Number theory

Syllabus

1. Arithmetic modulo p, Gauss sums, quadratic reciprocity, Riemann zeta function: meromorphic continuation, functional equation, Dirichlet L-functions
2. Galois theory
3. Arithmetic in number fields: ideals, units
4. Discriminants, ideal class groups
5. Splitting of ideals, computation of ideal class groups
6. Pell's equation, Dirichlet's unit theorem
7. Fermat's last theorem for regular primes, Selmer's equation
8. Hilbert 90, Kummer theory, main theorem of abelian class field theory, 3-rank of ideal class groups of quadratic fields
9. Zeroes of the Riemann zeta function
10. Tauberian theorems, Prime Number Theorem, primes in arithmetic progressions
11. Convexity bounds
12. Height zeta functions, algebraic aspects
13. Height zeta functions, analytic aspects

Projects

1. Primes:
   • Agrawal, M., Kayal, N., Saxena, N. PRIMES is in P, Ann. of Math. (2) 160 (2004), no. 2, 781--793.
2. Inverse Galois problem:
   • Serre, J.-P. Topics in Galois theory, Second edition. With notes by Henri Darmon. Research Notes in Mathematics, 1. A K Peters, Ltd., Wellesley, MA, 2008.
3. Class groups:
   • Cohen, H., Lenstra, H. W., Jr. Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), 33--62, Lecture Notes in Math., 1068, Springer, Berlin, 1984.
4. Discriminants:
   • Bhargava, M. The density of discriminants of quartic rings and fields, Ann. of Math. (2) 162 (2005), no. 2, 1031--1063.
   • Ellenberg, J., Venkatesh, A. The number of extensions of a number field with fixed degree and bounded discriminant, Ann. of Math. (2) 163 (2006), no. 2, 723--741.
5. Gauss class number one problem:
   • Goldfeld, D., The Gauss Class Number problem for Imaginary Quadratic Fields
   • Stark, H. M., The Gauss class-number problems, Analytic number theory, 247--256, Clay Math. Proc., 7, Amer. Math. Soc., Providence, RI, 2007.

Books