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\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 surface {\bf 81A}}
\psfrag{H}{\small images of {\bf 27A}: $y^2 + y = x^3 - 7$}
\includegraphics[width=30em, height=55ex]{243b.eps}
\vspace{2ex}\\
The Jacobian of $X_0(243)$
\end{center}
The function field is generated by
$x_1,\ldots,x_{k-1}$
with the cubic relations
$(z_j^3-1) (x_{j-1}^3-1) = 1$
where
$z_j = \frac{x_j+2}{x_j-1}.$
Here $x_1 = \xi(\tau) = 1 + (\eta(\tau)/\eta(9\tau))^3/3$
is a rational parameter for $X_0(9)$, and $x_j=\xi(3^{j-1}\tau)$.
\comment{
\vspace{-45ex}
\begin{itemize}
\item[---]
C.~Breuil, B.~Conrad, F.~Diamond, and R.~Taylor,
\emph{On the modularity of elliptic curves over $\mathbf{Q}$}.
\item[---]
B.~Conrad, F.~Diamond, and R.~Taylor,
\emph{Modularity of certain potentially {B}arsotti-{T}ate {G}alois
representations},
J.~Amer.~Math. Soc.~\textbf{12} (1999), 521--567.
\item[---] R.\thinspace{}L. Taylor, A.~Wiles,
\emph{Ring-theoretic properties of certain Hecke algebras},
Annals of Math.\ \textbf{141} (1995),
553--572.
\item[---] A.~Wiles, \emph{Modular elliptic curves and Fermat's
Last Theorem.} Annals of Math.\ \textbf{141} (1995), 443--551.
\end{itemize}
}
\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]{icosahedronb.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}
\newpage
\begin{center}
\Large
\bf
References
\end{center}
\begin{itemize}
\item[{\bf[BCDT]}] C.~Breuil, B.~Conrad, F.~Diamond, and R.~Taylor,
\emph{On the modularity of elliptic curves over
$\mathbf{Q}$}.
\item[{\bf[BDST]}] K.~Buzzard, M.~Dickinson, N.~Shepherd-Barron,
and R.~Taylor,
\emph{On icosahedral {A}rtin representations}.
\end{itemize}
\end{document}
