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