% hot topics t-shirt, MSRI 1999, William A. Stein
\documentclass{article}
\newcommand{\Gal}{\mbox{\rm Gal}}
\newcommand{\GL}{\mbox{\rm GL}}
\newcommand{\C}{\mathbf{C}}
\newcommand{\Q}{\mathbf{Q}}
\newcommand{\Qbar}{\overline{\Q}}
\usepackage{graphicx}
\usepackage{psfrag}
\pagestyle{empty}
\textwidth=1.02\textwidth
\begin{document}
\large
\begin{center}
\Large
\sc
Modularity of Elliptic Curves and Beyond\\
\LARGE MSRI 1999
\end{center}
\vspace{3ex}
\noindent{\bf M\normalsize{}ODULARITY THEOREM:}
{\em Every elliptic curve over~$\mathbf{Q}$
is \nobreak{modular}.}\vspace{1ex}\\
{\sc Proof.}
The proof follows a program initiated by Wiles
and Taylor-Wiles. See C.~Breuil,
B.~Conrad, F.~Diamond, and R.~Taylor,
\emph{On the modularity of elliptic curves over $\mathbf{Q}$}.
\vspace{.5ex}
\begin{center}
\psfrag{2}{$2$}
\psfrag{3}{$3$}
\psfrag{A}{\small{\bf 243A}: $y^2+y=x^3-1$}
\psfrag{B}{\small{\bf 243B}: $y^2+y=x^3+2$}
\psfrag{C}{\small{\bf 243C}: surface}
\psfrag{D}{\small{\bf 243D}: surface}
\psfrag{E}{\small{\bf 243E}: three-fold}
\psfrag{F}{\small{\bf 243F}: three-fold}
\psfrag{G}{\small images of {\bf 81A}: four-fold}
\psfrag{H}{\small images of {\bf 27A} ($y^2 + y = x^3 - 7$): three-fold}
\includegraphics[width=30em, height=55ex]{243.eps}
\vspace{2ex}\\
The Jacobian of $X_0(243)$
\end{center}
\newpage
\noindent{\bf A\normalsize{}RTIN'S CONJECTURE:}
The $L$-series of any continuous representation
$\Gal(\Qbar/\Q)\rightarrow\GL_n(\C)$ is entire, except
possibly at~$1$.
\vspace{1ex}
\begin{center}
\includegraphics[width=20em]{icosahedron.eps}
\end{center}
\vspace{-53ex}
\noindent{\sc Results:}
\begin{itemize}
\item[---] E.~Artin,
\emph{{\"U}ber eine neue {A}rt von {L}-Reihen},
Abh.\ Math.\ Sem.\ Univ.\ Hamburg
\textbf{3} (1924), 89--108.
\item[---] J.~Buhler,
\emph{Icosahedral Galois representations},
LNM 654, 1978.
\item[---] K.~Buzzard, M.~Dickinson, N.~Shepherd-Barron, and R.~Taylor,
\emph{On icosahedral {A}rtin representations}.
\item[---] G.~Frey, \emph{On Artin's conjecture for odd $2$-dimensional
representations}, LNM 1585, 1994.
\item[---] E.~Hecke, \emph{Eine neue Art von Zetafunktionen und ihre
Beziehungen zur Verteilung der Primzahlen},
Math.~Z.\ \textbf{6} (1920), 11--51.
\item[---] R.\thinspace{}P. Langlands,
\emph{Base change for $\GL(2)$},
Princeton University Press, Princeton, 1980.
\item[---] J.\ Tunnell,
\emph{Artin's conjecture for representations of
octahedral type},
Bull.\ AMS \textbf{5} (1981), 173--175.
\end{itemize}
\end{document}
