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\textbf{\large{ORAL QUALIFYING EXAM}}\\[1cm] 

 September 30   2003\\[0.7cm]

Satadal Ganguly \\

(email: satadal@math.rutgers.edu)\\[0.6cm]

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Committee (in alphabetical order) : R. T. Bumby, H. Iwaniec (chair),
S. D. Miller, J. Tunnell\\[1cm]




\textbf{Analytic Number Theory}\\ 

a) Analytic properties of Riemann Zeta function and Dirichlet L-functions\\

b) Dirichlet's Theorem on primes in arithmetic progressions\\

c) Prime Number Theorem. The proof by Hadamard and de la Vallee Poussin\\

d) Zero-free regions of Dirichlet L-functions and the exceptional zero problem\\[0.6cm]

\textbf{Algebraic Number Theory}\\

a) Ring of integers in a number field. Dedekind's Ideal theory\\

b) Decomposition of a prime ideal in a finite seperable extension.
 Hilbert's Ramification theory\\

c) Class group and finiteness of class number\\ 

d) Dirichlet's theorem on units\\[0.6cm]

\textbf{Modular Forms}\\

a) Modular forms for the full modular group and its congruence subgroups\\

b) Eisenstein series\\

c) Structure of the ring of modular forms\\

d) Hecke operators\\[0.6cm]

\textbf{Elliptic Curves}\\

a) Elleptic curves over $\mathbb{C}$. Elliptic functions, j-invariant\\

b) Elliptic curves over finite fields. Hasse's theorem\\

c) Elliptic curves over global fields. Mordell's theorem.\\

d) Hasse-Weil L-functions\\

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