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\title{Oral Qualifying Examination Syllabus}
\author{Sa\v{s}a Radomirovi\'c}
\date{}

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Committee: J. Tunnell, H. Iwaniec, S. Miller
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\section*{Algebraic Number Theory}

\begin{enumerate}
\item Invariants of number fields: embeddings, discriminants and orders
\item Arithmetic of orders in number fields: fractional ideals, splittings of primes, ramification
\item Ideal Class Group of number fields
\item Structure of units in orders of number fields
\end{enumerate}

\section*{Elliptic Curves}

\begin{enumerate}
\item The group law for adding points on an elliptic curve 
\item The group of rational solutions to $E : y^2 = x^3 + Ax + B$ form a
  finitely generated group (the Mordell-Weil group of $E$) 
\item Effective bounds on the rank of the Mordell-Weil group
\item Conjectural effective algorithms to find all rational solutions to 
   $y^2 = x^3 + Ax + B$ 
\item $L$--series of elliptic curves and relations to modular curves
\item Applications to classical Diophantine problems 
\end{enumerate}


\section*{Modular Forms}

\begin{enumerate}
\item Elliptic Modular Curves as Riemann Surfaces
\item Elliptic Functions
\item Modular Functions and Modular Forms
\item Hecke Operators
\end{enumerate}

\section*{Automorphic Forms}

\begin{enumerate}
\item Discontinuous groups
\item Automorphic Forms
\item Eisenstein series 
\item Spectral theorem
\item Trace formula
\item Estimates for the Fourier coefficients of cusp forms
\item Automorphic $L$--functions
\end{enumerate}

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