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{\bf Oral Qualifying Exam Syllabus}\\
Goran Djankovi\'c \\
 April 2006


\bigskip
\raggedright Committee (in alphabetical order):  H. Iwaniec, S.
Miller, J.Tunnell, C. Weibel

\bigskip

\begin{enumerate}


\item { \textbf{Automorphic Forms and Spectral Theory}}

\begin{enumerate}
\item The hyperbolic plane as a homogeneous space and the
classification of motions
\item Fuchsian groups and fundamental domains
\item Congruence subgroups
\item The double coset decomposition; Kloosterman  sums
\item Laplace and Maass operators
\item Automorphic forms- holomorphic and Maass; \\Fourier expansions
of automorphic forms
\item Eisenstein and Poincar\'e series (holomorphic and spectral)
\item Hilbert space of holomorphic cusp forms; $\mathcal{C}({\Gamma} \backslash \mathbb{H})=\mathcal{L}^2({\Gamma} \backslash \mathbb{H})\ominus \mathcal{E}({\Gamma} \backslash \mathbb{H})$
\item Petersson's formula
\item The Riemann-Roch theorem and the dimension of the linear space
of holomorphic modular forms
\item Hecke operators; Atkin-Lehner theory
\item Automorphic $L$-functions - classical $GL(2)$-theory
\item Automorphic Green function
\item Analytic continuation and functional equations for the Eisenstein series
\item The spectral theorem
\item Selberg's trace formula



\end{enumerate}



\item \textbf{Analytic Number Theory}

\begin{enumerate}
\item Primes in arithmetic progressions-Dirichlet's theorem
\item The Riemann zeta function
\item Dirichlet $L-$functions
\item Counting zeros of $L-$functions
\item The zero-free region; Landau-Siegel theorem
\item Explicit formulas
\item Prime number theorem and PNT for arithmetic progressions
\item Bilinear forms and the large sieve inequalities
\item Bombieri-A.I. Vinogradov theorem

\end{enumerate}

\bigskip

\item \textbf{Algebraic Number Theory}

\begin{enumerate}


\item Lattices and orders in number fields; discriminant of a
lattice in a number field
\item Minkowski's lattice point theorem; Minkowski's bound;
Hermite's theorem
\item Class group
\item Dirichlet's unit theorem
\item The different and ramification
\item Splitting of primes in Galois extensions


\end{enumerate}

\bigskip

\item  \textbf{Elliptic Curves}
\begin{enumerate}
\item Elliptic curves- group law and isogenies;
\item Elliptic functions and elliptic curves over $\mathbb {C}$
\item Elliptic curves over finite fields; Hasse's theorem
\item Hasse-Weil $L$-functions of elliptic curves
\end{enumerate}



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