\documentclass[11pt]{article}
\voffset=-0.1\textheight
\textheight=1.2\textheight
\input{macros}
%\newcommand{\zmod}[1]{\Z/#1\Z}
\newcommand{\zmod}[1]{\F_{#1}}

\title{Math 168: Homework Assignment 6}
\author{William Stein}
\date{\bf Due: Wednesday, Nov 9, 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:grpact}
Let $k$ be an integer, and for any function
$f:\h^*\to \C$ and $\gamma=\abcd{a}{b}{c}{d}\in\GL_2(\Q)$,
set $f|[\gamma]_k(z) = (cz+d)^{-k} f(\gamma(z))$. 
Prove that if $\gamma_1, \gamma_2 \in \SL_2(\Z)$,
then for all $z\in \h$ we have 
$$
 f|[\gamma_1 \gamma_2]_k(z)  = ((f|[\gamma_1]_k)|[\gamma_2]_k)(z).
$$
(I mostly did this on the blackboard in class.)

\item \label{ex:sl2ztrans}
Prove that for any $\alpha, \beta\in\P^1(\Q)$, there exists
  $\gamma\in\SL_2(\Z)$ such that $\gamma(\alpha) = \beta$.

\item \label{ex:expeis} By hand write down the coefficients of $1$,
  $q$, $q^2$, and $q^3$ of the Eisenstein series $E_8$.

\item \label{ex:vm} Explicitly compute the Victor Miller basis for
$M_{28}(\SL_2(\Z))$.

\item Let $f = E_4 E_6 \Delta$. 
\begin{enumerate}
\item What is the weight of $f$?
\item Determine the location of all the zeros of $f$. 
\end{enumerate}

\end{enumerate}
\end{document}
%%% Local Variables: 
%%% mode: latex 
%%% TeX-master: t
%%% End: 
