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\title{Math 168: Homework Assignment 5}
\author{William Stein}
\date{\bf Due: Wednesday, Nov 2, 2005}
\begin{document}
\maketitle

\noindent{\em The problems have equal point value, and multi-part
  problems are of the same value.}

\section{Problems}

\begin{enumerate}

\item\label{ex:upperhalfpres}
Suppose $\gamma = \abcd{a}{b}{c}{d}$ is a matrix with real entries
and positive determinant.  Prove that if $z\in\C$ is a complex
number with positive imaginary part, then the imaginary
part of $\gamma(z) = (az+b)/(cz+d)$ is also positive.

\item\label{ex:meromorphic}
\begin{enumerate}
\item Prove that a polynomial is an analytic function on $\C$.
\item Prove that a rational function (quotient of two polynomials)
is a meromorphic function on $\C$.
\end{enumerate}

\item \label{ex:wmfprod}
Suppose $f$ and $g$ are weakly modular functions with $f\neq 0$.
\begin{enumerate}
\item Prove that the product $fg$ is a weakly modular function.
\item Prove that $1/f$ is a weakly modular function.
\item If $f$ and $g$ are modular functions, show that $fg$
is a modular function.
\item If $f$ and $g$ are modular forms, show that $fg$
is a modular form.
\end{enumerate}

\item \label{ex:nomodformodd}
Suppose $f$ is a weakly modular function of odd weight $k$.
Show that $f=0$.

\item \label{ex:conggamma1}
\begin{enumerate}
\item Prove that $\Gamma_1(N)$ is a group.
\item Prove that $\Gamma_1(N)$ has finite index in $\SL_2(\Z)$
(Hint: it contains the kernel of the homomorphism
$\SL_2(\Z) \to \SL_2(\Z/N\Z)$.)
\end{enumerate}

    
\end{enumerate}
\end{document}
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