\documentclass{article}

\newcommand{\blank}[1]{\leavevmode\hbox to#1em{\hrulefill}}


\begin{document} \pagestyle{empty}

\begin{center}
  {\large\bfseries{}Outline for Preliminary Examination}
\end{center}

\noindent Examiners:   Brian Conrad \& Chris Skinner

\noindent Student: Bryden Cais

\noindent Date: {\bf August 10, 2004}

\noindent Time: 2--4 pm.

\subsection*{Algebraic number theory}

\begin{itemize}
  \item Basic algebraic number theory (local and global fields)
  \item Local class field theory
  \item Id\'elic and ideal-theoretic formulation of global class field theory
  \item Ray class groups: definitions and descriptions in the above formulations
  \item Kronecker-Weber theorem
  \item Chebotarev Density Theorem
  \item Gr\"o\ss{}encharaktere
  \item Artin L-functions (definitions)
  \item Examples (quadratic fields, cyclotomic fields)
  \item $\mathbf{Z}_p$-extensions
\end{itemize}

\subsection*{Algebraic geometry}

\begin{itemize}
  \item Basics of varieties
  \item Basics of sheaves and schemes
  \item Coherent cohomology of schemes
  \item Curves (genus, Riemann-Roch Theorem, Hurwitz genus formula, Picard group, etc.)
  \item Curves of genus zero (over any field)
  \item Elliptic curves (over any field)
\end{itemize}

\subsection*{Abelian Varieties}

\begin{itemize}
	\item Basics of abelian varieties
	\item Isogeny-invariance of BSD (no restriction on ground field) as in Milne's ADT, \S 7. 
\end{itemize}

\vfill

\noindent\begin{tabular}{@{}c}
  \blank{13} \\
  Brian Conrad \\
\end{tabular} \hspace{\fill}
\begin{tabular}{c@{}}
  \blank{13} \\
  Chris Skinner \\
\end{tabular}

\end{document}
