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\begin{document}

\centering
{\bf Oral Qualifying Exam Syllabus}\\
Roman Holowinsky \\
romanh@math.rutgers.edu\\
October 24, 2003\\
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\raggedright
Committee (in alphabetical order): R. Bumby, H. Iwaniec, S. Miller, J. Tunnell.

\bigskip

\begin{enumerate}

\item Modular Forms
\begin{enumerate}
\item Modular Forms for the full modular group and its congruence subgroups
\item Eisenstein series 
\item Structure of the ring of modular forms
\item Mellin transform
\item Hecke operators
\end{enumerate}

\bigskip

\item Elliptic Curves
\begin{enumerate}
\item Elliptic functions and the j-invariant
\item Elliptic curves over the complex field
\item Elliptic curves over finite fields, Hasse-Weil Theorem
\item Hasse-Weil $L$-functions of elliptic curves
\item Mordell-Weil Theorem and descent on elliptic curves
\end{enumerate}

\bigskip

\item Analytic Number Theory
\begin{enumerate}
\item Analytic properties of $L$-functions and the Riemann zeta functions
\item Primes in arithmetic progression
\item Siegel zero problem
\item Prime number theorem and prime number theorem for arithmetic progressions
\end{enumerate}

\bigskip

\item Algebraic Number Theory
\begin{enumerate} 
\item Invariants of number fields: rings of integers, discriminants and orders
\item Arithmetic of number fields: splitting of primes, ramification
\item Class groups
\item Structure of units in number rings
\end{enumerate}

\end{enumerate}

\end{document}







