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\begin{document}
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\title{\bf{Construction of Vector Valued Modular Forms of positive dimension Part II}}
\author{Pepe Gimenez}

\date{\today}
\maketitle

\noindent
{\setlength{\baselineskip}%
{1.5\baselineskip}
\newtheorem{thm}{Theorem}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\section{Construction of a vector-valued modular form of positive dimension $r$}
So far we have that for $M = \left(%
\begin{array}{cc}
  \alpha & \beta \\
  \gamma & \delta \\
\end{array}%
\right)\in \Gamma (1)$
\begin{equation}
\varepsilon^{-1}(M)\rho^{-1}(M)\left(-i(\gamma\tau + \delta )
\right)^rF_\nu(M\tau)- F_\nu(\tau)= P_{M,\nu}(\tau, r,
\varepsilon, \rho)
\end{equation}
where $P_{M,\nu}(\tau, r, \varepsilon, \rho)$ is a column vector
of polynomials of degree at most $r$. which automatically implies
that
\begin{equation}
\varepsilon^{-1}(M)\rho^{-1}(M)\left(-i(\gamma\tau + \delta )
\right)^rF(M\tau)- F(\tau)= Q_{M}(\tau, r, \varepsilon, \rho)
\end{equation}
where $Q_{M}(\tau, r, \varepsilon, \rho)$ is a column vector of
polynomials of degree at most $r$, since
\begin{equation}\label{Eq:defftauslkdfj}
F(\tau)=\left(%
\begin{array}{c}
  \sum_{\nu=1}^\mu b_\nu(1)e^{2\pi i(m_1-\nu)\tau}+ \sum_{m=0}^\infty a_m(1)e^{2 \pi i(m+m_1) \tau} \\
  \vdots \\
  \sum_{\nu=1}^\mu b_\nu(p)e^{2\pi i(m_p-\nu)\tau}+ \sum_{m=0}^\infty a_m(p)e^{2 \pi i(m+m_p) \tau} \\
\end{array}%
\right)=\left(%
\begin{array}{c}
  \sum_{\nu=1}^\mu b_\nu(1)F_\nu(\tau)(1) \\
  \vdots \\
  \sum_{\nu=1}^\mu b_\nu(p)F_\nu(\tau)(p) \\
\end{array}%
\right),
\end{equation}
now let $S=\left(%
\begin{array}{cc}
  1 & 1 \\
  0 & 1 \\
\end{array}%
\right)$ and $T=\left(%
\begin{array}{cc}
  0 & -1 \\
  1 & 0 \\
\end{array}%
\right)$. By the definition of $F(\tau)$  it is easy to see that
for $n \in \mathbb{Z}$ we have
\begin{equation}
F(S^n \tau)= \left(%
\begin{array}{ccc}
  e^{2 \pi i n m_1} &  &  \\
    & \ddots &  \\
    &   & e^{2 \pi i n m_p} \\
\end{array}%
\right)F(\tau)=\varepsilon(S^n)\rho(S^n)\left(-i \right)^rF(\tau)
\end{equation}
and therefore $Q_{S^n}(\tau, r, \varepsilon, \rho)\equiv 0$, and
since all the elements of $\Gamma (1)$ can be written as a product
of $S^n$ and $T$ for $n \in \mathbb{Z}$, now we want to find
$b_1,\ldots, b_\mu$ such that
$\varepsilon^{-1}(T)\rho^{-1}(T)\left(-i \tau \right)^rF(T\tau)-
F(\tau)=0$, or what is the same
\begin{eqnarray}\label{Eq:def of qttau}
\varepsilon^{-1}(T)\rho^{-1}(T)\left(-i \tau
\right)^rF(T\tau)- F(\tau)&=&\left(%
\begin{array}{c}
  \sum_{\nu=1}^\mu b_\nu(1)P_{T, \nu}(\tau, r, \varepsilon, \rho)(1) \\
  \vdots \\
  \sum_{\nu=1}^\mu b_\nu(p)P_{T, \nu}(\tau, r, \varepsilon, \rho)(p) \\
\end{array}%
\right)\\
&=& \left(%
\begin{array}{c}
  Q_{T}(\tau, r, \varepsilon, \rho)(1) \\
  \vdots \\
   Q_{T}(\tau, r, \varepsilon, \rho)(p) \\
\end{array}%
\right)\equiv \left(%
\begin{array}{c}
  0 \\
  \vdots \\
  0 \\
\end{array}%
\right)
\end{eqnarray}
because if we do so we will have a function $F(\tau)$ with the
transformation law
\begin{equation}
F(\tau)=\varepsilon^{-1}(M)\rho^{-1}(M)\left(-i(\gamma\tau +
\delta ) \right)^rF(M\tau)
\end{equation}
and since it is regular in $\mathcal{H}$, and has the expansion at
$\infty$ (\ref{Eq:defftauslkdfj}), then $F(\tau)$ is a modular
form of dimension $r$.

Now if we replace $\tau$ by $T \tau$ in (\ref{Eq:def of qttau}),
we see that
\begin{equation}
\varepsilon^{-1}(T)\rho^{-1}(T)\left(-i \tau \right)^r\left(%
\begin{array}{c}
  Q_{T}(\frac{-1}{\tau}, r, \varepsilon, \rho)(1) \\
  \vdots \\
   Q_{T}(\frac{-1}{\tau}, r, \varepsilon, \rho)(p) \\
\end{array}%
\right)=\left(%
\begin{array}{c}
  -Q_{T}(\tau, r, \varepsilon, \rho)(1) \\
  \vdots \\
   -Q_{T}(\tau, r, \varepsilon, \rho)(p) \\
\end{array}%
\right)
\end{equation}
and if $Q_{T}(\tau_o, r, \varepsilon, \rho) = 0$ then we have an
homogeneous system of $p$ equations in the $p$ unknowns
$Q_{T}(\frac{-1}{\tau_o}, r, \varepsilon, \rho)(1),\ldots,
Q_{T}(\frac{-1}{\tau_o}, r, \varepsilon, \rho)(p)$, and since
$det(\rho^{-1}(T))\neq 0$, we have that $Q_{T}(\frac{-1}{\tau_o},
r, \varepsilon, \rho)(j)= 0$ for $1 \leq j \leq p$. Therefore the
zeros of $Q_{T}(\tau, r, \varepsilon, \rho)$ occur in pairs except
for $\tau=\pm i$ now suppose that exists a set $\tau_1, \ldots,
\tau_n$ of distinct roots of $Q_{T}(\tau, r, \varepsilon, \rho)$,
such that $n=[r/2] +1$, also that $-1/\tau_j$ is not included in
that set for any $1 \leq j \leq n$ and neither does $\pm i$, then
since $Q_{T}(\tau_j, r, \varepsilon, \rho)=0$ for all the elements
in the set, the number of zeros of each of the p polynomials will
be $2([r/2] +1)>r$, and therefore $Q_{T}(\tau, r, \varepsilon,
\rho)\equiv 0$.

Now consider the homogeneous linear system corresponding to the
$k^{th}$ element in $Q_{T}(\tau_j, r, \varepsilon, \rho)$
\begin{equation}\label{Eq: homogsystem}
\sum_{\nu=1}^\mu b_\nu (k)P_{T, \nu}(\tau_j, r, \varepsilon,
\rho)(k) =0
\end{equation}
of  $2([r/2] +1)$ equations in the  $\mu$ unknowns $b_1(k),\ldots,
b_\mu(k)$, and if $\mu$ is greater than $2([r/2] +1)$, we can find
solutions for $b_1(k),\ldots, b_\mu(k)$. Therefore we can state
the following theorem

\begin{thm}Let $\mu$ be an integer greater than $2([r/2] +1)$. If
we define $F(\tau)$ as in (\ref{Eq:defftauslkdfj}) with $r>2
\alpha$ and $b_1,\ldots, b_\mu$ being column vectors of dimension
$p$ such that the $k^{th}$ element satisfies (\ref{Eq:
homogsystem}) for each $1 \leq k \leq p$ then $F(\tau)$ is a
vector-valued modular form of dimension r
\end{thm}
\section{The supplementary series}
Let $m_s'$ and $\nu'$ be defined by
\begin{equation}
%
\begin{array}{ccc}
  m_s'= 1 -m_s & \nu'= 1- \nu & \mbox{if }m_s>0 \\
  m_s'= 0 & \nu'= - \nu & \mbox{if }m_s=0 \\
\end{array}%
\end{equation}
Further we can define
\begin{equation}
\varepsilon'(V)=e^{i \pi r}\varepsilon^{-1}(V), \quad \mbox{and}
\quad \rho'(V)=\overline{\rho(V)}
\end{equation}
since $r$ is an integer and $\varepsilon$ is a multiplier system
for $\Gamma$, it follows that $\varepsilon'$ is also a multiplier
system for $\Gamma$. On the other hand since $\rho$ is a
representation for $\Gamma$, we have that $\rho'$ is also a
representation. Now let $a_m(\nu', r, \varepsilon',\rho')$ be
defined as in (2, Part I), therefore
\begin{equation}
a_m(\nu', r, \varepsilon', \rho')=2\pi
\sum_{c=1}^{\infty}\frac{1}{c}
 \sum_{\tiny{\begin{array}{c}
  d,c \\
  0 \leq d < c \\
  (d,c)=1 \\
\end{array}}}e^{-2
\pi i \frac{dm+d'\nu'}{c}}\varepsilon'^{-1}(V_{c,d})
\rho'^{-1}(V_{c,d})\left(%
\begin{array}{c}
  e^{2 \pi i m'_1
\frac{d'-d}{c}}B_{c,\nu',m, \rho', \varepsilon', r}^1 \\
  \vdots \\
   e^{2 \pi i m'_p
\frac{d'-d}{c}}B_{c,\nu',m, \rho', \varepsilon', r}^1 \\
\end{array}%
\right)
\end{equation}
now we define the supplementary series $\widehat{F}_{\nu'}(\tau)$
of the series $F_\nu(\tau)$ by
\begin{equation}\label{Eq:supplseries definition}
\widehat{F}_{\nu'}(\tau)=\left(%
\begin{array}{c}
  e^{2 \pi i(m'_1- \nu')\tau} \\
  \vdots \\
   e^{2 \pi i(m'_p- \nu')\tau} \\
\end{array}%
\right)+\left(%
\begin{array}{c}
  \sum_{m=0}^\infty a_m(\nu', r, \varepsilon', \rho')(1)e^{2 \pi i(m +m'_1)\tau}   \\
  \vdots \\
  \sum_{m=0}^\infty a_m(\nu', r, \varepsilon', \rho')(p)e^{2 \pi i(m +m'_p)\tau}  \\
\end{array}%
\right)
\end{equation}
we can see using exactly the same arguments as before that
\begin{enumerate}
\item$\widehat{F}_{\nu'}(\tau)$ is regular for $\tau \in
\mathcal{H}$ \item
$\widehat{F}_{\nu'}(\tau)-\varepsilon'^{-1}(M)\rho'^{-1}(M)\left(-i(\gamma\tau
+ \delta ) \right)^r\widehat{F}_{\nu'}(M\tau)=\widehat{P}_{M,
\nu'}(\tau, r, \varepsilon', \rho')$, for every $M = \left(%
\begin{array}{cc}
  \alpha & \beta \\
  \gamma & \delta \\
\end{array}%
\right)\in \Gamma (1)$ \item The column vector of polynomials
$\widehat{P}_{M, \nu'}(\tau, r, \varepsilon', \rho')$ is given by
the following formula
\begin{eqnarray}
\widehat{P}_{M, \nu'}(\tau, r, \varepsilon',
\rho')&=&\widehat{H}_{M, \nu'}(\tau, r, \varepsilon', \rho')+
\widehat{K}_(\tau, M,\widehat{a}_{0,\nu}^*)
\end{eqnarray}
where $\widehat{K}(\tau,M,\widehat{a}_{0,\nu}^*)$ is given by (50
part I)
\begin{equation}
\widehat{K}(\tau,M,\widehat{a}_{0,\nu}^*)=\left(
I-\left(-i(\gamma\tau + \delta ) \right)^r\varepsilon'^{-1}(M)
\rho'^{-1}(M)\right)\frac{1}{2}\widehat{a}_{0,\nu}^*
\end{equation}
where
\begin{equation}
\widehat{a}_{0,\nu}^*= 2\pi\sum_{c=1}^{\infty}\frac{1}{c}
\sum_{\tiny{\begin{array}{c}
  d,c \\
  0 \leq d < c \\
  (d,c)=1 \\
\end{array}}}\varepsilon'^{-1}(V_{c,d}) \rho'^{-1}(V_{c,d})\left(%
\begin{array}{c}
  e^{2 \pi i
\frac {d'(m'_1-\nu')}{c}}\left(\frac{2 \pi \nu'}{c} \right)^{r+1}\frac{1}{(r+1)!}[1-m'_1] \\
  \vdots \\
  e^{2 \pi i
\frac {d'(m'_p-\nu')}{c}}\left(\frac{2 \pi \nu'}{c} \right)^{r+1}\frac{1}{(r+1)!}[1-m'_p] \\
\end{array}%
\right)
\end{equation}
here $[x]$ is the greatest integer less than or equal to $x$. And
$\widehat{H}_{M, \nu'}(\tau, r, \varepsilon', \rho')$ is given by
(104 part I)
\begin{eqnarray}
\lefteqn{ \widehat{H}_{M, \nu'}(\tau, r, \varepsilon', \rho')}\\
&&= \widehat{G}_{\nu'}(\tau)-\varepsilon'^{-1}(M) \rho'^{-1}(M)
\left(-i(\gamma \tau + \delta ) \right)^r \widehat{G}_{\nu'}(M\tau)\\
&&=\widehat{F}_{\nu'}(\tau)-\varepsilon'^{-1}(M) \rho'^{-1}(M)
\left(-i(\gamma \tau + \delta ) \right)^r
\widehat{F}_{\nu'}(M\tau)-\widehat{K}_{\nu',r,\varepsilon',\rho'}(\tau,M,a_{0,\nu}^*(\nu',
r, \varepsilon',\rho'))
\\
&&=\lim_{N\rightarrow \infty} (\sum_{c=1}^{\infty}
\sum_{\tiny{\begin{array}{c}
  (c,d) \in \mathcal{J}_M(N) \\
   d \in D^c \\
\end{array}}}\varepsilon'^{-1}(V_{c,d}M)\rho'^{-1}(V_{c,d}M)\left(-i(\gamma \tau + \delta ) \right)^r \\
&& \left(-i(cM\tau + d ) \right)^r
 \left(%
\begin{array}{c}
  e^{2 \pi i
\frac {d'(m'_1-\nu')}{c}}\sum_{p=0}^r \frac{1}{p!}
\left(\frac{2 \pi i (\nu' -m'_1) }{c(c M\tau + d )} \right)^p  \\
  \vdots \\
  e^{2 \pi i
\frac {d'(m'_p-\nu')}{c}} \sum_{p=0}^r \frac{1}{p!}
\left(\frac{2 \pi i (\nu' -m'_p) }{c(c M\tau + d )} \right)^p \\
\end{array}%
\right)\\
 &&
 - \sum_{\tiny{\begin{array}{c}
  c \in \mathbb{Z} \\
   0< c\leq tN \\
\end{array}}} \sum_{\tiny{\begin{array}{c}
  d \in D^c \\
    |d| \leq N\\
\end{array}}}\varepsilon'^{-1}(V_{c,d})\rho'^{-1}(V_{c,d}) \left(-i(c\tau + d )
\right)^r\\
&&
 \left(%
\begin{array}{c}
  e^{2 \pi i
\frac {d'(m'_1-\nu')}{c}}\sum_{p=0}^r \frac{1}{p!}
\left(\frac{2 \pi i (\nu' -m'_1) }{c(c \tau + d )} \right)^p  \\
  \vdots \\
  e^{2 \pi i
\frac {d'(m'_p-\nu')}{c}} \sum_{p=0}^r \frac{1}{p!}
\left(\frac{2 \pi i (\nu' -m'_p) }{c(c \tau + d )} \right)^p \\
\end{array}%
\right))
\end{eqnarray}
\end{enumerate}
From now on we will use the notation $a_{0,\nu}^*$ and
$K(\tau,M,a_{0,\nu}^*)$ for the same formulas but using
$\varepsilon$, $\rho$ and $\nu$ instead of $\varepsilon'$, $\rho'$
and $\nu'$. It is easy to see from the formulas that for all $M
\in \Gamma$ we have
\begin{enumerate}
\item $\widehat{H}_{M, \nu'}(\tau, r, \varepsilon',
\rho')=\overline{H_{M, \nu}(\overline{\tau}, r, \varepsilon,
\rho)}$ \item $\widehat{a}_{0,\nu}^*=-\overline{a_{0,\nu}^*}$
\item
$\widehat{K}(\tau,M,\widehat{a}_{0,\nu}^*)=-\overline{K(\overline{\tau},M,a_{0,\nu}^*)}$
\item $\widehat{P}_{M, \nu'}(\tau, r, \varepsilon',
\rho')=\overline{P_{M, \nu}(\overline{\tau}, r, \varepsilon,
\rho)}-2\overline{K(\overline{\tau},M,a_{0,\nu}^*)}$
\end{enumerate}

Now let $b_1,\ldots , b_{\mu}$ be a set of column vectors such
that $b_j \in \mathbb{C}^p$, $b_{\mu}\neq 0$, $\rho:\Gamma
\longrightarrow GL(p,\mathbb{C})$ a $p$-dimensional complex
representation , $\varepsilon$ a Multiplier System , and $r>
2\alpha$, $r,\mu \in \mathbb{Z^+}$ and let $F(\tau)$ be defined as
in (41 part I)
\begin{equation}
F(\tau)=\left(%
\begin{array}{c}
  \sum_{\nu=1}^\mu b_\nu(1)F_\nu(\tau)(1) \\
  \vdots \\
  \sum_{\nu=1}^\mu b_\nu(p)F_\nu(\tau)(p) \\
\end{array}%
\right)
\end{equation}
then we can define the supplementary series $\widehat{F}(\tau)$ of
$F(\tau)$ by
\begin{equation}
\widehat{F}(\tau)=\left(%
\begin{array}{c}
  \sum_{\nu'=-\mu}^{-1} \overline{b}_{\nu'}(1)\widehat{F}_{\nu'}(\tau)(1) \\
  \vdots \\
  \sum_{\nu'=-\mu}^{-1} \overline{b}_{\nu'}(p)\widehat{F}_{\nu'}(\tau)(p) \\
\end{array}%
\right)
\end{equation}
For $M \in \Gamma(1)$ we have that
\begin{equation}
F(\tau)-\varepsilon^{-1}(M)\rho^{-1}(M)\left(-i(\gamma\tau +
\delta ) \right)^rF(M\tau)= Q_{M}(\tau, r, \varepsilon, \rho)
\end{equation}
where
\begin{equation}
Q_{M}(\tau, r, \varepsilon, \rho)=\left(%
\begin{array}{c}
  \sum_{\nu=1}^\mu b_\nu(1)P_{M, \nu}(\tau, r, \varepsilon,
\rho)(1) \\
  \vdots \\
  \sum_{\nu=1}^\mu b_\nu(p)P_{M, \nu}(\tau, r, \varepsilon,
\rho)(p) \\
\end{array}%
\right)
\end{equation}
and
\begin{equation}
\widehat{F}(\tau)-\varepsilon'^{-1}(M)\rho'^{-1}(M)\left(-i(\gamma\tau
+ \delta ) \right)^r\widehat{F}(M\tau)= \widehat{Q}_{M}(\tau, r,
\varepsilon', \rho')
\end{equation}
where
\begin{equation}
\widehat{Q}_{M}(\tau, r, \varepsilon', \rho')=\left(%
\begin{array}{c}
  \sum_{\nu=-\mu}^{-1} \overline{b}_{\nu'}(1)\widehat{P}_{M, \nu'}(\tau, r, \varepsilon',
\rho')(1) \\
  \vdots \\
  \sum_{\nu=-\mu}^{-1} \overline{b}_{\nu'}(p)\widehat{P}_{M, \nu'}(\tau, r, \varepsilon',
\rho')(p) \\
\end{array}%
\right)
\end{equation}
Now let $\tau_0 \in \mathcal{H}$ be such that $\left(
I-\left(-i\tau_0
\right)^r\varepsilon'^{-1}(T) \rho'^{-1}(T)\right)$ is invertible and let $L_\nu=\left(%
\begin{array}{c}
  L_\nu(1) \\
  \vdots \\
  L_\nu(p) \\
\end{array}%
\right)$ be the unknowns in the following  system of equations
\begin{eqnarray}\label{Eq: rel L and b}
\lefteqn{\left( I-\left(-i\tau_0
\right)^r\varepsilon'^{-1}(T) \rho'^{-1}(T)\right)\left(%
\begin{array}{ccc}
  L_\nu(1) &   &   \\
    & \ddots &   \\
    &   & L_\nu(p) \\
\end{array}%
\right)\widehat{a}_{0,\nu}^*}\\
&&=\left(%
\begin{array}{ccc}
  \overline{b}_\nu(1) &   &   \\
    & \ddots &   \\
    &   & \overline{b}_\nu(p) \\
\end{array}%
\right)\left( I-\left(-i\tau_0  \right)^r\varepsilon'^{-1}(T)
\rho'^{-1}(T)\right)\widehat{a}_{0,\nu}^*
\end{eqnarray}
then we have that
\begin{eqnarray}
\lefteqn{\left(%
\begin{array}{ccc}
  L_\nu(1) &   &   \\
    & \ddots &   \\
    &   & L_\nu(p) \\
\end{array}%
\right)}\\
&&=\left( I-\left(-i\tau_0 \right)^r\varepsilon'^{-1}(T)
\rho'^{-1}(T)\right)^{-1}\left(%
\begin{array}{ccc}
  \overline{b}_\nu(1) &   &   \\
    & \ddots &   \\
    &   & \overline{b}_\nu(p) \\
\end{array}%
\right)\left( I-\left(-i\tau_0  \right)^r\varepsilon'^{-1}(T)
\rho'^{-1}(T)\right)
\end{eqnarray}
now since both sides of (\ref{Eq: rel L and b}) are column vectors
of polynomials of degree at most $r$, if we choose $\tau_j,\ldots,
\tau_{r+1}$ such that $\left( I-\left(-i\tau_j
\right)^r\varepsilon'^{-1}(T) \rho'^{-1}(T)\right)$ is invertible
for every $0 \leq j \leq r+1$ we can extend the result for every
$\tau \in \mathcal{H}$. And therefore we have that
\begin{eqnarray}
\lefteqn{\left( I-\left(-i\tau
\right)^r\varepsilon'^{-1}(T) \rho'^{-1}(T)\right)\left(%
\begin{array}{ccc}
  \overline{b}_\nu(1) &   &   \\
    & \ddots &   \\
    &   & \overline{b}_\nu(p) \\
\end{array}%
\right)\widehat{a}_{0,\nu}^*}\\
&&=\left(%
\begin{array}{ccc}
  \overline{b}_\nu(1) &   &   \\
    & \ddots &   \\
    &   & \overline{b}_\nu(p) \\
\end{array}%
\right)\left( I-\left(-i\tau  \right)^r\varepsilon'^{-1}(T)
\rho'^{-1}(T)\right)\widehat{a}_{0,\nu}^*
\end{eqnarray}
we can show an analogous result for $\varepsilon$ and  $\rho$.
\begin{thm}
\begin{equation}
F( \tau ) \in \{ \Gamma(1), r, \varepsilon, \rho \}
\Leftrightarrow \widehat{F}( \tau )=\left(%
\begin{array}{c}
  \sum_{\nu=1}^\mu \overline{b_\nu}(1)\widehat{a}_{0,\nu}^*(1)\\
  \vdots \\
  \sum_{\nu=1}^\mu \overline{b_\nu}(p)\widehat{a}_{0,\nu}^*(p) \\
\end{array}%
\right)
\end{equation}
\end{thm}
\begin{proof}First let us see the relationship between $\widehat{Q}_{T}(\tau, r, \varepsilon',
\rho')$ and $Q_{T}(\tau, r, \varepsilon, \rho)$ for every $M \in
\Gamma (1)$

\begin{eqnarray}
\lefteqn{\widehat{Q}_{T}(\tau, r, \varepsilon', \rho')}\\
&&=\left(%
\begin{array}{c}
  \sum_{\nu=1}^\mu \overline{b_\nu}(1)\widehat{P}_{T, \nu'}(\tau, r, \varepsilon',
\rho')(1) \\
  \vdots \\
  \sum_{\nu=1}^\mu \overline{b_\nu}(p)\widehat{P}_{T, \nu'}(\tau, r, \varepsilon',
\rho')(p) \\
\end{array}%
\right)\\
&&=\left(%
\begin{array}{c}
  \sum_{\nu=1}^\mu \overline{b_\nu}(1)\left( \overline{P_{T, \nu}(\overline{\tau}, r, \varepsilon,
\rho)}(1)-2\overline{K(\overline{\tau},T,a_{0,\nu}^*)}(1)\right) \\
  \vdots \\
  \sum_{\nu=1}^\mu \overline{b_\nu}(p)\left( \overline{P_{T, \nu}(\overline{\tau}, r, \varepsilon,
\rho)}(p)-2\overline{K(\overline{\tau},T,a_{0,\nu}^*)}(p)\right) \\
\end{array}%
\right)\\
&&=\overline{Q_{T}(\overline{\tau}, r, \varepsilon, \rho)}-\left(%
\begin{array}{c}
  2\sum_{\nu=1}^\mu \overline{b_\nu}(1)\overline{K(\overline{\tau},T,a_{0,\nu}^*)}(1)\\
  \vdots \\
  2\sum_{\nu=1}^\mu \overline{b_\nu}(p)\overline{K(\overline{\tau},T,a_{0,\nu}^*)}(p) \\
\end{array}%
\right)\\
&&=\overline{Q_{T}(\overline{\tau}, r, \varepsilon, \rho)}+\left(
I-\left(-i\tau  \right)^r\varepsilon'^{-1}(T)
\rho'^{-1}(T)\right)\left(%
\begin{array}{c}
  \sum_{\nu=1}^\mu \overline{b_\nu}(1)\widehat{a}_{0,\nu}^*(1)\\
  \vdots \\
  \sum_{\nu=1}^\mu \overline{b_\nu}(p)\widehat{a}_{0,\nu}^*(p) \\
\end{array}%
\right)
\end{eqnarray}
therefore if $\widehat{F}( \tau )=\left(%
\begin{array}{c}
  \sum_{\nu=1}^\mu \overline{b_\nu}(1)\widehat{a}_{0,\nu}^*(1)\\
  \vdots \\
  \sum_{\nu=1}^\mu \overline{b_\nu}(p)\widehat{a}_{0,\nu}^*(p) \\
\end{array}%
\right)$, we have that $\overline{Q_{T}(\overline{\tau}, r,
\varepsilon, \rho)}=0$, and so does $Q_{T}(\tau, r, \varepsilon,
\rho)$, and therefore $F(\tau) \in \{ \Gamma(1), r, \varepsilon,
\rho \}$.

On the other hand if $F(\tau) \in \{ \Gamma(1), r, \varepsilon,
\rho \}$, then we can consider the function $\widehat{G}(\tau)=\widehat{F}( \tau )-\left(%
\begin{array}{c}
  \sum_{\nu=1}^\mu \overline{b_\nu}(1)\widehat{a}_{0,\nu}^*(1)\\
  \vdots \\
  \sum_{\nu=1}^\mu \overline{b_\nu}(p)\widehat{a}_{0,\nu}^*(p) \\
\end{array}%
\right)$, then we have that
\begin{equation}
\widehat{G}(\tau)-\varepsilon'^{-1}(T)\rho'^{-1}(T)\left(-i(\gamma\tau
+ \delta )
\right)^r\widehat{G}(T\tau)=\overline{Q_{T}(\overline{\tau}, r,
\varepsilon, \rho)}\equiv 0
\end{equation}
Therefore $\widehat{G}(\tau)$ is a vector-valued modular form of
dimension $r$, but since $\widehat{G}(\tau)$ is bounded at
infinity, we have that $\widehat{G}(\tau) \equiv 0$, and the proof
is complete.
\end{proof}





\begin{equation}
\end{equation}

\begin{eqnarray}
\lefteqn{}
\end{eqnarray}

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\end{document}