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.
- J.-P. Serre, “A Course in Arithmetic”, Graduate Texts in Mathematics 7, Springer-Verlag, 1973.
- K. Ireland, M. Rosen, “A Classical Introduction to Modern Number Theory”, Graduate Texts in Mathematics 84, Springer-Verlag, 1990.
- H. Hasse, “Number Theory”, Classics in Mathematics, Springer-Verlag, 2002.
- W. Narkiewicz, “Elementary and Analytic Theory of Algebraic Numbers”, Springer Monographs in Mathematics, Springer-Verlag, 2004.
- F. Q. Gouvˆea, “p-adic Numbers”, Universitext, Springer-Verlag, 1997.