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\begin{document}

\centering
{\bf Oral Qualifying Exam Syllabus}\\
John Bryk \\
jbryk@math.rutgers.edu\\
Week of Tuesday, January 17, 2006(tentative)\\
\bigskip
\raggedright
Committee (in alphabetical order): J. Beck, H. Iwaniec, J. Tunnell, C. Weibel.

\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 Hecke operators
\item Automorphic $L$-series
\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 Elliptic curves over local fields
\item Elliptic curves over the rational numbers, Mordell-Weil Theorem, and descent on elliptic curves
\item Hasse-Weil $L$-functions of elliptic curves
\end{enumerate}

\bigskip

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

\bigskip

\item Algebraic Number Theory
\begin{enumerate} 
\item Invariants of number fields: rings of integers, different, and discriminant
\item Dedekind domains
\item Arithmetic of number fields: splitting and ramification of primes
\item Class groups, finiteness of class number
\item Dirichlet's Unit Theorem
\end{enumerate}

\end{enumerate}

\begin{thebibliography}{9999}

\bibitem[TA1]{Apostol1}
T. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.

\bibitem[TA2]{Apostol2}
T. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1976.

\bibitem[HI]{Iwaniec}
H. Iwaniec, prime numbers course notes, Fall, 2005.

\bibitem[SL]{Lang}
S. Lang, Algebraic Number Theory, Springer-Verlag, 1986.

\bibitem[TM]{Miyake}
T. Miyake, Modular Forms, Springer-Verlag, 1989. (selected topics)

\bibitem[JS]{Silverman}
J. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1986. (basic parts)

\bibitem[JT1]{Tunnell1}
J. Tunnell, elliptic curves course notes, Fall, 2004.

\bibitem[JT2]{Tunnell2}
J. Tunnell, modular forms course notes, Spring, 2005.

\end{thebibliography}

\end{document}







