Course Archives Theoretical Statistics and Mathematics Unit
Course: Advanced Number Theory
Level: Postgraduate
Time: Currently not offered
Past Exams

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).

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Past Exams
11.pdf 19.pdf
11.pdf 19.pdf
Supplementary and Back Paper

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