Syllabus: Review of finite fields;
Polynomial equations over finite fields: theorems of Chevalley and
Warning; Quadratic Forms over prime fields. Review of the law of
quadratic reciprocity.
The ring of p-adic integers; the field of p-adic numbers; completion; padic equations and Hensels
lemma; Quadratic Forms with p-adic coefficients. Hilberts symbol.
Dirichlet series: abscissa of convergence and absolute convergence. Riemann Zeta function
and Dirichlet L-functions. Dirichlets theorem on primes in arithmetic progression. Functional
equation and Euler product for L-functions.
Modular forms and the modular group SL(2,ℝ). Eisenstein series. Zeros and poles of modular
functions. Dimensions of the spaces of modular forms. The j-invariant and Picards Theorem.
L-function and Ramanujans γ-function.
Suggested Texts :
1. J. P. Serre: A Course in Arithmetic, Springer-Verlag (1973).
2. Z. Borevich and I. Shafarevich: Number Theory (chapter 1), Academic Press (1966).
3. K. Chandrasekharan: Introduction to Analytic Number Theory, Springer- Verlag (1968).
https://www.isibang.ac.in/~adean/infsys/database/MMath/AdNT.html
- Teacher: Ramesh Sreekantan