\phi_2$.

\[lem:contx5\] The contribution of $x_5\cdot H(\Q)$ to $I^1(\Phi,\phi_1,\phi_2)$ is zero.

Set $$\mathscr{A}_2:=\{  (u_1E_{12},u_2E_{11}+u_3E_{22}+cE_{12}) \mid u_1,u_2,u_3\in F^\times, \;\; c\in F \},$$ $$\mathscr{A}_3:=\{  (u_2E_{11}+u_3E_{22}+cE_{12},u_1E_{12}) \mid u_1,u_2,u_3\in F^\times, \;\; c\in F \},$$ The contribution of $x_5\cdot H(\Q)$ equals $\mathscr{Y}_4(0)+\mathscr{Y}_5(0)$, where $$\mathscr{Y}_4(s):=-\int_{H(F)\bsl H(\A)^1}  \phi_1(g_1) \, \phi_2(g_2)  \sum_{\gamma\in (B(F)\bsl \GL_2(F))^3} e^{-s\sum_{j=1}^3H_B(\gamma_j g_j)} \sum_{w\in\mathscr{A}_2} \Phi(w\cdot \rho(\gamma h))  \, \d^1 h ,$$ $$\mathscr{Y}_5(s):=\int_{H(F)\bsl H(\A)^1}  \phi_1(g_1) \, \phi_2(g_2)  \sum_{\gamma\in (B(F)\bsl \GL_2(F))^3} e^{-s\sum_{j=1}^3H_B(\gamma_j g_j)} \sum_{w\in\mathscr{A}_3} \widehat{\Phi}(w\cdot\rho^\vee(\gamma h))   \, \d^1 h$$ where $\gamma=(\gamma_1,\gamma_2,\gamma_3)$. Let $N$ denote the unipotent radical of $B$, and let $\bK$ denote the standard maximal compact subgroup in $H(\A)$, that is, $\bK=K\times K\times K$. Set $B(\A)^1:=\{ n\diag(a_1,a_2)\in B(\A) \mid n\in N(\A)$, $a_1,a_2\in\A^1\}$. The function $\mathscr{Y}_4(s)$ is transformed into $$\begin{gathered}
\int_{Z(\A)^1 B(F)\bsl B(\A)^1 }\d b_1 \int_{Z(\A)^1 B(F)\bsl B(\A)^1 }\d b_2 \, \int_\bK \d k  \, \int_0^\inf \frac{\d t_1}{t_1} \, \int_0^\inf \frac{\d t_2}{t_2} \, \int_0^\inf \frac{\d t_3}{t_3} \\
\int_\A \d c    \, \int_{F^\times\bsl \A^1} \d^1 a_1  \, \int_{F^\times\bsl \A^1} \d^1 a_2    \,  (t_1t_2t_3)^{-1-2s} \, \phi_1(b_1\mathfrak{t}_1k_1) \, \phi_2(b_2\mathfrak{t}_2k_2)\, \\
\sum_{u_1,u_2,u_3\in F^\times} \Phi(( t_1^{-1} t_2^{-1}t_3 u_1 E_{12},  t_1^{-1}t_2t_3^{-1}u_2E_{11} +  t_1t_2^{-1}t_3^{-1}u_3E_{22}+cE_{12}) \cdot \rho((\mathfrak{t}_1^{-1} b_1\mathfrak{t}_1,\mathfrak{t}_2^{-1} b_2\mathfrak{t}_2,\mathfrak{a}_3)k) \end{gathered}$$ up to constant, where $k=(k_1,k_2,k_3)\in \bK$, $\mathfrak{t}_1=\diag(t_1^{1/2},t_1^{-1/2})$, $\mathfrak{t}_2=\diag(t_2^{1/2},t_2^{-1/2})$, $\mathfrak{a}_3=\diag(a_1,a_2)$. Hence, one can prove that $\mathscr{Y}_4(s)$ is absolutely convergent for any $s\in\C$ and entire on $\C$ by using Lemma \[lem:conv0816\] and dividing the integration domain into $t_1t_2t_3<1$ and $t_1t_2t_3>1$. Since $\mathscr{Y}_5(s)$ also satisfies a similar transformation, it follows that $\mathscr{Y}_5(s)$ is absolutely convergent for any $s\in\C$ and entire on $\C$. Furthermore, for $\Re(s)>0$, $ \mathscr{Y}_4(s)$ is equal to $$\begin{gathered}
\int_{ \A} \d c  \,  \int_{\A^\times}\d^\times a_1  \,  \int_{\A^\times}\d^\times a_2  \,  \int_{\A^\times}\d^\times a_3 \, \int_{N(F)\bsl N(\A)}\d n_1 \, \int_{N(F)\bsl N(\A)}\d n_2 \, \int_\bK \d k \\ 
|a_1a_2a_3|^{2s+1} \phi_1(n_1 \diag(a_1^{-1},1)k_1) \, \phi_2(n_2\diag(a_1^{-1}a_3^{-1}a_2,1)k_2) \, \Phi((a_1E_{12} ,a_2E_{11}+a_3E_{22}+cE_{12} ) \cdot \rho(k))  \end{gathered}$$ up to constant, where $k=(k_1,k_2,k_3)\in\bK$, and $\mathscr{Y}_5(s)$ also satisfies a similar equality. Hence, $\mathscr{Y}_4(s)$ and $\mathscr{Y}_5(s)$ are identically zero.

By Lemmas \[lem:contx2\], \[lem:contx34\], and \[lem:contx5\], we obtain Theorem \[thm:global\] for the case $D=M_2(F)$.

Essential factorization {#sec:fac}
-----------------------

We apply Saito’s formula [@Saito1; @Saito2] to $Z(\Phi,\phi_1,\phi_2,s)$. Notice that the Hasse principle holds for $H$, since it is necessary for his formula. First of all, we fix several measures along the lines of Saito [@Saito2 p.13–14].

Let $\fo$ denote the integer ring of $F$, and $\fO$ a maximal order in $D$. An integral structure on $V$ is given by $\fO$. We choose an integral basis $x_1,\dots,x_8$ on $V$, and then we have a gauge form $$\d X:= \d x_1\, \d x_2 \, \cdots \d x_8.$$ A Haar measure $\d X_v$ on $V(F_v)$ is obtained from $\d X_v$ and the Haar measure on $F_v$ given in § \[sec:notation\]. We may suppose that $\det$ on $D$ is defined over $\fo$ without loss of generality, namely, $P$ is in $\fo[V]$. Choose a gauge form $\omega$ on $H$ over $F$. Then, the Tamagawa measure $\d h$ on $H(\A)$ is given by $$\d h:= c_F^{-1}\, |\Delta_F|^{-5}\prod_{v\in\Sigma} c_v \, \omega_v$$ where $\omega_v$ is the measure on $G(F_v)$ associated with $\omega$, and $$c_v:=\begin{cases} 1 & \text{if $v|\inf$}, \\ (1-q_v^{-1})^{-1}  &  \text{if $v<\inf$}. \end{cases}$$ For each $E \in X(D)$, we have a $F$-rational regular point $x(\delta)=x(\delta_E)\in V(F)$ as in Proposition \[prop:orbits\], and set $$H_{x(\delta)}:=Z_\rho\bsl G_{x(\delta)} \quad \text{and} \quad  H_{x(\delta)}^0:=Z_\rho\bsl G_{x(\delta)}^0 .$$ A mapping $\mu_{x(\delta)}$ is defined by $$\mu_{x(\delta)}:  \, Y_{x(\delta)}:=H_{x(\delta)}^0 \bsl H\ni H_{x(\delta)}^0 g \mapsto H_{x(\delta)} g   \in H_{x(\delta)} \bsl H \cong V^0.$$ Set $\d Y=\mu_{x(\delta)}^* \d X$, and we have $$\omega_\delta=(\mu_{x(\delta)}^*P)(Y)^{-2} \d Y$$ is a $H$-invariant gauge on $Y_{x(\delta)}$. Let $\xi=\omega/\lambda_{x(\delta)}^*\omega_\delta$ denote the gauge form on $H_{x(\delta)}^0$ determined by $\omega$ and $\lambda_{x(\delta)}^*\omega_\delta$, where $\lambda_{x(\delta)} :H\to Y$, $\lambda_{x(\delta)}(g)=H_{x(\delta)}^0g$. Let $\eta=\otimes_v \eta_v$ denote the Hecke character on $F^\times\R_{>0}\bsl \A^\times$ corresponding to $E$, $L(s,\eta)$ denote the Hecke $L$-function, and we set $\mathfrak{d}_v:=  (1-\eta_v(\varpi_v) q_v^{-1})^{-2}$ if $v<\inf$ and $\eta_v$ is unramified, and $\mathfrak{d}_v:= 1$ otherwise. Then, we obtain the Tamagawa measure $$L(1,\eta)^{-2}|\Delta_F|^{-1}\prod_{v\in\Sigma} \mathfrak{d}_v \, \xi_v$$ on $H_{x(\delta)}^0(\A)$. Note that $L(s, \eta)$ has a simple pole at $s=1$ when $E=F\oplus F$. By abuse of notation, we let $$L(1,\eta):=c_F \, (=\mathrm{Res}_{s=1}L(s,\eta))$$ in this case. We take the measure $\d y$ on $Y_{x(\delta)}(\A)$ as $$\d y:=c_F^{-1}\, L(1,\eta)^2\, |\Delta_F|^{-4}\prod_{v\in\Sigma} c_v \mathfrak{d}_v^{-1} \omega_{\delta,v} .$$

Suppose that $\phi_1$ and $\phi_2$ are in $\pi$. We obviously find $$\label{eq:20200105e1}
Z(\Phi,\phi_1,\phi_2,s)=\sum_{E \in X(D)}Z_E(\Phi,\phi_1,\phi_2,s),$$ where $$Z_E(\Phi,\phi_1,\phi_2,s):=\int_{H(F)\bsl H(\A)} |\omega(h)|^s \phi_1(g_1)\, \phi_2(g_2)\sum_{x\in x(\delta_E)\cdot H(F)} \Phi(x\cdot \rho(h))\, \d h .$$ Set $$\phi_j^h(x):=\phi_j(xh) \qquad (h,x\in H_j(\A), \quad j=1,2).$$

\[lem:Saito\] $$Z_E(\Phi,\phi_1,\phi_2,s)=\frac{1}{2} \int_{Y_{x(\delta)}(\A)} \cP_E(\phi_1^{g_1})\, \cP_E(\phi_2^{g_2})   \,  |P(\mu_{x(\delta)}(y))|^s \Phi(\mu_{x(\delta)}(y)) \, \d y$$ where $y=H_{x(\delta)}^0 h=H_{x(\delta)}^0(g_1,g_2,g_3)\in H_{x(\delta)}^0\bsl H(\A)=Y_{x(\delta)}(\A)$.

See [@Saito1 Proof of Theorem 2.1]. We write some datum concerned with his formula: $|H_{x(\delta)}^0\bsl H_{x(\delta)}(F)|=2$, $|A(H_{x(\delta)}^0)|=1$ if $\delta\in (F^\times)^2$, $|A(H_{x(\delta)}^0)|=4$ if $\delta\not\in (F^\times)^2$, and $\mathrm{Ker}(\iota^0_{x(d),A})$ is trivial. See [@Saito1 p.598] and [@HW Section 4.2] for the computation.

From now, we put $$\phi_1=\phi \quad  \text{and} \quad \phi_2=\overline{\phi},$$ and suppose that $\pi$ is not one dimensional. This assumption is necessary only for Theorems \[thm:Wald\] in the following argument. We are always supposing that $\pi$ is cuspidal. Let $\pi=\otimes_v\pi_v$ and $\phi=\otimes_v\phi_v$, and set $$\phi_v^{g_v}:=\pi_v(g_v)\phi_v \quad (g_v \in D_v^\times) .$$ Assume that $\Phi=\otimes_v \Phi_v$, and for each place $v$. Let $X(D_v)$ denote the set of quadratic étale algebras over $F_v$ embedded in $D_v$. For each $E_v\in X(D_v)$, we set $$V_{E_v}(F_v):=\{ x\in V(F_v) \mid P(x)\in d_{E_v} (F_v^\times)^2  \}=x(\delta_v)\cdot \rho(G(F_v)),$$ where we have chosen elements $\delta_{E_v}\in E_v$ and $d_{E_v}=\delta_{E_v}^2\in F_v^\times$ as in § \[sec:orbits\]. For each place $v\in\Sigma$, a local zeta function $Z_{E_v}(\phi_v,\Phi_v,s)$ is defined by $$\label{eq:local422}
Z_{E_v}(\phi_v,\Phi_v,s):=  \frac{2\, c_v\, L(1,\pi_v,\mathrm{Ad}) }{L(1,\eta_v) \, \zeta_{F_v}(2)\, L(1/2,\pi_{E,v}) } \int_{V_{E_v}(F_v)}  \alpha_{E_v}(\phi_v^{g_{1,v}},\phi_v^{g_{2,v}})   \,   |P(X_v)|^{s-2} \, \Phi_v(X_v) \, \d X_v$$ where $X_v=x(\delta)\cdot \rho(g_{1,v},g_{2,v},g_{3,v})$. By Theorems \[thm:Wald\] and Lemma \[lem:Saito\], we have the Euler product $$\label{eq:20200105e2}
Z_E(\Phi,\phi,\overline{\phi},s)= \frac{L(1,\eta) \, \zeta_F(2) \, L(1/2,\pi_E)}{8 \, |\Delta_F|^4  \, c_F \, L(1,\pi,\mathrm{Ad})} \prod_{v\in\Sigma} Z_{E_v}(\phi_v,\Phi_v,s).$$

We note on the fact $$|Y_{x(\delta)}(F_v)/H(F_v)|=1$$ by [@Saito1 Proposition (2)], $V_{E}(F_v)=x(\delta)\cdot H(F_v)$ and $|H_{x(\delta)}/H^0_{x(\delta)}(F_v)|=2$. In addition, for the element $(\gamma,\gamma,\diag(1,-1))\in G_{x(\delta)}(F)\setminus G_{x(\delta)}^0(F)$ given in Proposition \[prop:orbits\], we have $$\alpha_{E_v}(\phi_v^{\gamma g_{1,v}},\phi_v^{\gamma g_{2,v}})= \alpha_{E_v}(\phi_v^{ g_{1,v}},\phi_v^{ g_{2,v}}),$$ since $\gamma^{-1} h_v\gamma$ is the conjugate of $h_v$ for any $h_v\in E_v^\times \subset D_v^\times$. It follows from these facts that the factor $\alpha_{E_v}(\phi_v^{g_{1,v}},\phi_v^{g_{2,v}})$ is uniquely determined by the point $x(\delta)\cdot \rho(g_{1,v},g_{2,v},g_{3,v})$ on $V(F_v,d)$.

Explicit formula {#sec:exp}
----------------

Let $\pi=\otimes_{v\in\Sigma}\pi_v$ be an automorphic representation of $D_\A^\times$ with trivial central character, and choose an automorphic form $\phi=\otimes \phi_v\in\pi$. Suppose that $\pi$ is cuspidal if $D=M_2$, and $\pi$ is not one dimensional if $D$ is division. Choose a finite set $S$ of places of $F$, and set $F_S:=\prod_{v\in S}F_v$. Assume that $S$ contains all infinite places of $F$, any finite place $v$ such that $\pi_v$ is ramified, and all finite places dividing $2$. We also suppose that $\phi_v$ is spherical and $D_v$ is split for any $v\notin S$. Put $\lambda_v:=q_v^{1/2} (\alpha_v+\alpha_v^{-1})$ where $\alpha_v$ denotes the Satake parameter of $\pi_v$. Set $$\mathcal{D}_E^S(\pi,s):=\prod_{v\notin S} (1+\mathcal{R}_{E_v}(\pi_v,s) \, q_v^{-2s+1})$$ where $$\label{eq:REv}
\mathcal{R}_{E_v}(\pi_v,s):= \begin{cases} 1+q_v^{-1}+q_v^{-2s} -2\eta_v(\varpi_v)\, q_v^{-1}\lambda_v & \text{if $\eta_v$ is unramified}, \\ 1& \text{if $\eta_v$ is ramified}. \end{cases}$$ Set $\Phi:=\otimes_v \Phi_v$. Assume that $\Phi_v$ denotes the characteristic function of $V(\fo_v)$ for each $v\notin S$. Let $D_S:=\prod_{v\in S}D_v$, and $X(D_S):=\prod_{v\in S}X(D_v)$. Choose an element $\cE_S=\prod_{v\in S}\cE_v$ in $X(D_S)$. We set $$X(D,\pi,\cE_S):=\{   E\in X(D) \mid  E_v=\cE_v \;\; (\forall v\in S) , \;\; \cP_E\neq 0 \}.$$ Let $N(\mathfrak{f}_{\eta_v})$ denote the norm of the conductor of $\eta_v$, and we set $N(\mathfrak{f}_\eta^S):=\prod_{v\notin S}N(\mathfrak{f}_{\eta_v})$. Now, define a Dirichlet series $\xi(s,D,\phi,\cE_S)$ as $$\begin{aligned}
\xi(s,D,\phi,\cE_S):=& \frac{ \zeta_F(2) }{8 \, |\Delta_F|^4  \, c_F \, L(1,\pi,\mathrm{Ad})}  \frac{\zeta^S_F(2s-1) \, L^S(2s-1,\pi,\mathrm{Ad}) }{\zeta^S_F(2)^3} \\
& \sum_{E\in X(D,\pi,\cE_S)} \frac{\alpha_E^\#(\phi,S)\, L(1,\eta) \, L(1/2,\pi_E) \, \mathcal{D}_E^S(\pi,s)}{N(\mathfrak{f}_\eta^S)^{s-1}},\end{aligned}$$ where $\alpha_E^\#(\phi,S):= \prod_{v\not\in S} \alpha_{E_v}^\#(\phi_v,\phi_v)$, and $\zeta_F^S(s)$ and $L^S(s,\pi,\mathrm{Ad})$ are defined by the Euler products outside $S$ for $\zeta_F(s)$ and $L(s,\pi,\mathrm{Ad})$ respectively. The following is an explicit formula of the zeta function $Z(\Phi,\phi,\overline{\phi},s)$.

\[thm:20200105t1\] Suppose that $\pi$ is cuspidal if $D=M_2(F)$, and $\pi$ is not one dimensional if $D$ is division. Let $\phi\in\pi$ and $\phi\neq 0$. For sufficiently large $\Re(s)>0$, $\xi(s,D,\phi,\cE_S)$ is absolutely convergent, and we have $$Z(\Phi,\phi,\overline{\phi},s)= \sum_{\cE_S \in X(D_S)}   \prod_{v\in S} Z_{\cE_v}(\phi_v,\Phi_v,s)  \times  \xi(s,D,\phi,\cE_S).$$ If the support of $\otimes_{v\in S} \Phi_v$ is contained in $\prod_{v\in S}V_{\cE_v}(F_v)$, then we have for sufficiently large $\Re(s)>0$ $$Z(\Phi,\phi,\overline{\phi},s)=   \prod_{v\in S} Z_{\cE_v}(\phi_v,\Phi_v,s)  \times   \xi(s,D,\phi,\cE_S)$$

The absolute convergence is ensured by Lemma \[lem:abs\]. The equalities follow from , and Theorem \[thm:localzeta\].

\[lem:202001251\] Choose a place $v\in\Sigma$ and an element $\cE_v\in X(D_v)$. Let $\phi_v\in\pi_v$, $\phi_v\neq 0$, and $\alpha_{\cE_v}(\; ,\; )\neq 0$. Then, there exists a test function $\Phi_v\in \cS(V(F_v))$ such that the support of $\Phi_v$ is contained in $V_{\cE_v}(F_v)$, and $Z_{\cE_v}(\phi_v,\Phi_v,s)$ is entire on $\C$ and $Z_{\cE_v}(\phi_v,\Phi_v,1)\neq 0$.

We may suppose $\alpha_{\cE_v}(\phi_v,\phi_v)\neq 0$ without loss of generality, because $\alpha_{\cE_v}(\phi_v^{g_1},\phi_v^{g_2})\neq 0$ holds for some $(g_1,g_2,1)\in G(F_v)$. Recall the topological isomorphism $G_{x(\delta_v)}(F_v) \bsl G(F_v)\cong V_{\cE_v}(F_v)$, where $\cE_v=F_v(\delta_v)$. Hence, if $v<\inf$ and we choose a sufficiently small neighborhood $x(\delta_v)+\varpi_v^MV(\fo_v)$ at $x(\delta_v)$, whose characteristic function is denoted by $\Phi_v$, then $Z_{\cE_v}(\phi_v,\Phi_v,s)$ is entire and $Z_{\cE_v}(\phi_v,\Phi_v,1)\neq 0$. When $v|\inf$, we can get the same assertion for a suitable test function.

\[prop:merozeta\] $\xi(s,D,\phi,\cE_S)$ is meromorphically continued to the whole $s$-plane.

This follows from Theorems \[thm:global\] and \[thm:20200105t1\] and Lemma \[lem:202001251\].

Non-vanishing theorems
----------------------

Let $\pi=\otimes_v \pi_v$ be an automoprhic representation of $D_\A^\times$ with trivial central character. When $D$ is division, we suppose that $\pi$ is not $1$-dimensional. When $D=M(2,F)$, we suppose that $\pi$ is cuspidal. From the proof of the next theorem, we see that the existence of the poles of the zeta functions implies nonvanishing of periods.

\[thm:nonvanishing423\] If $L(1/2,\pi)\neq 0$, then there exists an element $\cE_S\in X(D_S)$ such that $X(D,\pi,\cE_S)$ is not empty.

This follows from Proposition \[prop:202001241\] and Lemma \[lem:Saito\].

\[thm:infinite\] If $X(D,\pi,\cE_S)$ is not empty, then $X(D,\pi,\cE_S)$ is an infinite set.

Let us consider the right hand side $(*)$ of the formula of Theorem \[thm:20200105t1\]. Assume that $X(D,\phi,\cE_S)$ is finite. Choose an element $\delta'\in X(D,\phi,\cE_S)$. Then, it is obviously possible to choose a finite set $S'\supset S$ so that $X(D,\phi,\cE_{S'})$ consists of the single element $E'$ such that $\cE_{S'}=(E'_v)_{v\in S'}$. By Theorem \[thm:20200105t1\] and Lemma \[lem:202001251\] for $\cE_{S'}$, we can choose $\Phi=\otimes_{v\in\Sigma} \Phi_v$ so that we have $$(*)=c'\times  \prod_{v\in S'} Z_{E_v'}(\phi_v,\Phi_v,s)  \times  \frac{\zeta^{S'}_F(2s-1) \, L^{S'}(2s-1,\pi,\mathrm{Ad}) \, \mathcal{D}_{E'}^{S'}(\pi,s)}{N(\mathfrak{f}_{\eta'}^{S'})^{s-1}}$$ for some constant $c'\neq 0$, $Z_{E_v'}(\phi_v,\Phi_v,s)$ is entire and $Z_{E_v'}(\phi_v,\Phi_v,1)\neq 0$ $(\forall v\in S')$. Hence, it follows from Proposition \[prop:merozeta\] that $\mathcal{D}_{E'}^{S'}(\pi,s)$ is meromorphically continued to $\C$. In addition, $\mathcal{D}_{E'}^{S'}(\pi,s,)$ is absolutely convergent for $\Re(s)>1$ and we have $\Re(\mathcal{D}_{E'}^{S'}(\pi,s))$ tends to $+\inf$ $(s\to 1+0)$, because we may suppose that $S'$ contains all small finite places $v$ without loss of generality and we find $\mathcal{R}_v(\pi_v,s,\delta)=1+O(q_v^{-1/2})$ $(q_v\to\inf)$ uniformly for $s>1-\epsilon$ by a bound of $\lambda_v$, see [@Kim] and [@BB]. Thus, $\mathcal{D}_{E'}^{S'}(\pi,s)$ has a pole at $s=1$, that is, $(*)$ has a pole at $s=1$. This is a contradiction for the holomorphy of $Z(\Phi,\phi,\overline{\phi},s)$ at $s=1$, see Theorem \[thm:global\].

Functional equations and an expected refinement {#sec:20200424}
-----------------------------------------------

What is required for extending our main theorem to cover Theorem \[thm:FH\] is a close study on the local functional equations at “bad” places. They are established in the research of Wen-Wei Li which is currently in progress, see [@Li4]. Here, we give an overview of the local functional equations and how we will apply them to the study of nonvanishing of periods.

Take a cuspidal automorphic representation $\pi=\otimes_v \pi_v$ of $D_\A^\times$ with trivial central character of dimension greater than $1$ and an automorphic form $\phi=\otimes_v \phi_v\in\pi$. Define the local zeta function $Z_{\cE_v}^\vee(\phi_v,\Phi_v,s)$ for the contragredient representation $\rho^\vee$ of $\rho$ in the same manner for $Z_{\cE_v}(\phi_v,\Phi_v,s)$ (see ). Li’s local functional equation [@Li4 Theorem 5.6] states that there exist meromorphic functions $\gamma_{\cE_v,\cR_v}(s,\phi_v)$ on $\C$ such that the local functional equation $$\label{eq:glfunct}
Z_{\cE_v}^\vee(\phi_v,\widehat{\Phi_v},2-s)=\sum_{\cR_v\in X(D_v)} \gamma_{\cE_v,\cR_v}(s,\phi_v)\, Z_{\cR_v}( \phi_v,\Phi_v,s)$$ holds for any Schwartz function $\Phi_v$ on $V(F_v)$, where $\widehat{\Phi_v}$ denotes its Fourier transform. Notice that, in order to derive from [@Li4 Theorem 5.6], we need $\dim \mathrm{Hom}_{\cE_v^\times}(\pi_v,\trep)\leq 1$. Combining and and the global functional equation $Z(\Phi,\phi,\overline{\phi},s)=Z^\vee(\widehat\Phi,\phi,\overline{\phi},2-s)$ (Corollary \[cor:funct\]), we get $$\label{eq:xifunct}
\xi(s,D,\phi,\cE_S)  =\sum_{\cR_S\in X(D_S)} \prod_{v\in S}\gamma_{\cR_v,\cE_v}(s,\phi_v) \times   \xi(2-s,D,\overline{\phi},\cR_S)   .$$ The explicit computation of the gamma factors $\gamma_{\cR_v,\cE_v}(s,\phi_v)$ is carried out in [@Sato2] and [@Sato3] for some representations of real groups. In the following proposition, we will show a nonvanishing result at the real place by using the functional equation and an analytic property of $\gamma_{\cR_v,\cE_v}(s,\phi_v)$, which follows from [@Sato3 Theorem 2].

Suppose $F=\Q$ and $D_\inf=M_2(\R)$. Let $\pi=\otimes_v \pi_v$ be a cuspidal automorphic representation of $D_\A^\times$ with trivial central character such that $\pi_\inf$ is a spherical principal series. Assume $L(1/2, \pi)\neq 0$. Then, $X(D,\pi,\R\oplus\R)$ (resp. $X(D,\pi,\C)$) is infinite, that is, there are infinitely many real (resp. imaginary) quadratic fields $E\in X(D)$ such that $\pi$ is $E^\times$-distinguished.

By Theorem \[thm:infinite\] it is sufficient to prove that $X(D,\pi,\R\oplus \R)$ and $X(D,\pi,\C)$ are not empty. By $L(1/2, \pi)\neq 0$ and Theorem \[thm:nonvanishing423\], $X(D,\pi,\R\oplus \R)$ or $X(D,\pi,\C)$ is not empty. Hence, we will duduce a contradiction under the assumption that only one of $X(D,\pi,\R\oplus \R)$ and $X(D,\pi,\C)$ is empty. We discuss only the case that $X(D,\pi,\R\oplus \R)\neq\emptyset$ and $X(D,\pi,\C)=\emptyset$, since the proof for the other case is completely the same.

Assume that $X(D,\pi,\R\oplus \R)\neq\emptyset$ and $X(D,\pi,\C)=\emptyset$. Set $S_0:=S\setminus \{\inf\}$. By the assumption and Proposition \[prop:202001241\], there exists an element $\cR_{S_0}\in X(D_{S_0})$ such that $\xi(s,D,\phi,(\R\oplus\R,\cR_{S_0}))$ is not identically zero. Besides, by $X(D,\pi,\C)=\emptyset$, we have $\xi(s,D,\phi,(\C,\cR_{S_0}))\equiv 0$. Let $\phi_\inf$ be a spherical vector of $\pi_\inf$ and put $$\Gamma(s,\phi_\inf):=\begin{pmatrix} \gamma_{\R\oplus\R,\R\oplus\R}(s,\phi_v) & \gamma_{\C,\R\oplus\R}(s,\phi_v)\\ \gamma_{\R\oplus\R,\C}(s,\phi_v) & \gamma_{\C,\C}(s,\phi_v)\end{pmatrix}.$$ Then, it follows from the functional equation that $$\label{eq:funct0423}
\begin{pmatrix} \xi(s,D,\phi,(\R\oplus\R,\cR_{S_0})) \\ \xi(s,D,\phi,(\C,\cR_{S_0})) \end{pmatrix}= \Gamma(s,\phi_\inf)  \begin{pmatrix} \Xi_{\R\oplus \R}(\phi,s) \\ \Xi_{\C}(\phi,s) \end{pmatrix},$$ where, for $\mathcal{Q}=\R\oplus \R$ or $\C$, we set $$\Xi_{\mathcal{Q}}(\phi,s):=\sum_{\cE_S\in X(D_S), \; \cE_\inf=\mathcal{Q}}\prod_{v\in S_0}\gamma_{\cE_v,\cR_v}(s,\phi_v) \times\xi(2-s,D,\overline{\phi},\cE_S).$$ It follows from [@Sato3 Theorem 2] that all entries of $\Gamma(s,\phi_\inf)$ are not identically zero. Hence, we get $\Xi_{\R\oplus\R}(\phi,s)\not\equiv 0$ or $\Xi_{\C}(\phi,s)\not\equiv 0$ by $\xi(s,D,\phi,(\R\oplus\R,\cR_{S_0}))\not\equiv 0$ and . Furthermore, by $\xi(s,D,\phi,(\R\oplus\R,\cR_{S_0}))\equiv 0$ and we find that $\Xi_{\R\oplus\R}(\phi,s)$ and $\Xi_{\C}(\phi,s)$ are not identically zero. This implies that $X(D,\pi,\C)$ is not empty, and so this is a contradiction.

From this, we see that the gamma factors know something about nonvanishing of periods at “bad places”. Thus, in order to extend our results to cover Theorem \[thm:FH\], we should clarify the analytic properties of the gamma factors $\gamma_{\cE_v,\cR_v}(s,\phi_v)$ in .

One dimensional representations
-------------------------------

The proof of Theorem \[thm:infinite\] tells us that we may be able to show nonvanishing of infinitely many periods from the explicit formula of the zeta functions with infinite dimensional automorphic representations. We will also mention the zeta functions with 1-dimensional representations because they can be applied to density theorems and trace formulas according to the previous works, see, e.g., [@Taniguchi2], [@HW]. Let $\pi$ be a one dimensional unitary representation of $H_j(F)\bsl H_j(\A)$ $(j=1,2)$. Define a zeta function $Z(\Phi,s,\pi)$ by $$Z(\Phi,s,\pi):=\int_{H(F)\bsl H(\A)}  |\omega(h)|^s \, \pi(g_1^{-1}g_2) \sum_{x\in V^0(F), \, P(x)\not\in (F^\times)^2} \Phi(x\cdot \rho(h))\, \d h .$$ When $D$ is division, $Z(\Phi,s,\pi)$ agrees with $Z(\Phi,\phi_1,\phi_2,s)$ up to constant. The following explicit formulas and were essentially proved by Saito, see [@Saito0 Theorem 3.4 (2) $n=4$, $r=2$] and [@Saito1].

### Trivial representation

We set $$\xi(s,\cE_S):=  \frac{\zeta_F^S(2s-2) \, \zeta_F^S(2s-1)^2 \, \zeta_F^S(2s) }{2 \, |\Delta_F|^4  \, c_F\, \zeta^S_F(2)^3}  \sum_{E\in X(D,\trep,\cE_S), \, E\neq F\oplus F} \frac{ L(1,\eta)^2 \, \mathcal{D}_E^S(\trep,s)}{N(\mathfrak{f}_\eta^S)^{s-1}},$$ and it follows that $$\label{eq:onedimtri}
Z(\Phi,s,\trep)=   \sum_{\cE_S\in X(D_S)}   \prod_{v\in S} Z_{\cE_v}(\Phi_v,s)  \times  \xi(s,\cE_S)$$ where $Z_{\cE_v}(\Phi_v,s):=\int_{V_{\cE_v}(F_v)}|P(x)|^{s-2}\Phi_v(x) \, \d x$. Hence, $Z(\Phi,s,\trep)$ is expressed as an infinite sum of Euler products, since $X(D,\trep,\cE_S)$ is an infinite set.

### Non-trivial representation

Consider the case $\pi=\eta\circ\det$ for some non-trivial quadratic character $\eta=\otimes_v \eta_v$ on $F^\times\bsl \A^\times$. We treat only this special case for simplicity, though one should consider Hecke characters on narrow ideal classes in general. There exists a unique quadratic extension $E$ over $F$ such that $E\neq F\oplus F$ and $\eta$ corresponds to $E$ via the class field theory. It follows from the strong multiplicity one theorem that the set $X(D,\pi,\cE_S)$ consists of only $E$ if $E_v=\cE_v$ $(\forall v\in S)$, and it is the empty set otherwise. Hence, we have $$\label{eq:onedim}
Z(\Phi,s,\eta\circ\det)=   \prod_{v\in S} Z_{E_v}(\Phi_v,s,\eta_v)  \times  \frac{L(1,\eta)^2}{ 2\, |\Delta_F|^4  \, c_F  \zeta^S_F(2)^3 \, N(\mathfrak{f}_\eta^S)^{s-1}} \times \zeta_F^S(2s-1)\, \zeta_F^S(2s-2) ,$$ where $Z_{E_v}(\Phi_v,s,\eta_v):=Z_{\cE_v}(\Phi_v,s):=\int_{V_{\cE_v}(F_v)}|P(x)|^{s-2}\Phi_v(x) \, \eta_v(\det(h_1^{-1}h_2)) \, \d x$ $(x=x(\delta_v)\cdot h)$, since $\eta_v$ is unramified for any $v\not\in S$ and $\mathcal{D}_E^S(\pi,s)=\zeta^S_F(2s-1)^{-1}\zeta^S_F(2s)^{-1}$.

Local zeta functions with Waldspurger’s model over $p$-adic fields {#sec:localzeta424}
==================================================================

Notation {#sec:notation2}
--------

Let $F$ denote a $p$-adic field, $\fo$ the integer ring of $F$, $\varpi$ a prime element, and $q:=\# (\fo/\varpi\fo)$. We set $\F_q:=\fo/\varpi\fo$ which is a finite field of order $q$. Throughout this section, we suppose that $2$ does not divide $q$, and $D$ is split over $F$. Hence, we use the notations given in § \[sec:space\] by setting $D=M_2$. We also use the notations and normalizations given in § \[sec:notation\] by taking a finite place $v$, and we omit $v$ for simplification. For example, we have $|\varpi|=q$, $\int_\fo \d x=1$, and so on.

Let $E$ be a étale quadratic algebra over $F$. In the same manner as in § \[sec:orbits\], we choose elements $\delta=\delta_E\in E$ such that $E=F(\delta)$ and $\Tr(\delta)=0$. Put $d=d_E:=\delta^2$, and we can suppose $$d_E\in\fo^\times\sqcup\pi\fo^\times$$ without loss of generality. Write $\eta=\eta_E$ for the quadratic character on $F^\times$ corresponding to $E$, that is, $\eta(x)=(x,d)$, where $(\; , \; )$ means the Hilbert symbol on $F^\times\times F^\times$. We write $\ff_\eta$ for the conductor of $\eta$, and put $N(\ff_\eta):=\#(\fo/\ff)$. Set $L(s,\eta):=(1-\eta(\varpi)q^{-s})^{-1}$ if $\eta$ is unramified, and $L(s,\eta):=1$ otherwise. We write $\trep$ for the trivial representation on $F^\times$.

Let $W:=\{x\in M_2 \mid x={}^t\!x \}$, and $\left(\begin{smallmatrix} x_1& x_{12}/2 \\ x_{12}/2 & x_2 \end{smallmatrix}\right)\in W(F)$ is denoted by $(x_1,x_{12},x_2)$ below. Define equivariant maps $\cF_j:V(F)\to W(F)$ $(j=1,2)$ as, for $x=\left( \left( \begin{smallmatrix}x_{11}&x_{12} \\ x_{21}&x_{22} \end{smallmatrix}\right),\left(\begin{smallmatrix}y_{11}&y_{12} \\ y_{21}&y_{22} \end{smallmatrix}\right) \right)\in V(F)$, $$\cF_1(x):=\left(\det\begin{pmatrix}x_{11}&x_{12} \\ y_{11}&y_{12} \end{pmatrix},  \det\begin{pmatrix}x_{11}&x_{12} \\ y_{21}&y_{22} \end{pmatrix}+\det\begin{pmatrix}x_{21}&x_{22} \\ y_{11}&y_{12} \end{pmatrix}    , \det\begin{pmatrix}x_{21}&x_{22} \\ y_{21}&y_{22} \end{pmatrix} \right),$$ $$\cF_2(x):=\left(\det\begin{pmatrix} x_{11}&y_{11} \\ x_{21}&y_{21} \end{pmatrix},  \det\begin{pmatrix}x_{11}&y_{12} \\ x_{21}&y_{22} \end{pmatrix}+\det\begin{pmatrix}x_{12}&y_{11} \\ x_{22}&y_{12} \end{pmatrix}    , \det\begin{pmatrix}x_{12}&y_{12} \\ x_{22}&y_{22} \end{pmatrix}\right) .$$ For $g=(g_1,g_2,g_3)\in G$ and $x\in V$, we have $$\cF_1(x\, \rho(g))=\det(g_2) \, \det(g_3) \, g_1^{-1}\cF_1(x)\, {}^t\!g_1^{-1}, \quad \cF_2(x\, \rho(g))=\det(g_1)^{-1} \, \det(g_3) \, {}^t\! g_2\cF_2(x)\, g_2.$$ Notice that $P(x)=-4\det(\cF_j(x))$ holds for $j=1$, $2$, and $\cF_1$ is essentially same as $P_0$ given in § \[sec:space\]. It is well-known that these maps play an important role in Bhargava’s theory.

Let $\pi$ be an irreducible unitary unramified representation of $\GL_2(F)$ with trivial central character. This can be written uniquely as a quotient of a principal series representation $\chi\times\chi^{-1}$ with an unramified character $\chi$ of $F^\times$． The local $L$-factors of $\pi$ we need in this work is as follows:

-   $L(s, \pi)=(1-\chi(\varpi)q^{-s})^{-1}(1-\chi^{-1}(\varpi)q^{-s})^{-1}$: the standard $L$-factor

-   $L(s, \pi, \operatorname{Ad})=(1-q^{-s})^{-1}(1-\chi^2(\varpi)q^{-s})^{-1}(1-\chi^{-2}(\varpi)q^{-s})^{-1}$: the adjoint $L$-factor.

Throughout this section, we fix the representation $\pi$ as above.

Waldspurger’s model
-------------------

We fix $\GL_2(F)$-invariant Hermitian pairing $\l\,,\r$ on $\pi$ and define the $E^\times\times E^\times$-invriant bilinear form $\al_E$ on $\pi\boxtimes\bar{\pi}$ by $$\al_E(\phi_1, \phi_2)=\int_{F^\times\bs E^\times}\l\pi(t)\phi_1, \phi_2\r\,d^\times t.$$ Though this integral is over a non-compact region when $E^\times$ is a split torus, it converges absolutely. Let $\phi$ be the $K$-spherical vector of $\pi$ of norm 1. Assume that $\al_E(\phi, \phi)$ is not zero． For nonnegative integers $l_1$ and $l_2$, we set $$\be_E(l_1):=\al_{\pi,\delta}(\pi(\diag(\varpi^{-l_1}, 1))\phi,\phi)\,  \al_E(\phi, \phi)^{-1},$$ and $$\be_E(l_1, l_2):=\al_E(\pi(\diag(\varpi^{-l_1}, 1))\phi, \pi(\diag(1, \varpi^{l_2}))\phi) \,  \al_E(\phi, \phi)^{-1}.$$ Note that $$\beta_E(0,0)=1, \quad  \beta_E(l,0)=\beta_E(l), \quad \beta_E(l_1,l_2)=\overline{\beta_E(l_2,l_1)}.$$ Set $\alpha:=\chi(\varpi)\in\C^\times$ for $\pi=\chi\boxtimes\chi^{-1}$. Since $\pi$ is unitary, one has $$q^{-1/2}\leq |\alpha|\leq q^{1/2}, \quad \lambda:=q^{1/2}(\alpha+\alpha^{-1})\in\R,$$ see [@Tadic]. Using the notation $\lambda$, the $L$-factors are described as $$L(s,\pi)=(1-q^{-1/2}\lambda q^{-s}+q^{-2s})^{-1},\quad L(s,\pi,\mathrm{Ad})=\{1-(q^{-1}\lambda^2-1) q^{-s}+(q^{-1}\lambda^2-1)q^{-2s}-q^{-3s} \}^{-1}.$$ By [@KP Lemmas 3.2, 3.3], one has the relation $$\label{eq:re1}
q\beta_E(l+2,m) - \lambda \cdot \beta_E(l+1,m) + \beta_E(l,m) = 0$$ for $l,m\in\Z_{\geq0}$. In addition $$\label{eq:re3}
\beta_E(1)=\frac{\lambda-2}{q-1} \;\; \text{if $d_E\in(\fo^\times)^2$}, \quad \frac{\lambda}{q+1}\;\; \text{if $d_E\in\fo^\times\setminus(\fo^\times)^2$}, \quad \frac{\lambda-1}{q}\;\; \text{if $d_E\in\varpi\fo^\times$.}$$

Let $l_1$ and $ l_2$ be nonnegative integers. We have $$\label{eq:re5}
\be_E(l_1, l_2)=\be_E(l_1) \, \be_E(l_2).$$

For $x$ in $\GL_2(F)$, two linear forms $$\varphi\mapsto\al_E(\varphi, \phi), \hspace{10pt} \varphi\mapsto\al_E(\varphi, \pi(x)\phi)$$ on $\pi$ are both $H$-invariant. From the multiplicity one theorem proven in [@G97], there exists a constant $c_\phi(x)$ such that $$\label{eq0}
\al_E(\varphi, \pi(x)\phi)=c_\phi(x)\al_E(\varphi, \phi)$$ for any $\varphi\in\pi$. We compute $c_\phi(\diag(1, \varpi^{l_2}))$.

Substituting $x=\diag(1, \varpi^{l_2})$ and $\varphi=\phi$ in (\[eq0\]), we see that $$c_\phi(1, \diag(\varpi^{l_2}))=\ol{\beta_E(l_2)}.$$ Here, we used the fact that $\al_E(\phi, \phi)$ is a positive real number and the obvious equation $$\ol{\al_E(\varphi, \pi(x)\phi)}=\al_E(\pi(x)\phi, \varphi).$$

From , we know that $\be_\pi(l_2)$ is a real number. Thus $c_\phi(\diag(1, \varpi^{l_2}))=\be_\pi(l_2)$. Again substituting $\varphi=\pi(\diag(\varpi^{-l_1}, 1))\phi$ and $x=\diag(1, \varpi^{l_2})$ in (\[eq0\]), we obtain $$\be_E(l_1, l_2)=\be_E(l_2)\al_E(\pi(\diag(\varpi^{-l_1}, 1))\phi, \phi)\, \,  \al_E(\phi, \phi)^{-1}=\be_E(l_1) \, \be_E(l_2).$$

Some formulas for $\beta_E(l_1,l_2)$
------------------------------------

We will give some formulas for $\beta_E(l_1,l_2)$ by using the techniques of [@BFF] and [@KP]. They will be used for the proof of Theorem \[thm:localzeta\]. Throughout this subsection, $x$, $y$, $z$ are variables, and suppose $j\in \Z_{\geq 0}$. Set $$A_j(x):=\sum_{k=0}^\inf \beta_E(j+k) \, x^k.$$ From we have $$q A_{j+2}(x) - \lambda A_{j+1}(x) + A_j(x) =0.$$ Further, by $x A_{j+1}(x) = A_j(x) - \beta_E(j)$, we find $$q\left( A_j(x)-\beta_E(j)-\beta_E(j+1)x \right)- x\lambda \left( A_j(x)-\beta_E(j)\right)+ x^2 A_j(x) = 0.$$ Hence, $$\label{eq:re6}
A_j(x)=\frac{ \beta_E(j)+\beta_E(j+1)x-q^{-1}\lambda\cdot \beta_E(j)x  }{1-\lambda q^{-1}x+q^{-1}x^2}=\frac{ \beta_E(j)-\beta_E(j-1)\, q^{-1}x  }{1-\lambda q^{-1}x+q^{-1}x^2},$$ cf. [@KP Proposition 3.4]. We set $$B_j(x):= \sum_{l=0}^\inf \beta_E(l+j)^2 \, x^l , \quad 
C_j(x):= \sum_{l=0}^\inf \beta_E(l+j+1) \, \beta_E(l+j) \, x^l .$$

\[lem:relation\] $$C_j(x)=\frac{q^{-1}\lambda}{1+q^{-1}x}B_j(x) + \frac{ \beta_E(j) \, \beta_E(j+1)-q^{-1}\lambda \cdot \beta_E(j)^2 }{1+q^{-1}x}.$$

By one gets $$q C_{j+1}(x) - \lambda B_{j+1}(x) + C_j(x)=0 .$$ By $xB_{j+1}(x)=B_j(x)-\beta_E(j)^2$ and $xC_{j+1}(x)=C_j(x)-\beta_E(j+1)\, \beta_E(j)$ one finds $$(C_j(x)-\beta_E(j+1)\, \beta_E(j))-q^{-1}\lambda(B_j(x)-\beta_E(j)^2)+q^{-1}xC_j(x)=0.$$

\[prop:410x\] We have $$\begin{gathered}
B_j(x)=D(x) \times \\
\Big[\beta_E(j)^2 + x\{ \beta_E(j+1)^2-q^{-1}(q^{-1}\lambda^2-1)\, \beta_E(j)^2\} + q^{-1}x^2\{ \beta_E(j+1)-q^{-1}\lambda\cdot \beta_E(j)\}^2\Big],\end{gathered}$$ where $$D(x):=\{1-(q^{-1}\lambda^2-1)\, q^{-1}x +(q^{-1}\lambda^2-1)\, q^{-2}x^2 -q^{-3}x^3 \}^{-1} .$$

By one gets $$q^2B_{j+2}(x)-\lambda^2B_{j+1}(x)+2\lambda C_j(x)-B_j(x)=0.$$ By Lemma \[lem:relation\] and $xB_{j+1}(x)=B_j(x)-\beta_E(j)^2$, we have $$\begin{gathered}
(B_j(x)-\beta_E(j)^2-x\beta_E(j+1)^2)-q^{-2}\lambda^2x(B_j(x)-\beta_E(j)^2)-q^{-2}x^2B_j(x) \\
+2q^{-2}x^2\lambda \left\{ \frac{q^{-1}\lambda}{1+q^{-1}x}B_j(x) + \frac{ \beta_E(j) \, \beta_E(j+1)-q^{-1}\lambda \cdot \beta_E(j)^2 }{1+q^{-1}x}  \right\}=0.\end{gathered}$$ Hence, $$\begin{gathered}
(1-(q^{-1}\lambda^2-1)\, q^{-1}x +(q^{-1}\lambda^2-1)\, q^{-2}x^2 -q^{-3}x^3)B_j(x)  \\
= (\beta_E(j)^2+x\beta_E(j+1)^2   -q^{-2}\lambda^2x\beta_E(j)^2)(1+q^{-1}x)- 2q^{-2}x^2\lambda(\beta_E(j) \, \beta_E(j+1)-q^{-1}\lambda \cdot \beta_E(j)^2)  .\end{gathered}$$

Set $$U_j(x,y):=\sum_{l_1=0}^\inf \sum_{l_2=j}^\inf \beta_E(l_1+l_2+1) \, \beta_E(l_2) \, x^{l_1} y^{l_2} .$$

\[prop:412x\] $$U_j(x,y) = y^j \times \frac{ (q^{-1}\lambda-q^{-1}x- q^{-2}xy)B_j(y) + \beta_E(j)\beta_E(j+1)-q^{-1}\lambda \cdot \beta_E(j)^2}{(1-\lambda q^{-1}x+q^{-1}x^2)(1+q^{-1}y)}.$$

It follows from that $$U_j(x,y) =y^j \sum_{l=0}^\inf A_{l+j+1}(x)\, \beta_E(l+j)\, y^l =\frac{y^j}{1-\lambda q^{-1}x+q^{-1}x^2} \left( C_j(y)-q^{-1}x \, B_j(y)    \right).$$ Hence, the assertion is proved by Lemma \[lem:relation\].

\[lem:20191230\] $$\sum_{k=0}^\inf \sum_{u=0}^\inf  x^k y^u \beta_E(k+u+1)^2 =D(y) \times \{ B_0(x) \, f_1(x,y)+f_2(x,y) \},$$ $$\begin{aligned}
f_1(x,y)&=  x^{-2}y+x^{-1}+q^{-1}x^{-1}y-q^{-2}\lambda^2 x^{-1}y+q^{-3}y^2  ,\\
f_2(x,y)&=  -x^{-2}y-x^{-1}y\, \beta_E(1)^2 -x^{-1}-q^{-1}x^{-1}y+q^{-2}\lambda^2 x^{-1}y  .\end{aligned}$$

$$\begin{aligned}
& \sum_{k=0}^\inf \sum_{u=0}^\inf  x^k y^u \beta_E(k+u+1)^2 = \sum_{k=0}^\inf x^k B_{k+1}(y) \\
&=D(y)\sum_{k=0}^\inf x^k \{ \beta_E(k+1)^2   + y(\beta_E(k+2)^2-(q^{-2}\lambda^2 -q^{-1})\beta_E(k+1)^2)+q^{-3}y^2\beta_E(k)^2  \}\\
&= D(y)\, \{  yB_2(x)+ (1+q^{-1}y-q^{-2}\lambda^2)B_1(x)+q^{-3}y^2B_0(x)  \}.\end{aligned}$$

Hence, the assertion follows from $xB_1(x)=B_0(x)-1$ and $x^2B_2(x)=B_0(x)-1-x\beta_E(1)^2$.

\[lem:0409l\] $$\begin{gathered}
(1+q^{-1}z)\, (1-\lambda q^{-1}y+q^{-1}y^2) \sum_{k=1}^\inf \sum_{l=0}^\inf \sum_{u=0}^\inf x^k y^l z^u \beta_E(k+u,k+l+u+1) \\
=(q^{-1}\lambda-q^{-1}y-q^{-2}yz) x \sum_{k=0}^\inf x^k B_{k+1}(z) - q^{-1}x\, C_0(x).\end{gathered}$$

This equality can be proved by and Lemma \[lem:relation\].

Main result
-----------

We will consider the special case for the local zeta function given in § \[sec:fac\]. Set $$\cK:=\GL_2(\fo)\times \GL_2(\fo)\times \GL_2(\fo), \qquad V_E(\fo):= \{x\in V(\fo) \mid P(x)\in d_E(F^\times)^2\},$$ $$g(l_1,l_2,m_1,m_2,c):=\left( \begin{pmatrix} \varpi^{-l_1} & 0 \\ 0& 1\end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0& \varpi^{l_2}\end{pmatrix}, \begin{pmatrix} \varpi^{m_1} & c \\ 0& \varpi^{m_2}\end{pmatrix} \right) \in  G(F),$$ $$S_E(l_1,l_2,m_1,m_2,c):=x(\delta_E)\cdot \rho\left( g(l_1,l_2,m_1,m_2,c)\cK\right).$$ Our purpose is to explicitly calculate the local zeta function $$Z_E(\pi,s):=\int_{V_E(\fo)} \beta_E(l_1,l_2) \, |P(x)|^{s-2} \, \d x \qquad (s\in\C ),$$ where $(l_1,l_2)$ is determined by the condition that $x$ belongs to $S(E,l_1,l_2,m_1,m_2,c)$ for some $m_1$, $m_2\in\Z$ and some $c\in F$, see .

\[thm:localzeta\] Let $\pi$ be any irreducible unitary spherical representation of $\PGL_2(F)$. Then, we obtain $$Z_E(\pi,s)=\frac{1}{2} \frac{L(1,\eta)^2 \,L(2s-1,\trep)\,L(2s-1,\pi,\mathrm{Ad}) }{L(1,\trep) \, L(2,\trep)^3 \, N(\ff_\delta)^{s-1}} \times \left(1+\mathcal{R}_E(\pi,s) \, q^{-2s+1}\right),$$ where $\mathcal{R}_E(\pi,s)$ was defined in , that is, $\mathcal{R}_E(\pi,s) :=1+q^{-1}+q^{-2s} -2 \eta(\varpi) \, q^{-1} \lambda$ if $d_E \in \fo^\times$, and $\mathcal{R}_E(\pi,s) := 1$ if $d_E \in \varpi \fo^\times$.

Decompositions {#sec:binaryquad}
--------------

Let $X(D)$ denote the set of all étale quadratic algebras over $F$. This notation $X(D)$ is the same as $X(D_v)$ as in § \[sec:fac\] if we put $D=M_2$. Set $Q(x):=-4\det(x)$ $(x\in W(F))$ and $W(\fo):=\left\{(x_1, x_{12}x_2) \mid x_1,x_{12},x_2\in\fo  \right\}$. For each $E\in X(D)$, we set $$W_E(F):=\{   x\in W(F) \mid Q(x)\in d_E(F^\times)^2   \} , \quad W_E(\fo):=W(\fo)\cap W_E(F).$$ Let $\qG:=\GL_1(F)\times \GL_2(F)$, and a rational representation $\rho_W$ of $\qG$ on $W$ is defined by $$x \, \rho_W(a,h)=\frac{a}{\det(h)} \, {}^t\! h x h ,\qquad (a,h)\in \qG, \quad x\in W(F).$$ It is obvious that $\mathrm{Ker}\rho_W=\{ (1,a I_2)\in \qG \mid a\in F^\times \}\cong F^\times$. An orbit decomposition of $W^0(F):=\{x\in W(F) \mid Q(x)\neq 0\}$ is given by $$W^0(F)= \bigsqcup_{E\in X(D)}W_E(F),\quad W(F)= y(\delta_E) \, \rho_W(\qG), \qquad  y(\delta_E):=\diag( 1,-d_E)\in W(\fo) .$$ The stabilizer $\qG_E$ of $y(\delta)$ in $\qG$ and its connected component $\qG_E^0$ at $1$ are given by $$\qG_E^0=\{(1,g)\in \qG \mid g\in \mathrm{GO}_{2,E}^0(F)\}, \quad 
 \qG_E=\qG_E^0 \sqcup \diag( 1,-1) \qG_E^0 .$$ Let $\qK=\GL_1(\fo)\times\GL_2(\fo)$, and set $S_E(l_1,l_2):=\varpi^{l_1} \, \diag( 1 ,-d_E \varpi^{2l_2})\, \rho_W(\qK)$. Then $$W_E(F)= \bigsqcup_{l_1\in\Z} \bigsqcup_{l_2\in\Z_{\geq 0}} S_E(l_1,l_2), \quad W_E(\fo)= \bigsqcup_{l_1\in\Z_{\geq 0}} \bigsqcup_{l_2\in\Z_{\geq 0}} S_E(l_1,l_2).$$ From this we have $$\label{eq:906e1}
\qG_E^0\bsl \qG/\qK=\qG_E\bsl \qG/\qK=\bigsqcup_{m_1\in\Z} \bigsqcup_{m_2\in\Z_{\geq 0}} \qG_E^0 (\varpi^{m_1}, \diag(1,\varpi^{m_2})) \qK.$$ In addition, $$\label{eq:sym2}
\operatorname{vol}( S_E(l_1,l_2)) = \frac{1}{2} \, \frac{1}{L(1,\trep)\, L(2,\trep)} \begin{cases} q^{-3l_1-2l_2} & \text{if $d_E\in \fo^\times$ and $l_2>0$}, \\ q^{-3l_1-2l_2-1} & \text{if $d_E\in \varpi \fo^\times$},  \\ q^{-3l_1}L(1,\eta) & \text{if $d_E\in \fo^\times$ and $l_2=0$}, \end{cases}$$ under the normalization $\operatorname{vol}(W(\fo))=1$. See [@IS Lemma 3.2] or [@Saito0 Proposition 2.8] for the proof.

By we have $$G_{x(\delta)}^0(F)\bsl G(F)/\cK=\bigsqcup_{l_1,l_2\in \Z_{\geq 0}}\bigsqcup_{m_1,m_2\in \Z , \; c\in F/\varpi^{m_1}\fo}   G_{x(\delta)}^0(F)\, g(l_1,l_2,m_1,m_2,c) \cK  .$$ Further, putting $l_3:=\min(l_1,l_2)$, we get $$\label{eq:decoV}
V_E(\fo)=\bigsqcup_{l_1,l_2\in\Z_{\geq 0}} \bigsqcup_{m_1\geq -l_3}\bigsqcup_{m_2\in\Z_{\geq 0} } \bigsqcup_{c\in \varpi^{-l_3}\fo/\varpi^{m_1}\fo}  S_E(l_1,l_2,m_1,m_2,c).$$ The parameter $(l_1,l_2)$ in the definition of $Z_E(\pi,s)$ is determined by the condition that $x$ belongs to $\bigcup_{m_1,m_2\in\Z, \; c\in F/\varpi^{m_1}\fo} S_E(l_1,l_2,m_1,m_2,c) $.

\[lem:param\] For each $x\in V(\fo,E)$ and the parameter $(l_1,l_2)$, we have the relation $$|d_E| \, q^{-2l_j} \times \left(\max \cF_j(x)\right)^2  =|P(x)|   \qquad (j=1, \, 2),$$ where $\max \cF_j(x)$ means the maximum among $\{ |y_1|, \;\; |y_{12}| , \;\; |y_2| \}$ if $\cF_j(x)=(y_1,y_{12},y_2)$.

This follows from $$\cF_1(x(\delta)\cdot\rho(g(l_1,l_2,m_1,m_2,c)))=(\delta\varpi^{2l_1},0,-1) \times \varpi^{l_2+m_1+m_2},$$ $$\cF_2(x(\delta)\cdot\rho(g(l_1,l_2,m_1,m_2,c)))=(1,0,-\delta\varpi^{2l_2}) \times \varpi^{l_1+m_1+m_2}.$$

Proof of Theorem \[thm:localzeta\]
----------------------------------

Set $\sG:=G(\F_q)$. By Propositions \[prop:sor1\] and \[prop:orbits\], we have a list of $\mathscr{G}$-orbits in $V(\F_q)$. We use the same representative elements $x_j$ $(0\leq j\leq 5)$ given in Proposition \[prop:sor1\]. By a direct calculation, one obains the following lemma.

\[lem:card\] A list of cardinalities of orbits is as follows: $$\# \sG = \sG_{x_0}=q^{12}(1-q^{-1})^3(1-q^{-2})^3, \quad \#(x_0 \cdot \rho(\sG))=1,$$ $$\# \mathscr{G}_{x_1}=q^8 (1-q^{-1})^5, \quad \#(x_1 \cdot \rho(\mathscr{G}))= q^4(1+q^{-1})^2 (1-q^{-2}),$$ $$\# \mathscr{G}_{x_j}=q^7 (1-q^{-1})^3(1-q^{-2}), \quad \#(x_j \cdot \rho(\mathscr{G}))= q^5 (1-q^{-2})^2 \quad(j=2,3,4),$$ $$\# \mathscr{G}_{x_5}=q^5 (1-q^{-1})^3, \quad \#(x_5 \cdot \rho(\mathscr{G}))= q^7(1-q^{-2})^3,$$ $$\# \mathscr{G}_{x_6}=2q^4 (1-q^{-1})^4, \quad \#(x_6 \cdot \rho(\mathscr{G}))=\frac{1}{2} q^8(1+q^{-1})(1-q^{-2})^2,$$ $$\# \mathscr{G}_{x_7}=2q^4 (1-q^{-2})^2, \quad \#(x_7 \cdot \rho(\mathscr{G}))=\frac{1}{2} q^8(1-q^{-1})^3(1-q^{-2}),$$ where we have chosen regular elements $x_6$ and $x_7$ in $V(\F_q)$ such that $P(x_6)\in (\F_q^\times)^2$ and $P(x_7)\in \F_q^\times\setminus (\F_q^\times)^2$.

For each $x_j$ $(0\leq j\leq 7)$, we set $$Z_{E,j}(\pi,s):=\int_{(x_j\rho(G(\fo/\varpi\fo)) + \varpi V(\fo))\cap V_E(\fo)} \beta_E(l_1,l_2) \, |P(x)|^{s-2} \, \d x.$$ For any $x\in V_E(\fo)$ and any $m\in\Z$, $\varpi^m x$ has the same parameter $(l_1,l_2)$ as that of $x$ by Lemma \[lem:param\]. Hence, we obtain $$\label{eq:1} 
(1-q^{-4s})Z_E(\pi,s)=\int_{V_E(\fo)\setminus \varpi V_E(\fo)}\beta_E(l_1,l_2) \, |P(x)|^{s-2} \, \d x =  \sum_{j=1}^7 Z_{E,j}(\pi,s).$$ Therefore, the explicit calculation for $Z_E(\pi,s)$ is reduced to those of $Z_{E,j}(\pi,s)$. This is a standard method by Igusa for computations of local zeta functions, see [@Igusa].

### Contribution of $x_2\cdot \rho(\sG)$

Any element in $x_2\cdot \rho(\sG)+\varpi V(\fo)$ can be reduced to the form $$X_2:= \left( \begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}, \begin{pmatrix} x&z \\ y&-x \end{pmatrix}\right) \quad (x,y,z\in\varpi\fo)$$ by the action of $\sG$, and then the integral $Z_{E,2}(\pi,s)$ is $$\#(x_2 \cdot \rho(\sG)) \times q^{-5} \times \int_{(\varpi\fo)^{\oplus 3},\, P(X_2)\in d_E(F^\times)^2}  \beta_E(l_1,l_2) \, |P(X_2)|^{s-2} \d x \, \d y \, \d z .$$ In addition, we have $\cF_1(X_2)= (z,-2x,-y)$ and $\cF_2(X_2)=  (y,-2x,-z)$. Hence, by , Lemma \[lem:card\] and the fact that $\rho(g,g,1)$ acts on $Y=\left(\begin{smallmatrix} -y&x \\ x&z \end{smallmatrix}\right)$ as $Y\cdot\rho(g,g,1)=\det(g)^{-1}{}^t\!gYg$, one finds that $Z_{E,2}(\pi,s)$ equals $$\frac{1}{2} \frac{L(2s-1,\trep)}{L(1,\trep) \, L(2,\trep)^3} \, q^{-2s+1} \, \left( L(1,\eta)+ \sum_{l=1}^\inf \beta_E(l,l) \, q^{(-2s+2)l}  \right) \qquad \text{if $d_E\in\fo^\times$},$$ $$\frac{1}{2} \frac{L(2s-1,\trep)}{L(1,\trep) \, L(2,\trep)^3}\, q^{-3s+2}  \sum_{l=0}^\inf \beta_E(l,l) \, q^{(-2s+2)l} \qquad \text{if $d_E\in\varpi\fo^\times$}.$$

### Contribution of $x_3\cdot\rho(\sG)$

Any element in $x_3\cdot \rho(\sG)+\varpi V(\fo)$ is reduced to the form $$X_3:=\left( \begin{pmatrix} 1&0 \\ x&z \end{pmatrix}, \begin{pmatrix} 0&1 \\ y&-x \end{pmatrix}\right) \quad (x,y,z\in\varpi\fo)$$ by the action of $\sG$, and the integral $Z_{E,3}(\pi,s)$ is $$\#(x_3 \cdot \rho(\sG)) \times q^{-5} \times \int_{(\varpi\fo)^{\oplus 3},\, P(X_3)\in d_E(F^\times)^2}  \beta_E(l_1,l_2) \, |P(X_3)|^{s-2} \d x \, \d y \, \d z .$$ In addition, we have $\cF_1(X_3)= (1,0,- x^2- y z)$ and $\cF_2(X_3)=  (y,-2x,-z)$. By and the fact that $\rho(1,g,{}^t\!g^{-1})$ acts on $Y=\left(\begin{smallmatrix} -y&x \\ x&z \end{smallmatrix}\right)$ as $Y\cdot \rho(1,g,{}^t\!g^{-1})= \det(g)^{-1}\, {}^t\!gYg$, one finds that $Z_{E,3}(\pi,s)$ equals $$\begin{gathered}
\frac{1}{2} \frac{1}{L(1,\trep) \, L(2,\trep)^3} \, q^{-2s+1}  \Big( L(1,\eta)\sum_{l_1=0}^\inf \beta_E(l_1+1,0) q^{(-2s+1)l_1} \\
+ \sum_{l_1=0}^\inf \sum_{l_2=1}^\inf \beta_E(l_1+l_2+1,l_2) \, q^{(-2s+1)l_1} q^{(-2s+2)l_2}  \Big) \qquad \text{if $d_E\in\fo^\times$},\end{gathered}$$ $$\frac{1}{2} \frac{1}{L(1,\trep) \, L(2,\trep)^3}\, q^{-3s+2} \sum_{l_1=0}^\inf \sum_{l_2=0}^\inf \beta_E(l_1+l_2+1,l_2) \, q^{(-2s+1)l_1} q^{(-2s+2)l_2}   \qquad \text{if $d_E\in\varpi\fo^\times$}.$$

### Contribution of $x_4\cdot \rho(\sG)$

Any element in $x_4\cdot \rho(\sG)+\varpi V(\fo)$ is reduced to the form $$X_4:=\left( \begin{pmatrix} 1&-x \\ 0&y \end{pmatrix}, \begin{pmatrix} 0&z \\ 1&x \end{pmatrix} \right) \quad (x,y,z\in\varpi\fo)$$ by the action of $\sG$, and the integral $Z_{E,4}(\pi,s)$ is $$\#(x_4 \cdot \rho(\sG)) \times q^{-5} \times \int_{(\varpi\fo)^{\oplus 3},\, P(X_4)\in d_E(F^\times)^2}  \beta_E(l_1,l_2) \, |P(X_4)|^{s-2} \d x \, \d y \, \d z .$$ In addition, we have $\cF_1(X_4)= (z,-2x,-y)$ and $\cF_2(X_4)= (1,0,-x^2- y z)$. By and the fact that $\rho(g,1,g)$ acts on $Y=\left( \begin{smallmatrix} -y&x \\ x&z \end{smallmatrix} \right)$ as $Y\cdot \rho(g,1,g) = \det(g)^{-1} \, {}^t\!gYg$, one finds that $Z_{E,4}(\pi,s)$ equals $$\begin{gathered}
\frac{1}{2} \frac{1}{L(1,\trep) \, L(2,\trep)^3} \, q^{-2s+1}  \Big( L(1,\eta)\sum_{l_1=0}^\inf \beta_E(0,l_1+1) q^{(-2s+1)l_1} \\
+ \sum_{l_1=0}^\inf \sum_{l_2=1}^\inf \beta_E(l_2,l_1+l_2+1) \, q^{(-2s+1)l_1} q^{(-2s+2)l_2}  \Big) \qquad \text{if $d_E\in\fo^\times$},\end{gathered}$$ $$\frac{1}{2} \frac{1}{L(1,\trep) \, L(2,\trep)^3}\, q^{-3s+2} \sum_{l_1=0}^\inf \sum_{l_2=0}^\inf \beta_E(l_2,l_1+l_2+1) \, q^{(-2s+1)l_1} q^{(-2s+2)l_2}   \qquad \text{if $d_E\in\varpi\fo^\times$}.$$

### Contribution of $x_5\cdot \rho(\sG)$

Any element in $x_5\cdot \rho(\sG)+\varpi V(\fo)$ is reduced to $$X_5:=\left( \begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix},\begin{pmatrix} 0&1 \\ y&0 \end{pmatrix} \right) \quad (y\in\varpi\fo)$$ by the action of $\sG$, and the integral $Z_{E,5}(\pi,s)$ is $$\#(x_5 \, \rho(\sG)) \times q^{-7} \times \int_{\varpi\fo,\, P(X_5)\in d_E(F^\times)^2}  \beta_{\pi,\delta}(l_1,l_2) \, |P(X_5)|^{s-2} \d y .$$ In addition, we have $\cF_1(X_5)=  (1,0,- y)$ and $\cF_2(X_5)= ( y,0,-1)$. Therefore, $Z_{E,5}(\pi,s)$ equals $$\frac{1}{2} \frac{1}{L(1,\trep) \, L(2,\trep)^3}\sum_{l=1}^\inf \beta_E(l,l) \, q^{(-2s+2)l} \qquad \text{if $d_E\in\fo^\times$},$$ $$\frac{1}{2} \frac{1}{L(1,\trep) \, L(2,\trep)^3}\, q^{-s+1}  \sum_{l=0}^\inf \beta_E(l,l) \, q^{(-2s+2)l} \qquad \text{if $d_E\in\varpi\fo^\times$}.$$

### Contribution $x_6\cdot \rho(\sG)$

By Lemma \[lem:card\], one finds $$Z_{E,6}(\pi,s) = \frac{1}{2}\frac{L(1,\eta)^2}{L(1,\trep) \, L(2,\trep)^3} \quad \text{if $d_E\in (\fo^\times)^2$,} \quad \text{and $=0$ if otherwise}.$$

### Contribution $x_7\cdot \rho(\sG)$

By Lemma \[lem:card\], one gets $$Z_{E,7}(\pi,s)= \frac{1}{2}\frac{L(1,\eta)^2}{L(1,\trep) \, L(2,\trep)^3} \quad \text{if $d_E\in \fo^\times \setminus (\fo^\times)^2$,} \quad \text{and $=0$ otherwise}.$$

### Summand except $Z_{E,1}(\pi,s)$

It follows from Propositions \[prop:410x\] and \[prop:412x\] and the above mentioned equalities of $Z_{E,j}(\pi,s,\delta)$ $(2\leq j\leq 7)$ that $$\sum_{j=2}^7 Z_{E,j}(\pi,s) = \frac{1}{2} \frac{L(1,\eta)^2 \, L(2s-1,\trep)\,L(2s-1,\pi,\mathrm{Ad}) }{L(1,\trep) \, L(2,\trep)^3 \, N(\ff_\delta)^{s-1}} \times \mathscr{T}_{E,0}(\pi,s),$$ where $$\begin{gathered}
\mathscr{T}_{E,0}(\pi,s)=  1 + q^{-2s+1} - q^{-2s} + 3 q^{-4s+1} - q^{-4s} + 3 q^{-6s+2} + q^{-6s+1} - q^{-8s+3}  \\
+ 2 q^{-8s+2} - 2 q^{-2s} \lambda - 2 q^{-4s+1} \lambda +  2 q^{-4s} \lambda - 6 q^{-6s+1} \lambda + q^{-4s} \lambda^2 + q^{-6s+1} \lambda^2  \qquad \text{if $d_E\in(\fo^\times)^2$},\end{gathered}$$ $$\begin{gathered}
\mathscr{T}_{E,0}(\pi,s) =  1+q^{-2s+1}+q^{-2s}+q^{-4s+1}-q^{-4s} +q^{-6s+2}+q^{-6s+1} +q^{-8s+3} +2q^{-8s+2} \\
+ 2q^{-2s}\lambda  - 2q^{-4s+1}\lambda- 2q^{-4s}\lambda + 2q^{-6s+1}\lambda  - q^{-4s}\lambda^2 - q^{-6s+1}\lambda^2  \qquad \text{if $d_E\in\fo^\times\setminus(\fo^\times)^2$},\end{gathered}$$ $$\mathscr{T}_{E,0}(\pi,s) =  1+q^{-2s+1}-q^{-2s}+q^{-4s+1}-2q^{-4s+1}\lambda+2q^{-6s+2}  \qquad \text{if $d_E \in \varpi\fo^\times$}.$$ The explicit computation of $Z_{E,1}(\pi,s)$ will be done in the next subsection § \[sec:Zx1\]． We will combine the formula resulting from that computation with the above sum to get Theorem \[thm:localzeta\].

Calculations for the contribution of $x_1\cdot \rho(\sG)$ {#sec:Zx1}
---------------------------------------------------------

In what follows, we prove $$\label{eq:Zx1}
Z_{E,1}(\pi,s)=\frac{1}{2} \frac{L(1,\eta)^2 \, L(2s-1,\trep)\,L(2s-1,\pi,\mathrm{Ad}) }{L(1,\trep) \, L(2,\trep)^3\, N(\ff_\delta)^{s-1}} \times \mathscr{T}_{E,1}(\pi,s),$$ where $$\begin{gathered}
\mathscr{T}_{E,1}(\pi,s)=   2q^{-2s} - 2 q^{-4s+1} - 3 q^{-6s+2} - 2 q^{-6s+1} - q^{-6s} + q^{-8s+3} - 2 q^{-8s+2} - q^{-8s+1}\\
 +  2 q^{-4s+1} \lambda - 2 q^{-4s} \lambda + 6 q^{-6s+1} \lambda +  2 q^{-6s} \lambda - q^{-4s} \lambda^2 - q^{-6s+1} \lambda^2 \qquad \text{if $d_E \in  (\fo^\times)^2$},\end{gathered}$$ $$\begin{gathered}
\mathscr{T}_{E,1}(\pi,s) = -q^{-6s+2} - 2 q^{-6s+1} - q^{-6s} - q^{-8s+3} - 2 q^{-8s+2} - q^{-8s+1}\\
 + 2 q^{-4s+1}\lambda +  2 q^{-4s} \lambda - 2 q^{-6s+1} \lambda - 2 q^{-6s} \lambda + q^{-4s} \lambda^2 +  q^{-6s+1} \lambda^2 \qquad \text{if $d_E \in\fo^\times\setminus (\fo^\times)^2$},\end{gathered}$$ $$\mathscr{T}_{E,1}(\pi,s) = q^{-2s}-q^{-4s+1}-q^{-4s}+2q^{-4s+1}\lambda-2q^{-6s+2}-q^{-6s+1}   \qquad \text{if $d_E \in\varpi\fo^\times$}.$$

To prove , we begin with reduction of elements as we did for other terms. Any element in $x_1\cdot\rho(\sG)+\varpi V(\fo)$ can be reduced to the form $$X_1:= \left( \begin{pmatrix} 1&0 \\ 0&x \end{pmatrix}, \begin{pmatrix} 0&y \\ z&w \end{pmatrix} \right)  \quad (x,y,z,w\in\varpi\fo)$$ by the action of $\sG$, and then the integral $Z_{E,1}(\pi,s)$ equals $$\#(x_1 \cdot \rho(\sG)) \times q^{-4} \times \int_{(\varpi\fo)^{\oplus 4}, \, P(X_1)\in d_E(F^\times)^2}  \beta_E(l_1,l_2) \, |P(X_1)|^{s-2} \d x \, \d y \, \d z \, \d w.$$ From the above equality, we deduce $$\label{eq:04091}
Z_{E,1}(\pi,s)=(1+q^{-1})(1-q^{-2})^2\, q^{-2s} \sum_{k=0}^\inf q^{-k} \tZ_E(\pi,s,k),$$ where $$\tZ_E(\pi,s,k):=\int_{\fo^{\oplus 3} , \, x_{12}^2+\varpi^{k+1}x_1x_2\in d_E(F^\times)^2} \beta_E(l_1,l_2) \, |x_{12}^2+\varpi^{k+1}x_1x_2|^{s-2} \, \d x_1\, \d x_{12}\, \d x_2 ,$$ and it follows from Lemma \[lem:param\] that $(l_1, l_2)$ is determined by $$q^{-2l_1}|d_E|=\frac{|x_{12}^2+\varpi^{k+1}x_1x_2|}{\max(|x_1|,|x_{12}|, q^{-k-1}|x_2|)^2},\quad q^{-2l_2}|d_E|=\frac{|x_{12}^2+\varpi^{k+1}x_1x_2|}{\max(|x_2|,|x_{12}|,q^{-k-1}|x_1|)^2}.$$ For $k,j\in\Z_{\geq 0}$, we set $$\Xi_E(\pi,s,k,j):=\int_{\fo\oplus\fo}\beta_E(l_1,l_2) \, |x_{12}^2+\varpi^k x_1|^{s-2} \d x_{12} \, \d x_1$$ where $x_{12}^2+\varpi^k x_1\in d_E(F^\times)^2$ and $(l_1, l_2)$ is determined by $$q^{-2l_1}|d_E|=\frac{ |x_{12}^2+\varpi^k x_1|}{\max(|x_1|,|x_{12}|, q^{-k})^2},\quad q^{-2l_2}|d_E|=q^{-2-2j}|x_{12}^2+\varpi^{k}x_1|.$$ For $k\in\Z_{\geq -1}$, $j,m\in\Z_{\geq 0}$ we set $$\Omega_E(\pi,s,k,j,m):= \int_{\fo^\times}\d x_2 \int_{\fo}\d x_{12} \, \beta_E(l,l+m) \, \, |x_{12}^2+\varpi^k x_2|^{s-2}$$ where $x_{12}^2+\varpi^k x_2\in d_E (F^\times)^2$ and $l$ is determined by $q^{-2l}|d_E|=q^{-2-2j}|x_{12}^2+\varpi^{k}x_2|$. We also set $$\Delta_{E,1}:=\begin{cases} 1  & \text{if  $d_E\in (\fo^\times)^2$,} \\ 0  & \text{otherwise,} \end{cases} \quad 
\Delta_{E,2}:=\begin{cases} 1  & \text{if  $d_E \in \fo^\times\setminus(\fo^\times)^2$,} \\ 0  & \text{otherwise,} \end{cases} \quad \Delta_{E,3}:=\begin{cases} 1  & \text{if  $d_E \in \varpi\fo^\times$,} \\ 0  & \text{otherwise.} \end{cases}$$ Dividing the domain of integration into the five cases (i) $x_{12}\in\fo^\times$, $x_1\in\fo$, $x_2\in\fo$, (ii) $x_{12}\in\varpi\fo$, $x_1\in\fo^\times$, $x_2\in\fo^\times$, (iii) $x_{12}\in\varpi\fo$, $x_1\in\fo^\times$, $x_2\in\varpi\fo$, (iv) $x_{12}\in\varpi\fo$, $x_1\in\varpi\fo$, $x_2\in\fo^\times$, (v) $x_{12}\in\varpi\fo$, $x_1\in\varpi\fo$, $x_2\in\varpi\fo$, we have $$\label{eq:04092}
\frac{1-q^{-2s+1}}{1-q^{-1}}\, \tZ_E(\pi,s,k) = \Delta_{E,1} +  q^{-2s+3} \Omega_E(\pi,s,k-1,0,0)+2q^{-2s+2}\Xi_E(\pi,s,k,0)  .$$

\[lem:04091\] $$\Omega_E(\pi,s,-1,j,m)=\frac{1}{2}q^{s-2}(1-q^{-1})\, \beta_E(j,j+m) \, \Delta_{E,3},$$ $$\begin{gathered}
\Omega_E(\pi,s,0,j,m)=\frac{1}{2}(1-q^{-1})\, \left\{ 1-(1+\eta(\varpi))q^{-1} \right\}\, \beta_E(1+j,1+j+m) \{ \Delta_{E,1}+\Delta_{E,2}\} \\
+(1-q^{-1})\, q^{-s+1} \int_{\varpi x_2\in \varpi \fo\cap d_E(F^\times)^2} \beta_E(l,l+m)\,  |x_2|^{s-2} \, \d x_2\end{gathered}$$ where $l$ is determined by $q^{-2l}|\delta|=q^{-3-2j}|x_2|$, and for $k\geq 1$ we have $$\Omega_E(\pi,s,k,j,m)=(1-q^{-1})^2 \beta_E(1+j,1+j+m)\,  \Delta_{E,1} + q^{-2s+3}\Omega_E(\pi,s,k-2,j+1,m).$$

\[lem:04092\] For $k\geq 1$ we have $$\begin{gathered}
\Xi_E(\pi,s,k,j)=(1-q^{-1}) \beta_E(0,1+j)\,  \Delta_{E,1} \\
+ q^{-2s+3}\Omega_E(\pi,s,k-2,0,j+1) + q^{-2s+2}\Xi_E(\pi,s,k-1,j+1).\end{gathered}$$ In addition, $$\Xi_E(\pi,s,0,j)=\int_{\fo\cap d_E(F^\times)^2} \beta_E(l,l+j+1)\, |x|^{s-2} \, \d x$$ where $q^{-2l}|d_E|=|x|$.

By , , Lemmas \[lem:04091\] and \[lem:04092\], now we can see our way clear to computing $Z_{E,1}(\pi,s)$ explicitly. What remains to be done is just routine computation which we do by dividing into cases.

### The case $d_E\in\varpi\fo^\times$ ($E$ is ramified over $F$)

In this case, means $$\frac{1-q^{-2s+1}}{1-q^{-1}}\, \tZ_E(\pi,s,k) = q^{-2s+3} \Omega_E(\pi,s,k-1,0,0)+2q^{-2s+2}\Xi_E(\pi,s,k,0)  .$$ By Lemmas \[lem:04091\] and \[lem:04092\], we have $$\frac{1-q^{-2s+1}}{1-q^{-1}}\, \tZ_E(\pi,s,k) =\mathcal{A}_1(k) + \mathcal{A}_2(k),$$ $$\mathcal{A}_1(k):=q^{-2s+3} \Omega_E(\pi,s,k-1,0,0)  + 2q^{-2s+3}\sum_{u=1}^k \Omega_E(\pi,s,k-u-1,0,u) \, q^{u(-2s+2)},$$ $$\mathcal{A}_2(k):= q^{-3s+3}(1-q^{-1})\,  q^{k(-2s+2)}  \sum_{u=0}^\inf q^{u(-2s+2)}\beta_E(u,u+k+1).$$ It is easy to get $$\label{eq:a21113}
 q^{-2s} \sum_{k=0}^\inf q^{-k} \mathcal{A}_2(k)=   \frac{L(2s-1,\pi,\mathrm{Ad})}{ L(1,\trep) \, N(\ff_\eta)^{s-1}} \times \{ -q^{-4s+1}+q^{-4s+1}\lambda -q^{-6s+2}  \}$$ by Proposition \[prop:412x\]. Hence, our task is to compute the sum $q^{-2s} \sum_{k=0}^\inf q^{-k} \mathcal{A}_1(k)$.

It follows from Lemma \[lem:04091\] that $$\Omega_E(\pi,s,2k-1,0,m)=\frac{1}{2}q^{(-2s+3)k}q^{s-2}(1-q^{-1})\, \beta_E(2k,2k+m),$$ $$\Omega_E(\pi,s,2k,0,m)=\frac{1}{2}q^{(-2s+3)k}q^{-s+1}(1-q^{-1})^2 \sum_{u=0}^\inf q^{-2u(s-1)} \beta_E(2k+u+1,2k+u+1+m).$$ Hence, $$\begin{aligned}
\mathcal{A}_1(2k)=&\frac{1}{2}q^{-s+1}(1-q^{-1})\, q^{k(-2s+3)}  \beta_E(k,k) \\
& + (1-q^{-1})\,q^{-s+1}\sum_{u=0}^{k-1} \beta_E(u , -u+2k) \, q^{u(2s-1)}q^{2k(-2s+2)} \\
& + (1-q^{-1})^2\,q^{-s+2} \sum_{u=0}^{k-1} \sum_{m=0}^\inf  q^{u(2s-1)}q^{m(-2s+2)} q^{2k(-2s+2)} \beta_E(u+m+1,-u+2k+m),\end{aligned}$$ $$\begin{gathered}
\mathcal{A}_1(2k+1)=\frac{1}{2}q^{(-2s+3)k}q^{-s+2}(1-q^{-1})^2 \sum_{u=1}^\inf q^{u(-2s+2)} \beta_E(k+u,k+u) \\
 + q^{-s+2}(1-q^{-1})^2\sum_{u=0}^{k-1} \sum_{m=0}^\inf q^{(2s-1)u} q^{m(-2s+2)} q^{(2k+1)(-2s+2)}\, \beta_E(u+m+1,-u+2k+m+1) \\
 + q^{-s+1}(1-q^{-1})\sum_{u=0}^{k}  q^{u(2s-1)} q^{(2k+1)(-2s+2)}\, \beta_E(u,-u+2k+1) .\end{gathered}$$ Therefore, $$\label{eq:20191115a}
 q^{-2s} \sum_{k=0}^\inf q^{-k} \mathcal{A}_1(k)= (1-q^{-1})  \,  q^{-3s+1} \times \{     \eqref{eq:a3l1} + \eqref{eq:a3l3} + \eqref{eq:a3l2}+\eqref{eq:a3l4}       \} ,$$ $$\label{eq:a3l1}
\frac{1}{2} \sum_{k=0}^\inf q^{k(-2s+1)}  \beta_E(k,k)=\frac{1}{2}B_0(q^{-2s+1}) ,$$ $$\label{eq:a3l3}
\sum_{l_1=1}^\inf  \sum_{l_2=0}^\inf q^{l_2(-2s+1)}q^{l_1(-2s+1)} \beta_E(l_2 , l_1+l_2)=q^{-2s+1}U_0( q^{-2s+1},q^{-2s+1}),$$ $$\label{eq:a3l2}
\frac{1}{2}(1-q^{-1}) \sum_{k=0}^\inf   \sum_{u=1}^\inf q^{k(-2s+1)} q^{u(-2s+2)} \beta_E(k+u,k+u) ,$$ $$\label{eq:a3l4}
(q-1)\sum_{l_1=2}^\inf \sum_{l_2=0}^\inf \sum_{m=0}^\inf  q^{l_2(-2s+1)} q^{l_1(-2s+1)} q^{m(-2s+2)} \beta_E(l_2+m+1,l_1+l_2+m).$$ , , can be computed by Proposition \[prop:410x\], Proposition \[prop:412x\], Lemma \[lem:20191230\] respectively. It follows from Lemmas \[lem:relation\], \[lem:20191230\] and \[lem:0409l\] that $$\begin{gathered}
\eqref{eq:a3l4}= - (1+q^{-2s+1})^{-1}\, L(2s-\frac{1}{2},\pi)\, (1-q^{-1})\, q^{-4s+2} \, (1+q^{-2s})^{-1}\Big\{ q^{-1}\lambda B_0(q^{-2s+1})  - q^{-1} \Big\} \\
+  (1+q^{-2s+1})^{-1}\, L(2s-\frac{1}{2},\pi)\, (1-q^{-1})\, q^{-4s+3} (q^{-1}\lambda-q^{-2s}-q^{-4s+1})L(2s-1,\pi,\mathrm{Ad}) \\
\times \Big\{  B_0(q^{-2s+1})(q^{-4s+1}+q^{2s}+q^{2s-1}+1-q^{-1}\lambda^2) -q^{2s}-q^{2s-1} +2q^{-1}\lambda-2\Big\} .\end{gathered}$$ Thus, $$\eqref{eq:20191115a}= \frac{1}{2} \frac{L(2s-1,\pi,\mathrm{Ad}) }{L(1,\trep)  \, N(\ff_\eta)^{s-1}} q^{-2s}(1+q^{-2s+1})(1-q^{-2s}).$$ The explicit formula derives from this and $Z_{E,1}(\pi,s)=L(2s-1,\trep)\, L(2,\trep)^{-3}\times \{\eqref{eq:a21113}+\eqref{eq:20191115a}\}$.

### The case $d_E \in\fo^\times\setminus (\fo^\times)^2$ ($E$ is unramified over $F$)

It follows from Lemma \[lem:04091\] that we have $\Omega_E(\pi,s,k,j,m)=0$ when $k$ is odd. Hence, by and Lemma \[lem:04092\] $$\frac{1-q^{-2s+1}}{1-q^{-1}}\, \tZ_E(\pi,s,2k+1) = q^{-2s+3} \Omega_E(\pi,s,2k,0,0)+2(q^{-2s+2})^2 \Xi_E(\pi,s,2k,1)  ,$$ $$\frac{1-q^{-2s+1}}{1-q^{-1}}\, \tZ_E(\pi,s,2k)  =   2 q^{-2s+2} \Xi_E(\pi,s,2k ,0)  .$$ By Lemma \[lem:04091\] we obtain $$\begin{gathered}
\Omega_E(\pi,s,2k,0,m)= \frac{1}{2}q^{-1}(1-q^{-1}) q^{(-2s+3)k} \beta_E(k+1, k+1+m)  \\
+\frac{1}{2}(1-q^{-1})^2 q^{(-2s+3)k} \sum_{u=0}^\inf q^{-2u(s-1)} \beta_E(k+u+1, k+u+1+m).\end{gathered}$$ From this and Lemma \[lem:20191230\] we deduce $$\begin{gathered}
\label{eq:20191230e1}
\sum_{k=0}^\inf q^{-2k-1}q^{-2s+3} \Omega_E(\pi,s,2k,0,0)= \frac{1}{2}(1-q^{-1})q^{-2s+1}B_1(q^{-2s+1})  \\
+\frac{1}{2}(1-q^{-1})^2 q^{-2s+2}\times L(2s-1,\pi,\mathrm{Ad}) \\
\times \Big\{  B_0(q^{-2s+1})(q^{-4s+1}+q^{2s}+q^{2s-1}+1-q^{-1}\lambda^2) -q^{2s}-q^{2s-1}-1 +q^{-1}\lambda^2 -q^{-1}\lambda^2(1+q^{-1})^{-2} \Big\} .\end{gathered}$$ Furthermore, by Lemma \[lem:04092\], $$\begin{gathered}
\label{eq:20191230e2}
\sum_{k=0}^\inf q^{-2k} \cdot 2q^{-2s+2} \Xi_{2k,0}(\pi,s,\delta) + \sum_{k=0}^\inf q^{-2k-1} \cdot 2q^{-4s+4} \Xi_{2k,1}(\pi,s,\delta) \\
=  q^{-1}(1-q^{-1}) q^{-2s+2} \sum_{k=1}^\inf \sum_{l=0}^\inf q^{(-2s+1)k} q^{(-2s+1)l} \beta_E(k,k+l+1)\\
+ (1-q^{-1})^2 q^{-2s+2} \sum_{k=1}^\inf \sum_{l=0}^\inf \sum_{u=0}^\inf q^{(-2s+1)k} q^{(-2s+1)l} q^{(-2s+2)u}\beta_E(k+u,k+l+u+1) \\
+ (1-q^{-1}) q^{-2s+2} \sum_{k=0}^\inf \sum_{u=0}^\inf q^{(-2s+1)k}  q^{(-2s+2)u}\beta_E(u,u+k+1).\end{gathered}$$ It follows from Lemmas \[lem:20191230\] and \[lem:0409l\] that $$\begin{gathered}
\eqref{eq:20191230e2} =  q^{-1}(1-q^{-1}) q^{-2s+2}U_1(q^{-2s+1},q^{-2s+1}) \\
+(1-q^{-1})^2 (1+q^{-2s+1})^{-1} q^{-4s+3} (q^{-1}\lambda-q^{-2s}-q^{-4s+1})\, L(2s-\frac{1}{2},\pi)\, L(2s-1,\pi,\mathrm{Ad}) \\
\times \Big\{  B_0(q^{-2s+1})(q^{-4s+1}+q^{2s}+q^{2s-1}+1-q^{-1}\lambda^2) -q^{2s}-q^{2s-1}-1 +q^{-1}\lambda^2 -q^{-1}\lambda^2(1+q^{-1})^{-2} \Big\} \\
- (1-q^{-1})^2 (1+q^{-2s+1})^{-1}L(2s-\frac{1}{2},\pi)\, q^{-4s+2} C_0(q^{-2s+1}) \\
+ (1-q^{-1}) q^{-2s+2} U_0(q^{-2s+1},q^{-2s+2}).\end{gathered}$$ Thus, we obtain $$\begin{gathered}
\label{eq:2020.1.1}
\frac{1-q^{-2s+1}}{1-q^{-1}} \sum_{k=0}^\inf q^{-k} \tZ_E(\pi,s,k)= \eqref{eq:20191230e1} + \eqref{eq:20191230e2} \\
=  \frac{1}{2} q^{-2s+1}(1-q^{-1})(1+q^{-1})^{-2}L(2s-1,\pi,\mathrm{Ad}) \times (-q^{-2s+1} - 2 q^{-2s} - q^{-2s-1} - q^{-4s+2} \\
- 2 q^{-4s+1} - q^{-4s} + 2 \lambda +  2 q^{-1} \lambda - 2 q^{-2s} \lambda - 2 q^{-2s-1} \lambda + q^{-1} \lambda^2 +  q^{-2s} \lambda^2),\end{gathered}$$ which implies the assertion.

### The case $d_E\in  (\fo^\times)^2$ ($E=F\oplus F$)

In this case, means $$\frac{1-q^{-2s+1}}{1-q^{-1}}\, \tZ_E(\pi,s,k) = 1 +  q^{-2s+3} \Omega_E(\pi,s,k-1,0,0)+2q^{-2s+2}\Xi_E(\pi,s,k,0)  .$$ By Lemma \[lem:04091\] we have $$\Omega_E(\pi,s,2k-1,0,m)= (1-q^{-1})^2 \sum_{t=1}^{k}q^{(-2s+3)(t-1)} \beta_E(t, t+m) ,$$ $$\begin{gathered}
\Omega_E(\pi,s,2k,0,m)=  (1-q^{-1})^2 \sum_{t=1}^{k}q^{(-2s+3)(t-1)} \beta_E(t, t+m) \\
-\frac{1}{2}q^{-1}(1-q^{-1}) q^{(-2s+3)k} \beta_E(k+1, k+1+m)  \\
+\frac{1}{2}(1-q^{-1})^2 q^{(-2s+3)k} \sum_{u=0}^\inf q^{-2u(s-1)} \beta_E(k+u+1, k+u+1+m).\end{gathered}$$ From this we have $$\begin{gathered}
\label{eq:20200104e1}
\sum_{m=0}^\inf q^{-m} q^{-2s+3} \Omega_E(\pi,s,m-1,0,0)= \frac{1}{2}(1-q^{-1})q^{-2s+1}\sum_{k=0}^\inf q^{(-2s+1)k} \beta_E(k+1, k+1)  \\
+\frac{1}{2}(1-q^{-1})^2 q^{-2s+2}\sum_{k=0}^\inf \sum_{u=0}^\inf q^{(-2s+1)k}q^{(-2s+2)u}\beta_E(k+u+1,k+u+1)\end{gathered}$$ since $\sum_{m=2}^\inf q^{-m}   \sum_{t=1}^{[m/2]}q^{(-2s+3)(t-1)} \beta_E(t, t)= (1-q^{-1})^{-1}  q^{-2}\sum_{t=0}^{\inf}q^{(-2s+1)t} \beta_E(t+1, t+1)$. By Lemma \[lem:20191230\] we have $$\begin{gathered}
\eqref{eq:20200104e1} = \frac{1}{2}(1-q^{-1})q^{-2s+1}B_1(q^{-2s+1})  +\frac{1}{2}(1-q^{-1})^2 q^{-2s+2}\times L(2s-1,\pi,\mathrm{Ad}) \\
\times \Big\{  B_0(q^{-2s+1})(q^{-4s+1}+q^{2s}+q^{2s-1}+1-q^{-1}\lambda^2) -q^{2s}-q^{2s-1}-1 +q^{-1}\lambda^2 -q^{-1}(\lambda-2)^2(1-q^{-1})^{-2} \Big\} .\end{gathered}$$ It follows from Lemma \[lem:04092\] that $$\Xi_E(\pi,s,0,j)=\frac{1}{2}(1-q^{-1})\sum_{u=0}^\inf q^{u(-2s+2)} \beta_E(u , u+j+1),$$ and we get for $k\geq 1$ $$\begin{gathered}
\Xi_E(\pi,s,k , j)= (1-q^{-1})\sum_{t=1}^{k} q^{(-2s+2)(t-1)} \beta_E(0,j+t) \\
+ q^{-2s+3}\sum_{l=1}^{k} q^{(-2s+2)(l-1)}\Omega_E(\pi,s,k-l-1 , 0, j+l)+  q^{(-2s+2)k} \Xi_E(\pi,s,0, j+k) .\end{gathered}$$ Therefore, $$\begin{gathered}
\label{eq:20200104e2}
2q^{-2s+2}\sum_{m=0}^\inf  q^{-m}  \Xi_E(\pi,s,m,0) = 2q^{-2s+1}\sum_{t=0}^\inf q^{(-2s+1)t}\beta_E(t+1) \\
+q^{-1}(1-q^{-1}) q^{-2s+2} \sum_{k=1}^\inf \sum_{l=0}^\inf q^{(-2s+1)k} q^{(-2s+1)l} \beta_E(k,k+l+1)\\
+ (1-q^{-1})^2 q^{-2s+2} \sum_{k=1}^\inf \sum_{l=0}^\inf \sum_{u=0}^\inf q^{(-2s+1)k} q^{(-2s+1)l} q^{(-2s+2)u}\beta_E(k+u,k+l+u+1) \\
+ (1-q^{-1}) q^{-2s+2} \sum_{k=0}^\inf \sum_{u=0}^\inf q^{(-2s+1)k}  q^{(-2s+2)u}\beta_E(u,u+k+1).\end{gathered}$$ By Lemmas \[lem:20191230\] and \[lem:0409l\] we have $$\begin{gathered}
\eqref{eq:20200104e2} =2q^{-2s+1} A_1(q^{-2s+1}) +  q^{-1}(1-q^{-1}) q^{-2s+2}U_1(q^{-2s+1},q^{-2s+1}) \\
+(1-q^{-1})^2 (1+q^{-2s+1})^{-1} q^{-4s+3} (q^{-1}\lambda-q^{-2s}-q^{-4s+1})\, L(2s-\frac{1}{2},\pi)\, L(2s-1,\pi,\mathrm{Ad}) \\
\times \Big\{  B_0(q^{-2s+1})(q^{-4s+1}+q^{2s}+q^{2s-1}+1-q^{-1}\lambda^2) -q^{2s}-q^{2s-1}-1 +q^{-1}\lambda^2 -q^{-1}(\lambda-2)^2(1-q^{-1})^{-2} \Big\} \\
- (1-q^{-1})^2 (1+q^{-2s+1})^{-1}L(2s-\frac{1}{2},\pi)\, q^{-4s+2} C_0(q^{-2s+1}) \\
+ (1-q^{-1}) q^{-2s+2} U_0(q^{-2s+1},q^{-2s+2}).\end{gathered}$$ Thus, we obtain $$\begin{gathered}
\frac{1-q^{-2s+1}}{1-q^{-1}} \sum_{k=0}^\inf q^{-k} \tZ_E(\pi,s,k)= (1-q^{-1})^{-1}  +\eqref{eq:20191230e1} + \eqref{eq:20191230e2} \\
=  \frac{1}{2} (1-q^{-1})^{-1}L(2s-1,\pi,\mathrm{Ad}) \times (2 - 2 q^{-2s+1} - 3 q^{-4s+2} - 2 q^{-4s+1} - q^{-4s} + q^{-6s+3} - 2 q^{-6s+2} - q^{-6s+1}\\
 +  2 q^{-2s+1} \lambda - 2 q^{-2s} \lambda + 6 q^{-4s+1} \lambda +  2 q^{-4s} \lambda - q^{-2s} \lambda^2 - q^{-4s+1} \lambda^2).\end{gathered}$$ The proof is completed.
