\documentclass[9pt]{article} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsmath} \usepackage{graphicx} \usepackage{color} \usepackage[colorlinks]{hyperref} \usepackage{newlfont} \usepackage{mathrsfs} \usepackage[active]{srcltx} % SRC Specials for DVI Searching \begin{document} \title{\bf{Construction of Vector Valued Modular Forms of positive dimmension}} \author{Pepe Gimenez} \date{\today} \maketitle \noindent {\setlength{\baselineskip}% {1.5\baselineskip} \newtheorem{thm}{Theorem} \begin{thm}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 $r> 2\alpha$ (\ref{eq:alpha}), $r,\mu \in \mathbb{Z^+}$ then if \begin{equation} 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), \end{equation} where \begin{equation} a_m(\nu ,r,\varepsilon)=2 \pi \sum_{c=1}^{\infty}\frac {1}{c}A_{c,\nu,m, \rho, \varepsilon}B_{c,\nu,m, \rho, \varepsilon, r} \end{equation} which is a column vector in which the jth component is given by \begin{equation} a_m(\nu ,r,\varepsilon)(j)=2 \pi \sum_{c=1}^{\infty}\frac{1}{c} \left[ \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) \left( \sum_{s=1}^p x_{js}(c,d) e^{2 \pi i m_s \frac{d'-d}{c}}\right) e^{-2 \pi i \frac{dm+d'\nu}{c}} \right] B_{c,\nu,m, \rho, \varepsilon, r}^j \end{equation} where \begin{equation} \left( \rho^{-1}(V_{c,d})\right)^t=\left(% \begin{array}{ccc} x_{11}(c,d) & \cdots & x_{1p}(c,d) \\ \vdots & \ddots & \vdots \\ x_{p1}(c,d) & \cdots & x_{pp}(c,d) \\ \end{array}% \right) \end{equation} where \begin{equation} V_{c,d}=\left(% \begin{array}{cc} d' & - \frac{dd'+1}{c} \\ c & -d \\ \end{array}% \right), \quad dd' \equiv -1 \pmod{c}, \quad c>0, \quad 0\leq d,d'< c \end{equation} \begin{eqnarray}\label{Eq:defBjknum} \lefteqn{B_{c, \nu,m, \rho, \varepsilon, r}^j}\\ &&=\left( \frac{\nu - m_j}{m+m_j}\right)^{\frac{r+1}{2}}I_{r+1}\left(\frac{4 \pi}{c}(\nu - m_j)^{\frac{1}{2}}(m+m_j)^{\frac{1}{2}}\right)\quad if \quad m+m_j > 0\\ &&= \frac{2 \pi}{(r+1)!} \left(\frac{2 \pi \nu}{c} \right)^{r+1} \quad if \quad m=m_j=0 \end{eqnarray} \begin{equation} a_m=\left(% \begin{array}{c} \sum_{\nu=1}^\mu b_\nu (1) a_m(\nu ,r,\varepsilon)(1) \\ \vdots \\ \sum_{\nu=1}^\mu b_\nu (p) a_m(\nu ,r,\varepsilon)(p) \\ \end{array}% \right)=\left(% \begin{array}{c} a_m (1) \\ \vdots \\ a_m (p) \\ \end{array}% \right) \end{equation} then \begin{enumerate} \item $F(\tau)$ is regular in the complex upper half-plane $ \mathcal{H}$ \item $F(\tau)$ satisfies \begin{equation} F(\tau)- \varepsilon^{-1}(M)(-i(c \tau +d))^r \rho^{-1} (M) F(M\tau)= p_M(\tau) \end{equation}, for all \begin{equation} M=\left(% \begin{array}{cc} a & b \\ c & d \\ \end{array}% \right)\in \Gamma \end{equation} \end{enumerate} where $p_M(\tau)$ is a column vector of polynomials on $\tau$ of degree at most $r$. \end{thm} \newtheorem{lem}[thm]{Lemma} \begin{lem} For $r>2 \alpha$ as $m\rightarrow \infty$, we have \begin{equation} |a_m(\nu ,r,\varepsilon)|=O \left( (m+\kappa_m)^{-\frac{3}{4}- \frac{r}{2}} e^{4 \pi(\nu -\kappa_m)^{\frac{1}{2}}(m+\kappa_M)^{\frac{1}{2}}} \right) \end{equation} where \begin{equation} \kappa_m= \min{m_1,\ldots,m_p} \quad and \quad \kappa_M= \max{m_1,\ldots,m_p} \end{equation} \end{lem} \begin{proof}In \cite {[12]} Marvin Knopp proves that given $0 \leq m_s <1$ and $q \geq 0$ \begin{equation} c^{-q} A_{c,\nu,m_s}(m)=O \left( c^{\frac{2}{3}+ \varepsilon }\right) \end{equation} where \begin{equation} A_{c,\nu,m_s}(m)=\sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}} \varepsilon^{-1}(V_{c,d}) e^{2 \pi i m_s \frac{d'-d}{c}}e^{-2 \pi i \frac{dm+d'\nu}{c}} \end{equation} the strategy is the same as the one in the proof of Knopp first we can fix $s$ and show that \begin{equation} \begin{split}2 \pi \sum_{c=2}^{\infty}\frac{1}{c} \left[ \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) \left( x_{js}(c,d) e^{2 \pi i m_s \frac{d'-d}{c}}\right) e^{-2 \pi i \frac{dm+d'\nu}{c}} \right] B_{c,\nu,m, r}^s \\ \leq C(m+m_s)^{-\frac{1}{2}}e^{2 \pi (m+m_s)^{\frac{1}{2}}(\nu-m_s)^{\frac{1}{2}}}\end{split} \end{equation} and show that the summations behaves as the first term as $m\rightarrow \infty$. Here $B_{c,\nu,m, r}^s $ can be understood as $B_{c,\nu,m, \rho, \varepsilon, r}^s$, with a MS $\varepsilon$ and a representation $\rho$ such that \begin{equation} \varepsilon \left(% \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array}% \right) \rho \left(% \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array}% \right)=\left(% \begin{array}{ccccc} e^{2 \pi i m_1} & & s^{th}cloumn & & \\ & \ddots & & & \\ s^{th}row & & e^{2 \pi i m_s} & & \\ & & & \ddots & \\ & & & & e^{2 \pi i m_p} \\ \end{array}% \right) \end{equation} Also in the previous paper I showed that $|\rho(V_{c,d})|=O(c^{2 \alpha})$, which means that $|x_{ij}|=O(c^{2 \alpha})$. And since $r> 2 \alpha$, we can write $r=q+\alpha$ where $q>0$ In the same fashion as the mentioned proof we can see that \begin{eqnarray} \lefteqn{\left|2 \pi \sum_{c=2}^{\infty}\frac{1}{c} \left[ \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) \left( x_{js}(c,d) e^{2 \pi i m_s \frac{d'-d}{c}}\right) e^{-2 \pi i \frac{dm+d'\nu}{c}} \right] B_{c,\nu,m, r}^s\right|} \\ &&\leq C_1 \frac{(\nu-m_s)^{r+\frac{1}{2}}}{(m+m_s)^{\frac{1}{2}}}\sinh \left( 2 \pi (m+m_s)^{\frac{1}{2}}(\nu-m_s)^{\frac{1}{2}}\right) \sum_{c=2}^\infty x_{js}(c,d)c^{-r-2}A_{c,\nu,m_s}(m)\\ &&\leq C_2(m+m_s)^{-\frac{1}{2}}e^{2 \pi (m+m_s)^{\frac{1}{2}}(\nu-m_s)^{\frac{1}{2}}}\sum_{c=2}^\infty c^{-\frac{4}{3}+\varepsilon}\\ &&\leq C_3(m+m_s)^{-\frac{1}{2}}e^{2 \pi (m+m_s)^{\frac{1}{2}}(\nu-m_s)^{\frac{1}{2}}} \end{eqnarray} the first term is \begin{equation} \begin{split}2 \pi \varepsilon^{-1}(V_{1,0})x_{1,s}(1,0) e^{2 \pi i m_s (d'-d)}e^{-2 \pi i (dm+d'\nu)}\left( \frac{\nu - m_s}{m+m_s}\right)^{\frac{r+1}{2}}I_{r+1}\left(4 \pi(\nu - m_s)^{\frac{1}{2}}(m+m_s)^{\frac{1}{2}}\right)\\ =O\left((m+m_s)^{-\frac{r}{2}- \frac{3}{4}}e^{4 \pi (m+m_s)^{\frac{1}{2}}(\nu-m_s)^{\frac{1}{2}}}\right)\end{split} \end{equation} therefore we can see that the series \begin{equation} 2 \pi \sum_{c=2}^{\infty}\frac{1}{c} A_{c,\nu,m_s}(m) B_{c,\nu,m, r}^s\ \end{equation} converges uniformly on compacts and therefore absolutely. So can conclude that \begin{eqnarray} a_m(\nu ,r,\varepsilon)(j)&=&2 \pi \sum_{c=1}^{\infty}\frac{1}{c} \left[ \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) \left( \sum_{s=1}^p x_{js}(c,d) e^{2 \pi i m_s \frac{d'-d}{c}}\right) e^{-2 \pi i \frac{dm+d'\nu}{c}} \right] B_{c,\nu,m, \rho, \varepsilon, r}^j\\ &&=2 \pi \sum_{s=1}^p\sum_{c=1}^{\infty}\frac{1}{c} \left[ \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) \left( x_{js}(c,d) e^{2 \pi i m_s \frac{d'-d}{c}}\right) e^{-2 \pi i \frac{dm+d'\nu}{c}} \right] B_{c,\nu,m, \rho, \varepsilon, r}^j\\ &&=O \left( (m+\kappa_m)^{-\frac{3}{4}- \frac{r}{2}} e^{4 \pi(\nu -\kappa_m)^{\frac{1}{2}}(m+\kappa_M)^{\frac{1}{2}}} \right) \end{eqnarray} \end{proof} \newtheorem{cor}[thm]{Corollary} \begin{cor}The following series converges uniformly on $I_w=\{\tau : \mathcal{I}(\tau) > w > 0 \}$ \begin{equation} \sum_{m=0}^\infty a_m(\nu ,r,\varepsilon)(j)e^{2 \pi i(m+m_j) \tau} \end{equation} \end{cor} \begin{proof} \begin{eqnarray} \lefteqn{\left|a_m(\nu ,r,\varepsilon)(j)e^{2 \pi i(m+m_j) \tau} \right|}\\ && \leq C_1 \left|(m+\kappa_m)^{-\frac{3}{4}- \frac{r}{2}} e^{4 \pi(\nu -\kappa_m)^{\frac{1}{2}}(m+\kappa_M)^{\frac{1}{2}}}e^{2 \pi i(m+m_j)\tau}\right|\\ && \leq C_2m^{-\frac{3}{4}- \frac{r}{2}}e^{-2 \pi m w+4 \pi \mu^{\frac{1}{2}} (m+1)^{\frac{1}{2}}}\\ &&\leq C_3m^{-\frac{3}{4}} \quad \mbox{for $m$ large enough} \end{eqnarray} \end{proof} \begin{proof}[Proof of the Theorem] Let $F_\nu(\tau)$ be a column function in which the $j^{th}$ component is given by \begin{equation} F_\nu(\tau)(j)=e^{2\pi i(m_j-\nu)\tau}+ \sum_{m=0}^\infty a_m(\nu ,r,\varepsilon)(j)e^{2 \pi i(m+m_j) \tau} \end{equation} By the previous corollary we have that $F(\tau)(j)$ converges absolutely and therefore we can write $F(\tau)$ in the following form \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} also since the series converges, $F(\tau)$ is regular on $\mathcal{H}$. We will prove the result for $\tau=iy$ and $y>0$, and by analytic continuation the result will follow for $\tau$ on $\mathcal{H}$. Now let us rewrite the function $F_\nu(\tau)(j)$ in the following manner \begin{eqnarray*} \lefteqn{F_\nu(\tau)(j) - e^{2\pi i(m_j-\nu)\tau}}\\ &&= 2 \pi \sum_{m=0}^\infty \sum_{s=1}^p\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}) \left( x_{js}(c,d) e^{2 \pi i m_s \frac{d'-d}{c}}\right) \\ && \quad \quad e^{-2 \pi i \frac{dm+d'\nu}{c}}B_{c,\nu,m, \rho, \varepsilon, r}^je^{2\pi i(m+m_j)\tau}\\ &&=2 \pi \sum_{m=0}^\infty \sum_{s=1}^p\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}) x_{js}(c,d) e^{2 \pi i \frac {d'(m_s-\nu)}{c}- \frac {d(m_s-m_j)}{c}} \\ && \quad \quad B_{c,\nu,m, \rho, \varepsilon, r}^je^{2\pi i(m+m_j)(\tau - \frac{d}{c})} \end{eqnarray*} to continue with the proof we will need the following lemma \begin{lem}If $m_j>0$ we have \begin{equation} \sum_{m=0}^\infty B_{c,\nu,m, \rho, \varepsilon, r}^je^{2\pi i(m+m_j)(\tau - \frac{d}{c})}= \sum_{q=-\infty}^\infty e^{2\pi i m_j q}\left(-i(c\tau + d -cq) \right)^r \sum_{p=r+1}^\infty \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_j)}{c^2 \tau +c d -cq} \right)^p \end{equation} and if $m_j=0$ we have \begin{equation}\begin{split} \sum_{m=0}^\infty B_{c,\nu,m, \rho, \varepsilon, r}^je^{2\pi i(m+m_j)(\tau - \frac{d}{c})}= \frac{1}{2} \left( \frac{2 \pi \nu}{c}\right)^{r+1} \frac{1}{(r+1)!} + \\ \lim_{N\rightarrow \infty} \sum_{q=-N}^N \left(-i(c\tau + d -cq) \right)^r \sum_{p=r+1}^\infty \frac{1}{p!} \left(\frac{2 \pi i \nu }{c^2 \tau +c d -cq} \right)^p \end{split} \end{equation} \end{lem} \begin{proof}It is a simply application of the Lipschitz formula and the power series expression for $I_{r+1}$ \end{proof} The above result implies that \begin{eqnarray*} \lefteqn{F_\nu(\tau)(j) - e^{2\pi i(m_j-\nu)\tau}}\\ &&=\frac{1}{2} a_0(\nu ,r,\varepsilon) + 2 \pi \sum_{s=1}^p\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}) x_{js}(c,d) e^{2 \pi i \frac {d'(m_s-\nu)}{c}- \frac {d(m_s-m_j)}{c}} \\ && \quad \quad \lim_{N\rightarrow \infty} \sum_{q=-N}^N \left(-i(c\tau + d -cq) \right)^r \sum_{p=r+1}^\infty \frac{1}{p!} \left(\frac{2 \pi i \nu }{c^2 \tau +c d -cq} \right)^p\quad \mbox{if } m_j=0\\ &&=2 \pi \sum_{s=1}^p\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}) x_{js}(c,d) e^{2 \pi i \frac {d'(m_s-\nu)}{c}- \frac {d(m_s-m_j)}{c}}\\ && \quad \quad \sum_{q=-\infty}^\infty e^{2\pi i m_j q}\left(-i(c\tau + d -cq) \right)^r \sum_{p=r+1}^\infty \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_j)}{c^2 \tau +c d -cq} \right)^p \quad \mbox{if } m_j>0 \end{eqnarray*} \end{proof} \begin{equation} \end{equation} \begin{eqnarray} \lefteqn{} \end{eqnarray} \begin{thebibliography}{99} \bibitem{[12]} M. ~Knopp, Automorphic Forms of Nonnegative dimension and exponential sums, Michigan Mathematical Journal. 7 (1960) 257-287. \bibitem{[3]} M. ~Knopp and G. Mason, On vector-valued modular forms and their Fourier coefficients, Acta Arith. 110 (2003), no. 2, 117–124. \bibitem{[4]} M. ~Knopp and G. Mason, Vector-Valued Modular forms and Poincare Series, Illinois Journal of Mathematics 48 (2004), 1345-1366. \bibitem{[11]} H. ~Rademacher and H. Zuckerman, On the the Fourier coefficients of certain modular forms of positive dimension. Ann. Math. 39 (1938), 433-462. \end{thebibliography} \end{document}