# On a Rankin-Selberg integral of three Hermitian cusp forms

Thanasis Bouganis and Rafail Psyroukis  Department of Mathematical Sciences
Durham University
South. Rd.
Durham, DH1 3LE, U.K<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $\displaystyle K=\mathbb{Q}(i)$. In this work, we study the Petersson
inner product of a Hermitian Eisenstein series of Siegel type on the unitary
group $\displaystyle U_{5}(K)$, diagonally-restricted on $\displaystyle
U_{2}(K)\times U_{2}(K)\times U_{1}(K)$, against two Hermitian cuspidal
eigenforms $\displaystyle F,G$ of degree $\displaystyle 2$ and an elliptic
cuspidal eigenform $\displaystyle h$ (seen as a Hermitian modular form of
degree 1), all having weight $\displaystyle k\equiv 0\pmod{4}$. This
consideration gives an integral representation of a certain Dirichlet series,
which will then have an analytic continuation and functional equation, due to
the one of the Eisenstein series. By taking $\displaystyle F$ to belong in the
Maass space, we are able to show that the Dirichlet series possesses an Euler
product. Moreover, its $\displaystyle p-$factor for an inert prime
$\displaystyle p$ can be essentially identified with the twist by
$\displaystyle h$ of a degree six Euler factor attached to $\displaystyle G$
by Gritsenko. The question of whether the same holds for the primes that split
remains unanswered here, even though we make considerable steps in that
direction too. Our paper is inspired by work of Heim, who considered a similar
question in the case of Siegel modular forms.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Hecke algebras and $\displaystyle L-$functions
  4. 4 Integral Representation and Dirichlet Series
  5. 5 Inert primes
    1. 5.1 Hecke operators and weak rationality theorems
    2. 5.2 Calculation of the Dirichlet Series
  6. 6 Split Primes
    1. 6.1 Relations between operators in $\displaystyle\textup{GL}_{4}$ and factorisation
    2. 6.2 Hecke Operators and weak rationality theorems
    3. 6.3 Calculation of the Dirichlet series - First Part
    4. 6.4 Calculation of the Dirichlet Series - Second Part
    5. 6.5 Calculation of the Dirichlet Series - Third Part
    6. 6.6 Final expression for the Dirichlet series
  7. 7 Euler Product

##  1\. Introduction

The spinor $\displaystyle L-$function attached to a cuspidal (holomorphic)
Siegel eigenform $\displaystyle G$ of degree two has been an object of intense
study in the literature. It was the seminal work of Andrianov in [1] who first
obtained an integral representation of such an $\displaystyle L-$function and
from there derived a functional equation and established its analytic
properties. However, the integral representation obtained by Andrianov does
not allow one (or at least it is not known how) to obtain algebraicity
properties of the critical values. The difficulty seems to be related to the
fact that the integral representation involves Eisenstein series which are
defined over symmetric spaces which do not have a structure of a Shimura
variety. Later work, such as the one of Kohnen and Skoruppa in [15], obtained
other integral representations using Eisenstein series over the Siegel upper
space but of weight zero and hence not holomorphic (or nearly holomorphic).
One should also mention here that if on the other hand $\displaystyle G$ is
taken non-holomorphic, there is recent work of Loeffler, Pilloni, Skinner and
Zerbes in [18], where they obtain algebraicity results and construct
$\displaystyle p-$adic $\displaystyle L-$functions.

It is perhaps surprising that a seemingly more complicated object, the twist
of the spinor $\displaystyle L-$function of $\displaystyle G$ (from now on
always assumed holomorphic) by an elliptic cusp form $\displaystyle h$ (that
is $\displaystyle\textup{Sp}_{4}\times\textup{GL}_{2}$) does afford an
integral representation, which allows one to not only study analytic
properties but also algebraic (and $\displaystyle p-$adic). Actually, there
are (at least) two different such integral representations. The one obtained
by Furusawa in [4], uses an Eisenstein series over a unitary group and its
restriction to $\displaystyle\textup{Sp}_{4}$. There is a series of works
based on this idea, most notably by Saha in [19], generalising the work of
Furusawa.

The other integral representation of the twist was obtained by Heim in [13] in
his effort to answer a question posed by Garrett on the possibility of
extending the doubling method to more copies of the group (here the
symplectic). Indeed, Heim managed to show that the twisted spinor
$\displaystyle L-$function can be obtained by integrating the restriction of a
Siegel type Eisenstein series of degree $\displaystyle 5$ to
$\displaystyle\mathbb{H}_{2}\times\mathbb{H}_{2}\times\mathbb{H}_{1}$ against
the (degree two) cusp form $\displaystyle G$, another degree two cusp form
$\displaystyle F$ in the Maass space and an elliptic cusp form $\displaystyle
h$, all being Hecke eigenforms for their corresponding Hecke algberas. This
integral expression was later exploited systematically by Böcherer and Heim in
[3], in order to establish various algebraicity properties and lift various
restrictions on the weights of the Siegel and elliptic modular forms by the
use of differential operators.

Shortly after the work of Andrianov on the spinor $\displaystyle L-$function
in [1], Gritsenko, in a series of papers, extended Andrianov’s approach of the
use of parabolic Hecke algebras to the study of a degree 6 $\displaystyle
L-$function attached to a cuspidal Hermitian eigenform of degree two, where
the underlying imaginary quadratic field is taken to be the field of Gaussian
numbers $\displaystyle K:=\mathbb{Q}(i)$. Indeed in [11], Gritsenko first
defined such an $\displaystyle L-$function and in the later work of [8], he
obtained the analogue construction of Kohnen and Skoruppa but with a quite
different approach. Both integral representations allowed him to obtain a
functional equation and study the analytic properties. However, as in the case
of the symplectic group, neither of the above integral representations could
be used to to derive algebraicity properties, due to the Eisenstein series
involved (only of real analytic nature).

In this paper, we ask whether the phenomenon observed in the case of the
twisted Siegel spinor $\displaystyle L-$function carries over to the unitary
one, namely whether the degree 6 $\displaystyle L-$function considered by
Gritsenko, twisted by an elliptic cusp form $\displaystyle h$, affords an
integral representation which allows one to study algebraic properties of the
twisted $\displaystyle L-$function. Here the elliptic cusp form $\displaystyle
h$ is seen as a Hermitian form of degree one (i.e. of $\displaystyle U_{1}$).
Given the similarities between $\displaystyle\textup{Sp}_{4}$ and
$\displaystyle U_{2}$, we investigate the possibility of extending the idea of
Heim to the unitary setting. That is, we study the Petersson inner product of
a Hermitian Eisenstein series of Siegel type on the unitary group
$\displaystyle U_{5}(K)$, diagonally-restricted on $\displaystyle
U_{2}(K)\times U_{2}(K)\times U_{1}(K)$, against two Hermitian cusp forms
$\displaystyle F,G$ of degree $\displaystyle 2$ and a Hermitian cusp form
$\displaystyle h$ of degree $\displaystyle 1$, all being Hecke eigenforms for
their corresponding Hecke algebras. This consideration gives an integral
representation of a certain Dirichlet series, which will then have an analytic
continuation and functional equation, due to the one of the Eisenstein series.
By taking $\displaystyle F$ to belong in the Maass space, we are able to show
in Theorem 7.1 that the Dirichlet series possesses an Euler product. Moreover,
its $\displaystyle p-$factor for an inert prime $\displaystyle p$ can be
essentially identified with the degree $\displaystyle 12$ $\displaystyle
p-$factor of $\displaystyle Z_{G\otimes h}$, the degree $\displaystyle 6$
$\displaystyle L-$function attached to $\displaystyle G$, twisted by the
Satake parameters of $\displaystyle h$. This is shown in Theorem 5.13. The
question of whether the same holds for the primes that split in $\displaystyle
K$ remains unanswered here, even though we make considerable steps in that
direction too. In particular, we have obtained all the essential ingredients,
i.e. the factorization of polynomials in parabolic Hecke algebras, the
necessary rationality theorems, relations between Hecke operators as well as
the main computations of the Dirichlet series. However, performing the last
few calculations seems very complicated. Our progress is summarised in Theorem
6.21. The case of the only ramified prime, namely $\displaystyle 2$, is not
discussed here but we should say that our calculations for the case of inert
give essentially all ideas required to compute the Euler factor also in this
case.

The reader may have already recognised that the choice of $\displaystyle
U_{2}$ is not as random. Indeed, thanks to the so called accidental
isomorphisms between “small” orthogonal groups and other classical groups, the
spinor $\displaystyle L-$function of $\displaystyle\textup{Sp}_{4}$ can be
identified with the standard $\displaystyle L-$function attached to a
holomorphic modular form of $\displaystyle\textup{SO}(2,3)$. Similarly, the
$\displaystyle L-$function studied by Gritsenko is closely related to the
standard $\displaystyle L-$function of an orthogonal group of signature
$\displaystyle(2,4)$. Therefore, the twist we are studying here is nothing
else than the two dimensional twist of this standard $\displaystyle
L-$function. Such twists have been studied before in [5] but these methods can
only be used for analytic results. If, on the other hand, one is interested in
algebraicity results of special values, these methods cannot (or at least is
not known how) used to obtain such results because of the use of orthogonal
groups which do not correspond to Shimura varieties. On the other hand, the
approach taken here (as a triple product) is known to give algebraicity
results (see Proposition 4.2 and the discussion there), very much in the way
that Heim and Böcherer obtained their algebraicity results in the case of the
two dimensional twist of the spinor $\displaystyle L-$function attached to a
degree two Siegel modular form. This possibility of obtaining algebraicity
results is the main motivation for the present work.

######  Notation.

In the following we will always use the following notation. We will denote by
$\displaystyle K=\mathbb{Q}(i)$, the Gaussian field. Let also
$\displaystyle\mathcal{O}_{K}=\mathbb{Z}[i]$ denote its ring of integers. Let
also

$J_{n}=\begin{pmatrix}0_{n}&-1_{n}\\\ 1_{n}&0_{n}\end{pmatrix}.$

For a complex number $\displaystyle z$, we denote by $\displaystyle
e(z):=e^{2\pi iz}$. We will use the bracket notation $\displaystyle
A[B]:=\overline{B}^{t}AB$ for complex matrices $\displaystyle A,B$. Finally,
for a polynomial $\displaystyle U$ with coefficients Hecke operators and
$\displaystyle G$ a Hecke eigenform, we write $\displaystyle U_{G}$ for the
polynomial obtained when we substitute the operators with the corresponding
eigenvalues.

##  2\. Preliminaries

We start with some definitions. Everything below can be found in [17] or more
generally in [21]. For the material on Jacobi forms we refer to [8].

######  Definition 2.1.

Let $\displaystyle R$ be either $\displaystyle K$,
$\displaystyle\mathcal{O}_{K}$ or $\displaystyle\mathbb{C}$ and fix an
embedding $\displaystyle R\hookrightarrow\mathbb{C}$. We write $\displaystyle
U_{n}(R)$ for the $\displaystyle R-$points of the unitary group of degree
$\displaystyle n\geq 1$. That is,

$U_{n}(R)=\\{g\in\textup{GL}_{2n}(R)\mid J_{n}[g]=J_{n}\\}.$

Hence, for an element $\displaystyle\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\in
U_{n}(R)$ with $\displaystyle n\times n$ matrices $\displaystyle A,B,C,D$,
these satisfy the relations

$\overline{A}^{t}C=\overline{C}^{t}A,\overline{D}^{t}B=\overline{B}^{t}D,A\overline{D}^{t}-\overline{B}^{t}C=1_{n}.$

######  Definition 2.2.

The upper Hermitian plane of degree $\displaystyle n$ is defined by

$\mathbb{H}_{n}=\\{Z=X+iY\in\textup{M}_{n}(\mathbb{C})|\overline{X}^{t}=X,\overline{Y}^{t}=Y>0\\}.$

We fix an embedding $\displaystyle K\hookrightarrow\mathbb{C}$. Then, an
element $\displaystyle g=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\in
U_{n}(K)\hookrightarrow U_{n}(\mathbb{C})$ of the unitary group acts on the
above upper half plane via the action

$Z\longmapsto g\langle Z\rangle:=(AZ+B)(CZ+D)^{-1}.$

The usual factor of automorphy is denoted by $\displaystyle
j(g,Z)=\textup{det}(CZ+D)$.

We will be writing $\displaystyle\Gamma_{n}$ for the Hermitian modular group,
that is $\displaystyle\Gamma_{n}=U_{n}(\mathcal{O}_{K})$.

######  Definition 2.3.

By a Hermitian modular form of weight $\displaystyle k\in\mathbb{Z}$, we mean
a holomorphic function $\displaystyle
F:\mathbb{H}_{n}\longrightarrow\mathbb{C}$ such that

$F\left(g\langle Z\rangle\right)=j(g,Z)^{-k}F(Z),$

for all $\displaystyle g\in\Gamma_{n}$.

If $\displaystyle n=1$ we further require that $\displaystyle F$ is
holomorphic at infinity.

It is well known that the set of all such forms constitutes a finite
dimensional space, which we call $\displaystyle M_{n}^{k}$.

Thanks to our assumption for $\displaystyle n=1$ and because of the Köcher
principle for $\displaystyle n\geq 2$, each such $\displaystyle F$ admits a
Fourier expansion

$F(Z)=\sum_{N}a(N)e(\textup{tr}(NZ))$

where $\displaystyle N$ runs through all the semi-integral non-negative
Hermitian matrices

$N\in\left\\{(n_{ij})\geq 0\mid
n_{ii}\in\mathbb{Z},n_{ij}=\overline{n_{ji}}\in\frac{1}{2}\mathcal{O}_{K}\right\\}.$

$\displaystyle F$ is called a cusp form if $\displaystyle a(N)\neq 0$ only for
$\displaystyle N$ positive definite. We denote the space of cusp forms by
$\displaystyle S_{n}^{k}$.

######  Definition 2.4.

The Petersson inner product for two Hermitian modular forms $\displaystyle
F,G$ is given by

$\int_{\Gamma_{n}\backslash\mathbb{H}_{n}}{F(Z)\overline{G(Z)}(\det
Y)^{k}d^{*}Z}$

where $\displaystyle d^{*}Z=(\det Y)^{-2n}dXdY$ with $\displaystyle Z=X+iY$.

It is well known that such an integral is always defined when one of the
$\displaystyle F$ or $\displaystyle G$ is a cusp form. By now partitioning

$Z=\begin{pmatrix}\tau&z_{1}\\\ z_{2}&\omega\end{pmatrix},$

with $\displaystyle\tau\in\mathbb{H}_{n-1},\omega\in\mathbb{H}_{1}$ and
$\displaystyle z_{1},z_{2}\in\mathbb{C}$, we can consider the Fourier
expansion of $\displaystyle F$ with respect to the variable
$\displaystyle\omega$:

$F(Z)=\sum_{m=1}^{\infty}\phi_{m}(\tau,z_{1},z_{2})e(m\omega).$

The functions
$\displaystyle\phi_{m}:\mathbb{H}_{1}\times\mathbb{C}^{2}\longrightarrow\mathbb{C}$
are called the Fourier$\displaystyle-$Jacobi coefficients of $\displaystyle
F$.
For $\displaystyle R$ as above we consider the following subgroups of
$\displaystyle U_{n}(R)$:

$P_{n,r}(R)=\left\\{\begin{pmatrix}*&*\\\ 0_{n-r,n+r}&*\end{pmatrix}\in
U_{n}(R)\right\\},$ $C_{n,r}(R)=\left\\{\begin{pmatrix}*&*\\\
0_{n+r,n-r}&*\end{pmatrix}\in U_{n}(R)\right\\}.$

In particular when $\displaystyle R=\mathcal{O}_{K}$, we write $\displaystyle
P_{n,r},C_{n,r}$. Let also $\displaystyle\Gamma_{n}=U_{n}\cap
M_{2n}(\mathcal{O}_{K})$.

We will be particularly interested in the parabolic subgroup $\displaystyle
P_{n,n-1}(K)\leq U_{n}(K)$, mainly because we can treat
Fourier$\displaystyle-$Jacobi forms as a special type of modular forms with
respect to this subgroup. Let us make this more explicit. We first start with
the definition of the slash operator.

######  Definition 2.5.

Let $\displaystyle n\geq 1$ and $\displaystyle k$ be any integer. Then, for
any function $\displaystyle F$ on $\displaystyle\mathbb{H}_{n+1}$ and a matrix
$\displaystyle g=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\in U_{n+1}(K)$, we
define

$(F\mid_{k}g)(Z):=\textup{det}(CZ+D)^{-k}F(g\langle Z\rangle).$

We now write $\displaystyle\Gamma_{n,1}=P_{n+1,n}(\mathcal{O}_{K})$, the group
of integral points of the parabolic $\displaystyle P_{n+1,n}$. We then have
the following definition

######  Definition 2.6.

Let $\displaystyle n\geq 1$. A holomorphic function $\displaystyle F$ on
$\displaystyle\mathbb{H}_{n+1}$ is a modular form of weight $\displaystyle k$
with respect to the parabolic subgroup $\displaystyle\Gamma_{n,1}$ if the
following conditions hold:

  * •

$\displaystyle F\mid_{k}M=F$ for any $\displaystyle M\in\Gamma_{n,1}$

  * •

The function $\displaystyle F(Z)$ is bounded in the domain
$\displaystyle\textup{Im}(Z)\geq c>0$.

We note here that we can relax the second condition if $\displaystyle n\geq
2$. This again follows by Köcher’s principle. The space of all such forms will
be denoted by $\displaystyle M_{n,1}^{k}$. Again, each such $\displaystyle F$
has a Fourier expansion as above and we call $\displaystyle F$ a cusp form if
$\displaystyle a(N)\neq 0$ only for positive definite matrices $\displaystyle
N$.

We can now give the definition of Fourier$\displaystyle-$Jacobi forms:

######  Definition 2.7.

A holomorphic function $\displaystyle\phi$ on
$\displaystyle\mathbb{H}_{n}\times\mathbb{C}^{n}\times\mathbb{C}^{n}$ is said
to be a Jacobi form of genus $\displaystyle n$, weight $\displaystyle k$ and
index $\displaystyle m$ if the function
$\displaystyle\tilde{\phi}\left(\begin{pmatrix}\tau&z_{1}\\\
z_{2}^{t}&\omega\end{pmatrix}\right):=\phi(\tau,z_{1},z_{2})e(m\omega)$, where
$\displaystyle\omega\in\mathbb{H}_{1}$ is chosen so that
$\displaystyle\begin{pmatrix}\tau&z_{1}\\\
z_{2}^{t}&\omega\end{pmatrix}\in\mathbb{H}_{n+1}$, is a modular form with
respect to the group $\displaystyle\Gamma_{n,1}$. The space of such forms is
denoted by $\displaystyle J_{k,m}^{n}$ and we will call
$\displaystyle\tilde{\phi}$ a $\displaystyle P-$form.

Let us now restrict ourselves to the case $\displaystyle n=2$:

######  Definition 2.8.

The Petersson inner product of two Fourier$\displaystyle-$Jacobi forms
$\displaystyle\phi_{m},\psi_{m}\in J_{k,m}^{2}$ is defined as

$\langle\phi_{m},\psi_{m}\rangle=\int_{\mathcal{F}^{J}}\phi_{m}(\tau,z_{1},z_{2})\overline{\psi_{m}(\tau,z_{1},z_{2})}v^{k}e^{-\pi
m|z_{1}-\overline{z_{2}}|^{2}/v}d\mu,$

where $\displaystyle d\mu=v^{-4}dudvdx_{1}dy_{1}dx_{2}dy_{2}$ with
$\displaystyle\tau=u+iv$, $\displaystyle z_{j}=x_{j}+iy_{j}$ for
$\displaystyle j=1,2$ and $\displaystyle\mathcal{F}^{J}$ is a fundamental
domain for the action of $\displaystyle P_{2,1}$ on
$\displaystyle\mathbb{H}_{1}\times\mathbb{C}^{2}$.

We also have the following definition:

######  Definition 2.9.

Let $\displaystyle\phi_{m},\psi_{m}\in J_{k,m}^{2}$ and denote by
$\displaystyle\tilde{\phi}_{m},\tilde{\psi}_{m}$ the $\displaystyle P-$forms
obtained as in Definition 2.7. We then define

$\langle\tilde{\phi}_{m},\tilde{\psi}_{m}\rangle_{\mathcal{A}}=\int_{\mathcal{Q}_{1,1}}F(Z)\overline{G(Z)}(\det{Y})^{k}d^{*}Z$

where $\displaystyle d^{*}Z=(\det{Y})^{-4}dXdY$ is the invariant element for
the action of the unitary group on $\displaystyle\mathbb{H}_{2}$ and

$\mathcal{Q}_{1,1}=\\{Z\in\mathbb{H}_{2}|(\tau,z_{1},z_{2})\in\mathcal{F}^{J}\textup{
and }|x_{\omega}|\leq 1/2\\}.$

There is a relation between the two inner products above, given in the
following Lemma:

######  Lemma 2.10.

Let $\displaystyle\phi_{m},\psi_{m}\in J_{k,m}^{2}$ and denote by
$\displaystyle\tilde{\phi}_{m},\tilde{\psi}_{m}$ the corresponding
$\displaystyle P-$forms. Then

$\langle\phi_{m},\psi_{m}\rangle=\beta_{k}t^{k-3}\langle\tilde{\phi}_{m},\tilde{\psi}_{m}\rangle_{\mathcal{A}}$

some constant $\displaystyle\beta_{k}$.

###### Proof.

We have

$\langle F,G\rangle_{\mathcal{A}}=\int_{Q_{1,1}}\phi(\tau,z_{1},z_{2})e^{2\pi
it\omega}\overline{\psi(\tau,z_{1},z_{2})}e^{-2\pi
it\overline{\omega}}(\det{Y})^{k-4}dXdY.$

Let now
$\displaystyle\tilde{y}_{\omega}=y_{\omega}-|z_{1}-\overline{z_{2}}|^{2}/4y_{\tau}$.
Then $\displaystyle\det{Y}=y_{\tau}\tilde{y}_{\omega}$. Hence, the above
integral can be written as

$\int_{\tilde{y}_{\omega}>0}\int_{\mathcal{F}^{J}}\int_{x_{\omega}\pmod{1}}\phi(\tau,z_{1},z_{2})e^{-4\pi
t(\tilde{y}_{\omega}+|z_{1}-\overline{z_{2}}|^{2}/4y_{\tau})}\overline{\psi(\tau,z_{1},z_{2})}(y_{\tau}\tilde{y}_{\omega})^{k-4}d\tau
dz_{1}dz_{2}d\tilde{y}_{\omega}dx_{\omega}$
$=\langle\phi,\psi\rangle\int_{\tilde{y}_{\omega}>0}e^{-4\pi
t\tilde{y}_{\omega}}\tilde{y}_{\omega}^{k-4}d\tilde{y}_{\omega}=(4\pi
t)^{3-k}\Gamma(k-3)\langle\phi,\psi\rangle$

so the result follows with
$\displaystyle\beta_{k}=(4\pi)^{k-3}\Gamma(k-3)^{-1}$. ∎

##  3\. Hecke algebras and $\displaystyle L-$functions

In this section, we give an account of a general Hecke theory we will need. We
follow Gritsenko in [8]. In order to simplify the exposition, we restrict
ourselves to the case $\displaystyle n=2$. We start by defining the groups of
similitude:

$S^{2}=\\{g\in M_{4}(K)\mid J_{2}[g]=\mu(g)J_{2},\mu(g)>0\\}$
$S^{2}_{p}=\\{g\in S^{2}\cap
M_{4}(\mathcal{O}[p^{-1}])\mid\mu(g)=p^{\delta},\delta\in\mathbb{Z}\\}.$

It is then well-known that the sets $\displaystyle
H(\Gamma_{2},S^{2}),H(\Gamma_{2},S^{2}_{p})$ are Hecke pairs and we can define
the corresponding Hecke rings, which we will denote by $\displaystyle H^{2}$
and $\displaystyle H_{p}^{2}$. From [8, Corollary $\displaystyle 2.2$], we can
decompose the global Hecke ring into the tensor product of $\displaystyle
p-$rings as follows:

$H(\Gamma_{2},S^{2})=\bigotimes_{p}H(\Gamma_{2},S^{2}_{p})$

We start with a very general lemma regarding the embeddings of Hecke algebras.

######  Lemma 3.1.

Let $\displaystyle(\Gamma_{0},S_{0})$ and $\displaystyle(\Gamma,S)$ be two
Hecke pairs. We assume that

$\Gamma_{0}\subset\Gamma,\,\,\,\Gamma S_{0}=S,\,\,\,\Gamma\cap
S_{0}S_{0}^{-1}\subset\Gamma_{0}$

Then, given an arbitrary element $\displaystyle X\in H(\Gamma,S)$, according
to the second condition, we can write it as

$X=\sum_{i}a_{i}(\Gamma g_{i})$

with $\displaystyle g_{i}\in S_{0}$. Then, if we set

$\epsilon(X)=\sum_{i}a_{i}(\Gamma_{0}g_{i})$

then $\displaystyle\epsilon$ does not depend on the selection of the elements
$\displaystyle g_{i}\in S_{0}$ and is an embedding (as a ring homomorphism) of
the Hecke algebra $\displaystyle H(\Gamma,S)$ to $\displaystyle
H(\Gamma_{0},S_{0})$.

###### Proof.

See page $\displaystyle 2890$ in [8]. ∎

It is also well-known that each $\displaystyle p-$ring is isomorphic to the
Hecke ring over the corresponding local field, and the structure of these
rings depends on the decomposition of the prime $\displaystyle p$ in
$\displaystyle\mathcal{O}_{K}$. In order to work locally, we give the
following definitions:

$K_{p}=K\otimes\mathbb{Q}_{p},\mathcal{O}_{p}=\mathcal{O}_{K}\otimes\mathbb{Z}_{p},\Phi_{p}=(2i)^{-1}\begin{pmatrix}0&-E_{2}\\\
E_{2}&0\end{pmatrix}$

which denote the algebra over $\displaystyle\mathbb{Q}_{p}$, the maximal
lattice and a Hermitian form on the vector space $\displaystyle K_{p}$
respectively. We also define the unitary group $\displaystyle G_{p}^{2}$ and a
maximal compact subgroup $\displaystyle U_{p}^{2}$ by

$G_{p}^{2}=\\{g\in\textup{GL}_{4}(K_{p})\mid
g^{*}\Phi_{p}g=\mu(g)\Phi_{p},\mu(g)\in\mathbb{Q}_{p}^{*}\\}$
$U_{p}^{2}=\\{g\in G_{p}^{2}\cap
M_{4}(\mathcal{O}_{p})\mid\mu(g)\in\mathbb{Z}_{p}^{*}\\}$

where $\displaystyle g^{*}=(g_{ji})^{\sigma}$ with $\displaystyle\sigma$ is
the canonical involution of the algebra $\displaystyle K_{p}$, determined by
the behaviour of the prime $\displaystyle p$ in $\displaystyle K$ (split,
inert or ramified).

We then have the following proposition:

######  Proposition 3.2.

The local Hecke ring $\displaystyle H(U_{p}^{2},G_{p}^{2})$ is isomorphic to
the $\displaystyle p-$ring $\displaystyle H(\Gamma_{2},S_{p}^{2})$.

###### Proof.

See [8, Proposition $\displaystyle 2.3$]. ∎

Let us now recall the definition of the so-called spherical or Satake mapping.
We follow [8], [9] and [20]. We need to distinguish between the cases
$\displaystyle p$ is inert or $\displaystyle p=2$ and $\displaystyle p$
splits.
In the first case, we know that given $\displaystyle g\in G_{p}^{2}$ we have
the double coset decomposition

$U_{p}^{(2)}gU_{p}^{(2)}=\sum_{i}U_{p}^{(2)}M^{(m_{i})}N_{i},$

where $\displaystyle N_{i}$ is a unipotent matrix, $\displaystyle
m_{i}=(m_{i_{1}},m_{i_{2}};m_{i_{0}})$ and

$M^{(m_{i})}=\begin{pmatrix}p^{m_{i_{0}}}(\overline{D}^{t})^{-1}&0\\\
0&D\end{pmatrix},D=\textup{diag}(\pi^{m_{i_{1}}},\pi^{m_{i_{2}}}),$

with $\displaystyle\pi=p$ if $\displaystyle p$ is inert or
$\displaystyle\pi=(1+i)$ if $\displaystyle p=2$. We then define

$\Phi:H(U_{p}^{(2)},G_{p}^{(2)})\longrightarrow\mathbb{Q}^{W_{4}}[x_{0}^{\pm
1},x_{1}^{\pm 1},x_{2}^{\pm 1}]$

via

$\Phi(U_{p}^{(2)}gU_{p}^{(2)})=\sum_{i}x_{0}^{m_{i_{0}}}\prod_{j=1}^{2}(x_{j}q^{-j})^{m_{i_{j}}}$

where the ring $\displaystyle\mathbb{Q}^{W_{2}}[x_{0}^{\pm 1},x_{1}^{\pm
1},x_{2}^{\pm 1}]$ denotes the ring of polynomials invariant with respect to
the permutation of the variables $\displaystyle x_{0},x_{1},x_{2}$ under the
transformations $\displaystyle w^{(j)},j=1,2$, defined by

$x_{0}\longmapsto p^{-1}x_{0}x_{i}^{e},x_{i}\longmapsto
p^{2/e}x_{i}^{-1},x_{j}\longmapsto x_{j},(i\neq j)$

with $\displaystyle q$ denoting the number of elements in the residue field
$\displaystyle\mathbb{Q}(i)\otimes\mathbb{Q}_{p}$ and $\displaystyle e$ is the
ramification index of the prime $\displaystyle p$.

For the case of decomposable $\displaystyle p$, the definition of the
spherical mapping is different. In particular, there is an isomorphism

$\rho:H(U_{p}^{2},G_{p}^{2})\longrightarrow
H(\textup{GL}_{4}(\mathbb{Z}_{p}),\textup{GL}_{4}(\mathbb{Q}_{p}))[x^{\pm 1}]$

(see [8, Proposition 2.4]). We can then define the Satake mapping
$\displaystyle\Omega$ for $\displaystyle
H(\textup{GL}_{4}(\mathbb{Z}_{p}),\textup{GL}_{4}(\mathbb{Q}_{p}))$ in an
analogous way as for the case $\displaystyle p$ inert or $\displaystyle p=2$.
Let us describe it here. Given an element $\displaystyle X\in
H(\textup{GL}_{4}(\mathbb{Z}_{p}),\textup{GL}_{4}(\mathbb{Q}_{p}))$, we know
that we can write it as

$X=\sum_{i}a_{i}\textup{GL}_{4}(\mathbb{Z}_{p})g_{i}$

where $\displaystyle g_{i}=\begin{pmatrix}p^{d_{i1}}&*&\cdots&*\\\
0&p^{d_{i2}}&\cdots&*\\\ \vdots&\vdots&\vdots&\vdots\\\
0&0&\cdots&p^{d_{i4}}\end{pmatrix}$ and $\displaystyle a_{i}\in\mathbb{C}$.

Then the mapping $\displaystyle\Omega$ given by

$\Omega(X)=\sum_{i}\prod_{j=1}^{4}(x_{j}p^{-j})^{d_{ij}}$

defines an isomorphism between $\displaystyle
H(\textup{GL}_{4}(\mathbb{Z}_{p}),\textup{GL}_{4}(\mathbb{Q}_{p}))$ and
$\displaystyle\mathbb{Q}^{\textup{sym}}[x_{1}^{\pm 1},\cdots,x_{4}^{\pm 1}]$
of symmetric polynomials.

We then define the Satake mapping $\displaystyle\Phi$ in this case as the
composition $\displaystyle\Phi=\Omega\circ\rho$.

Let us now define the parabolic Hecke algebras we will need. Let
$\displaystyle S^{1,1},S_{p}^{1,1},\Gamma_{1,1}$ denote the intersection of
the groups $\displaystyle S^{2},S_{p}^{2},\Gamma_{2}$ with the parabolic
subgroup $\displaystyle P_{2,1}$ respectively. Again, the pairs
$\displaystyle(\Gamma_{1,1},S^{1,1})$ and
$\displaystyle(\Gamma_{1,1},S_{p}^{1,1})$ are Hecke pairs and we can then
define the Hecke rings

$H^{1,1}=H_{\mathbb{Q}}(\Gamma_{1,1},S^{1,1}),H_{p}^{1,1}=H_{\mathbb{Q}}(\Gamma_{1,1},S_{p}^{1,1}).$

Since $\displaystyle\Gamma_{2}S_{p}^{1,1}=S_{p}^{2}$ and after writing an
element $\displaystyle X\in H_{p}^{2}$ as

$X=\sum_{i}a_{i}\Gamma_{2}g_{i}$

with $\displaystyle g_{i}\in S_{p}^{2}$ we can define an embedding

$X\longmapsto\epsilon(X)=\sum_{i}a_{i}\Gamma_{1,1}g_{i}$

using Lemma 3.1.

Given the way the Fourier-Jacobi forms are defined, these will be acted upon
by elements of the parabolic Hecke algebra $\displaystyle
H(\Gamma_{1,1},S^{1,1})$. The structure of these rings again depends on the
decomposition of the prime $\displaystyle p$ in
$\displaystyle\mathcal{O}_{K}$.

If $\displaystyle p$ is inert or $\displaystyle p=2$, then the structure of
the Hecke ring is constructed in a similar way as the corresponding ring for
the symplectic group of genus $\displaystyle 2$, see [13], [6] for example.

In the case of a decomposable $\displaystyle p$, however, the situation is
quite different. This follows from the fact that

$H(U_{p}^{2},G_{p}^{2})\cong
H(\textup{GL}_{4}(\mathbb{Z}_{p}),\textup{GL}_{4}(\mathbb{Q}_{p}))[x^{\pm 1}]$

This gives that the corresponding $\displaystyle p-$ring of the parabolic
Hecke algebra is isomorphic to the ring of polynomials of one variable with
coefficients from the Hecke ring of the parabolic subgroup

$P_{1,2,1}=\left\\{\begin{pmatrix}\pm 1&*&*\\\ 0&g&*\\\ 0&0&\pm
1\end{pmatrix}\in\textup{GL}_{4}(\mathbb{Z}_{p})\mid
g\in\textup{GL}_{2}(\mathbb{Z}_{p})\right\\}$

Properties of this ring have been investigated in [10] and that’s the ring
where our calculations about Hecke operators are going to occur.

Finally, let us now describe the action of elements of the above Hecke
algebras on modular forms. Let $\displaystyle F$ denote any modular form with
respect to the parabolic subgroup $\displaystyle\Gamma_{1,1}$ and let
$\displaystyle g=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\in S^{1,1}$. We then
define

$(F\mid g)(Z):=\mu(g)^{2k-4}\textup{det}(CZ+D)^{-k}F(g\langle Z\rangle).$

Let now $\displaystyle X=\Gamma_{1,1}\begin{pmatrix}*&*&*&*\\\ *&a&*&*\\\
*&0&*&*\\\ 0&0&0&b\end{pmatrix}\Gamma_{1,1}=\sum_{i}\Gamma_{1,1}g_{i}\in
H^{1,1}$, with $\displaystyle g_{i}\in S^{1,1}$. Then

$(F\mid X)(Z)=\sum_{i}(F\mid g_{i})(Z).$

Gritsenko gave the following very convenient definition of the signature.

######  Definition 3.3.

The signature of $\displaystyle X$ is defined as $\displaystyle s(X)=b/a$.

Using the signature $\displaystyle s:=s(X)$ of $\displaystyle X$ we can now
define its action on Fourier-Jacobi forms. This is the content of the
following Proposition.

######  Proposition 3.4.

Let $\displaystyle\phi\in J_{k,m}^{2}$ denote a Fourier-Jacobi form of index
$\displaystyle m$. Then, for $\displaystyle Z=\begin{pmatrix}\tau&z_{1}\\\
z_{2}&\omega\end{pmatrix}\in\mathbb{H}_{2}$, we define the action of
$\displaystyle X$ on $\displaystyle J_{k,m}^{2}$ via

$(\phi\mid X)(\tau,z_{1},z_{2}):=(\tilde{\phi}\mid
X)(Z)e\left(-\frac{m}{s}\right)$

with $\displaystyle\tilde{\phi}(Z):=\phi(\tau,z_{1},z_{2})e(m\omega)$ and
$\displaystyle\omega\in\mathbb{C}$ with
$\displaystyle\begin{pmatrix}\tau&z_{1}\\\
z_{2}&\omega\end{pmatrix}\in\mathbb{H}_{2}$. Then $\displaystyle\phi\mid X$
belongs to $\displaystyle J_{k,m/s}^{2}$ if $\displaystyle m/s$ is an integer
and is $\displaystyle 0$ otherwise.

###### Proof.

See [8, Lemma $\displaystyle 4.1$]. ∎

Now, if $\displaystyle F$ is a Hermitian modular form, we can write

$F\left(\begin{pmatrix}\tau&z_{1}\\\
z_{2}&\omega\end{pmatrix}\right)=\sum_{m=0}^{\infty}\phi_{m}(\tau,z_{1},z_{2})e(m\omega)$

For $\displaystyle X\in H^{1,1}$ as above, we have that $\displaystyle F\mid
X$ is a modular form with respect to $\displaystyle\Gamma_{1,1}$ and so we can
write

$(F\mid X)\left(\begin{pmatrix}\tau&z_{1}\\\
z_{2}&\omega\end{pmatrix}\right)=\sum_{m=0}^{\infty}\psi_{m}(\tau,z_{1},z_{2})e(m\omega).$

Therefore, there is an action of Hecke operators from $\displaystyle H^{1,1}$
on the Fourier-Jacobi forms coming from a Hermitian modular form
$\displaystyle F$ via

$\phi_{m}^{(F)}\mid\mid X=\psi_{m}^{(F\mid X)}.$

We note here that this action is extended to $\displaystyle P-$forms in the
obvious way.

Given now the definitions of the Hecke algebras above, we assume that
$\displaystyle G\in S_{2}^{k}$ is a Hecke eigenform for $\displaystyle H^{2}$,
i.e. it is an eigenfunction for all Hecke operators in $\displaystyle H^{2}$.
We remind the reader here that for a polynomial $\displaystyle U$ with
coefficients Hecke operators and $\displaystyle G$ a Hecke eigenform, we write
$\displaystyle U_{G}$ for the polynomial obtained when we substitute the
operators with their corresponding eigenvalues.

######  Definition 3.5.

The standard $\displaystyle L-$function attached to $\displaystyle G$ (see
also [21, Paragraph 20.6]) is defined as

$Z_{G}^{(2)}(s)=\prod_{p\textup{ inert or
}p=2}Z_{p,G}^{(2)}(p^{-2s})^{-1}\prod_{p=\pi\overline{\pi}}Z_{\pi,G}^{(2)}(p^{-s})^{-1}Z_{\overline{\pi},G}^{(2)}(p^{-s})^{-1}$

where for each inert prime $\displaystyle p$ or $\displaystyle p=2$,
$\displaystyle Z_{p,G}^{(2)}(t)=\Phi^{-1}(z_{p,G}^{(2)}(t))$ and for
$\displaystyle p=\pi\overline{\pi}$, $\displaystyle
Z_{\pi,G}^{(2)}(t)=\Phi^{-1}(z_{\pi,G}^{(2)}(t))$ and $\displaystyle
Z_{\overline{\pi},G}^{(2)}(t)=\Phi^{-1}(z_{\overline{\pi},G}^{(2)}(t))$, where

$z_{p,G}^{(2)}(t)=\begin{cases}\prod_{i=1}^{2}(1-p^{2}x_{i,p}t)(1-p^{4}x_{i,p}^{-1}t)&\textup{
if $\displaystyle p$ inert}\\\
\prod_{i=1}^{2}(1-px_{i}t)(1-p^{2}x_{i}^{-1}t)&\textup{ if $\displaystyle
p=2$}\end{cases}$

and

$z_{\pi,G}^{(2)}(t)=\prod_{i=1}^{4}(1-p^{-1}x_{i,p}t),z_{\overline{\pi},G}^{(2)}(t)=\prod_{i=1}^{4}(1-p^{4}x_{i,p}^{-1}t)$

where $\displaystyle x_{i,p}$ are the Satake parameters of $\displaystyle G$.

######  Definition 3.6.

The $\displaystyle L-$function attached to $\displaystyle G$ by Gritsenko in
[8] is defined as

$Q_{G}^{(2)}(s)=\prod_{p\textup{
inert}}(1+p^{k-2-s})^{-2}Q_{p,G}^{(2)}(p^{-s})^{-1}\prod_{p\textup{ splits or
}p=2}Q_{p,G}^{(2)}(p^{-s})^{-1}$

where $\displaystyle Q_{p,G}^{(2)}(t)=\Phi^{-1}(q_{p,G}^{(2)}(t))$ with

$q_{p,G}^{(2)}(t)=\begin{cases}(1-x_{0,p}t)\prod_{r=1}^{2}\prod_{1\leq
i_{1}<i_{2}\leq 2}(1-p^{-r}x_{i_{1},p}\cdots x_{i_{r},p}x_{0,p}t)&\textup{ if
$\displaystyle p$ is inert}\\\ (1-x_{0,p}t)\prod_{r=1}^{2}\prod_{1\leq
i_{1}<i_{2}\leq 2}(1-p^{-r}(x_{i_{1},p}\cdots
x_{i_{r},p})^{2}x_{0,p}t)&\textup{ if $\displaystyle p=2$}\\\ \prod_{1\leq
i<j\leq 4}(1-p^{-3}x_{i,p}x_{j,p}xt)&\textup{ if $\displaystyle p$
splits}\end{cases}$

Let us now define the so-called Maass space for the case of Hermitian cusp
forms. We mainly follow [7] and for the definition we will use is [7, Lemma
$\displaystyle 2.4$].

######  Definition 3.7.

The Maass space is the space

$\left\\{F\left(\begin{pmatrix}\tau&z_{1}\\\
z_{2}&\omega\end{pmatrix}\right)=\sum_{m=1}^{\infty}\left(\phi(\tau,z_{1},z_{2})|_{k}T_{-}(m)\right)e^{2\pi
im\omega}m^{3-k}|\phi\in J_{k,1}^{2}\right\\}$

where $\displaystyle
T_{-}(m)=\sum_{a|b,ab=m}\Gamma_{1,1}\textup{diag}(a,m,b,1)\Gamma_{1,1}$ is an
element of the parabolic Hecke algebra $\displaystyle H^{1,1}$.

The main property of the Maass space is the following:

######  Proposition 3.8.

Let $\displaystyle F\in S_{2}^{k}$ belong in the Maass space defined and
assume $\displaystyle F$ is an eigenfunction for the Hecke algebra
$\displaystyle H^{2}$. Then, there exists $\displaystyle f\in
S_{k-1}\left(\Gamma_{0}(4),\left(\frac{-4}{*}\right)\right)$, which is also an
eigenfunction for its corresponding Hecke algebra, such that

$Q_{F}^{(2)}(s)=\zeta(s-k+1)L\left(s-k+2,\left(\frac{-4}{*}\right)\right)\zeta(s-k+3)R_{f}(s)$

where $\displaystyle R_{f}(s)$ denotes the symmetric square function of
$\displaystyle f$ defined as follows. Let $\displaystyle f(\tau)=\sum_{n\geq
1}a(n)e(n\tau)$ be the Fourier expansion of $\displaystyle f$ and assume we
write

$1-a(p)t+\left(\frac{-4}{p}\right)p^{k-2}t^{2}=(1-\alpha_{p}t)\left(1-\beta_{p}\left(\frac{-4}{p}\right)t\right)$

We then define

$R_{f}(s)=(1-\alpha_{2}2^{-s})^{-1}(1-\beta_{2}2^{-s})^{-1}\prod_{p\neq
2}\left[(1-\alpha_{p}^{2}p^{-s})\left(1-\left(\frac{-4}{p}\right)\alpha_{p}\beta_{p}p^{-s}\right)(1-\beta_{p}^{2}p^{-s})\right]^{-1}$

We call $\displaystyle F$ the Maass lift of $\displaystyle f$.

###### Proof.

See [7] or the Appendix in [8]. ∎

We end this section by giving a Lemma regarding the correspondence of elliptic
cusp forms and Hermitian modular forms of degree $\displaystyle 1$, both as
analytic objects as well as Hecke eigenforms.

######  Lemma 3.9.

A Hermitian modular form of degree $\displaystyle 1$ and weight $\displaystyle
k$ with $\displaystyle k\equiv 0\pmod{4}$ can be considered as a classical
modular form of the same weight(i.e. for the group $\displaystyle
SL_{2}(\mathbb{Z})$) and vice versa. Also, a classical modular form which is a
normalised eigenform for the Hecke algebra $\displaystyle
H(\textup{GL}_{2}(\mathbb{Z}),\textup{GL}_{2}(\mathbb{Q}))$ is also a
normalised eigenform for $\displaystyle H(\Gamma_{1},S^{1})$, when it is
considered as a Hermitian modular form and vice versa.

###### Proof.

We have that $\displaystyle\Gamma_{1}=SL_{2}(\mathbb{Z})\cdot\\{\alpha\cdot
1_{2}\mid\alpha\in\mathcal{O}_{K}^{\times}\\}$ and the corresponding upper
half planes are the same. So, holomorphicity is equivalent(including
infinity). Now, for the invariance condition, the one direction is trivial, as
$\displaystyle SL_{2}(\mathbb{Z})\subseteq\Gamma_{1}$. For the other one, let
$\displaystyle\gamma\in\Gamma_{1}$ and write
$\displaystyle\gamma=\alpha\delta$ with
$\displaystyle\delta=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
SL_{2}(\mathbb{Z})$ and $\displaystyle\alpha\in\mathcal{O}_{K}^{\times}$. Then

$(F|_{k}\gamma)(Z)=(\alpha cz+\alpha d)^{-k}F(Z)=(F|_{k}\delta)(Z)$

as $\displaystyle k\equiv 0\pmod{4}$ and
$\displaystyle\mathcal{O}_{K}^{\times}=\\{\pm 1,\pm i\\}$.

Assume now that we start with a normalised (i.e $\displaystyle a(1)=1$ in the
Fourier expansion) Hermitian cusp form $\displaystyle h$ of degree
$\displaystyle 1$, which we further take to be an eigenform for the
corresponding Hecke algebra. The canonical embedding of
$\displaystyle\textup{GL}_{2}(\mathbb{Q})$ into $\displaystyle S^{1}$, the
group of similitude of degree $\displaystyle 1$, induced from the embedding
$\displaystyle\mathbb{Q}\hookrightarrow K$, allows us to see $\displaystyle h$
as a classical normalised Hecke eigenform.

But the converse is also true, that is, if we start with $\displaystyle h$ a
classical normalised Hecke eigenform, then it is also a normalised Hermitian
eigenform of degree $\displaystyle 1$. Indeed, since the Hecke operators of
the Hermitian Hecke algebra are normal, we know that the space of Hermitian
cusp forms is diagonalizable with a basis of normalised eigenforms. But then
this basis has to coincide with the basis derived by diagonalizing the action
of the classical Hecke algebra, thanks to the multiplicity one theorem for the
classical Hecke algebra. ∎

######  Remark.

From now on, we will use the terms elliptic modular form(or classical modular
form) and Hermitian modular form of degree $\displaystyle 1$ interchangeably.

##  4\. Integral Representation and Dirichlet Series

Let $\displaystyle n\geq 1$. We start with the following definition of the
Eisenstein series we will need:

######  Definition 4.1.

Let $\displaystyle F\in S_{r}^{k}$ with $\displaystyle k\equiv 0\pmod{4}$. The
Klingen-type Eisenstein series with respect to the parabolic subgroup
$\displaystyle C_{n,r}$ attached to $\displaystyle F$ is given by:

$E_{n,r}^{k}(Z,F;s)=\sum_{\gamma\in
C_{n,r}\backslash\Gamma_{n}}F(\gamma\langle
Z\rangle_{*})j(\gamma,Z)^{-k}\left(\frac{\det\operatorname{Im}\gamma\langle
Z\rangle}{\det{\operatorname{Im}\gamma\langle Z\rangle_{*}}}\right)^{s}$

where $\displaystyle*$ denotes the lower right $\displaystyle r\times r$ part
of the matrix.

This series is absolutely and uniformly convergent for $\displaystyle
k+\textup{Re}(s)>2(n+r)$ (see [17, page 152].

Now, given $\displaystyle F,G\in S_{n}^{k}$ and $\displaystyle h\in
S_{1}^{k}$, which are all eigenforms for their corresponding Hecke algebras,
the expression of interest is

$\Phi(F,G,h;s)=\left\langle\left\langle\left\langle
E_{2n+1,0}^{k}\left(\begin{pmatrix}z_{1}&&\\\ &z_{2}&\\\
&&z_{3}\end{pmatrix};s\right),F(z_{3})\right\rangle,G(z_{2})\right\rangle,h(z_{1})\right\rangle$

We have the following algebraic result:

######  Proposition 4.2.

With notation as above and assume further that $\displaystyle
F$,$\displaystyle G$ and $\displaystyle h$ have algebraic Fourier
coefficients. For $\displaystyle k>4n+2$ we have

$\displaystyle\frac{\Phi(F,G,h,0)}{\langle F,F\rangle\langle G,G\rangle\langle
h,h\rangle}\in\overline{\mathbb{Q}}.$

###### Proof.

This can be shown exactly as [13, Theorem 1.9]. In the proof there, a result
of Böcherer is used on the algebraic decomposition of the space of modular
forms as an orthogonal product of the space of cusps forms and of the
Eisenstein series, i.e.

$\displaystyle
M_{n}^{k}(\overline{\mathbb{Q}})=S_{n}^{k}(\overline{\mathbb{Q}})\oplus\textup{Eis}_{n}^{k}(\overline{\mathbb{Q}}).$

Such a result is also available for unitary groups [21, Theorem $\displaystyle
27.14$]. ∎

Actually, one can give an even stronger statement of the proposition above,
namely establish even a reciprocity law on the action of the absolute Galois
group. The statement is similar to [13, Theorem $\displaystyle 1.9$] of Heim.
The main point here is that we can establish an algebraicity result for
special values of $\displaystyle L-$functions if we can relate the expression
above to an Euler product expression. This is the main motivation of the
present paper.

By using the well-known doubling method for unitary groups, as for example is
studied in [21, Equation 24.29(a)], we know that the first inner product is
related to a Klingen-type Eisenstein series as defined above. That is,

$\left\langle E_{2n+1,0}^{k}\left(\begin{pmatrix}z_{1}&&\\\ &z_{2}&\\\
&&z_{3}\end{pmatrix};s\right),F(z_{3})\right\rangle=\nu(s)\frac{Z_{F}^{(n)}(2s+k-n)}{\prod_{i=0}^{2n-1}L(4s+2k,\chi^{i})}E_{n+1,n}^{k}\left(\begin{pmatrix}z_{1}&0\\\
0&z_{2}\end{pmatrix},F;s\right)$

where $\displaystyle Z_{F}^{(n)}$ is the standard $\displaystyle L-$function
attached to $\displaystyle F$, $\displaystyle\chi$ is the non-trivial
quadratic character attached to the extension $\displaystyle K/\mathbb{Q}$ and
$\displaystyle\nu(s)$ is an expression involving Gamma factors (the explicit
expression is given in [21]). So, our focus shifts to computing

$\left\langle\left\langle E_{n+1,n}^{k}\left(\begin{pmatrix}W&0\\\
0&Z\end{pmatrix},F;s\right),G(Z)\right\rangle,h(W)\right\rangle.$

Given the definition of the Eisenstein series, we will start by finding
representatives for $\displaystyle C_{n+1,n}\backslash\Gamma_{n+1}$. We begin
by first finding representatives for $\displaystyle C_{n+1,n}(K)\backslash
U_{n}(K)$.

######  Proposition 4.3.

The left coset space $\displaystyle C_{n+1,n}(K)\backslash U_{n}(K)$ has
representatives

$S_{1}=C_{1,0}(K)\backslash U_{1}(K)\times 1_{2n}$
$S_{2}=\pi_{n}\cdot(1_{2}\times C_{n,n-1}(K)\backslash U_{n}(K))$
$S_{3}=\xi_{n}\cdot(C_{1,0}(K)\backslash U_{1}(K)\times((T\times
1_{2(n-1)})\cdot C_{n,n-1}(K)\backslash U_{n}(K))$

where

$\pi_{n}=\begin{pmatrix}0&1\\\ 1&0\\\ &&1_{n-1}\\\ &&&0&1\\\ &&&1&0\\\
&&&&&1_{n-1}\end{pmatrix},\xi_{n}=\begin{pmatrix}1&0\\\ 1&1\\\ &&1_{n-1}\\\
&&&1&-1\\\ &&&0&1\\\ &&&&&1_{n-1}\end{pmatrix},T=\left\\{\begin{pmatrix}a&0\\\
0&\overline{a}^{-1}\end{pmatrix}|a\in K^{\times}\right\\}$

###### Proof.

The proof proceeds in the same way as in [13, Proposition $\displaystyle
2.1$]. ∎

We now want to pull these representatives back to representatives for
$\displaystyle C_{n+1,n}\backslash\Gamma_{n+1}$.

######  Corollary 4.4.

The left coset space $\displaystyle C_{n+1,n}\backslash\Gamma_{n+1}$ has
representatives

$T_{1}=C_{1,0}\backslash\Gamma_{1}\times 1_{2n}$
$T_{2}=\pi_{n}\cdot(1_{2}\times C_{n,n-1}\backslash\Gamma_{n})$
$T_{3}=\bigsqcup_{p,q}(\xi^{p,q}\times
1_{2(n-1)})\cdot(C_{1,0}\backslash\Gamma_{1}\times
C_{n,n-1}\backslash\Gamma_{n})$

where $\displaystyle p,q\in\mathbb{Z}[i]\backslash\\{0\\}$ with
$\displaystyle(p,q)=1$ and $\displaystyle q=u+iv,u>0,v\geq 0$ and
$\displaystyle\xi^{p,q}=\begin{pmatrix}*&*&0&0\\\ q&p&0&0\\\
0&0&\overline{p}&-\overline{q}\\\ 0&0&*&*\end{pmatrix}$, with
$\displaystyle\xi^{p,q}\times 1_{2(n-1)}\in\Gamma_{n+1}$.

###### Proof.

There is one to one correspondence between $\displaystyle C_{n,r}(K)\backslash
U_{n}(K)$ with $\displaystyle C_{n,r}\backslash\Gamma_{n}$ for all
$\displaystyle r,n$, due to the fact that $\displaystyle K$ has class number
$\displaystyle 1$. It suffices then to pull $\displaystyle S_{3}$ back.

We therefore need to find a matrix $\displaystyle p_{a}$ in
$\displaystyle\bigcap_{r\leq n}C_{n+1,r}(K)$ such that

$p_{a}\cdot\xi_{n}\cdot\left(1_{2}\times\begin{pmatrix}a&0\\\
0&\overline{a}^{-1}\end{pmatrix}\times 1_{2(n-1)}\right)\in\Gamma_{n+1}$

We parametrize $\displaystyle K$ as

$\left\\{\frac{p}{q}\mid(p,q)=1,q=u+iv,u>0,v\geq 0\right\\}.$

All these elements are different as $\displaystyle p,q$ are in
$\displaystyle\mathbb{Z}[i]$ with the above conditions and their union is
$\displaystyle K$.

For $\displaystyle a=p/q$ as above, we define

$p_{p,q}=\begin{pmatrix}p^{-1}&y&0&0\\\ 0&q&0&0\\\ 0&0&\overline{p}&0\\\
0&0&l&\overline{q}^{-1}\end{pmatrix}\times 1_{2(n-1)}$

with $\displaystyle l,q$ chosen so that $\displaystyle l\overline{q}\equiv
1\pmod{\overline{p}}$ and then $\displaystyle y=-q\overline{l}/p$. We can then
show that this actually belongs to $\displaystyle U_{n+1}$ and clearly to
$\displaystyle C_{n+1,n}(K)$. Hence, by computing the product, we obtain the
required representatives. ∎

Our aim is to first show an analogue of [13, Theorem $\displaystyle 2.3$].

We denote by $\displaystyle
Y=\operatorname{Im}Z=\frac{1}{2i}(Z-^{t}\overline{Z})$. Then
$\displaystyle\operatorname{Im}M\langle Z\rangle=Y[(CZ+D)^{-1}]$, so

$\det\operatorname{Im}M\langle
Z\rangle=|\det(CZ+D)|^{-2}\det\operatorname{Im}Z=|j(M,Z)|^{-2}\det\operatorname{Im}Z$

######  Proposition 4.5.

Let $\displaystyle k\equiv 0\pmod{4}$ and $\displaystyle\textup{Re}(s)\gg 0$.
Let also $\displaystyle F\in S_{n}^{k}$, $\displaystyle z_{1}\in\mathbb{H}$
and $\displaystyle z_{2}\in\mathbb{H}_{n}$. We then have

$E_{n+1,n}^{k}\left([z_{1},z_{2}],F;s\right)=E_{1}^{k}(z_{1},s)F(z_{2})+E_{n,n-1}^{k}(z_{2},F_{z_{1}};s)+$
$+\delta(z_{1})^{s}\delta(z_{2})^{s}\sum_{p,q}\sum_{\begin{subarray}{c}\gamma_{1}\in
C_{1,0}\backslash\Gamma_{1}\\\ \gamma_{2}\in
C_{n,n-1}\backslash\Gamma_{n}\end{subarray}}\chi^{k,s}(\gamma_{1},z_{1})\chi^{k,s}(\gamma_{2},z_{2})F\left(\begin{pmatrix}N(q)\gamma_{1}\langle
z_{1}\rangle&&0\\\ 0&&0\end{pmatrix}+(\gamma_{2}\langle
z_{2}\rangle)\left[\begin{pmatrix}\overline{p}&&0\\\
0&&1_{n-1}\end{pmatrix}\right]\right)\times$
$\times\delta\left(\begin{pmatrix}N(q)\gamma_{1}\langle z_{1}\rangle&&0\\\
0&&0\end{pmatrix}+(\gamma_{2}\langle
z_{2}\rangle)\left[\begin{pmatrix}\overline{p}&&0\\\
0&&1_{n-1}\end{pmatrix}\right]\right)^{-s}$

where $\displaystyle F_{z_{1}}(\tau)=F([z_{1},\tau])\in S_{n-1}^{k}$,
$\displaystyle\chi^{k,s}(M,Z)=j(M,Z)^{-k}|j(M,Z)|^{-2s}$,
$\displaystyle\delta(Z)=\det\left(\operatorname{Im}Z\right)$ and
$\displaystyle p,q$ are summed as in Corollary 4.4.

###### Proof.

We can write

$E_{n+1,n}^{k}\left([z_{1},z_{2}],F;s\right)=\delta(z_{1})^{s}\delta(z_{2})^{s}\sum_{M\in
C_{n+1,n}\backslash\Gamma_{n+1}}\chi^{k,s}(M,[z_{1},z_{2}])F(M\langle[z_{1},z_{2}]\rangle_{*})\delta\left(M\langle[z_{1},z_{2}]\rangle_{*}\right)^{-s}.$

For the representatives of $\displaystyle T_{1},T_{2}$ the proof is exactly
the same as in [13, Theorem $\displaystyle 2.3$].

For a representative $\displaystyle M$ of $\displaystyle T_{3}$, write
$\displaystyle M=(\xi\times 1_{2(n-1)})(\gamma_{1}\times\gamma_{2})$ with
$\displaystyle\gamma_{1}\in C_{1,0}\backslash\Gamma_{1}$ and
$\displaystyle\gamma_{2}\in C_{n,n-1}\backslash\Gamma_{n}$. We write
$\displaystyle\gamma_{2}\langle z_{2}\rangle=\begin{pmatrix}x_{1}&x_{2}\\\
x_{3}&x_{4}\end{pmatrix}$ and we then have

$(M\langle[z_{1},z_{2}]\rangle)_{*}=((\xi\times
1_{2(n-1)})[\gamma_{1}z_{1},\gamma_{2}z_{2}])_{*}=\left(\begin{pmatrix}*&*&0\\\
q&p&0\\\ 0&0&1\end{pmatrix}\begin{pmatrix}\gamma_{1}\langle
z_{1}\rangle&0&0\\\ 0&x_{1}&x_{2}\\\
0&x_{3}&x_{4}\end{pmatrix}\begin{pmatrix}*&\overline{q}&0\\\
0&\overline{p}&0\\\ 0&0&1\end{pmatrix}\right)_{*}=$
$=\begin{pmatrix}N(q)\gamma_{1}\langle z_{1}\rangle+N(p)x_{1}&&px_{2}\\\
\overline{p}x_{3}&&x_{4}\end{pmatrix}=\begin{pmatrix}N(q)\gamma_{1}\langle
z_{1}\rangle&&0\\\ 0&&0\end{pmatrix}+(\gamma_{2}\langle
z_{2}\rangle)\left[\begin{pmatrix}\overline{p}&&0\\\
0&&1_{n-1}\end{pmatrix}\right]$

Also,

$j((\xi\times
1_{2(n-1)})(\gamma_{1}\times\gamma_{2}),[z_{1},z_{2}])=j(\xi\times
1_{2(n-1)},[\gamma_{1}z_{1},\gamma_{2}z_{2}])j(\gamma_{1}\times\gamma_{2},[z_{1},z_{2}])$

and $\displaystyle j(\xi\times
1_{2(n-1)},[\gamma_{1}z_{1},\gamma_{2}z_{2}])=\det(D)$, where we denote
$\displaystyle\xi$ by $\displaystyle\begin{pmatrix}A&0\\\ 0&D\end{pmatrix}$.

By unitarity, we have $\displaystyle D\overline{A}^{t}=1$ so
$\displaystyle\det D\cdot\overline{\det A}=1$ so $\displaystyle N(\det(D))=1$
and $\displaystyle\det(D)$ is in $\displaystyle\mathbb{Z}[i]$, which shows
that $\displaystyle\det(D)\in\\{\pm 1,\pm i\\}$. As $\displaystyle k\equiv
0\pmod{4}$, we get

$\chi^{k,s}((\xi\times
1_{2(n-1)})(\gamma_{1}\times\gamma_{2}),[z_{1},z_{2}])=\chi^{k,s}(\gamma_{1},z_{1})\chi^{k,s}(\gamma_{2},z_{2})$

and so the proposition follows. ∎

From now on we work with $\displaystyle n=2$. Write $\displaystyle
Z=\begin{pmatrix}\tau&z_{1}\\\ z_{2}&\omega\end{pmatrix}$ and consider the
Fourier-Jacobi expansion of $\displaystyle F,G$ and the Fourier expansion of
$\displaystyle h$ as follows:

$F(Z)=\sum_{m=1}^{\infty}\phi_{m}(\tau,z_{1},z_{2})e^{2\pi
im\omega},G(Z)=\sum_{m=1}^{\infty}\psi_{m}(\tau,z_{1},z_{2})e^{2\pi im\omega}$
$h(W)=\sum_{n=1}^{\infty}a_{n}e^{2\pi inW}$

We state and prove the main theorem of this section, which will give us the
Dirichlet series with which we are going to work with.

######  Theorem 4.6.

Let $\displaystyle k\equiv 0\pmod{4}$. Let $\displaystyle F,G\in S_{2}^{k}$
and $\displaystyle h\in S_{1}^{k}$. Then, for $\displaystyle Re(s)$ large
enough so that the Eisenstein series converges absolutely, we have

$\left\langle\left\langle E_{3,2}^{k}\left(\begin{pmatrix}W&0\\\
0&Z\end{pmatrix},F;s\right),G(Z)\right\rangle,h(W)\right\rangle=(4\pi)^{-(2k+s-4)}\frac{\Gamma(2k+s-4)\Gamma(k+s-3)\Gamma(k+s-1)}{\Gamma(2k+2s-4)}D_{F,G,h}(s)$

where we define the Dirichlet series

$D_{F,G,h}(s)=\sum_{p,q}\sum_{m=1}^{\infty}\langle\phi_{m}|U_{p},\psi_{mN(p)}\rangle\overline{a_{mN(q)}}N(p)^{-(k+s-3)}N(q)^{-(k+s-1)}m^{-(2k+s-4)}$

with $\displaystyle U_{p}$ the operator defined by

$\displaystyle\displaystyle U_{p}:J_{k,m}^{n}$
$\displaystyle\displaystyle\longrightarrow J_{k,mp}^{n}$
$\displaystyle\displaystyle\phi_{m}(\tau,z_{1},z_{2})$
$\displaystyle\displaystyle\longmapsto\phi_{m}(\tau,\overline{p}z_{1},pz_{2})$

and $\displaystyle p,q$ summed as in Corollary 4.4.

###### Proof.

We will start dealing with the third part of the inner product, using the
decomposition of Proposition 4.5. This can be written as(the summations are as
above):

$\int_{\Gamma_{1}\backslash\mathbb{H}_{1}}\int_{\Gamma_{2}\backslash\mathbb{H}_{2}}\delta(W)^{k+s}\delta(Z)^{k+s}\sum_{p,q}\sum_{\gamma_{1},\gamma_{2}}\chi^{k,s}(\gamma_{1},W)\chi^{k,s}(\gamma_{2},Z)\times$
$\times F\left(\begin{pmatrix}N(q)\gamma_{1}\langle W\rangle&0\\\
0&0\end{pmatrix}+(\gamma_{2}\langle
Z\rangle)\left[\begin{pmatrix}\overline{p}&0\\\
0&1\end{pmatrix}\right]\right)\delta\left(\begin{pmatrix}N(q)\gamma_{1}\langle
W\rangle&0\\\ 0&0\end{pmatrix}+(\gamma_{2}\langle
Z\rangle)\left[\begin{pmatrix}\overline{p}&0\\\
0&1\end{pmatrix}\right]\right)^{-s}\times$
$\overline{G(Z)}\overline{h(W)}d^{*}Wd^{*}Z$

Now, using the automorphy condition for $\displaystyle G,h$, we have

$G(Z)=(G|_{k}\gamma_{2})(Z)=j(\gamma_{2},Z)^{-k}G(\gamma_{2}Z)$
$h(W)=(h|_{k}\gamma_{1})(W)=j(\gamma_{1},W)^{-k}h(\gamma_{1}W)$

Also $\displaystyle\delta(\gamma_{2}Z)=|j(\gamma_{2},Z)|^{-2}\delta(Z)$,
$\displaystyle\delta(\gamma_{1}W)=|j(\gamma_{1},W)|^{-2}\delta(W)$, so

$\overline{G(Z)}\delta(Z)^{k+s}\chi^{k,s}(\gamma_{2},Z)=\overline{j(\gamma_{2},Z)}^{-k}\overline{G(\gamma_{2}Z)}\delta(\gamma_{2}Z)^{k+s}|j(\gamma_{2},Z)|^{2(k+s)}j(\gamma_{2},Z)^{-k}|j(\gamma_{2},Z)|^{-2s}$
$=\overline{G(\gamma_{2}Z)}\delta(\gamma_{2}Z)^{k+s}$

and similarly for $\displaystyle h$. Hence, by the usual "unfolding" trick, we
obtain that the above integral equals

$\int_{C_{1,0}\backslash\mathbb{H}_{1}}\int_{C_{2,1}\backslash\mathbb{H}_{2}}\sum_{p,q}F\left(\begin{pmatrix}N(q)W&0\\\
0&0\end{pmatrix}+Z\left[\begin{pmatrix}\overline{p}&0\\\
0&1\end{pmatrix}\right]\right)\delta\left(\begin{pmatrix}N(q)W&0\\\
0&0\end{pmatrix}+Z\left[\begin{pmatrix}\overline{p}&0\\\
0&1\end{pmatrix}\right]\right)^{-s}\times$
$\times\overline{G(Z)}\overline{h(W)}\delta(Z)^{k+s}\delta(W)^{k+s}d^{*}Zd^{*}W$

We now rewrite the integral in the following form:

$\int_{P_{1,0}\backslash\mathbb{H}_{1}}\int_{P_{2,1}\backslash\mathbb{H}_{2}}\sum_{p,q}F\left(\begin{pmatrix}0&0\\\
0&N(q)W\end{pmatrix}+Z\left[\begin{pmatrix}1&0\\\
0&\overline{p}\end{pmatrix}\right]\right)\delta\left(\begin{pmatrix}0&0\\\
0&N(q)W\end{pmatrix}+Z\left[\begin{pmatrix}1&0\\\
0&\overline{p}\end{pmatrix}\right]\right)^{-s}\times$
$\times\overline{G(Z)}\overline{h(W)}\delta(Z)^{k+s}\delta(W)^{k+s}d^{*}Zd^{*}W$

Now, a fundamental domain for the action of $\displaystyle P_{1,0}$ on
$\displaystyle\mathbb{H}_{1}$ is

$\mathcal{F}=\\{z\in\mathbb{H}_{1}|z=x+iy,0\leq x\leq 1\\}$

while a fundamental domain for the action of $\displaystyle P_{2,1}$ on
$\displaystyle\mathbb{H}_{2}$

$\left\\{\begin{pmatrix}\tau&z_{1}\\\
z_{2}&\omega\end{pmatrix}|(\tau,z_{1},z_{2})\in\mathcal{F}^{J},y_{\omega}>|z_{1}-\overline{z_{2}}|^{2}/4y_{\tau},0\leq
x_{\omega}\leq 1\right\\}$

(as we can see for example in page $\displaystyle 2907$ of [8]). Hence, the
above inner product equals

$\sum_{p,q}\int_{\mathcal{F}^{J}}d\tau
dz_{1}dz_{2}\int_{y_{\omega}>|z_{1}-\overline{z_{2}}|^{2}/4y_{\tau}}dy_{\omega}\int_{0}^{1}dx_{\omega}\int_{0}^{1}dx_{W}\int_{0}^{\infty}dy_{W}\delta(Z)^{k+s-4}\times$
$\times\sum_{m=1}^{\infty}\phi_{m}(\tau,\overline{p}z_{1},pz_{2})e^{2\pi
im(N(q)W+N(p)\omega)}\sum_{n=1}^{\infty}\overline{a_{n}}e^{-2\pi
in\overline{W}}\sum_{k=1}^{\infty}\overline{\psi_{k}(\tau,z_{1},z_{2})}e^{-2\pi
ik\overline{\omega}}\times$ $\times\delta\left(\begin{pmatrix}0&0\\\
0&N(q)W\end{pmatrix}+Z\left[\begin{pmatrix}1&0\\\
0&\overline{p}\end{pmatrix}\right]\right)^{-s}$

We now first perform the integration over $\displaystyle x_{\omega}$ and
$\displaystyle x_{W}$. For $\displaystyle x_{\omega}$ we have

$\int_{0}^{1}e^{2\pi imN(p)x_{\omega}-2\pi ikx_{\omega}}dx_{\omega}$

which is zero, unless $\displaystyle k=mN(p)$, in which case the integral is
$\displaystyle 1$.
Similarly for $\displaystyle x_{W}$ the integral we need to compute is

$\int_{0}^{1}e^{2\pi imN(q)x_{W}-2\pi inx_{W}}dx_{W}$

which is again zero unless $\displaystyle n=mN(q)$, in which case it is
$\displaystyle 1$.

These are the only terms we need to integrate as the real parts of
$\displaystyle\omega$ and $\displaystyle W$ do not appear as arguments of
$\displaystyle\delta$ by definition. We now substitute $\displaystyle
t=y_{\omega}-|z_{1}-\overline{z_{2}}|^{2}/4y_{\tau}$ and compute

$\delta(Z)=\det\left(\frac{1}{2i}\left(Z-^{t}\overline{Z}\right)\right)=\cdots=y_{\tau}t$

and

$\delta\left(\begin{pmatrix}0&0\\\
0&N(q)W\end{pmatrix}+Z\left[\begin{pmatrix}1&0\\\
0&\overline{p}\end{pmatrix}\right]\right)=\delta\left(\begin{pmatrix}\tau&\overline{p}z_{1}\\\
pz_{2}&N(q)W+N(p)\tau^{\prime}\end{pmatrix}\right)=\cdots=y_{\tau}(N(q)y_{W}+N(p)t)$

So, the integral becomes

$\sum_{p,q}\int_{\mathcal{F}^{J}}\sum_{m=1}^{\infty}\phi_{m}(\tau,\overline{p}z_{1},pz_{2})\overline{\psi_{mN(p)}(\tau,z_{1},z_{2})}\overline{a_{mN(q)}}y_{\tau}^{k-4}e^{-\pi
m(|z_{1}-\overline{z_{2}}|^{2}/y_{\tau})}d\tau dz_{1}dz_{2}\times$
$\times\int_{0}^{\infty}dt\int_{0}^{\infty}dy_{W}t^{k+s-4}(N(q)y_{W}+N(p)t)^{-s}e^{-4\pi
m(N(q)y_{W}+N(p)t)}y_{W}^{s+k-2}=$
$=\sum_{p,q}\sum_{m=1}^{\infty}\langle\phi_{m}|U_{p},\psi_{mN(p)}\rangle\overline{a_{mN(q)}}\int_{0}^{\infty}\int_{0}^{\infty}\frac{dt}{t}\frac{dy_{W}}{y_{W}}t^{k+s-3}y_{W}^{s+k-1}(N(q)y_{W}+N(p)t)^{-s}e^{-4\pi
m(N(q)y_{W}+N(p)t)}=$
$=(4\pi)^{-(2k+s-4)}\frac{\Gamma(2k+s-4)\Gamma(k+s-3)\Gamma(k+s-1)}{\Gamma(2k+2s-4)}\times$
$\times\sum_{p,q}\sum_{m=1}^{\infty}\langle\phi_{m}|U_{p},\psi_{mN(p)}\rangle\overline{a_{mN(q)}}N(p)^{-(k+s-3)}N(q)^{-(k+s-1)}m^{-(2k+s-4)}$

by using the fact that

$\int_{0}^{\infty}\int_{0}^{\infty}\frac{dx}{x}\frac{dy}{y}x^{\alpha}y^{\beta}\left(\frac{xy}{x+y}\right)^{\gamma}e^{-(x+y)}=\frac{\Gamma(\alpha+\beta+\gamma)\Gamma(\alpha+\gamma)\Gamma(\beta+\gamma)}{\Gamma(\alpha+\beta+2\gamma)}$

and substituting $\displaystyle x=4\pi mN(p)t$, $\displaystyle y=4\pi
mN(q)y_{W}$, $\displaystyle\alpha=k-3$, $\displaystyle\beta=k-1$,
$\displaystyle\gamma=s$.

Let us now deal with the second part of the integral. We want to compute

$\int_{\Gamma_{1}\backslash\mathbb{H}_{1}}\int_{\Gamma_{2}\backslash\mathbb{H}_{2}}\sum_{\gamma\in
C_{2,1}\backslash\Gamma_{2}}j(\gamma,Z)^{-k}F_{W}\left(\gamma\langle
Z\rangle_{*}\right)\left(\frac{\delta(\gamma\langle
Z\rangle)}{\delta(\gamma\langle
Z\rangle_{*})}\right)^{s}\overline{G(Z)}\overline{h(W)}\delta(Z)^{k}\delta(W)^{k}d^{*}Zd^{*}W$

and using again the automorphy condition for $\displaystyle G$, we obtain by
unfolding the integral

$\int_{\Gamma_{1}\backslash\mathbb{H}_{1}}\int_{C_{2,1}\backslash\mathbb{H}_{2}}F_{W}(Z_{*})\overline{G(Z)}\overline{h(W)}\delta(Z)^{k+s}\delta(Z_{*})^{-s}d^{*}Zd^{*}W$

We now write $\displaystyle Z=\begin{pmatrix}\tau&z_{1}\\\
z_{2}&\omega\end{pmatrix}$ and rewrite the inner integral as

$\int_{P_{2,1}\backslash\mathbb{H}_{2}}F\left(\begin{pmatrix}\tau&0\\\
0&W\end{pmatrix}\right)\overline{G(Z)}\delta(Z)^{k+s}y_{\tau}^{-s}d^{*}Z$

and by using the Fourier-Jacobi expansions and the fundamental domains
mentioned above we get

$\int_{\mathcal{F}^{J}}d\tau
dz_{1}dz_{2}\int_{y_{\omega}>{|z_{1}-\overline{z_{2}}|^{2}/4y_{\tau}}}\int_{0}^{1}dx_{\omega}\sum_{m=1}^{\infty}\phi_{m}(\tau,0,0)e^{2\pi
imW}\sum_{n=1}^{\infty}\overline{\psi_{m}(\tau,z_{1},z_{2})}e^{-2\pi
im\overline{\omega}}\delta(Z)^{k+s-4}y_{\tau}^{-s}$

and we can see that this is zero by calculating the integral over
$\displaystyle x_{\omega}$, i.e.

$\int_{0}^{1}e^{-2\pi imx_{\omega}}dx_{\omega}=0$

Finally, we want to show that the first part of the integral is zero. We have

$\langle\langle E_{1}(W;s)F(Z),G(Z)\rangle,h(W)\rangle=\langle
F(Z),G(Z)\rangle\langle E_{1}(W;s),h(W)\rangle$

and we will show the second inner product is zero. We have

$\langle
E_{1}(W;s),h(W)\rangle=\int_{\Gamma_{1}\backslash\mathbb{H}_{1}}\sum_{\gamma\in
C_{1,0}\backslash\Gamma_{1}}j(\gamma,W)^{-k}\delta(\gamma
W)^{s}\overline{h(W)}\delta(W)^{k}d^{*}W$

and by the usual unfolding trick

$\int_{C_{1,0}\backslash\mathbb{H}_{1}}\delta(W)^{k+s}\overline{h(W)}d^{*}W=\int_{x=0}^{1}\int_{y=0}^{\infty}\sum_{n=1}^{\infty}\overline{a_{n}}e^{-2\pi
in(x-iy)}y^{k-2}dxdy=0$

by looking at the integral

$\int_{0}^{1}e^{-2\pi inx}dx=0$

for $\displaystyle n\geq 1$. Combining all the above, the result follows. ∎

##  5\. Inert primes

This is the case which is closer to the situation considered by Heim [13].
Most of the ideas below are based on similar ideas appearing in Heim’s work.

###  5.1. Hecke operators and weak rationality theorems

Throughout this section, $\displaystyle p$ is assumed to be a rational prime
which remains prime in $\displaystyle\mathcal{O}_{K}$.
Let us make a list of Hecke operators and relations between them that we are
going to use in this case.

  * •

$\displaystyle T_{p}=\Gamma_{2}\textup{diag}(1,1,p,p)\Gamma_{2}$.

  * •

$\displaystyle T_{1,p}=\Gamma_{2}\textup{diag}(1,p,p^{2},p)\Gamma_{2}$.

  * •

$\displaystyle T^{J}(p)=\Gamma_{1,1}\textup{diag}(1,p,p^{2},p)\Gamma_{1,1}$.

  * •

$\displaystyle\Lambda_{-}(p)=\Gamma_{1,1}\textup{diag}(p,p^{2},p,1)\Gamma_{1,1}=\Gamma_{1,1}\textup{diag}(p,p^{2},p,1)$.

  * •

$\displaystyle\Lambda_{+}(p)=\Gamma_{1,1}\textup{diag}(p,1,p,p^{2})\Gamma_{1,1}$.

  * •

$\displaystyle T_{-}(p)=\Gamma_{1,1}\textup{diag}(1,p,p,1)\Gamma_{1,1}$.

  * •

$\displaystyle T_{+}(p)=\Gamma_{1,1}\textup{diag}(1,1,p,p)\Gamma_{1,1}$.

  * •

$\displaystyle\nabla_{p}=\sum_{a\in\mathbb{Z}/p\mathbb{Z}}\Gamma_{1,1}\begin{pmatrix}p&0&0&0\\\
0&p&0&a\\\ 0&0&p&0\\\
0&0&0&p\end{pmatrix}=\sum_{a\in\mathbb{Z}/p\mathbb{Z}}\nabla{a}$

  * •

$\displaystyle\Delta_{p}=\Gamma_{1,1}\begin{pmatrix}p&0&0&0\\\ 0&p&0&0\\\
0&0&p&0\\\
0&0&0&p\end{pmatrix}\Gamma_{1,1}=\Gamma_{1,1}\begin{pmatrix}p&0&0&0\\\
0&p&0&0\\\ 0&0&p&0\\\ 0&0&0&p\end{pmatrix}$

Also, for any operator $\displaystyle X(p)$, we write $\displaystyle X^{r}(p)$
to denote $\displaystyle\Delta_{p}^{-1}X(p)$.
We start with giving the images of some of the above operators under the
Satake mapping, in the form of the following proposition:

######  Proposition 5.1.

Let $\displaystyle\Phi$ denote the Satake mapping for the inert prime
$\displaystyle p$. We have

  * •

$\displaystyle\Phi(T_{p})=x_{0}+p^{-1}x_{0}x_{1}+p^{-1}x_{0}x_{2}+p^{-2}x_{0}x_{1}x_{2}=x_{0}(1+p^{-1}x_{1})(1+p^{-1}x_{2})$.

  * •

$\displaystyle\Phi(T_{1,p})=p^{-2}x_{0}^{2}x_{1}+p^{-2}x_{0}^{2}x_{2}+p^{-4}x_{0}^{2}x_{1}^{2}x_{2}+p^{-4}x_{0}^{2}x_{1}x_{2}^{2}+p^{-6}(p^{2}+1)(p-1)x_{0}^{2}x_{1}x_{2}$.

  * •

$\displaystyle\Phi(\Delta_{p})=p^{-6}x_{0}^{2}x_{1}x_{2}$.

###### Proof.

Given the decomposition given in p. $\displaystyle 677$ of Krieg’s paper [16],
or comparing the decomposition of [8, Lemma $\displaystyle 3.6$] and using
[14, Theorem $\displaystyle 1$], the proposition follows. ∎

Let now $\displaystyle D_{p}^{(2)}(X)=Z_{p}^{(2)}(p^{-3}X)$. We will now give
the form and factorization of $\displaystyle D_{p}^{(2)}(X)$.

######  Proposition 5.2.

We have

$D_{p}^{(2)}(X)=1-B_{1}X+B_{2}X^{2}-B_{1}X^{3}+X^{4}$

where

$\displaystyle\displaystyle
B_{1}=p^{-3}\Delta_{p}^{-1}(T_{1,p}-(p^{2}+1)(p-1)\Delta_{p})$
$\displaystyle\displaystyle
B_{2}=p^{-4}\Delta_{p}^{-1}(T_{p}^{2}-2pT_{1,p}-2p(p^{2}-p+1)\Delta_{p})$

###### Proof.

This follows by direct verification, after applying the Satake isomorphism. We
remind the reader here that $\displaystyle
Z_{p}^{(2)}(X)=\Phi^{-1}(z_{p}^{(2)}(X))$, where

$z_{p}^{(2)}(X)=\prod_{i=1}^{2}(1-p^{4}x_{i,p}^{-1}X)(1-p^{2}x_{i,p}X)$

This also gives the $\displaystyle\Phi-$image of $\displaystyle D_{p}^{(2)}$.
∎

We now have the following proposition regarding the factorisation of
$\displaystyle D_{p}^{(2)}$.

######  Proposition 5.3.

We have the following factorisation over $\displaystyle H_{p}^{1,1}[X]$

$D_{p}^{(2)}(X)=(1-p^{-3}\Delta_{p}^{-1}\Lambda_{-}(p)X)S^{(2)}(X)(1-p^{-3}\Delta_{p}^{-1}\Lambda_{+}(p)X)$

where

$S^{(2)}(X)=S_{0}-S_{1}X+S_{2}X^{2}-S_{3}X^{3}$

with

  * •

$\displaystyle S_{0}=1$

  * •

$\displaystyle S_{1}=p^{-3}(T^{J,r}+\nabla_{p}^{r}-p(p^{2}-p+1))$

  * •

$\displaystyle
S_{2}=p^{-4}\Delta_{p}^{-1}T_{+}T_{-}-p^{-3}T^{J,r}-2p^{-3}\nabla_{p}^{r}-p^{-2}(p-2)$

  * •

$\displaystyle S_{3}=p^{-3}(\nabla_{p}^{r}-p)$

###### Proof.

This can be verified directly, by using the following identities/relations
which can be found in [8], or be proved directly.

  * •

$\displaystyle\epsilon(T_{1,p})=T^{J}(p)+\Lambda_{-}(p)+\Lambda_{+}(p)+\nabla_{p}-\Delta_{p}$

  * •

$\displaystyle\epsilon(T_{p})=T_{-}(p)+T_{+}(p)$

  * •

$\displaystyle T_{-}(p)T_{+}(p)=pT^{J}(p)+(p^{3}+p^{4})\Delta_{p}$

  * •

$\displaystyle\Lambda_{-}(p)T_{+}(p)=p^{3}\Delta_{p}T_{-}(p)$

  * •

$\displaystyle T_{-}(p)\Lambda_{+}(p)=p^{3}\Delta_{p}T_{+}(p)$

  * •

$\displaystyle\Lambda_{-}(p)\Lambda_{+}(p)=p^{6}\Delta_{p}^{2}$

  * •

$\displaystyle\Lambda_{-}^{r}\nabla_{p}^{r}=p\Lambda_{-}^{r}$

  * •

$\displaystyle\nabla_{p}^{r}\Lambda_{+}^{r}=p\Lambda_{+}^{r}$

where $\displaystyle\epsilon$ denotes the embedding of $\displaystyle
H(\Gamma_{2},S^{2})$ to $\displaystyle H(\Gamma_{1,1},S^{1,1})$, as described
in 3.1. ∎

We now have the following weak rationality theorems:

######  Proposition 5.4.

Let $\displaystyle F\in S_{2}^{k}$ be a Hecke eigenform and $\displaystyle
m\geq 1$. Then

$\displaystyle\displaystyle Q_{p,F}^{(2)}(X)\sum_{\delta\geq
0}\phi_{mp^{\delta}}|T_{+}(p^{\delta})X^{\delta}=\left(\phi_{m}-\phi_{\frac{m}{p}}|T_{-}(p)X+p\phi_{\frac{m}{p^{2}}}|\Lambda_{-}(p)X^{2}\right)\mid(1+p(\nabla_{p}-p\Delta_{p})X^{2})$

where $\displaystyle\phi_{m}\mid(1+p(\nabla_{p}-p\Delta_{p})X^{2})=$
$\displaystyle\begin{cases}\phi_{m}&\textup{if }p|m\\\
(1-p^{2k-6}X^{2})\phi_{m}&\textup{otherwise}\end{cases}$

###### Proof.

We follow the same proof of [12, Corollary]. Then, the result follows from [8,
Proposition $\displaystyle 3.2$]. We will just show the computations for the
last claim of our proposition. We have

$(F|\Delta_{p})(Z)=(p^{2})^{2k-4}(p^{2})^{-k}F(Z)=p^{2k-8}F(Z)$

and so $\displaystyle\phi_{m}\mid\Delta_{p}=p^{2k-8}\phi_{m}$. Also,

$(F|\nabla{a})(Z)=(p^{2})^{2k-4}(p^{2})^{-k}F\left(\begin{pmatrix}\tau&z_{1}\\\
z_{2}&\tau^{\prime}+a/p\end{pmatrix}\right)=p^{2k-8}\sum_{m=1}^{\infty}\phi_{m}(\tau,z_{1},z_{2})e^{2\pi
im\tau^{\prime}}e^{2\pi ima/p}$

so $\displaystyle\phi_{m}\mid\nabla_{p}=\left(p^{2k-8}\sum_{a=0}^{p-1}e^{2\pi
ia/p}\right)\phi_{m}=\begin{cases}0&\textup{if }(m,p)=1\\\
p^{2k-7}\phi_{m}&\textup{otherwise}\end{cases}$
from which the result follows. ∎

######  Proposition 5.5.

Let $\displaystyle F\in S_{2}^{k}$ be a Hecke eigenform and $\displaystyle
m\geq 1$. Then

$D_{p,F}^{(2)}(X)\sum_{\delta\geq
0}\phi_{mp^{2\delta}}\mid_{k}(\Delta_{p^{\delta}}^{-1}\Lambda_{+}(p^{\delta}))(p^{-3}X)^{\delta}=\phi_{m}\mid_{k}S^{(2)}(X)-\phi_{m/p^{2}}\mid_{k}(\Delta_{p}^{-1}\Lambda_{-}(p)S^{(2)}(X))p^{-3}X$

###### Proof.

This follows from the proposition 5.3 using the same techniques as in [12,
Corollary]. ∎

In a similar fashion to Heim in [13] now, we have that the action of the
operators $\displaystyle T_{+}(p),\Lambda_{+}(p),\nabla_{p}^{r}(p)$ on
Fourier-Jacobi forms of index coprime to $\displaystyle p$ is identical to
zero. This leads to the definition of the following polynomials:

$S^{(2)}(X)^{\textup{factor}}=1-(p^{-3}T^{J,r}-p^{-2}+p^{-1})X+p^{-2}X^{2}$
$S^{(2)}(X)^{\textup{prim}}=1-(p^{-3}T^{J,r}-p^{-2}(p^{2}-p+1))X+(-p^{-3}T^{J,r}-p^{-2}(p-2))X^{2}+p^{-2}X^{3}$
$=S^{(2)}(X)^{\textup{factor}}(1+X)$

Hence, $\displaystyle\phi|S^{(2)}(X)=\phi|S^{(2)}(X)^{\textup{prim}}$ if
$\displaystyle\phi$ has index relatively prime to $\displaystyle p$. We now
have the following lemma:

######  Lemma 5.6.

$\displaystyle\phi|S^{(2)}(X)T_{+}(p)=\phi|T_{+}(p)S^{(2)}(X)^{\textup{factor}}$
if $\displaystyle\phi$ has index $\displaystyle p$.

###### Proof.

The proof follows by the following results:

  * •

$\displaystyle\phi\mid\nabla_{p}^{r}=p\phi$

  * •

$\displaystyle\phi\mid T_{+}(p)\nabla_{p}^{r}=0$ because
$\displaystyle\phi\mid T_{+}(p)$ will have index $\displaystyle 1$.

  * •

$\displaystyle\phi\mid\left[T^{J,r},T_{+}(p)\right]=\phi\mid(p^{3}T_{+}(p)-\nabla_{p}^{r}T_{+}(p)=(p^{3}-p)\phi\mid
T_{+}(p)$ by the first point.

Here
$\displaystyle\left[T^{J,r},T_{+}(p)\right]:=T^{J,r}T_{+}(p)-T_{+}(p)T^{J,r}$
denotes the commutator. We will now give the proof of the third point. By the
proof of 5.3, we have

  * •

$\displaystyle\epsilon(T_{1,p})=T^{J}(p)+\Lambda_{-}(p)+\Lambda_{+}(p)+\nabla_{p}-\Delta_{p}$

  * •

$\displaystyle\epsilon(T_{p})=T_{-}(p)+T_{+}(p)$

and by looking the elements whose product has signature $\displaystyle p$, we
obtain

$T^{J}(p)T_{+}(p)+\Lambda_{+}(p)T_{-}(p)+(\nabla_{p}-\Delta_{p})T_{+}(p)=$
$T_{+}(p)T^{J}(p)+T_{-}(p)\Lambda_{+}(p)+T_{+}(p)(\nabla_{p}-\Delta_{p})$

from which the result follows as $\displaystyle\phi\mid\Lambda_{+}(p)=0$ for
$\displaystyle\phi$ of index $\displaystyle p$ and $\displaystyle
T_{-}(p)\Lambda_{+}(p)=p^{3}\Delta_{p}T_{+}(p)$. ∎

###  5.2. Calculation of the Dirichlet Series

In what follows, we will assume that $\displaystyle F$ is in the Maass space,
as we have defined in 3.7.

From now on we will always assume that $\displaystyle h$ has totally real
Fourier coefficients. This is a technical assumption which could be lifted. We
can rewrite $\displaystyle D_{F,G,h}$ as

$\displaystyle\displaystyle D_{F,G,h}(s)$
$\displaystyle\displaystyle=4\sum_{l,\epsilon,m}\langle
m^{3-k}\phi|T_{-}(m)U_{l},\psi_{mN(l)}\rangle_{\mathcal{A}}a_{mN(\epsilon)}N(l)^{-(k+s-3)}N(\epsilon)^{-(k+s-1)}m^{-(2k+s-4)}$
$\displaystyle\displaystyle=4\beta_{k}\sum_{l,\epsilon,m}\langle\tilde{\phi}_{1}|T_{-}(m)U_{l},\tilde{\psi}_{mN(l)}\rangle_{\mathcal{A}}a_{mN(\epsilon)}N(l)^{-s}N(\epsilon)^{-(k+s-1)}m^{-(2k+s-4)}$

with $\displaystyle l,\epsilon\in\mathbb{Z}[i]$ coprime with their real parts
positive and imaginary parts non-negative and $\displaystyle m\in\mathbb{N}$.
Then, if $\displaystyle\phi$ is a Fourier-Jacobi form, by the above, we get

$\phi|_{k}\Lambda_{-}(p)=p^{3k-8}\tilde{\phi}\left(\begin{pmatrix}\tau&pz_{1}\\\
pz_{2}&p^{2}\tau^{\prime}\end{pmatrix}\right)e^{-2\pi
mp^{2}\tau^{\prime}}=p^{3k-8}\phi(\tau,pz_{1},pz_{2})=p^{3k-8}\phi|_{k}U_{p}$

We now define the $\displaystyle p-$part of the Dirichlet series

$\displaystyle\displaystyle D_{F,G,h}^{(p)}(s)$
$\displaystyle\displaystyle=\sum_{l,\epsilon,m\geq
0}\langle\tilde{\phi}_{1}|T_{-}(p^{m})U_{p^{l}},\tilde{\psi}_{p^{m+2l}}\rangle_{\mathcal{A}}a_{p^{m+2\epsilon}}p^{-2sl}p^{-2(k+s-1)\epsilon}p^{-(2k+s-4)m}$
$\displaystyle\displaystyle=\sum_{l,\epsilon,m\geq
0}\langle\tilde{\phi}_{1}|T_{-}(p^{m})\Lambda_{-}(p^{l}),\tilde{\psi}_{p^{m+2l}}\rangle_{\mathcal{A}}a_{p^{m+2\epsilon}}p^{-(3k+2s-8)l}p^{-2(k+s-1)\epsilon}p^{-(2k+s-4)m}$

together with the condition that $\displaystyle\min(l,\epsilon)=0$. The last
line is obtained using the relation between $\displaystyle
U_{p},\Lambda_{-}(p)$.

Now, with respect to the inner product of Jacobi forms, we have by [8,
Proposition $\displaystyle 5.1$] that
$\displaystyle\Lambda_{-}^{\textup{adj}}(p)=p^{2k-6}\Lambda_{+}(p)$ and
$\displaystyle T_{-}^{\textup{adj}}(p)=p^{k-3}T_{+}(p)$. This then gives that
the adjoint of $\displaystyle\Lambda_{-}(p)$ is $\displaystyle\Lambda_{+}(p)$
for the inner product on $\displaystyle P-$forms and similarly the
$\displaystyle P-$form adjoint for $\displaystyle T_{-}$ is $\displaystyle
T_{+}$.

Let now $\displaystyle X=p^{-(k+s-1)}$ and $\displaystyle N=p^{k-1}$. Consider
the Satake parameters $\displaystyle\alpha_{1},\alpha_{2}$ of $\displaystyle
h$ such that $\displaystyle\alpha_{1}+\alpha_{2}=a_{p}$ and
$\displaystyle\alpha_{1}\alpha_{2}=p^{k-1}$. Let also $\displaystyle
X_{i}=\alpha_{i}p^{-(2k+s-4)},i=1,2$.

We write

$D_{F,G,h}^{(p)}(s)=D_{(\epsilon)}(s)+D_{(l)}(s)-D_{(\epsilon,l)}(s)$

where the corresponding index means that this variable(or both) is
$\displaystyle 0$. Using the fact that

$a_{p^{m}}=\frac{\alpha_{1}^{m+1}-\alpha_{2}^{m+1}}{\alpha_{1}-\alpha_{2}}$

and properties for the adjoint operators we mentioned above, we obtain:

$\displaystyle\displaystyle D_{(\epsilon)}(s)(\alpha_{1}-\alpha_{2})$
$\displaystyle\displaystyle=\alpha_{1}\sum_{l,m=0}^{\infty}\langle\tilde{\phi}_{1},\tilde{\psi}_{p^{m+2l}}|T_{+}(p^{m})\Lambda_{+}(p^{l})\rangle_{\mathcal{A}}p^{-(3k+2s-8)l}(\alpha_{1}p^{-(2k+s-4)})^{m}$
$\displaystyle\displaystyle-\alpha_{2}\sum_{l,m=0}^{\infty}\langle\tilde{\phi}_{1},\tilde{\psi}_{p^{m+2l}}|T_{+}(p^{m})\Lambda_{+}(p^{l})\rangle_{\mathcal{A}}p^{-(3k+2s-8)l}(\alpha_{2}p^{-(2k+s-4)})^{m}$
$\displaystyle\displaystyle D_{(l)}(s)(\alpha_{1}-\alpha_{2})$
$\displaystyle\displaystyle=\alpha_{1}\sum_{\epsilon,m=0}^{\infty}\langle\tilde{\phi}_{1},\tilde{\psi}_{p^{m}}|T_{+}(p^{m})\rangle_{\mathcal{A}}(\alpha_{1}p^{-(k+s-1)})^{2\epsilon}(\alpha_{1}p^{-(2k+s-4)})^{m}$
$\displaystyle\displaystyle-\alpha_{2}\sum_{\epsilon,m=0}^{\infty}\langle\tilde{\phi}_{1},\tilde{\psi}_{p^{m}}|T_{+}(p^{m})\rangle_{\mathcal{A}}(\alpha_{2}p^{-(k+s-1)})^{2\epsilon}(\alpha_{2}p^{-(2k+s-4)})^{m}$
$\displaystyle\displaystyle D_{(\epsilon,l)}(s)(\alpha_{1}-\alpha_{2})$
$\displaystyle\displaystyle=\alpha_{1}\sum_{m=0}^{\infty}\langle\tilde{\phi}_{1},\tilde{\psi}_{p^{m}}|T_{+}(p^{m})\rangle_{\mathcal{A}}(\alpha_{1}p^{-(2k+s-4)})^{m}$
$\displaystyle\displaystyle-\alpha_{2}\sum_{m=0}^{\infty}\langle\tilde{\phi}_{1},\tilde{\psi}_{p^{m}}|T_{+}(p^{m})\rangle_{\mathcal{A}}(\alpha_{1}p^{-(2k+s-4)})^{m}$

######  Proposition 5.7.

We have

$D_{(l)}(s)-D_{(\epsilon,l)}(s)=\frac{\langle\tilde{\phi}_{1},\tilde{\psi}_{1}\rangle_{\mathcal{A}}}{\alpha_{1}-\alpha_{2}}\left(\frac{\alpha_{1}^{3}X^{2}}{Q_{p,F}^{(2)}(X_{1})}-\frac{\alpha_{2}^{3}X^{2}}{Q_{p,F}^{(2)}(X_{2})}\right)$

###### Proof.

This follows by the above and the Proposition 5.4 with $\displaystyle m=1$. We
have

$\sum_{m=0}^{\infty}\langle\tilde{\phi}_{1},\tilde{\psi}_{p^{m}}|T_{+}(p^{m})X_{1}^{m}\rangle_{\mathcal{A}}=\langle\tilde{\phi}_{1},Q_{p,F}^{(2)}(X_{1})^{-1}(1-p^{2k-6}X_{1}^{2})\tilde{\phi}_{1}\rangle_{\mathcal{A}}=(1-p^{2k-6}X_{1}^{2})Q_{p,F}^{(2)}(X_{1})^{-1}\langle\tilde{\phi}_{1},\tilde{\psi}_{1}\rangle_{\mathcal{A}}$

Also,

$\sum_{\epsilon=0}^{\infty}(\alpha_{1}p^{-(k+s-1)})^{2\epsilon}=\frac{1}{1-\alpha_{1}^{2}p^{-2(k+s-1)}}$

So, the first part of the difference we are interested in is

$\alpha_{1}\frac{\langle\tilde{\phi}_{1},\tilde{\psi}_{1}\rangle_{\mathcal{A}}}{\alpha_{1}-\alpha_{2}}Q_{p,F}^{(2)}(X_{1})^{-1}(1-p^{2k-6}X_{1}^{2})\left(\frac{1}{1-\alpha_{1}^{2}p^{-2(k+s-1)}}-1\right)=\frac{\langle\tilde{\phi}_{1},\tilde{\psi}_{1}\rangle_{\mathcal{A}}}{\alpha_{1}-\alpha_{2}}\frac{\alpha_{1}^{3}X^{2}}{Q_{p,F}^{(2)}(X_{1})}$

and similarly for the second part. ∎

######  Proposition 5.8.

Let $\displaystyle Y=p^{2}NX^{2}$ and $\displaystyle l\geq 0$. Then,

$\displaystyle\displaystyle\sum_{n=0}^{\infty}\tilde{\psi}_{p^{m+2l}}|_{k}T_{+}(p^{n})\Lambda_{+}(p^{l})X_{1}^{m}(X^{2}N^{-1}p^{5})^{l}=$
$\displaystyle\displaystyle
Q_{p,F}^{(2)}(X_{1})^{-1}(\tilde{\psi}_{p^{2l}}-\tilde{\psi}_{p^{2l-1}}|_{k}T_{-}(p)X_{1}+p\tilde{\psi}_{p^{2l-2}}|\Lambda_{-}(p)X_{1}^{2})$
$\displaystyle\displaystyle|_{k}\left((1+p(\nabla_{p}-p\Delta_{p})X_{1}^{2})\Delta_{p^{l}}^{-1}\Lambda_{+}(p^{l})(p^{-3}Y)^{l}\right)$

###### Proof.

The proof just uses Proposition 5.4, together with the fact that
$\displaystyle F|_{k}\Delta_{p^{l}}=(p^{2k-8})^{l}F$. ∎

######  Proposition 5.9.

We have

$\alpha_{1}Q_{p,G}^{(2)}(X_{1})^{-1}\sum_{l=0}^{\infty}\langle\tilde{\phi}_{1},\tilde{\psi}_{p^{2l}}|_{k}(1+p(\nabla_{p}-p\Delta_{p})X_{1}^{2})\Delta_{p^{l}}^{-1}\Lambda_{+}(p^{l})(p^{-3}Y)^{l}\rangle_{\mathcal{A}}-$
$-\alpha_{2}Q_{p,G}^{(2)}(X_{2})^{-1}\sum_{l=0}^{\infty}\langle\tilde{\phi}_{1},\tilde{\psi}_{p^{2l}}|_{k}(1+p(\nabla_{p}-p\Delta_{p})X_{2}^{2})\Delta_{p^{l}}^{-1}\Lambda_{+}(p^{l})(p^{-3}Y)^{l}\rangle_{\mathcal{A}}$
$=\left(\alpha_{1}Q_{p,G}^{(2)}(X_{1})^{-1}-\alpha_{2}Q_{p,G}^{(2)}(X_{2})^{-1}\right)\frac{\langle\tilde{\phi}_{1};\tilde{\psi}_{1}|_{k}S^{(2)}(Y)\rangle_{\mathcal{A}}}{D_{p,G}^{(2)}(Y)}+\left(\frac{\alpha_{2}^{3}X^{2}}{Q_{p,G}^{(2)}(X_{2})}-\frac{\alpha_{1}^{3}X^{2}}{Q_{p,G}^{(2)}(X_{1})}\right)\langle\tilde{\phi}_{1},\tilde{\psi}_{1}\rangle_{\mathcal{A}}$

###### Proof.

We first observe
$\displaystyle\tilde{\psi}_{p^{2l}}|_{k}(1+p(\nabla_{p}-p\Delta_{p})X_{1}^{2})$
$\displaystyle=\begin{cases}(1-p^{2k-6}X_{1}^{2})\tilde{\psi}_{1}&\textup{if
}l=0\\\ \tilde{\psi}_{p^{2l}}&\textup{if }l\geq 1\end{cases}$ ∎

by using the result of Proposition 5.4. Hence, by Proposition 5.3 we obtain

$\sum_{l=0}^{\infty}\langle\tilde{\phi}_{1},\tilde{\psi}_{p^{2l}}|_{k}(1+p(\nabla_{p}-p\Delta_{p})X_{2}^{2})\Delta_{p^{l}}^{-1}\Lambda_{+}(p^{l})(p^{-3}Y)^{l}\rangle_{\mathcal{A}}=(1-p^{2k-6}X_{1}^{2})\tilde{\psi}_{1}+\sum_{l=1}^{\infty}\tilde{\psi}_{p^{2l}}|_{k}\Delta_{p^{l}}^{-1}\Lambda_{+}(p^{l})(p^{-3}Y)^{l}=$
$=\sum_{l=0}^{\infty}\tilde{\psi}_{p^{2l}}|_{k}\Delta_{p^{l}}^{-1}\Lambda_{+}(p^{l})(p^{-3}Y)^{l}-p^{2k-6}X_{1}^{2}\tilde{\psi}_{1}=\tilde{\psi}_{1}|_{k}S^{(2)}(Y)D_{p,G}^{(2)}(Y)^{-1}-\alpha_{1}^{2}X^{2}\tilde{\psi}_{1}$

and from this the result follows.

######  Proposition 5.10.

We have

$\alpha_{1}Q_{p,G}^{(2)}(X_{1})^{-1}\sum_{l=0}^{\infty}p\langle\tilde{\phi}_{1},\tilde{\psi}_{p^{2l-2}}|_{k}\Lambda_{-}(p)X_{1}^{2}(1+p(\nabla_{p}-p\Delta_{p})X_{1}^{2})\Delta_{p^{l}}^{-1}\Lambda_{+}(p^{l})(p^{-3}Y)^{l}\rangle_{\mathcal{A}}-$
$-\alpha_{2}Q_{p,G}^{(2)}(X_{2})^{-1}\sum_{l=0}^{\infty}p\langle\tilde{\phi}_{1},\tilde{\psi}_{p^{2l-2}}|_{k}\Lambda_{-}(p)X_{2}^{2}(1+p(\nabla_{p}-p\Delta_{p})X_{2}^{2})\Delta_{p^{l}}^{-1}\Lambda_{+}(p^{l})(p^{-3}Y)^{l}\rangle_{\mathcal{A}}$
$=\left(\frac{\alpha_{1}^{3}Np^{4}X^{4}}{Q_{p,G}^{(2)}(X_{1})}-\frac{\alpha_{2}^{3}Np^{4}X^{4}}{Q_{p,G}^{(2)}(X_{2})}\right)\frac{\langle\tilde{\phi}_{1},\tilde{\psi}_{1}|_{k}S^{(2)}(Y)\rangle_{\mathcal{A}}}{D_{p,G}^{(2)}(Y)}$

###### Proof.

We use the identities $\displaystyle\Lambda_{-}(p)(\nabla_{p}-p\Delta_{p})=0$
and $\displaystyle\Lambda_{-}(p)\Lambda_{+}(p)=p^{6}(\Delta_{p})^{2}$. We have

$\sum_{l=0}^{\infty}p\tilde{\psi}_{p^{2l-2}}|_{k}\Lambda_{-}(p)X_{1}^{2}(1+p(\nabla_{p}-p\Delta_{p})X_{1}^{2})\Delta_{p^{l}}^{-1}\Lambda_{+}(p^{l})(p^{-3}Y)^{l}$
$=p\sum_{l=0}^{\infty}\tilde{\psi}_{p^{2l-2}}|_{k}\Lambda_{-}(p)X_{1}^{2}\Delta_{p^{l}}^{-1}\Lambda_{+}(p^{l})(p^{-3}Y)^{l}=p\sum_{l=0}^{\infty}\tilde{\psi}_{p^{2l-2}}|_{k}p^{6}(\Delta_{p})^{2}\Delta_{p^{l}}^{-1}\Lambda_{+}(p^{l-1})(p^{-3}Y)^{l-1}X_{1}^{2}$
$=p^{4}\sum_{l=0}^{\infty}\tilde{\psi}_{p^{2l-2}}|\Delta_{p^{l-1}}^{-1}\Lambda_{+}(p^{l-1})(p^{-3}Y)^{l-1}\Delta_{p}X_{1}^{2}Y=p^{2k-4}D_{p,G}^{(2)}(Y)^{-1}\tilde{\psi}_{1}|S^{(2)}(Y)X_{1}^{2}Y$

from which the result then follows. ∎

######  Proposition 5.11.

$\alpha_{1}Q_{p,G}^{(2)}(X_{1})^{-1}\sum_{l=0}^{\infty}\langle\tilde{\phi}_{1},\tilde{\psi}_{p^{2l-1}}|_{k}T_{-}(p)X_{1}(1+p(\nabla_{p}-p\Delta_{p})X_{1}^{2})\Delta_{p^{l}}^{-1}\Lambda_{+}(p^{l})(p^{-3}Y)^{l}\rangle_{\mathcal{A}}-$
$-\alpha_{2}Q_{p,G}^{(2)}(X_{2})^{-1}\sum_{l=0}^{\infty}\langle\tilde{\phi}_{1},\tilde{\psi}_{p^{2l-1}}|_{k}T_{-}(p)X_{2}(1+p(\nabla_{p}-p\Delta_{p})X_{2}^{2})\Delta_{p^{l}}^{-1}\Lambda_{+}(p^{l})(p^{-3}Y)^{l}\rangle_{\mathcal{A}}$
$=\left(\frac{\alpha_{1}^{2}p^{4}X^{3}\lambda_{p}}{Q_{p,G}^{(2)}(X_{1})(1+Y)}-\frac{\alpha_{2}^{2}p^{4}X^{3}\lambda_{p}}{Q_{p,G}^{(2)}(X_{2})(1+Y)}\right)\frac{\langle\tilde{\phi}_{1},\tilde{\psi}_{1}\rangle
S_{F}^{(2)}(Y)^{\textup{prim}}}{(\alpha_{1}-\alpha_{2})D_{p,G}^{(2)}(Y)}$

###### Proof.

The proof is exactly the same as the proof in [13, Proposition $\displaystyle
4.5$](using also the above claim that
$\displaystyle\phi|_{k}S^{(2)}(X)T_{+}(p)=\phi|_{k}T_{+}(p)S^{(2)}(X)^{\textup{factor}}$,
if $\displaystyle\phi$ has index $\displaystyle p$.) Here
$\displaystyle\lambda_{p}$ is the eigenvalue given by
$\displaystyle\tilde{\phi}_{p}|T_{+}(p)=\lambda_{p}\tilde{\phi}_{1}$. ∎

By combining all the above, we obtain that

$D_{F,G,h}^{(p)}(s)=\frac{\langle\tilde{\phi}_{1},\tilde{\psi}_{1}\rangle
S_{F}^{(2)}(Y)^{\textup{prim}}}{(\alpha_{1}-\alpha_{2})D_{p,G}^{(2)}(Y)}\times$
$\times\left(\frac{\alpha_{1}}{Q_{p,G}^{(2)}(X_{1})}-\frac{\alpha_{2}}{Q_{p,G}^{(2)}(X_{2})}+\frac{\alpha_{1}^{3}Np^{4}X^{4}}{Q_{p,G}^{(2)}(X_{1})}-\frac{\alpha_{2}^{3}Np^{4}X^{4}}{Q_{p,G}^{(2)}(X_{2})}-\frac{\alpha_{1}^{2}p^{4}X^{3}\lambda_{p}}{Q_{p,G}^{(2)}(X_{1})(1+Y)}+\frac{\alpha_{2}^{2}p^{4}X^{3}\lambda_{p}}{Q_{p,G}^{(2)}(X_{2})(1+Y)}\right)$

Let us now look at the expression in the big bracket. The numerator equals

$((\alpha_{1}+\alpha_{1}^{3}Np^{4}X^{4})(1+Y)-\alpha_{1}^{2}p^{4}X^{3}\lambda_{p})Q_{p,G}^{(2)}(X_{2})-((\alpha_{2}+\alpha_{2}^{3}Np^{4}X^{4})(1+Y)-\alpha_{2}^{2}p^{4}X^{3}\lambda_{p})Q_{p,G}^{(2)}(X_{1})$

Here

$Q_{p,G}^{(2)}(t)=1-\lambda_{p}t+(p\lambda_{T_{1,p}}+p(p^{3}+p^{2}-p+1)p^{2k-8})t^{2}-p^{4}p^{2k-8}\lambda_{p}t^{3}+p^{4k-8}t^{4}$

where $\displaystyle\lambda_{T_{1,p}}$ is the eigenvalue corresponding to the
operator $\displaystyle T_{1,p}$. Denote by $\displaystyle
A_{2}=p\lambda_{T_{1,p}}+p(p^{3}+p^{2}-p+1)p^{2k-8}$. By then performing the
very lengthy calculation, and grouping in powers of $\displaystyle Y$, we
obtain that the above numerator equals

$(\alpha_{1}-\alpha_{2})(1-Y)(1-Y(A_{2}p^{2}N^{-2}-2)+Y^{2}(p^{2}N^{-2}\lambda_{p}^{2}-2A_{2}p^{2}N^{-2}+2)-Y^{3}(A_{2}p^{2}N^{-2}-2)+Y^{4})$
$=(\alpha_{1}-\alpha_{2})(1-Y)D_{p,G}^{(2)}(Y)$

using the relations of equation $\displaystyle(2)$. Hence,

$D_{F,G,h}^{(p)}(s)=\frac{\langle\tilde{\phi}_{1},\tilde{\psi}_{1}\rangle_{\mathcal{A}}S_{F}^{(2)}(Y)^{\textup{factor}}(1-Y)}{Q_{p,G}^{(2)}(X_{1})Q_{p,G}^{(2)}(X_{2})}$

Let us now explore the connection of $\displaystyle
S_{F}^{(2)}(Y)^{\textup{factor}}$ with known $\displaystyle L-$functions.

######  Proposition 5.12.

We have

$S_{F}^{(2)}(Y)^{\textup{factor}}=L_{p}(f,k+s-2)L_{p}\left(f,k+s-2,\left(\frac{-4}{p}\right)\right)$

where $\displaystyle f\in
S_{k-1}\left(\Gamma_{0}(4),\left(\frac{-4}{\cdot}\right)\right)$ is the
modular form whose Maass lift is $\displaystyle F$ as in 3.8.

###### Proof.

We first write

$f\mid_{k-1}T(p)=a(p)f$

for the standard operator $\displaystyle
T(p)=\Gamma_{0}(4)\textup{diag}(1,p)\Gamma_{0}(4)=T(1,p)$. By standard
relations between Hecke operators, we then have

$T(p^{2})=T(p)^{2}-\left(\frac{-4}{p}\right)p^{k-2}$

where $\displaystyle T(p^{2})=T(1,p^{2})+\left(\frac{-4}{p}\right)p^{k-3}$.
This then implies

$f\mid_{k-1}T(1,p^{2})=(a(p)^{2}+p^{k-2}+p^{k-3})f$

Using now [7, Lemma 3.3] we obtain that

$\tilde{\phi}_{1}\mid_{k}T^{J}(p)=p^{k-4}(a(p)^{2}+p^{k-2}+p^{k-3})\tilde{\phi}_{1}$

Hence

$S_{F}^{(2)}(Y)=1-p^{1-k}(a(p)^{2}+2p^{k-2})Y+p^{-2}Y^{2}$

We now define the Satake parameters of $\displaystyle f$ as follows:

$1-a(p)p^{-s}+\left(\frac{-4}{p}\right)p^{k-2}p^{-2s}=(1-\alpha_{p}p^{-s})(1-\beta_{p}p^{-s})$

We now have $\displaystyle Y=p^{2}NX^{2}$, $\displaystyle N=p^{k-1}$ and
$\displaystyle X=p^{-(k+s-1)}$ and so we obtain

$S_{F}^{(2)}(Y)=1-p^{4-2k}(\alpha_{p}^{2}+\beta_{p}^{2})p^{-2s}+p^{-2k+4-2s}=(1-p^{4-2k-2s}\alpha_{p}^{2})(1-p^{4-2k-2s}\beta_{p}^{2})=$
$=(1-p^{2-k-s}\alpha_{p})(1-p^{2-k-s}\beta_{p})(1+p^{2-k-s}\alpha_{p})(1+p^{2-k-s}\beta_{p})$
$=L_{p}(f,k+s-2)L_{p}\left(f,k+s-2,\left(\frac{-4}{p}\right)\right)$

∎

Hence, in total, we obtain the following Theorem:

######  Theorem 5.13.

Let $\displaystyle F,G\in S_{2}^{k}$ and $\displaystyle h\in S_{1}^{k}$ be
Hecke eigenforms, with $\displaystyle h$ having totally real Fourier
coefficients, and $\displaystyle F$ belonging in the Maass space, with
corresponding $\displaystyle f\in
S_{k-1}\left(\Gamma_{0}(4),\left(\frac{-4}{\cdot}\right)\right)$. Let also
$\displaystyle\phi_{1},\psi_{1}$ denote the first Fourier-Jacobi coefficients
of $\displaystyle F,G$ and $\displaystyle X=p^{-(k+s-1)},N=p^{k-1}$ and
$\displaystyle Y=pNX^{2}$. We then have for $\displaystyle\textup{Re}(s)$
large enough

$D_{F,G,h}^{(p)}(s)=\frac{\langle\tilde{\phi}_{1},\tilde{\psi}_{1}\rangle_{\mathcal{A}}L_{p}(f,k+s-2)L_{p}\left(f,k+s-2,\left(\frac{-4}{p}\right)\right)(1-Y)}{Q_{p,G}^{(2)}(X_{1})Q_{p,G}^{(2)}(X_{2})}$

######  Remark.

The factor $\displaystyle Q_{p,G}^{2}(X_{1})Q_{p,G}^{2}(X_{2})$ is of degree
8, whereas the twisted $\displaystyle L-$function should have degree 12.
Looking at the function $\displaystyle Z_{G}^{*}(s)$ (the one with the
symmetric functional equation) in the notation of [8, Theorem $\displaystyle
5.1$], we see that the Euler factor at an inert prime of this function is
given by:

$\displaystyle(1+p^{k-2-s})^{-1}(1-p^{k-2-s})^{-1}Q^{2}_{p,G}(p^{-s})^{-1}$

In particular if we want to introduce the missing factors in the twisted
series, we need to multiply the Dirichlet series $\displaystyle D_{F,G,h}(s)$
with $\displaystyle L(h,s-k+2)L(h,\chi,s-k+2)$. But this will create some
extra factors at the split prime. Namely at a split prime we have the factor:

$\displaystyle\prod_{i=1}^{2}(1-\alpha_{i}p^{k-2-s})^{-1}(1-\alpha_{i}p^{k-2-s})^{-1}=\prod_{i=1}^{2}(1-\alpha_{i}p^{k-2-s})^{-2}$

These extra factors should cancel. That is, we should find them in the
numerator of the Dirichlet series for the split primes, which we consider in
the next section.

##  6\. Split Primes

We will now consider the case where the rational prime $\displaystyle p$
splits. That is, we have that $\displaystyle p=\pi\overline{\pi}$ for some
prime element $\displaystyle\pi\in\mathcal{O}_{K}$. Our aim in this section is
to prove weak rationality theorems analogous to Propositions 5.4 and 5.5. In
order to do that, we will first have to factorize the polynomials which serve
as the $\displaystyle p-$factors of the standard and Gritsenko’s
$\displaystyle L-$function attached to a Hermitian Hecke eigenform in the
parabolic Hecke algebra $\displaystyle H_{p}^{1,1}$, as defined in Section 3.
The factorisation of the latter polynomial has been done by Gritsenko in [8].
Our aim, therefore, is to factorize the standard Hecke polynomial. However, as
we mentioned in Section 3, $\displaystyle H^{1,1}_{p}$ is isomorphic to the
ring of polynomials of one variable with coefficients from the Hecke ring of
the parabolic subgroup

$P_{1,2,1}=\left\\{\begin{pmatrix}\pm 1&*&*\\\ 0&g&*\\\ 0&0&\pm
1\end{pmatrix}\in\textup{GL}_{4}(\mathbb{Z}_{p})\mid
g\in\textup{GL}_{2}(\mathbb{Z}_{p})\right\\}$

Hence, we will first investigate Hecke algebras of the general linear group
and then use the above isomorphism to translate the relations back to
$\displaystyle H_{p}^{1,1}$.

###  6.1. Relations between operators in $\displaystyle\textup{GL}_{4}$ and
factorisation

For this section, we denote by
$\displaystyle\Gamma_{n}=\textup{GL}_{n}(\mathbb{Z}_{p})$ and let

$P_{m_{1},\cdots,m_{l}}^{(n)}=\left\\{\begin{pmatrix}A_{1}&*&\cdots&*\\\
0&A_{2}&\cdots&*\\\ \vdots&\vdots&\vdots&\vdots\\\
0&0&\cdots&A_{l}\end{pmatrix},A_{i}\in\textup{GL}_{m_{i}}(\mathbb{Q}_{p}),m_{1}+\cdots+m_{l}=n\right\\}$

be a parabolic subgroup of $\displaystyle\textup{GL}_{n}(\mathbb{Q}_{p})$.
Denote by
$\displaystyle\Gamma_{P}=\Gamma_{P}^{(n)}=\Gamma_{m_{1},\cdots,m_{l}}^{(n)}$,
the group of $\displaystyle\mathbb{Z}_{p}-$points of $\displaystyle
P_{m_{1},\cdots,m_{l}}^{(n)}$. Let also $\displaystyle
H_{n}=H(\Gamma_{n},\textup{GL}_{n}(\mathbb{Q}_{p}))$ be the full Hecke algebra
and $\displaystyle
H_{m_{1},\cdots,m_{l}}=H(\Gamma_{m_{1},\cdots,m_{l}}^{(n)},P_{m_{1},\cdots,m_{l}}^{(n)})$
denote the corresponding parabolic Hecke algebra.

Let us explain the isomorphism we mentioned above explicitly, as Gritsenko
describes in [8, Proposition $\displaystyle 2.4$].

We fix an identification $\displaystyle
K_{p}:=K\otimes\mathbb{Q}_{p}\cong\mathbb{Q}_{p}\times\mathbb{Q}_{p}$ and
denote by $\displaystyle(\mu,-\mu)$ the image of the element
$\displaystyle(2i)^{-1}$. Let also $\displaystyle e=(1,0),e^{\sigma}=(0,1)\in
K_{p}$. We perform a change of variables with the matrix

$\displaystyle C=\begin{pmatrix}eI_{2}&-\mu e^{\sigma}I_{2}\\\ \mu
e^{\sigma}E_{2}&eE_{2}\end{pmatrix}=\left(\begin{pmatrix}I_{2}&0\\\
0&E_{2}\end{pmatrix},\begin{pmatrix}0&-\mu I_{2}\\\ \mu
E_{2}&0\end{pmatrix}\right)\in\textup{GL}_{4}(\mathbb{Q}_{p})\times\textup{GL}_{4}(\mathbb{Q}_{p}).$

where $\displaystyle I_{2}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$ and
$\displaystyle E_{2}$ is the identity. We remark that for $\displaystyle n=2$:

$\displaystyle\begin{pmatrix}I_{2}&0\\\
0&E_{2}\end{pmatrix}\textup{diag}(a_{1},a_{2},a_{3},a_{4})\begin{pmatrix}I_{2}&0\\\
0&E_{2}\end{pmatrix}^{-1}=\textup{diag}(a_{2},a_{1},a_{3},a_{4})$

We then have that $\displaystyle G_{p}^{(2)}$ (see Section 3) is identified by

$\tilde{G}_{p}^{(2)}=\\{(X,Y)\in\textup{GL}_{4}(\mathbb{Q}_{p})\times\textup{GL}_{4}(\mathbb{Q}_{p}):Y^{t}X=cE_{4},c\in\mathbb{Q}_{p}^{\times}\\}$

and $\displaystyle U_{p}^{(2)}$ by

$\tilde{U}_{p}^{(2)}=\\{(\gamma,\alpha(\gamma^{-1})^{t}),\gamma\in\textup{GL}_{4}(\mathbb{Z}_{p}),\alpha\in\mathbb{Z}_{p}^{\times}\\}$

The identification $\displaystyle
G_{p}^{2}=\textup{GL}_{4}(\mathbb{Q}_{p})\times\mathbb{Q}_{p}^{\times}$ is
obtained by projection to the first component induced by $\displaystyle
K_{p}=\mathbb{Q}_{p}\times\mathbb{Q}_{p}$.

The double coset of $\displaystyle(M,c(M^{-1})^{t})\in\tilde{G}_{p}^{(2)}$
with respect to $\displaystyle\tilde{U}_{p}^{(2)}$ is determined by the double
coset of $\displaystyle M$ with respect to
$\displaystyle\textup{GL}_{4}(\mathbb{Z}_{p})$ and by the order
$\displaystyle\delta$ of the ideal $\displaystyle c\mathbb{Z}_{p}$. We will
denote such a coset by $\displaystyle(M,\delta)_{\tilde{U}_{p}^{(2)}}$. Note
that here there is a choice, namely whether we have
$\displaystyle\pi\mapsto(pu,v)$ with $\displaystyle
u,v\in\mathbb{Z}_{p}^{\times}$ or
$\displaystyle\overline{\pi}\mapsto(pu^{\prime},v^{\prime})$ with
$\displaystyle u^{\prime},v^{\prime}\in\mathbb{Z}_{p}^{\times}$. In the
following, we always choose the first identification.

The above identification makes the following diagram commutative:

$\displaystyle{H_{p}^{2}}$$\displaystyle{H_{4}}$$\displaystyle{H_{p}^{1,1}}$$\displaystyle{H_{1,2,1}}$

We will now prove [9, Lemma 2], which is given without proof. We note here
that the Hecke ring of interest in [9] is the so called $\displaystyle p-$ring
of $\displaystyle\textup{GL}_{n}$, but this is canonically isomorphic to the
$\displaystyle p-$adic Hecke ring.

######  Lemma 6.1.

We set $\displaystyle n=n_{1}+n_{2}$ with $\displaystyle
n_{1},n_{2}\in\mathbb{N}$ and let $\displaystyle
A_{i}\in\textup{GL}_{n_{i}}(\mathbb{Q}_{p})$. Then we have the coset
decomposition

$\displaystyle\Gamma_{n_{1},n_{2}}\begin{pmatrix}A&0\\\
0&D\end{pmatrix}\Gamma_{n_{1},n_{2}}=\sum\Gamma_{n_{1},n_{2}}\begin{pmatrix}A&B\\\
0&D\end{pmatrix}\begin{pmatrix}\mu&0\\\ 0&\nu\end{pmatrix},$

where the sum is over
$\displaystyle\mu\in\Gamma_{n_{1}}^{A}\setminus\Gamma_{n_{1}}$,
$\displaystyle\nu\in\Gamma_{n_{2}}^{D}\setminus\Gamma_{n_{2}}$ and
$\displaystyle B\in V(A,D)$ with $\displaystyle\Gamma_{m}^{R}:=\Gamma_{m}\cap
R^{-1}\Gamma_{m}R$ for a given matrix $\displaystyle R$ and

$\displaystyle V(A,D)=\\{AY\,\,|\,\,Y\in
M_{n_{1},n_{2}}(\mathbb{Z}_{p})\pmod{D^{\times}}\\}$

that is $\displaystyle AY_{1}\equiv AY_{2}\pmod{D^{\times}}$ if and only if
$\displaystyle AY_{1}D^{-1}-AY_{2}D^{-1}\in M_{n_{1},n_{2}}(\mathbb{Z}_{p})$.

###### Proof.

We let $\displaystyle X\in\Gamma_{n_{1},n_{2}}\begin{pmatrix}A&0\\\
0&D\end{pmatrix}\Gamma_{n_{1},n_{2}}$. That is

$\displaystyle X\in\Gamma_{n_{1},n_{2}}\begin{pmatrix}A&0\\\
0&D\end{pmatrix}\begin{pmatrix}R_{1}&R_{2}\\\
0&R_{3}\end{pmatrix}=\Gamma_{n_{1},n_{2}}\begin{pmatrix}A&AR_{2}R_{3}^{-1}\\\
0&D\end{pmatrix}\begin{pmatrix}R_{1}&0\\\ 0&R_{3}\end{pmatrix}$

Setting $\displaystyle Y:=R_{2}R_{3}^{-1}$ and $\displaystyle\mu=R_{1}$ and
$\displaystyle\nu=R_{3}$ shows that $\displaystyle X$ belongs to a right coset
of the stated form. We now show the equivalence.

We assume that

$\displaystyle\Gamma_{n_{1},n_{2}}\begin{pmatrix}A&B_{1}\\\
0&D\end{pmatrix}\begin{pmatrix}\mu_{1}&0\\\
0&\nu_{1}\end{pmatrix}=\Gamma_{n_{1},n_{2}}\begin{pmatrix}A&B_{2}\\\
0&D\end{pmatrix}\begin{pmatrix}\mu_{2}&0\\\ 0&\nu_{2}\end{pmatrix}$

This gives us,

$\displaystyle\begin{pmatrix}A\mu_{1}&B_{1}\nu_{1}\\\
0&D\nu_{1}\end{pmatrix}\begin{pmatrix}A\mu_{2}&B_{2}\nu_{2}\\\
0&D\nu_{2}\end{pmatrix}^{-1}\in\Gamma_{n_{1},n_{2}}$

or equivalently

$\displaystyle\begin{pmatrix}A\mu_{1}&B_{1}\nu_{1}\\\
0&D\nu_{1}\end{pmatrix}\begin{pmatrix}(A\mu_{2})^{-1}&-(A\mu_{2})^{-1}B_{2}\nu_{2}(D\nu_{2})^{-1}\\\
0&(D\nu_{2})^{-1}\end{pmatrix}\in\Gamma_{n_{1},n_{2}}$

It then follows that $\displaystyle\mu_{1}\mu_{2}^{-1}\in
A^{-1}\Gamma_{n_{1}}A$ and $\displaystyle\nu_{1}\nu_{2}^{-1}\in
D^{-1}\Gamma_{n_{2}}D$. Moreover, if we write $\displaystyle B_{1}=AY_{1}$ and
$\displaystyle B_{2}=AY_{2}$ we obtain that

$\displaystyle-A\mu_{1}\mu_{2}^{-1}Y_{2}D^{-1}+AY_{1}\nu_{1}\nu_{2}^{-1}D^{-1}\in
M_{n_{1},n_{2}}(\mathbb{Z}_{p}).$

We now write $\displaystyle\mu_{1}\mu_{2}^{-1}=A^{-1}\gamma_{1}A$ and
$\displaystyle\nu_{1}\nu_{2}^{-1}=D^{-1}\gamma_{2}D$, with
$\displaystyle\gamma_{1}\in\Gamma_{n_{1}}$ and
$\displaystyle\gamma_{2}\in\Gamma_{n_{2}}$. We have
$\displaystyle\mu_{1}\mu_{2}^{-1}\in A^{-1}\Gamma_{n_{1}}A$ and as
$\displaystyle\mu\in\Gamma_{n_{1}}$ we get $\displaystyle\mu$ runs in a set of
representatives for $\displaystyle
A^{-1}\Gamma_{n_{1}}A\cap\Gamma_{n_{1}}\backslash\Gamma_{n_{1}}$ and similarly
for $\displaystyle\nu$. If we now first fix such sets of representatives for
$\displaystyle\mu,\nu$, we get that $\displaystyle\mu_{1}\mu_{2}^{-1}=I$ and
$\displaystyle\nu_{1}\nu_{2}^{-1}=I$ because $\displaystyle\mu_{1},\mu_{2}$
must take values from a finite set of representatives, with every two
different of them different $\displaystyle\pmod{A^{-1}\Gamma_{n_{1}}A}$ as
well. Similarly for $\displaystyle\nu$. The result then follows. ∎

We now generalise the above lemma in the situation of a parabolic where on the
main diagonal we have three block matrices. That is

######  Lemma 6.2.

We set $\displaystyle n=n_{1}+n_{2}+n_{3}$ with $\displaystyle
n_{1},n_{2},n_{3}\in\mathbb{N}$. Then, with $\displaystyle
A_{i}\in\textup{GL}_{n_{i}}(\mathbb{Q}_{p})$, we have the coset decomposition

$\displaystyle\Gamma_{n_{1},n_{2},n_{3}}\begin{pmatrix}A_{1}&0&0\\\
0&A_{2}&0\\\ 0&0&A_{3}\\\
\end{pmatrix}\Gamma_{n_{1},n_{2},n_{3}}=\sum\Gamma_{n_{1},n_{2},n_{3}}\begin{pmatrix}A_{1}&B_{1}&C\\\
0&A_{2}&B_{2}\\\ 0&0&A_{3}\end{pmatrix}\begin{pmatrix}\mu_{1}&0&0\\\
0&\mu_{2}&0\\\ 0&0&\mu_{3}\end{pmatrix},$

where the sum is over
$\displaystyle\mu_{i}\in\Gamma_{n_{i}}^{A_{i}}\setminus\Gamma_{n_{i}}$, for
$\displaystyle i=1,2,3$, $\displaystyle B_{1}\in V(A_{1},A_{2})$,
$\displaystyle B_{2}\in V(A_{2},A_{3})$ and $\displaystyle C\in
V(A_{1},A_{3})$

###### Proof.

The fact that any element $\displaystyle
X\in\Gamma_{n_{1},n_{2},n_{3}}\begin{pmatrix}A_{1}&0&0\\\ 0&A_{2}&0\\\
0&0&A_{3}\\\ \end{pmatrix}\Gamma_{n_{1},n_{2},n_{3}}$ belongs to a right coset
of the above form, follows from the decomposition
$\displaystyle\Gamma_{n_{1},n_{2},n_{3}}=U_{n_{1},n_{2},n_{3}}T_{n_{1},n_{2},n_{3}}$
where

$\displaystyle
U_{n_{1},n_{2},n_{3}}=\left\\{\begin{pmatrix}1&M_{n_{1},n_{2}}(\mathbb{Z}_{p})&M_{n_{1},n_{3}}(\mathbb{Z}_{p})\\\
0&1&M_{n_{2},n_{3}}(\mathbb{Z}_{p})\\\
0&0&1\end{pmatrix}\right\\},\,\,\text{and}\,\,\,T_{n_{1},n_{2},n_{3}}=\left\\{\text{diag}[\gamma_{1},\gamma_{2},\gamma_{3}],\,\,\,\gamma_{i}\in\Gamma_{i}\right\\}.$

For the equivalence relations we assume

$\displaystyle\Gamma_{n_{1},n_{2},n_{3}}\begin{pmatrix}A_{1}&B_{1}&C\\\
0&A_{2}&B_{2}\\\ 0&0&A_{3}\end{pmatrix}\begin{pmatrix}\mu_{1}&0&0\\\
0&\mu_{2}&0\\\
0&0&\mu_{3}\end{pmatrix}=\Gamma_{n_{1},n_{2},n_{3}}\begin{pmatrix}A_{1}&\tilde{B}_{1}&\tilde{C}\\\
0&A_{2}&\tilde{B}_{2}\\\
0&0&\tilde{A}_{3}\end{pmatrix}\begin{pmatrix}\tilde{\mu}_{1}&0&0\\\
0&\tilde{\mu}_{2}&0\\\ 0&0&\tilde{\mu}_{3}\end{pmatrix}$

This gives

$\displaystyle\begin{pmatrix}A_{1}&B_{1}&C\\\ 0&A_{2}&B_{2}\\\
0&0&A_{3}\end{pmatrix}\begin{pmatrix}\mu_{1}\tilde{\mu}_{1}^{-1}&0&0\\\
0&\mu_{2}\tilde{\mu}_{2}^{-1}&0\\\
0&0&\mu_{3}\tilde{\mu}_{3}^{-1}\end{pmatrix}\begin{pmatrix}A_{1}&\tilde{B}_{1}&\tilde{C}\\\
0&A_{2}&\tilde{B}_{2}\\\
0&0&A_{3}\end{pmatrix}^{-1}\in\Gamma_{n_{1},n_{2},n_{3}},$

or equivalently

$\displaystyle\begin{pmatrix}A_{1}\mu_{1}\tilde{\mu}_{1}^{-1}&B_{1}\mu_{2}\tilde{\mu}_{2}^{-1}&C\mu_{3}\tilde{\mu}_{3}^{-1}\\\
0&A_{2}\mu_{2}\tilde{\mu}_{2}^{-1}&B_{2}\mu_{3}\tilde{\mu}_{3}^{-1}\\\
0&0&A_{3}\mu_{3}\tilde{\mu}_{3}^{-1}\end{pmatrix}\begin{pmatrix}A_{1}&\tilde{B}_{1}&\tilde{C}\\\
0&A_{2}&\tilde{B}_{2}\\\
0&0&A_{3}\end{pmatrix}^{-1}\in\Gamma_{n_{1},n_{2},n_{3}},$

The block-diagonal entries give us the relations
$\displaystyle\mu_{i}\tilde{\mu}^{-1}_{i}\in
A_{i}^{-1}\Gamma_{n_{i}}A_{i}\cap\Gamma_{n_{i}}$ while the “$\displaystyle B$”
entries give us the relations on $\displaystyle B_{1}$ and $\displaystyle
B_{2}$. We now show the relation on $\displaystyle C$. We write

$\displaystyle\begin{pmatrix}A_{1}&\tilde{B}_{1}&\tilde{C}\\\
0&A_{2}&\tilde{B}_{2}\\\
0&0&\tilde{A}_{3}\end{pmatrix}^{-1}=\begin{pmatrix}A_{1}^{-1}&*&Z\\\
0&A_{2}^{-1}&X\\\ 0&0&\tilde{A}_{3}^{-1}\end{pmatrix}$

with $\displaystyle
Z=-A_{1}^{-1}(-\tilde{B}_{1}A_{2}^{-1}\tilde{B}_{2}+\tilde{C})A_{3}^{-1}$ and
$\displaystyle X=-A_{2}^{-1}\tilde{B}_{2}A_{3}^{-1}$. We then have that

$\displaystyle
A_{1}\mu_{1}\tilde{\mu}_{1}^{-1}Z+B_{1}\mu_{2}\tilde{\mu}_{2}^{-1}X+C\mu_{3}\tilde{\mu}_{3}^{-1}A_{3}^{-1}\in
M_{n_{1},n_{3}}(\mathbb{Z}_{p}).$

We now write
$\displaystyle\mu_{i}\tilde{\mu}_{i}^{-1}=A_{i}^{-1}\gamma_{i}A_{i}$ for
$\displaystyle\gamma_{i}\in\Gamma_{n_{i}}$, $\displaystyle B_{1}=A_{1}Y_{1}$
and $\displaystyle C=A_{1}Y$ and get

$\gamma_{1}A_{1}Z+A_{1}Y_{1}A_{2}^{-1}\gamma_{2}A_{2}Y_{1}+A_{1}YA_{3}^{-1}\gamma_{3}\in
M_{(n_{1},n_{3})}(\mathbb{Z}_{p})$

Arguing as before we may set $\displaystyle\mu_{i}=\tilde{\mu}_{i}$ to obtain

$\displaystyle\tilde{B}_{1}A_{2}^{-1}\tilde{B}_{2}A_{3}^{-1}-B_{1}A_{2}^{-1}\tilde{B}_{2}A_{3}^{-1}-\tilde{A}_{3}^{-1}+CA_{3}^{-1}\in
M_{n_{1},n_{3}}(\mathbb{Z}_{p})$

Using the same argument we may even set $\displaystyle\tilde{B}_{1}=B_{1}$ and
hence obtain $\displaystyle CA_{3}^{-1}\in M_{n_{1},n_{3}}(\mathbb{Z}_{p})$
which concludes the proof.

∎

In the following, we will make use of the special case of the above lemma with
$\displaystyle n_{1}=n_{3}=1$ and $\displaystyle n_{2}=2$. In particular, we
get the following corollary:

######  Corollary 6.3.

Let $\displaystyle a,b\in\mathbb{Q}_{p}^{\times}$ and $\displaystyle
A\in\textup{GL}_{2}(\mathbb{Q}_{p})$. Then we have

$\Gamma_{1,2,1}\begin{pmatrix}a&*&*\\\ 0&A&*\\\
0&0&b\end{pmatrix}\Gamma_{1,2,1}=\sum\Gamma_{1,2,1}\begin{pmatrix}a&B&D\\\
0&A&C\\\ 0&0&b\end{pmatrix}\begin{pmatrix}1&0&0\\\ 0&N&0\\\
0&0&1\end{pmatrix}$

where $\displaystyle N\in\Gamma_{2}^{A}\backslash\Gamma_{2}$ and
$\displaystyle B\in V(a,A),D\in V(a,b),C\in V(A,b)$ with

$V(A,D)=\\{AY/\sim\mid AY=AY^{\prime}\iff A(Y-Y^{\prime})D^{-1}\in
M_{n}(\mathbb{Z}_{p})\\}$

We now denote by $\displaystyle T(a,b,c,d)$ the element of the Hecke algebra
$\displaystyle
H(\textup{GL}_{4}(\mathbb{Q}_{p}),\textup{GL}_{4}(\mathbb{Z}_{p}))$ defined by

$T(a,b,c,d)=\textup{GL}_{4}(\mathbb{Z}_{p})\textup{diag}(a,b,c,d)\textup{GL}_{4}(\mathbb{Z}_{p})$

We also have the standard elements of $\displaystyle H_{4}$:

$T_{1}:=T(1,1,1,p),T_{2}:=T(1,1,p,p),T_{3}:=T(1,p,p,p)$

The decomposition of these elements into right cosets can be found in [2,
Lemma $\displaystyle 3.2.18$].

We now turn our focus into the parabolic Hecke algebra $\displaystyle
H_{1,2,1}$ and our aim is to compute the images of the above standard Hecke
operators under the embedding described in Lemma 3.1. We note here that the
conditions of the lemma for the Hecke algebras $\displaystyle H_{4},H_{1,2,1}$
hold, as explained in page $\displaystyle 2890$ of [10]. In a similar fashion
to $\displaystyle H_{4}$, we denote by

$T_{0}(a,b,c,d)=\Gamma_{1,2,1}\textup{diag}(a_{1},a_{2},a_{3},a_{4})\Gamma_{1,2,1}$

for an element of $\displaystyle H_{1,2,1}$. Let us now introduce some useful
elements of $\displaystyle H_{1,2,1}$.

  * •

$\displaystyle\Lambda_{+}^{1,3}:=T_{0}(1,p,p,p)$

  * •

$\displaystyle\Lambda_{+}^{3,1}:=T_{0}(1,1,1,p)$

  * •

$\displaystyle\Lambda_{-}^{1,3}:=T_{0}(p,1,1,1)$

  * •

$\displaystyle\Lambda_{-}^{3,1}:=T_{0}(p,p,p,1)$

  * •

$\displaystyle T_{-}(p):=T_{0}(p,1,p,1)$

  * •

$\displaystyle T_{+}(p):=T_{0}(1,1,p,p)$

The right coset decomposition can be now computed by 6.3.

######  Proposition 6.4.

Let $\displaystyle\epsilon$ denote the embedding of the Hecke algebra
$\displaystyle H_{4}$ into $\displaystyle H_{1,2,1}$ as described in 3.1. We
then have the following images of the elements $\displaystyle T_{i}$:

  * •

$\displaystyle\epsilon(T_{1})=\Lambda_{-}^{1,3}+T_{0}(1,1,p,1)+\Lambda_{+}^{3,1}$

  * •

$\displaystyle\epsilon(T_{2})=T_{-}(p)+T_{+}(p)+T_{0}(1,p,p,1)+T_{0}(p,1,1,p)$

  * •

$\displaystyle\epsilon(T_{3})=\Lambda_{-}^{3,1}+T_{0}(p,1,p,p)+\Lambda_{+}^{1,3}$

###### Proof.

These follow directly by the right coset decompositions and the definition of
the embedding $\displaystyle\epsilon$. We note here a typo in page
$\displaystyle 2879$ of Gritsenko’s paper [10] in the
$\displaystyle\Lambda_{+}$ component(it appears we need to swap
$\displaystyle\Lambda_{+}^{1,3}$ and $\displaystyle\Lambda_{+}^{3,1}$). ∎

We are now in a position to give the factorisation of the standard Hecke
polynomial $\displaystyle Q_{4}$, as this is defined in [10, Example
$\displaystyle 2$].

######  Theorem 6.5.

Let $\displaystyle Q_{4}(t)=1-T_{1}t+pT_{2}t^{2}-p^{3}T_{3}t^{3}+p^{6}\Delta
t^{4}\in H^{4}[t]$. Then, in $\displaystyle H_{1,2,1}$, we have the
factorisation

$Q_{4}(t)=(1-\Lambda_{-}^{1,3}t)(1-X_{1}t+pX_{2}t^{2})(1-\Lambda_{+}^{3,1}t)$

where $\displaystyle X_{1}=T_{0}(1,1,p,1)$ and $\displaystyle
X_{2}=T_{0}(1,p,p,1)$.

###### Proof.

The factorisation follows by the above embeddings as well as a series of
relations, which follow from the above right coset decompositions:

  * •

$\displaystyle\Lambda_{-}^{1,3}X_{1}=pT_{-}(p)$

  * •

$\displaystyle X_{1}\Lambda_{+}^{3,1}=pT_{+}(p)$

  * •

$\displaystyle\Lambda_{-}^{1,3}\Lambda_{+}^{3,1}=pT_{0}(p,1,1,p)$

  * •

$\displaystyle X_{2}\Lambda_{+}^{3,1}=p^{2}\Lambda_{+}^{1,3}$

  * •

$\displaystyle\Lambda_{-}^{1,3}X_{2}=p^{2}\Lambda_{-}^{3,1}$

  * •

$\displaystyle\Lambda_{-}^{1,3}X_{1}\Lambda_{+}^{3,1}=p^{3}T_{0}(p,1,p,p)$

  * •

$\displaystyle\Lambda_{-}^{1,3}X_{2}\Lambda_{+}^{3,1}=p^{5}\Delta$

∎

###  6.2. Hecke Operators and weak rationality theorems

We will now translate the results above back to the Hecke algebras
$\displaystyle H_{p}^{2}$ and $\displaystyle H_{p}^{1,1}$ of the unitary
group. Let us first give the correspondence between the Hecke operators of
$\displaystyle H_{p}^{1,1}$ and $\displaystyle H_{1,2,1}$:

  * •

$\displaystyle\Lambda_{+}^{1,3}\longleftrightarrow\Lambda_{+}(\pi):=\Gamma_{1,1}\textup{diag}(\pi,1,\pi,p)\Gamma_{1,1}$

  * •

$\displaystyle\Lambda_{+}^{3,1}\longleftrightarrow\Lambda_{+}(\overline{\pi}):=\Gamma_{1,1}\textup{diag}(\overline{\pi},1,\overline{\pi},p)\Gamma_{1,1}$

  * •

$\displaystyle\Lambda_{-}^{3,1}\longleftrightarrow\Lambda_{-}(\pi):=\Gamma_{1,1}\textup{diag}(\pi,p,\pi,1)\Gamma_{1,1}$

  * •

$\displaystyle\Lambda_{-}^{1,3}\longleftrightarrow\Lambda_{-}(\overline{\pi}):=\Gamma_{1,1}\textup{diag}(\overline{\pi},p,\overline{\pi},1)\Gamma_{1,1}$

  * •

$\displaystyle T_{0}(1,1,p,1)\longleftrightarrow
T(\overline{\pi}):=\Gamma_{1,1}\textup{diag}(1,\overline{\pi},p,\overline{\pi})\Gamma_{1,1}$

  * •

$\displaystyle T_{0}(p,1,p,p)\longleftrightarrow
T(\pi):=\Gamma_{1,1}\textup{diag}(1,\pi,p,\pi)\Gamma_{1,1}$

  * •

$\displaystyle T_{0}(1,p,p,1)\longleftrightarrow
T(\pi,\overline{\pi}):=\Gamma_{1,1}\textup{diag}(\pi,\overline{\pi},\pi,\overline{\pi})\Gamma_{1,1}$

  * •

$\displaystyle T_{0}(p,1,1,p)\longleftrightarrow
T(\overline{\pi},\pi):=\Gamma_{1,1}\textup{diag}(\overline{\pi},\pi,\overline{\pi},\pi)\Gamma_{1,1}$

  * •

$\displaystyle T_{-}(p)\longleftrightarrow
T_{-}(p):=\Gamma_{1,1}\textup{diag}(1,p,p,1)\Gamma_{1,1}$

  * •

$\displaystyle T_{+}(p)\longleftrightarrow
T_{+}(p):=\Gamma_{1,1}\textup{diag}(1,1,p,p)\Gamma_{1,1}$

We also denote
$\displaystyle\Delta_{\pi}:=\Gamma_{1,1}\textup{diag}(\pi,\pi,\pi,\pi)\Gamma_{1,1}$
and similarly for $\displaystyle\Delta_{\overline{\pi}}$ and
$\displaystyle\Delta_{p}$.

In order to make clear how the isomorphism described in the beginning of
subsection 6.1 works, let us describe it in the case
$\displaystyle\Lambda_{+}(\pi)$. We have by sending
$\displaystyle\pi\longmapsto p$

$\textup{diag}(\pi,1,\pi,p)\longmapsto\textup{diag}(p,1,p,p)\longmapsto\textup{diag}(1,p,p,p)$

where the second arrow follows by applying the isomorphism $\displaystyle C$
described in subsection 6.1. Also, since
$\displaystyle\mu(\textup{diag}(\pi,1,\pi,p))=p$,
$\displaystyle\Lambda_{+}(\pi)$ gets mapped to
$\displaystyle(\Lambda_{+}^{1,3},1)_{\tilde{U}_{p}^{(2)}}$ but in general we
will not keep account of the second coordinate. The only case in which this
plays a difference is in the identification of $\displaystyle\Delta_{\pi}$ and
$\displaystyle\Delta_{p}$, which both get mapped to
$\displaystyle\textup{diag}(p,p,p,p)$ but their factors of similitude are
$\displaystyle 1,2$ respectively. The reason why factors of
$\displaystyle\Delta_{\overline{\pi}}$ appear in the relations below is to
compensate for the second coordinate, as
$\displaystyle\textup{diag}(\overline{\pi},\overline{\pi},\overline{\pi},\overline{\pi})\longmapsto\textup{diag}(1,1,1,1)$.

The table below shows some relations between the above Hecke operators. These
can be obtained by translating back to $\displaystyle H_{1,2,1}$ and using the
right coset decompositions. The way to read the table is that we first read an
operator $\displaystyle X$ in the first row and an operator $\displaystyle Y$
in the first column and the result is $\displaystyle XY$. We write "comm" to
mean that the operators commute.

| $\displaystyle\Lambda_{+}(\pi)$ | $\displaystyle\Lambda_{+}(\overline{\pi})$ | $\displaystyle\Lambda_{-}(\pi)$ | $\displaystyle\Lambda_{-}(\overline{\pi})$ | $\displaystyle T(\overline{\pi})$ | $\displaystyle T(\pi)$ | $\displaystyle T(\pi,\overline{\pi})$ | $\displaystyle T(\overline{\pi},\pi)$ | $\displaystyle T_{-}(p)$ | $\displaystyle T_{+}(p)$
---|---|---|---|---|---|---|---|---|---|---
$\displaystyle\Lambda_{+}(\pi)$ |  | comm | $\displaystyle p\Delta_{\pi}T(\pi,\overline{\pi})$ | $\displaystyle p^{3}\Delta_{p}$ | comm | $\displaystyle p\Delta_{\pi}T_{+}(p)$ | comm | $\displaystyle p^{2}\Delta_{\pi}\Lambda_{+}(\overline{\pi})$ | $\displaystyle p^{2}\Delta_{\pi}T(\overline{\pi})$ | comm
$\displaystyle\Lambda_{+}(\overline{\pi})$ | comm |  | $\displaystyle p^{3}\Delta_{p}$ | $\displaystyle p\Delta_{\overline{\pi}}T(\overline{\pi},\pi)$ | $\displaystyle p\Delta_{\overline{\pi}}T_{+}(p)$ | comm | $\displaystyle p^{2}\Delta_{\overline{\pi}}\Lambda_{+}(\pi)$ | comm | $\displaystyle p^{2}\Delta_{\overline{\pi}}T(\pi)$ | comm
$\displaystyle\Lambda_{-}(\pi)$ |  |  |  |  |  |  |  |  |  |
$\displaystyle\Lambda_{-}(\overline{\pi})$ |  |  |  |  |  |  |  |  |  |
$\displaystyle T(\overline{\pi})$ |  |  | comm | $\displaystyle p\Delta_{\overline{\pi}}T_{-}(p)$ |  |  | comm |  |  |
$\displaystyle T(\pi)$ |  |  | $\displaystyle p\Delta_{\pi}T_{-}(p)$ | comm |  |  |  | comm |  |
$\displaystyle T(\pi,\overline{\pi})$ |  |  | comm | $\displaystyle p^{2}\Delta_{\overline{\pi}}\Lambda_{-}(\pi)$ |  |  |  |  |  |
$\displaystyle T(\overline{\pi},\pi)$ |  |  | $\displaystyle p^{2}\Delta_{\pi}\Lambda_{-}(\overline{\pi})$ | comm |  |  |  |  |  |
$\displaystyle T_{-}(p)$ |  |  |  |  |  |  |  |  |  |
$\displaystyle T_{+}(p)$ |  |  | $\displaystyle p^{2}\Delta_{\pi}T(\overline{\pi})$ | $\displaystyle p^{2}\Delta_{\overline{\pi}}T(\pi)$ |  |  |  |  |  |

Table 1. Relations of Hecke Operators for split primes

######  Proposition 6.6.

Let

$D_{\pi}^{(2)}(t)=1-T(\overline{\pi})t+p\Delta_{\overline{\pi}}T(p)t^{2}-p^{3}\Delta_{\overline{\pi}}^{2}T(\pi)t^{3}+p^{6}\Delta_{\overline{\pi}}^{3}\Delta_{\pi}t^{4}$

and

$D_{\overline{\pi}}^{(2)}(t)=1-T(\pi)t+p\Delta_{\pi}T(p)t^{2}-p^{3}\Delta_{\pi}^{2}T(\overline{\pi})t^{3}+p^{6}\Delta_{\pi}^{3}\Delta_{\overline{\pi}}t^{4}$

We remark here that $\displaystyle
Z_{\pi}(t)=D_{\pi}(\Delta_{\overline{\pi}}^{-1}t)$ and $\displaystyle
Z_{\overline{\pi}}(t)=D_{\overline{\pi}}(\Delta_{\pi}^{-1}t)$ where
$\displaystyle Z_{\pi},Z_{\overline{\pi}}$ are the standard polynomials
defined in Section 3. This can be seen by computing the images under the
Satake mapping of the above coefficients, as can be found in [8, Lemma 3.7].
Let also

$S_{\pi}(t)=1-T(\overline{\pi})t+p\Delta_{\overline{\pi}}T(\pi,\overline{\pi})t^{2}$

and

$S_{\overline{\pi}}(t)=1-T(\pi)t+p\Delta_{\pi}T(\overline{\pi},\pi)t^{2}$

We then have the following factorisations

$D_{\pi}^{(2)}(t)=(1-\Lambda_{-}(\overline{\pi})t)S_{\pi}(t)(1-\Lambda_{+}(\overline{\pi})t)$
$D_{\overline{\pi}}^{(2)}(t)=(1-\Lambda_{-}(\pi)t)S_{\overline{\pi}}(t)(1-\Lambda_{+}(\pi)t)$

###### Proof.

This follows from Theorem 6.5 after pulling back to the parabolic algebra of
the unitary group. ∎

We will need a lemma regarding the decomposition of a Hecke operator in
$\displaystyle H^{1,1}$ into left cosets.

######  Lemma 6.7.

Let

$\displaystyle
M=\textup{diag}(\pi^{a_{1}}\overline{\pi}^{b_{1}},\pi^{a_{2}}\overline{\pi}^{b_{2}},\pi^{a_{3}}\overline{\pi}^{b_{3}},\pi^{a_{4}}\overline{\pi}^{b_{4}})\in
S_{p}^{2}$

with $\displaystyle a_{i},b_{i}\geq 0$. Then

$\displaystyle\Gamma_{1,1}M\Gamma_{1,1}=\sum_{l,q,r^{\prime},\gamma}\Gamma_{1,1}M\begin{pmatrix}1&0&0&l\\\
-\overline{q}&1&\overline{l}&r^{\prime}-l\overline{q}\\\ 0&0&1&q\\\
0&0&0&1\end{pmatrix}\imath(\gamma),$

where $\displaystyle l,q,r^{\prime}$ runs over elements in
$\displaystyle\mathcal{O}_{K}$ that satisfy $\displaystyle
r^{\prime}\in\mathbb{Z}$ and they give representatives of

$l\in\mathcal{O}_{K}/\pi^{a_{4}-a_{1}}\overline{\pi}^{b_{4}-b_{1}},q\in\mathcal{O}_{K}/\pi^{a_{4}-a_{3}}\overline{\pi}^{b_{4}-b_{3}},r^{\prime}\in\mathbb{Z}/\pi^{a_{4}-a_{2}}$

For the above we understand that if any of the $\displaystyle a_{4}-a_{i}$ or
$\displaystyle b_{4}-b_{i}$ is zero or negative then the corresponding group
is set to the trivial one i.e.
$\displaystyle\mathcal{O}_{K}/\pi^{a_{4}-a_{i}}=0$ and
$\displaystyle\mathcal{O}_{K}/\overline{\pi}^{b_{4}-b_{i}}=0$.

Finally $\displaystyle\gamma$ runs over elements of $\displaystyle S_{p}^{1}$
such that

$\displaystyle\Gamma_{1}\textup{diag}(\pi^{a_{1}}\overline{\pi}^{b_{1}},\pi^{a_{3}}\overline{\pi}^{b_{3}})\Gamma_{1}=\sum_{\gamma}\Gamma_{1}\textup{diag}(\pi^{a_{1}}\overline{\pi}^{b_{1}},\pi^{a_{3}}\overline{\pi}^{b_{3}})\gamma$

###### Proof.

We write $\displaystyle H_{1,1}=\left\\{\begin{pmatrix}1&0&0&l\\\
-\overline{q}&1&\overline{l}&r\\\ 0&0&1&q\\\ 0&0&0&1\end{pmatrix}\in
S_{p}^{2}\mid l,q,r\in\mathcal{O}_{K}\right\\}$ for the (integral) Heisenberg
part of the Klingen parabolic. We then claim that

$\displaystyle
H_{1,1}MH_{1,1}=\sum_{q,l,r^{\prime}}H_{1,1}M\begin{pmatrix}1&0&0&l\\\
-\overline{q}&1&\overline{l}&r^{\prime}+q\overline{l}\\\ 0&0&1&q\\\
0&0&0&1\end{pmatrix}\,\,\,\,\,\,(*),$

where $\displaystyle q,l,r^{\prime}$ as in the statement of the proposition.

To see this we first set $\displaystyle h(l,q,r):=\begin{pmatrix}1&0&0&l\\\
-\overline{q}&1&\overline{l}&r\\\ 0&0&1&q\\\ 0&0&0&1\end{pmatrix}$ and
$\displaystyle M=\textup{diag}(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})$
and calculate

$\displaystyle h(l_{1},q_{1},r_{1})Mh(l_{2},q_{2},r_{2})=$
$\displaystyle\begin{pmatrix}\alpha_{1}&0&0&\alpha_{1}l_{2}+l_{1}\alpha_{4}\\\
-\overline{q}_{1}\alpha_{1}-\alpha_{2}\overline{q}_{2}&\alpha_{2}&\alpha_{2}\overline{l}_{2}+\overline{l}_{1}\alpha_{3}&*\\\
0&0&\alpha_{3}&\alpha_{3}q_{2}+q_{1}\alpha_{4}\\\
0&0&0&\alpha_{4}\end{pmatrix}$

where
$\displaystyle*=-\overline{q}_{1}\alpha_{1}l_{2}+\alpha_{2}r_{2}+\overline{l}_{1}\alpha_{3}q_{2}+r_{1}\alpha_{4}$.
We first look at the upper right entry. We have

$\displaystyle\alpha_{1}l_{2}+l_{1}\alpha_{4}=\pi^{a_{1}}\overline{\pi}^{b_{1}}l_{2}+l_{1}\pi^{a_{4}}\overline{\pi}^{b_{4}}$

If $\displaystyle a_{1}\geq a_{4}$ and $\displaystyle b_{1}\geq b_{4}$ we may
write

$\displaystyle\alpha_{1}l_{2}+l_{1}\alpha_{4}=\left(\pi^{a_{1}-a_{4}}\overline{\pi}^{b_{1}-b_{4}}l_{2}+l_{1}\right)\pi^{a_{4}}\overline{\pi}^{b_{4}}$

and so no need for right cosets in this case. In the other cases, we write
$\displaystyle l_{2}=x+y\pi^{a_{4}-a_{1}}\overline{\pi}^{b_{4}-b_{1}}$ with
the understanding that we set $\displaystyle\pi^{i}=1$ and
$\displaystyle\overline{\pi}^{j}=1$ if $\displaystyle i,j\leq 0$. Then we have
that

$\displaystyle\alpha_{1}l_{2}+l_{1}\alpha_{4}=\pi^{a_{1}}\overline{\pi}^{b_{1}}x+(l_{1}+y)\pi^{a_{4}}\overline{\pi}^{b_{4}}$

For example when $\displaystyle a_{4}>a_{1}$ and $\displaystyle b_{1}\geq
b_{4}$ we write $\displaystyle l_{2}=x+y\pi^{a_{4}-a_{1}}$ and obtain

$\displaystyle\alpha_{1}l_{2}+l_{1}\alpha_{4}=\pi^{a_{1}}\overline{\pi}^{b_{1}}x+y\pi^{a_{4}}\overline{\pi}^{b_{1}}+l_{1}\pi^{a_{4}}\overline{\pi}^{b_{4}}=\pi^{a_{1}}\overline{\pi}^{b_{1}}x+(y\overline{\pi}^{b_{1}-b_{4}}+l_{1})\pi^{a_{4}}\overline{\pi}^{b_{4}}$

Similarly looking at the entry

$\displaystyle\alpha_{2}\overline{l}_{2}+\overline{l}_{1}\alpha_{3}=\pi^{a_{2}}\overline{\pi}^{b_{2}}\overline{l}_{2}+\overline{l}_{1}\pi^{a_{3}}\overline{\pi}^{b_{3}}$

With $\displaystyle l_{2}=x+y\pi^{a_{4}-a_{1}}\overline{\pi}^{b_{4}-b_{1}}$ as
above, we obtain,

$\displaystyle\alpha_{2}\overline{l}_{2}+\overline{l}_{1}\alpha_{3}=\pi^{a_{2}}\overline{\pi}^{b_{2}}\overline{x}+\overline{y}\pi^{a_{2}}\overline{\pi}^{b_{2}}\overline{\pi}^{a_{4}-a_{1}}\pi^{b_{4}-b_{1}}+\overline{l}_{1}\pi^{a_{3}}\overline{\pi}^{b_{3}}=\pi^{a_{2}}\overline{\pi}^{b_{2}}\overline{x}+(\overline{y}+\overline{l_{1}})\pi^{a_{3}}\overline{\pi}^{b_{3}}$

where we have used the fact that $\displaystyle a_{2}+b_{4}=b_{1}+a_{3}$ and
$\displaystyle a_{4}+b_{2}=a_{1}+b_{3}$ since $\displaystyle M\in S_{p}^{2}$.

In particular for these entries it is enough to consider the entry
$\displaystyle l_{2}$ modulo
$\displaystyle\pi^{a_{4}-a_{1}}\overline{\pi}^{b_{4}-b_{1}}$ (with our
convention). Similarly we can argue for the entries
$\displaystyle-\overline{q}_{1}\alpha_{1}-\alpha_{2}\overline{q}_{2}$ and
$\displaystyle\alpha_{3}q_{2}+q_{1}\alpha_{4}$.

We are now left with the $\displaystyle*-$ entry. Using first the condition
that each representative has to be in the group $\displaystyle S_{p}^{2}$, we
get by the condition $\displaystyle\overline{B}^{t}D=\overline{D}^{t}B$ that

$\overline{\alpha_{1}}\alpha_{3}\overline{l_{2}}q_{2}+\alpha_{4}\overline{\alpha_{4}}\overline{l_{1}}q_{1}+\alpha_{4}\overline{\alpha}_{2}\overline{r_{2}}+\alpha_{4}\overline{\alpha_{4}}\overline{r_{1}}\in\mathbb{R}\cap\mathcal{O}_{K}=\mathbb{Z}$

We now set this expression to be $\displaystyle 0$ and by the conditions on
the matrix $\displaystyle
M$($\displaystyle\overline{\alpha_{1}}\alpha_{3}=\alpha_{4}\overline{\alpha_{2}}$)
we obtain

$r_{2}=-l_{2}\overline{q_{2}}-p^{a_{4}-a_{2}}(r_{1}+l_{1}\overline{q_{1}})$

Using the fact that we can write $\displaystyle
r_{1}=r^{\prime}_{1}+q_{1}\overline{l}_{1}$ and $\displaystyle
r_{2}=r^{\prime}_{2}+q_{2}\overline{l}_{2}$ for some
$\displaystyle\tilde{r}_{i}\in\mathbb{Z}$ we have that the only part of the
$\displaystyle*-$ entry which is not determined by our choices of the
$\displaystyle q_{1},q_{2},l_{1},l_{2}$ is
$\displaystyle\alpha_{2}\tilde{r}_{2}+\tilde{r}_{1}\alpha_{4}$. That is

$\displaystyle\alpha_{2}r^{\prime}_{2}+r^{\prime}_{1}\alpha_{4}=\pi^{a_{2}}\overline{\pi}^{b_{2}}r^{\prime}_{2}+r^{\prime}_{1}\pi^{a_{4}}\overline{\pi}^{b_{4}}$

Arguing as above and using the fact that $\displaystyle
a_{4}-a_{2}=b_{4}-b_{2}$, we see that the element $\displaystyle
r^{\prime}_{2}$ needs to be selected from integers modulo
$\displaystyle\pi^{a_{4}-a_{2}}\overline{\pi}^{b_{4}-b_{2}}=p^{a_{4}-a_{2}}$.

This establishes our claim $\displaystyle(*)$. The rest of the proof is
identical to the symplectic case as done by Gritsenko in [6].

∎

We are now finally ready to obtain the rationality theorems, as in the case of
an inert prime.

######  Proposition 6.8.

Let $\displaystyle F\in S_{2}^{k}$ be a Hecke eigenform and $\displaystyle
m\geq 1$. Then

$\displaystyle\displaystyle Q_{p,F}^{(2)}(X)\sum_{\delta\geq
0}\phi_{mp^{\delta}}|T_{+}(p^{\delta})X^{\delta}=\left(\phi_{m}-\phi_{\frac{m}{p}}|T_{-}(p)X+p\phi_{\frac{m}{p^{2}}}|\Lambda_{-}(p)X^{2}\right)\mid
K(X)$

where $\displaystyle K$ is as given in pages $\displaystyle 2898,2899$ in [8]
and such that

$\phi_{m}\mid K(X)=(1-p^{k-3}X)^{2}(1-p^{2k-4}X^{2})\phi_{m}.$

if $\displaystyle(p,m)=1$.

###### Proof.

The proof is given by Gritsenko in [8]. ∎

######  Proposition 6.9.

For $\displaystyle m\geq 1$

$D_{\pi,F}^{(2)}(X)\sum_{\delta\geq
0}\phi_{mp^{\delta}}|\Lambda_{+}(\overline{\pi}^{\delta})X^{\delta}=\left(\phi_{m}-\phi_{\frac{m}{p}}|\Lambda_{-}(\overline{\pi})X\right)\mid
S_{\pi}(X)$

Similarly,

$D_{\overline{\pi},F}^{(2)}(X)\sum_{\delta\geq
0}\phi_{mp^{\delta}}|\Lambda_{+}(\pi^{\delta})X^{\delta}=\left(\phi_{m}-\phi_{\frac{m}{p}}|\Lambda_{-}(\pi)X\right)\mid
S_{\overline{\pi}}(X)$

with notation obtained by exchanging $\displaystyle\pi$ and
$\displaystyle\overline{\pi}$.

###### Proof.

This can be established as the corresponding statement for $\displaystyle
Q^{(2)}_{p}(X)$, as long as we establish that (using notation from Section 3)

$\displaystyle\phi_{m}||_{k}\Lambda_{+}(\overline{\pi})=\phi_{mp}|_{k}\Lambda_{+}(\overline{\pi}).$

We do know from [10, Lemma 4.1] that the right hand side is a Jacobi form of
index $\displaystyle m$. To see the equality above we recall that by Lemma 6.7
we have

$\displaystyle\Lambda_{+}(\overline{\pi})=\sum_{l,q\in\mathcal{O}/\pi,r\in\mathbb{Z}/p}\Gamma_{1,1}\begin{pmatrix}\overline{\pi}&0&0&0\\\
0&1&0&0\\\ 0&0&\overline{\pi}&0\\\ 0&0&0&p\end{pmatrix}h(q,l,r)$

where $\displaystyle h(l,q,r):=\begin{pmatrix}1&0&0&l\\\
-\overline{q}&1&\overline{l}&r\\\ 0&0&1&q\\\ 0&0&0&1\end{pmatrix}$.

Writing $\displaystyle Z=\begin{pmatrix}\tau&z_{1}\\\
z_{2}&\omega\end{pmatrix}$, we observe that the $\displaystyle\omega$ entry of
the action $\displaystyle\begin{pmatrix}\overline{\pi}&0&0&0\\\ 0&1&0&0\\\
0&0&\overline{\pi}&0\\\ 0&0&0&p\end{pmatrix}h(q,l,r)\cdot Z$ is given as

$\displaystyle\frac{\overline{q}\tau
q-z_{2}q-\overline{l}q-\overline{q}z_{1}+r}{p}+\frac{\omega}{p}=:a(\tau,z_{1},z_{2},q,r,l)+\frac{\omega}{p}.$

Moreover, we know that the other three entries do not depend on
$\displaystyle\omega$. If now

$F(Z)=\sum_{m\geq 1}\phi_{m}(\tau,z_{1},z_{2})e(m\omega)$

is the Fourier-Jacobi expansion of $\displaystyle F$, we have

$\displaystyle(F|_{k}\Lambda_{+}(\overline{\pi}))(Z)=\sum_{m}\sum_{l,q,r}\left(\phi_{m}(\tau,z_{1},z_{2})e(m\omega)|_{k}\begin{pmatrix}\overline{\pi}&0&0&0\\\
0&1&0&0\\\ 0&0&\overline{\pi}&0\\\ 0&0&0&p\end{pmatrix}h(q,l,r)\right)$

(note the factor of automorphy is the same so it is fine to pass the action
inside the expansion).

From the remark above we may write this as:

$\displaystyle
F|_{k}\Lambda_{+}(\overline{\pi})=p^{k-4}\sum_{m}\left(\sum_{l,q,r}\phi_{m}(\tau_{l,q,r},z_{1,l,q,r}z_{2,l,q,r})e(ma(\tau,z_{1},z_{2},q,r,l))\right)e\left(\frac{m}{p}\omega\right)$

This is the Fourier expansion of $\displaystyle
F|_{k}\Lambda_{+}(\overline{\pi})$ (the terms inside the parentheses do not
involve the $\displaystyle\omega$ entry). In particular,

$\displaystyle\phi_{m}||_{k}\Lambda_{+}(\overline{\pi})=p^{k-4}\sum_{l,q,r}\phi_{mp}(\tau_{l,q,r},z_{1,l,q,r}z_{2,l,q,r})e(mpa(\tau,z_{1},z_{2},q,r,l))$

But by definition:

$\displaystyle
a(\tau,z_{1},z_{2},q,r,l)=\omega\left(\begin{pmatrix}\overline{\pi}&0&0&0\\\
0&1&0&0\\\ 0&0&\overline{\pi}&0\\\ 0&0&0&p\end{pmatrix}h(q,l,r)\cdot
Z\right)-\frac{\omega}{p}=:\omega_{q,l,r}(Z)-\frac{\omega}{p}$

where the first term denotes the $\displaystyle\omega$-entry after the action.
That is,

$\displaystyle\phi_{m}||_{k}\Lambda_{+}(\overline{\pi})=\left(p^{k-4}\sum_{l,q,r}\phi_{mp}(\tau_{l,q,r},z_{1,l,q,r}z_{2,l,q,r})e(mp\omega_{q,l,r}(Z))\right)e(-m\omega)$

But now the expression inside the parentheses is nothing else than
$\displaystyle\phi_{mp}(\tau,z_{1},z_{2})e(mp\omega)|_{k}\Lambda_{+}(\overline{\pi})$.
In particular we obtain that

$\displaystyle\phi_{m}||_{k}\Lambda_{+}(\overline{\pi})=\phi_{mp}|_{k}\Lambda_{+}(\overline{\pi}).$

∎

To end this section, we will prove a couple of lemmas, which will be useful
later, when we are dealing with the Dirichlet series of interest.

######  Lemma 6.10.

Denote by $\displaystyle\Lambda_{-}(\pi)^{\textup{adj}}$ the adjoint of the
operator $\displaystyle\Lambda_{-}(\pi)$ with respect to the inner product of
Fourier-Jacobi forms. Then

$\Lambda_{-}(\pi)^{\textup{adj}}=p^{k-3}\Lambda_{+}(\overline{\pi})$

###### Proof.

We observe from the lemma above that
$\displaystyle\Lambda_{-}(\pi)=\Gamma_{1,1}\textup{diag}(\pi,p,\pi,1)$. We
also note that the Jacobi form

$\displaystyle\phi_{l}|_{k}\textup{diag}(\pi,p,\pi,1)$

is of index $\displaystyle pl$ for the group
$\displaystyle\Gamma_{-}=U_{1}\times\mathcal{O}_{K}\times\mathcal{O}_{K}\left\\{\begin{pmatrix}a&0&b&\mu\\\
*&1&*&*\\\ c&0&d&\lambda\\\ 0&0&0&1\end{pmatrix}\in\Gamma_{1,1}\right\\}$. In
particular, we may write

$\displaystyle\langle\phi_{l}|_{k}\Lambda_{-}(\pi),\psi_{lp}\rangle=\frac{p^{2k-4}\pi^{-k}}{\left[\Gamma_{1,1}:\Gamma_{-}\right]}\int_{\Gamma_{-}\setminus\mathbb{H}_{1}^{J}}\phi(\tau,\pi
z_{1},\overline{\pi}z_{2})\overline{\psi(\tau,z_{1},z_{2})}\exp\left(-\pi
lp\frac{|z_{1}-\overline{z_{2}}|^{2}}{y}\right)y^{k-4}dxdydx_{1}dy_{1}dx_{2}dy_{2}$

We now perform the change of variables $\displaystyle
z_{1}\mapsto\pi^{-1}z_{1}$ and $\displaystyle
z_{2}\mapsto\overline{\pi}^{-1}z_{2}$. This is equivalent to the action of the
matrix $\displaystyle\textup{diag}(\overline{\pi},1,\overline{\pi},p)$ on
$\displaystyle\mathbb{H}_{1}\times\mathbb{C}\times\mathbb{C}$. Now

$\psi_{lp}\mid_{k}(\tau,z_{1},z_{2})=p^{2k-4}\overline{\pi}^{-k}\psi_{lp}(p\tau,z_{1},z_{2})$

is a Jacobi form of weight $\displaystyle k$ and index $\displaystyle l$ with
respect to the group

$\Gamma_{+}=U_{1}\times\overline{\pi}\mathcal{O}_{K}\times\overline{\pi}\mathcal{O}_{K}=\left\\{\begin{pmatrix}a&0&b&\mu\\\
*&1&*&*\\\ c&0&d&\lambda\\\
0&0&0&1\end{pmatrix}\in\Gamma_{1,1}\mid\mu,\lambda\equiv
0\pmod{\overline{\pi}}\right\\}$

(this is obtained by considering the group
$\displaystyle\textup{diag}(\overline{\pi},1,\overline{\pi},p)^{-1}\Gamma_{-}\textup{diag}(\overline{\pi},1,\overline{\pi},p)$).

We then obtain the integral

$\displaystyle\frac{p^{2k-6}\pi^{-k}}{\left[\Gamma_{1,1}:\Gamma_{+}\right]}\int_{\Gamma_{+}\setminus\mathbb{H}_{1}^{J}}\phi(\tau,z_{1},z_{2})\overline{\psi(\tau,\pi^{-1}z_{1},\overline{\pi}^{-1}z_{2})}\exp\left(-\pi
l\frac{|z_{1}-\overline{z_{2}}|^{2}}{y}\right)y^{k-4}dxdydx_{1}dy_{1}dx_{2}dy_{2}.$

On the other hand, we have by the Lemma 6.7 that

$\Lambda_{+}(\overline{\pi})=\sum_{l,q\in\mathcal{O}/\pi,r\in\mathbb{Z}/p}\Gamma_{1,1}\begin{pmatrix}\overline{\pi}&0&0&0\\\
0&1&0&0\\\ 0&0&\overline{\pi}&0\\\ 0&0&0&p\end{pmatrix}h(q,l,r)$

and so by using the property
$\displaystyle\langle\phi|_{k}h(q,l,r)^{-1},\psi\rangle=\langle\phi,\psi|_{k}\
h(q,l,r)\rangle$ we obtain

$\langle\phi_{l},\psi_{lp}\mid_{k}\Lambda_{+}(\overline{\pi})\rangle=$
$=p^{3}\frac{p^{k-4}\pi^{-k}}{[\Gamma_{1,1}:\Gamma_{+}]}\int_{\Gamma_{+}\backslash\mathbb{H}}\phi_{l}(\tau,z_{1},z_{2})\overline{\psi_{pl}(\tau,\pi^{-1}z_{1},\overline{\pi}^{-1}z_{2})}e\left(-\pi
l\frac{|z_{1}-\overline{z_{2}}|^{2}}{y}\right)y^{k-4}dxdydx_{1}dy_{1}dx_{2}dy_{2}$

as $\displaystyle h(q,l,r)^{-1}\in\Gamma_{1,1}$ and hence they act trivially.
Therefore,

$\langle\phi_{l}\mid_{k}\Lambda_{-}(\pi),\psi_{pl}\rangle=p^{k-5}\frac{[\Gamma_{1,1}:\Gamma_{+}]}{[\Gamma_{1,1}:\Gamma_{-}]}\langle\phi_{l},\psi_{lp}\mid\Lambda_{+}(\overline{\pi})\rangle=p^{k-3}\langle\phi_{l},\psi_{lp}\mid_{k}\Lambda_{+}(\overline{\pi})\rangle$

as the ratio of indices is $\displaystyle p^{2}$. ∎

Finally, knowing the action of the operators $\displaystyle
T(\pi,\overline{\pi})$ and $\displaystyle T(\overline{\pi},\pi)$ will prove
helpful in the following, so we give the following lemma:

######  Lemma 6.11.

Let $\displaystyle X=\Gamma_{1,1}\begin{pmatrix}\overline{\pi}&0&0&0\\\
0&\pi&0&0\\\ 0&0&\overline{\pi}&0\\\ 0&0&0&\pi\end{pmatrix}\Gamma_{1,1}$. We
have

$\phi_{1}\mid_{k}X=p^{k-3}\phi_{1}$

###### Proof.

Using Gritsenko’s decomposition in [7], we have

$\Gamma_{1,1}\begin{pmatrix}\overline{\pi}&0&0&0\\\ 0&\pi&0&0\\\
0&0&\overline{\pi}&0\\\
0&0&0&\pi\end{pmatrix}\Gamma_{1,1}=\sum_{\gamma\in\mathcal{O}_{K}/\pi,\beta\in\mathcal{O}_{K}/\overline{\pi}}\Gamma_{1,1}\begin{pmatrix}\overline{\pi}&0&0&\overline{\pi}\overline{\beta}\\\
-\overline{\gamma}&\pi&\pi\beta&-\overline{\gamma}\overline{\beta}\\\
0&0&\overline{\pi}&\gamma\\\ 0&0&0&\pi\end{pmatrix}$

Now, the result of the action of a representative on the element
$\displaystyle\begin{pmatrix}\tau&z_{1}\\\ z_{2}&\omega\end{pmatrix}$ is

$\begin{pmatrix}\tau&-\frac{\gamma}{\pi}\tau+\frac{\overline{\pi}}{\pi}z_{1}+\frac{\overline{\pi}}{\pi}\overline{\beta}\\\
-\frac{\overline{\gamma}}{\overline{\pi}}\tau+\frac{\pi}{\overline{\pi}}z_{2}+\frac{\pi}{\overline{\pi}}\beta&\frac{|\gamma|^{2}}{p}\tau-\frac{\gamma}{\overline{\pi}}z_{2}-\frac{\overline{\gamma}}{\pi}z_{1}+\omega-
tr\left(\frac{\overline{\beta}\overline{\gamma}}{\pi}\right)\end{pmatrix}$

But $\displaystyle\phi$ has a theta expansion given by

$\phi(\tau,z_{1},z_{2})=\sum_{h\pmod{2}}\phi_{h}(\tau)\sum_{n\in\mathcal{O}}e(\tau|n+h/2|^{2}+(n+h/2)z_{1}+(\overline{n}+\overline{h/2})z_{2})$

So, after the action we obtain

$p^{k-4}\sum_{h}\phi_{h}(\tau)\sum
e\left(\tau|n+h/2|^{2}+(n+h/2)\left(-\frac{\gamma}{\pi}\tau+\frac{\overline{\pi}}{\pi}z_{1}+\frac{\overline{\pi}}{\pi}\overline{\beta}\right)+(\overline{n}+\overline{h/2})\left(\\-\frac{\overline{\gamma}}{\overline{\pi}}\tau+\frac{\pi}{\overline{\pi}}z_{2}+\frac{\pi}{\overline{\pi}}\beta\right)\right)\times$
$\times
e\left(\frac{|\gamma|^{2}}{p}\tau-\frac{\gamma}{\overline{\pi}}z_{2}-\frac{\overline{\gamma}}{\pi}z_{1}-tr\left(\frac{\overline{\beta}\overline{\gamma}}{\pi}\right)\right)$

For fixed $\displaystyle n,h,\gamma$ we observe that the sum

$\sum_{\beta\in\mathcal{O}_{K}/\overline{\pi}}e\left(\textup{tr}\left(\frac{(n+h/2)\overline{\pi}-\overline{\gamma}}{\pi}\right)\overline{\beta}\right)$

is $\displaystyle 0$ unless
$\displaystyle\pi\mid(2n+h)\overline{\pi}-2\overline{\gamma}$, in which case
is $\displaystyle p$([7], p. $\displaystyle 76$). Now, the terms which contain
$\displaystyle z_{1},z_{2}$ are

$e\left(\frac{(n+h/2)\overline{\pi}-\overline{\gamma}}{\pi}z_{1}+\frac{(\overline{n}+\overline{h/2})\pi-\gamma}{\overline{\pi}}z_{2}\right)$

and if we write
$\displaystyle\frac{(n+h/2)\overline{\pi}-\overline{\gamma}}{\pi}=m+l/2$ for
$\displaystyle m\in\mathcal{O}_{K},l\in\mathcal{O}_{K}/2$ we have the form
wanted. Similarly for $\displaystyle\tau$, we have

$e\left(\tau|n+h/2|^{2}-\frac{(n+h/2)\gamma}{\pi}-\frac{(\overline{n}+\overline{h/2})\overline{\gamma}}{\overline{\pi}}+\frac{|\gamma|^{2}\tau}{p}\right)$

and the inside expression can be written as

$\left(\frac{(n+h/2)\overline{\pi}-\overline{\gamma}}{\pi}\right)\left(\frac{(\overline{n}+\overline{h/2})\pi-\gamma}{\overline{\pi}}\right)=|m+l/2|^{2}$

with the notation before. Hence, putting all together we finally get the
factor $\displaystyle p^{k-3}$. ∎

######  Remark.

The same is true for the operator
$\displaystyle\Gamma_{1,1}\begin{pmatrix}\pi&0&0&0\\\ 0&\overline{\pi}&0&0\\\
0&0&\pi&0\\\ 0&0&0&\overline{\pi}\end{pmatrix}\Gamma_{1,1}$.

###  6.3. Calculation of the Dirichlet series - First Part

We recall that

$D_{F,G,h}(s)=4\beta_{k}\sum_{l,\epsilon,m}\langle\tilde{\phi}_{1}|T_{-}(m)U_{l},\tilde{\psi}_{mN(l)}\rangle_{\mathcal{A}}a_{mN(\epsilon)}N(l)^{-s}N(\epsilon)^{-(k+s-1)}m^{-(2k+s-4)}$

with $\displaystyle l,\epsilon\in\mathbb{Z}[i]$ coprime with their real parts
positive and imaginary parts non-negative and $\displaystyle m\in\mathbb{N}$.
In the case of a split prime $\displaystyle p=\pi\overline{\pi}$, we define
the $\displaystyle p-$part by

$D_{F,G,h}^{(p)}(s)=\sum_{l_{1},l_{2},\epsilon_{1},\epsilon_{2},m\geq
0}\langle\tilde{\phi}_{1}|T_{-}(p^{m})U_{\pi^{l_{1}}}U_{\overline{\pi}^{l_{2}}},\tilde{\psi}_{p^{m+l_{1}+l_{2}}}\rangle_{\mathcal{A}}a_{p^{m+\epsilon_{1}+\epsilon_{2}}}p^{-s(l_{1}+l_{2})}p^{-(k+s-1)(\epsilon_{1}+\epsilon_{2})}p^{-(2k+s-4)m}$

with the conditions that $\displaystyle\min(\epsilon_{1},l_{1})=0$ and
$\displaystyle\min(\epsilon_{2},l_{2})=0$.

Consider now the Hecke operator
$\displaystyle\Lambda_{-}(\pi)=\Gamma_{1,1}\textup{diag}(\pi,p,\pi,1)\Gamma_{1,1}=\Gamma_{1,1}\textup{diag}(\pi,p,\pi,1)$,
by Lemma 6.7. Then, if $\displaystyle\phi$ is a Fourier-Jacobi, we get

$\phi|_{k}\Lambda_{-}(\pi)=p^{2k-4}\pi^{-k}\tilde{\phi}\left(\begin{pmatrix}\tau&\pi
z_{1}\\\
\overline{\pi}z_{2}&p\tau^{\prime}\end{pmatrix}\right)e^{-2\pi\frac{m}{p}\tau^{\prime}}=p^{2k-4}\pi^{-k}\phi(\tau,\pi
z_{1},\overline{\pi}z_{2})=p^{2k-4}\pi^{-k}\phi|_{k}U_{\pi}$

Hence, we can rewrite the series as:

$D_{F,G,h}^{(p)}(s)=\sum_{\begin{subarray}{c}l_{1},l_{2},\\\
\epsilon_{1},\epsilon_{2},m\geq