1% hot topics t-shirt, MSRI 1999, William A. Stein
2\documentclass{article}
3\newcommand{\Gal}{\mbox{\rm Gal}}
4\newcommand{\GL}{\mbox{\rm GL}}
5\newcommand{\C}{\mathbf{C}}
6\newcommand{\Q}{\mathbf{Q}}
7\newcommand{\Qbar}{\overline{\Q}}
8\usepackage{graphicx}
9\usepackage{psfrag}
10\pagestyle{empty}
11\textwidth=1.02\textwidth
12\begin{document}
13\large
14
15\begin{center}
16\Large
17\sc
18Modularity of Elliptic Curves and Beyond\\
19\LARGE MSRI 1999
20\end{center}
21\vspace{3ex}
22
23\noindent{\bf M\normalsize{}ODULARITY THEOREM:}
24{\em Every elliptic curve over~$\mathbf{Q}$
25is \nobreak{modular}.}\vspace{1ex}\\
26The proof follows a program initiated by R.~Taylor and A.~Wiles.
27For further details see C.~Breuil, B.~Conrad, F.~Diamond, and R.~Taylor,
28\emph{On the modularity of elliptic curves over $\mathbf{Q}$}.
29\vspace{.5ex}
30
31\begin{center}
32\psfrag{2}{$2$}
33\psfrag{3}{$3$}
34\psfrag{A}{\small{\bf 243A}: $y^2+y=x^3-1$}
35\psfrag{B}{\small{\bf 243B}: $y^2+y=x^3+2$}
36\psfrag{C}{\small{\bf 243C}: surface}
37\psfrag{D}{\small{\bf 243D}: surface}
38\psfrag{E}{\small{\bf 243E}: three-fold}
39\psfrag{F}{\small{\bf 243F}: three-fold}
40\psfrag{G}{\small images of {\bf 81A}: four-fold}
41\psfrag{H}{\small images of {\bf 27A} ($y^2 + y = x^3 - 7$): three-fold}
42\includegraphics[width=30em, height=55ex]{243.eps}
43\vspace{2ex}\\
44The Jacobian of $X_0(243)$
45\end{center}
46
47\newpage
48\noindent{\bf A\normalsize{}RTIN'S CONJECTURE:}
49The $L$-series of any continuous representation
50$\Gal(\Qbar/\Q)\rightarrow\GL_n(\C)$ is entire, except
51possibly at~$1$.
52\vspace{1ex}
53
54\begin{center}
55\includegraphics[width=20em]{icosahedron.eps}
56\end{center}
57
58\vspace{-53ex}
59\noindent{\sc Results:}
60\begin{itemize}
61 \item[---] E.~Artin,
62            \emph{{\"U}ber eine neue {A}rt von {L}-Reihen},
63            Abh.\ Math.\ Sem.\ Univ.\ Hamburg
64            \textbf{3} (1924), 89--108.
65 \item[---] J.~Buhler,
66              \emph{Icosahedral Galois representations},
67               LNM 654, 1978.
68 \item[---] K.~Buzzard, M.~Dickinson, N.~Shepherd-Barron, and R.~Taylor,
69              \emph{On icosahedral {A}rtin representations}.
70 \item[---] G.~Frey, \emph{On Artin's conjecture for odd $2$-dimensional
71              representations}, LNM 1585, 1994.
72 \item[---] E.~Hecke, \emph{Eine neue Art von Zetafunktionen und ihre
73              Beziehungen zur Verteilung der Primzahlen},
74              Math.~Z.\ \textbf{6} (1920), 11--51.
75 \item[---] R.\thinspace{}P. Langlands,
76              \emph{Base change for $\GL(2)$},
77              Princeton University Press, Princeton, 1980.
78 \item[---] J.\ Tunnell,
79              \emph{Artin's conjecture for representations of
80              octahedral type},
81              Bull.\ AMS \textbf{5} (1981), 173--175.
82\end{itemize}
83\end{document}
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