# On analytic properties of the standard zeta function attached to a vector
valued modular form

Oliver Stein  Fakultät für Informatik und Mathematik
Ostbayerische Technische Hochschule Regensburg
Galgenbergstrasse 32
93053 Regensburg
Germany<EMAIL_ADDRESS>
###### Abstract.

We proof a Garrett-Böcherer decomposition of a vector valued Siegel Eisenstein
series $E_{l,0}^{2}$ of genus 2 transforming with the Weil representation of
$\operatorname{Sp}_{2}(\mathbb{Z})$ on the group ring
$\mathbb{C}[(L^{\prime}/L)^{2}]$. We show that the standard zeta function
associated to a vector valued common eigenform $f$ for the Weil representation
can be meromorphically continued to the whole $s$-plane and that it satisfies
a functional equation. The proof is based on an integral representation of
this zeta function in terms of $f$ and $E_{l,0}^{2}$.

###### 2020 Mathematics Subject Classification:

11F27 11F25 11F66 11M41

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

##  1\. Introduction

Vector valued modular forms transforming with the Weil representation play a
prominent role in the theory of Borcherds products, see e. g. [Bor] or [Br1].
Since then, a lot of research regarding this type of modular forms has been
done. Among other things, a theory of newforms was developed ([Br2]). In [BS]
the foundations of a theory of Hecke operators was laid. The Fourier
coefficients of Eisenstein series were calculated in [BK] and the analytic
properties of their non-holomorphic versions were studied in detail in [St].
Also, several relations between these modular forms and scalar valued modular
forms have been established (see e. g. [Sch1]). One important part of the
theory of modular forms that has not yet been addressed (to the best of
knowledge) is their relation to $L$-functions. We can find in the literature
several ways to associate an $L$-function to a modular form (see e. g. [Bo1],
[BM], [PSR]). Such $L$-functions are a central object in number theory and
were studied extensively in several aspects among them their analytic
properties. The present paper can be seen as a contribution to these
investigations. Its main objective is to examine the analytic properties of a
certain $L$-function of a vector valued Hecke eigenform.

To describe the main results more closely, we introduce some notation. Let $L$
be non-degenerate even lattice of type $(b^{+},b^{-})$ and level $N$, equipped
with a bilinear form and associated quadratic form $q$. By $L^{\prime}$ we
denote the dual lattice of $L$. We assume further that the rank and the
signature $\operatorname{sig}(L)=b^{+}-b^{-}$ of $L$ is even. The Weil
representation $\rho_{L,1}$ associated to $L$ is a unitary representation of
$\Gamma_{1}=\operatorname{SL}_{2}(\mathbb{Z})$ on the group ring
$\mathbb{C}[L^{\prime}/L]$,

$\rho_{L,1}:\Gamma_{1}\rightarrow\operatorname{GL}(\mathbb{C}[L^{\prime}/L]).$

For reasons, which will become apparent later, we introduce a more general
Weil representation $\rho_{L,n}$ of the symplectic group
$\Gamma_{n}=\operatorname{Sp}(n,\mathbb{Z})$ on the group ring
$\mathbb{C}[(L^{\prime}/L)^{n}]$ (cf. Definition 3.1 in Chapter 3).

Now let $l\in\mathbb{Z}$ be even. A holomorphic function
$f:\mathbb{H}\rightarrow\mathbb{C}$ is called a vector valued modular form of
weight $k$ and type $\rho_{L,1}$ for the group $\Gamma_{1}$ if

$f(\gamma\tau)=(c\tau+d)^{k}\rho_{L,1}(\gamma)f(\tau)$

for all $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma$, and $f$ is holomorphic at the cusp
$\infty$. The space of all such functions is denoted with $M_{l}(\rho_{L,1})$.
For the subspace of cusp forms we write $S_{l}(\rho_{L,1})$. For the details
see Chapter 4.

As already mentioned above, in [BS] a Hecke operator $T(M)$ for $M\in
GL_{2}^{+}(\mathbb{Q})$ was introduced by the action of a suitable Hecke
algebra. Based on this definition, it is possible to recover the main results
of the classical Hecke theory. In particular, it is proved that
$S_{l}(\rho_{L,1})$ possesses a basis of common Hecke eigenforms of all Hecke
operators $T\left(\begin{smallmatrix}d^{2}&0\\\ 0&1\end{smallmatrix}\right)$
for all $d\in\mathbb{N}$ coprime to the level $N$ of $L$. For such an
eigenform we have

$f\mid_{k,L}T\left(\begin{smallmatrix}d^{2}&0\\\
0&1\end{smallmatrix}\right)=\lambda_{d}(f)f$

for $(d,N)=1$. The authors in [BS] proposed to associate to $f$ an an
$L$-function of the form

$L^{N}(s,f)=\sum_{\begin{subarray}{c}d\in\mathbb{N}\\\
(d,N)=1\end{subarray}}\lambda_{d}(f)d^{-s}.$

$L$-functions of this type were considered in many places in the literature
(cf. for example [Bo1], [Ar] or [BM]). It was suggested in [BS] to establish
the usual analytic properties, that is, the meromorphic continuation to the
whole $s$-plane and a functional equation of $L^{N}(s,f)$ by means of a
variant of the doubling method ([Bo1], [Ga] or [PSR]). This was the starting
point of the present paper. Its main content is the proof of the meropmorphic
continuation and a function equation of a modified version of $L^{N}(s,f)$.

More precisely, we assume that there is a common Hecke eigenform $f$ for all
Hecke operators $T\left(\begin{smallmatrix}d^{2}&0\\\
0&1\end{smallmatrix}\right)$, $d\in\mathbb{N}$. This assumption is justified
if a multiplicity one theorem (see e. g. [Mi], $\S$4.6) is in place. The
question under what conditions such a theorem for vector valued modular forms
for the Weil representation holds was investigated in [We]. It turns out that
a multiplicity one theorem is valid if the Weil representation decomposes into
irreducible subrepresentations, each subrepresentation occurring with
multiplicity one. In Remark 6.1 we discuss more detailed under what
circumstances $\rho_{L,1}$ allows such a decomposition. Under these
assumptions we assign to $f$ the $L$-function

(1.1) $Z(s,f)=\sum_{d\in\mathbb{N}}\lambda_{d}(f)d^{-s},$

which we call according to [Ar] a standard zeta function.

To study the analytic properties of (1.1), we develop a doubling method along
the lines of [Bo1] and [Ga], tailored to the vector valued setting. Note that
there are papers addressing this topic for vector valued Siegel modular forms
([Ko1], [Ko2]). However, our approach deals with $\rho_{L,1}$ as
representation requiring methods, specifically adapted to this representation,
causing several technical difficulties along the way. We obtain the following
pullback formula:

######  Theorem 1.1.

Let $d\in\mathbb{Z}$ be a positive integer, $D=\left(\begin{smallmatrix}d&0\\\
0&d^{-1}\end{smallmatrix}\right)$ and $E_{k,0}^{2*}$ be as in Definition 4.3.
Then for all $\tau,\zeta\in\mathbb{H}$

$\displaystyle E_{l,0}^{2*}(\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right),s)=$ $\displaystyle
E_{l,0}^{1*}(\tau,s)\otimes
E_{l,0}^{1*}(\zeta,s)+\frac{e(\operatorname{sig}(L)/8)}{|L^{\prime}/L|^{1/2}}\sum_{d\geq
1}\frac{g_{d}(L)}{g(L)}d^{-l-2s}\mathscr{P}^{+}_{l}(-\tau,\zeta,D,s).$

Here $E_{l,0}^{2*}$ is the non-holomorphic vector valued Siegel Eisenstein
series of type $\rho^{*}_{L,2}$, $\mathscr{P}^{+}_{L}$ means a certain vector
valued Poincaré series attached to the discriminant form $(L^{\prime}/L)^{2}$
(analogous to the Poincaré series in [Bo1]) and $E_{l,0}^{1*}$ is the non-
holomorphic vector valued Eisenstein series of type $\rho^{*}_{L,1}$, where
$\rho^{*}_{L,1}$ and $\rho^{*}_{L,2}$ denotes the dual representation of
$\rho_{L,1}$ and $\rho_{L,2}$, respectively.

Based on this theorem, we then provide an integral representation of $Z(s,f)$
in terms of the Siegel Eisenstein series $E_{l,0}^{2}$:

######  Theorem 1.2.

Let $l\in 2\mathbb{Z}$, $l\geq 3$, satisfy $2l+\operatorname{sig}(L)\equiv
0\bmod{4}$. Let $f\in S_{l}(\rho_{L,1})$ a common eigenform of Hecke operators
$T\left(\begin{smallmatrix}d^{2}&0\\\ 0&1\end{smallmatrix}\right)$ and
$E_{l,0}^{2}$ the Eisenstein series in Definition 4.3 below. If
$l+2\operatorname{Re}(s)>3$, then, for any $\zeta\in\mathbb{H}$,

$\displaystyle\sum_{\lambda\in
L^{\prime}/L}\left(\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},E_{l,0}^{2}(\left(\begin{smallmatrix}\tau&0\\\
0&-\overline{\zeta}\end{smallmatrix}\right),\overline{s})\rangle_{2}\operatorname{Im}(\tau)^{l}d\mu(\tau)\right)\mathfrak{e}_{\lambda}$
$\displaystyle=K(l,s)\left(\sum_{d\in\mathbb{N}}\lambda_{d}(f)d^{-l-2s}\right)f(\zeta).$

Now the analytic properties of $E_{l,0}^{2}$ with respect to $s$ can be
transferred to $Z(s,f)$. Note that for the Siegel Eisenstein series
$E_{l,0}^{n}$ of genus $n$ and type $\rho_{L,n}$ these properties were
established in [St]. The functional equation for $E_{l,0}^{n}$ is rather
complicated. Corollary 4.6 specializes it to the case $n=2$, leading to a
quite clean formula.

We finish this section with an overview of the subsequent chapters: After some
notations and preliminaries, we introduce in Chapter 3 the Weil representation
$\rho_{L,n}$ for the Siegel modular group $\operatorname{Sp}(n,\mathbb{Z})$.
We prove several relations between $\rho_{L,2}$ and $\rho_{L,1}$, which are
vital for the pullback formula above. The next chapter deals with vector
valued Siegel modular forms transforming with $\rho_{L,n}$, the Eisenstein
series $E_{l,0}^{n}$ and certain vector valued Poincaré series, the main
ingredients of the above stated formulas. After some general remarks, we
switch to the case $n=1$ and give a brief account to Hecke operators and
explain how to extend their definition to double cosets
$\Gamma_{1}\left(\begin{smallmatrix}d&0\\\
0&d^{-1}\end{smallmatrix}\right)\Gamma_{1}$ if $d$ is not coprime to the level
$N$. The second part of this chapter defines and studies the Eisenstein series
$E_{l,0}^{n}$. In particular, we prove a functional equation for $n=2$ and
explain how to assign to $E_{l,0}^{n}$ a value on the Siegel lower half space,
which is a necessary technical detail to deduce the main theorems of the
paper. Finally, two vector valued Poincaré series are introduced, one of them
analogous to a series introduced in [Bo1], the other related to the first. The
main results here are the reproducing formulas in Theorem 4.10. In Chapter 5
we derive the Garrett-Böcherer pullback formula in our setting. The proof
follows the one of [Ga]. In each relevant step the corresponding calculations
for $\rho_{L,2}$ have to be included, causing several technical difficulties.
The last chapter then collects all results established before and presents the
desired analytic properties of $Z(s,f)$.

##  2\. Notation and preliminaries

We use the symbol $e(x)$, $x\in\mathbb{C}$ as an abbreviation for $e^{2\pi
ix}$. As usual, by $\overline{z}$ we mean the conjugate complex number of $z$.
In this paper, $n$, $m$ are always natural numbers. For any ring $R$,
$M_{m,n}(R)$, $R^{n}$ and $\operatorname{Sym}_{n}(R)$ are the set of $m\times
n$ matrices, the set of row vectors of size $n$ and the set of symmetric
matrices in $M_{n,n}(R)$. We write $1_{n}$ and $0_{n}$ for the unit matrix and
zero matrix of size n, respectively. Moreover, by $A^{t}$ and
$\operatorname{tr}(A)$ we denote the transposed matrix and the trace of $A$.
Also, for any matrix $S\in\operatorname{Sym}_{n}(\mathbb{R})$ we write $S>0$
(resp. $S\geq 0)$ if $S$ is positive definite (resp. semi positive definite).
By $\operatorname{Sp}(n)$ we denote the symplectic group of genus $n$. We use
the symbol $\Gamma_{n}$ for $\operatorname{Sp}(n,\mathbb{Z})$. In particular,
we have $\Gamma_{1}=\operatorname{SL}_{2}(\mathbb{Z})$. For $N\in\mathbb{N}$

$\displaystyle\Gamma_{n}(N):=\left\\{\gamma\in\Gamma_{n}\;|\;\gamma\equiv
1_{n}\bmod{N}\right\\}$

is the principal congruence subgroup of $\Gamma_{n}$. The subgroups

(2.1) $\Gamma_{n,r}:=\left\\{\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma_{n}\;:\;a=\left(\begin{smallmatrix}a_{1}&0\\\
a_{3}&a_{4}\end{smallmatrix}\right),b=\left(\begin{smallmatrix}b_{1}&b_{2}\\\
b_{3}&b_{4}\end{smallmatrix}\right),\;c=\left(\begin{smallmatrix}c_{1}&0\\\
0&0\end{smallmatrix}\right),\;d=\left(\begin{smallmatrix}d_{1}&d_{2}\\\
0&d_{4}\end{smallmatrix}\right)\right\\},$

where $0\leq r\leq n$ and $*_{1}\in M_{r,r}(\mathbb{Z}),*_{2}\in
M_{r,n-r}(\mathbb{Z}),*_{3}\in M_{n-r,r}(\mathbb{Z}),$ and $*_{4}\in
M_{n-r,n-r}(\mathbb{Z})$ are part of the definition of Siegel Eisenstein
series. As special cases we have $\Gamma_{n,n}=\Gamma_{n}$ and

(2.2) $\Gamma_{n,0}=\left\\{\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma_{n}\;|\;c=0_{n}\right\\}.$

Further, by $\Gamma_{1}\left(\begin{smallmatrix}0&d^{-1}\\\
d&0\end{smallmatrix}\right)$ we mean the subgroup

$\displaystyle\Gamma_{1}\left(\begin{smallmatrix}0&d^{-1}\\\
d&0\end{smallmatrix}\right)=\begin{pmatrix}0&d^{-1}\\\
d&0\end{pmatrix}\Gamma_{1}\begin{pmatrix}0&d^{-1}\\\
d&0\end{pmatrix}\cap\Gamma_{1},$

where $d$ is a positive integer. Note that
$\Gamma_{1}\left(\begin{smallmatrix}0&d^{-1}\\\ d&0\end{smallmatrix}\right)$
is isomorphic to

(2.3) $\Gamma_{1}(d)=\left(\begin{smallmatrix}d&0\\\
0&d^{-1}\end{smallmatrix}\right)^{-1}\Gamma_{1}\left(\begin{smallmatrix}d&0\\\
0&d^{-1}\end{smallmatrix}\right)\cap\Gamma_{1}$

and as a consequence, $\Gamma_{1}(\left(\begin{smallmatrix}0&d^{-1}\\\
d&0\end{smallmatrix}\right))\backslash\Gamma_{1}\cong\Gamma_{1}(d)\backslash\Gamma_{1}$.
The isomorphism is given by

(2.4) $\gamma\mapsto\ell(\gamma)=\left(\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right)\gamma^{-1}\left(\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\right).$

It is immediate that $\ell$ is an involution.

The group $\Gamma_{m}\times\Gamma_{n}$ can be embedded into $\Gamma_{n+m}$ by
the map

(2.5) $l_{m,n}:\Gamma_{m}\times\Gamma_{n}\rightarrow\Gamma_{n+m},\quad
l_{m,n}\left(\begin{pmatrix}a&b\\\
c&d\end{pmatrix},\begin{pmatrix}a^{\prime}&b^{\prime}\\\
c^{\prime}&d^{\prime}\end{pmatrix}\right)=\begin{pmatrix}a&0&b&0\\\
0&a^{\prime}&0&b^{\prime}\\\ c&0&d&0\\\
0&c^{\prime}&0&d^{\prime}\end{pmatrix}.$

Via $l_{m,n}$ we can embed $\Gamma_{m}$ and $\Gamma_{n}$ into $\Gamma_{n+m}$:

(2.6) $0\uparrow:\Gamma_{m}\rightarrow\Gamma_{n+m},\quad\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\mapsto\begin{pmatrix}a&b\\\
c&d\end{pmatrix}^{\uparrow}=l_{m,n}\left(\begin{pmatrix}a&b\\\
c&d\end{pmatrix},\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\right),$ (2.7)
$\downarrow:\Gamma_{n}\rightarrow\Gamma_{n+m},\quad\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\mapsto\begin{pmatrix}a&b\\\
c&d\end{pmatrix}^{\downarrow}=l_{n,m}\left(\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\right).$

Denote with $\mathbb{H}_{n}:=\\{\tau\in
M_{n,n}(\mathbb{C})\;:\;\tau^{t}=\tau\;\text{ and
}\operatorname{Im}(\tau)>0\\}$ the Siegel upper half space of genus $n$. It is
well known that the group $\Gamma_{n}$ acts on $\mathbb{H}_{n}$ by

$\left(\begin{pmatrix}a&b\\\
c&d\end{pmatrix},\tau\right)\mapsto\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\tau:=(a\tau+b)\cdot(c\tau+d)^{-1}.$

If $n=1$, $\mathbb{H}_{n}=\mathbb{H}_{1}$ specializes to the usual upper half
plane $\mathbb{H}$. Moreover, $\mathbb{H}\times\mathbb{H}$ can be embedded
into $\mathbb{H}_{2}$ by

$(\tau_{1},\tau_{2})\mapsto\begin{pmatrix}\tau_{1}&0\\\
0&\tau_{2}\end{pmatrix}.$

Let $g^{\prime},g=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{Sp}(n,\mathbb{R})$ and $\tau\in
H_{n}$. In order to define Siegel modular forms, we also need an automorphy
factor $j_{n}$: We set

$j_{n}(g,\tau):=\det(c\tau+d).$

The automorphy factor satisfies the usual cocycle relation

$j_{n}(gg^{\prime},\tau)=j_{n}(g,g^{\prime}\tau)j_{n}(g^{\prime},\tau).$

According to [Fr], Satz 1.4, we have the following identity

$\operatorname{Im}(g\tau)=^{t}(c\tau+d)^{-1}\operatorname{Im}(\tau)\overline{c\tau+d}^{-1},$

which yields

(2.8)
$\det(\operatorname{Im}(g\tau))=|j_{n}(g,\tau)|^{-2}\det(\operatorname{Im}(\tau)).$

For later purposes we introduce the function

(2.9) $w\mapsto\varphi_{l,s}(w)=w^{-l}|w|^{-2s}$

and denote with $|A|$ the order of any finite set $A$.

##  3\. The Weil representation

In this section we introduce a finite dimensional representation $\rho_{L,n}$
of the symplectic modular group $\Gamma_{n}$. It is isomorphic to a
subrepresentation of the Weil representation (see e. g. [St], Chapter 3.3) as
defined originally in [Wei], which explains the name “Weil representation”. We
then specialize to the cases of $n=2$ and $n=1$ and study some relations
between $\rho_{L,2}$ and $\rho_{L,1}$, which will be crucial for the Garret-
Böcherer decomposition of the vector valued Siegel Eisenstein series
$E_{l,0}^{2}$.

The following notation will be used this way throughout the whole paper: Let
$L$ be a lattice of rank $m$ equipped with a symmetric $\mathbb{Z}$-valued
bilinear form $(\cdot,\cdot)$ such that the associated quadratic form

$q(x):=\frac{1}{2}(x,x),\quad x\in L,$

takes values in $\mathbb{Z}$. We assume that $m$ is even, $L$ is non-
degenerate and denote its type by $(b^{+},b^{-})$ and its signature
$b^{+}-b^{-}$ by $\operatorname{sig}(L)$. Note that $\operatorname{sig}(L)$ is
also even. We stick with these assumptions on $L$ for the rest of this paper
unless we state it otherwise. Further, let

$L^{\prime}:=\\{x\in V=L\otimes\mathbb{Q}\;:\;(x,y)\in\mathbb{Z}\quad\text{
for all }\;y\in L\\}$

be the dual lattice of $L$. Since $L\subset L^{\prime}$, the elementary
divisor theorem implies that $L^{\prime}/L$ is a finite group. The modulo 1
reduction of both, the bilinear form $(\cdot,\cdot)$ and the associated
quadratic form, defines a $\mathbb{Q}/\mathbb{Z}$-valued bilinear form
$(\cdot,\cdot)$ with corresponding $\mathbb{Q}/\mathbb{Z}$-valued quadratic
form on $L^{\prime}/L$. For any two elements $\mu,\nu\in(L^{\prime}/L)^{n}$ we
define

(3.1)
$\begin{split}&(\mu,\nu)=((\mu_{i},\nu_{j}))_{i,j}\in\operatorname{Sym}_{n}(\mathbb{Q}),\\\
&Q[\mu]=\frac{1}{2}(\mu,\mu)\in\operatorname{Sym}_{n}(\mathbb{Q}).\end{split}$

It can be easily verified that

(3.2) $\operatorname{tr}(\mu,\nu)=\sum_{i=1}^{n}(\mu_{i},\nu_{i})$

defines a $\mathbb{Q}/\mathbb{Z}$-valued bilinear form on $(L^{\prime}/L)^{n}$
with associated quadratic form

(3.3) $\operatorname{tr}(Q[\mu])=\frac{1}{2}\sum_{i=1}^{n}q(\mu_{i}).$

We call $((L^{\prime}/L)^{n},\operatorname{tr}(\mu,\nu))$ a finite quadratic
module or a discriminant form. We call the discriminant form
$(L^{\prime}/L,(\cdot,\cdot))$ anisotropic, if $q(\mu)=0$ holds only for
$\mu=0$.

The Weil representation $\rho_{L,n}$ is a representation on the group ring
$\mathbb{C}[(L^{\prime}/L)^{n}]$. We denote its standard basis by
$\\{\mathfrak{e}_{\lambda}\\}_{\lambda\in(L^{\prime}/L)^{n}}$. The standard
scalar product on $\mathbb{C}[(L^{\prime}/L)^{n}]$ is given by

(3.4)
$\left\langle\sum_{\lambda\in(L^{\prime}/L)^{n}}a_{\lambda}\mathfrak{e}_{\lambda},\sum_{\lambda\in(L^{\prime}/L)^{n}}b_{\lambda}\mathfrak{e}_{\lambda}\right\rangle_{n}=\sum_{\lambda\in(L^{\prime}/L)^{n}}a_{\lambda}\overline{b_{\lambda}}.$

As $\Gamma_{n}$ is generated by the matrices

(3.5) $S_{n}=\begin{pmatrix}0&-1_{n}\\\ 1_{n}&0\end{pmatrix},\quad
T_{n}(b)=\begin{pmatrix}1_{n}&b\\\ 0&1_{n}\end{pmatrix},$

where $b\in\operatorname{Sym}_{n}(\mathbb{Z})$, it is sufficient to define
$\rho_{L,n}$ by the action on these generators. Note that we will use these
symbols in the case $n=1$ without the subscript $n$. Clearly, $T(b)$ is equal
to $T^{b}$ in this case.

######  Definition 3.1.

The representation $\rho_{L,n}$ of $\Gamma_{n}$ on
$\mathbb{C}[(L^{\prime}/L)^{n}],$ defined by

(3.6)
$\begin{split}&\rho_{L,n}(T_{n}(b))\mathfrak{e}_{\lambda}:=e(\operatorname{tr}(bQ[\lambda]))\mathfrak{e}_{\lambda},\\\
&\rho_{L,n}(S_{n}))\mathfrak{e}_{\lambda}:=\frac{e(-\frac{n\operatorname{sig}(L)}{8})}{|L^{\prime}/L|^{n/2}}\sum_{\mu\in(L^{\prime}/L)^{n}}e(-\operatorname{tr}(\mu,\lambda))\mathfrak{e}_{\mu},\end{split}$

is called Weil representation.

######  Remark 3.2.

  1. (1)

The action of $m_{n}(a)=\begin{pmatrix}a&0\\\
0&(a^{-1})^{t}\end{pmatrix}\in\Gamma_{n}$ with
$a\in\operatorname{GL}_{n}(\mathbb{Z})$ via the Weil representation will be
needed later on. It is given by

(3.7)
$\rho_{L,n}(m_{n}(a))\mathfrak{e}_{\lambda}=\chi_{V}(\det(a))\mathfrak{e}_{\lambda
a^{-1}},$

where $\chi_{V}(\det(a))=\det(a)^{\operatorname{sig}(L)/2}$, see [Zh], Def.
2.2. For further details, e. g. regarding the connection of $\rho_{L,n}$ to
the Weil representation in [Wei], see [Zh] or [St], p. 10.

  2. ii)

We denote by $N$ the level of the lattice $L$. It is the smallest positive
integer such that $Nq(\lambda)\in\mathbb{Z}$ for all $\lambda\in L^{\prime}$.
One can prove that the Weil representation $\rho_{L,n}$ is trivial on
$\Gamma_{n}(N)$, the principal congruence subgroup of level $N$, cf. [R],
Theorem 2.4. Therefore, $\rho_{L,n}$ factors over the finite group

$\Gamma_{n}/\Gamma_{n}(N)\cong\operatorname{Sp}(n,\mathbb{Z}/N\mathbb{Z}).$

  3. iii)

We denote by $\rho^{*}_{L,n}$ the dual representation of $\rho_{L,n}$. Since
the Weil representation is unitary, we obtain $\rho^{*}_{L,n}$ from
$\rho_{L,n}$ just by replacing the quadratic module
$((L^{\prime}/L)^{n},\operatorname{tr}(\cdot,\cdot))$ with
$((L^{\prime}/L)^{2},-\operatorname{tr}(\cdot,\cdot))$. Therefore, any result
involving $\rho_{L,n}$ carries over to $\rho^{*}_{L,n}$. Since $\rho_{L,n}$ is
unitary with respect to $\langle\cdot,\cdot\rangle_{n}$, its dual
representation $\rho^{*}_{L,n}$ is equal to the complex conjugate of
$\rho_{L,n}$ (interpreted as matrices), that is,

$\rho^{*}_{L,n}(g)=\rho^{-1}_{L,n}(g)=\overline{\rho_{L,n}(g)}$

for all $g\in\Gamma_{n}$. In particular,

$\displaystyle\rho^{*}_{L,n}(g)\mathfrak{e}_{\lambda}$
$\displaystyle=\sum_{\mu\in(L^{\prime}/L)^{n}}\langle\mathfrak{e}_{\lambda},\rho^{*}_{L,n}(g)\mathfrak{e}_{\mu}\rangle_{n}\mathfrak{e}_{\mu}=\sum_{\mu\in(L^{\prime}/L)^{n}}\overline{\langle\mathfrak{e}_{\lambda},\rho_{L,n}(g)\mathfrak{e}_{\mu}\rangle}_{n}\mathfrak{e}_{\mu}.$

  4. iv)

There is an isomorphism on $\Gamma_{n}$:

(3.8) $\gamma\mapsto\widetilde{\gamma}=\left(\begin{smallmatrix}-1_{n}&0\\\
0&1_{n}\end{smallmatrix}\right)\gamma\left(\begin{smallmatrix}-1_{n}&0\\\
0&1_{n}\end{smallmatrix}\right).$

For the generators of $\Gamma_{n}$ we have

$\displaystyle\widetilde{S_{n}}=S_{n}^{-1}\text{ and
}\widetilde{T_{n}(b)}=T_{n}(b)^{-1}$

Since the map (3.8) is an isomorphism, it follows immediately

(3.9)
$\rho_{L,n}(\widetilde{\gamma})\mathfrak{e}_{\lambda}=\rho_{L,n}^{-1}(\gamma)=\rho^{*}_{L,n}(\gamma)$

for any $\gamma\in\Gamma_{n}$, where the last equation is due to part iii) of
this remark.

We now concentrate on the special cases $n=1$ and $n=2$. It is well known that
$\rho_{L,2}$ and $\rho_{L,1}$ are closely related. To describe this relation
more explicitly for some elements of $\Gamma_{n}$ we first recall some facts
see from [BY], p. 647 and [SV], Chapter 5, which relate the group rings
$\mathbb{C}[(L^{\prime}/L)^{2}]$ and $\mathbb{C}[L^{\prime}/L]$ and as a
consequence $\rho_{L,2}$ and $\rho_{L,1}$: The map

(3.10)
$\mathbb{C}[(L^{\prime}/L)^{2}]\rightarrow\mathbb{C}[L^{\prime}/L]\otimes\mathbb{C}[L^{\prime}/L],\quad\mathfrak{e}_{(\mu,\nu)}\mapsto\mathfrak{e}_{\mu}\otimes\mathfrak{e}_{\nu}$

defines an isomorphism, which induces an isomorphism of the representations
$\rho_{L,2}$ and $\rho_{L,1}\otimes\rho_{L,1}$. It is also a known fact that
we can equip $\mathbb{C}[L^{\prime}/L]\otimes\mathbb{C}[L^{\prime}/L]$ with an
inner product by

(3.11)
$\langle\mathfrak{e}_{\lambda_{1}}\otimes\mathfrak{e}_{\mu_{1}},\mathfrak{e}_{\lambda_{2}}\otimes\mathfrak{e}_{\mu_{2}}\rangle_{2}=\langle\mathfrak{e}_{\lambda_{1}},\mathfrak{e}_{\lambda_{2}}\rangle_{1}\cdot\langle\mathfrak{e}_{\mu_{1}},\mathfrak{e}_{\mu_{2}}\rangle_{1}.$

It is then easily checked that the scalar product (3.11) coincides with (3.4)
and the map (3.10) becomes an isometry.

Finally, note that $\mathbb{C}[L^{\prime}/L]$ can be embedded into
$\mathbb{C}[L^{\prime}/L]\otimes\mathbb{C}[L^{\prime}/L]$ and therefore into
$\mathbb{C}[(L^{\prime}/L)^{2}]$ via the map

(3.12)
$\mathfrak{e}_{\lambda}\mapsto\mathfrak{e}_{\lambda}\otimes\mathfrak{e}_{\lambda}.$

The image of $\mathbb{C}[L^{\prime}/L]$ with respect to this map is isomorphic
to $\mathbb{C}[L^{\prime}/L]$.

Using the formulas in (3.6) and the isomorphism (3.10), we can express
$\rho_{L,2}$ on the generators of $\Gamma_{2}$ explicitly in terms of
$\rho_{L,1}$.

######  Lemma 3.3.

We have the following relation between $\rho_{L,2}$ and $\rho_{L,1}$ on the
generators of $\Gamma_{2}$ and $\Gamma_{1}$, respectively:

(3.13)
$\begin{split}&\rho_{L,2}(S_{2})\mathfrak{e}_{(\lambda_{1},\lambda_{2})}=\rho_{L,1}(S)\mathfrak{e}_{\lambda_{1}}\otimes\rho_{L,1}(S)\mathfrak{e}_{\lambda_{2}}\\\
&\rho_{L,2}(T_{2}\left(\begin{smallmatrix}a_{1}&0\\\
0&a_{2}\end{smallmatrix}\right))\mathfrak{e}_{(\lambda_{1},\lambda_{2})}=\rho_{L}(T^{a_{1}})\mathfrak{e}_{\lambda_{1}}\otimes\rho_{L}(T^{a_{2}})\mathfrak{e}_{\lambda_{2}},\end{split}$

where $\left(\begin{smallmatrix}a_{1}&0\\\ 0&a_{2}\end{smallmatrix}\right)\in
M_{2}(\mathbb{Z})$. Clearly, an analogous formula holds for $S_{2}^{-1}$.

###### Proof.

These identities can be checked by a straightforward computation involving the
formulas (3.6) for $n=2$ and $n=1$ and the isomorphism (3.10). ∎

The results of the next lemma will be crucial in the proof of a Garret-
Böcherer decomposition of the Siegel Eisenstein series $E_{l,0}^{2}$. By
identifying $\mathbb{C}[(L^{\prime}/L)^{2}]$ with
$\mathbb{C}[L^{\prime}/L]\otimes\mathbb{C}[L^{\prime}/L]$ they provide
formulas that link $\rho_{L,2}(\gamma^{\uparrow})$ and
$\rho_{L,2}(\gamma^{\downarrow})$ with $\rho_{L,1}(\gamma)$.

######  Lemma 3.4.

The following formulas for the Weil representation $\rho_{L,2}$ hold:

  1. i)

(3.14)
$\rho_{L,2}(S^{\uparrow})\mathfrak{e}_{(\lambda_{1},\lambda_{2})}=\rho_{L,1}(S)\mathfrak{e}_{\lambda_{1}}\otimes\mathfrak{e}_{\lambda_{2}},$
(3.15)
$\rho_{L,2}(S^{\downarrow})\mathfrak{e}_{(\lambda_{1},\lambda_{2})}=\mathfrak{e}_{\lambda_{1}}\otimes\rho_{L,1}(S)\mathfrak{e}_{\lambda_{2}},$
(3.16)
$\rho_{L,2}(T^{\uparrow})\mathfrak{e}_{(\lambda_{1},\lambda_{2})}=\rho_{L,1}(T)\mathfrak{e}_{\lambda_{1}}\otimes\mathfrak{e}_{\lambda_{2}},$

and

(3.17)
$\rho_{L,2}(T^{\downarrow})\mathfrak{e}_{(\lambda_{1},\lambda_{2})}=\mathfrak{e}_{\lambda_{1}}\otimes\rho_{L,1}(T)\mathfrak{e}_{\lambda_{2}}.$

  2. ii)

For any $\gamma\in\operatorname{SL}_{2}(\mathbb{Z})$ we have

(3.18)
$\rho_{L,2}(\gamma^{\uparrow})\mathfrak{e}_{(\lambda_{1},\lambda_{2})}=\rho_{L}(\gamma)\mathfrak{e}_{\lambda_{1}}\otimes\mathfrak{e}_{\lambda_{2}}$

and

(3.19)
$\rho_{L,2}(\gamma^{\downarrow})\mathfrak{e}_{(\lambda_{1},\lambda_{2})}=\mathfrak{e}_{\lambda_{1}}\otimes\rho_{L}(\gamma)\mathfrak{e}_{\lambda_{2}}.$

###### Proof.

  1. $i)$

For arbitrary $a\in\operatorname{Sym}_{2}(\mathbb{Z})$ we use the notation

(3.20) $U_{2}(a)=\begin{pmatrix}1_{2}&0\\\ a&1_{2}\end{pmatrix}\in\Gamma_{2}.$

One can easily verify the decomposition

$\displaystyle S^{\uparrow}=U_{2}(b)T_{2}(-b)U_{2}(b)\text{ and }$
$\displaystyle S^{\downarrow}=U_{2}(c)T_{2}(-c)U_{2}(c),$

where $b=\left(\begin{smallmatrix}1&0\\\ 0&0\end{smallmatrix}\right)$ and
$c=\left(\begin{smallmatrix}0&0\\\ 0&1\end{smallmatrix}\right)$. The matrix
$U_{2}(a)$ can be written as product of the generators (3.5)

$\displaystyle U_{2}(a)=S_{2}T_{2}(-a)S_{2}^{-1}$

and therefore

$\displaystyle
S^{\uparrow}=S_{2}T_{2}(-b)S_{2}^{-1}T_{2}(-b)S_{2}T_{2}(-b)S_{2}^{-1},\quad
S^{\downarrow}=S_{2}T_{2}(-c)S_{2}^{-1}T_{2}(-c)S_{2}T_{2}(-c)S_{2}^{-1}.$

If we employ the identities (3.13) successively for all constituents of the
decomposition above, (3.6) applied to $T_{2}(-b)$ and $T_{2}(-c)$, and the
bilinearity of the tensor product, we may transfer the evaluation of
$\rho_{L,2}(S^{\uparrow})$ to the evaluation of $\rho_{L,1}$ on the
corresponding element in $\Gamma_{1}$:

(3.21)
$\begin{split}\rho_{L,2}(S^{\uparrow})\mathfrak{e}_{(\lambda_{1},\lambda_{2})}&=\rho_{L,1}(ST^{-1}S^{-1}T^{-1}ST^{-1}S^{-1})\mathfrak{e}_{\lambda_{1}}\otimes\rho_{L,1}(SS^{-1}SS^{-1})\mathfrak{e}_{\lambda_{2}}\\\
&=\rho_{L,1}(ST^{-1}S^{-1}T^{-1}ST^{-1}S^{-1})\mathfrak{e}_{\lambda_{1}}\otimes\mathfrak{e}_{\lambda_{2}}\end{split}$

and accordingly

(3.22)
$\begin{split}\rho_{L,2}(S^{\downarrow})\mathfrak{e}_{(\lambda_{1},\lambda_{2})}&=\rho_{L,1}(SS^{-1}SS^{-1})\mathfrak{e}_{\lambda_{1}}\otimes\rho_{L,1}(ST^{-1}S^{-1}T^{-1}ST^{-1}S^{-1})\mathfrak{e}_{\lambda_{2}}\\\
&=\mathfrak{e}_{\lambda_{1}}\otimes\rho_{L,1}(ST^{-1}S^{-1}T^{-1}ST^{-1}S^{-1})\mathfrak{e}_{\lambda_{2}}.\end{split}$

We can further simplify the expressions above by means of [McG], Lemma 4.6. It
states for integers $a,d$ satisfying $ad\equiv 1\bmod{N}$ that

$\rho_{L,1}(ST^{d}S^{-1}T^{a}ST^{d})\mathfrak{e}_{\lambda}=\frac{g_{d}(L)}{g(L)}\mathfrak{e}_{d\lambda},$

where $g_{d}(L)=\sum_{\lambda\in L^{\prime}/L}e(dq(\lambda))$ and
$g(L)=g_{1}(L)$. If we apply this result for $a=d=-1$, we obtain

$\displaystyle\rho_{L,1}(ST^{-1}S^{-1}T^{-1}ST^{-1}S^{-1})\mathfrak{e}_{\lambda}=\frac{e\left(\frac{\operatorname{sig}(L)}{8}\right)}{\sqrt{|L^{\prime}/L|}}\sum_{\mu\in
L^{\prime}/L}e((\mu,\lambda))\frac{g_{-1}(L)}{g(L)}\mathfrak{e}_{-\mu}.$

By Milgram’s formula,

$g(L)=\sqrt{|L^{\prime}/L|}e(\operatorname{sig}(L)/8),$

we have

$\displaystyle e(\operatorname{sig}(L)/8)\frac{g_{-1}(L)}{g(L)}$
$\displaystyle=e(-\operatorname{sig}(L)/8),$

which implies

$\rho_{L,1}(ST^{-1}S^{-1}T^{-1}ST^{-1}S^{-1})\mathfrak{e}_{\lambda}=\rho_{L,1}(S)\mathfrak{e}_{\lambda}.$

The identities (3.16) and (3.17) are due to Lemma 3.3 and the fact that

$T^{\uparrow}=T_{2}(b)\text{ and }T^{\downarrow}=T_{2}(c),$

where $b,c$ have same meaning as before in the proof.

  2. $ii)$

This follows easily from the fact that the embeddings (2.6) and (2.7) are
group homomorphisms combined with the corresponding formulas for the
generators $S$ and $T$ of $\Gamma_{1}$.

∎

We end this chapter with the evaluation of the Weil representation
$\rho_{L,2}$ on the element $U_{2}(D)\in\Gamma_{2}$, where $D$ is equal to
$\left(\begin{smallmatrix}0&d\\\ d&0\end{smallmatrix}\right)$ and $d$ is a
positive integer. This evaluation will also contribute to the proof of a
Garret-Böcherer decomposition of the Siegel Eisenstein series $E_{l,0}^{2}$.

######  Lemma 3.5.

Let $d\in\mathbb{N}$, $D=\left(\begin{smallmatrix}0&d\\\
d&0\end{smallmatrix}\right)\in\operatorname{Sym}_{2}(\mathbb{Z})$ and
$U_{2}(D)\in\Gamma_{2}$ defined as in (3.20).

Then

(3.23)
$\begin{split}\rho_{L,2}^{-1}(U_{2}(D))\mathfrak{e}_{\lambda_{1}}\otimes\mathfrak{e}_{\lambda_{2}}&=\frac{1}{|L^{\prime}/L|}\sum_{\mu,\nu\in
L^{\prime}/L}e((\mu,\lambda_{2}-\nu))\mathfrak{e}_{d\mu+\lambda_{1}}\otimes\mathfrak{e}_{\nu}\\\
\end{split}$

###### Proof.

Clearly, $U_{2}(D)^{-1}=U_{2}(-D)=S_{2}T_{2}(D)S_{2}^{-1}$. A straightforward
calculation using Lemma 3.3 and (3.6) yields

$\displaystyle\rho_{L,2}(U_{2}(-D))\mathfrak{e}_{\lambda_{1}}\otimes\mathfrak{e}_{\lambda_{2}}$
$\displaystyle=\frac{1}{|L^{\prime}/L|^{2}}\sum_{\nu_{1},\nu_{2}\in
L^{\prime}/L}\sum_{\mu_{2}\in
L^{\prime}/L}e((\mu_{2},\lambda_{2}-\nu_{2}))\sum_{\mu_{1}\in
L^{\prime}/L}e((\mu_{1},d\mu_{2}+\lambda_{1}-\nu_{1}))\mathfrak{e}_{\nu_{1}}\otimes\mathfrak{e}_{\nu_{2}}.$

The sum over $\mu_{1}$ is equal to $|L^{\prime}/L|$ if
$\nu_{1}=d\mu_{2}+\lambda_{1}$ and 0 otherwise. Hence,

$\displaystyle\rho_{L,2}(U_{2}(D)^{-1})\mathfrak{e}_{\lambda_{1}}\otimes\mathfrak{e}_{\lambda_{2}}=\frac{1}{|L^{\prime}/L|}\sum_{\nu_{2}\in
L^{\prime}/L}\sum_{\mu_{2}\in
L^{\prime}/L}e((\mu_{2},\lambda_{2}-\nu_{2}))\mathfrak{e}_{d\mu_{2}+\lambda_{1}}\otimes\mathfrak{e}_{\nu_{2}}.$

as claimed.

∎

##  4\. Vector valued Siegel modular forms and Eisenstein series

One major goal of the present paper is to study the analytic properties of the
standard zeta function of a certain vector valued modular form of type
$\rho_{L,1}$ (a common eigenform of all Hecke operators $T(m^{2})$). The main
tool in this regard is the Garret-Böcherer decomposition of a non-holomorphic
vector valued Siegel Eisenstein series $E_{l,0}^{2}$. In this section we
provide the necessary background to prove the Garret-Böcherer decomposition
and to define and study the before mentioned standard zeta function. To this
end, we collect some facts about vector valued Siegel modular forms of type
$\rho_{L,n}$ and introduce the non-holomorphic vector valued Siegel Eisenstein
series $E_{l,0}^{2}$ transforming with $\rho_{L,n}$, which plays a key role in
determining the analytic properties of the standard zeta function. As a matter
of fact, the zeta function inherits these properties from $E_{l,0}^{2}$, which
is the reason we state and discuss them in some detail in this section. We
follow [Zh] and [St].

###  4.1. Vector valued Siegel modular forms of type $\rho_{L,n}$

Let $l\in\mathbb{Z}$. For
$f:\mathbb{H}_{n}\longrightarrow\mathbb{C}[(L^{\prime}/L)^{n}]$ and
$g\in\Gamma_{n}$ we define the usual Petersson slash operator by

$(f\mid^{n}_{l,L}g)(\tau):=j_{n}(g,\tau)^{-l}\rho_{L,n}(g)^{-1}f(g\tau).$

The Petesson slash operator applied to the dual Weil representation
$\rho^{*}_{L,n}$ will be denoted with $\mid^{n*}_{l,L}$. For $n=1$ we drop the
superscript and subscript $n$ and simply write $j(g,\tau)$, $\mid_{l,L}$ or
$\mid_{l,L}^{*}$.

######  Definition 4.1.

Let $l\in\mathbb{Z}$. A function
$f:\mathbb{H}_{n}\longrightarrow\mathbb{C}[(L^{\prime}/L)^{n}]$ is called a
Siegel modular form of weight $l$ with respect to $\Gamma_{n}$ and
$\rho_{L,n}$ if

  1. (1)

$f$ is holomorphic on $\mathbb{H}_{n}$,

  2. (2)

$f\mid^{n}_{l,L}g=f\;$ for all $g\in\Gamma_{n}$,

  3. (3)

$f$ has a Fourier expansion of the form

(4.1)
$f(\tau)=\sum_{\lambda\in(L^{\prime}/L)^{n}}\sum_{\begin{subarray}{c}t\in\Lambda_{n}\\\
t+Q[\lambda]\geq
0\end{subarray}}a(\lambda,t+Q[\lambda])e(\operatorname{tr}((t+Q[\lambda])\tau))\mathfrak{e}_{\lambda},$

where $\Lambda_{n}$ is the set of half-integral $(n\times n)$-matrices.

We denote the space of these modular forms with $M_{l}(\rho_{L,n})$. Note that
the vanishing condition on the Fourier coefficients is automatically fulfilled
for $n\geq 2$ by the Koecher principle. Moreover, if
$a(\lambda,t+Q[\lambda])=0$ for all $t\in\Lambda_{n}$ with $t+Q[\lambda]=0$,
then $f$ is called a cusp form. We use the notation $S_{l}(\rho_{L,n})$ for
the subspace of all cusp forms.

######  Remark 4.2.

  1. i)

We could have defined the modular forms in Definition 4.1 also for half
integral weight $l$. The following argument shows that
$M_{l}(\rho_{L,n})=\\{0\\}$, if the condition

(4.2) $2l\equiv\operatorname{sig}(L)\bmod{2}$

is not satisfied.

For
$e_{n}=\operatorname{diag}(-1,1,\dots,1)\in\operatorname{GL}_{n}(\mathbb{Z})$
and $f\in M_{l}(\rho_{L,n})$ by (3.7) we have

$\displaystyle
f(\tau)=(f\mid^{n}_{l,L}m_{n}(e_{n}))(\tau)=(-1)^{-l}(-1)^{\operatorname{sig}(L)/2}f(\tau)$

for all $\tau\in\mathbb{H}_{n}$. Thus $2l$ and $\operatorname{sig}(L)$ must
either be both even or odd if $f$ is a non-trivial form in
$M_{l}(\rho_{L,n})$. Since we have assumed that the rank of $L$ is even, so is
the signature $\operatorname{sig}(L)$ and therefore there are only non-trivial
integral weight modular forms.

  2. ii)

Since $\rho_{L,n}$ is trivial on $\Gamma_{n}(N)$, it follows directly from
Definition 4.1 that the component functions $f_{\lambda}$ of
$f=\sum_{\lambda\in(L^{\prime}/L)^{n}}f_{\lambda}\mathfrak{e}_{\lambda}$ are
Siegel modular forms of weight $l$ for $\Gamma_{n}(N)$.

####  4.1.1. Petersson scalar product and Hecke operators

For $f,g\in M_{l}(\rho_{L,n})$, one of which is a cusp form, we define the
Petersson scalar product of $f$ and $g$ analogously to [Br1], Section 1.2.2.,
by

(4.3) $(f,g)_{n}=\int_{\Gamma_{n}\backslash\mathbb{H}_{n}}\langle
f(\tau),g(\tau)\rangle_{n}\det(\operatorname{Im}(\tau))^{l}d\mu_{n}(\tau).$

Here $d\mu_{n}(\tau)=\det(y)^{-n-1}dxdy$ with $\tau=x+iy$ is the usual
symplectic volume element. If $n=1$, we omit the subscript $n$. The same
arguments as in [Br1] show that the integral (4.3) is well defined.

Let $D=\left(\begin{smallmatrix}d&0\\\ 0&d^{-1}\end{smallmatrix}\right)$,
$D^{\prime}=\left(\begin{smallmatrix}d^{2}&0\\\ 0&1\end{smallmatrix}\right)$
and $R_{d}=\left(\begin{smallmatrix}d&0\\\ 0&d\end{smallmatrix}\right)$. We
now briefly describe the definition of Hecke operators $T(D^{\prime})$ and
$T(D)$ on $M_{l}(\rho_{L,1})$ (to the best of my knowledge there is no theory
of Hecke operators for $M_{l}(\rho_{L,n})$, although it might be developed in
a similar manner). We follow [BS]. All details can be found therein. According
to our assumptions on $L$, we only need to consider the case of even
signature. Recall that $N$ is the level of $L$. We have to distinguish between
the cases $(d,N)=1$ and $(d,N)>1$.

Let

(4.4)
$\begin{split}\mathcal{G}(N)=&\left\\{M\in\operatorname{GL}_{2}^{+}(\mathbb{Q});\quad\exists
n\in\mathbb{Z}\text{ with }(n,N)=1\text{ such that }nM\in
M_{2}(\mathbb{Z})\right.\\\ &\left.\text{ and
}(\det(nM),N)=1\right\\}.\end{split}$

If $d$ is coprime to $N$, $(D,1)$ and $(D^{\prime},d)$ belong to the group

(4.5)
$\mathcal{Q}(N)=\left\\{(M,r)\in\mathcal{G}(N)\times(\mathbb{Z}/N\mathbb{Z})^{*};\quad\det(M)\equiv
r^{2}\pmod{N}\right\\}.$

We identify $\Gamma_{1}$ with a subgroup of $\mathcal{Q}(N)$ via the map
$\gamma\mapsto(\gamma,1)$. In [BS] an extension of $\rho_{L,1}$ to the group
$\mathcal{Q}(N)$ is introduced. The condition that the determinant of any
element of this group is coprime to the level is crucial. Based on this
extension we now define Hecke operators in terms of the Hecke algebra given by
the pair of groups $(\mathcal{Q}(N),\Gamma_{1})$. For any
$(g,r)\in\mathcal{Q}(N)$ and $f\in M_{l}(\rho_{L,1})$ we set

(4.6)
$f\mid_{l,L}T(g,r)=\det(g)^{l/2-1}\sum_{M\in\Gamma_{1}\backslash\Gamma_{1}g\Gamma_{1}}f\mid_{l,L}(M,r).$

Instead of $T(D,1)$ and $T(D^{\prime},d)$ we write $T(D)$ and $T(D^{\prime})$,
respectively. Taking [BS], (3.5), into account, it can be easily verified that
$\rho_{L,1}(D)$ and $\rho_{L,1}(D^{\prime})$ are connected via

(4.7)
$\rho_{L,1}(D^{\prime})=\rho_{L,1}(R_{d})\rho_{L,1}(D)=\frac{g(L)}{g_{d}(L)}\rho_{L,1}(D).$

This immediately yields a relation between $T(D)$ and $T(D^{\prime})$:

(4.8) $T(D)=\frac{g(L)}{g_{d}(L)}T(D^{\prime}).$

Replacing $\rho_{L,1}$ with its dual representation gives the corresponding
relation

(4.9) $T(D)=\frac{g_{d}(L)}{g(L)}T(D^{\prime}).$

If $(d,N)>1$, $D$ and $D^{\prime}$ do not belong to $\mathcal{Q}(N)$ and the
Weil representation cannot be extended to a suitable subgroup of
$\operatorname{GL}_{2}^{+}(\mathbb{Q})$ as before. However, it can be extended
to the double coset $\Gamma_{1}D^{\prime}\Gamma_{1}$ in the following way
([BS], Section 5):

(4.10)
$\rho_{L,1}^{-1}(D^{\prime})\mathfrak{e}_{\lambda}=\mathfrak{e}_{d\lambda}$

and for $\delta=\gamma
D^{\prime}\gamma^{\prime}\in\Gamma_{1}D^{\prime}\Gamma_{1}$ we put

(4.11)
$\rho_{L,1}^{-1}(\delta)\mathfrak{e}_{\lambda}=\rho_{L,1}^{-1}(\gamma^{\prime})\rho_{L,1}^{-1}(D^{\prime})\rho_{L,1}^{-1}(\gamma)\mathfrak{e}_{\lambda}.$

Note that (4.10) is compatible with the definition of $\rho_{L,1}$ in the case
of $(d,N)=1$ (see [BS], Lemma 3.6). It can be shown (see [BS], Section 5) that
(4.11) is independent of the decomposition of $\delta$. With these definitions
in mind, we define the Hecke operator $T(D^{\prime})$ exactly as in (4.6). In
order to define the operator $T(D)$, we set in consistency with [BS], (3.5)

(4.12)
$\rho_{L,1}^{-1}(R_{d})\mathfrak{e}_{\lambda}=\frac{g_{d}(L)}{g(L)}\mathfrak{e}_{\lambda}\text{
and
}\rho_{L,1}(R_{d})\mathfrak{e}_{\lambda}=\frac{g(L)}{g_{d}(L)}\mathfrak{e}_{\lambda}.$

Accordingly we define

(4.13)
$\rho_{L,1}^{-1}(D)\mathfrak{e}_{\lambda}=\frac{g_{d}(L)}{g(L)}\rho_{L,1}^{-1}(D^{\prime})\mathfrak{e}_{\lambda}=\frac{g_{d}(L)}{g(L)}\mathfrak{e}_{d\lambda}.$

Since the $\rho_{L,1}^{-1}(D)$ and $\rho_{L,1}^{-1}(D^{\prime})$ differ just
by a constant factor,

(4.14) $\rho_{L,1}^{-1}(\gamma
D\gamma^{\prime})=\frac{g_{d}(L)}{g(L)}\rho_{L,1}^{-1}(\gamma
D^{\prime}\gamma^{\prime})$

together with (4.11) defines an action of the double coset
$\Gamma_{1}D\Gamma_{1}$. Again, we find that the relation (4.8) holds.

Finally, note that the analogous identity to $(5.5)$ in [BS] also holds for
$D$. It can easily be confirmed that

(4.15)
$\langle\mathfrak{b},\rho_{L,1}^{-1}(D)\mathfrak{a}\rangle_{1}=\langle\rho_{L,1}^{-1}(D^{-1})\mathfrak{b},\mathfrak{a}\rangle_{1}$

for any $\mathfrak{a},\mathfrak{b}\in\mathbb{C}[L^{\prime}/L]$.

If $d$ is coprime to $N$, one can show that $\frac{g(L)}{g_{d}(L)}$ is real
and thus $T(D)$ is self-adjoint with respect to $(\cdot,\cdot)_{1}$. If
$(d,N)>1$, it is in general not true that $\frac{g(L)}{g_{d}(L)}$ is a real
number. In fact, it can be shown that it is an 8-th root of unity (see [Sch],
Section 3, for these assertions) In this case, in light of [BS], Theorem 5.6,
we may only conclude that

(4.16) $(f,g\mid_{l,L}T(D))_{1}=\kappa_{d}(f\mid_{l,L}T(D),g)_{1}$

holds, where

$\kappa_{d}=\overline{\frac{g(L)}{g_{d}(L)}}\frac{g_{d}(L)}{g(L)}.$

In both cases $T(D)$ preservers the space $S_{l}(\rho_{L,1})$.

###  4.2. Vector valued Eisenstein series

Eisenstein series and Poincaré series are important examples of the vector
valued Siegel modular forms in Definition 4.1. In this subsection we introduce
non-holomorphic versions of these series and study the analytic properties of
the real analytic Eisenstein series $E_{l,0}^{n}$ for $\Gamma_{n}$
transforming with $\rho_{L,n}$.

######  Definition 4.3.

Let $l\in 2\mathbb{Z},\;l\geq n+1,$ satisfy the condition
$2l+\operatorname{sig}(L)\equiv 0\bmod{4}.$ For $\tau\in\mathbb{H}_{n}$ and
$s\in\mathbb{C}$ we define by

(4.17)
$\begin{split}E_{l,0}^{n}(\tau,s)&=\sum_{\gamma\in\Gamma_{n,0}\backslash\Gamma_{n}}\det(\operatorname{Im}(\tau))^{s}\mathfrak{e}_{0}\mid^{n}_{l,L}\gamma\\\
&=\det(\operatorname{Im}(\tau))^{s}\sum_{\gamma\in\Gamma_{n,0}\backslash\Gamma_{n}}|j_{n}(\gamma,\tau)|^{-2s}j_{n}(\gamma,\tau)^{-l}\rho_{L,n}^{-1}(\gamma)\mathfrak{e}_{0}.\end{split}$

a vector valued non-holomorphic Siegel Eisenstein series of weight $k$
transforming with $\rho_{L,n}$.

######  Remark 4.4.

  1. i)

From the formulas (3.6), (3.7) and the assumption
$2l\equiv-\operatorname{sig}(L)\bmod{4}$ follows that
$\det(\operatorname{Im}(\tau))^{s}\mathfrak{e}_{0}$ is
$\Gamma_{n,0}$-invariant with respect to the slash operator.

  2. ii)

From [St], Lemma 3.14, follows that $E_{l,0}^{n}(\tau,s)$ converges absolutely
for all $\tau\in\mathbb{H}_{n}$ and all $s\in\mathbb{C}$ with
$\operatorname{Re}(s)>\frac{n+1-k}{2}$. The usual argument then shows that
$E_{l,0}^{n}$ is with respect to $\mid_{k,L}$ a $\Gamma_{n}$-invariant real
analytic function. In particular, $\tau\mapsto E_{l,0}^{n}(\tau,0)$ is a
vector valued Siegel modular form in $M_{l}(\rho_{L,n})$.

  3. iii)

Although $E_{l,0}^{n}$ is defined on $\mathbb{H}_{n}$, we can also assign to
$E_{l,0}^{n}(-\tau)$, $\tau\in\mathbb{H}_{n}$, a meaningful value: To this
end, let $E_{l,0}^{n*}$ be the Eisenstein series in Definition 4.3 with
respect to the dual Weil representation $\rho^{*}_{L,n}$. Then, since
$\gamma\mapsto\widetilde{\gamma}$ is an isomorphism which preservers
$\Gamma_{n,0}$,

$\displaystyle
E_{l,0}^{n*}(\tau,s)=\det(\operatorname{Im}(\tau))^{s}\sum_{\gamma\in\Gamma_{n,0}\backslash\Gamma_{n}}|j_{n}(\widetilde{\gamma},\tau)|^{-2s}j_{n}(\widetilde{\gamma},\tau)^{-l}\rho_{L,n}^{*-1}(\widetilde{\gamma})\mathfrak{e}_{0},$

where $\widetilde{\gamma}$ is given by (3.8). Clearly,
$j_{n}(\widetilde{\gamma},\tau)=j_{n}(\gamma,-\tau)$ and by Remark 3.2, iv),

$\rho_{L,n}^{*-1}(\widetilde{\gamma})=\rho_{L,n}^{-1}(\gamma).$

Therefore, we put

(4.18)
$\begin{split}&E_{l,0}^{n}(-\tau,s)=((-1)^{n})^{s}E_{l,0}^{n*}(\tau,s)\text{
and }\\\
&E_{l,0}^{n}(\overline{\tau},s)=((-1)^{n})^{s}E_{l,0}^{n*}(-\overline{\tau},s).\end{split}$

With the same arguments

(4.19)
$\begin{split}&E_{l,0}^{n*}(-\tau,s)=((-1)^{n})^{s}E_{l,0}^{n}(\tau,s)\text{
and }\\\
&E_{l,0}^{n*}(\overline{\tau},s)=((-1)^{n})^{s}E_{l,0}^{n}(-\overline{\tau},s)\end{split}$

is justified.

The analytic properties of $E_{l,0}^{n}$ with respect to $s$ were investigated
in [St]. We summarize the results therein and specialize them to the case
$n=2$. In particular, the functional equation of $E_{l,0}^{n}$ is for general
$n$ quite complicated, yet for $n=2$ it is much simpler and quite clean. To
state the functional equation, we need quite a bit of notation, which we
describe here for convenience of the reader in full detail. Most of the
notation is needed for the transformation behaviour regarding the primes $p$
dividing $|L^{\prime}/L|$:

By $S_{n}$ we mean the symmetric group acting on $I=\\{1,\dots,n\\}$.
Associated to $\sigma\in S_{n}$ we have the quantities

$\displaystyle c_{1}(\sigma)=|\\{i\in I\;|\;\sigma(i)=i\\}|,\quad
c_{2}(\sigma)=|\\{i\in I\;|\;\sigma(i)\not=i\\}|.$

For any partition $I=I_{0}\cup\dots\cup I_{s}$ of $I$ into disjoint
$\sigma$-stable subsets, we put

(4.20) $\begin{split}&n_{r}=|I_{r}|,\\\
&n(r)=\frac{n^{(r)}(n^{(r)}+1)}{2},\end{split}$

where $n^{(r)}=n_{r}+n_{r+1}+\cdots+n_{s}$ and $r\geq 0$. Here we understand
$n^{(r)}$ to be zero if $r>s$. Also, the terms

(4.21) $\begin{split}&t(\sigma,\\{I_{i}\\})=\sum_{r=0}^{s}|\\{(i,j)\in
I_{r}\times I_{r}\;|\;i<j<\sigma(i),\;\sigma(j)<\sigma(i)\\}|,\\\
&\tau(\left\\{I_{i}\right\\})=\sum_{r=1}^{s}|\\{(i,j)\in
I_{r}\times(I_{0}\cup\cdots\cup I_{r-1})\;|\;j<i\\}|,\\\
&c_{1}^{(r)}(\sigma)=|\\{i\in I_{r}\cup\cdots\cup
I_{s}\;|\;\sigma(i)=i\\}|,\;l\geq 0\\\ &\text{ and }\\\
&c_{1,r}(\sigma)=|\\{i\in I_{r}\;|\;\sigma(i)=i\\}|,\\\
&c_{2,r}(\sigma)=|\\{i\in I_{r}\;|\;\sigma(i)>i\\}|\end{split}$

will occur. Since the latter quantities are part of complicated formulas, we
drop the $(\sigma)$-part and write from now on $c_{1,r}$ and $c_{2,r}$. In
terms of $c_{1,r}$ and $c_{2,r}$ the following abbreviations will also be used

$\displaystyle A_{i}({\bf
I_{r}})=\sum_{j=0}^{i-1}(\frac{1}{2}c_{1,j}+c_{2,j},\quad B_{i}({\bf
I_{r}})=-mA_{i}({\bf I_{r}})+n(0)-n(i),\;i=1,\dots,k.$

They appear as argument of the local zeta functions
$\zeta_{p}(s)=(1-p^{-s})^{-1}$:

(4.22) $D_{p,j}(s)=\zeta_{p}(4A_{j}({\bf I_{r}})(s-s_{0})-2B_{j}({\bf
I_{r}})),$

with $s_{0}=\frac{m}{2}-\rho_{n}$ and $\rho_{n}=\frac{n+1}{2}$. Attached to a
positive integer $k\leq n$ and the set

$\\{\nu\\}_{k}=\\{(\nu_{0}=-\sum_{i=1}^{k-1}\nu_{i}-t,\nu_{1},\dots,\nu_{k-1})\in\mathbb{Z}\times\mathbb{N}^{k-1}\;|\;t\in\mathbb{N}\\}$

there are several further quantities: Apart from

(4.23)
$\begin{split}&S^{(k)}_{r}(\nu)=\sum_{i=0}^{r}\nu_{i}=-t-\sum_{i=r+1}^{k-1}\nu_{i},\quad
0\leq r\leq k-1,\\\ &o(\nu)_{k}=\\{0\leq r\leq k-1\;|\;S_{r}^{(k)}(\nu)\equiv
1\bmod{2}\\},\\\ &e(\nu)_{k}=\\{0\leq r\leq k-1\;|\;S_{r}^{(k)}(\nu)\equiv
0\bmod{2}\\}\end{split}$

we will need the subsets

$\\{\nu_{i_{1},\dots,i_{d}}\\}_{k},\quad 0\leq i_{1}<i_{2}<\dots<i_{d}\leq
k-1$

of $\\{\nu\\}_{k}$, which consist of all tuples $\nu$, where the components
$\nu_{i_{j}},\;j=1,\dots,d$ run over all odd integers and the remaining
components over all even integers. To emphasize the dependency of $o(\nu)_{k}$
and $e(\nu)_{k}$ on the parity of the components of $\nu$ we will later write
$o(i_{1},\dots,i_{d})_{k}$ and $e(i_{1},\dots,i_{d})$ respectively if
$\nu\in\\{\nu_{i_{1},\dots,i_{d}}\\}_{k}$. Note that there is a misprint in
[St] regarding these sets. Incorrectly, $n$ is given as the upper bound for
the indices $i_{j}$. Whenever the indices $i_{j}$ appear in the original
paper, $n$ should be replaced by $k-1$ as the upper bound for $i_{j}$. Based
on the notation above, we use the abbreviation

(4.24) $\sum_{(\sigma,I,k)}=\sum_{\begin{subarray}{c}\sigma\in S_{n}\\\
\sigma^{2}=\operatorname{id}\end{subarray}}\sum_{I=I_{0}\cup\cdots\cup
I_{s}}\sum_{k=0}^{s+1},$

put

(4.25)
$\begin{split}\kappa_{p}(\sigma,I,k)&=2^{-c_{1}(\sigma)}(1-p^{-1})^{c_{1}(\sigma)+c_{2}(\sigma)}p^{-c_{2}(\sigma)}p^{-\tau(\\{I_{i}\\})-t(\sigma,\\{I_{i}\\})}\\\
&\times\frac{2^{c_{1}^{(k)}(\sigma)}2^{\sum_{r=0}^{k-1}c_{1,r}}(1-p^{-1})^{c_{1}^{(k)(\sigma)}}p^{-\sum_{r=k+1}^{s}n(r)}p^{\sum_{r=0}^{k-1}(c_{1,r}+2c_{2,r})}}{\prod_{r=k}^{s}(1-p^{-n(r)})}\end{split}$

and further introduce $\Delta_{p}(\sigma,q,k)=1$ for $k=0$ and

(4.26) $\begin{split}\Delta_{p}(\sigma,q,k)&=\sum_{0\leq i_{1}\leq\dots\leq
i_{d}\leq k-1}(-1)^{\sum_{r\in e(i_{1},\dots,i_{d})_{k}}c_{1,r}}\\\
&\times\left(\frac{-1}{p}\right)^{\frac{(m-2)}{2}\sum_{r\in
o(i_{1},\dots,i_{d})_{k}}c_{1,r}}\left(\frac{2}{p}\right)^{\sum_{r\in
e(i_{1},\dots,i_{d})_{k}}c_{1,r}}\\\ &\times p^{\sum_{j=1}^{d}2A_{i_{j}}({\bf
I_{r}})s-B_{i_{j}}({\bf I_{r}})},\end{split}$

if $L^{\prime}/L\cong(\mathbb{Z}/p\mathbb{Z})^{2}$ and zero if
$L^{\prime}/L\cong\mathbb{Z}/p\mathbb{Z}$ for $k\geq 1$. Here the sum
$\sum_{0\leq i_{1}\leq\dots\leq i_{d}\leq k-1}$ runs over all subsets of
$\\{0,\dots,k-1\\}$ indicating the positions of the odd components of $\nu$.
This includes the empty set, which covers the case that all components of
$\nu$ are even. Note that the parameter $k$ in the symbol
$\Delta_{p}(\sigma,q,k)$ in [St] is missing, which could give the impression
that it does not depend on this parameter. This is reason why we add it here.

The primes $p$ not dividing $|L^{\prime}/L|$ are collected in the set
$P=\\{q\text{ prime }\;|\;q\nmid|L^{\prime}/L|\\}$. For each such prime we put

(4.27)
$\begin{split}&a_{n,p}(s)=L_{p}(s+\rho_{n}-n,\chi_{V,p})\prod_{k=1}^{[\frac{n}{2}]}\zeta_{p}(2s-n+2k),\\\
&b_{n,p}(s)=L_{p}(s+\rho_{n},\chi_{V,p})\prod_{k=1}^{[\frac{n}{2}]}\zeta_{p}(2s+n-2k+1),\\\
\end{split}$

and form the products

(4.28) $\begin{split}&a_{n}^{P}(s)=\prod_{p\in P}a_{n,p}(s),\quad
b_{n}^{P}(s)=\prod_{p\in P}b_{n,p}(s).\end{split}$

Here $L_{p}(s,\chi_{V,p}))$ means the local $L$-function given by
$(1-\chi_{V,p}(p)p^{-s})$. The constant $\xi(s,n,k,P)$ then gathers the
contribution of the places belonging to the primes $p\in P$ and the
archimedian place:

(4.29)
$\begin{split}&\xi(s,n,k,P)=\frac{(-1)^{k/2}2^{(1-s)n}\pi^{\frac{n(n+1)}{2}}}{|L^{\prime}/L|^{n/2}}\frac{\Gamma_{n}(s)}{\Gamma_{n}(\alpha)\Gamma_{n}(\beta)}\frac{a_{n}^{P}(s)}{b_{n}^{P}(s)},\end{split}$

where $\alpha=\frac{1}{2}(s+\rho_{n}+l)$ and
$\beta=\frac{1}{2}(s+\rho_{n}-l)$. We are now ready to state the analytic
properties of $E_{l,0}^{n}$.

######  Theorem 4.5.

Let $E_{l,0}^{n}(\tau,s)$ be the Eisenstein series as in Definition 4.3 with
respect to $\rho_{L,n}$. Then $E_{l,0}^{n}(\tau,s)$ has a meromorphic
continuation in $s$ to the whole complex plane. If $L^{\prime}/L$ is
anisotropic and $|L^{\prime}/L|$ odd, it satisfies the functional equation

(4.30)
$\begin{split}E_{l,0}^{n}(\tau,s-\frac{l}{2})&=\xi(2s-\rho_{n},n,l,P)\prod_{p\mid|L^{\prime}/L|}\sum_{(\sigma,I,k)}\kappa_{p}(\sigma,I,k)\Delta_{p}(\sigma,q,k)\\\
&\times\prod_{j=1}^{k}(D_{p,j}(2s-\rho_{n})-1)E_{l,0}^{n}(\tau,\rho_{n}-s-\frac{l}{2}).\end{split}$

In this formula we interpret any occurring sum or product with lower index
bigger than the upper index to be zero or one, respectively.

As a corollary we phrase now the corresponding theorem for the case $n=2$. To
this end, we introduce some quantities, which comprise the contribution of the
primes $p$ dividing $|L^{\prime}/L|$: We write
$S_{2}=\\{\operatorname{id},\sigma\\}$ and use the symbol $I_{0}$ in two ways.
On the one hand, it means $I_{0}=\\{1,2\\}$ if the partition of $I$ consists
only of $I_{0}$. On the other hand, it means $I_{0}=\\{1\\}$ if the partition
of $I$ is of the form $I=I_{0}\cup I_{1}=\\{1\\}\cup\\{2\\}$. Further, by
$(L^{\prime}/L)_{p}$ we mean the $p$-component of $L^{\prime}/L$. For $\tau\in
S_{2}$, any ($\tau$-stable) decomposition $I^{\prime}$ of $I=\\{1,2\\}$ and
$k\in\\{0,1,2\\}$ we give explicit expressions for

(4.31)
$K_{p}(\tau,I^{\prime},k)=\kappa_{p}(\tau,I^{\prime},k)\Delta_{p}(\tau,q,k)\prod_{j=1}^{k}(D_{p,j}(2s-\frac{3}{2})-1):$
(4.32)
$\begin{split}&K_{p}(\operatorname{id},I_{0},0)=\frac{(1-p^{-1})^{4}}{1-p^{-3}},\\\
&K_{p}(\operatorname{id},I_{0}\cup
I_{1},0)=\frac{p^{-1}(1-p^{-1})^{3}}{1-p^{-3}},\\\
&K_{p}(\sigma,I_{0},0)=\frac{p^{-2}(1-p^{-1})^{2}}{1-p^{-3}}.\end{split}$

For $k=1$ and $k=2$ we have to distinguish the cases
$(L^{\prime}/L)_{p}\cong(\mathbb{Z}/p\mathbb{Z})^{2}$ and
$(L^{\prime}/L)_{p}\cong\mathbb{Z}/p\mathbb{Z}$. In the latter case we have
$K_{p}(\tau,I^{\prime},k)=0$ for all listed quantities in (4.33). Therefore,
the following identities hold only for
$(L^{\prime}/L)_{p}\cong(\mathbb{Z}/p\mathbb{Z})^{2}$.

(4.33)
$\begin{split}&K_{p}(\operatorname{id},I_{0},1)=2(1-p^{-1})^{4}(\zeta_{p}(8s+6)-1),\\\
&K_{p}(\sigma,I_{0},1)=2(1-p^{-1})^{2}(\zeta_{p}(16s-6)-1),\\\
&K_{p}(\operatorname{id},I_{0}\cup
I_{1},1)=p(1-p^{-1})^{2}((-1)\left(\frac{2}{p}\right)+\left(\frac{-1}{p}\right)^{(m-2)/2})(\zeta_{p}(8s-6)-1),\\\
&K_{p}(\operatorname{id},I_{0}\cup I_{1},2)=\\\
&4p^{2}(1-p^{-1})^{3}(1+(-1)\left(\frac{-1}{p}\right)^{(m-2)/2}\left(\frac{2}{p}\right)p^{s+\frac{m}{2}-2})(\zeta_{p}(4s-4)-1)(\zeta_{p}(8s-6)-1).\end{split}$

######  Corollary 4.6.

Let $E_{l,0}^{2}(\tau,s)$ be the Eisenstein series as in Definition 4.3 with
respect to $\rho_{L,2}$. Then $E_{l,0}^{2}(\tau,s)$ has a meromorphic
continuation in $s$ to the whole complex plane. If $L^{\prime}/L$ is
anisotropic and $|L^{\prime}/L|$ odd, it satisfies the functional equation

(4.34)
$E_{l,0}^{2}(\tau,s-\frac{l}{2})=\xi(2s-\frac{3}{2},2.l,P)\prod_{p\mid|L^{\prime}/L|}\left(C_{p}(\operatorname{id},I_{0})+C_{p}(\operatorname{id},I_{0}\cup
I_{1})+C_{p}(\sigma,I_{0})\right)E_{l,0}^{2}(\tau,\frac{3}{2}-s-\frac{l}{2}),$

Here

$\xi(2-\frac{3}{2},2.l,P)=\frac{(-1)^{l/2}4^{1-s}\pi^{3}}{|L^{\prime}/L|}\frac{\Gamma_{2}(s)}{\Gamma_{2}(\alpha)\Gamma_{2}(\beta)}\frac{a_{2}^{P}(s)}{b_{2}^{P}(s)}$

with the same meaning as above and

$\displaystyle
C_{p}(\operatorname{id},I_{0})=K_{p}(\operatorname{id},I_{0},0)+K_{p}(\operatorname{id},I_{0},1),$
$\displaystyle C_{p}(\operatorname{id},I_{0}\cup
I_{1})=K_{p}(\operatorname{id},I_{0}\cup
I_{1},0)+K_{p}(\operatorname{id},I_{0}\cup
I_{1},1)+K_{p}(\operatorname{id},I_{0}\cup I_{1},2),$ $\displaystyle
C_{p}(\sigma,I_{0})=K_{p}(\sigma,I_{0},0)+K_{p}(\sigma,I_{0},1).$

###### Proof.

As indicated before, for the statement of Corollary 4.6 we have to calculate
$K_{p}(\tau,I^{\prime},k)$ in (4.31) for all $\tau\in S_{2}$, any
$\tau$-stable partition $I^{\prime}$ of $I=\\{1,2\\}$ and all
$k\in\\{0,1,2\\}$. Since $\Delta_{p}(\tau,q,0)=1$ and the product
$\prod_{j=1}^{0}(D_{p,j}(2s-\frac{3}{2})-1)$ is empty and thus equal to 1, we
have

$K_{p}(\tau,I^{\prime},0)=\kappa_{p}(\tau,I^{\prime},0).$

By the definition of $\Delta_{p}(\tau,q,0)$, this identity holds for both
possible cases of $(L^{\prime}/L)_{p}$. The expressions for
$K_{p}(\tau,I^{\prime},0)$ in (4.32) can be verified by a direct but tedious
calculation. Again, the definition of $\Delta_{p}(\tau,q,k)$ for $k=1,2$
yields $K_{p}(\tau,I^{\prime},k)=0$ for all $\tau\in S_{2}$, all partitions
$I^{\prime}$ of $I$ and $(L^{\prime}/L)_{p}\cong\mathbb{Z}/p\mathbb{Z}$. It
remains to confirm the equations in (4.33). This is also a straightforward but
tedious calculation. For all these computations we have to take the convention
in Theorem 4.5 into account. ∎

######  Remark 4.7.

Note that by Remark 3.18 in [St] the functional equation (4.34) remains valid
for the Eisenstein series $E_{l,0}^{2*}$.

Since the case $n=2$ lies mainly in our interest, we stick with it for the
rest of this section and introduce now a vector valued analogue of the
Poincaré series in [Bo1], Section 2.2, and a related Poincaré series. To this
end, let $\mathcal{H}=\mathbb{H}\cup\mathbb{H}^{-}$, where $\mathbb{H}^{-}$ is
the lower complex half-plane.

######  Definition 4.8.

Let $d\in\mathbb{Z}$ be a positive integer, $D=\left(\begin{smallmatrix}d&0\\\
0&d^{-1}\end{smallmatrix}\right)\in\operatorname{GL}_{2}(\mathbb{Q})^{+}$,
$l\in\mathbb{Z}$ and $s\in\mathbb{C}$ with
$\operatorname{Re}(s)>\frac{3-k}{2}$.

  1. i)

$P_{l}:\mathbb{H}\times\mathbb{H}\rightarrow\mathbb{C}[L^{\prime}/L]\otimes\mathbb{C}[L^{\prime}/L]$
is defined by

(4.35) $P_{l}(\tau,\zeta,s)=\sum_{\lambda\in
L^{\prime}/L}\sum_{\gamma\in\Gamma_{1}}\det\left(\operatorname{Im}\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right)\right)^{s}(\tau+\zeta)^{-l}|\tau+\zeta|^{-2s}\mathfrak{e}_{\lambda}\otimes\mathfrak{e}_{\lambda}\mid_{l,L}\gamma_{(\tau,1)},$

where the subscript $(\tau,1)$ means that $\gamma$ acts on the first component
and with respect to the variable $\tau$. We further define a variant of
$P_{l}$, which occurs in the course of the present paper. It already appeared
in slightly less general scalar valued form in Poincaré’s work (see [Le]).

$P_{l}^{+}:\mathcal{H}\times\mathcal{H}\rightarrow\mathbb{C}[L^{\prime}/L]\otimes\mathbb{C}[L^{\prime}/L]$
with

(4.36) $P^{+}_{l}(\tau,\zeta,s)=\sum_{\lambda\in
L^{\prime}/L}\sum_{\gamma\in\Gamma_{1}}\det\left(\operatorname{Im}\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right)\right)^{s}(\tau-\zeta)^{-l}|\tau-\zeta|^{-2s}\mathfrak{e}_{\lambda}\otimes\mathfrak{e}_{\lambda}\mid_{l,L}\gamma_{(\tau,1)}.$

  2. ii)

Associated to $P_{l}$ and its variant we define

(4.37)
$\begin{split}\mathscr{P}_{l}:\mathbb{H}\times\mathbb{H}\rightarrow\mathbb{C}[L^{\prime}/L]\otimes\mathbb{C}[L^{\prime}/L],\quad\mathscr{P}_{l}(\tau,\zeta,D,s)=\sum_{M\in\Gamma_{1}\backslash\Gamma_{1}D\Gamma_{1}}P_{l}(\tau,\zeta,s)\mid_{l,L}^{*}M_{(\zeta,2)},\end{split}$

where the subscript $(\zeta,2)$ indicates that $M$ acts on the second
component and with respect to the variable $\zeta$ and $\rho^{*}_{L,1}$.
Corresponding to $P_{l}^{+}$ we put

(4.38)
$\begin{split}\mathscr{P}^{+}_{l}:\mathcal{H}\times\mathcal{H}\rightarrow\mathbb{C}[L^{\prime}/L]\otimes\mathbb{C}[L^{\prime}/L],\quad\mathscr{P}^{+}_{l}(\tau,\zeta,D,s)=\sum_{M\in\Gamma_{1}\backslash\Gamma_{1}D\Gamma_{1}}P^{+}_{l}(\tau,\zeta,s)\mid_{l,L}^{*}M_{(\zeta,2)}.\end{split}$

######  Remark 4.9.

  1. i)

The series $P_{l}$ can be understood as vector valued version of the Poincaré
series $P_{r}^{k}(Z,W,g,s)$ introduced in [Bo1], Section 2.2. It can also be
interpreted as the non-holomorphic variant of the series $h_{\beta,n}$ defined
in [RS], Def. 5.1. In terms of the coefficients of $\rho_{L,1}$ the component
function $P_{l}(\tau,\zeta,s)_{\mu}$ can be written as

(4.39) $\det(\operatorname{Im}\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right))^{s}\sum_{\gamma\in\Gamma_{1}}\langle\rho_{L,1}^{-1}(\gamma)\mathfrak{e}_{\lambda},\mathfrak{e}_{\mu}\rangle
j(\gamma,\tau)^{-l}|j(\gamma,\tau)|^{-2s}(\gamma\tau+\zeta)^{-l}|\gamma\tau+\zeta|^{-2s}.$

Since $\rho_{L,1}$ factors through the finite group
$\operatorname{SL}_{2}(\mathbb{Z}/N\mathbb{Z})$, the coefficient
$\rho_{\lambda,\mu}^{-1}(\gamma)$ is bounded on $\Gamma_{1}$. Thus, for
matters of convergence of (4.39) it suffices to study the scalar valued
Poincaré series

$\det(\operatorname{Im}\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right))^{s}\sum_{\gamma\in\Gamma_{1}}j(\gamma,\tau)^{-l}|j(\gamma,\tau)|^{-2s}(\gamma\tau+\zeta)^{-l}|\gamma\tau+\zeta|^{-2s},$

which is exactly the series $P_{1}^{l}(\tau,\zeta,g,s)$ in [Bo1], Section 2.2,
for $g\in\Gamma_{1}$. It follows from [Bo1], Sec. 2.2, that each component
function of $P_{l}(\tau,\zeta,s)_{\mu}$ is absolutely and uniformly convergent
for all $s\in\mathbb{C}$ with $\operatorname{Re}(s)>\frac{3-l}{2}$ on any
product $V_{1}(\delta)\times V_{1}(\delta)$, where

$V_{1}(\delta)=\\{z=x+iy\in\mathbb{H}\;|\;y\geq\delta\text{ and
}x^{2}\leq\frac{1}{\delta}\\}$

with $\delta>0$, and thereby represents a real analytic function on
$\mathbb{H}^{2}$. Thus, the usual argument shows that $P_{l}$ transforms under
the action of $\Gamma_{1}$ with respect to the variable $\tau$ like a vector
valued modular form of weight $l$ and type $\rho_{L,1}$.

Since the sum $\sum_{M\in\Gamma_{1}(D)\backslash\Gamma_{1}}$ is finite, the
same holds for the Poincaré series $\mathscr{P}_{l}$. In terms of the function
$\varphi_{l,s}$ (see (2.9)), we may write $\mathscr{P}_{l}$ in the following
more explicit form

(4.40) $\begin{split}&\sum_{\lambda\in
L^{\prime}/L}\sum_{M\in\Gamma_{1}(D)\backslash\Gamma_{1}}\\\
&\times\sum_{\gamma\in\Gamma_{1}}\det(\operatorname{Im}\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right))^{s}\varphi_{l,s}(j(\gamma,\tau)j(M,\zeta)(\gamma\tau+M\zeta))\rho_{L,1}^{-1}(\gamma)\mathfrak{e}_{\lambda}\otimes\rho_{L,1}^{*-1}(M)\mathfrak{e}_{\lambda}.\end{split}$

  2. ii)

The series $P_{l}^{+}$ has similar properties as $P_{l}$. More specifically,
it converges absolutely and uniformly on compact subsets of
$\mathcal{H}\times\mathcal{H}$. It is thereby real-analytic on
$\mathcal{H}\times\mathcal{H}$, leaving out possible poles. Indeed, if
$\tau,\zeta$ are both in the upper or lower half-plane, $P_{l}^{+}$ has a pole
at $\tau=\zeta$. In all other cases no poles can occur. Also, the usual
argument shows that $P_{l}^{+}$ transforms with respect to $\tau\in\mathbb{H}$
like a vector-valued modular form.

If $\tau$ is an element of $\mathbb{H}$, the before mentioned statements can
be deduced from Theorem 3A and Section 3C in Chapter V of [Le].

If $\tau\in\mathbb{H}^{-}$ and $\zeta\in\mathbb{H}$, $P_{l}^{+}$ inherits the
analytic properties of $P_{l}$. To be specific, by employing the same argument
as in i), we may estimate each of the component functions of $P_{l}^{+}$ by

$\displaystyle\sum_{\gamma\in\Gamma_{1}}|j(\gamma,\tau)|^{-2s-l}|(\gamma\tau-\zeta)|^{-l-2s}$
$\displaystyle=\sum_{\gamma\in\Gamma_{1}}|j(\gamma,-z)|^{-2s-l}|(\gamma(-z)-\zeta)|^{-l-2s}$
$\displaystyle=\sum_{\gamma\in\Gamma_{1}}|j(\widetilde{\gamma},z)|^{-2s-l}|(\widetilde{\gamma}z+\zeta)|^{-l-2s},$

where we replaced $\tau\in\mathbb{H}^{-}$ with $-z$, $z\in\mathbb{H}$ and used
the fact that $\gamma(-z)=-(\widetilde{\gamma}z)$. The latter sum is up to a
constant nothing else but an estimate of the series (4.39). If $\tau,\zeta$
are both in $\mathbb{H}^{-}$, the same reasoning as before shows that
$P_{l}^{+}$ is bounded by

$\sum_{\gamma\in\Gamma_{1}}|j(\widetilde{\gamma},z)|^{-2s-l}|(\widetilde{\gamma}z+\zeta)|^{-l-2s}$

with $z\in\mathbb{H}$ and $\zeta\in\mathbb{H}^{-}$. This is up to a constant
an estimate of $P^{+}_{l|\mathbb{H}\times\mathbb{H}^{-}}$. Again, in all
considered cases the Poincaré series $\mathscr{P}_{l}^{+}$ shares the same
properties as $P_{l}^{+}$.

The following theorem provides some further properties of $P_{l}^{+}$ and
$\mathscr{P}_{l}^{+}$, which will be vital later on.

######  Theorem 4.10.

Let $l\in\mathbb{Z}$ and $s\in\mathbb{C}$ with
$\operatorname{Re}(s)>\frac{3-l}{2}$, $f\in S_{l}(\rho_{L,1})$ and

(4.41)
$C(l,s)=(-1)^{\frac{l}{2}}2^{2-2s-k}\pi\frac{\Gamma(s+1)}{\Gamma(s+2)}.$

  1. i)

Then we have

(4.42) $\sum_{\lambda\in
L^{\prime}/L}\left(\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},P^{+}_{l}(\tau,\overline{\zeta},\overline{s})\rangle_{2}\operatorname{Im}(\tau)^{l}d\mu(\tau)\right)\mathfrak{e}_{\lambda}=(-1)^{-s}C(l,s)f(\zeta)$

for all $\zeta\in\mathbb{H}$.

  2. ii)

Let $D=\left(\begin{smallmatrix}d&0\\\
0&d^{-1}\end{smallmatrix}\right)\in\operatorname{GL}_{2}(\mathbb{Q})^{+}$ and
$T(D)$ the Hecke operator defined in Subsection 4.1.1. Then

(4.43) $\sum_{\lambda\in
L^{\prime}/L}\left(\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},\mathscr{P}^{+}_{l}(\tau,\overline{\zeta},D,\overline{s})\rangle_{2}\operatorname{Im}(\tau)^{l}d\mu(\tau)\right)\mathfrak{e}_{\lambda}=(-1)^{-s}C(l,s)(f\mid_{l,L}T(D))(\zeta)$

for all $\zeta\in\mathbb{H}$.

###### Proof.

$i)$: It can easily be confirmed by direct calculation using the relation
(3.11) that

(4.44) $\begin{split}&\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},P_{l}(\tau,\overline{\zeta},s)\rangle_{2}=\\\
&\langle f(\tau),\sum_{\mu\in
L^{\prime}/L}\sum_{\gamma\in\Gamma_{1}}\det\left(\operatorname{Im}\left(\begin{smallmatrix}\tau&0\\\
0&\overline{\zeta}\end{smallmatrix}\right)\right)^{\overline{s}}(\tau-\overline{\zeta})^{-l}|\tau-\overline{\zeta}|^{-2\overline{s}}\mathfrak{e}_{\mu}\mid_{l,L}\gamma\rangle_{1}\langle\mathfrak{e}_{\lambda}\otimes\mathfrak{e}_{\mu}\rangle_{1}\end{split}$

The last expression shows that the integral in (4.42) as Petersson scalar
product $(\cdot,\cdot)_{1}$ is well defined since $f$ is a cusp form and
$P^{+}_{l}$ transforms like a modular form in $M_{l}(\rho_{L,1})$ with respect
to $\tau$. Using (4.44) the left-hand side of (4.42) becomes

(4.45) $\sum_{\lambda\in
L^{\prime}/L}\left(\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau),\sum_{\gamma\in\Gamma_{1}}\det\left(\operatorname{Im}\left(\begin{smallmatrix}\tau&0\\\
0&\overline{\zeta}\end{smallmatrix}\right)\right)^{\overline{s}}(\tau-\overline{\zeta})^{-l}|\tau-\overline{\zeta}|^{-2\overline{s}}\mathfrak{e}_{\lambda}\mid_{l,L}\gamma\rangle_{1}\operatorname{Im}(\tau)^{l}d\mu(\tau)\right)\mathfrak{e}_{\lambda},$

which can be seen as non-holomorphic version of the kernel operator $K_{1}$
defined in [RS], Definition 5.3. To prove the stated assertion, we may use the
same unfolding trick as laid out in the proof of Theorem 5.6 in [RS]. In doing
so, we obtain

(4.46)
$\begin{split}&(-1)^{-s}\operatorname{Im}(\zeta)^{s}\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau),\sum_{\gamma\in\Gamma_{1}}\varphi_{l,\overline{s}}(j(\gamma,\tau)(\tau-\overline{\zeta}))\rho_{L,1}^{-1}(\gamma)\mathfrak{e}_{\lambda}\rangle_{1}\operatorname{Im}(\tau)^{l+s}d\mu(\tau)\\\
&=(-1)^{-s}\operatorname{Im}(\zeta)^{s}\int_{\mathbb{H}}\langle
f(\tau),(\tau-\overline{\zeta})^{-l}|\tau-\overline{\zeta}|^{-2\overline{s}}\mathfrak{e}_{\lambda}\rangle_{1}\operatorname{Im}(\tau)^{l+s}d\mu(\tau).\end{split}$

Since each component of $f$ is a cusp form for $\Gamma(N)$, the latter
integral in (4.46) can be evaluated explicitly by means of the proposition in
[Bo1], Section 2.2:

(4.47)
$(-1)^{-s}\operatorname{Im}(\zeta)^{s}\int_{\mathbb{H}}f_{\lambda}(\tau)(\overline{\tau}-\zeta)^{-l}|\tau-\overline{\zeta}|^{-2s}\operatorname{Im}(\tau)^{l+s}d\mu(\tau)=(-1)^{-s}C(l,s)f_{\lambda}(\zeta)$

for any $\zeta\in\mathbb{H}$.

$ii)$: Similar to $i)$ we have

(4.48) $\begin{split}&\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},\mathscr{P}^{+}_{l}(\tau,\overline{\zeta},D,\overline{s})\rangle_{2}=\sum_{M\in\Gamma_{1}\backslash\Gamma_{1}D\Gamma_{1}}\sum_{\mu\in
L^{\prime}/L}\sum_{\gamma\in\Gamma_{1}}\times\\\ &\langle
f(\tau),\det(\operatorname{Im}\left(\begin{smallmatrix}\tau&0\\\
0&\overline{\zeta}\end{smallmatrix}\right))^{\overline{s}}\varphi_{l,\overline{s}}(j(\gamma,\tau)j(M,\overline{\zeta})(\gamma\tau-M\overline{\zeta}))\rho_{L,1}^{-1}(\gamma)\mathfrak{e}_{\mu}\rangle_{1}\langle\mathfrak{e}_{\lambda},\rho_{L,1}^{*-1}(M)\mathfrak{e}_{\mu}\rangle_{1}.\end{split}$

Thus, as before,

(4.49) $\begin{split}&\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},\mathscr{P}^{+}_{l}(\tau,\overline{\zeta},D,\overline{s})\rangle_{2}\operatorname{Im}(\tau)^{l}d\mu(\tau)\\\
&=(-1)^{-s}\sum_{M\in\Gamma_{1}(D)\backslash\Gamma_{1}}\operatorname{Im}(M\zeta)^{s}j(M,\zeta)^{-l}\sum_{\mu\in
L^{\prime}/L}\langle\mathfrak{e}_{\lambda},\rho_{L,1}^{*-1}(M)\mathfrak{e}_{\mu}\rangle_{1}\\\
&\times\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau),\sum_{\gamma\in\Gamma_{1}}\varphi_{l,\overline{s}}(j(\gamma,\tau)(\gamma\tau-M\overline{\zeta}))\rho_{L,1}^{-1}(\gamma)\mathfrak{e}_{\mu}\rangle_{1}\operatorname{Im}(\tau)^{l+s}d\mu(\tau).\end{split}$

Unfolding the latter integral and applying (4.47) subsequently, yields

(4.50) $\begin{split}&\sum_{\lambda\in
L^{\prime}/L}\left(\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},\mathscr{P}^{+}_{l}(\tau,\overline{\zeta},D,\overline{s})\rangle_{2}\operatorname{Im}(\tau)^{l}d\mu(\tau)\right)\mathfrak{e}_{\lambda}\\\
&=(-1)^{-s}C(l,s)\sum_{M\in\Gamma_{1}\backslash\Gamma_{1}D\Gamma_{1}}\sum_{\mu\in
L^{\prime}/L}j(M,\zeta)^{-l}f_{\mu}(M\zeta)\sum_{\lambda\in
L^{\prime}/L}\langle\mathfrak{e}_{\lambda},\rho^{*-1}_{L,1}(M)\mathfrak{e}_{\mu}\rangle_{1}\mathfrak{e}_{\lambda}.\\\
\end{split}$

The right-hand side of (4.50) is nothing else than
$(-1)^{-s}C(l,s)(f\mid_{l,L}T(D))(\zeta)$.

∎

##  5\. Garrett-Böcherer decomposition of vector-valued Siegel Eisenstein
series

In this section we present a decomposition of the Siegel Eisenstein series
$E^{2}_{l,0}$ in terms of $E^{1}_{l,0}$ and the Poincaré series
$\mathscr{P}^{+}_{l}$. Such a decomposition was developed by Garrett ([Ga])
and Böcherer ([Bo1]) for scalar valued holomorphic and non-holomorphic Siegel
Eisenstein series, respectively. It is based on an explicit system of
representatives of
$\Gamma_{n+m,0}\backslash\Gamma_{n+m}/l_{n+m}(\Gamma_{m}\times\Gamma_{n})$
determined by Garrett. Since we are dealing with the same groups for $n=m=1$,
we may use Garrett’s results. They become considerably easier in this special
case, so we summarize them here in the following two theorems. Subsequently,
we will use these theorems to state and prove our vector valued version of the
Garrett-Böcherer decomposition.

######  Theorem 5.1.

A complete set of representatives for the double coset
$\Gamma_{2,0}\backslash\Gamma_{2}/l_{1,1}(\Gamma_{1}\times\Gamma_{1})$ is
given by

$\bigcup_{r=0}^{1}\mathscr{M}_{r},$

where

(5.1) $\mathscr{M}_{0}=\\{U_{2}(0_{2})\\}\text{ and
}\mathscr{M}_{1}=\left\\{U_{2}\left(\begin{smallmatrix}0&d\\\
d&0\end{smallmatrix}\right)\;|\;d\in\mathbb{N}\right\\}.$

Let $g_{d}\in\mathscr{M}_{i},\;d\in\mathbb{N}_{0}$. Tailored to the case
$n=m=1$ the theorem in [Ga], $\S$3, specifies a complete set of left coset
representatives of
$\Gamma_{2,0}\backslash\Gamma_{2,0}\;g_{d}\;l_{1,1}(\Gamma_{1}\times\Gamma_{1}).$

######  Theorem 5.2.

Let $d\in\mathbb{N}_{0},\;r\in\\{0,1\\}$ and $g_{d}\in\mathscr{M}_{r}$. Then a
complete set of representatives of the $\Gamma_{2,0}$-left cosets in
$\Gamma_{2,0}\;g_{d}\;l_{1,1}(\Gamma_{1}\times\Gamma_{1})$ is given by

(5.2) $\left\\{g_{d}\;(\gamma
g)^{\uparrow}(Mh)^{\downarrow}\;|\;\gamma\in\Gamma_{r},\;g\in\Gamma_{1,r}\backslash\Gamma_{1}\;,M\in\Gamma_{r}\left(\begin{smallmatrix}0&d^{-1}\\\
d&0\end{smallmatrix}\right)\backslash\Gamma_{r},\;h\in\Gamma_{1,r}\backslash\Gamma_{1}\right\\}.$

Note that we use the convention that $\Gamma_{0}=\\{1_{2}\\}$ and accordingly
$\Gamma_{0}\left(\begin{smallmatrix}0&d\\\
d^{-1}&0\end{smallmatrix}\right)\backslash\Gamma_{0}=\\{1_{2}\\}$. We also
have that $\Gamma_{1,1}\backslash\Gamma_{1}=\\{1_{2}\\}$.

Based on these theorems a vector valued variant of the pullback formula in
[Ga], $\S 5$, and [Bo1], involving the Siegel Eisenstein series (4.17), can be
given.

######  Theorem 5.3.

Let $d\in\mathbb{Z}$ be a positive integer, $D=\left(\begin{smallmatrix}d&0\\\
0&d^{-1}\end{smallmatrix}\right)$ and $E_{k,0}^{2*}$ be defined as in
Definition 4.3. Then for all $\tau,\zeta\in\mathbb{H}$

(5.3) $\begin{split}&E_{l,0}^{2*}(\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right),s)=\\\ &E_{l,0}^{1*}(\tau,s)\otimes
E_{l,0}^{1*}(\zeta,s)+\frac{e(\operatorname{sig}(L)/8)}{|L^{\prime}/L|^{1/2}}\sum_{d\geq
1}\frac{g_{d}(L)}{g(L)}d^{-l-2s}\mathscr{P}^{+}_{l}(-\tau,\zeta,D,s).\end{split}$

######  Remark 5.4.

Since $E_{l,0}^{2}$ is absolutely and uniformly convergent on compact subsets
of $\mathbb{H}^{2}$, the same holds for any subseries occurring on the right-
hand side of (5.3). In particular,
$\sum_{d\in\mathbb{N}}\frac{g_{d}(L)}{g(L)}d^{-l-2s}\mathscr{P}^{+}_{l}(-\tau,\zeta,D,s)$
is normally convergent for $\operatorname{Re}(s)>\frac{3-l}{2}$.

###### Proof.

The proof is an adaption of the one given in [Ga]. All steps concerning the
factor of automorphy $j_{2}$ carry over immediately. The parts in
$E_{l,0}^{2*}$ coming from the Weil representation $\rho_{L,2}^{*}$ have to be
treated separately.

According to Theorem 5.1 and Theorem 5.2 we have

(5.4) $\begin{split}E_{l,0}^{2*}(\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right),s)&=\sum_{g\in\Gamma_{1,0}\backslash\Gamma_{1}}\sum_{h\in\Gamma_{1,0}\backslash\Gamma_{1}}\det\left(\operatorname{Im}\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right)\right)^{s}\mathfrak{e}_{0}\mid^{2*}_{l,L}g^{\uparrow}h^{\downarrow}\\\
&+\sum_{d\in\mathbb{N}}\sum_{M\in\Gamma_{1}(d)\backslash\Gamma_{1}}\sum_{\gamma\in\Gamma_{1}}\det\left(\operatorname{Im}\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right)\right)^{s}\mathfrak{e}_{0}\mid^{2*}_{l,L}U_{2}\left(\begin{smallmatrix}0&d\\\
d&0\end{smallmatrix}\right)\gamma^{\uparrow}M^{\downarrow}.\end{split}$

Note that we have replaced the left cosets over
$\Gamma_{1}\left(\begin{smallmatrix}0&d^{-1}\\\
d&0\end{smallmatrix}\right)\backslash\Gamma_{1}$ with
$\Gamma_{1}(d)\backslash\Gamma_{1}$. This amounts to replace $M$ with
$\ell(M)$ (see Section 2 for the notation), which only alters the order of
summation but not the whole expression. We consider both of the summands above
separately.

Starting with first summand, note that a by direct computation we find

(5.5) $j_{2}(g^{\uparrow}h^{\downarrow},\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right))=j(g,\tau)j(h,\zeta).$

For the part involving the Weil representation $\rho_{L,2}^{*}$ we exploit
(3.18) and (3.19) and infer that

(5.6)
$\begin{split}\rho_{L,2}^{*-1}(g^{\uparrow}h^{\downarrow})\mathfrak{e}_{(0,0)}&=\rho_{L,2}^{*-1}(g^{\uparrow})(\mathfrak{e}_{0}\otimes\rho_{L,1}^{*-1}(h)\mathfrak{e}_{0})\\\
&=\rho_{L,1}^{*-1}(g)\mathfrak{e}_{0}\otimes\rho_{L}^{*-1}(h)\mathfrak{e}_{0}.\end{split}$

If we insert the right-hand side of (5.5) and (5.6), we may write in terms of
(2.9)

$\displaystyle\sum_{g\in\Gamma_{\infty}\backslash\Gamma_{1}}\sum_{h\in\Gamma_{\infty}\backslash\Gamma_{1}}\det\left(\operatorname{Im}\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right)\right)^{s}\mathfrak{e}_{0}\otimes\mathfrak{e}_{0}\mid^{2*}_{l,L}g^{\uparrow}h^{\downarrow}$
$\displaystyle=\sum_{g\in\Gamma_{\infty}\backslash\Gamma_{1}}\sum_{h\in\Gamma_{\infty}\backslash\Gamma_{1}}\det\left(\operatorname{Im}\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right)\right)^{s}\varphi_{l,s}(j(g,\tau)j(h,\zeta))\rho_{L,1}^{*-1}(g)\mathfrak{e}_{0}\otimes\rho_{L}^{*-1}(h)\mathfrak{e}_{0}$
$\displaystyle=E_{l,0}^{1*}(\tau,s)\otimes E_{l,0}^{1*}(\zeta,s).$

For the second summand of (5.4), a straightforward calculation, using the
cocycle relation of $j_{2}$ twice, yields

$\displaystyle j_{2}\left((U_{2}\left(\begin{smallmatrix}0&d\\\
d&0\end{smallmatrix}\right)\gamma^{\uparrow}M^{\downarrow},\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right)\right)=\left(1-(\gamma\tau)\cdot
d^{2}(M\zeta)\right)j(\gamma,\tau)j(M,\zeta).$

On the other hand, by means of Lemma 3.4 and Lemma 3.5 we have

(5.7) $\begin{split}\rho_{L,2}^{*-1}(U_{2}\left(\begin{smallmatrix}0&d\\\
d&0\end{smallmatrix}\right)\gamma^{\uparrow}M^{\downarrow})\mathfrak{e}_{0}\otimes\mathfrak{e}_{0}&=\frac{1}{|L^{\prime}/L|}\sum_{\mu,\nu\in
L^{\prime}/L}e(-(\mu,\nu))\rho_{L,2}^{*-1}(M^{\downarrow})\rho_{L,2}^{*-1}(\gamma^{\uparrow})\mathfrak{e}_{d\mu}\otimes\mathfrak{e}_{\nu}\\\
&=\frac{1}{|L^{\prime}/L|}\sum_{\mu,\nu\in
L^{\prime}/L}e(-(\mu,\nu))\rho_{L,2}^{*-1}(M^{\downarrow})(\rho_{L,1}^{*-1}(\gamma)\mathfrak{e}_{d\mu}\otimes\mathfrak{e}_{\nu})\\\
&=\frac{1}{|L^{\prime}/L|}\sum_{\mu,\nu\in
L^{\prime}/L}e(-(\mu,\nu))\rho_{L,1}^{*-1}(\gamma)\mathfrak{e}_{d\mu}\otimes\rho_{L,1}^{*-1}(M)\mathfrak{e}_{\nu}.\end{split}$

The transformation $\gamma\mapsto S^{-1}\widetilde{\gamma}$ (see (3.8) for the
definition of $\widetilde{\gamma}$) leaves the sum over $\gamma\in\Gamma_{1}$
and therefore the latter summand of (5.4) invariant. The subsequent
calculations can be easily verified

$\displaystyle j_{2}\left((U_{2}\left(\begin{smallmatrix}0&d\\\
d&0\end{smallmatrix}\right)(S\widetilde{\gamma})^{\uparrow}M^{\downarrow},\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right)\right)$
$\displaystyle=\left(1-(S^{-1}\widetilde{\gamma}\tau)\cdot
d^{2}(M\zeta)\right)j(S^{-1}\gamma,\tau)j(M,\zeta)$
$\displaystyle=(\gamma(-\tau)-d^{2}M\zeta)j(\gamma,-\tau)j(M,\zeta)$
$\displaystyle=d(\gamma(-\tau)-(\left(\begin{smallmatrix}d&0\\\
0&d^{-1}\end{smallmatrix}\right)M)\zeta)j(\gamma,-\tau)j(\left(\begin{smallmatrix}d&0\\\
0&d^{-1}\end{smallmatrix}\right)M,\zeta).$

Taking (5.7) and (3.6) into account, the corresponding calculations on the
level of the Weil representation yield

$\displaystyle\rho_{L,2}^{*-1}(U_{2}\left(\begin{smallmatrix}0&d\\\
d&0\end{smallmatrix}\right)\gamma^{\uparrow}M^{\downarrow})\mathfrak{e}_{0}\otimes\mathfrak{e}_{0}$
$\displaystyle=\frac{1}{|L^{\prime}/L|}\sum_{\mu,\nu\in
L^{\prime}/L}e(-(\mu,\nu))\rho_{L,1}^{*-1}(\widetilde{\gamma})\rho_{L,1}^{*}(S)\mathfrak{e}_{d\mu}\otimes\rho^{*-1}_{L,1}(M)\mathfrak{e}_{\nu}$
$\displaystyle=\frac{e(\operatorname{sig}(L)/8)}{|L^{\prime}/L|^{3/2}}\sum_{\varrho,\nu\in
L^{\prime}/L}\sum_{\mu\in
L^{\prime}/L}e((\mu,d\varrho-\nu))\rho^{-1}_{L,1}(\gamma)\mathfrak{e}_{\varrho}\otimes\rho_{L,1}^{*-1}(M)\mathfrak{e}_{\nu}$
$\displaystyle=\frac{e(\operatorname{sig}(L)/8)}{|L^{\prime}/L|^{1/2}}\sum_{\varrho\in
L^{\prime}/L}\rho^{-1}_{L,1}(\gamma)\mathfrak{e}_{\varrho}\otimes\rho_{L,1}^{*-1}(M)\mathfrak{e}_{d\varrho}.$

For the second equation we have additionally used (3.9).

With (4.13) in mind we may then rewrite the last expression in the form

(5.8)
$\frac{e(\operatorname{sig}(L)/8)}{|L^{\prime}/L|^{1/2}}\frac{g_{d}(L)}{g(L)}\sum_{\varrho\in
L^{\prime}/L}\rho^{-1}_{L,1}(\gamma)\mathfrak{e}_{\varrho}\otimes\rho_{L,1}^{*-1}(\left(\begin{smallmatrix}d&0\\\
0&d^{-1}\end{smallmatrix}\right)M)\mathfrak{e}_{\varrho}.$

Putting together the rearrangements above, we obtain

$\displaystyle\sum_{d\in\mathbb{N}}\sum_{M\in\Gamma_{1}(d)\backslash\Gamma_{1}}\sum_{\gamma\in\Gamma_{1}}\det\left(\operatorname{Im}\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right)\right)^{s}\mathfrak{e}_{0}\mid^{2}_{l,L}U_{2}\left(\begin{smallmatrix}0&d\\\
d&0\end{smallmatrix}\right)\gamma^{\uparrow}M^{\downarrow}=$
$\displaystyle\sum_{d\in\mathbb{N}}\sum_{R\in\Gamma_{1}\backslash\Gamma_{1}\left(\begin{smallmatrix}d&0\\\
0&d^{-1}\end{smallmatrix}\right)\Gamma_{1}}\sum_{\gamma\in\Gamma_{1}}\det\left(\operatorname{Im}\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right)\right)^{s}d^{-l-2s}\varphi_{l,s}(j(\gamma,-\tau)j(R,\zeta)(\gamma(-\tau)-R\zeta))\times$
$\displaystyle\frac{e(\operatorname{sig}(L)/8)}{|L^{\prime}/L|^{1/2}}\frac{g_{d}(L)}{g(L)}\sum_{\varrho\in
L^{\prime}/L}\rho^{-1}_{L,1}(\gamma)\mathfrak{e}_{\varrho}\otimes\rho_{L,1}^{*-1}(R)\mathfrak{e}_{\varrho}.$

∎

##  6\. Standard zeta function of an eigenform

In [BS], p. 251, it was proposed to associate a zeta function

$\sum_{\stackrel{{\scriptstyle
d\in\mathbb{N}}}{{(d,N)=1}}}\lambda_{d}(f)d^{-s}$

to a common eigenform $f$ of all Hecke operators
$T\left(\begin{smallmatrix}d^{2}&0\\\ 0&1\end{smallmatrix}\right),\;(d,N)=1$,
where

$f\mid_{l,L}T\left(\begin{smallmatrix}d^{2}&0\\\
0&1\end{smallmatrix}\right)=\lambda_{d}(f)f.$

In this chapter we study the analytic properties of a slightly different zeta
function. Attached to an Eigenform $f$ of all Hecke operators
$T\left(\begin{smallmatrix}d^{2}&0\\\ 0&1\end{smallmatrix}\right)$, we define

(6.1) $Z(s,f)=\sum_{d\in\mathbb{N}}\lambda_{d}(f)d^{-s}.$

In consistency with [Bo1] the zeta function $Z(s,f)$ can be viewed as the
standard zeta function of $f$.

######  Remark 6.1.

  1. i)

As was already stated in [BS], $Z(s,f)$ converges for $\operatorname{Re}(s)$
sufficiently large. We obtain this result as a by-product of the subsequent
studies of the analytic properties of $Z(s,f)$.

  2. ii)

Under certain assumptions on the discriminant form one can prove the existence
of a common eigenform of all Hecke operators
$T\left(\begin{smallmatrix}d^{2}&0\\\
0&1\end{smallmatrix}\right),\;d\in\mathbb{N}$. This is possible if a
multiplicity one theorem for $S_{l}(\rho_{L,1})$ holds. In [We], Theorem 41,
conditions for the validity of such a theorem are stated. These conditions
depend heavily on the decomposition of the Weil representation $\rho_{L,1}$
into irreducible subrepresentations. The multiplicities of these
subrepresentations encode the dimension of the common eigenspace for a set of
eigenvalues $\lambda_{d}$ for all Hecke operators
$T\left(\begin{smallmatrix}d^{2}&0\\\ 0&1\end{smallmatrix}\right),\;(d,N)=1$.
If all occurring irreducible subrepresentations have multiplicity one, the
same holds for the before mentioned dimension of common eigenspaces. The
decomposition of $\rho_{L,1}$ into irreducible subrepresentations is well
known, see e.g. [No], [NW]. Among other things, it depends on the structure of
the discriminant form $L^{\prime}/L$. If for example each $p$-group of
$L^{\prime}/L$ consists of a single Jordan block of the form
$(\mathbb{Z}/p^{\lambda}\mathbb{Z},\frac{rx^{2}}{p^{\lambda}})$, $\rho_{L,1}$
decomposes into irreducible subrepresentations of multiplicity one.

###  6.1. Analytic properties of $Z(s,f)$

In this section we will prove that $Z(s,f)$ can be continued meromorphically
to the whole $s$-plane. Also, we are able to establish a functional equation
of $Z(s,f)$ under the assumption that the discriminant form is anisotropic.
The proof is adaptation of the corresponding result in [Bo1] to the vector
valued setting. A first step is

######  Theorem 6.2.

Let $f\in S_{l}(\rho_{L,1})$ be a cusp form with Fourier expansion

$f(\tau)=\sum_{\mu\in
L^{\prime}/L}\sum_{\begin{subarray}{c}n\in\mathbb{Z}+q(\mu)\\\
n>0\end{subarray}}a(\mu,n)e(n\tau).$

Then for $\frac{l}{2}+\operatorname{Re}(s)>1$ we have

(6.2) $\sum_{\lambda\in
L^{\prime}/L}\left(\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},E_{l,0}^{1*}(-\tau,\overline{s})\otimes
E_{l,0}^{1*}(\overline{\zeta},\overline{s})\rangle_{2}\operatorname{Im}(\tau)^{l}d\mu(\tau)\right)\mathfrak{e}_{\lambda}=0.$

Here $E_{l,0}^{1*}(-\tau,\overline{s})$ and
$E_{l,0}^{1*}(\overline{\zeta},\overline{s})$ are defined by (4.19).

###### Proof.

Similar to (4.44) and (4.48) a straightforward calculation using the relation
(3.11) gives

$\displaystyle\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},E_{l,0}^{1*}(-\tau,\overline{s})\otimes
E_{l,0}^{1*}(\overline{\zeta},\overline{s})\rangle_{2}$
$\displaystyle=(-1)^{2s}\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},E_{l,0}^{1}(\tau,\overline{s})\otimes
E_{l,0}^{1}(-\overline{\zeta},\overline{s})\rangle_{2}$
$\displaystyle=(-1)^{2s}\langle
f(\tau),E_{l,0}^{1}(\tau,\overline{s})\rangle_{1}\langle\mathfrak{e}_{\lambda},E_{l,0}^{1}(-\overline{\zeta},\overline{s})\rangle_{1}.$

Replacing this with the integrand in (6.2), we find that the integral in (6.2)
is equal to

(6.3) $(-1)^{2s}\left(\sum_{\lambda\in
L^{\prime}/L}\langle\mathfrak{e}_{\lambda},E_{l,0}^{1}(-\overline{\zeta},\overline{s})\rangle_{1}\mathfrak{e}_{\lambda}\right)\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau),E_{l,0}^{1}(\tau,\overline{s})\rangle_{1}\operatorname{Im}(\tau)^{l}d\mu(\tau).$

The integral in (6.3) is the Petesson scalar product of $f$ and $E_{l,0}^{1}$.
As such, it is well defined. For its evaluation we adapt the calculations in
the proof of [Br1], Proposition 1.5, to our situation. The usual unfolding
argument (see the proof of Thm. 5.6 in [RS]) yields

$\displaystyle\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau),E_{l,0}^{1}(\tau,\overline{s})\rangle_{1}\operatorname{Im}(\tau)^{l}d\mu(\tau)$
$\displaystyle=\int_{\Gamma_{1}\backslash\mathbb{H}}\sum_{g\in\Gamma_{\infty}\backslash\Gamma_{1}}\langle
f(g\tau),\mathfrak{e}_{0}\rangle_{1}\operatorname{Im}(g\tau)^{l+s}d\mu(\tau)$
$\displaystyle=\int_{\Gamma_{\infty}\backslash\mathbb{H}}\langle
f(\tau),\mathfrak{e}_{0}\rangle_{1}\operatorname{Im}(\tau)^{l+s-2}dxdy.$

Inserting the Fourier expansion of $f_{0}$ into the last integral, we find

$\displaystyle\int_{0}^{\infty}\int_{0}^{1}\sum_{\begin{subarray}{c}n\in\mathbb{Z}\\\
n>0\end{subarray}}a(0,n)e(n\tau)\operatorname{Im}(\tau)^{l+s-2}dxdy$
$\displaystyle=\int_{0}^{\infty}\sum_{\begin{subarray}{c}n\in\mathbb{Z}\\\
n>0\end{subarray}}a(0,n)e^{2\pi ny}y^{l+s-2}dy\left(\int_{0}^{1}e^{2\pi
inx}dx\right)$ $\displaystyle=0.$

∎

In order to study the analytic properties of the standard zeta function, we
express it basically as a Petersson scalar product of $f$ with the restricted
Eisenstein series $E_{l,0}^{2}(\left(\begin{smallmatrix}\tau&0\\\
0&\zeta\end{smallmatrix}\right),s)$. This approach is weĺl known and has been
applied in several settings, see e. g. [Bo1], [Ar] or [BM].

######  Definition 6.3.

Let $f\in S_{l}(\rho_{L,1})$ be a cusp form and $E_{l,0}^{2}$ be the
Eisenstein series as in Definition 4.3. For $\zeta\in\mathbb{H}$ and
$l+2\operatorname{Re}(s)>3$ we define the integral

(6.4) $\sum_{\lambda\in
L^{\prime}/L}\left(\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},E_{l,0}^{2}(\left(\begin{smallmatrix}\tau&0\\\
0&-\overline{\zeta}\end{smallmatrix}\right),\overline{s})\rangle_{2}\operatorname{Im}(\tau)^{l}d\mu(\tau)\right)\mathfrak{e}_{\lambda}.$

######  Remark 6.4.

In view of Theorem 5.3 we may write the integral in (6.4) in the form

(6.5) $\begin{split}&\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},E_{l,0}^{2*}(\left(\begin{smallmatrix}-\tau&0\\\
0&\overline{\zeta}\end{smallmatrix}\right),\overline{s})\rangle_{2}\operatorname{Im}(\tau)^{l}d\mu(\tau)\\\
&=\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},E_{l,0}^{1*}(-\tau,\overline{s})\otimes
E_{l,0}^{1*}(\overline{\zeta},\overline{s})\rangle_{2}\operatorname{Im}(\tau)^{l}d\mu(\tau)\\\
&+\frac{e(\operatorname{sig}(L)/8)}{|L^{\prime}/L|^{1/2}}\sum_{d\in\mathbb{N}}\frac{g_{d}(L)}{g(L)}d^{-l-2s}\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},\mathscr{P}^{+}_{l}(\tau,\overline{\zeta},\left(\begin{smallmatrix}d&0\\\
0&d^{-1}\end{smallmatrix}\right),\overline{s}\rangle_{2}\operatorname{Im}(\tau)^{l}d\mu(\tau).\end{split}$

It follows from Remark 5.4, (4.49) and (6.3) that the integral in (6.4) is
well defined.

The pullback formula (5.3) combined with Theorem 6.2 and Theorem 4.10 give
rise to the before mentioned integral formula of the standard zeta function of
a common Hecke eigenform $f$.

######  Theorem 6.5.

Let $l\in 2\mathbb{Z}$, $l\geq 3$, satisfy $2l+\operatorname{sig}(L)\equiv
0\bmod{4}$. Let $f\in S_{l}(\rho_{L,1})$ and $E_{l,0}^{2}$ the Eisenstein
series in Definition 4.3. If $l+2\operatorname{Re}(s)>3$, then, for any
$\zeta\in\mathbb{H}$,

(6.6) $\begin{split}&\sum_{\lambda\in
L^{\prime}/L}\left(\int_{\Gamma_{1}\backslash\mathbb{H}}\langle
f(\tau)\otimes\mathfrak{e}_{\lambda},E_{l,0}^{2}(\left(\begin{smallmatrix}\tau&0\\\
0&-\overline{\zeta}\end{smallmatrix}\right),\overline{s})\rangle_{2}\operatorname{Im}(\tau)^{l}d\mu(\tau)\right)\mathfrak{e}_{\lambda}\\\
&=K(l,s)\sum_{d\in\mathbb{N}}d^{-l-2s}\left(f\mid_{k,L}T\left(\begin{smallmatrix}d^{2}&0\\\
0&1\end{smallmatrix}\right)\right)(\zeta),\end{split}$

where
$K(l,s)=\frac{e(\operatorname{sig}(L)/8)}{|L^{\prime}/L|^{1/2}}(-1)^{-s}C(l,s)$
and $C(l,s)$ is specified in (4.41). Moreover, if $f$ is common eigenform of
all Hecke operators $T\left(\begin{smallmatrix}d^{2}&0\\\
0&1\end{smallmatrix}\right)$, the right-hand side of the above identity
coincides with

(6.7)
$K(l,s)\left(\sum_{d\in\mathbb{N}}\lambda_{d}(f)d^{-l-2s}\right)f(\zeta).$

###### Proof.

This is routine work. We have just to collect the results we established
before and put them together. Equation (6.5) in Remark 6.4 combined with
Theorem 6.2 and Theorem 4.10 allows us to replace the left-hand side of the
above stated identity with

$\frac{e(\operatorname{sig}(L)/8)}{|L^{\prime}/L|^{1/2}}(-1)^{-s}C(l,s)\sum_{d\in\mathbb{N}}\frac{g_{d}(L)}{g(L)}d^{-l-2s}\left(f\mid_{k,L}T\left(\begin{smallmatrix}d&0\\\
0&d^{-1}\end{smallmatrix}\right)\right)(\zeta).$

Employing the relation (4.8) afterwards, gives the desired result. ∎

We are now in a position to prove a result concerning the analytic properties
of $Z(s,f)$. Corollary 4.6 together with (6.6) gives us the means to transfer
the analytic properties of $E_{l,0}^{2}$ to the standard zeta function. We use
the same notation as in Corollary 4.6.

######  Theorem 6.6.

Let $l\in 2\mathbb{Z},\,l\geq 3$, satisfy $2l+\operatorname{sig}(L)\equiv
0\bmod{4}$ and $f\in S_{l}(\rho_{L,1})$ a common eigenform of Hecke operators
$T\left(\begin{smallmatrix}d^{2}&0\\\ 0&1\end{smallmatrix}\right)$. Then the
Dirichlet series $Z(s,f)$ can be continued meromorphically to the $s$-plane.
If additionally $L^{\prime}/L$ is anisotropic and $|L^{\prime}/L|$ odd, then

$\mathscr{Z}(f,s)=K(l,s)Z(f,2s+l)$

satisfies the following functional equation

(6.8)
$\mathscr{Z}(f,s-\frac{l}{2})=\xi(2s-\frac{3}{2},2.l,P)\prod_{p\mid|L^{\prime}/L|}\left(C_{p}(\operatorname{id},I_{0})+C_{p}(\operatorname{id},I_{0}\cup
I_{1})+C_{p}(\sigma,I_{0})\right)\mathscr{Z}(f,\frac{3}{2}-s-\frac{l}{2}).$

###### Proof.

First, we note that

$\displaystyle\overline{E_{l,0}^{2}(\left(\begin{smallmatrix}\tau&0\\\
0&-\overline{\zeta}\end{smallmatrix}\right),\overline{s})}$
$\displaystyle=E_{l,0}^{2*}(\left(\begin{smallmatrix}\overline{\tau}&0\\\
0&-\zeta\end{smallmatrix}\right),s).$
$\displaystyle=E_{l,0}^{2}(\left(\begin{smallmatrix}-\overline{\tau}&0\\\
0&\zeta\end{smallmatrix}\right),s),$

where the last equation is due to (4.19). Clearly, this identity holds for
each component function of $E_{l,0}^{2}$:

(6.9) $\overline{E_{l,0}^{2}(\left(\begin{smallmatrix}\tau&0\\\
0&-\overline{\zeta}\end{smallmatrix}\right),\overline{s})}_{\mu}=E_{l,0}^{2}(\left(\begin{smallmatrix}-\overline{\tau}&0\\\
0&\zeta\end{smallmatrix}\right),s)_{\mu}$

for any $\mu\in(L^{\prime}/L)^{2}$. From this follows the first assertion.

For the functional equation make use of the fact that (4.34) is valid for each
component function of $E_{l,0}^{2}$, which can be immediately read off the
proof of Theorem 3.16 of [St] and [BY], Section 2.2 on pages 641 and 642.
Taking this into account, we obtain the claimed functional equation. ∎

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