Gamma factors for the Asai cube representation

TL;DR

This paper establishes an equality between gamma factors for the Asai cube representation and local zeta integrals, leading to new insights into the analytic properties of associated automorphic L-functions.

Contribution

It proves the equality of gamma factors for the Asai cube representation using Weil-Deligne and local zeta integrals, advancing understanding of their analytic behavior.

Findings

Gamma factors for Asai cube representation are equal to local zeta integrals.

Derived analytic properties of automorphic L-functions for the Asai cube.

Enhanced understanding of local-global compatibility in automorphic forms.

Abstract

We prove an equality between the gamma factors for the Asai cube representation of ${\rm R}_{E/F}{\rm GL}_2$ defined by the Weil$-$Deligne representations and the local zeta integrals of Ikeda and Piatetski-Shapiro$-$Rallis, where $E$ is an \'etale cubic algebra over a local field $F$ of characteristic zero. As an application we obtain the analytic properties of the automorphic $L$-functions for the Asai cube representation.

Equations

$$\mathrm{As}:{}^L({\mathrm{R}}_{E/F}{\mathrm{GL}}_2)⟶\mathrm{GL}({\mathbb{C}}^2\otimes {\mathbb{C}}^2\otimes {\mathbb{C}}^2).$$

$$Z_{\tilde{\alpha }}(f_s,W)={\int }_{F^{\times }{U}_0(F)\backslash \mathbf{G}(F)}{f}_s({\iota }_{\tilde{\alpha }}(\eta g))W(g)dg.$$

$${\gamma }_{\mathrm{PSR}}(s,\mathrm{As}\Pi ,\psi ,\alpha )={\omega }_{\Pi }({\Delta }_{E/F}(\alpha ))|{\Delta }_{E/F}(\alpha ){|}_F^{2s-1}{\omega }_{\mathbb{K}/F}(-1)\gamma (s,\mathrm{As}\Pi ,\psi ).$$

$$L_{\mathrm{PSR}}(s,\mathrm{As}\Pi )$$

$${\epsilon }_{\mathrm{PSR}}(s,\mathrm{As}\Pi ,\psi ,\alpha )$$

$$L(s,\mathrm{As}\Pi )=\prod _{v}L(s,\mathrm{As}{\Pi }_v),\epsilon (s,\mathrm{As}\Pi )=\prod _v\epsilon (s,\mathrm{As}{\Pi }_v,{\psi }_v)$$

$$\begin{cases}E=F\times F\times F&\mbox{ Case 1},\\ E=F^{\prime}\times F\mbox{ for some quadratic extension $F^{\prime}$ of $F$}&\mbox{ Case 2},\\ E\mbox{ is a field}&\mbox{ Case 3}.\end{cases}$$

$$L(s,\mathrm{As}\Pi )=\epsilon (s,\mathrm{As}\Pi )L(1-s,\mathrm{As}{\Pi }^{\vee }).$$

$$L(s,{\rm As}\,\Pi)=\zeta_{K}(s)\begin{cases}L(s,\chi_{1}^{-1}\chi_{1}^{\sigma})L(s,\chi_{2}^{-1}\chi_{2}^{\sigma})L(s,\chi_{3}^{-1}\chi_{3}^{\sigma})&\mbox{ in Case 1 and $\chi=(\chi_{1},\chi_{2},\chi_{3})$},\\ L(s,\chi_{1}^{-1}\chi_{1}^{\sigma})L(s,\chi_{2}^{-1}\chi_{2}^{\sigma})&\mbox{ in Case 2 and $\chi=(\chi_{1},\chi_{2})$},\\ L(s,\chi^{-1}\chi^{\sigma})&\mbox{ in Case 3}.\end{cases}$$

$$\{\begin{array}{ll}\mathrm{Gal}(K/F)& \text{mbox}inCase1,\\ \mathrm{Gal}(F^{\prime }K/F^{\prime })& \text{mbox}inCase2,\\ \mathrm{Gal}(EK/E)& \text{mbox}inCase3.\end{array}$$

$$Z_{\tilde{\alpha }}(M_{\mathrm{w}}^{\ast }{f}_s,W^{\vee })={\omega }_{\Pi }({\Delta }_{E/F}(\alpha ))|{\Delta }_{E/F}(\alpha ){|}_F^{2s-1}{\omega }_{\mathbb{K}/F}(-1)\gamma (s,\mathrm{As}\Pi ,\psi )Z_{\tilde{\alpha }}(f_s,W).$$

$${\mathcal{Z}}_1(s,\lambda )=\frac{Z_{\tilde{\alpha }}(M_{\mathrm{w}}^{\ast }{f}_{-s+1/2,\lambda },W_{\lambda }^{\vee })}{L(s+1/2,\mathrm{As}{\Pi }_{\lambda }^{\vee })}$$

$${\mathcal{Z}}_2(s,\lambda )={\omega }_{{\Pi }_{\lambda }}({\Delta }_{E/F}(\alpha ))|{\Delta }_{E/F}(\alpha ){|}_F^{2s}{\omega }_{\mathbb{K}/F}(-1)\epsilon (s+1/2,\mathrm{As}{\Pi }_{\lambda },\psi ,\alpha )\frac{Z_{\tilde{\alpha }}(f_{s+1/2,\lambda },W_{\lambda })}{L(s+1/2,\mathrm{As}{\Pi }_{\lambda })}$$

$$\{(s,\lambda )\in \mathbb{C}\times \mathcal{D}(\Pi )|\mathrm{Re}(s)>|\lambda {|}_{\Pi }-1/2\}.$$

$${\mathcal{Z}}_1(-s,\lambda )={\mathcal{Z}}_2(s,\lambda )$$

$$\{\lambda \in \mathcal{D}(\Pi )|{\Pi }_{\lambda }\text{mbox}istempered\}.$$

$$\mathbb{C}\times \{\lambda \in \mathcal{D}(\Pi )||\lambda {|}_{\Pi }<1/2\}$$

$$\mathbf{a}(\nu )=\left(\begin{array}{cc}\nu & 0\\ 0& 1\end{array}\right),\mathbf{d}(\nu )=\left(\begin{array}{cc}1& 0\\ 0& \nu \end{array}\right),\mathbf{m}(t)=\left(\begin{array}{cc}t& 0\\ 0& t^{-1}\end{array}\right),\mathbf{n}(x)=\left(\begin{array}{cc}1& x\\ 0& 1\end{array}\right),w=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$$

$$K=\begin{cases}\operatorname{GL}_{2}(\mathfrak{o}_{F})&\mbox{ if $F$ is non-archimedean},\\ {\rm O}(2)&\mbox{ if $F={\mathbb{R}}$},\\ {\rm U}(2)&\mbox{ if $F={\mathbb{C}}$},\end{cases}$$

$$W(\mathbf{n}(x)g)=\psi (x)W(g).$$

$$W^{\vee }(g)={\omega }_{\pi }(\det (g){)}^{-1}W(g).$$

$$l(\pi)=\begin{cases}{\rm wt}(\omega_{\pi})/2&\mbox{ if $\pi$ is essentially square-integrable},\\ \min\{{\rm wt}(\chi_{1}),{\rm wt}(\chi_{2})\}&\mbox{ if $\pi={\rm Ind}_{B(F)}^{\operatorname{GL}_{2}(F)}(\chi_{1}\otimes\chi_{2})$}.\end{cases}$$

$${\pi }_{\lambda }={\mathrm{Ind}}_{B(F)}^{{\mathrm{GL}}_2(F)}({\chi }_1|\text{mbox}{|}_F^{{\lambda }_1}\otimes {\chi }_2|\text{mbox}{|}_F^{{\lambda }_2}).$$

$$\mathcal{D}(\pi)=\begin{cases}{\mathbb{C}}&\mbox{ if $\pi$ is essentially square-integrable},\\ \{\lambda\in{\mathbb{C}}^{2}\,|\,\pi_{\lambda}\mbox{ is irreducible}\}&\mbox{ if $\pi$ is a principal series representation}.\end{cases}$$

$$|\lambda|_{\pi}=\begin{cases}|{\rm wt}(\omega_{\pi})/2+{\rm Re}(\lambda)|&\mbox{ if $\pi$ is essentially square-integrable},\\ \max\{|{\rm wt}(\chi_{1})+{\rm Re}(\lambda_{1})|,|{\rm wt}(\chi_{2})+{\rm Re}(\lambda_{2})|\}&\mbox{ if $\pi={\rm Ind}_{B(F)}^{\operatorname{GL}_{2}(F)}(\chi_{1}\otimes\chi_{2})$}.\end{cases}$$

$$\mathcal{D}(\pi )⟶C_0^{\infty }(N(F)\backslash {\mathrm{GL}}_2(F),\psi ),\lambda ⟼W_{\lambda }$$

$${\omega }_{\psi }(\mathbf{m}(t))\phi (x,y)=|t{|}_F\phi (tx,ty).$$

$${\omega }_{\psi }(\mathbf{n}(b))\phi (x,y)=\psi (bxy)\phi (x,y).$$

$${\omega }_{\psi }(w)\phi (x,y)={\int }_{F^2}\phi (u,v)\psi (uy+vx)dudv,$$

$${\omega }_{\psi }(\mathbf{a}(\nu ))\phi (x,y)=\phi (\nu x,y).$$

$${\mathcal{S}}_{\psi }(F^2)$$

Full text

1. 1 Introduction
2. 2 Holomorphic family of Whittaker functions
3. 3 Asai cube factors via the Weil$-$Deligne representations
4. 4 Asai cube factors via the local zeta integrals
5. 5 Proof of Main Results

1. Introduction

1.1. Main results

Let $F$ be a local field of characteristic zero and $E$ an étale cubic algebra over $F$. Denote by $\mathbb{K}$ the quadratic discriminant algebra of $E$. Let $\alpha$ be a basis of $E$ over $F$ and $\psi$ a non-trivial additive character of $F$. Let $\Pi$ be an irreducible generic admissible representation of $\operatorname{GL}_{2}(E)$. Denote by $\phi_{\Pi}:W_{F}^{\prime}\rightarrow{}^{L}({\rm R}_{E/F}\operatorname{GL}_{2})$ the $L$-parameter associated to $\Pi$ via the local Langlands correspondence. Here $W_{F}^{\prime}$ is the Weil$-$Deligne group of $F$ and ${}^{L}({\rm R}_{E/F}\operatorname{GL}_{2})$ is the Langlands dual group of ${\rm R}_{E/F}\operatorname{GL}_{2}$. We have the Asai cube representation (see § 3 for the precise definition)

[TABLE]

Let $L(s,{\rm As}\,\Pi)$, $\varepsilon(s,{\rm As}\,\Pi,\psi)$, and $\gamma(s,{\rm As}\,\Pi,\psi)$ be the $L$-factor, $\varepsilon$-factor, and $\gamma$-factor, respectively, associated to the admisible representation ${\rm As}\circ\phi_{\Pi}$ of $W_{F}^{\prime}$. On the other hand, associated to $\Pi$ we can define the local factors $L_{\rm PSR}(s,{\rm As}\,\Pi)$, $\varepsilon_{\rm PSR}(s,{\rm As}\,\Pi,\psi,\alpha)$, and $\gamma_{\rm PSR}(s,{\rm As}\,\Pi,\psi,\alpha)$ via the local zeta integrals of Ikeda and Piatetski-Shapiro$-$Rallis introduced in [Ike89] and [PSR87b]. The local zeta integrals are of the form

[TABLE]

Here, $\tilde{\alpha}$ is a symplectic basis of the symplectic space $V(F)=E\oplus E$ equipped with the trace form, $\iota_{\tilde{\alpha}}$ is the isomorphism between $\operatorname{GSp}(V)(F)$ and $\operatorname{GSp}_{6}(F)$ induced by $\tilde{\alpha}$, ${\bf G}(F)=\{g\in\operatorname{GL}_{2}(E)\,|\,\det(g)\in F^{\times}\}\subseteq\operatorname{GSp}(V)(F)$, $U_{0}(F)$ is a unipotent subgroup of ${\bf G}(F)$, $\eta\in\operatorname{GSp}(V)(F)$ satisfies certain conditions, $f_{s}$ is a section in certain degenerate principal series representation of $\operatorname{GSp}_{6}(F)$, and $W$ is a Whittaker function of $\Pi$ with respect to $\psi_{E}=\psi\circ{\rm tr}_{E/F}$. We refer to § 4 for more detail.

The main results of this paper are as follows:

Theorem 1.1**.**

We have

[TABLE]

Here $\omega_{\Pi}$ is the central character of $\Pi$, $\Delta_{E/F}(\alpha)$ is the relative discriminant of $\alpha$ for $E/F$, and $\omega_{\mathbb{K}/F}$ is the quadratic character associated to $\mathbb{K}/F$ by local class field theory.

Remark 1.2**.**

When $E$ is unramified over $F$ and $\Pi$ is unramified, the identity follows from the explicit calculation due to Piatetski-Shapiro and Rallis in [PSR87b]. When $E=F\times F\times F$, the identity was established by Ikeda and Ramakrishnan in [Ike89] and [Ram00], respectively. When $E=F^{\prime}\times F$ for some quadratic extension $F^{\prime}$ of $F$, the assertion was partially proved by Chen$-$Cheng$-$Ishikawa in [CCI18].

Corollary 1.3**.**

Assume $\max\{|L(\Pi)|,|L(\Pi^{\vee})|\}<1/2$. Then

[TABLE]

Here $L(\Pi)$ and $L(\Pi^{\vee})$ are defined in (2.3).

Now we switch to a global setting. Let $F$ be a number field and $E$ an étale cubic algebra over $F$. Denote by ${\mathbb{A}}_{E}$ and ${\mathbb{A}}_{F}$ the rings of adeles of $E$ and $F$, respectively. Let $\psi$ be a non-trivial additive character of ${\mathbb{A}}_{F}/F$. Let $\Pi$ be an irreducible unitary cuspidal automorphic representation of $\operatorname{GL}_{2}({\mathbb{A}}_{E})$ with central character $\omega_{\Pi}$. Write $\omega=\omega_{\Pi}|_{{\mathbb{A}}_{F}^{\times}}$. Let

[TABLE]

be the automorphic $L$-function and $\varepsilon$-factor associated to $\Pi$ and the Asai cube representation. We have the following three cases:

[TABLE]

Combining Corollary 1.3 and the results in [Ike92], [Ram00], and [KS02], we have the following description of the analytic properties of $L(s,{\rm As}\,\Pi)$.

Theorem 1.4**.**

The $L$-function $L(s,{\rm As}\,\Pi)$ is absolutely convergent for ${\rm Re}(s)\geq 3/2$, admits meromorphic continuation to $s\in{\mathbb{C}}$, bounded in vertical strips of finite width, and satisfies the functional equation

[TABLE]

If either $\omega^{2}$ is not principal or $\omega$ is principal, then $L(s,{\rm As}\,\Pi)$ is entire. If $\omega^{2}$ is principal and $\omega$ is not principal, we may assume $\omega^{2}=1$ and $\omega\neq 1$ and let $K$ be the quadratic extension of $F$ associated to $\omega$ by class field theory. Then $L(s,{\rm As}\,\Pi)$ is not entire if and only if $K\neq F^{\prime}$ in Case 2 and there exists a unitary Hecke character $\chi$ of ${\mathbb{A}}_{E\otimes_{F}K}^{\times}$ with $\chi|_{{\mathbb{A}}_{K}^{\times}}=1$ such that $\Pi={\rm Ind}_{E\otimes_{F}K}^{E}(\chi)$. In this case, we have

[TABLE]

Here ${\rm Ind}_{E\otimes_{F}K}^{E}(\chi)$ is the automorphic induction of $\chi$ from ${\mathbb{A}}_{E\otimes_{F}K}^{\times}$ to $\operatorname{GL}_{2}({\mathbb{A}}_{E})$, $\zeta_{K}(s)$ is the completed Dedekind zeta function of $K$, and $\sigma$ is the generator of

[TABLE]

1.2. An outline of the proof

We sketch the proof of Theorem 1.1. Since the assertion is known when $F$ is archimedean, we assume $F$ is non-archimedean. Let $f_{s}$ be a good section of $I(\omega,s)$ and $W$ a Whittaker function of $\Pi$ with respect to $\psi_{E}$. By the functional equation of the local zeta integrals, it suffices to prove

[TABLE]

We extend $\Pi$ to a family of irreducible generic admissible representations $\Pi_{\lambda}$ of $\operatorname{GL}_{2}(E)$ defined in (2.5). Here $M_{\rm w}^{*}$ is the normalized intertwining operator (4.5), $W^{\vee}(g)=\omega_{\Pi}(\det(g))^{-1}W(g)$, and $\lambda$ varies in the domain $\mathcal{D}(\Pi)$ defined in (2.4). Write $\omega_{\lambda}=\omega_{\Pi_{\lambda}}|_{F^{\times}}$. Let $f_{s,\lambda}$ be a good section of $I(\omega_{\lambda},s)$ and $W_{\lambda}$ a holomorphic family of Whittaker functions of $\Pi_{\lambda}$ with respect to $\psi_{E}$ extending $f_{s}$ and $W$, respectively (cf. § 2). Let

[TABLE]

and

[TABLE]

be meromorphic functions on ${\mathbb{C}}\times\mathcal{D}(\Pi)$. By the uniform asymptotic estimate for $W_{\lambda}$ proved in Lemma 2.2, we show in Lemma 4.2 below that $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ define holomorphic functions on the domain

[TABLE]

Here $|\lambda|_{\Pi}\in{\mathbb{R}}_{\geq 0}$ is the absolute value of $\lambda$ with respect to $\Pi$ defined in (2.3). Now we use the limit multiplicity method, which is a global-to-local argument (for example, see [BP18, § 3.8] and [CI19, § 5.3]). More precisely, based on the following ingredients:

• •

the limit multiplicity property for the principal congruence subgroups of $\operatorname{GL}_{2}$ proved in [FLM15],

• •

the known cases for (1.1) recalled in Lemma 4.3 and Corollary 4.5,

• •

the equality between the Asai cube $\gamma$-factors defined by the Weil$-$Deligne representation and the Langlands$-$Shahidi method proved in [HL18],

we prove that the functional equation

[TABLE]

holds for $s\in{\mathbb{C}}$ and $\lambda$ in a dense subset of

[TABLE]

Note that $\Pi_{\lambda}$ is tempered if and only if $|\lambda|_{\Pi}=0$. It then follows from the continuity and holomorphicity that $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ define holomorphic functions on the domain

[TABLE]

and satisfy the functional equation (1.2). Finally, we are in the position to apply [BP18, Proposition 2.8.1], which together with Lemma 4.2 imply that $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ admit holomorphic continuation to ${\mathbb{C}}\times\mathcal{D}(\Pi)$ and satisfy the functional equation (1.2). In particular, (1.1) follows.

1.3. Notation

Let $F$ be a local field of characteristic zero. When $F$ is non-archimedean, let $\mathfrak{o}_{F}$ be the ring of integers of $F$, $\varpi_{F}$ a uniformizer of $\mathfrak{o}_{F}$, $q_{F}$ the cardinality of $\mathfrak{o}_{F}/\varpi_{F}\mathfrak{o}_{F}$, $|\mbox{ }|_{F}$ the absolute value on $F$ normalized so that $|\varpi_{F}|_{F}=q_{F}^{-1}$, and ${\rm ord}_{F}$ the valuation on $F$ normalized so that ${\rm ord}_{F}(\varpi_{F})=1$. When $F$ is archimedean, let $|\mbox{ }|_{{\mathbb{R}}}=|\mbox{ }|$ be the usual absolute value on ${\mathbb{R}}$ and $|z|_{\mathbb{C}}=z\overline{z}$ on ${\mathbb{C}}$.

An additive character $\psi$ of $F$ is a continuous homomorphism $\psi:F\rightarrow{\mathbb{C}}^{\times}$. For $a\in F^{\times}$, let $\psi^{a}$ be the additive character defined by $\psi^{a}(x)=\psi(ax)$.

A character $\chi$ of $F^{\times}$ is a continuous homomorphism $\chi:F^{\times}\rightarrow{\mathbb{C}}^{\times}$. For a character $\chi$ of $F^{\times}$, let ${\rm wt}(\chi)\in{\mathbb{R}}$ defined so that $|\chi|=|\mbox{ }|_{F}^{{\rm wt}(\chi)}$ and denote by $L(s,\chi)$, $\varepsilon(s,\chi,\psi)$, and $\gamma(s,\chi,\psi)$ the $L$-factor, $\varepsilon$-factor, and $\gamma$-factor of $\chi$, respectively, with respect to an additive character $\psi$ of $F$ defined in [Tat79].

Let $B$ be the standard Borel subgroup of $\operatorname{GL}_{2}$ consisting of upper triangular matrices and $N$ its unipotent radical. We put

[TABLE]

for $\nu,t\in\mathbb{G}_{m}$ and $x\in\mathbb{G}_{a}$.

2. Holomorphic family of Whittaker functions

Let $F$ be a local field of characteristic zero and $\psi$ a non-trivial additive character of $F$. Let

[TABLE]

be a maximal compact subgroup of $\operatorname{GL}_{2}(F)$. Denote by $C_{0}^{\infty}(N(F)\backslash\operatorname{GL}_{2}(F),\psi)$ the space of smooth functions $W:\operatorname{GL}_{2}(F)\rightarrow{\mathbb{C}}$ such that

• •

For $x\in F$ and $g\in\operatorname{GL}_{2}(F)$,

[TABLE]

• •

$W$ is right $K$-finite.

Let $\pi$ be an irreducible generic admissible representation of $\operatorname{GL}_{2}(F)$ with central character $\omega_{\pi}$. We denote by $\pi^{\vee}$ the contragredient representation of $\pi$ and by $\mathcal{W}(\pi,\psi)$ the Whittaker model of $\pi$ with respect to $\psi$. Recall that $\mathcal{W}(\pi,\psi)$ is the image of a non-zero intertwining map $\pi\rightarrow C_{0}^{\infty}(N(F)\backslash\operatorname{GL}_{2}(F),\psi)$. For $W\in\mathcal{W}(\pi,\psi)$, we define $W^{\vee}\in\mathcal{W}(\pi^{\vee},\psi)$ by

[TABLE]

We define $l(\pi)\in{\mathbb{R}}$ by

[TABLE]

If $\pi$ is essentially square-integrable and $\lambda\in{\mathbb{C}}$, we define $\pi_{\lambda}=\pi\otimes|\mbox{ }|_{F}^{\lambda}$. If $\pi={\rm Ind}_{B(F)}^{\operatorname{GL}_{2}(F)}(\chi_{1}\otimes\chi_{2})$ and $\lambda=(\lambda_{1},\lambda_{2})\in{\mathbb{C}}^{2}$, we define

[TABLE]

Let $\mathcal{D}(\pi)$ be the domain associated to $\pi$ defined by

[TABLE]

For $\lambda\in\mathcal{D}(\pi)$, define $|\lambda|_{\pi}\in{\mathbb{R}}_{\geq 0}$ by

[TABLE]

We call a map

[TABLE]

a holomorphic family of Whittaker functions of $\pi_{\lambda}$ with respect to $\psi$ if it satisfies the following conditions:

• •

The map $(\lambda,g)\mapsto W_{\lambda}(g)$ is continuous.

• •

For each $g\in\operatorname{GL}_{2}(F)$, the map $\lambda\mapsto W_{\lambda}(g)$ is holomorphic on $\mathcal{D}(\pi)$.

• •

For each $\lambda\in\mathcal{D}(\pi)$, the function $g\mapsto W_{\lambda}(g)$ belongs to $\mathcal{W}(\pi_{\lambda},\psi)$.

• •

$W_{\lambda}$ is right $K$-finite.

We recall a construction of holomorphic family of Whittaker functions. Let $\omega_{\psi}$ be the Weil representation of $\operatorname{GL}_{2}(F)$ on $\mathcal{S}(F^{2})$, the space of Schwartz functions on $F^{2}$, with respect to $\psi$ defined by the following rules:

• •

For $t\in F^{\times}$,

[TABLE]

• •

For $b\in F$,

[TABLE]

• •

[TABLE]

where $du$ and $dv$ are self-dual with respect to $\psi$.

• •

For $\nu\in F^{\times}$,

[TABLE]

Let

[TABLE]

If $\pi$ is essentially square-integrable and $W\in\mathcal{W}(\pi,\psi)$, then the map

[TABLE]

is a holomorphic family of Whittaker functions of $\pi_{\lambda}$ with respect to $\psi$. If $\pi={\rm Ind}_{B(F)}^{\operatorname{GL}_{2}(F)}(\chi_{1}\otimes\chi_{2})$ and $\varphi\in\mathcal{S}_{\psi}(F^{2})$, then the map

[TABLE]

is a holomorphic family of Whittaker functions of $\pi_{\lambda}$ with respect to $\psi$. Note that for any fixed $\lambda_{0}\in\mathcal{D}(\pi)$, any holomorphic family of Whittaker functions can be written as a linear combination, with holomorphic functions of $\lambda$ as coefficients, of holomorphic families of the form (2.1) or (2.2) in a neighborhood of $\lambda_{0}$.

We have the following uniform asymptotic estimate for holomorphic families of Whittaker functions.

Lemma 2.1**.**

Assume $F$ is non-archimedean. Let $W_{\lambda}$ be a holomorphic family of Whittaker functions of $\pi_{\lambda}$ with respect to $\psi$. Let $\epsilon>0$. There exist an integer $n$ independent of $\epsilon,\lambda$ and a constant $C_{\lambda,\epsilon}>0$ bounded uniformly as $\lambda$ varies in a compact set such that

[TABLE]

for $\nu\in F^{\times}$ and $k\in\operatorname{GL}_{2}(\mathfrak{o}_{F})$.

Proof.

Since we only consider the convergence for $\lambda$ varying in a compact set, it suffices to consider holomorphic families of the form (2.1) or (2.2). We assume $\pi={\rm Ind}_{B(F)}^{\operatorname{GL}_{2}(F)}(\chi_{1}\otimes\chi_{2})$ is a principal series representation. The other case follows from the asymptotic estimate for a single Whittaker function in [Jac72, Lemma 14.3]. By the formulae defining $\omega_{\psi}$, there exist an integer $n$ and a constant $C>0$ such that

[TABLE]

for $(x,y)\in F^{2}$, $\nu\in F^{\times}$, and $k\in\operatorname{GL}_{2}(\mathfrak{o}_{F})$. Let

[TABLE]

Then we have

[TABLE]

for $\nu\in F^{\times}$ and $k\in\operatorname{GL}_{2}(\mathfrak{o}_{F})$. Here

[TABLE]

It is cleat that $C_{\lambda}^{(1)}$ is bounded uniformly as $\lambda$ varies in a compact set and there exists a constant $C_{\lambda,\epsilon}^{(2)}>0$ bounded uniformly as $\lambda$ varies in a compact set such that

[TABLE]

for $\nu\in\varpi_{F}^{-2n}\mathfrak{o}_{F}$. This completes the proof. ∎

Let $E$ be a finite étale algebra of degree $d$ over $F$. Let $\psi_{E}$ the additive character of $E$ defined by $\psi_{E}=\psi\circ{\rm tr}_{E/F}$. We denote by $\mathcal{W}(\Pi,\psi_{E})$ the Whittaker model of $\Pi$ with respect to $\psi_{E}$. Let $\Pi$ be an irreducible generic admissible representation of $\operatorname{GL}_{2}(E)$ with central character $\omega_{\Pi}$. Assume $E=F_{1}\times\cdots\times F_{r}$ for some finite extension $F_{i}$ of degree $d_{i}$ over $F$. Then

[TABLE]

for some irreducible generic admissible representation $\pi_{i}$ of $\operatorname{GL}_{2}(F_{i})$. Define $L(\Pi)\in{\mathbb{R}}$ by

[TABLE]

Let $\mathcal{D}(\Pi)$ be the domain associated to $\Pi$ defined by

[TABLE]

For $\lambda=(\lambda_{1},\cdots,\lambda_{r})\in\mathcal{D}(\Pi)$, define

[TABLE]

Note that by definition we have

[TABLE]

and $\Pi_{\lambda}$ is tempered if and only if $|\lambda|_{\Pi}=0$. Let

[TABLE]

Similar to the case $E=F$, we have the notion of holomorphic families of Whittaker functions of $\Pi_{\lambda}$ with respect to $\psi_{E}$.

Lemma 2.2**.**

Assume $F$ is non-archimedean. Let $W_{\lambda}$ be a holomorphic family of Whittaker functions of $\Pi_{\lambda}$ with respect to $\psi_{E}$. Let $\epsilon>0$. There exist an integer $n$ independent of $\epsilon,\lambda$ and a constant $C_{\lambda,\epsilon}>0$ bounded uniformly as $\lambda$ varies in a compact set such that

[TABLE]

for $\nu=(\nu_{1},\cdots,\nu_{r})\in(F^{\times})^{r}$, $t\in\mathcal{C}$, and $k\in\operatorname{GL}_{2}(\mathfrak{o}_{E})$. Here $\mathcal{C}$ is a complete set of coset representatives for $(F_{1}^{\times}\times\cdots\times F_{r}^{\times})/(F^{\times})^{r}$.

Proof.

The assertion follows directly from Lemma 2.1 and the fact that $(F_{1}^{\times}\times\cdots\times F_{r}^{\times})/(F^{\times})^{r}$ is compact. ∎

3. Asai cube factors via the Weil$-$Deligne representations

Let $F$ be a local field of characteristic zero and $\psi$ a non-trivial additive character of $F$. Let $W_{F}^{\prime}$ be the Weil$-$Deligne group of $F$. We identify characters of $F^{\times}$ with one-dimensional admissible representations of $W_{F}^{\prime}$ by local class field theory.

Let $F^{\prime}$ be a finite extension of degree $d$ over $F$. We identify the Langlands dual group ${}^{L}({\rm R}_{F^{\prime}/F}\operatorname{GL}_{2})$ of ${\rm R}_{F^{\prime}/F}\operatorname{GL}_{2}$ with $\operatorname{GL}_{2}({\mathbb{C}})^{d}\rtimes\operatorname{Gal}(\overline{F}/F)$ (cf. [Bor79, § 5]), where the action of $\operatorname{Gal}(\overline{F}/F)$ on $\operatorname{GL}_{2}({\mathbb{C}})^{d}$ is the permutation of components induced by the natural homomorphism $\operatorname{Gal}(\overline{F}/F)\rightarrow\operatorname{Gal}(F^{\prime}/F)$. Let ${\rm As}$ be the Asai representation of ${}^{L}({\rm R}_{F^{\prime}/F}\operatorname{GL}_{2})$ on $({\mathbb{C}}^{2})^{\otimes d}={\mathbb{C}}^{2}\otimes\cdots\otimes{\mathbb{C}}^{2}$ so that the restriction of ${\rm As}$ to $\operatorname{GL}_{2}({\mathbb{C}})^{d}$ is defined by

[TABLE]

and the action of $\operatorname{Gal}(\overline{F}/F)$ on $({\mathbb{C}}^{2})^{\otimes d}$ is the permutation of components induced by the natural homomorphism $\operatorname{Gal}(\overline{F}/F)\rightarrow\operatorname{Gal}(F^{\prime}/F)$. Let $\pi$ be an irreducible admissible representation of $({\rm R}_{F^{\prime}/F}\operatorname{GL}_{2})(F)=\operatorname{GL}_{2}(F^{\prime})$ with central character $\omega_{\pi}$. Denote by $\phi_{\pi}:W_{F}^{\prime}\rightarrow^{L}({\rm R}_{F^{\prime}/F}\operatorname{GL}_{2})$ the $L$-parameter associated to $\pi$ via the local Langlands correspondence. Then we have a $2^{d}$-dimensional admissible representation

[TABLE]

Assume $d=2$ and $\pi={\rm Ind}_{B(F^{\prime})}^{\operatorname{GL}_{2}(F^{\prime})}(\chi_{1}\otimes\chi_{2})$ for some characters $\chi_{1}$ and $\chi_{2}$ of $(F^{\prime})^{\times}$. Then

[TABLE]

where $\sigma$ is the generator of $\operatorname{Gal}(F^{\prime}/F)$ and $\chi_{2}^{\sigma}(a)=\chi_{2}(\sigma(a))$.

Assume $d=3$ and $\pi={\rm Ind}_{B(F^{\prime})}^{\operatorname{GL}_{2}(F^{\prime})}(\chi_{1}\otimes\chi_{2})$ for some characters $\chi_{1}$ and $\chi_{2}$ of $(F^{\prime})^{\times}$. When $F^{\prime}$ is Galois over $F$, we have

[TABLE]

where $\sigma$ is a generator of $\operatorname{Gal}(F^{\prime}/F)$ and $\chi_{i}^{\sigma^{j}}(a)=\chi_{i}(\sigma^{j}(a))$. When $F^{\prime}$ is not Galois over $F$, we have

[TABLE]

Let $E$ be a finite étale algebra of degree $d$ over $F$. Let $\psi_{E}$ the additive character of $E$ defined by $\psi_{E}=\psi\circ{\rm tr}_{E/F}$. Let $\Pi$ be an irreducible admissible representation of $\operatorname{GL}_{2}(E)$ with central character $\omega_{\Pi}$. Assume $E=F_{1}\times\cdots\times F_{r}$ for some finite extension $F_{i}$ of degree $d_{i}$ over $F$. Then

[TABLE]

for some irreducible generic admissible representation $\pi_{i}$ of $\operatorname{GL}_{2}(F_{i})$. Let

[TABLE]

be the admissible representation of $W_{F}^{\prime}$ obtained by composing $({\rm As}\circ\phi_{\pi_{1}})\times\cdots\times({\rm As}\circ\phi_{\pi_{r}})$ with the tensor representation

[TABLE]

We denote by

[TABLE]

the $L$-factor and $\varepsilon$-factor associated to the admissible representation $({\rm As}\circ\phi_{\pi_{1}})\otimes\cdots\otimes({\rm As}\circ\phi_{\pi_{r}})$ defined as in [Tat79, § 3]. Let

[TABLE]

be the associated $\gamma$-factor. Since $({\rm As}\circ\phi_{\pi_{1}})\otimes\cdots\otimes({\rm As}\circ\phi_{\pi_{r}})$ has determinant $(\omega_{\Pi}|_{F^{\times}})^{2^{d-1}}$ and dimension $2^{d}$, we have

[TABLE]

for all $a\in F^{\times}$.

Lemma 3.1**.**

Let $E$ be an étale cubic algebra over $F$ and $\Pi$ an irreducible generic admissible representation of $\operatorname{GL}_{2}(E)$. Then the $L$-factor $L(s,{\rm As}\,\Pi)$ has no poles for

[TABLE]

Proof.

We assume $F$ is non-archimedean and $E$ is a field. The other cases can be proved in a similar way and we omit it. When $\Pi$ is essentially square-integrable, the representation $\Pi\otimes|\mbox{ }|_{E}^{-{\rm wt}(\omega_{\Pi})/2}$ is square-integrable. Thus the $L$-factor $L(s,{\rm As}\,(\Pi\otimes|\mbox{ }|_{E}^{-{\rm wt}(\omega_{\Pi})/2}))$ has no poles for ${\rm Re}(s)>0$. Since ${\rm As}\circ(\phi_{\Pi}\otimes|\mbox{ }|_{E}^{-{\rm wt}(\omega_{\Pi})/2})=({\rm As}\circ\phi_{\Pi})\otimes|\mbox{ }|_{F}^{-3{\rm wt}(\omega_{\Pi})/2}$, we deduce that $L(s,{\rm As}\,\Pi)$ has no poles for

[TABLE]

When $\Pi={\rm Ind}_{B(E)}^{\operatorname{GL}_{2}(E)}(\chi_{1}\otimes\chi_{2})$, the $L$-factor $L(s,{\rm As}\,\Pi)$ is equal to

[TABLE]

or

[TABLE]

depending on whether $E/F$ is Galois or not. In any case, we see that $L(s,{\rm As}\,\Pi)$ has no poles for

[TABLE]

This completes the proof. ∎

4. Asai cube factors via the local zeta integrals

4.1. Preliminaries

Recall the similitude symplectic group

[TABLE]

and its standard Siegel parabolic subgroup

[TABLE]

Denote by $U$ the unipotent radical of $P$.

Let $F$ be a field of characteristic zero and $E$ an étale cubic algebra over $F$. Let $\mathbb{K}$ be the quadratic discriminant algebra of $E$. Let

[TABLE]

be a linear algebraic group over $F$. Let $(V,\langle\,,\,\rangle)$ be the nondegenerate symplectic form over $F$ defined by

[TABLE]

for $x=(x_{1},x_{2}),y=(y_{1},y_{2})\in V$. Let

[TABLE]

be the similitude symplectic group associated to $\langle\,,\,\rangle$. Then it is easy to verify that ${\bf G}$ is a subgroup of $\operatorname{GSp}(V)$ and $\det(g)=\nu(g)$ for $g\in{\bf G}$.

Let $X$ and $X_{0}$ be two maximal isotropic subspaces of $V$ defined by

[TABLE]

Define an isomorphism between $X(F)$ and $X_{0}(F)$ by

[TABLE]

for $x\in F$ and $y\in E$ with ${\rm tr}_{E/F}(y)=0$. Fix $\eta\in\operatorname{GSp}(V)(F)$ such that

[TABLE]

We denote by $P_{0}$ and $R_{0}$ the stabilizers of $X_{0}$ in $\operatorname{GSp}(V)$ and ${\bf G}$, respectively. Let $U_{0}$ be the unipotent radical of $R_{0}$. Note that

[TABLE]

A symplectic basis of $V(F)$ is an ordered basis $\alpha=\{e_{1}^{*},e_{2}^{*},e_{3}^{*},e_{1},e_{2},e_{3}\}$ of $V(F)$ such that

[TABLE]

and $\{e_{1},e_{2},e_{3}\}$ is an ordered basis of $X(F)$. We write $\alpha_{X}=\{e_{1},e_{2},e_{3}\}$. With respect to a symplectic basis $\alpha$ of $V(F)$, we identify $V(F)$ with the space of row vectors $F^{6}$. The identification induces an isomorphism

[TABLE]

Let $\beta$ be another symplectic basis of $V(F)$ and $A_{\alpha,\beta}\in\operatorname{GL}_{3}(F)$ be the transition matrix from $\alpha_{X}$ to $\beta_{X}$. We recall that the transition matrix $A_{\alpha,\beta}=(a_{ij})_{1\leq i,j\leq 3}$ is defined so that

[TABLE]

where $\beta_{X}=\{e_{1}^{\prime},e_{2}^{\prime},e_{3}^{\prime}\}$. Then we have

[TABLE]

for $g\in\operatorname{GSp}(V)(F)$, where $p_{\alpha,\beta}\in P$ is of the form

[TABLE]

4.2. Local zeta integrals and local factors

Let $F$ be a local field of characteristic zero and $\psi$ be a non-trivial additive character of $F$. Let

[TABLE]

be a maximal compact subgroup of $\operatorname{GSp}_{6}(F)$.

Let $\omega$ be a character of $F^{\times}$. For $s\in{\mathbb{C}}$, let $\chi_{\omega,s}$ be the character of $P(F)$ defined by

[TABLE]

for $p=\begin{pmatrix}\nu{}^{t}A^{-1}&*\\ 0&A\end{pmatrix}$, where $\delta_{P}$ is the modulus character of $P$. Recall that $\delta_{P}(p)=|\nu^{6}\det(A)^{-4}|_{F}$. Denote by $I(\omega,s)$ the degenerate principal series representation

[TABLE]

and by $\rho$ the right translation action of $\operatorname{GSp}_{6}(F)$ on $I(\omega,s)$. Recall that $I(\omega,s)$ consisting of smooth functions $f:\operatorname{GSp}_{6}(F)\rightarrow{\mathbb{C}}$ such that

• •

For $p\in P(F)$ and $g\in\operatorname{GSp}_{6}(F)$,

[TABLE]

• •

$f$ is right $K$-finite.

Let ${\rm w}\in\operatorname{GSp}_{6}(F)$ be the Weyl element defined by

[TABLE]

where $J\in\operatorname{GL}_{3}$ is the anti-diagonal matrix with non-zero entries all equal to $1$. We define the intertwining operator

[TABLE]

The integral is absolutely convergent for ${\rm Re}(s)$ sufficiently large and can be meromorphically continued to $s\in{\mathbb{C}}$. Let $M_{{\rm w},\psi}^{*}(\omega):I(\omega,s)\longrightarrow I(\omega^{-1},1-s)$ be the normalized intertwining operator defined by

[TABLE]

Then $M_{{\rm w},\psi}^{*}(\omega^{-1})\circ M_{{\rm w},\psi}^{*}(\omega)$ is a scalar multiple of the identity map on $I(\omega,s)$. We normalize the Haar measure on $U(F)$ so that

[TABLE]

We write $M_{{\rm w},\psi}^{*}(\omega)=M_{\rm w}^{*}$ if there is no cause of confusion. A map

[TABLE]

is a holomorphic section of $I(\omega,s)$ if it satisfies the following conditions:

• •

For each $s\in{\mathbb{C}}$, the function $g\mapsto f_{s}(g)$ belongs to $I(\omega,s)$.

• •

For each $g\in\operatorname{GSp}_{6}(F)$, the function $s\mapsto f_{s}(g)$ is holomorphic.

• •

$f_{s}$ is right $K$-finite.

A map

[TABLE]

is a good section of $I(\omega,s)$ if it satisfies the following conditions:

• •

The map $(s,g)\mapsto L(2s+1,\omega)^{-1}L(4s,\omega^{2})^{-1}f_{s}(g)$ is a holomorphic section of $I(\omega,s)$.

• •

The map $(s,g)\mapsto L(3-2s,\omega^{-1})^{-1}L(4-4s,\omega^{-2})^{-1}M_{\rm w}^{*}f_{s}(g)$ is a holomorphic section of $I(\omega^{-1},1-s)$.

By [Ike92, Lemma 1.3], every holomorphic section is a good section.

Let $\Pi$ be an irreducible generic admissible representation of $\operatorname{GL}_{2}(E)$ with central character $\omega_{\Pi}$. Write $\omega=\omega_{\Pi}|_{F^{\times}}$. Recall $\mathcal{W}(\Pi,\psi_{E})$ is the space of Whittaker functions of $\Pi$ with respect to $\psi_{E}=\psi\circ{\rm tr}_{E/F}$. Let $f_{s}$ be a good section of $I(\omega,s)$ and $W\in\mathcal{W}(\Pi,\psi_{E})$. Define the local zeta integral

[TABLE]

where $\alpha$ is a symplectic basis of $V(F)$, $\iota_{\alpha}$ is the isomorphism (4.3), and $\eta\in\operatorname{GSp}(V)(F)$ satisfies (4.2). Note that the integral is independent of the choice of $\eta$. Indeed, if $\eta^{\prime}\in\operatorname{GSp}(V)(F)$ also satisfies (4.2), then $\iota_{\alpha}(\eta^{\prime}\eta^{-1})\in P(F)$ and $\nu(\eta^{\prime}\eta^{-1})=\det(\eta^{\prime}|_{X(F)}\eta|_{X(F)}^{-1})=1$. By the results in [PSR87b] and [Ike89], the integral is absolutely convergent for ${\rm Re}(s)$ sufficiently large and admits meromorphic continuation to $s\in{\mathbb{C}}$. Moreover, by [PSR87b, Proposition 3.1] and [Ike89, Proposition 4.2], there exists a unique meromorphic function $\gamma_{\rm PSR}(s,{\rm As}\,\Pi,\psi,\alpha)$, called the $\gamma$-factor, such that we have the functional equation

[TABLE]

Recall that $W^{\vee}\in\mathcal{W}(\Pi^{\vee},\psi_{E})$ is defined by

[TABLE]

By Lemma 4.1-(1) below, we see that $\gamma_{\rm PSR}(s,{\rm As}\,\Pi,\psi,\alpha)=\gamma_{\rm PSR}(s,{\rm As}\,\Pi,\psi,\beta)$ if $\alpha_{X}=\beta_{X}$. Therefore we also write $\gamma_{\rm PSR}(s,{\rm As}\,\Pi,\psi,\alpha_{X})=\gamma_{\rm PSR}(s,{\rm As}\,\Pi,\psi,\alpha)$. In particular, identifying $E$ with $X(F)$ via the isomorphism

[TABLE]

the notion $\gamma_{\rm PSR}(s,{\rm As}\,\Pi,\psi,\alpha)$ is defined for any basis $\alpha$ of $E$ over $F$.

Lemma 4.1**.**

(1) Let $\alpha,\beta$ be symplectic bases of $V(F)$. We have

[TABLE]

Here $A_{\alpha,\beta}\in\operatorname{GL}_{3}(F)$ is the transition matrix from $\alpha_{X}$ to $\beta_{X}$.

(2) Let $a\in F^{\times}$ and $\alpha$ be a symplectic basis of $V(F)$. We have

[TABLE]

Proof.

Let $f_{s}$ be a good section of $I(\omega,s)$ and $W\in\mathcal{W}(\Pi,\psi_{E})$. First we prove assertion (1). By (4.4),

[TABLE]

Similarly, we have

[TABLE]

Assertion (1) then follows from the functional equation (4.6).

To prove assertion (2), it follows from assertion (1) that we may assume

[TABLE]

for some $\delta_{1},\delta_{1}^{*},\delta_{2},\delta_{2}^{*}\in E$ with ${\rm tr}_{E/F}(\delta_{1})={\rm tr}_{E/F}(\delta_{2})={\rm tr}_{E/F}(\delta_{1}^{*})={\rm tr}_{E/F}(\delta_{2}^{*})=0$ and $\eta\in\operatorname{GSp}(V)(F)$ satisfying (4.2) is defined by

[TABLE]

Then it is easy to verify that $\iota_{\alpha}(\eta{\bf a}(a^{-1})\eta^{-1})={\rm diag}(1,a^{-1},a^{-1},a^{-1},1,1)$. Let $W^{\prime}\in\mathcal{W}(\Pi,\psi_{E}^{a})$ defined by $W^{\prime}(g)=W({\bf a}(a)g)$. We write $Z_{\alpha,\psi}(f_{s},W)=Z_{\alpha}(f_{s},W)$ and $M_{{\rm w},\psi}=M_{\rm w}$ to emphasis the dependence on $\psi$. We have

[TABLE]

Note that

[TABLE]

Similarly, we have

[TABLE]

Assertion (2) then follows from the functional equation (4.6). ∎

Assume $F$ is non-archimedean. By [PSR87b, Appendix 3 to § 3], the ${\mathbb{C}}[q_{F}^{s},q_{F}^{-s}]$-module generated by $Z_{\alpha}(f_{s},W)$ for good sections $f_{s}$ of $I(\omega,s)$ and $W\in\mathcal{W}(\Pi,\psi_{E})$ is a fractional ideal of ${\mathbb{C}}[q_{F}^{s},q_{F}^{-s}]$ containing $1$. Therefore, there is a unique generator of the form $P(q_{F}^{-s})$ with $P(X)\in{\mathbb{C}}[X]$ and $P(0)=1$. Define the $L$-factor and $\varepsilon$-factor as follows:

[TABLE]

Note that $\varepsilon$-factor is a unit in ${\mathbb{C}}[q_{F}^{s},q_{F}^{-s}]$ by the functional equation (4.6).

Assume $F$ is archimedean. Up to holomorphic functions without zeros, there exists a unique meromorphic function $L_{\rm PRS}(s,{\rm As}\,\Pi)$ without zeros, called the $L$-factor, satisfying the following conditions:

• •

$L_{\rm PRS}(s,{\rm As}\,\Pi)^{-1}Z(f_{s},W)$ is holomorphic for any good section $f_{s}$ of $I(\omega,s)$ and $W\in\mathcal{W}(\Pi,\psi_{E})$.

• •

For each $s_{0}\in{\mathbb{C}}$, there exist a good section $f_{s}$ and $W\in\mathcal{W}(\Pi,\psi_{E})$ such that $L_{\rm PRS}(s,{\rm As}\,\Pi)^{-1}Z(f_{s},W)$ is non-zero at $s=s_{0}.$

Define the $\varepsilon$-factor

[TABLE]

which is well-defined up to holomorphic function without zeros. By the properties characterizing the $L$-factor above and the functional equation (4.6), the $\varepsilon$-factor is a holomorphic function without zeros.

Recall the domain $\mathcal{D}(\Pi)$ associated to $\Pi$ defined in (2.4). Let $W_{\lambda}$ be a holomorphic family of Whittaker functions of $\Pi_{\lambda}$ with respect to $\psi_{E}$. Write $\omega_{\lambda}=\omega_{\Pi_{\lambda}}|_{F^{\times}}$. Similar to the case for $I(\omega,s)$, we define the notion of holomorphic sections and good sections of $I(\omega_{\lambda},s)$ for $(s,\lambda)$ varying in ${\mathbb{C}}\times\mathcal{D}(\Pi)$.

The following lemma is a variant of the estimation in [PSR87b, Proposition 3.2] and [Ike89, § 3.1] by replacing a single Whittaker function with a holomorphic family of Whittaker functions.

Lemma 4.2**.**

Assume $F$ is non-archimedean. Let $W_{\lambda}$ be a holomorphic family of Whittaker functions of $\Pi_{\lambda}$ with respect to $\psi_{E}$. Write $\omega_{\lambda}=\omega_{\Pi_{\lambda}}|_{F^{\times}}$. Let $f_{s,\lambda}$ be a good section of $I(\omega_{\lambda},s)$. Then the integral $Z_{\alpha}(f_{s,\lambda},W_{\lambda})$ is absolutely convergent for

[TABLE]

uniformly for $s$ and $\lambda$ varying in compact sets. In particular, the integral $Z_{\alpha}(f_{s,\lambda},W_{\lambda})$ defines a holomorphic function on the domain

[TABLE]

Here $L(\Pi_{\lambda})\in{\mathbb{R}}$ is defined as in (2.3).

Proof.

We have the following three cases:

[TABLE]

Then $\Pi=\pi_{1}\times\pi_{2}\times\pi_{3}$ for some irreducible generic admissible representations $\pi_{i}$ of $\operatorname{GL}_{2}(F)$ in Case 1, and $\Pi=\pi_{1}\times\pi_{2}$ for some irreducible generic admissible representations $\pi_{1}$ and $\pi_{2}$ of $\operatorname{GL}_{2}(F^{\prime})$ and $\operatorname{GL}_{2}(F)$, respectively, in Case 2. In Case 3, we write $\Pi=\pi_{1}$. Let

[TABLE]

Define $(d_{1},\cdots,d_{r})\in{\mathbb{Z}}^{r}$ by

[TABLE]

We write $\lambda=(\lambda_{1},\cdots,\lambda_{r})\in\mathcal{D}(\pi_{1})\times\cdots\times\mathcal{D}(\pi_{r})=\mathcal{D}(\Pi)$. Let $\mathcal{C}$ be a complete set of coset representatives for $E^{\times}/(F^{\times})^{r}$. Let $s_{\lambda}={\rm wt}(\omega_{\lambda})$ and $s_{i,\lambda}={\rm wt}(\omega_{({\pi_{i}})_{\lambda_{i}}}|_{F^{\times}})$ for $1\leq i\leq r$. Then $s_{\lambda}=\sum_{i=1}^{r}s_{i,\lambda}$. Note that

[TABLE]

by definition.

Let ${\bf K}={\bf G}(F)\cap\operatorname{GL}_{2}(\mathfrak{o}_{E})$. Let $f_{s,\lambda}^{o}$ be the $K$-invariant good section of $I(|\omega_{\lambda}|,s)$ normalized so that $f_{s,\lambda}^{o}(1)=1$. Since $E^{\times}/(F^{\times})^{r}$ is compact, by the $K$-finiteness of $f_{s,\lambda}$, there exists a constant $C_{s,\lambda}>0$ bounded uniformly as $s$ and $\lambda$ vary in a compact set such that

[TABLE]

for $g\in\operatorname{GSp}_{6}(F)$, $t\in\mathcal{C}$, and $k\in{\bf K}$. Let $\epsilon>0$. By Lemma 2.2, there exist an integer $n$ independent of $\epsilon$, $\lambda$ and a constant $C_{\lambda,\epsilon}>0$ bounded uniformly as $\lambda$ varies in a compact set such that

[TABLE]

for $\nu=(\nu_{1},\cdots,\nu_{r})\in(F^{\times})^{r}$, $t\in\mathcal{C}$, and $k\in{\bf K}$, where $\varphi\in\mathcal{S}(F^{r})$ is given by

[TABLE]

We have

[TABLE]

where

[TABLE]

Here ${\bf u}(x)\in{\bf G}(F)$ is defined by

[TABLE]

By (4.7), without lose of generality, we assume $\alpha=\{e_{1}^{*},e_{2}^{*},e_{3}^{*},e_{1},e_{2},e_{3}\}$ is given as follows:

• •

In Case 1,

[TABLE]

• •

In Case 2,

[TABLE]

for some $\delta,\delta^{*}\in F^{\prime}$ with ${\rm tr}_{F^{\prime}/F}(\delta)={\rm tr}_{F^{\prime}/F}(\delta^{*})=0$.

• •

In Case 3,

[TABLE]

for some $\delta_{1},\delta_{1}^{*},\delta_{2},\delta_{2}^{*}\in E$ with ${\rm tr}_{E/F}(\delta_{1})={\rm tr}_{E/F}(\delta_{2})={\rm tr}_{E/F}(\delta_{1}^{*})={\rm tr}_{E/F}(\delta_{2}^{*})=0$.

We define $\eta\in\operatorname{GSp}(V)(F)$ as follows:

• •

We have

[TABLE]

• •

In Case 1,

[TABLE]

• •

In Case 2,

[TABLE]

• •

In Case 3,

[TABLE]

It is easy to verify that $\eta$ satisfies (4.2). Then

[TABLE]

for $x\in F$ and $t=(t_{1},\cdots,t_{r})\in(F^{\times})^{r}$. Therefore,

[TABLE]

for $x\in F$ and $t=(t_{1},\cdots,t_{r})\in(F^{\times})^{r}$. We deduce from (4.8)-(4.10) that the integral $Z^{(0)}_{\alpha}(f_{s,\lambda},W_{\lambda})$ is majorized by

[TABLE]

The above integral is absolutely convergent for

[TABLE]

Moreover, it is clear that the above integral is uniformly convergent as $s$ and $\lambda$ vary in compact sets. Therefore we obtain a uniform estimate for $Z^{(0)}_{\alpha}(f_{s,\lambda},W_{\lambda})$. We have a similar estimate for $Z^{(1)}_{\alpha}(f_{s,\lambda},W_{\lambda})$. This completes the proof. ∎

In the following lemma, we recall the known cases of Theorem 1.1 in the literature.

Lemma 4.3**.**

Assume one of the following assumptions is satisfied:

• •

$F$* is archimedean.*

• •

$E=F\times F\times F$.

• •

$F$* is non-archimedean, $E$ is unramified over $F$, and $\Pi$ is unramified.*

• •

$E$* is not a field and $\Pi$ is unramified.*

We have

[TABLE]

for any basis $\alpha$ of $E$ over $F$ and non-trivial additive character $\psi$ of $F$. Here $\Delta_{E/F}(\alpha)$ is the relative discriminant of $\alpha$ for $E/F$ and $\omega_{\mathbb{K}/F}$ is the quadratic character associated to $\mathbb{K}/F$ by local class field theory.

Proof.

We assume that $F$ is non-archimedean, $E$ is unramified over $F$, and $\Pi$ is unramified. The rest of the cases follow from [Ike89, Theorem 3], [Ram00, Theorem 4.4.1], and [CCI18, Theorem B]. By (3.4) and Lemma 4.1, we may further assume that $\alpha=\{x_{1},x_{2},x_{3}\}$ is an integral basis of $\mathfrak{o}_{E}$ over $\mathfrak{o}_{F}$ and $\psi$ is of conductor $\mathfrak{o}_{F}$. Note that in this case $\omega_{\Pi}(\Delta_{E/F}(\alpha))|\Delta_{E/F}(\alpha)|_{F}^{2s-1}\omega_{\mathbb{K}/F}(-1)=1$ by assumption. Let $f_{s}^{o}$ be the $K$-invariant good section of $I(\omega,s)$ and $W^{o}\in\mathcal{W}(\Pi,\psi)$ the $\operatorname{GL}_{2}(\mathfrak{o}_{E})$-invariant Whittaker function normalized so that $f_{s}^{o}(1)=1$ and $W^{o}(1)=1$. Let $\alpha^{*}=\{x_{1}^{*},x_{2}^{*},x_{3}^{*}\}$ be the dual basis of $\alpha$ and $\tilde{\alpha}$ the symplectic basis of $V(F)$ defined by

[TABLE]

By [PSR87b, Theorem 3.1], we have

[TABLE]

Here the measure on $F^{\times}U_{0}(F)\backslash{\bf G}(F)$ is normalized so that ${\rm vol}(\mathfrak{o}_{F}^{\times}U_{0}(\mathfrak{o}_{F})\backslash{\bf G}(\mathfrak{o}_{F}))=1$. It follows from the functional equation (4.6) that

[TABLE]

Since $\Pi$ is unramified and $\psi$ is of order $\mathfrak{o}_{F}$, we have $\varepsilon(s,{\rm As}\,\Pi,\psi)=1$. This completes the proof. ∎

4.3. Unramified calculation

In this section, we prove a case of Theorem 1.1 in Corollary 4.5. This case will be needed in the proof of general case in § 5.2 below. Assume $F$ is non-archimedean, $E$ is a field, and there is a uniformizer $\varpi_{E}$ of $\mathfrak{o}_{E}$ such that $\varpi_{E}^{3}=\varpi_{F}$ is a uniformizer of $\mathfrak{o}_{F}$. In particular, the last assumption is satisfied if $E$ is totally tamely ramified over $F$. Note that $3\varpi_{E}^{2}\mathfrak{o}_{E}$ is the relative different ideal for $E/F$. Let $\alpha^{o}$ be the symplectic basis of $V(F)$ defined by

[TABLE]

Lemma 4.4**.**

Let $\Pi$ be an irreducible generic admissible representation of $\operatorname{GL}_{2}(E)$ with central character $\omega_{\Pi}$. Write $\omega=\omega_{\Pi}|_{F^{\times}}$. Assume $\Pi$ is unramified and $\psi$ is of conductor $\mathfrak{o}_{F}$. Let $f_{s}^{o}$ be the $K$-invariant good section of $I(\omega,s)$ and $W^{o}\in\mathcal{W}(\Pi,\psi_{E})$ the ${\bf a}(3^{-1}\varpi_{E}^{-2})\operatorname{GL}_{2}(\mathfrak{o}_{E}){\bf a}(3\varpi_{E}^{2})$-invariant Whittaker function normalized so that $f_{s}^{o}(1)=1$ and $W^{o}(1)=1$. We have

[TABLE]

Here the measure on $F^{\times}U_{0}(F)\backslash{\bf G}(F)$ is normalized so that

[TABLE]

Proof.

We write $\varpi=\varpi_{F}$ and $q=q_{F}$ for brevity. Assume $\Pi={\rm Ind}_{B(E)}^{\operatorname{GL}_{2}(E)}(\chi_{1}\otimes\chi_{2})$ for some unramified characters $\chi_{1}$ and $\chi_{2}$ of $E^{\times}$. Let $\alpha=\chi_{1}(\varpi_{E})$, $\beta=\chi_{2}(\varpi_{E})$, and $\chi_{1}\chi_{2}=|\mbox{ }|_{E}^{s_{0}}$ for some $s_{0}\in{\mathbb{C}}$. We have

[TABLE]

Also note that (cf. [PSR87b, p. 54])

[TABLE]

Since $\{1,\varpi_{E},\varpi_{E}^{2}\}$ is an integral basis of $\mathfrak{o}_{E}$ over $\mathfrak{o}_{F}$, one can easily verify that

[TABLE]

Therefore, by the invariance of $f_{s}^{o}$ and $W^{o}$, we have

[TABLE]

where

[TABLE]

Here the measures are normalized so that ${\rm vol}(\mathfrak{o}_{F})={\rm vol}(\mathfrak{o}_{E}^{\times})=1$. We assume $\eta\in\operatorname{GSp}(V)(F)$ satisfying (4.2) is defined by

[TABLE]

for $i=1,2$. For $x\in F$, $n\in{\mathbb{Z}}_{\geq 0}$, and $0\leq i\leq 2$, we have

[TABLE]

Therefore,

[TABLE]

It follows from (4.11)-(4.13) that

[TABLE]

Note that $f_{s}^{o}(\iota_{\alpha^{o}}({\bf a}(\varpi))g)=q^{-s-1/2}f_{s}^{o}(g)$. Similarly,

[TABLE]

By a direct calculation, we have

[TABLE]

Therefore,

[TABLE]

This completes the proof. ∎

Corollary 4.5**.**

Let $\Pi$ be an irreducible generic admissible representation of $\operatorname{GL}_{2}(E)$ with central character $\omega_{\Pi}$. Assume $\Pi$ is unramified. We have

[TABLE]

for any basis $\alpha$ of $E$ over $F$ and non-trivial additive character $\psi$ of $F$.

Proof.

We write $\omega=\omega_{\Pi}|_{F^{\times}}$. Assume $\Pi={\rm Ind}_{B(E)}^{\operatorname{GL}_{2}(E)}(\chi_{1}\otimes\chi_{2})$ for some unramified characters $\chi_{1}$ and $\chi_{2}$ of $E^{\times}$. We first prove the assertion for $\alpha=\alpha^{o}_{X}=\left\{(0,{1}/{3}),(0,{\varpi_{E}^{-1}}/{3}),(0,{\varpi_{E}^{-2}}/{3})\right\}$ and $\psi$ is of conductor $\mathfrak{o}_{F}$. Let $f_{s}^{o}$ be the $K$-invariant good section of $I(\omega,s)$ normalized so that $f_{s}^{o}(1)=1$ and $W^{o}\in\mathcal{W}(\Pi,\psi_{E})$ the ${\bf a}(3^{-1}\varpi_{E}^{-2})\operatorname{GL}_{2}(\mathfrak{o}_{E}){\bf a}(3\varpi_{E}^{2})$-invariant Whittaker function normalized so that $W^{o}(1)=1$. By the Gindikin-Karpelevich formula (cf. [PSR87a, Proposition 5.3]), we have

[TABLE]

Note that $(W^{o})^{\vee}\in\mathcal{W}(\Pi,\psi_{E})$ is the ${\bf a}(3^{-1}\varpi_{E}^{-2})\operatorname{GL}_{2}(\mathfrak{o}_{E}){\bf a}(3\varpi_{E}^{2})$-invariant Whittaker function with $(W^{o})^{\vee}(1)=1$. By Lemma 4.4, we have

[TABLE]

It follows from the functional equation that

[TABLE]

Since $\chi_{1}$ and $\chi_{2}$ are unramified, by (3.2) and (3.3), we have

[TABLE]

Also note that $\varepsilon(s,\chi_{1}|_{F^{\times}},\psi)=\varepsilon(s,\chi_{2}|_{F^{\times}},\psi)=1$ since $\psi$ is of conductor $\mathfrak{o}_{F}$. On the other hand,

[TABLE]

where $\lambda_{E/F}(\psi)$ is the Langlands constant for $E/F$ with respect to $\psi$ (cf. [BH06, § 30]). Since $\chi_{1}$ and $\chi_{2}$ are unramified and $\psi_{E}^{3^{-1}\varpi_{E}^{-2}}$ is of conductor $\mathfrak{o}_{E}$, we have

[TABLE]

Note that $\Delta_{E/F}(\alpha)=3^{-3}\varpi_{F}^{-2}$ and $\lambda_{E/F}(\psi)^{2}=\omega_{\mathbb{K}/F}(-1)$. Indeed, since $\omega_{\mathbb{K}/F}=\det({\rm Ind}_{W_{E}^{\prime}}^{W_{F}^{\prime}}(1))$, we have $\lambda_{E/F}(\psi)^{2}=\omega_{\mathbb{K}/F}(-1)$ by [BH06, (30.4.3)]. We conclude that

[TABLE]

The assertion for the special case is proved. It then follows from (3.4) and Lemma 4.1-(2) that the assertion holds for any non-trivial additive character $\psi$ of $F$ and $\alpha=\alpha^{o}_{X}$. Let $\beta$ be any basis of $E$ over $F$ and $A_{\alpha,\beta}\in\operatorname{GL}_{3}(F)$ the transition matrix from $\alpha$ to $\beta$. Then $\Delta_{E/F}(\beta)=\det(A_{\alpha,\beta})^{2}\Delta_{E/F}(\alpha)$. The assertion for $\beta$ then follows from Lemma 4.1-(1). This completes the proof. ∎

5. Proof of Main Results

5.1. Automorphic $L$-functions

In § 5.2 below, a key ingredient in the proof of Theorem 1.1 is a global-to-local argument. We recall in this section the automorphic $L$-functions for the Asai cube representation. Let $F$ be a number field and $E$ an étale cubic algebra over $F$. Let ${\mathbb{K}}$ be the quadratic discriminant algebra of $E$ and $\omega_{{\mathbb{K}}/F}$ the quadratic Hecke character associated to ${\mathbb{K}}/F$ by class field theory. Denote by ${\mathbb{A}}_{E}$ and ${\mathbb{A}}_{F}$ the rings of adeles of $E$ and $F$, respectively. Let $\psi$ be a non-trivial additive character of ${\mathbb{A}}_{F}/F$. Let $\Pi$ be an irreducible cuspidal automorphic representation of $\operatorname{GL}_{2}({\mathbb{A}}_{E})$ with central character $\omega_{\Pi}$. Write $\omega=\omega_{\Pi}|_{F^{\times}}$. For each place $v$ of $F$, let

[TABLE]

be the Asai cube factors associated to $\Pi_{v}$ defined in § 3 and

[TABLE]

the Asai cube $\gamma$-factor defined by the Langlands-Shahidi method [Sha90] (cf. [HL18, § 5]). By [Sha90, Theorem 3.5-(1)] for $v$ archimedean and [HL18, Theorem 1.1] for $v$ non-archimedean (see also [Ram00, Theorem 4.4.1], [Kim03, § 5], and [Kri03, § 6]), we have

[TABLE]

The automorphic $L$-function and $\varepsilon$-factor associated to $\Pi$ and the Asai cube representation are defined by

[TABLE]

Note that the product defining the $L$-function is absolutely convergent for ${\rm Re}(s)$ sufficiently large. For a set $S$ of places of $F$, we define the partial $L$-function

[TABLE]

We recall the integral representation of $L(s,{\rm As}\,\Pi)$ due to Garrett in [Gar87] (see also [PSR87b] and [Ike89]). Let $\alpha$ be a symplectic basis of $V(F)$ and

[TABLE]

the isomorphism defined by $\iota_{\alpha,{\mathbb{A}}_{F}}=\bigotimes_{v}\iota_{\alpha,v}$, where

[TABLE]

is the isomorphism (4.3) by regarding $\alpha$ as a symplectic basis of $V(F_{v})$. Let $f_{s}$ be a good section of $I(\omega,s)$ and $\phi\in\Pi$ a cusp form. Let $E(f_{s})$ be the Eisenstein series on $\operatorname{GSp}_{6}({\mathbb{A}}_{F})$ defined by

[TABLE]

for ${\rm Re}(s)$ sufficiently large, and by meromorphic continuation otherwise. Let $W_{\phi}$ be the Whittaker function of $\phi$ with respect to $\psi_{E}$ defined by

[TABLE]

By a standard unfolding argument (cf. [PSR87b, § 2]), the integral

[TABLE]

is equal to

[TABLE]

for ${\rm Re}(s)$ sufficiently large, where $\eta\in\operatorname{GSp}(V)(F)$ satisfies (4.2). By the functional equation of the Eisenstein series $E(f_{s})$ and the properties of the local zeta integrals in § 4.2, the integral $Z_{\alpha}(f_{s},W_{\phi})$ admits meromorphic continuation to $s\in{\mathbb{C}}$ and satisfies the functional equation

[TABLE]

Lemma 5.1**.**

Let $S$ be a finite set of places of $F$ containing all archimedean places such that for $v\notin S$, we have $E_{v}$ is unramified over $F_{v}$ and $\Pi_{v}$ is unramified. Then

[TABLE]

for any basis $\alpha$ of $E$ over $F$.

Proof.

Let $\alpha$ be a basis of $E$ over $F$. Let $T$ be a set of places containing $S$ such that for $v\notin T$, we have

• •

$\alpha$ is an integral basis of $\mathfrak{o}_{E_{v}}$ over $\mathfrak{o}_{F_{v}}$,

• •

$\psi_{v}$ is of conductor $\mathfrak{o}_{F_{v}}$,

• •

$E_{v}$ is unramified over $F_{v}$,

• •

$\Pi_{v}$ is unramified.

Write $\alpha=\{x_{1},x_{2},x_{3}\}$ and let $\alpha^{*}=\{x_{1}^{*},x_{2}^{*},x_{3}^{*}\}$ be the dual basis of $\alpha$. Let $\tilde{\alpha}$ be the symplectic basis of $V(F)$ defined by

[TABLE]

For each place $v$ of $F$, let $f_{s,v}$ be a good section of $I(\omega_{v},s)$ and $W_{v}\in\mathcal{W}(\Pi_{v},\psi_{E,v})$ satisfying the following conditions:

• •

For $v\notin T$, $f_{s,v}$ is the $\operatorname{GSp}_{6}(\mathfrak{o}_{F_{v}})$-invariant good section normalized so that $f_{s,v}(1)=1$ and $W_{v}$ the $\operatorname{GL}_{2}(\mathfrak{o}_{E_{v}})$-invariant Whittaker function normalized so that $W_{v}(1)=1$.

• •

For $v\in T$, we have $Z_{\tilde{\alpha}}(f_{s,v},W_{v})\neq 0$.

By [PSR87b, Theorem 3.1], for $v\notin T$ we have

[TABLE]

We conclude from the functional equations (4.6) and (5.2) that the partial $L$-function $L^{T}(s,{\rm As}\,\Pi)$ admits meromorphic continuation to $s\in{\mathbb{C}}$ and satisfies the functional equation

[TABLE]

On the other hand, by [Sha90, Theorem 3.5-(4)] and (5.1), we have the functional equation

[TABLE]

We deduce that

[TABLE]

By Lemma 4.3, for $v\in T\setminus S$, we have

[TABLE]

By assumption, for $v\notin T$, we have

[TABLE]

The assertion then follows from (5.3) and the product formula

[TABLE]

This completes the proof. ∎

5.2. Proof of Theorem 1.1

Let $F$ be a non-archimedean local field of characteristic zero and $E$ an étale cubic algebra over $F$. Let ${\mathbb{K}}$ be the quadratic discriminant algebra of $E$. Let $\Pi$ be an irreducible generic admissible representation of $\operatorname{GL}_{2}(E)$ with central character $\omega_{\Pi}$. Let $\psi$ be a non-trivial additive character of $F$ and $\alpha$ a basis of $E$ over $F$. Recall the domains $\mathcal{D}(\Pi)$ and $\mathcal{D}(\Pi)^{\circ}$ defined in (2.4) and (2.6), respectively. Consider the family of irreducible generic admissible representations $\Pi_{\lambda}$ of $\operatorname{GL}_{2}(E)$ for $\lambda$ varying in $\mathcal{D}(\Pi)$. We are going to prove that the identity

[TABLE]

holds for all $\lambda\in\mathcal{D}(\Pi)$. In particular, (5.4) holds for $\Pi_{\lambda}=\Pi$. We divide the proof into three steps as follows:

• Step 1.

Establish (5.4) for a dense subset $\mathcal{U}$ of $\mathcal{D}(\Pi)^{\circ}$.

• Step 2.

Establish (5.4) for all $\lambda\in\mathcal{D}(\Pi)$ with $|\lambda|_{\Pi}<1/2$.

• Step 3.

Establish (5.4) for all $\lambda\in\mathcal{D}(\Pi)$.

Fix a symplectic basis $\tilde{\alpha}$ of $V(F)$ such that $\tilde{\alpha}_{X}=\alpha$. Write $\omega_{\lambda}=\omega_{\Pi_{\lambda}}|_{F^{\times}}$.

Step 1. By Kranser’s lemma, there exist an étale cubic algebra ${\bm{E}}$ over a number field $\bm{F}$ and a place $v_{1}$ of $\bm{F}$ such that

• •

${\bm{E}}_{v_{1}}=E$ and ${\bm{F}}_{v_{1}}=F$,

• •

for any non-archimedean place $v\neq v_{1}$ and $v$ divides $3$, ${\bm{E}}_{v}$ is unramified over ${\bm{F}}_{v}$.

Let ${\bm{\alpha}}$ be a basis of ${\bm{E}}$ over $\bm{F}$, $\bm{K}$ the quadratic discriminant algebra of $\bm{E}$, and $\bm{\psi}$ a non-trivial additive character of ${\mathbb{A}}_{\bm{F}}/{\bm{F}}$. By (3.4) and Lemma 4.1, we may assume $\alpha={\bm{\alpha}}$ regarding as a basis of $E$ over $F$ and $\psi={\bm{\psi}}_{v}$. Fix a non-archimedean place $v_{2}$ of $\bm{F}$ such that ${\bm{E}}_{v_{2}}={\bm{F}}_{v_{2}}\times{\bm{F}}_{v_{2}}\times{\bm{F}}_{v_{2}}$. Let $U$ be an open subset of $\mathcal{D}(\Pi)^{\circ}$. By the limit multiplicity property for the principal congruence subgroups of $\operatorname{GL}_{2}$ proved in [FLM15, Theorem 1.3], there exists an irreducible cuspidal automorphic representation ${\bm{\Pi}}$ of $\operatorname{GL}_{2}({\mathbb{A}}_{\bm{E}})$ (see [BP18, Theorem 3.7.1] for the deduction) such that

• •

${\bm{\Pi}}_{v_{1}}=\Pi_{\lambda}$ for some $\lambda\in U$,

• •

${\bm{\Pi}}_{v}$ is unramified for all non-archimedean places $v\notin\{v_{1},v_{2}\}$.

Let $S$ be the finite set of places $v$ of $\bm{F}$ such that $v$ is archimedean or $v\in\{v_{1},v_{2}\}$ or $v$ is non-archimedean and ${\bm{E}}_{v}$ is ramified over ${\bm{F}}_{v}$. By Lemma 5.1, we have

[TABLE]

On the other hand, by Lemma 4.3 and Corollary 4.5, for $v\in S$ with $v\neq v_{1}$, we have

[TABLE]

It follows that (5.4) holds for some $\lambda\in U$.

Step 2. Let $\lambda_{0}\in\mathcal{D}(\Pi)$ with $|\lambda_{0}|_{\Pi}<1/2$. Let $0<\epsilon<1/2-|\lambda_{0}|_{\Pi}$. Let $f_{s}$ be a good section of $I(\omega_{\lambda_{0}},s)$ and $W\in\mathcal{W}(\Pi_{\lambda_{0}},\psi)$. We extend $f_{s}$ to a good section $f_{s,\lambda}$ of $I(\omega_{\lambda},s)$ and $W$ to a holomorphic family of Whittaker functions $W_{\lambda}$ of $\Pi_{\lambda}$ with respect to $\psi_{E}$. By Step 1,

[TABLE]

holds for $s\in{\mathbb{C}}$ and $\lambda\in\mathcal{U}$. On the other hand, by Lemma 4.2, the right-hand side and the left-hand side of (5.5) define holomorphic functions on

[TABLE]

and

[TABLE]

respectively. Since the set $\mathcal{U}$ is dense in $\mathcal{D}(\Pi)^{\circ}$, it follows that (5.5) holds for $(s,\lambda)$ in the domain

[TABLE]

We then deduce form the holomorphicity that (5.5) holds for all $s\in{\mathbb{C}}$ and $|\lambda|_{\Pi}<1/2-\epsilon$. In particular, it holds for $\lambda=\lambda_{0}$. Since $f_{s}$ and $W$ are arbitrary chosen, we see that $(\ref{E:main identity})$ holds for $\lambda=\lambda_{0}$.

Step 3. Let $f_{s,\lambda}$ be a good section of $I(\omega_{\lambda},s)$ and $W_{\lambda}$ a holomorphic family of Whittaker functions of $\Pi_{\lambda}$ with respect to $\psi_{E}$. Let

[TABLE]

and

[TABLE]

be partially defined functions on ${\mathbb{C}}\times\mathcal{D}(\Pi)$. By Lemma 4.2 and Step 2, $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ are holomorphic functions on the domain

[TABLE]

and satisfy the functional equation

[TABLE]

Note that it is clear that $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ are periodic of period $\log(q_{F})^{-1}2\pi\sqrt{-1}$ in the variable $s$. We claim that $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ admit holomorphic continuation to ${\mathbb{C}}\times\mathcal{D}(\Pi)$ and satisfy the above functional equation. To prove the claim, by [BP18, Proposition 2.8.1] with $M=\mathcal{D}(\Pi)$ and $U=\{\lambda\in\mathcal{D}(\Pi)\,|\,|\lambda|_{\Pi}<1/2\}$, it suffices to show that for every $C^{\prime}\geq 1/2$, there exists $C>0$ such that $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ admit holomorphic continuation to the domain

[TABLE]

By Lemma 4.2, we can take $C=C^{\prime}-1/2$. As $f_{s,\lambda}$ and $W_{\lambda}$ are arbitrary chosen, we conclude that (5.4) holds for all $\lambda\in\mathcal{D}(\Pi)$. This completes the proof of Theorem 1.1.

Remark 5.2**.**

In [BP18, Proposition 2.8.1], $U$ and $U^{\prime}$ are connected relatively compact open subsets of $M$. It is clear that any connected relatively compact open subset of $\mathcal{D}(\Pi)$ is contained in $\{\lambda\in\mathcal{D}(\Pi)\,|\,|\lambda|_{\Pi}<C\}$ for some $C>0$.

5.3. Proof of Corollary 1.3

By [Ike92, Lemma 2.1] and Lemma 3.1, $L(s,{\rm As}\,\Pi)$ and $L_{\rm PSR}(s,{\rm As}\,\Pi)$ have no poles for ${\rm Re}(s)\geq 1/2$. Similarly, $L(1-s,{\rm As}\,\Pi^{\vee})$ and $L_{\rm PSR}(1-s,{\rm As}\,\Pi^{\vee})$ have no poles for ${\rm Re}(s)\leq 1/2$. Also note that the $L$-functions have no zeros. We deduce that the poles of $L(s,{\rm As}\,\Pi)$, count with multiplicities, are equal to the poles of the meromorphic function $L(s,{\rm As}\,\Pi)L(1-s,{\rm As}\,\Pi^{\vee})^{-1}$. We have a similar conclusion for $L_{\rm PSR}(s,{\rm As}\,\Pi)$. Since the $\varepsilon$-factors have neither poles nor zeros, the assertion then follows from Theorem 1.1.

5.4. Proof of Theorem 1.4

By [KS02, Theorem C], we have $\max\{|L(\Pi_{v})|,|L(\Pi_{v}^{\vee})|\}<1/2$ for all places $v$ of $F$. Therefore, by Corollary 1.3, we have $L_{\rm PSR}(s,{\rm As}\,\Pi_{v})=L(s,{\rm As}\,\Pi_{v})$ for all places $v$ of $F$. It follows from [Ike92, Propositions 2.3-2.5] and the integral representation recalled in § 5.1 that $L(s,{\rm As}\,\Pi)$ is absolutely convergent for ${\rm Re}(s)\geq 3/2$, admits meromorphic continuation to $s\in{\mathbb{C}}$, entire if either $\omega^{2}\neq 1$ or $\omega=1$, and satisfies the functional equation

[TABLE]

Proceeding as in the proof of [Ram00, Theorem 3.4.1], we can prove that $L(s,{\rm As}\,\Pi)$ is bounded in vertical strips of finite width. Note that in our case the assumption that $F$ is totally complex in [Ram00, Theorem 3.4.1] is not necessary. Finally, the description of the poles at $s=0,1$ were established in [Ike92, Theorems 2.6-2.8].