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\begin{center} {\bf\large Oral Qualifying Examination Syllabus} \\
Liangyi Zhao \\ lzhao@math.rutgers.edu \\ \today \end{center}


\noindent
Committee(in alphabetical order): R. Bumby, H. Iwaniec, J. Tunnell.

\begin{enumerate}
\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,
Frobenius
\item Class groups
\item Structure of units in number rings
\item Binary quadratic forms
\end{enumerate}

\item Analytic Number Theory
\begin{enumerate}
\item Elementary estimates of standard arithmetic functions
\item Analytic properties of $L$-functions and the Riemann zeta functions
\item Primes in arthmetic progression
\item Siegel zero problem
\item Prime number theorem and Prime Number Theorem for Arithmetic Progressions
\item The large sieve inequality for multiplicative characters
\end{enumerate}

\item Elliptic Curves
\begin{enumerate}
\item Elliptic curves over the complex field: Elliptic functions and
the $j$-function
\item Elliptic curves over finite fields
\item Hasse-Weil $L$-functions of elliptic curves
\item Mordell-Weil Theorem and descent on elliptic curves
\end{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 transforms and Dirichlet $L$-series
\item Hecke operators
\item Modular forms of half integer weight
\end{enumerate}

\end{enumerate}

\begin{thebibliography}{9999}

\bibitem[JC]{Cassels}
J. Cassels, Lectures on Ellipitc Curves, Cambridge, 1991,

\bibitem[HD]{Davenport}
H. Davenport, Multiplicative Number Theory, Springer-Verlag, 2000.

\bibitem[HI]{Iwaniec}
H. Iwaniec, Course Notes.

\bibitem[NK]{Koblitz}
N. Kobliz, Introduction to Elliptic Curves and Modular Forms,
Springer, 1993.

\bibitem[DM]{Marcus} 
D. Marcus, Number Fields, Springer-Verlag, 1997.

\bibitem[HM]{Montgomery} 
H. Montgomery, Topics in Multiplicaitive Number Theory,
Springer-Verlag, 1997.

\bibitem[JT]{Tunnell}
J. Tunnell, course notes, Fall, 2000.

\end{thebibliography}

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