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\begin{document}
\title{\bf{Proposal}}
\author{Wissam Raji}
\date{June 2, 2005}
\maketitle

\noindent
{\setlength{\baselineskip}%
{1.5\baselineskip}C. Siegel introduced a new technique using
Residue Calculus to prove the transformation law of the Dedekind
eta function defined as
\begin{equation*}
\eta(\tau)=e^{\frac{\pi i \tau}{12}}\prod_{n=1}^\infty (1-e^2\pi i
n\tau)
\end{equation*}
where $\tau$ is in the upper half plane [7]. The transformation
law of the Dedekind eta function is given by
\begin{equation*}
\eta\left(\frac{-1}{\tau}\right)=(-i\tau)^{\frac{1}{2}}\eta(\tau)
\end{equation*}
\\
H. Rademacher [5] widened Siegel's method to derive the general
transformation law, given by
\begin{equation*}
\eta\left(\frac{a\tau + b}{c\tau +
d}\right)=\epsilon(a,b,c,d)\{-i(c\tau+d)^{\frac{1}{2}}\}\eta(\tau)
\end{equation*}
where
\begin{equation*}
\epsilon(a,b,c,d)=exp\left\{\pi i\left(\frac{a+d}{12c}+
s(-d,c)\right)\right\},
\end{equation*}

\begin{equation*}
s(h,k)=\sum_{r=1}^{k-1}\frac{r}{k}\left(\frac{hr}{k}-\left[\frac{hr}{k}\right]-\frac{1}{2}\right)
\end{equation*}
for all \[ \left(\begin{array}{lcr}
\ a  &b \\
\ c  &d \\
\end{array}\right) \in \Gamma\]\\
The first part of my thesis introduces a new proof of the
transformation law of the Jacobi $\theta_3(w,\tau)$ function
defined in the upper half plane.  The proof is inspired by
Siegel's proof and Rademacher's generalization. The Jacobi
function $\theta_3(w, \tau)$ is defined as follows,
\begin{equation*}
\theta_3(w,\tau)=\prod_{n=1}^{\infty}(1-q^{2n})(1+2 q^{2n-1}\cos
2w +q^{4n-2})\enspace
\end{equation*}
where $q=e^{\pi i \tau}$,$\tau$ is in the upper half plane and $w$
is any complex number. Letting $w=0$ in $\theta_3(w,\tau)$ leads
us to the very important function $\theta_3(\tau)$. M. Knopp
pointed out an important connection between $\theta_3(\tau)$ and
$\eta(\tau)$ [2]. For $\tau \in H$, we have
\begin{equation*}
\theta_3(\tau)=\eta^{2}\left(\frac{\tau+1}{2}\right)/\eta(\tau+1),
\end{equation*}
\\
the right-hand side being a simple example of an "eta-product."
\\
\\
Eta products appear in many areas of mathematics in which algebra
and analysis overlap [1]. M. Newman published a pair of well-known
papers aimed at using eta-product to construct forms on the group
$\Gamma_0(n)$ with the trivial multiplier system. In our work we
shall pay no attention to the multiplier system, but shall be
concerned only with the orders of the resulting eta product at the
cusps [3,4].
\\
\\
\\
Modular forms are frequently constructed as products of
transforms of the Dedekind eta function, where a transform means
function arising from a composition of eta with a linear
fractional transformation with integral coefficients. A. Biagioli
[1] investigated the limits of this technique, obtaining a
criterion for determining whether a modular form can be so
represented, for the family of all such modular forms that are
automorphic forms on a given subgroup of the modular group.
His methods involve consideration of the orders of the eta-product at the cusps.\\
\\
In investigating a class of functions formed by generalized
Dedekind eta products, S. Robins studied when those products live
on the Riemann surface $X_1(N)=\Gamma_1(N)/H$.  He obtained a
simple necessary and sufficient condition for this [6]. M. Newmann
and H. Stark have found similar results for Dedekind eta products
on $X_0(N)=\Gamma_0(N)/H$ [8] where \[
\Gamma_0(N)=\left\{\left(\begin{array}{lcr}
\ a  &b \\
\ c  &d \\
\end{array}\right): c\equiv 0 \mod N, ad-bc=1 \right\}, \]\\
and $\Gamma_1(N)$ is a subgroup of $\Gamma_0(N)$ satisfying
\[
\Gamma_1(N)=\left\{\left(\begin{array}{lcr}
\ a  &b \\
\ c  &d \\
\end{array}\right): c\equiv 0 \mod N, a\equiv d\equiv 1\mod N \right\}. \]\\
\\
The goal of my thesis is to:
\begin{itemize}
   \item
    Extend Siegel's method to a large class of modular forms defined as products.
    \item
    study the behavior of general eta
    products.
    \item
     attempt to determine which modular forms of
half integral or integral weight are eta products.
\end{itemize}
\newpage
\begin{thebibliography}{99}
    \bibitem{[1]}
A.~Biagioli, The construction of modular forms as products of
transforms of the Dedekind eta function, Acta Arithmetica LIV 4
(1990).
\bibitem{[2]} M.~Knopp , Modular Functions, AMS-Chelsea Publishing Company, 2nd edition 1993.
    \bibitem{[3]}
M.~Newman,Construction and application of a certain class of
modular functions, Proc. London Math. Soc., (3), 7(1956), 334-350.
    \bibitem{[4]}
M.~Newman,Construction and application of a certain class of
modular functions II, ibid. 9(1959), 373-387.
    \bibitem{[5]}
H.~Rademacher, On the transformation of $\log \eta(\tau)$. J.
Indian Math Soc. 19(1955), 25-30.
    \bibitem{[6]}
S.~Robins, Generalized Dedekind $\eta$-Products, J. of contem.
Math.,
    \bibitem{[7]}
C.~Siegel, A simple proof of
$\eta(-\frac{1}{\tau})=\sqrt{\frac{\tau}{i}}\eta(\tau)$, J.
Mathematika 1(1954), 4.
    \bibitem{[8]}
H. Stark, On the minimal level of modular forms, Analytic number
theory, Proc. of a conference in honor of Paul Bateman 85(1990),
479-491.
\end{thebibliography}
\end{document}