# A two variable Rankin–Selberg integral for ${\mathrm{GU}}(2,2)$ and the
degree 5 $L$-function of ${\mathrm{GSp}}_{4}$

Antonio Cauchi  Antonio Cauchi
Dept. of Mathematics, School of Science
Tokyo Institute of Technology
2-12-1 Ookayama, Meguro-ku
Tokyo, 152-8551 Japan<EMAIL_ADDRESS>and  Armando Gutierrez
Terradillos  Armando Gutierrez Terradillos
Morningside Center of Mathematics, Chinese Academy of Sciences
No. 55, Zhongguancun East Road, Haidian District, Beijing 100190, China
<EMAIL_ADDRESS>
###### Abstract.

We give a two-variable Rankin–Selberg integral for cusp forms on
$\mathrm{PGL}_{4}$ and $\mathrm{PGU}_{2,2}$ which represents a product of
exterior square $L$-functions. As a residue of our integral, we obtain an
integral representation on $\mathrm{PGU}_{2,2}$ of the degree 5 $L$-function
of ${\mathrm{GSp}}_{4}$ twisted by the quadratic character of $E/F$ of
cuspidal automorphic representations which contribute to the theta
correspondence for the pair
$(\mathrm{P}{\mathrm{GSp}}_{4},\mathrm{P}{\mathrm{GU}}_{2,2})$.

###### Key words and phrases:

Langlands $L$-functions, Multivariable Rankin–Selberg integrals, periods of
automorphic forms.

###### 2020 Mathematics Subject Classification:

11F66, 11F70, 22E55

###### Contents

  1. 1 Introduction
  2. 2 Groups and $L$-functions
  3. 3 The two variable Rankin–Selberg integral
  4. 4 Residues and the degree 5 $L$-function of ${\mathrm{GSp}}_{4}$

##  1\. Introduction

Automorphic $L$-functions are one of the cornerstones of the Langlands
program. They provide a bridge between number theory and representation
theory, dispensing a profound link between the arithmetic of algebraic objects
such as elliptic curves and harmonic analysis on reductive groups. Automorphic
$L$-functions are defined by the data of a cuspidal automorphic representation
$\pi$ on a reductive group $G$ and of a finite dimensional representation $r$
of the Langlands group ${}^{L}G$. Their zeros or poles are often linked to the
non-vanishing of certain periods of $\pi$ and to the participation of $\pi$ as
a Langlands functorial transfer or theta lift of an automorphic representation
of another reductive group. One of the main approaches to study the analytic
properties of automorphic $L$-functions are via Rankin–Selberg integrals.
Among those, the exotic family of Rankin–Selberg integrals in more than one
complex variable represent products of $L$-functions and have applications to
the study of their simultaneous poles. For instance, in [3] Bump, Friedberg,
and Ginzburg gave integrals in two complex variables on ${\mathrm{GSp}}_{4}$,
${\mathrm{GSp}}_{6}$, and ${\mathrm{GSp}}_{8}$, each representing the product
of the spin and the standard $L$-functions. Similarly, in [17] Pollack and
Shah gave a two-variable integral on ${\mathrm{GU}}(2,2)$, which represents
the product of the standard and the exterior square $L$-functions. In this
article, we consider a new two-variable Rankin–Selberg integral, which for a
globally generic cusp form on ${\mathrm{GU}}(2,2)$ or $\mathrm{GL}_{4}$,
represents a product of exterior square $L$-functions. As a residue of this
integral, we obtain an integral representation of a twist of the degree $5$
$L$-function of ${\mathrm{GSp}}_{4}$, which gives a new instance of the
curious phenomenon where a Rankin–Selberg integral of a cusp form on one group
is used to represent an $L$-function of a different cusp form on another
group.

We now describe our main results in more detail. Let $F$ be a number field and
denote by ${\bf A}$ its adeles. We let $E^{*}_{B}(g,s,z)$ be the normalized
two variable Borel Eisenstein series for ${\mathrm{GSp}}_{4/F}$. If $\Pi$ is a
generic cuspidal automorphic representation of either ${\mathrm{GU}}_{2,2/F}$
or $\mathrm{GL}_{4/F}$ with trivial central character, and $\varphi$ is a cusp
form in the space of $\Pi$, we define

$I^{*}(\varphi,s,z):=\int_{{\mathrm{GSp}}_{4}(F)Z_{{\mathrm{GSp}}_{4}}({\bf
A})\setminus{\mathrm{GSp}}_{4}({\bf
A})}\\!\\!\\!\\!\\!E^{*}_{B}(g,s,z)\varphi(g)dg.$

Let $\Sigma$ be a finite set of places of $F$ containing the ramified places
for $\Pi$, for $E/F$ (when working with ${\mathrm{GU}}_{2,2}$), and the
archimedean places of $F$. In Theorem 3.4 and (8), we prove the following.

######  Theorem 1.1.

We have

$I^{*}(\varphi,s,z)=L^{\Sigma}(s,\Pi,\wedge^{2})L^{\Sigma}(z,\Pi,\wedge^{2})I_{\Sigma}(\varphi,s,z),$

where $I_{\Sigma}(\varphi,s,z)$, which is the product of the local integrals
at places in $\Sigma$, can be made non-zero at $(s,z)$ for a certain choice of
$\varphi$.

Let now $\Pi$ be a globally generic cuspidal automorphic representation of
${\mathrm{GU}}_{2,2}({\bf A})$ and let $E^{*}_{P}(g,s)$ be the normalized
degenerate Siegel Eisenstein series for ${\rm GSp}_{4}$. As $E^{*}_{P}(g,s)$
can be obtained as the residue at $z=1$ of $E^{*}_{B}(g,s,z)$, the residue at
$z=1$ of $I^{*}(\varphi,s,z)$ is the integral

$J^{*}(\varphi,s)=\int_{{\rm GSp}_{4}(F)Z_{{\rm GSp}_{4}}({\bf
A})\backslash{\rm GSp}_{4}({\bf A})}E^{*}_{P}(g,s)\varphi(g)dg.$

We note that $J^{*}(\varphi,s)$ is the quasi-split counterpart of the integral
for the group ${\rm SO}_{3,3}$ (hence $\mathrm{PGL}_{4}$) studied in [2] and
can be regarded as the degenerate companion of the integral for ${\rm
SO}_{4,2}$ studied in [18] and [16]. $J^{*}(\varphi,s)$ unfolds to the Shalika
period of $\varphi$, which is non-zero if and only if the partial twisted
exterior square $L$-function $L(s,\Pi,\wedge^{2})$ has a simple pole at $s=1$
(cf. [7]). When any of these hold, $\Pi$ comes from a globally generic
cuspidal automorphic representation $\sigma$ of ${\rm GSp}_{4}({\bf A})$ with
trivial central character, by means of a theta correspondence for $({\rm
PGSp}_{4},{\rm PGU}_{2,2})$ (cf. [15, Theorem B]), and one has that

$L^{\Sigma}(s,\Pi,\wedge^{2})=L^{\Sigma}(s,\sigma,{\rm
std}\otimes\chi_{E/F})\zeta_{F}^{\Sigma}(s),$

where $L^{\Sigma}(s,\sigma,{\rm std}\otimes\chi_{E/F})$ denotes the partial
degree 5 $L$-function for ${\rm GSp}_{4}$ twisted by the quadratic Hecke
character associated to the quadratic extension $E/F$ which defines ${\rm
GU}_{2,2}$. Theorem 1.1 and an analysis of the integral $J^{*}(\varphi,s)$ at
the points where $E^{*}_{P}(g,s)$ might obtain a pole let us show the
following (cf. Theorems 4.10 and 4.15).

######  Theorem 1.2.

Suppose that $\Pi$ appears in the theta lift of $\sigma$ in the sense of [15,
Theorem B], where $\sigma$ is a globally generic irreducible cuspidal
automorphic representation of ${\rm GSp}_{4}({\bf A})$ with trivial central
character. For a given cusp form $\varphi$ in $\Pi$, there exist a cusp form
$\varphi^{\prime}$ in $\Pi$ and section $f$ defining the Borel Eisenstein
series such that

$J^{*}(\varphi,s)=L^{\Sigma}(s,\sigma,{\rm
std}\otimes\chi_{E/F})\cdot\prod_{v\in\Sigma_{\infty}}I(\varphi_{v}^{\prime},f_{v},s,1).$

where $\Sigma$ is a finite set of places containing the set $\Sigma_{\infty}$
of archimedean places and the ramified places for $\Pi$, $\sigma$, and $E/F$.
Furthermore, the integral $J^{*}(\varphi,s)$ and the partial $L$-function
$L^{\Sigma}(s,\sigma,{\rm std}\otimes\chi_{E/F})$ extend to holomorphic
functions for all $s$.

One novelty of our integral representation $J^{*}(\varphi,s)$ of the degree 5
$L$-function of $\mathrm{GSp}_{4}$ is that it might be used to address motivic
questions. Indeed, since $\mathrm{GSp}_{4}$ and $\mathrm{GU}_{2,2}$ have
Shimura varieties, our integral seems to have applications to the study of the
conjectural “standard” degree 5 motive associated to cohomological cuspidal
automorphic representations of $\mathrm{GSp}_{4}$, such as the construction of
an Euler system for the corresponding Galois representation, similar in spirit
to the one given in work in progress by Skinner and Sangiovanni Vincentelli
for the symmetric square of a modular form.

### Acknowledgements

A.C. was financially supported by the JSPS Postdoctoral Fellowship for
Research in Japan. A.G. was supported by the Morningside Center of Mathematics
(CAS). We are greatly indebted to the anonymous referee, whose insights on an
earlier draft of the manuscript have notably strengthened the main results of
the manuscript.

##  2\. Groups and $L$-functions

Let $F$ be a number field, with ring of integers $\mathcal{O}$, and denote by
${\bf A}$ its adeles. We let $\zeta_{F}$ denote the Dedekind zeta function of
$F$. When $v$ is a finite place of $F$, we let $F_{v}$ be the $v$-adic
completion of $F$ with ring of integers $\mathcal{O}_{v}$; we let $q_{v}$ be
the size of the quotient of $\mathcal{O}_{v}$ modulo its maximal ideal
$\mathfrak{p}_{v}$ and fix a uniformizer $\varpi_{v}$ of $\mathcal{O}_{v}$.
Accordingly, we normalize the norm $|\cdot|$ of $F_{v}$ so that
$|\varpi_{v}|=q_{v}^{-1}$. Here and throughout the article, if $G$ is a group
with a parabolic subgroup $P$, we will denote by ${\rm Ind}_{P}^{G}(\bullet)$
the normalized induction and by by $I_{P}^{G}(\bullet)$ the not normalized
one.

###  2.1. Groups

####  2.1.1. Definitions

Denote by $J_{2}$ the $2\times 2$ anti-diagonal matrix with all entries 1 and
$J=\left(\begin{smallmatrix}&J_{2}\\\ -J_{2}&\end{smallmatrix}\right)$. Fix a
quadratic extension $E/F$ and define $\mathbf{G}$ the group scheme over $F$
given by

$\mathbf{G}(R):={\mathrm{GU}}_{2,2}(R)=\\{(g,m_{g})\in\mathrm{GL}_{4,F}(R\otimes_{F}E)\times\mathrm{GL}_{1,F}(R):\bar{g}^{t}Jg=m_{g}J\\},$

where $\bar{\bullet}$ denotes the non-trivial automorphism of order 2 of $E/F$
and $R$ denotes any $F$-algebra. Denote by
$\nu:\mathbf{G}\to\mathrm{GL}_{1,F},\;g\mapsto m_{g}$ the similitude
character. Inside $\mathbf{G}$, we have the subgroup

$\mathbf{H}(R):={\mathrm{GSp}}_{4,F}(R)=\\{(g,m_{g})\in\mathrm{GL}_{4,F}(R)\times\mathrm{GL}_{1,F}(R):g^{t}Jg=m_{g}J\\},$

which embeds into $\mathbf{G}$ via the natural inclusion. Throughout the
manuscript, for any algebraic group ${\rm G}$, we let ${\rm PG}$ denote the
quotient of ${\rm G}$ by its center.

####  2.1.2. Parabolic groups

Let $B_{\mathbf{G}}=T_{\mathbf{G}}U_{\mathbf{G}}$ denote the upper-triangular
Borel subgroup of $\mathbf{G}$, with $T_{\mathbf{G}}$ the diagonal torus

$\left\\{\left(\begin{smallmatrix}a&&&\\\ &b&&\\\ &&\nu\bar{b}^{-1}&\\\
&&&\nu\bar{a}^{-1}\end{smallmatrix}\right)\;:a,b\in{\rm
Res}_{E/F}\mathrm{GL}_{1,F},\;\nu\in\mathrm{GL}_{1,F}\right\\}.$

The modular character $\delta_{B_{\mathbf{G}}}:B_{\mathbf{G}}(F_{v})\to{\bf
C}$ is given by

$\left(\begin{smallmatrix}a&&&\\\ &b&&\\\ &&\nu\bar{b}^{-1}&\\\
&&&\nu\bar{a}^{-1}\end{smallmatrix}\right)\mapsto\tfrac{|a\bar{a}|^{3}|b\bar{b}|}{|\nu|^{4}}.$

Let $Q_{\mathbf{G}}=M_{Q_{\mathbf{G}}}N_{Q_{\mathbf{G}}}$ denote the Klingen
parabolic of $\mathbf{G}$. It has Levi
$M_{Q_{\mathbf{G}}}\simeq{\mathrm{GU}}_{1,1}\times{\rm
Res}_{E/F}\mathrm{GL}_{1}$. Similarly, we let
$B_{\mathbf{H}}=T_{\mathbf{H}}U_{\mathbf{H}}=B_{\mathbf{G}}\cap\mathbf{H}$ be
the upper triangular Borel subgroup of $\mathbf{H}$, with diagonal torus

$T_{\mathbf{H}}=\left\\{t(a,b;\mu)=\left(\begin{smallmatrix}a&&&\\\ &b&&\\\
&&\mu b^{-1}&\\\ &&&\mu
a^{-1}\end{smallmatrix}\right)\;:a,b,\mu\in\mathrm{GL}_{1,F}\right\\}.$

Let $Q_{\mathbf{H}}$ denote the standard Klingen parabolic of $\mathbf{H}$
with Levi decomposition $M_{Q_{\mathbf{H}}}N_{Q_{\mathbf{H}}}$ where

$M_{Q_{\mathbf{H}}}=\left\\{\left(\begin{smallmatrix}a&&\\\ &r&\\\
&&d\end{smallmatrix}\right),\;r\in\mathrm{GL}_{2,F},a,d\in\mathrm{GL}_{1,F}\;:\;ad={\rm
det}(r)\right\\},$

and $N_{Q_{\mathbf{H}}}\simeq\mathbf{G}_{a,F}^{3}$. Denote by
$\delta_{Q_{\mathbf{H}}}$ the modulus character of ${Q_{\mathbf{H}}}$, given
by

$\displaystyle\delta_{Q_{\mathbf{H}}}$
$\displaystyle:\left(\begin{smallmatrix}a&&\\\ &r&\\\
&&d\end{smallmatrix}\right)\mapsto\left|\tfrac{a}{d}\right|^{2}.$

Let $P_{\mathbf{H}}$ be the standard Siegel parabolic of $\mathbf{H}$ with
Levi decomposition $M_{P_{\mathbf{H}}}N_{P_{\mathbf{H}}}$, with

$M_{P_{\mathbf{H}}}=\left\\{\left(\begin{smallmatrix}g&\\\ &\mu
J_{2}{}^{t}g^{-1}J_{2}\end{smallmatrix}\right),\;g\in\mathrm{GL}_{2,F},\mu\in\mathrm{GL}_{1,F}\right\\},$

and modulus character given by

$\displaystyle\delta_{P_{\mathbf{H}}}$
$\displaystyle:\left(\begin{smallmatrix}g&\star\\\ &\mu
J_{2}{}^{t}g^{-1}J_{2}\end{smallmatrix}\right)\mapsto\left|\tfrac{{\rm
det}(g)}{\mu}\right|^{3}.$

###  2.2. Dual groups and $L$-functions

####  2.2.1. Dual groups

Recall that the dual group of $\mathbf{G}$ is

${}^{L}\mathbf{G}=(\mathrm{GL}_{4}({\bf C})\times\mathrm{GL}_{1}({\bf
C}))\rtimes{\rm Gal}(E/F),$

with the action of the non-trivial element $\tau\in{\rm Gal}(E/F)$ given by
(cf. [1, §1.8(c)])

$(g,\lambda)\mapsto(\Phi_{4}{}^{t}g^{-1}\Phi_{4},\lambda{\rm det}(g)),\text{
where }\Phi_{4}=\left(\begin{smallmatrix}&&&1\\\ &&-1&\\\ &1&&\\\
-1&&&\end{smallmatrix}\right).$

The dual group of $\mathbf{P}\mathbf{G}$ is the derived subgroup of
${}^{L}\mathbf{G}$, namely

${}^{L}\mathbf{P}\mathbf{G}={\rm SL}_{4}({\bf C})\rtimes{\rm Gal}(E/F).$

Recall also that ${}^{L}\mathbf{P}\mathbf{H}={\rm Spin}_{5}({\bf
C})\simeq{\mathrm{Sp}}_{4}({\bf C})$. Under this exceptional isomorphism, the
standard representation $W_{4}$ of ${\mathrm{Sp}}_{4}({\bf C})$ coincides with
the spin representation of ${\rm Spin}_{5}({\bf C})$.

####  2.2.2. The twisted exterior square representation

The standard representation of $\mathrm{GL}_{4}({\bf C})$ on ${\bf C}^{4}$,
with basis $\\{e_{1},e_{2},e_{3},e_{4}\\}$, induces the representation
$\wedge^{2}$ on $\wedge^{2}{\bf C}^{4}\simeq{\bf C}^{6}$ by the rule
$(\wedge^{2}g)\cdot(e_{i}\wedge e_{j})=g\cdot e_{i}\wedge g\cdot e_{j}$.
Consider the six dimensional representation

$\mathrm{GL}_{4}({\bf C})\times\mathrm{GL}_{1}({\bf C})\to\mathrm{GL}_{6}({\bf
C})$

given by $(g,\lambda)\mapsto(\wedge^{2}g)\lambda$. By [13, Lemma 2.1],
$\wedge^{2}$ extends to a representation of ${}^{L}\mathbf{G}$. Indeed, notice
that both representations $\wedge^{2}$ and $\wedge^{2}\circ\tau$, with
$1\neq\tau\in{\rm Gal}(E/F)$, have same highest weight. Hence they are
isomorphic, i.e. there exists $A\in\mathrm{GL}_{6}({\bf C})$ such that
$(\wedge^{2}\circ\tau)(g,\lambda)=A^{-1}(\wedge^{2}(g,\lambda))A$. Choose $A$
with ${\rm Tr}(A)>0$ and extend $\wedge^{2}$ to ${}^{L}\mathbf{G}$ by sending

$(g,\lambda,1)\mapsto(\wedge^{2}g)\lambda,\,(1,1,\tau)\mapsto A.$

The resulting representation, which we still denote by
$\wedge^{2}:{}^{L}\mathbf{G}\to\mathrm{GL}_{6}({\bf C})$, is called the
(twisted) exterior square representation of ${}^{L}\mathbf{G}$. We still
denote by $\wedge^{2}:{}^{L}\mathbf{P}\mathbf{G}\to\mathrm{GL}_{6}({\bf C})$
the composition of ${}^{L}\mathbf{P}\mathbf{G}\hookrightarrow{}^{L}\mathbf{G}$
with $\wedge^{2}$.

If we restrict the exterior square representation along the embedding
${}^{L}\mathbf{P}\mathbf{H}\hookrightarrow{}^{L}\mathbf{P}\mathbf{G},\,g\mapsto(g,1)$,
we have

$\wedge^{2}_{|{\mathrm{Sp}}_{4}({\bf C})}={\rm std}\oplus\mathbf{1},$ (1)

where ${\rm std}:{\mathrm{Sp}}_{4}({\bf C})\to\mathrm{GL}_{5}({\bf C})$ is the
irreducible representation of highest weight $(1,1)$ and gives the composition
of the projection ${\mathrm{Sp}}_{4}({\bf C})\simeq{\rm Spin}_{5}({\bf
C})\to{\rm SO}_{5}({\bf C})$ with the standard representation of ${\rm
SO}_{5}({\bf C})$.

####  2.2.3. $L$-factors

Suppose that $v$ is an unramified place for $E/F$ and let $\Pi_{v}$ be an
unramified representation of $\mathbf{P}\mathbf{G}(F_{v})$ with Frobenius
conjugacy class $s_{\Pi_{v}}$. Define the local Euler factor of the exterior
square $L$-function for $\Pi_{v}$ by

$L(s,\Pi_{v},\wedge^{2}):={{\rm det}(1-\wedge^{2}(s_{\Pi_{v}})\cdot
q_{v}^{-s})}^{-1}.$

These Euler factors can be related to Euler factors of other $L$-functions on
$\mathbf{PH}$ or $\mathrm{PGL}_{4}$, as follows.

  * •

Suppose that $v$ splits in $E$, then the isomorphism $E\otimes_{F}F_{v}\simeq
F_{v}\times F_{v}$ induces an isomorphism
$\mathrm{GL}_{4}(E\otimes_{F}F_{v})\simeq\mathrm{GL}_{4}(F_{v})\times\mathrm{GL}_{4}(F_{v})$.
If $g$ maps to $(g_{1},g_{2})$, then $\bar{g}$ is sent to $(g_{2},g_{1})$.
Hence, we have

$\displaystyle\mathbf{G}(F_{v})\simeq\\{(g_{1},g_{2},m)\in\mathrm{GL}_{4}(F_{v})\times\mathrm{GL}_{4}(F_{v})\times\mathrm{GL}_{1}(F_{v})\,:\,g_{2}=mJ{}^{t}g_{1}^{-1}J^{-1}\\}$

Hence, projection to the first factor induces

$\displaystyle\mathbf{G}(F_{v})\simeq\mathrm{GL}_{4}(F_{v})\times\mathrm{GL}_{1}(F_{v}),\,(g_{1},g_{2},m)\mapsto(g_{1},m).$
(2)

This let us identify $\mathbf{P}\mathbf{G}(F_{v})$ with
$\mathrm{PGL}_{4}(F_{v})$ and thus $\Pi_{v}$ with an unramified representation
$\pi_{v}$ of $\mathrm{GL}_{4}(F_{v})$ with trivial central character. Then

$L(s,\Pi_{v},\wedge^{2})=L(s,\pi_{v},\wedge^{2}).$

Using the exceptional isomorphism
$\mathrm{PGL}_{4}(F_{v})\simeq\mathrm{PGSO}_{3,3}(F_{v})$, we can consider the
local theta correspondence for
$(\mathbf{P}\mathbf{H}(F_{v}),\mathrm{PGL}_{4}(F_{v}))$. If $\pi_{v}$ is the
small theta lift of a representation $\sigma_{v}$ of
$\mathbf{P}\mathbf{H}(F_{v})$, by functoriality of the theta lift and the
decomposition (1), we have

$L(s,\pi_{v},\wedge^{2})=L(s,\sigma_{v},{\rm std})\cdot{\zeta}_{v}(s).$

  * •

Let $v$ be inert in $E$ and denote again with $v$ the corresponding place of
$E$. If $\Pi_{v}$ is an unramified irreducible subquotient of the (normalized)
induction $\mathrm{Ind}_{B_{\mathbf{G}}(F_{v})}^{\mathbf{G}(F_{v})}\xi$,
denote by $\pi_{v}$ the unramified irreducible subquotient of
$\mathrm{Ind}_{B_{\mathbf{H}}(F_{v})}^{\mathbf{H}(F_{v})}\xi^{\prime},$ with
$\xi^{\prime}$ being the restriction of $\xi$ to $T_{\mathbf{H}}(F_{v})$.
Then, we have (cf. [17, Proposition 3.2])

$\displaystyle L(s,\Pi_{v},\wedge^{2})=L(s,\pi_{v},{\rm
Spin})\cdot{\zeta}_{v}(2s).$ (3)

Finally, let

$\mathbf{H}(F_{v})^{+}:=\\{(g,m_{g})\in\mathbf{H}(F_{v})\,:\,m_{g}\in
N_{E_{v}/F_{v}}(E_{v}^{\times})\\},$

and consider the local theta correspondence for
$(\mathbf{P}\mathbf{H}(F_{v})^{+},\mathbf{P}\mathbf{G}(F_{v}))$ studied in
[15]. If $\Pi_{v}$ is the small theta lift of a generic unramified
representation $\sigma_{v}^{+}$ of $\mathbf{P}\mathbf{H}(F_{v})^{+}$, and
$\sigma_{v}^{+}$ appears as an irreducible factor of an unramified
representation $\sigma_{v}$ of $\mathbf{H}(F_{v})$, then we have (cf [15,
Corollary 6.23])

$\displaystyle L(s,\Pi_{v},\wedge^{2})=L(s,\sigma_{v},{\rm
std}\otimes\chi_{E_{v}/F_{v}})\cdot{\zeta}_{v}(s),$ (4)

where

$\chi_{E_{v}/F_{v}}(x)=\begin{cases}-1&\text{if }x\not\in
N_{E_{v}/F_{v}}(E_{v}^{\times})\\\ 1&\text{otherwise.}\end{cases}$

##  3\. The two variable Rankin–Selberg integral

###  3.1. Global integral and unfolding

Let $s,z$ be complex variables. We denote by $I_{B_{\mathbf{H}}}(s,z)$ the
induced representation of $\mathbf{H}({\bf A})$ consisting of smooth functions
$f(\cdot,s,z):\mathbf{H}({\bf A})\to{\bf C}$ such that, $\forall n\in
U_{\mathbf{H}}({\bf A}),\forall t\in T_{\mathbf{H}}({\bf A})$, we have

$f(ntg,s,z)=\chi_{s,z}(t)f(g,s,z),$ (5)

for $\chi_{s,z}(t(a,b;\mu)):=|a|^{s+z+1}|b|^{s-z+1}|\mu|^{-s-1}$. This
representation has trivial central character. Moreover, the restriction to
$P_{\mathbf{H}}$ of $\chi_{s,0}$ is
$\delta_{P_{\mathbf{H}}}^{\tfrac{1}{3}(s+1)}$.

We let $f\in I_{B_{\mathbf{H}}}(s,z)$ be a standard section, which factorizes
as $\otimes_{v}f_{v}$, and define the two variable Borel Eisenstein series

$\displaystyle{\rm
Eis}_{B_{\mathbf{H}}}(g,s,z):={\sum_{\gamma\in{B_{\mathbf{H}}}(F)\backslash\mathbf{H}(F)}}f(\gamma
g,s,z).$ (6)

The sum converges absolutely for ${\rm Re}(s)$ and ${\rm Re}(z)$ big enough
and it extends to a meromorphic function on ${\bf C}^{2}$. Let $(\Pi,V_{\Pi})$
be a cuspidal automorphic representation of $\mathbf{PG}({\bf A})$. Given a
cusp form $\varphi\in V_{\Pi}$, we define

$I(\varphi,f,s,z):=\int_{\mathbf{H}(F)Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf A})}\\!\\!\\!\\!\\!{\rm
Eis}_{B_{\mathbf{H}}}(g,s,z)\varphi(g)dg.$

Let $\psi$ be a non-trivial additive character of ${\bf A}/F$. We define a
principal character $\chi:U_{\mathbf{G}}({\bf A})/U_{\mathbf{G}}(F)\to{\bf
C}^{\times}$ as follows:

$\chi\left(\left(\begin{smallmatrix}1&x&y&a\\\ &1&b&\bar{y}\\\ &&1&-\bar{x}\\\
&&&1\end{smallmatrix}\right)\right)=\psi(\mathrm{Tr}_{E/F}(\alpha x)-b),$

where $\alpha\in E$ is such that $\overline{\alpha}=-\alpha\neq 0$. We
consider the Whittaker model of a cusp form $\varphi$ of $\mathbf{G}$ with
respect to $\chi$, i.e.

$W_{\varphi}(g):=\int_{[U_{\mathbf{G}}]}\varphi(ng)\chi(n)dn.$

######  Proposition 3.1.

The zeta integral unfolds to

$I(\varphi,f,s,z)=\int_{N_{Q_{\mathbf{H}}}({\bf A})Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf A})}\\!\\!\\!\\!\\!\\!f(wg,s,z)W_{\varphi}(g)dg,$

where

$w=\left(\begin{smallmatrix}1&&&\\\ &&1&\\\ &-1&&\\\
&&&1\end{smallmatrix}\right).$

###### Proof.

We start by unfolding the Eisenstein series along the Klingen parabolic
$Q_{\mathbf{H}}$ to get

$\int_{M_{Q_{\mathbf{H}}}(F)N_{Q_{\mathbf{H}}}({\bf A})Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf
A})}\\!\\!\\!\\!\\!\\!\varphi_{N_{Q_{\mathbf{H}}}}(g)\cdot\\!\\!\\!\\!\\!\\!\sum_{\gamma\in{B_{\mathbf{H}}}(F)\backslash
Q_{\mathbf{H}}(F)}f(\gamma g,s,z)dg,$

where $\varphi_{N_{Q_{\mathbf{H}}}}(g)$ denotes the constant term of $\varphi$
along $N_{Q_{\mathbf{H}}}$. We now Fourier expand
$\varphi_{N_{Q_{\mathbf{H}}}}$ along $N_{Q_{\mathbf{H}}}\setminus
N_{Q_{\mathbf{G}}}$ to get

$\varphi_{N_{Q_{\mathbf{H}}}}(g)=\sum_{\chi:[N_{Q_{\mathbf{H}}}\setminus
N_{Q_{\mathbf{G}}}]\to{\bf C}^{\times}}\varphi_{\chi}(g),$

with
$\varphi_{\chi}(g)=\int_{[N_{Q_{\mathbf{G}}}]}\varphi(n^{\prime}g)\chi^{-1}(n^{\prime})dn^{\prime}$.
Any character $\chi:[N_{Q_{\mathbf{H}}}\setminus N_{Q_{\mathbf{G}}}]\to{\bf
C}^{\times}$ can be described as follows. Recall that the characters
$\chi:[N_{Q_{\mathbf{G}}}/[N_{Q_{\mathbf{G}}},N_{Q_{\mathbf{G}}}]]\to{\bf
C}^{\times}$ are of the form $n^{\prime}\mapsto\psi({\rm Tr}_{E/F}(\alpha
x+\beta y)),$ where $\alpha,\beta\in E$, $\psi$ is a non-trivial additive
character on $F\backslash{\bf A}$, and where we have written

$n^{\prime}\equiv\left(\begin{smallmatrix}1&x&y&0\\\ &1&&\bar{y}\\\
&&1&-\bar{x}\\\ &&&1\end{smallmatrix}\right)\;\text{ mod
}[N_{Q_{\mathbf{G}}},N_{Q_{\mathbf{G}}}],\;\text{ with }x,y\in{\bf A}_{E}.$

Such a character $\chi$ is trivial on $N_{Q_{\mathbf{H}}}({\bf A})$ if
$\alpha+\bar{\alpha}=\beta+\bar{\beta}=0$. We deduce that any character
$\chi:[N_{Q_{\mathbf{H}}}\setminus N_{Q_{\mathbf{G}}}]\to{\bf C}^{\times}$ is
of the form $n^{\prime}\mapsto\psi({\rm Tr}_{E/F}(\alpha x+\beta y))$, with
$\alpha,\beta\in E$ such that $\bar{\alpha}=-\alpha$ and $\bar{\beta}=-\beta$.
To remark the dependence by these data, denote such a character by
$\chi_{\alpha,\beta}$.

Since $M_{Q_{\mathbf{H}}}(F)$ acts on $N_{Q_{\mathbf{G}}}({\bf A})$ by
conjugation and this action preserves $N_{Q_{\mathbf{H}}}({\bf A})$,
$M_{Q_{\mathbf{H}}}(F)$ acts on the space of characters of
$[N_{Q_{\mathbf{H}}}\setminus N_{Q_{\mathbf{G}}}]$. Indeed, if
$m=\left(\begin{smallmatrix}a&&\\\ &g&\\\ &&d\end{smallmatrix}\right)\in
M_{Q_{\mathbf{H}}}(F)$ and $g^{-1}=(\tilde{g}_{i,j})$,

$\displaystyle\chi_{\alpha,\beta}(mn^{\prime}m^{-1})$
$\displaystyle=\psi(a\cdot{\rm
Tr}_{E/F}(\alpha(\tilde{g}_{1,1}x+\tilde{g}_{2,1}y)+\beta(\tilde{g}_{1,2}x+\tilde{g}_{2,2}y)))$
$\displaystyle=\psi({\rm
Tr}_{E/F}(a(\alpha\tilde{g}_{1,1}+\beta\tilde{g}_{1,2})x+a(\alpha\tilde{g}_{2,1}+\beta\tilde{g}_{2,2})y))$
$\displaystyle=\chi_{\alpha^{\prime},\beta^{\prime}}(n^{\prime}),$

with $\alpha^{\prime}=a(\alpha\tilde{g}_{1,1}+\beta\tilde{g}_{1,2})$ and
$\beta^{\prime}=a(\alpha\tilde{g}_{2,1}+\beta\tilde{g}_{2,2})$. Then,
$M_{Q_{\mathbf{H}}}(F)$ acts on the space of characters of
$[N_{Q_{\mathbf{H}}}\setminus N_{Q_{\mathbf{G}}}]$ with two orbits, the
trivial one and an open one. A representative of the open orbit is given by
the character $\chi_{\alpha,0}$ with $\bar{\alpha}=-\alpha\neq 0$. As the
constant term $\varphi_{N_{Q_{\mathbf{G}}}}(g)$ vanishes because of
cuspidality of $\varphi$, the integral can be written as

$\int_{M_{\alpha}(F)N_{Q_{\mathbf{H}}}({\bf A})Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf
A})}\\!\\!\\!\\!\\!\\!\varphi_{\chi_{\alpha,0}}(g)\cdot\\!\\!\\!\\!\\!\\!\sum_{\gamma\in{B_{\mathbf{H}}}(F)\backslash
Q_{\mathbf{H}}(F)}f(\gamma g,s,z)dg,$ (7)

where $M_{\alpha}(F)$ is the stabilizer of $\chi_{\alpha,0}$ in
$M_{Q_{\mathbf{H}}}(F)$:

$M_{\alpha}(F)=L_{\alpha}(F)N_{\alpha}(F)=\left\\{\left(\begin{smallmatrix}a&&&\\\
&a&&\\\ &&d&\\\ &&&d\end{smallmatrix}\right)\left(\begin{smallmatrix}1&&&\\\
&1&b&\\\ &&1&\\\ &&&1\end{smallmatrix}\right)\;a,d\in F^{\times},\,b\in
F\right\\}.$

By the Bruhat decomposition for $\mathrm{GL}_{2}$, we have that
${B_{\mathbf{H}}}(F)\backslash
Q_{\mathbf{H}}(F)={B_{\mathbf{H}}}(F)\cup{B_{\mathbf{H}}}(F)wN_{\alpha}(F)$.
This let us write (7) as a sum of two integrals, with the term corresponding
to the coset ${B_{\mathbf{H}}}(F)$ that vanishes because of the cuspidality of
$\varphi$:

$\eqref{auxFourierexp}=\int_{L_{\alpha}(F)N_{Q_{\mathbf{H}}}({\bf
A})Z_{\mathbf{H}}({\bf A})\setminus\mathbf{H}({\bf
A})}\\!\\!\\!\\!\\!\\!\varphi_{\chi_{\alpha,0}}(g)f(wg,s,z)dg.$

We now Fourier expand $\varphi_{\chi_{\alpha,0}}(g)$ over
$N_{Q_{\mathbf{G}}}\backslash U_{\mathbf{G}}$. $L_{\alpha}(F)$ acts on the
space of characters on $[N_{Q_{\mathbf{G}}}\backslash U_{\mathbf{G}}]$ with
one open orbit and the trivial one. The latter does not contribute to the
integral because of cuspidality of $\varphi$. As
$U_{\mathbf{G}}=N_{Q_{\mathbf{G}}}N_{\alpha}$, a representative of the open
orbit can be taken to be $b\mapsto\psi(b)$ on $N_{\alpha}$ and has stabilizer
$Z_{\mathbf{H}}(F)$. Since the character on
$[U_{\mathbf{G}}/[U_{\mathbf{G}},U_{\mathbf{G}}]]$, defined by taking
$b\mapsto\psi(-b)$ on $N_{\alpha}$ and by taking $\chi_{\alpha,0}$ on the one
parameter unipotent subgroup associated to the other simple root, is a
principal character, the function
$W_{\varphi}(g)=\int_{[N_{\alpha}]}\varphi_{\chi_{\alpha,0}}(n(b)g)\psi(-b)db$
is a Whittaker functional for $\varphi$ in $\Pi$. This concludes the proof. ∎

######  Remark 3.2.

Using the natural embedding of $\mathbf{H}\hookrightarrow\mathrm{GL}_{4}$, we
can define the analogous integral for cusp forms on $\mathrm{PGL}_{4}$ by
integrating their restriction to $\mathbf{H}({\bf A})$ against ${\rm
Eis}_{B_{\mathbf{H}}}(g,s,z)$. Then the unfolding follows from an almost
verbatim translation of the proof of Proposition 3.1. We leave the details to
the reader.

By the uniqueness of Whittaker functionals, we have the following.

######  Corollary 3.3.

Let $\varphi=\otimes_{v}\varphi_{v}$ and $f=\otimes_{v}f_{v}$ be factorizable
elements of $\Pi$ and $I_{B_{\mathbf{H}}}(s,z)$ respectively. Then
$I(\varphi,f,s,z)=\prod_{v}I(\varphi_{v},f_{v},s,z)$, with

$I(\varphi_{v},f_{v},s,z)=\int_{N_{Q_{\mathbf{H}}}(F_{v})Z_{\mathbf{H}}(F_{v})\setminus\mathbf{H}(F_{v})}\\!\\!\\!\\!\\!\\!f_{v}(wg_{v},s,z)W_{\varphi_{v}}(g_{v})dg_{v}.$

We are now ready to state our first main result. Let $(\Pi,V_{\Pi})$ be a
globally generic cuspidal automorphic representation of either
$\mathbf{P}\mathbf{G}({\bf A})$ or $\mathrm{PGL}_{4}({\bf A})$ and let
$\varphi=\otimes_{v}\varphi_{v}$ be a factorizable cusp form of $\Pi$. Let
$\Sigma$ be a finite set of places of $F$ containing the ramified places for
$\Pi$, for $E/F$ (when working with $\mathbf{P}\mathbf{G}$), and the
archimedean places of $F$. Fix a standard section $f=\otimes_{v}f_{v}\in
I_{B_{\mathbf{H}}}(s,z)$ such that $f_{v}$ is unramified at every
$v\not\in\Sigma$ with value $f_{v}(1,s,z)=1$.

######  Theorem 3.4.

We have

$I(\varphi,f,s,z)=\frac{L^{\Sigma}(s,\Pi,\wedge^{2})L^{\Sigma}(z,\Pi,\wedge^{2})}{\zeta_{F}^{\Sigma}(2z)\zeta_{F}^{\Sigma}(2s)\zeta_{F}^{\Sigma}(s+z)\zeta_{F}^{\Sigma}(s-z+1)}\prod_{v\in\Sigma}I(\varphi_{v},f_{v},s,z).$

Moreover, for given $s,z\in{\bf C}$ we can choose $\varphi_{v}$ and $f_{v}$ so
that $\prod_{v\in\Sigma}I(\varphi_{v},f_{v},s,z)$ is non-zero.

###### Proof.

This follows from Corollary 3.3, Theorem 3.7, Theorem 3.9, and Lemma 3.10. ∎

Finally, if we normalize
$f_{v}^{*}(g,s,z):=\zeta_{v}(2z)\zeta_{v}(2s)\zeta_{v}(s+z)\zeta_{v}(s-z+1)f_{v}(g,s,z)$
at every $v\not\in\Sigma$ and denote by $E^{*}_{B}(g,s,z)$ the corresponding
normalized Borel Eisenstein series, we can define

$I^{*}(\varphi,s,z):=\int_{\mathbf{H}(F)Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf
A})}\\!\\!\\!\\!\\!{E}^{*}_{B}(g,s,z)\varphi(g)dg.$

Theorem 3.4 shows that

$\displaystyle
I^{*}(\varphi,s,z)=L^{\Sigma}(s,\Pi,\wedge^{2})L^{\Sigma}(z,\Pi,\wedge^{2})\prod_{v\in\Sigma}I(\varphi_{v},f_{v},s,z).$
(8)

###  3.2. The unramified unipotent computation

Let $v$ be a finite place of $F$ which is unramified for $\Pi$, $f$, and
$E/F$. Recall that we have denoted $q_{v}$ the size of the quotient of
$\mathcal{O}_{v}$ modulo its maximal ideal $\mathfrak{p}_{v}$ and fixed a
uniformizer $\varpi_{v}$ of $\mathcal{O}_{v}$. The norm $|\cdot|$ of $F_{v}$
is normalized so that $|\varpi_{v}|=q_{v}^{-1}$. Let $W_{v}$ denote the vector
in the Whittaker model of $\Pi_{v}$ normalized so that it is trivial on
$\mathbf{G}(\mathcal{O}_{v})$; finally, recall that $f_{v}(1,s,z)=1$. Using
the Iwasawa decomposition
$\mathbf{H}(F_{v})=B_{\mathbf{H}}(F_{v})\mathbf{H}(\mathcal{O}_{v})$ and
identifying $N_{Q_{\mathbf{H}}}(F_{v})Z_{\mathbf{H}}(F_{v})\setminus
B_{\mathbf{H}}(F_{v})$ with

$L(F_{v}):=\left\\{\ell_{v}=n(b)m(a,d):=\left(\begin{smallmatrix}1&&&\\\
&1&b&\\\ &&1&\\\ &&&1\end{smallmatrix}\right)\left(\begin{smallmatrix}ad&&&\\\
&d&&\\\ &&1&\\\ &&&a^{-1}\end{smallmatrix}\right)\;a,d\in
F_{v}^{\times},\,b\in F_{v}\right\\},$

we write

$\displaystyle I(\varphi_{v},f_{v},s,z)$
$\displaystyle=\int_{L(F_{v})}\delta^{-1}_{B_{\mathbf{H}}}(\ell_{v})f_{v}(w\ell_{v},s,z)W_{v}(\ell_{v})d\ell_{v}$
$\displaystyle=\int_{(F_{v}^{\times})^{2}}|a|^{-4}|d|^{-3}W_{v}(m(a,d))\int_{F_{v}}f_{v}(w\,n(b)m(a,d),s,z)\psi_{v}(b)db\,d^{\times}a\,d^{\times}d$
$\displaystyle=\int_{(F_{v}^{\times})^{2}}|a|^{-4}|d|^{-2}W_{v}(m(a,d))\int_{F_{v}}f_{v}(w\,m(a,d)n(b),s,z)\psi_{v}(bd)db\,d^{\times}a\,d^{\times}d$
$\displaystyle=\int_{(F_{v}^{\times})^{2}}|a|^{s+z-3}|d|^{z-2}W_{v}(m(a,d))\int_{F_{v}}f_{v}(w\,n(b),s,z)\psi_{v}(bd)db\,d^{\times}a\,d^{\times}d.$

Note that $W_{v}(m(a,d))$ is zero unless the $v$-adic valuations of $a$ and
$d$ are non-negative. Moreover, if $b\neq 0$, the inner 2-by-2 block of
$w\,n(b)w^{-1}$ can be written as

$\left(\begin{smallmatrix}1&-b^{-1}\\\
&1\end{smallmatrix}\right)\left(\begin{smallmatrix}b^{-1}&\\\
&b\end{smallmatrix}\right)\left(\begin{smallmatrix}&1\\\
-1&b^{-1}\end{smallmatrix}\right).$

In particular, when $|b|>1$, it is right
$\mathrm{GL}_{2}(\mathcal{O}_{v})$-equivalent to
$\left(\begin{smallmatrix}1&-b^{-1}\\\
&1\end{smallmatrix}\right)\left(\begin{smallmatrix}b^{-1}&\\\
&b\end{smallmatrix}\right)$. Therefore

$f_{v}(w\,n(b),s,z)=\begin{cases}|b|^{z-s-1}&\text{ if }|b|>1\\\ 1&\text{ if
}|b|\leq 1.\end{cases}$

and the inner integral becomes

$\displaystyle 1+\int_{|b|>1}|b|^{z-s-1}\psi_{v}(bd)db$
$\displaystyle=1+\sum_{\ell=1}^{\infty}q_{v}^{\ell(z-s)}\int_{|b|=1}\psi_{v}(\varpi_{v}^{n-\ell}b)db$
$\displaystyle=1+{\rm
vol}(\mathcal{O}_{v}^{\times})\sum_{\ell=1}^{n}q_{v}^{\ell(z-s)}+\sum_{\ell=n+1}^{\infty}q_{v}^{\ell(z-s)}\int_{|b|=1}\psi_{v}(\varpi_{v}^{n-\ell}b)db.$

with $n$ being the $v$-adic valuation of $d$. As $\psi_{v}$ has conductor
$\mathcal{O}_{v}$, we have

$\int_{\mathfrak{p}_{v}^{i}}\psi_{v}(b)db=\begin{cases}{\rm
vol}(\mathfrak{p}_{v}^{i})&\text{ if }i\geq 0,\\\ 0&\text{
otherwise.}\end{cases}$

Therefore, in the last sum only the term for $\ell=n+1$ survives. As

$\int_{|b|=1}\psi_{v}(\varpi_{v}^{-1}b)db=\int_{\mathfrak{p}_{v}^{-1}}\psi_{v}(b)db-
q_{v}^{-1}\cdot\int_{\mathcal{O}_{v}}\psi_{v}(b)db=-q_{v}^{-1},$

we have

$\sum_{\ell=n+1}^{\infty}q_{v}^{\ell(z-s)}\int_{|b|=1}\psi_{v}(\varpi_{v}^{-1}b)db=-q_{v}^{-1}q_{v}^{(n+1)(z-s)}.$

This implies that the integral is

$\displaystyle=1+(1-q_{v}^{-1})\sum_{\ell=1}^{n}q_{v}^{\ell(z-s)}-q_{v}^{-1}q_{v}^{(n+1)(z-s)}$
$\displaystyle=1+(1-q_{v}^{-1})q_{v}^{z-s}\frac{1-q_{v}^{n(z-s)}}{1-q_{v}^{z-s}}-q_{v}^{-1}q_{v}^{(n+1)(z-s)}$
$\displaystyle=(1-q_{v}^{z-s-1})\frac{1-q_{v}^{(n+1)(z-s)}}{1-q_{v}^{z-s}}$
$\displaystyle=(1-q_{v}^{(n+1)(z-s)})\frac{\zeta_{v}(s-z)}{\zeta_{v}(s-z+1)}$

This reduces the local integral $I(\varphi_{v},f_{v},s,z)$ to

$\frac{\zeta_{v}(s-z)}{\zeta_{v}(s-z+1)}\sum_{m,n\geq
0}q_{v}^{-m(s+z-3)-n(z-2)}W_{v}(m(\varpi_{v}^{m},\varpi_{v}^{n}))(1-q_{v}^{(n+1)(z-s)}).$
(9)

Note that we have used that $W_{v}(m(\varpi_{v}^{m},\varpi_{v}^{n}))$ is zero
unless $m,n\geq 0$. In what follows, we use the Casselman–Shalika formula for
$W_{v}$ to calculate (9). We conveniently split the calculation into two cases
depending on whether $v$ is split or inert in $E$.

###  3.3. Unramified computation at split primes

Let $v\not\in\Sigma$ be a place which splits over $E$. As $\Pi_{v}$ has
trivial central character, we can assume that it is the unique unramified
subquotient of
$\mathrm{Ind}_{B_{\mathrm{GL}_{4}}(F_{v})}^{\mathrm{GL}_{4}(F_{v})}\chi$ with
$\chi$ an unramified character of $T_{\mathrm{GL}_{4}}(F_{v})$ such that
$\chi_{1}\chi_{2}\chi_{3}\chi_{4}=1$. As $\Pi_{v}$ has trivial central
character, we can write

$W_{v}(m(\varpi_{v}^{m},\varpi_{v}^{n}))=W_{v}({\rm
diag}(\varpi_{v}^{2m+n},\varpi_{v}^{m+n},\varpi_{v}^{m},1))$

Then, the Casselman–Shalika formula gives that

$\displaystyle W_{v}({\rm
diag}(\varpi_{v}^{2m+n},\varpi_{v}^{m+n},\varpi_{v}^{m},1))$
$\displaystyle=\delta_{B_{\mathrm{GL}_{4}}}^{1/2}({\rm
diag}(\varpi_{v}^{2m+n},\varpi_{v}^{m+n},\varpi_{v}^{m},1)){\rm
Tr}(s_{\Pi_{v}}|V_{(m,n,m)})$ $\displaystyle=q_{v}^{-3m-2n}{\rm
Tr}(s_{\Pi_{v}}|V_{(m,n,m)})$

where $V_{(m,n,m)}$ is the irreducible representation of ${\rm SL}_{4}({\bf
C})$ with highest weight $m\omega_{1}+n\omega_{2}+m\omega_{3}$. Here
$\omega_{1}$,$\omega_{2}$, and $\omega_{3}$ are the fundamental weights of
${\rm SL}_{4}({\bf C})$ and are the highest weights of the standard
representation, the exterior square representation, and the exterior cube
representation respectively.

Hence (9) looks like

$\frac{\zeta_{v}(s-z)}{\zeta_{v}(s-z+1)}\sum_{m,n\geq
0}q_{v}^{-m(s+z)-nz}\cdot{\rm
Tr}(s_{\Pi_{v}}|V_{(m,n,m)})(1-q_{v}^{(n+1)(z-s)}).$

If we let $X:=q_{v}^{-s}$ and $Y:=q_{v}^{-z}$, it reads as

(9) $\displaystyle=\frac{1}{\zeta_{v}(s-z+1)}\sum_{m,n\geq 0}{\rm
Tr}(s_{\Pi_{v}}|V_{(m,n,m)})X^{m}Y^{m+n}\frac{1-X^{n+1}Y^{-n-1}}{1-XY^{-1}}$
$\displaystyle=\frac{1}{\zeta_{v}(s-z+1)}\sum_{m,n\geq 0}{\rm
Tr}(s_{\Pi_{v}}|V_{(m,n,m)})X^{m}Y^{m+n}\sum_{\alpha=0}^{n}X^{\alpha}Y^{-\alpha}$
$\displaystyle=\frac{1}{\zeta_{v}(s-z+1)}\sum_{m,\alpha,\beta\geq 0}{\rm
Tr}(s_{\Pi_{v}}|V_{(m,\alpha+\beta,m)})X^{m+\alpha}Y^{m+\beta},$

where we did the change of variables writing $n=\alpha+\beta$. We manipulate
(9) further. Let $\boldsymbol{\zeta}_{v}(Z)$ denote $\sum_{k\geq 0}Z^{k}$. The
product between (9) and $\boldsymbol{\zeta}_{v}(XY)$ is

$\displaystyle=\frac{1}{\zeta_{v}(s-z+1)}\sum_{m,\alpha,\beta\geq
0}X^{m+\alpha}Y^{m+\beta}\sum_{a=0}^{m}{\rm
Tr}(s_{\Pi_{v}}|V_{(a,\alpha+\beta,a)})$
$\displaystyle=\frac{1}{\zeta_{v}(s-z+1)}\sum_{a,b,\alpha,\beta\geq 0}{\rm
Tr}(s_{\Pi_{v}}|V_{(a,\alpha+\beta,a)})X^{a+b+\alpha}Y^{a+b+\beta}$

After further multiplication by
$\boldsymbol{\zeta}_{v}(X^{2})\boldsymbol{\zeta}_{v}(Y^{2})$, we get

$\displaystyle\eqref{localform}=\frac{1}{\zeta_{v}(2z)\zeta_{v}(2s)\zeta_{v}(s+z)\zeta_{v}(s-z+1)}\sum_{a,b,c,d,e,f\geq
0}{\rm Tr}(s_{\Pi_{v}}|V_{(a,c+e,a)})X^{a+b+c+2d}Y^{a+b+e+2f}$ (10)

Let $\boldsymbol{L}(Z,\Pi_{v},\wedge^{2})$ denote $\sum_{k\geq 0}{\rm
Tr}(s_{\Pi_{v}}|{\rm Sym}^{k}(V_{(0,1,0)}))Z^{k}$. Our goal is to compare the
right hand side of (10) with the product

$\boldsymbol{L}(X,\Pi_{v},\wedge^{2})\boldsymbol{L}(Y,\Pi_{v},\wedge^{2}).$

We first need two lemmas.

######  Lemma 3.5.

For any $k\geq 0$, we have the following isomorphism

$\mathrm{Sym}^{k}\left(V_{(0,1,0)}\right)=\bigoplus_{i=0}^{\lfloor
k/2\rfloor}V_{(0,k-2i,0)}.$

###### Proof.

For all $k\geq 2$, there is a surjection

$\mathrm{Sym}^{k}\left(V_{(0,1,0)}\right)\longrightarrow\mathrm{Sym}^{k-2}\left(V_{(0,1,0)}\right)$

with kernel $V_{(0,k,0)}$ (cf. [6, Exercise 15.12]). The result follows from
applying the surjection recursively. ∎

######  Lemma 3.6.

For any $n,m\geq 0$, we have

$V_{(0,n,0)}\otimes
V_{(0,m,0)}=\bigoplus_{\begin{subarray}{c}m_{1},m_{2},m_{3}\geq 0\\\ \sum
m_{i}=m\\\ m_{2}+m_{3}\leq
n\end{subarray}}\\!\\!\\!\\!V_{(m_{2},n+m_{1}-m_{2}-m_{3},m_{2})}.$

###### Proof.

We apply the Littlewood–Richardson rule. If we let $D_{t}$ denote the Young
diagram with rows of size $(t,t)$, we have

$V_{(0,n,0)}\otimes V_{(0,m,0)}=\oplus_{\lambda}N_{n,m,\lambda}V_{\lambda},$

where $N_{n,m,\lambda}$ is the number of strict expansions of $D_{n}$ by
$D_{m}$ of shape the Young diagram of $V_{\lambda}$. In our case
$N_{n,m,\lambda}$ is either zero or one; hence we are left to determine the
set of strict expansions of $D_{n}$ by $D_{m}$. Recall that an expansion of
$D_{n}$ by $D_{m}$ is a Young diagram obtained by first adding $m$ boxes and
putting the integer $1$ in each of these so that no two 1’s are in the same
column and then adding other $m$ boxes with a 2 so that no two 2’s are in the
same column. Such expansion is called strict if when reading from right to
left, starting with the top row and working down, for any $1\leq t\leq 2m$ the
number of 1’s up to position $t$ is greater or equal to the number of 2’s up
to position $t$. It follows that the only way we can add the first $m$ boxes
is by getting the Young diagram with rows of size $(n+\ell_{1},n,\ell_{2})$
such that $\ell_{1}+\ell_{2}=m$ and $\ell_{2}\leq n$. If we let
$(m_{1},m_{2},m_{3})$ be any partition of $m$, the $m$ boxes labeled by $2$
can be added by appending $m_{1}$ boxes to the second row, $m_{2}$ to the
third, and $m_{3}$ to the fourth so that $\ell_{1}\geq m_{1}+m_{2}$,
$\ell_{2}\geq m_{3}$, $\ell_{2}+m_{2}\leq n$. This implies that
$\ell_{1}=m_{1}+m_{2}$ and $\ell_{2}=m_{3}$, hence any strict expansion of
$D_{n}$ by $D_{m}$ is determined by choosing a partition $(m_{1},m_{2},m_{3})$
of $m$ such that $m_{2}+m_{3}\leq n$. The irreducible representation of
$\mathrm{SL}_{4}({\bf C})$ corresponding to the strict expansion of $D_{n}$ by
$D_{m}$ associated to $(m_{1},m_{2},m_{3})$ is
$V_{(m_{2},n+m_{1}-m_{2}-m_{3},m_{2})}$. This proves the formula. ∎

######  Theorem 3.7.

We have that

$I(\varphi_{v},f_{v},s,z)=\frac{L(s,\Pi_{v},\wedge^{2})L(z,\Pi_{v},\wedge^{2})}{\zeta_{v}(2z)\zeta_{v}(2s)\zeta_{v}(s+z)\zeta_{v}(s-z+1)}.$

In particular, if we normalize
$f_{v}^{*}(g,s,z):=\zeta_{v}(2z)\zeta_{v}(2s)\zeta_{v}(s+z)\zeta_{v}(s-z+1)f_{v}(g,s,z)$,

$I(\varphi_{v},f_{v}^{*},s,z)=L(s,\Pi_{v},\wedge^{2})L(z,\Pi_{v},\wedge^{2}).$

###### Proof.

By Lemma 3.5, the coefficient of $X^{m}Y^{n}$ of
$\boldsymbol{L}(X,\Pi_{v},\wedge^{2})\boldsymbol{L}(Y,\Pi_{v},\wedge^{2})$ is
given by

$\sum_{i=0}^{\lfloor\frac{m}{2}\rfloor}\sum_{j=0}^{\lfloor\frac{n}{2}\rfloor}{\rm
Tr}(s_{\Pi_{v}}|V_{(0,m-2i,0)})\mathrm{Tr}(s_{\Pi_{v}}|V_{(0,n-2j,0)}).$

By Lemma 3.6, we thus have that

$\boldsymbol{L}(X,\Pi_{v},\wedge^{2})\boldsymbol{L}(Y,\Pi_{v},\wedge^{2})=\sum_{m,n\geq
0}X^{m}Y^{n}\sum_{i=0}^{\lfloor\frac{m}{2}\rfloor}\sum_{j=0}^{\lfloor\frac{n}{2}\rfloor}\sum_{\begin{subarray}{c}t_{1},t_{2},t_{3}\geq
0\\\ \sum t_{i}=n-2j\\\ t_{2}+t_{3}\leq m-2i\end{subarray}}{\rm
Tr}(s_{\Pi_{v}}|V_{(t_{2},m-2i-t_{2}-t_{3}+t_{1},t_{2})}).$

Do a change of variables by $m=r_{1}+t_{2}+t_{3}+2i$ and
$n=t_{1}+t_{2}+t_{3}+2j$ to get that

$\boldsymbol{L}(X,\Pi_{v},\wedge^{2})\boldsymbol{L}(Y,\Pi_{v},\wedge^{2})=\sum_{r_{1},t_{1},t_{2},t_{3},i,j\geq
0}{\rm
Tr}(s_{\Pi_{v}}|V_{(t_{2},r_{1}+t_{1},t_{2})})X^{r_{1}+t_{2}+t_{3}+2i}Y^{t_{1}+t_{2}+t_{3}+2j}.$

This equals the infinite sum in the right hand side of (10). The theorem
follows. ∎

###  3.4. Unramified computation at inert primes

Let $v\not\in\Sigma$ be a place which is inert over $E$ and let us assume that
$\Pi_{v}$ is the unique unramified subquotient of
$\mathrm{Ind}_{B_{\mathbf{G}}(F_{v})}^{\mathbf{G}(F_{v})}\chi$ with $\chi$ an
unramified character of $T_{\mathbf{G}}(F_{v})$. Denote by $\pi_{v}$ the
unique unramified subquotient of
$\mathrm{Ind}_{B_{\mathbf{H}}(F_{v})}^{\mathbf{H}(F_{v})}\chi_{|_{T_{\mathbf{H}}}}$.
As $\Pi_{v}$ has trivial central character, we can write

$W_{v}(m(\varpi_{v}^{m},\varpi_{v}^{n}))=W_{v}({\rm
diag}(\varpi_{v}^{2m+n},\varpi_{v}^{m+n},\varpi_{v}^{m},1)).$

As the relative root system of $\mathbf{G}_{/F_{v}}$ is that of
$\mathbf{H}_{/F_{v}}$, the Casselman–Shalika formula for the Whittaker
coefficient $W_{v}$ of $\Pi_{v}$ gives that (cf. [8, Proposition 5.2.1])

$\displaystyle W_{v}({\rm
diag}(\varpi_{v}^{2m+n},\varpi_{v}^{m+n},\varpi_{v}^{m},1))$
$\displaystyle=\delta_{B_{\mathbf{G}}}^{1/2}({\rm
diag}(\varpi_{v}^{2m+n},\varpi_{v}^{m+n},\varpi_{v}^{m},1)){\rm
Tr}(s_{\pi_{v}}|W_{(m,n)})$ $\displaystyle=q_{v}^{-3m-2n}{\rm
Tr}(s_{\pi_{v}}|W_{(m,n)})$

where $W_{(m,n)}$ is the irreducible representation of ${\rm Spin}_{5}({\bf
C})$ with highest weight $m\omega_{1}+n\omega_{2}$. Here $\omega_{1}$ and
$\omega_{2}$ are the fundamental weights of ${\rm Spin}_{5}({\bf C})$ and
correspond to the five-dimensional standard orthogonal representation and to
the four-dimensional spin representation, respectively. Hence, if we let
$X:=q_{v}^{-s}$ and $Y:=q_{v}^{-z}$, (9) is

$\frac{1}{\zeta_{v}(s-z+1)}\sum_{m,n\geq 0}{\rm
Tr}(s_{\pi_{v}}|W_{(m,n)})X^{m}Y^{n+m}\frac{1-X^{n+1}Y^{-n-1}}{1-XY^{-1}}.$

After multiplying and dividing by $\boldsymbol{\zeta}(XY)$, we have

(9)
$\displaystyle=\frac{1}{\zeta_{v}(z+s)\zeta_{v}(s-z+1)}\sum_{a,b,\alpha,\beta\geq
0}{\rm Tr}(s_{\pi_{v}}|W_{(a,\alpha+\beta)})X^{a+b+\alpha}Y^{a+b+\beta},$ (11)

where we have written $m=a+b$ and $n=\alpha+\beta$. Let
$\boldsymbol{L}(Z,\pi_{v},{\rm Spin})$ denote $\sum_{k\geq 0}{\rm
Tr}(s_{\Pi_{v}}|{\rm Sym}^{k}(W_{(0,1)}))Z^{k}$. We now compare the right hand
side of (11) with the product

$\boldsymbol{L}(X,\pi_{v},{\rm Spin})\boldsymbol{L}(Y,\pi_{v},{\rm Spin}),$

which we calculate as follows. Making use of the exceptional isomorphism
$\mathrm{Spin}_{5}({\bf C})\xrightarrow{\simeq}\mathrm{Sp}_{4}({\bf C})$, the
explicit branching law for

$\mathrm{Spin}_{5}({\bf C})\xrightarrow{\simeq}\mathrm{Sp}_{4}({\bf
C})\hookrightarrow{\mathbf{SL}}_{4}({\bf C})$

reads as (cf. [3, p. 739])

$\displaystyle V_{(u,v,w)}=\bigoplus_{s=0}^{v}\bigoplus_{t=0}^{{\rm
min}(w,u)}W_{(s+t,u+w-2t)},$ (12)

where $V_{(u,v,w)}$ denotes the irreducible representation of
${\mathbf{SL}}_{4}({\bf C})$ as in §3.3. As a special case, we see that ${\rm
Sym}^{n}(V_{(1,0,0)})=V_{(n,0,0)}=W_{(0,n)}={\rm Sym}^{n}(W_{(0,1)})$ for
every $n\geq 0$.

######  Lemma 3.8.

For any $n,m\geq 0$, we have

$W_{(0,n)}\otimes W_{(0,m)}=\bigoplus_{\begin{subarray}{c}m_{1},m_{2}\geq 0\\\
m_{1}+m_{2}=m\\\ m_{2}\leq
n\end{subarray}}\bigoplus_{s=0}^{m_{2}}W_{(s,n+m_{1}-m_{2})}.$

###### Proof.

We start by computing the tensor product of $V_{(n,0,0)}$ with $V_{(m,0,0)}$
by the Littlewood-Richardson rule. Let $Y_{t}$ denote the Young diagram with a
row of size $t$. Each strict expansion of $Y_{n}$ by $Y_{m}$ correspond to a
pair of positive integers $(m_{1},m_{2})$ such that $m_{1}+m_{2}=m$ and
$m_{2}\leq n$ and it gives the Young diagram of $V_{(m_{1}+n-m_{2},m_{2},0)}$.
Hence, we have

$V_{(n,0,0)}\otimes V_{(m,0,0)}=\bigoplus_{\begin{subarray}{c}m_{1},m_{2}\geq
0\\\ m_{1}+m_{2}=m\\\ m_{2}\leq
n\end{subarray}}\\!\\!\\!\\!V_{(n+m_{1}-m_{2},m_{2},0)}.$

This and (12) give the desired formula. ∎

######  Theorem 3.9.

We have that

$I(\varphi_{v},f_{v},s,z)=\frac{L(s,\Pi_{v},\wedge^{2})L(z,\Pi_{v},\wedge^{2})}{\zeta_{v}(2z)\zeta_{v}(2s)\zeta_{v}(s+z)\zeta_{v}(s-z+1)}.$

In particular, if we normalize
$f_{v}^{*}(g,s,z):=\zeta_{v}(2z)\zeta_{v}(2s)\zeta_{v}(s+z)\zeta_{v}(s-z+1)f_{v}(g,s,z)$,

$I(\varphi_{v},f_{v}^{*},s,z)=L(s,\Pi_{v},\wedge^{2})L(z,\Pi_{v},\wedge^{2}).$

###### Proof.

By Lemma 3.8, the coefficient of $X^{m}Y^{n}$ of
$\boldsymbol{L}(X,\pi_{v},{\rm Spin})\boldsymbol{L}(Y,\pi_{v},{\rm Spin})$ is
given by

$\sum_{\begin{subarray}{c}m_{1},m_{2}\geq 0\\\ m_{1}+m_{2}=m\\\ m_{2}\leq
n\end{subarray}}\sum_{s=0}^{m_{2}}{\rm
Tr}(s_{\pi_{v}}|W_{(s,n+m_{1}-m_{2})}).$

Write $m_{2}=s+r$, $n=t+s+r$, and $m=m_{1}+s+r$, then this implies that

$\boldsymbol{L}(X,\pi_{v},{\rm Spin})\boldsymbol{L}(Y,\pi_{v},{\rm
Spin})=\sum_{m_{1},s,r,t\geq 0}{\rm
Tr}(s_{\pi_{v}}|W_{(s,m_{1}+t)})X^{s+r+m_{1}}Y^{s+r+t}.$

This is exactly the infinite sum in the right hand side of (11). Then the
theorem follows from (3), which says that, for $s^{\prime}\in\\{s,z\\}$

$L(s^{\prime},\Pi_{v},\wedge^{2})=\zeta_{v}(2s^{\prime})L(s^{\prime},\pi_{v},{\rm
Spin}).$

∎

###  3.5. Ramified computations

Let $\Sigma_{\infty}\subseteq\Sigma$ denote the set of archimedean places of
$F$.

######  Lemma 3.10.

  1. (1)

For any place $v$, there exists $\varepsilon>0$ such that
$I(\varphi_{v},f_{v},s,z)$ converges absolutely for ${\rm
Re}(z)>1-\varepsilon$ and ${\rm Re}(s)>{\rm Re}(z)-4$. Moreover,
$\prod_{v\in\Sigma}I(\varphi_{v},f_{v},s,z)$ has meromorphic continuation in
$s$ and $z$.

  2. (2)

For any finite place $v$, there exist $\varphi_{v}$ and $f_{v}(g,s,z)$ such
that $I(\varphi_{v},f_{v},s,z)$ is a non-zero constant independent of $s$ and
$z$.

  3. (3)

For any place $v\in\Sigma_{\infty}$, given $s_{0},z_{0}\in{\bf C}$, one can
choose $\varphi_{v}$ and $f_{v}(g,s,z)$ such that $I(\varphi_{v},f_{v},s,z)$
is non-zero at $(s_{0},z_{0})$.

###### Proof.

We will reduce our first claim to the study in [7, §3.1] of the local
integrals [7, (3.1)]. To do so, we choose the section defining the Borel
Eisenstein series as follows. Let $v$ be any place of $F$. Let $W_{4}$ be the
standard representation of $\mathbf{H}(F_{v})$, with basis
$\\{e_{1},e_{2},f_{2},f_{1}\\}$ and let $W_{1,1}$ be the five-dimensional
representation of $\mathbf{H}(F_{v})$ of highest weight $(1,1)$. It sits
inside the exterior square of $W_{4}$ and in particular contains the vector
$f_{2}\wedge f_{1}$. For a Schwartz–Bruhat function $\Phi_{P}$ on $W_{1,1}$,
define the function on $\mathbf{H}(F_{v})$ given by

$f_{P_{\mathbf{H}}}(\Phi_{P},g,s-z+1):=|\mu(g)|^{s-z+1}\int_{F^{\times}_{v}}\Phi_{P}(tf_{2}\wedge
f_{1}g)|t|^{2(s-z+1)}dt\in
I_{P_{\mathbf{H}}(F_{v})}^{\mathbf{H}(F_{v})}(\delta_{P_{\mathbf{H}}}^{\frac{1}{3}(s-z+1)}).$

Similarly, for a Schwartz–Bruhat function $\Phi_{Q}$ on $M_{1\times
4}(F_{v})$, where we let $\mathbf{H}(F_{v})$ acts on it by right
multiplication, define the function on $\mathbf{H}(F_{v})$ given by

$f_{Q_{\mathbf{H}}}(\Phi_{Q},g,z):=|\mu(g)|^{z}\int_{F^{\times}_{v}}\Phi_{Q}(tf_{1}g)|t|^{2z}dt\in
I_{Q_{\mathbf{H}}(F_{v})}^{\mathbf{H}(F_{v})}(\delta_{Q_{\mathbf{H}}}^{\frac{1}{2}z}).$

Here
$I_{P_{\mathbf{H}}(F_{v})}^{\mathbf{H}(F_{v})}(\delta_{P_{\mathbf{H}}}^{\frac{1}{3}(s-z+1)})$
and
$I_{Q_{\mathbf{H}}(F_{v})}^{\mathbf{H}(F_{v})}(\delta_{Q_{\mathbf{H}}}^{\frac{1}{2}z})$
are not normalized inductions. It’s easy to see that
$f_{v}(g,s,z):=f_{P_{\mathbf{H}}}(\Phi_{P},g,s-z+1)f_{Q_{\mathbf{H}}}(\Phi_{Q},g,z)\in
I_{B_{\mathbf{H}}(F_{v})}(s,z)$. Call $\tilde{P}$ the subgroup of
$P_{\mathbf{H}}$ given by $\mathrm{GL}_{2}N_{P_{\mathbf{H}}}$, with
$\mathrm{GL}_{2}\hookrightarrow M_{P_{\mathbf{H}}}$ via $h\mapsto
h^{*}:=\left(\begin{smallmatrix}h&\\\ &{\rm
det}(h)J_{2}{}^{t}h^{-1}J_{2}\end{smallmatrix}\right)$. With this choice of
section and collapsing the sum with respect to
$\tilde{P}_{w}(F_{v}):=w^{-1}\tilde{P}(F_{v})w$, the local integral

$\displaystyle I(\varphi_{v},f_{v},s,z)$
$\displaystyle=\int_{\tilde{P}_{w}(F_{v})\setminus\mathbf{H}(F_{v})}\int_{N_{Q_{\mathbf{H}}}(F_{v})Z_{\mathbf{H}}(F_{v})\setminus\tilde{P}_{w}(F_{v})}\\!\\!\\!\\!\\!\\!f_{v}(wp_{w}h,s,z)W_{\varphi_{v}}(p_{w}h)dp_{w}dh$
$\displaystyle=\int_{\cdots}\int_{N_{Q_{\mathbf{H}}}(F_{v})Z_{\mathbf{H}}(F_{v})\setminus\tilde{P}_{w}(F_{v})}\\!\\!\\!\\!\\!\\!f_{P_{\mathbf{H}}}(\Phi_{P},pwh,s-z+1)f_{Q_{\mathbf{H}}}(\Phi_{Q},p_{w}h,z)W_{\varphi_{v}}(p_{w}h)dp_{w}dh$
$\displaystyle=\int_{\cdots}f_{P_{\mathbf{H}}}(\Phi_{P},wh,s-z+1)\int_{N_{Q_{\mathbf{H}}}(F_{v})Z_{\mathbf{H}}(F_{v})\setminus\tilde{P}_{w}(F_{v})}\\!\\!\\!\\!\\!\\!f_{Q_{\mathbf{H}}}(\Phi_{Q},p_{w}h,z)W_{\varphi_{v}}(p_{w}h)dp_{w}dh,$

where we have written $p_{w}=w^{-1}pw$ for $p\in\tilde{P}(F_{v})$, we have
used that $w$ acts trivially on $f_{Q_{\mathbf{H}}}$ and in the last equality
that the modulus character of the Siegel parabolic is trivial on
$\tilde{P}(F_{v})$. We now look at the inner integral. The domain
$N_{Q_{\mathbf{H}}}(F_{v})Z_{\mathbf{H}}(F_{v})\setminus\tilde{P}_{w}(F_{v})$
is isomorphic to
$\left(Z_{\mathrm{GL}_{2}}(F_{v})U_{\mathrm{GL}_{2}}(F_{v})\setminus\mathrm{GL}_{2}(F_{v})\right)\cdot
F_{v}$, where $\mathrm{GL}_{2}(F_{v})$ and $F_{v}$ embed into
$\mathbf{H}(F_{v})$ as

$\iota:\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\mapsto\left(\begin{smallmatrix}a&&b&\\\ &a&&b\\\
c&&d&\\\
&c&&d\end{smallmatrix}\right),\,u:\alpha\mapsto\left(\begin{smallmatrix}1&&&\\\
&1&&\\\ &-\alpha&1&\\\ &&&1\end{smallmatrix}\right).$

This let us write the inner integral as

$\displaystyle\int_{Z_{\mathrm{GL}_{2}}(F_{v})U_{\mathrm{GL}_{2}}(F_{v})\setminus\mathrm{GL}_{2}(F_{v})}\int_{F_{v}}f_{Q_{\mathbf{H}}}(\Phi_{Q},\iota(g)h,z)W_{\varphi_{v}}(u(\alpha)\iota(g)h)d\alpha\,dg,$
(13)

where we used that $u(\alpha)$ is in the Levi of the Klingen parabolic and its
image via $\delta_{Q_{\mathbf{H}}}$ is trivial, so that it acts trivially on
$f_{Q_{\mathbf{H}}}$. Note that for certain $\Phi_{Q}$, (13) coincides with
the local zeta integral of [7, (3.1)]. Let $P_{w}:=w^{-1}P_{\mathbf{H}}w$; by
the Iwasawa decomposition, there is a compact subgroup $K_{w}$ of
$\mathbf{H}(F_{v})$ such that $P_{w}(F_{v})K_{w}=\mathbf{H}(F_{v})$. Moreover,
the quotient $\mathrm{GL}_{1}\simeq\tilde{P}_{w}\backslash P_{w}$ via the map

$\iota^{\prime}:r\mapsto\left(\begin{smallmatrix}1&&&\\\ &r&&\\\ &&1&\\\
&&&r\end{smallmatrix}\right).$

Note that $\iota^{\prime}(r)\iota(g)=\iota(g)\iota^{\prime}(r)$. Thus, using
this and doing the change of variables
$u(\alpha)\mapsto\iota^{\prime}(r)^{-1}u(\alpha)\iota^{\prime}(r)$, we write
$I(\varphi_{v},f_{v},s,z)$ as

$\displaystyle\int_{K_{w}}\\!\\!\\!\\!f_{P_{\mathbf{H}}}(\Phi_{P},wk,s-z+1)\int_{Z_{2}U_{2}\setminus\mathrm{GL}_{2}}\int_{F_{v}}\\!\\!\\!\\!f_{Q_{\mathbf{H}}}(\Phi_{Q},\iota(g)k,z)\int_{F_{v}^{\times}}\\!\\!\\!\\!\\!\\!|r|^{-s-2}W_{\varphi_{v}}(\iota^{\prime}(r)u(\alpha)\iota(g)k)dr\,d\alpha\,dg\,dk.$

We now apply Dixmier–Malliavin theorem to the maximal unipotent subgroup $N$
of the Levi of $Q_{\mathbf{H}}$ to write
$\varphi_{v}=\sum_{j}\xi_{j}*\varphi_{v,j}$ where $\varphi_{v,j}$ are smooth
vectors of $\Pi_{v}$ and $\xi_{j}\in C^{\infty}_{c}(N(F_{v}))$. Using the
defining property of the Whittaker functional, we have

$\int_{F_{v}^{\times}}\\!\\!\\!\\!\\!\\!|r|^{-s-2}W_{\varphi_{v}}(\iota^{\prime}(r)u(\alpha)\iota(g)k)dr=\sum_{j}\int_{F_{v}^{\times}}\\!\\!\\!\\!\\!\\!|r|^{-s-2}\hat{\xi}_{j}(r^{-1})W_{\varphi_{v,j}}(\iota^{\prime}(r)u(\alpha)\iota(g)k)dr,$

where

$\hat{\xi}_{j}(r)=\int_{N(F_{v})}\xi_{j}\left(\left(\begin{smallmatrix}1&n\\\
&1\end{smallmatrix}\right)\right)\psi_{v}(rn)dn.$

Note that $\hat{\xi}_{j}$ are compactly supported functions on
$F_{v}^{\times}$. Our first claim now follows from [7, Proposition 3.2].
Indeed, using the Iwasawa decomposition for
$\mathrm{GL}_{2}(F_{v})=U_{2}TK_{2}$ where
$T=\left\\{\left(\begin{smallmatrix}t&\\\
&1\end{smallmatrix}\right)\right\\}$, and [7, Proposition 3.1], a direct
computation shows that each integral is majorated by

$\left(\int_{F_{v}^{\times}}|\hat{\xi}_{j}(r^{-1})||r|^{-s+z-4}dr\right)\cdot\left(\int_{K_{w}}\int_{K_{2}}\int_{F_{v}^{\times}}\int_{F_{v}}\\!\\!\\!\\!|t|^{z-1}|W_{\varphi_{v,j}}(u(\alpha)\iota(t)k_{2}k)|d\alpha\,dt\,dk_{2}\,dk\right).$

The first converges for ${\rm Re}(s-z+4)>0$, while by [7, Proposition 3.2]
there exists $\varepsilon>0$ such that the second converges for ${\rm
Re}(z)>1-\varepsilon$. This proves the first statement of (1). The meromorphic
continuation of $\prod_{v\in\Sigma}I_{v}(\varphi_{v},f_{v},s,z)$ follows from
our unramified computations as $I(\varphi,f,s,z)$ and the $L$-function
$L^{\Sigma}(s,\Pi,\wedge^{2})$ have meromorphic continuation (cf. [13, §7]).
The statements (2) and (3) follow from applying the Dixmier–Malliavin theorem
as in [9, Lemma 9.1]. ∎

##  4\. Residues and the degree 5 $L$-function of ${\mathrm{GSp}}_{4}$

###  4.1. A first-term identity between Eisenstein series for parabolics of
${\mathrm{GSp}}_{4}$

####  4.1.1. Siegel Eisenstein series on ${\mathrm{GSp}}_{4}$

We define the Siegel Eisenstein series associated to $\mathbf{H}$ and
recollect some of their properties (for instance, see [5, §5]). Recall from
§2.1.2 the Siegel parabolic $P_{\mathbf{H}}$ of $\mathbf{H}$ with Levi
decomposition $M_{P_{\mathbf{H}}}N_{P_{\mathbf{H}}}$ and modulus character
$\delta_{P_{\mathbf{H}}}$. Let $s$ be a complex variable and let
$I_{P_{\mathbf{H}}}(s)$ be the degenerate principal series representation of
$\mathbf{H}({\bf A})$ consisting of smooth functions $f$ on $\mathbf{H}({\bf
A})$ such that

$f(nmg,s)=\delta_{P_{\mathbf{H}}}^{\tfrac{1}{3}(s+1)}(m)f(g,s),\;\forall n\in
N_{P_{\mathbf{H}}}({\bf A}),\forall m\in M_{P_{\mathbf{H}}}({\bf A}).$

Given a standard holomorphic section $f\in I_{P_{\mathbf{H}}}(s)$, we define
the degenerate Eisenstein series

${\rm
Eis}_{P_{\mathbf{H}}}(g,s,f)=\sum_{\gamma\in{P_{\mathbf{H}}}(F)\backslash\mathbf{H}(F)}f(\gamma
g,s),$

which is absolutely convergent in the half-plane ${\rm Re}(s)>2$ and admits
analytic continuation to a meromorphic function on ${\bf C}$ which satisfies a
functional equation

${\rm Eis}_{P_{\mathbf{H}}}(g,s,f)={\rm Eis}_{P_{\mathbf{H}}}(g,1-s,M_{s}f),$

where $M_{s}$ is a certain intertwining operator. At a finite place $v$, let
$f_{v}^{0}$ be the function in $I_{P_{\mathbf{H}}(F_{v})}(s)$ such that
$f_{v}^{0}(k,s)=1$ for all $k\in\mathbf{H}(\mathcal{O}_{v})$. We assume that
$f$ is a pure tensor and denote by $f_{v}$ its $v$-adic component, which
equals $f_{v}^{0}$ almost everywhere. Let $S$ be the finite set of places
containing the archimedean places and all $v$ such that $f_{v}\neq f_{v}^{0}$.
We then define the normalized Siegel Eisenstein series to be equal to

$E^{*}_{P_{\mathbf{H}}}(g,s):=\zeta^{S}(s+1)\zeta^{S}(2s){\rm
Eis}_{P_{\mathbf{H}}}(g,s,f).$ (14)

It follows from [14, Theorem 1.1 and Theorem 4.12] that
$E^{*}_{P_{\mathbf{H}}}(g,s)$ has at most simple poles at the points $s=1,2$.
While $s=2$ is the end of the critical strip and so the residue of the
Eisenstein series is the constant representation, i.e. there exists $f\in
I_{P_{\mathbf{H}}}(s)$ such that ${\rm Res}_{s=2}E_{P_{\mathbf{H}}}^{*}(g,s)$
is constant, the residue of $E^{*}_{P_{\mathbf{H}}}(g,s)$ at $s=1$ is related
to a value of a Klingen Eisenstein series by a result of Ikeda described in
Proposition 4.1.

####  4.1.2. The Klingen Eisenstein series and the first-term identity

The Klingen parabolic $Q_{\mathbf{H}}=M_{Q_{\mathbf{H}}}N_{Q_{\mathbf{H}}}$
with modulus character $\delta_{Q_{\mathbf{H}}}$ was introduced in §2.1.2. Let
$I_{Q_{\mathbf{H}}}(s)$ be the degenerate principal series representation of
$\mathbf{H}({\bf A})$ consisting of smooth functions $f^{\prime}$ on
$\mathbf{H}({\bf A})$ such that

$f^{\prime}(nmg,s)=\delta_{Q_{\mathbf{H}}}^{\tfrac{1}{2}(s+1/2)}(m)f^{\prime}(g,s),\;\forall
n\in N_{Q_{\mathbf{H}}}({\bf A}),\forall m\in M_{Q_{\mathbf{H}}}({\bf A}).$

Given a smooth section $f^{\prime}\in I_{Q_{\mathbf{H}}}(s)$, we define the
degenerate (Klingen) Eisenstein series

${\rm
Eis}_{Q_{\mathbf{H}}}(g,s,f^{\prime})={\sum_{\gamma\in{Q_{\mathbf{H}}}(F)\backslash\mathbf{H}(F)}}f^{\prime}(\gamma
g,s).$

In the study of the poles of our integral we will employ the following result
of Ikeda.

######  Proposition 4.1.

There exists a section $f_{1/2}^{\prime}\in I_{Q_{\mathbf{H}}}(1/2)$ such that

${\rm Res}_{s=1}E_{P_{\mathbf{H}}}^{*}(g,s)=\frac{{\rm
Res}_{s=1}(\zeta^{S}_{F}(s))\zeta^{S}_{F}(3)}{2\zeta^{S}_{F}(2)}{\rm
Eis}_{Q_{\mathbf{H}}}(g,1/2,f_{1/2}^{\prime}).$

###### Proof.

This is basically [12, Proposition 1.10]. Notice that the result of _loc.cit._
is for ${\mathrm{Sp}}_{4}$-Eisenstein series. In order to use it, we consider
the following. By [10, p. 11], we have a factorization

$\mathbf{H}({\bf A})=\mathbf{H}(F)Z_{\mathbf{H}}({\bf
A}){\mathrm{Sp}}_{4}({\bf A})\mathbf{H}({\bf A}_{f}).$

Then, given any Eisenstein series $E(g,s,f)$ defined over $\mathbf{H}({\bf
A})$, we may write

$E(g,s,f)=E(g_{0}zg_{1}g_{f},s,f)=E(g_{1}g_{f},s,f)=E(g_{1},s,r(g_{f})f),$

by the invariant properties of the Eisenstein series, and where $r(g_{f})$
denotes the right multiplication action of $g_{f}$ on $f$. In particular, the
values of the Eisenstein series $E_{P_{\mathbf{H}}}(g,s,f)$ and ${\rm
Eis}_{Q_{\mathbf{H}}}(g,s,f^{\prime})$ at $g$ equal to the values at $g_{1}$
of the Eisenstein series $E_{P_{{\mathrm{Sp}}_{4}}}(g_{1},s,r(g_{f})f)$,
$E_{Q_{{\mathrm{Sp}}_{4}}}(g_{1},s,r(g_{f})f^{\prime})$ for
${\mathrm{Sp}}_{4}$ associated respectively to the parabolics
$P_{{\mathrm{Sp}}_{4}}=P_{\mathbf{H}}\cap{\mathrm{Sp}}_{4}$ and
$Q_{{\mathrm{Sp}}_{4}}=Q_{\mathbf{H}}\cap{\mathrm{Sp}}_{4}$, and to the
elements $r(g_{f})f\in I_{P({\bf A})}^{{\mathrm{Sp}}_{4}({\bf
A})}\nu_{P}^{s+1}$, $r(g_{f})f^{\prime}\in I_{Q({\bf
A})}^{{\mathrm{Sp}}_{4}({\bf A})}\nu_{Q}^{2s+1}$, where the inductions are not
normalized. Here we denoted

$\displaystyle\nu_{P}:$ $\displaystyle\left(\begin{smallmatrix}g&\star\\\
&J_{2}{}^{t}g^{-1}J_{2}\end{smallmatrix}\right)\mapsto\left|{\rm
det}(g)\right|,$ $\displaystyle\nu_{Q}:$
$\displaystyle\left(\begin{smallmatrix}a&&\\\ &g&\\\
&&a^{-1}\end{smallmatrix}\right)\mapsto\left|a\right|.$

With respect to [12], the Eisenstein series
$E_{P_{{\mathrm{Sp}}_{4}}}(g_{1},s,r(g_{f})f)$, resp.
$E_{Q_{{\mathrm{Sp}}_{4}}}(g_{1},s,r(g_{f})f^{\prime})$ have complex variable
$s$ shifted by $-1/2$. We now apply [12, Proposition 1.10], which says that,
given $\omega$ the longest element of the quotient of Weyl groups
$W_{M_{P_{\mathbf{H}}}}\backslash W_{\mathbf{H}}$ and $M_{\omega}$ the
corresponding intertwining operator, then

${\rm
Res}_{s=1}E_{P_{{\mathrm{Sp}}_{4}}}(g_{1},s,r(g_{f})f)=\tfrac{1}{2}E_{Q_{{\mathrm{Sp}}_{4}}}(g_{1},1/2,{\rm
Res}_{s=1}M_{\omega}(r(g_{f})f)).$

As $M_{\omega}$ is an intertwining map, the action of $g_{f}$ commutes with
it, hence we have

${\rm
Res}_{s=1}E_{P_{{\mathrm{Sp}}_{4}}}(g_{1},s,r(g_{f})f)=\tfrac{1}{2}E_{Q_{{\mathrm{Sp}}_{4}}}(g_{1},1/2,{\rm
Res}_{s=1}r(g_{f})M_{\omega}(f)).$

This implies that ${\rm Res}_{s=1}{\rm Eis}_{P_{\mathbf{H}}}(g,s,f)$ equals to
$\tfrac{1}{2}{\rm Eis}_{Q_{\mathbf{H}}}(g,s,{\rm Res}_{s=1}M_{\omega}(f))$,
i.e. $f_{1/2}^{\prime}$ can be taken to be ${\rm Res}_{s=1}M_{\omega}(f)$.
After normalizing $f$, we thus get the desired result for the normalized
Eisenstein series $E_{P_{\mathbf{H}}}^{*}(g,s)$ (see also [12, Proposition
1.8]). ∎

####  4.1.3. The constant term of the Borel Eisenstein series

The following informal discussion will not be used in the article, but we add
it as a motivation for the normalization of the Borel Eisenstein series and
the existence of its poles along the line $z=1$.

Let $W_{\mathbf{H}}$ denote the Weyl group of $\mathbf{H}$; also let
$\Lambda_{s,z}:=(2z-1)\beta_{1}+(s-z)\beta_{2}\in X^{*}(T_{\mathbf{H}})$,
where $\beta_{1},\;\beta_{2}$ denote the fundamental weights of $\mathbf{H}$.
Let $S$ be a finite set of places such that the defining datum of the
Eisenstein series $E_{B_{\mathbf{H}}}(g,s,z)$ is unramified outside $S$. The
analytic properties of $E_{B_{\mathbf{H}}}(g,s,z)$ are reflected in its
constant term, that is equal to

$\sum_{\omega\in W_{\mathbf{H}}}M(\omega,\Lambda_{s,z})f(g,s,z),$

where
$M(\omega,\Lambda_{s,z})f(g,s,z)=\int_{(U_{\mathbf{H}}\cap\omega\overline{U}_{\mathbf{H}}\omega^{-1})({\bf
A})}f(\omega^{-1}ug,s,z)du$, with $\overline{U}_{\mathbf{H}}$ the unipotent
radical of the lower-triangular Borel, is the well known intertwining map. The
Gindikin-Karpelevic formula shows that the poles of $M(\omega,\Lambda_{s,z})$
are exactly the ones of the following functions:

$\prod_{\begin{subarray}{c}\alpha>0\\\
\omega\alpha<0\end{subarray}}\zeta^{S}_{F}(\left<\Lambda_{s,z},\alpha^{\vee}\right>),$

where $\alpha$ ranges through the positive roots for $\mathbf{H}$ such that
$\omega\alpha$ is negative. Therefore, the only possible poles are located
along the lines determined by $\left<\Lambda_{s,z},\alpha^{\vee}\right>=1$.
Concretely, these are $\\{z=1\\}$, $\\{s=1\\}$, $\\{s-z=1\\}$, and
$\\{z+s=2\\}$. In fact, the constant term is a combination of the following
zeta functions:

$\zeta^{S}_{F}(2z-1),\;\;\zeta^{S}_{F}(s-z),\;\;\zeta^{S}_{F}(s+z-1),\;\;\zeta^{S}_{F}(2s-1).$

####  4.1.4. Residues of the Borel Eisenstein series

In the following, we relate the residue of the Borel Eisenstein series along
the line $z=1$ to the Siegel Eisenstein series. Let $f\in
I_{B_{\mathbf{H}}}(s,z)$ be the section defining
$\mathrm{Eis}_{B_{\mathbf{H}}}(g,z,s)$, then

$\mathrm{Res}_{z=1}\mathrm{Eis}_{B_{\mathbf{H}}}(g,z,s)=\sum_{\gamma^{\prime}\in
P_{\mathbf{H}}(F)\setminus\mathbf{H}(F)}r(f)(\gamma^{\prime}g),$

where $r(f)(g)=\mathrm{Res}_{z=1}\left(\sum_{\gamma\in
B_{\mathbf{H}}(F)\setminus P_{\mathbf{H}}(F)}f(\gamma g,s,z)\right)$. Since
$B_{\mathbf{H}}\backslash P_{\mathbf{H}}\simeq
B_{\mathrm{GL}_{2}}\backslash\mathrm{GL}_{2}$, we have

$f\in I_{P_{\mathbf{H}}({\bf A})}^{\mathbf{H}({\bf
A})}\left(\delta_{P_{\mathbf{H}}}^{\frac{1}{3}(s+1)}I_{B_{\mathrm{GL}_{2}}({\bf
A})}^{\mathrm{GL}_{2}({\bf A})}\delta_{B_{\mathrm{GL}_{2}}}^{z}\right).$

As the residue of degenerate Eisenstein series on $\mathrm{GL}_{2}$ is
constant, we get that $r(f)(g)\in I_{P_{\mathbf{H}}}(s)$. Summing up, we have
obtained the following:

######  Proposition 4.2.

For $f$ and $r(f)$ as above we have that
$\mathrm{Res}_{z=1}\mathrm{Eis}_{B_{\mathbf{H}}}(g,z,s)={\rm
Eis}_{P_{\mathbf{H}}}(g,s,r(f))$.

Similarly, we have a formula for the residue along $s-z=1$:

$\displaystyle\mathrm{Res}_{s-z=1}\mathrm{Eis}_{B_{\mathbf{H}}}(g,z,s)$
$\displaystyle=\mathrm{Eis}_{Q_{\mathbf{H}}}(g,2z+1/2,r^{\prime}(f)),$

where $r^{\prime}(f)(g)=\mathrm{Res}_{s-z=1}\left(\sum_{\gamma\in
B_{\mathbf{H}}(F)\setminus Q_{\mathbf{H}}(F)}f(\gamma g,s,z)\right)$.

###  4.2. Shalika models and a theta correspondence

The Shalika subgroup of $\mathbf{G}$ is defined as follows. For any
$F$-algebra $R$, $S$ is given by

$S(R)=\left\\{\left(\begin{smallmatrix}a&b&&\\\ c&d&&\\\ &&a&-b\\\
&&-c&d\end{smallmatrix}\right)\left(\begin{smallmatrix}1&&\alpha&x\\\
&1&y&\overline{\alpha}\\\ &&1&\\\
&&&1\end{smallmatrix}\right)\;:\;\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\mathrm{GL}_{2}(R),x,y\in\mathbb{G}_{a}(R),\;\alpha\in{\rm
Res}_{E/F}\mathbb{G}_{a}(R)\right\\}.$ (15)

Note that the Shalika group $S$ is isomorphic to
$\mathrm{GL}_{2}N_{P_{\mathbf{G}}}$, where $N_{P_{\mathbf{G}}}$ is the
unipotent radical of the Siegel parabolic $P_{\mathbf{G}}$ of $\mathbf{G}$,
while $\mathrm{GL}_{2}$ embeds into the Levi of the Siegel parabolic
$M_{P_{\mathbf{H}}}$ of $\mathbf{H}$ via the embedding $h\mapsto
h^{*}=\left(\begin{smallmatrix}h&\\\ &{\rm
det}(h)J_{2}{}^{t}h^{-1}J_{2}\end{smallmatrix}\right)$. Notice that the
modulus character $\delta_{S}$ of $S$ is trivial.

Recall that we have fixed a non-degenerate character $\psi:F\backslash{\bf
A}\to{\bf C}^{\times}$. Given $\eta\in E^{\times}$ such that
$\bar{\eta}=-\eta$, we define

$\chi_{\eta}:N_{P_{\mathbf{G}}}(F)\backslash N_{P_{\mathbf{G}}}({\bf
A})\longrightarrow{\bf C}^{\times},\,\left(\begin{smallmatrix}1&&\alpha&x\\\
&1&y&\overline{\alpha}\\\ &&1&\\\
&&&1\end{smallmatrix}\right)\mapsto\psi(\eta\alpha-\eta\overline{\alpha}).$

By a slight abuse of notation, denote also by $\chi_{\eta}$ the character on
$S(F)\backslash S({\bf A})$ given by $h^{*}n\mapsto\chi_{\eta}(n)$. Let $\Pi$
be a cuspidal automorphic representation of $\mathbf{G}({\bf A})$ with trivial
central character.

######  Definition 4.3.

For a cusp form $\varphi$ in the space of $\Pi$, the unitary Shalika period is
defined by

$\mathcal{S}_{\varphi}^{\eta}(g):=\int_{{\bf
A}^{\times}\mathrm{GL}_{2}(F)\backslash\mathrm{GL}_{2}({\bf
A})}\int_{N_{P_{\mathbf{G}}}(F)\backslash N_{P_{\mathbf{G}}}({\bf
A})}\varphi\left(nh^{*}g\right)\chi_{\eta}^{-1}(n)dndg.$

The representation $\Pi$ is said to have a Shalika model if there exist
$\varphi$ in $\Pi$ and $g\in\mathbf{G}({\bf A})$ such that
$\mathcal{S}_{\varphi}^{\eta}(g)\neq 0$.

In [7], Furusawa and Morimoto characterized the representations $\Pi$ with
unitary Shalika model in terms of the existence of a pole at $s=1$ of the
partial twisted exterior square $L$-function. Namely, they showed the
following.

######  Theorem 4.4 ([7, Theorem 4.1]).

Let $\Pi$ be an irreducible cuspidal automorphic representation of
$\mathbf{G}({\bf A})$ with trivial central character. Then the following two
conditions are equivalent:

  1. (1)

The Shalika period $\mathcal{S}_{\varphi}^{\eta}(1)$ does not vanish on the
space of $\Pi$.

  2. (2)

$\Pi$ is globally generic and the partial $L$-function
$L^{\Sigma}(s,\Pi,\wedge^{2})$ has a simple pole at $s=1$.

Motivated by this, in [15], Morimoto gave a characterization of cuspidal
automorphic representations of $\mathbf{G}({\bf A})$ with Shalika models by
means of the theta correspondence $\Theta$ of the pair
$(\mathbf{P}\mathbf{H}({\bf A})^{+},\mathbf{P}\mathbf{G}({\bf A}))$, with

$\mathbf{H}({\bf A})^{+}:=\\{(g,m_{g})\in\mathbf{H}({\bf A})\,:\,m_{g}\in
N_{E/F}({\bf A}_{E}^{\times})\\}.$

######  Theorem 4.5 ([15, Theorem 3.6 & (3.6)]).

Let $\Pi$ be an irreducible cuspidal automorphic representation of
$\mathbf{G}({\bf A})$ with trivial central character. Then $\Pi$ has a Shalika
model if and only if $\Pi$ appears as a subquotient of $\Theta(\sigma^{+})$,
for an irreducible cuspidal automorphic representation $\sigma^{+}$ of
$\mathbf{H}({\bf A})^{+}$ which appears as a subrepresentation of a globally
generic cuspidal automorphic representation $\sigma$ of $\mathbf{H}({\bf A})$
with trivial central character. If any of these conditions hold, we have

$L^{\Sigma}(s,\Pi,\wedge^{2})=L^{\Sigma}(s,\sigma,{\rm
std}\otimes\chi_{E/F})\zeta^{\Sigma}_{F}(s),$

where $\chi_{E/F}$ is the Hecke character associated to the extension $E/F$
and $\Sigma$ is a finite set of places which contains all the ramified places
of $\Pi$, $\sigma$, and $E/F$.

###  4.3. The representation of the standard $L$-function of
${\mathrm{GSp}}_{4}$

Let $\Pi$ be a cuspidal automorphic representation of $\mathbf{G}({\bf A})$
with trivial central character. Let $\Sigma$ be a finite set of places
containing the archimedean places and all the ramified places for $\Pi$ and
$E/F$. We take a cusp form $\varphi\in\Pi$, which we assume to be a pure
tensor and consider the following integral:

$J(\varphi,f,s):=\int_{\mathbf{H}(F)Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf
A})}\mathrm{Eis}_{P_{\mathbf{H}}}(h,s,f)\varphi(h)dh,$

where $\mathrm{Eis}_{P_{\mathbf{H}}}(h,s,f)$ is the Siegel Eisenstein series
for $\mathbf{H}$ introduced in §4.1.1 associated to a factorizable standard
holomorphic section $f$ unramified outside $\Sigma$. Note that this integral
converges absolutely as the restriction to $\mathbf{H}({\bf A})$ of $\varphi$
is rapidly decreasing. Furthermore, due to the analytic properties of the
Eisenstein series, it defines a meromorphic function.

######  Proposition 4.6.

The integral $J(\varphi,f,s)$ unfolds to

$\int_{\mathrm{GL}_{2}({\bf A})N_{P_{\mathbf{H}}}({\bf
A})\setminus\mathbf{H}({\bf A})}f(h,s)\mathcal{S}_{\varphi}^{\eta}(h)dh.$

###### Proof.

We start by unfolding the Eisenstein series; for ${\rm Re}(s)$ big enough, we
get

$\int_{P_{\mathbf{H}}(F)Z_{\mathbf{H}}({\bf A})\setminus\mathbf{H}({\bf
A})}f(h,s)\varphi(h)dh.$

Collapse the sum over the unipotent radical $N_{P_{\mathbf{H}}}$ of
$P_{\mathbf{H}}$ to get

$\displaystyle\int_{M_{P_{\mathbf{H}}}(F)N_{P_{\mathbf{H}}}({\bf
A})Z_{\mathbf{H}}({\bf A})\setminus\mathbf{H}({\bf
A})}\int_{N_{P_{\mathbf{H}}}(F)\backslash N_{P_{\mathbf{H}}}({\bf
A})}f(nh,s)\varphi(nh)dndh=\int_{\cdots}f(h,s)\int_{N_{P_{\mathbf{H}}}(F)\backslash
N_{P_{\mathbf{H}}}({\bf A})}\varphi(nh)dndh,$

as $f\in I_{P_{\mathbf{H}}}(s)$. Now Fourier expanding over
$[N_{P_{\mathbf{G}}}/N_{P_{\mathbf{H}}}]$, with $N_{P_{\mathbf{G}}}$ the
unipotent radical of the Siegel parabolic of $\mathbf{G}$, the integral equals

$\int_{M_{P_{\mathbf{H}}}(F)N_{P_{\mathbf{H}}}({\bf A})Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf A})}f(h,s)\sum_{\chi}\varphi_{\chi}(h)dh,$

where the sum runs over the characters
$\chi:[N_{P_{\mathbf{G}}}/N_{P_{\mathbf{H}}}]\to{\bf C}^{\times}$ and where we
have denoted

$\varphi_{\chi}(h)=\int_{N_{P_{\mathbf{G}}}(F)\backslash
N_{P_{\mathbf{G}}}({\bf
A})}\varphi(n^{\prime}h)\chi^{-1}(n^{\prime})dn^{\prime}.$

Any character $\chi:[N_{P_{\mathbf{G}}}]\to{\bf C}^{\times}$ is of the form
$n^{\prime}\mapsto\psi({\rm Tr}(A\cdot\left(\begin{smallmatrix}\alpha&x\\\
y&\bar{\alpha}\end{smallmatrix}\right))),$ where $A\in M_{2\times 2}(E)$ is
such that $\overline{A}^{t}=J_{2}AJ_{2}$, $\psi$ is a non-trivial additive
character on $F\backslash{\bf A}$, and where we have written

$n^{\prime}=\left(\begin{smallmatrix}1&&\alpha&x\\\ &1&y&\bar{\alpha}\\\
&&1&\\\ &&&1\end{smallmatrix}\right),\;\text{ with }x,y\in{\bf
A},\alpha\in{\bf A}_{E}.$

If $A=(a_{i,j})$, the associated character $\chi$ is trivial on
$N_{P_{\mathbf{H}}}({\bf A})$ if $a_{1,1}+a_{2,2}=0$ and $a_{1,2}=a_{2,1}=0$.
We deduce that any character
$\chi:[N_{P_{\mathbf{G}}}/N_{P_{\mathbf{H}}}]\to{\bf C}^{\times}$ is of the
form

$n^{\prime}\mapsto\psi({\rm Tr}(\left(\begin{smallmatrix}\eta&\\\
&-\eta\end{smallmatrix}\right)\cdot\left(\begin{smallmatrix}\alpha&x\\\
y&\bar{\alpha}\end{smallmatrix}\right)))=\psi(\eta\alpha-\eta\bar{\alpha}),$

with $\eta\in E^{\times}$ such that $\bar{\eta}=-\eta$. To emphasize its
dependence on $\eta$, we denote this character by $\chi_{\eta}$.

As $M_{P_{\mathbf{H}}}$ acts on $N_{P_{\mathbf{G}}}$ by conjugation and this
action preserves $N_{P_{\mathbf{H}}}$, $M_{P_{\mathbf{H}}}(F)$ acts on the
space of characters of $[N_{P_{\mathbf{G}}}/N_{P_{\mathbf{H}}}]$. Indeed, if
$m=\left(\begin{smallmatrix}g&\\\ &\mu
J_{2}{}^{t}g^{-1}J_{2}\end{smallmatrix}\right)\in M_{P_{\mathbf{H}}}(F)$ ,

$\chi_{\eta}(mn^{\prime}m^{-1})=\chi_{\eta^{\prime}}(n^{\prime}),\;\text{ with
}\eta^{\prime}=\mu^{-1}{\rm det}(g)\eta.$

Then, $M_{P_{\mathbf{H}}}(F)$ acts on the space of characters of
$[N_{P_{\mathbf{G}}}/N_{P_{\mathbf{H}}}]$ with two orbits, the trivial one and
an open one associated to any $\chi_{\eta}$ with $\eta\neq 0$. Using this
action, we can write the integral as

$\displaystyle\int_{M_{P_{\mathbf{H}}}(F)N_{P_{\mathbf{H}}}({\bf
A})Z_{\mathbf{H}}({\bf A})\setminus\mathbf{H}({\bf
A})}f(h,s)\int_{[N_{P_{\mathbf{G}}}]}\varphi(n^{\prime}h)dn^{\prime}dh+\int_{M_{\eta}(F)N_{P_{\mathbf{H}}}({\bf
A})Z_{\mathbf{H}}({\bf A})\setminus\mathbf{H}({\bf
A})}f(h,s)\varphi_{\chi_{\eta}}(h)dh,$ (16)

where $M_{\eta}(F)$ denotes the stabiliser in $M_{P_{\mathbf{G}}}(F)$ of
$\chi_{\eta}$. Precisely, $m\in M_{\eta}(F)$ if $\mu={\rm det}(g)$, hence we
can fix an isomorphism $\mathrm{GL}_{2}(F)\simeq M_{\eta}(F)$, sending
$g\mapsto m(g,{\rm det}(g))$. As the first integral of (16) vanishes because
of cuspidality of $\varphi$ along $[N_{P_{\mathbf{G}}}]$,

$J(\varphi,f,s)=\int_{\mathrm{GL}_{2}(F)N_{P_{\mathbf{H}}}({\bf
A})Z_{\mathbf{H}}({\bf A})\setminus\mathbf{H}({\bf
A})}f(h,s)\varphi_{\chi_{\eta}}(h)dh.$

Collapse the sum over $[\mathrm{GL}_{2}]$, to get

$\displaystyle J(\varphi,f,s)$
$\displaystyle=\int_{\mathrm{GL}_{2}(F)N_{P_{\mathbf{H}}}({\bf
A})Z_{\mathbf{H}}\setminus\mathbf{H}({\bf
A})}f(h,s)\varphi_{\chi_{\eta}}(h)dh$
$\displaystyle=\int_{\mathrm{GL}_{2}({\bf A})N_{P_{\mathbf{H}}}({\bf
A})\setminus\mathbf{H}({\bf A})}\int_{\mathrm{GL}_{2}(F)Z_{\mathbf{H}}({\bf
A})\backslash\mathrm{GL}_{2}({\bf A})}f(m(g,{\rm
det}(g))h,s)\varphi_{\chi_{\eta}}(m(g,{\rm det}(g))h)dgdh$
$\displaystyle=\int_{\mathrm{GL}_{2}({\bf A})N_{P_{\mathbf{H}}}({\bf
A})\setminus\mathbf{H}({\bf
A})}f(h,s)\int_{\mathrm{GL}_{2}(F)Z_{\mathbf{H}}({\bf
A})\backslash\mathrm{GL}_{2}({\bf A})}\varphi_{\chi_{\eta}}(gh)dgdh$
$\displaystyle=\int_{\mathrm{GL}_{2}({\bf A})N_{P_{\mathbf{H}}}({\bf
A})\setminus\mathbf{H}({\bf A})}f(h,s)\mathcal{S}_{\varphi}^{\eta}(h)dh,$

where we have used that $\delta_{P_{\mathbf{H}}}(m(g,{\rm
det}(g)))=\left|\tfrac{{\rm det}(g)}{{\rm det}(g)}\right|^{3}=1$. ∎

######  Corollary 4.7.

The integral $J(\varphi,f,s)$ is identically zero unless $\Pi$ is globally
generic and $L^{\Sigma}(s,\Pi,\wedge^{2})$ has a simple pole at $s=1$.

###### Proof.

If $J(\varphi,f,s)$ is not identically zero, then the Shalika period does not
vanish on $\Pi$. The result then follows from Theorem 4.4. ∎

Corollary 4.7 is coherent with the following.

######  Lemma 4.8.

For $f$ and $r(f)$ as in §4.1.4, we have that

$\mathrm{Res}_{z=1}I(\varphi,f,s,z)=J(\varphi,r(f),s).$

###### Proof.

It follows immediately from Proposition 4.2. ∎

By Lemma 4.8 and Theorem 3.4, we obtain that

$\displaystyle J(\varphi,r(f),s)$ $\displaystyle={\rm
Res}_{z=1}\left(\frac{L^{\Sigma}(s,\Pi,\wedge^{2})L^{\Sigma}(z,\Pi,\wedge^{2})}{\zeta_{F}^{\Sigma}(2z)\zeta_{F}^{\Sigma}(2s)\zeta_{F}^{\Sigma}(s+z)\zeta_{F}^{\Sigma}(s-z+1)}\prod_{v\in\Sigma}I(\varphi_{v},f_{v},s,z)\right)$
$\displaystyle=\frac{L^{\Sigma}(s,\Pi,\wedge^{2})}{\zeta_{F}^{\Sigma}(2)\zeta_{F}^{\Sigma}(2s)\zeta_{F}^{\Sigma}(s+1)\zeta_{F}^{\Sigma}(s)}{\rm
Res}_{z=1}\left(L^{\Sigma}(z,\Pi,\wedge^{2})\prod_{v\in\Sigma}I(\varphi_{v},f_{v},s,z)\right)$
(17)

Let $\Sigma_{\infty}\subseteq\Sigma$ denote the set of archimedean places of
$F$.

######  Proposition 4.9.

Given a cusp form $\varphi$ in $\Pi$, there exists a section $f$ such that

$\displaystyle
J(\varphi,r(f),s)=\frac{L^{\Sigma}(s,\Pi,\wedge^{2})}{\zeta_{F}^{\Sigma}(2)\zeta_{F}^{\Sigma}(2s)\zeta_{F}^{\Sigma}(s+1)\zeta_{F}^{\Sigma}(s)}\prod_{v\in\Sigma}I(\varphi_{v},f_{v},s,1){\rm
Res}_{z=1}\left(L^{\Sigma}(z,\Pi,\wedge^{2})\right).$

Moreover, if $\Pi$ has a non trivial Shalika period there exists a cusp form
$\varphi^{\prime}$ in $\Pi$ and a section $f$ such that

$\displaystyle J(\varphi,r(f),s)=\frac{L^{\Sigma}(s,\sigma,{\rm
std}\otimes\chi_{E/F})}{\zeta_{F}^{\Sigma}(2s)\zeta_{F}^{\Sigma}(s+1)}\prod_{v\in\Sigma_{\infty}}I(\varphi_{v}^{\prime},f_{v},s,1),$

with $\sigma$ a globally generic cuspidal automorphic representation of
$\mathbf{H}({\bf A})$ as in Theorem 4.5.

###### Proof.

By Lemma 3.10, we can choose $\varphi_{v}$ and $f_{v}$ such that
$\prod_{v\in\Sigma-\Sigma_{\infty}}I(\varphi_{v},f_{v},s,z)$ is a non-zero
constant $C$ independent of $s$ and $z$ and there exists $\varepsilon>0$ such
that $\prod_{v\in\Sigma_{\infty}}I(\varphi_{v},f_{v},s,z)$ converges
absolutely for ${\rm Re}(z)>1-\varepsilon$ and ${\rm Re}(s)>{\rm Re}(z)-4$.
This and (17) prove the first equality. Now suppose that $\Pi$ has a Shalika
period. By Theorems 4.4 and 4.5, ${\rm
Res}_{z=1}\left(L^{\Sigma}(z,\Pi,\wedge^{2})\right)\neq 0$ and there exists a
globally generic cuspidal automorphic representation $\sigma$ of
$\mathbf{H}({\bf A})$ with trivial central character such that, up to
enlarging $\Sigma$,

$L^{\Sigma}(s,\Pi,\wedge^{2})=L^{\Sigma}(s,\sigma,{\rm
std}\otimes\chi_{E/F})\zeta^{\Sigma}_{F}(s).$

This implies that

$\displaystyle J(\varphi,r(f),s)=\frac{C\cdot L^{\Sigma}(s,\sigma,{\rm
std}\otimes\chi_{E/F})}{\zeta_{F}^{\Sigma}(2)\zeta_{F}^{\Sigma}(2s)\zeta_{F}^{\Sigma}(s+1)}\prod_{v\in\Sigma_{\infty}}I(\varphi_{v},f_{v},s,1)L^{\Sigma}(1,\sigma,{\rm
std}\otimes\chi_{E/F}){\rm Res}_{z=1}\zeta_{F}^{\Sigma}(z).$

For a given $v\in\Sigma_{\infty}$, define $\varphi^{\prime}_{v}:=[C\cdot
L^{\Sigma}(1,\sigma,{\rm std}\otimes\chi_{E/F}){\rm
Res}_{z=1}\zeta_{F}^{\Sigma}(z)]^{-1}\zeta_{F}^{\Sigma}(2)\cdot\varphi_{v}$
and let $\varphi_{v}^{\prime}=\varphi_{v}$ otherwise. Then we have

$\displaystyle J(\varphi,r(f),s)=\frac{L^{\Sigma}(s,\sigma,{\rm
std}\otimes\chi_{E/F})}{\zeta_{F}^{\Sigma}(2s)\zeta_{F}^{\Sigma}(s+1)}\prod_{v\in\Sigma_{\infty}}I(\varphi_{v}^{\prime},f_{v},s,1),$

as desired. ∎

Let

$J^{*}(\varphi,s):=\int_{\mathbf{H}(F)Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf A})}E_{P_{\mathbf{H}}}^{*}(h,s)\varphi(h)dh,$

where $E_{P_{\mathbf{H}}}^{*}(h,s)$ denotes the normalized Siegel Eisenstein
series of (14). By Proposition 4.9, after normalizing the section $r(f)$, we
get the following.

######  Theorem 4.10.

Given a cusp form $\varphi$ in $\Pi$, there exists a cusp form
$\varphi^{\prime}$ and section $f$ such that

$J^{*}(\varphi,s)=L^{\Sigma}(s,\sigma,{\rm
std}\otimes\chi_{E/F})\cdot\prod_{v\in\Sigma_{\infty}}I(\varphi_{v}^{\prime},f_{v},s,1).$

###  4.4. Properties of the Rankin–Selberg integrals

In the following, we show that the integral $J^{*}(\varphi,s)$ is holomorphic
for all $s$.

####  4.4.1. Vanishing of a zeta integral

In view of Proposition 4.1 and Proposition 4.2, we are interested in
calculating the integral

$J^{\prime}(\varphi,f^{\prime},s):=\int_{\mathbf{H}(F)Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf A})}{\rm
Eis}_{Q_{\mathbf{H}}}(h,s,f^{\prime})\varphi(h)dh,$ (18)

where $\varphi$ is a cusp form in $\Pi$.

######  Proposition 4.11.

The integral $J^{\prime}(\varphi,f^{\prime},s)$ is identically zero.

######  Remark 4.12.

Notice that if the induced datum of the Klingen Eisenstein series is not
degenerate, the corresponding integral calculates the $\wedge^{2}\times{\rm
Std}$ $L$-function on ${\mathrm{GU}}_{2,2}\times\mathrm{GL}_{2}$, as we show
in forthcoming work.

###### Proof of Proposition 4.11.

We start by unfolding the Eisenstein series. For big enough ${\rm Re}(s)$, we
get

$\int_{{Q_{\mathbf{H}}}(F)Z_{\mathbf{H}}({\bf A})\setminus\mathbf{H}({\bf
A})}f^{\prime}(h,s))\varphi(h)dh.$

Collapse the sum over the unipotent radical $N_{Q_{\mathbf{H}}}$ of
${Q_{\mathbf{H}}}$ to get

$\displaystyle\int_{M_{Q_{\mathbf{H}}}(F)N_{Q_{\mathbf{H}}}({\bf
A})Z_{\mathbf{H}}({\bf A})\setminus\mathbf{H}({\bf
A})}f^{\prime}(h,s)\int_{N_{Q_{\mathbf{H}}}(F)\backslash
N_{Q_{\mathbf{H}}}({\bf A})}\varphi(nh)dndh.$

We now Fourier expand over $[N_{Q_{\mathbf{G}}}/N_{Q_{\mathbf{H}}}]$. The
integral equals

$\int_{M_{Q_{\mathbf{H}}}(F)N_{Q_{\mathbf{H}}}({\bf A})Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf
A})}f^{\prime}(h,s)\sum_{\chi}\varphi_{\chi}(h)dh,$

where the sum runs over the characters
$\chi:[N_{Q_{\mathbf{G}}}/N_{Q_{\mathbf{H}}}]\to{\bf C}^{\times}$. As shown in
the proof of Proposition 3.1, any character
$\chi:[N_{Q_{\mathbf{G}}}/N_{Q_{\mathbf{H}}}]\to{\bf C}^{\times}$ is of the
form $\chi_{\alpha,\beta}:n\mapsto\psi({\rm Tr}_{E/F}(\alpha x+\beta y))$,
with $\alpha,\beta\in E$ such that $\bar{\alpha}=-\alpha$ and
$\bar{\beta}=-\beta$. Again from the proof of Proposition 3.1, we know that
$M_{Q_{\mathbf{H}}}(F)$ acts on the space of characters of
$[N_{Q_{\mathbf{G}}}/N_{Q_{\mathbf{H}}}]$ with two orbits, the trivial one and
an open one, with a representative given by the character $\chi_{\alpha,0}$
with $\bar{\alpha}=-\alpha\neq 0$. By cuspidality of $\varphi$ along
$N_{Q_{\mathbf{G}}}$, we have

$\displaystyle
J^{\prime}(\varphi,f^{\prime},s)=\int_{M_{\alpha}(F)N_{Q_{\mathbf{H}}}({\bf
A})Z_{\mathbf{H}}({\bf A})\setminus\mathbf{H}({\bf
A})}f^{\prime}(h,s)\varphi_{\chi_{\alpha,0}}(h)dh,$ (19)

where $M_{\alpha}(F)=\left\\{\left(\begin{smallmatrix}a&&&\\\ &a&b&\\\ &&d&\\\
&&&d\end{smallmatrix}\right)\;\right\\}=L_{\alpha}(F)N_{\alpha}(F)$ denotes
the stabiliser in $M_{Q_{\mathbf{H}}}(F)$ of $\chi_{\alpha,0}$, with Levi part
(resp. unipotent part) $L_{\alpha}$ (resp. $N_{\alpha}$). Collapse the sum
over $[N_{\alpha}]$ to get

$\displaystyle J^{\prime}(\varphi,f^{\prime},s)$
$\displaystyle=\int_{L_{\alpha}(F)N_{\alpha}({\bf A})N_{Q_{\mathbf{H}}}({\bf
A})Z_{\mathbf{H}}\setminus\mathbf{H}({\bf
A})}f^{\prime}(h,s)\int_{[N_{\alpha}]}\varphi_{\chi_{\alpha,0}}(nh)dndh.$

Notice that the inner integral
$\int_{[N_{\alpha}]}\varphi_{\chi_{\alpha,0}}(nh)dn$ equals the Fourier
coefficient of $\varphi$ with respect to the unipotent radical of the Borel
subgroup of $\mathbf{G}$ and the choice of a character on it which is
degenerate (as it’s only supported on the entry (1,2)). As a consequence,
$\int_{[N_{\alpha}]}\varphi_{\chi_{\alpha,0}}(nh)dn$ contains, as an inner
integral, the period of $\varphi$ over the unipotent radical of the Siegel
parabolic of $\mathbf{G}$, which vanishes by cuspidality of $\varphi$. This
completes the proof. ∎

######  Remark 4.13.

Proposition 4.11 has potential applications to the study of the archimedean
regulator of the cohomology classes introduced in [4]. Indeed, we expect the
vanishing of this integral to imply that the image via the archimedean
regulator of the motivic incarnation of the classes of [4, Definition 6.6]
contributes entirely to Eisenstein cohomology.

####  4.4.2. Periods over ${\mathrm{GSp}}_{4}$

For any cusp form $\varphi$ on $\mathbf{G}({\bf A})$, we consider

$\varphi^{\mathbf{H}}(g):=\int_{\mathbf{H}(F)Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf A})}\varphi(hg)dh.$

######  Lemma 4.14.

Let $\Pi$ be a globally generic cuspidal automorphic representation of
$\mathbf{P}\mathbf{G}({\bf A})$. For any cusp form $\varphi$ in $\Pi$ and
$g\in\mathbf{G}({\bf A})$, the period $\varphi^{\mathbf{H}}(g)=0$.

###### Proof.

Since ${\mathrm{Sp}}_{4}$ is a normal subgroup of $\mathbf{H}$, the result is
implied by showing that, for any $\varphi$ in $\Pi$ and $g\in\mathbf{G}({\bf
A})$,

$\varphi^{{\mathrm{Sp}}_{4}}(g):=\int_{[{\mathrm{Sp}}_{4}]}\varphi(hg)dh=0.$

We show the latter by contradiction. Suppose that there exists $\varphi$ in
$\Pi$ and $g\in\mathbf{G}({\bf A})$ for which
$\varphi^{{\mathrm{Sp}}_{4}}(g)\neq 0$. This defines a non-trivial
${\mathrm{Sp}}_{4}$-functional on $\Pi$. Let $v$ be an unramified place for
$\Pi$ which splits in $E$. We can further assume that $g_{v}=1$. Since $v$
splits, we can identify $\Pi_{v}$ with an unramified generic representation
$\pi_{v}$ of $\mathrm{GL}_{4}(F_{v})$ with trivial central character. As the
linear span $R_{v}$ of the $\textbf{P}\mathbf{G}(F_{v})$-right translates of
$\varphi$ is isomorphic to $\pi_{v}$, the map

$R_{v}\ni\varphi^{\prime}\mapsto\left(g^{\prime}\mapsto\int_{[{\mathrm{Sp}}_{4}]}\varphi^{\prime}(hgg^{\prime})dh\right)$

defines an injective intertwining map in ${\rm
Hom}_{\mathrm{GL}_{4}(F_{v})}(\pi_{v},{C}_{c}^{\infty}({\mathrm{Sp}}_{4}(F_{v})\backslash\mathrm{GL}_{4}(F_{v})))$.
This contradicts the genericity of $\pi_{v}$ (cf. [11, Theorem 3.2.2]). The
Lemma follows. ∎

####  4.4.3. Location of poles

######  Theorem 4.15.

Let $\Pi$ be a globally generic cuspidal automorphic representation of
$\mathbf{G}({\bf A})$, with trivial central character and let $\varphi$ be a
cusp form in $\Pi$. The integral $J^{*}(\varphi,s)$ is holomorphic for all
$s$. In particular, if $\Pi$ is the small theta lift of a $\mathbf{H}({\bf
A})^{+}$-factor of a globally generic cuspidal automorphic representation
$\sigma$ of $\mathbf{H}({\bf A})$, then $L^{\Sigma}(s,\sigma,{\rm
std}\otimes\chi_{E/F})$ is holomorphic for all $s$, with $\Sigma$ a finite set
of places containing all the ramified places of $\Pi$, $\sigma$, and $E/F$.

###### Proof.

Recall that the normalized Eisenstein series $E_{P_{\mathbf{H}}}^{*}(g,s)$
achieves poles at $s=1,2$ of order at most one, thus $J^{*}(\varphi,s)$ may
have a pole at these two values.

At $s=1$, the residue of the integral equals to

$\displaystyle{\rm Res}_{s=1}J^{*}(\varphi,s)$ $\displaystyle={\rm
Res}_{s=1}\int_{\mathbf{H}(F)Z_{\mathbf{H}}({\bf A})\setminus\mathbf{H}({\bf
A})}E^{*}_{P_{\mathbf{H}}}(h,s)\varphi(h)dh$
$\displaystyle=\int_{\mathbf{H}(F)Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf A})}{\rm
Res}_{s=1}E^{*}_{P_{\mathbf{H}}}(h,s)\varphi(h)dh$
$\displaystyle=C\cdot\int_{\mathbf{H}(F)Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf A})}{\rm
Eis}_{Q_{\mathbf{H}}}(g,1/2,f_{1/2}^{\prime})\varphi(h)dh$
$\displaystyle=C\cdot J^{\prime}(\varphi,f_{1/2}^{\prime},1/2),$

where the third equality follows from Proposition 4.1 and $C=\frac{{\rm
Res}_{s=1}(\zeta^{\Sigma}_{F}(s))\zeta^{\Sigma}_{F}(3)}{2\zeta^{\Sigma}_{F}(2)}$.
The latter integral vanishes by Proposition 4.11, hence $J^{*}(\varphi,s)$ is
holomorphic at $s=1$.

At $s=2$ there exists $f\in I_{P_{\mathbf{H}}}(s)$ such that ${\rm
Res}_{s=2}E_{P_{\mathbf{H}}}^{*}(g,s)$ is constant. Therefore

$\displaystyle{\rm Res}_{s=2}J^{*}(\varphi,s)$
$\displaystyle=\int_{\mathbf{H}(F)Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf A})}{\rm
Res}_{s=2}E^{*}_{P_{\mathbf{H}}}(h,s)\varphi(h)dh$
$\displaystyle=C\cdot\int_{\mathbf{H}(F)Z_{\mathbf{H}}({\bf
A})\setminus\mathbf{H}({\bf A})}\varphi(h)dh.$

Hence, $J^{*}(\varphi,s)$ achieves a pole at $s=2$ if
$\varphi^{\mathbf{H}}(1)\neq 0$. However, the latter is always zero because of
Lemma 4.14. This proves the first statement. The second one follows from this
and from Theorem 4.10. ∎

######  Remark 4.16.

Since the theta lift of $\sigma$ to ${\rm PGSO}_{4,2}\simeq\mathbf{PG}$ is
non-zero and cuspidal by hypothesis, Rallis tower property implies that the
theta lift of $\sigma$ to ${\rm GSO}_{3,1}$ and ${\rm GSO}_{2,0}$ is zero.
Thus, by using [14, Theorem 7.2.5], one could already deduce that the partial
$L$-function $L^{\Sigma}(s,\sigma,{\rm std}\otimes\chi_{E/F})$ is entire.

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