M471 - Advanced Number Theory

Course No: 
M207, M307, M308
Review of Finite fields, Gauss Sums and Jacobi Sums, Cubic and biquadratic reciprocity, Polynomial equations over finite fields, Theorems of Chevally and Warning, Quadratic forms over prime fields. Ring of p-adic integers, Field of p-adic numbers, completion, p-adic equations, Hensel’s lemma, Hilbert symbol, Quadratic forms with p-adic coefficients. Dirichlet series: Abscissa of convergence and absolute convergence, Riemann Zeta function and Dirichlet L-functions. Dirichlet’s theorem on primes in arithmetic progression. Functional equation and Euler product for L-functions. Modular Forms and the Modular Group, Eisenstein series, Zeros and poles of modular functions, Dimensions of the spaces of modular forms, The j-invariant L-function associated to modular forms, Ramanujan τ function.
Reference Books: 
  1. J.-P. Serre, “A Course in Arithmetic”, Graduate Texts in Mathematics 7, Springer-Verlag, 1973.
  2. K. Ireland, M. Rosen, “A Classical Introduction to Modern Number Theory”, Graduate Texts in Mathematics 84, Springer-Verlag, 1990.
  3. H. Hasse, “Number Theory”, Classics in Mathematics, Springer-Verlag, 2002.
  4. W. Narkiewicz, “Elementary and Analytic Theory of Algebraic Numbers”, Springer Monographs in Mathematics, Springer-Verlag, 2004.
  5. F. Q. Gouvˆea, “p-adic Numbers”, Universitext, Springer-Verlag, 1997.

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School of Mathematical Sciences

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