Course Archives Theoretical Statistics and Mathematics Unit

Course: Advanced Number Theory Level: Postgraduate Time: Currently not offered

Syllabus 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). Top of the page Past Exams

Midterm 11.pdf 19.pdf Semestral 11.pdf 19.pdf 21.pdf Supplementary and Back Paper Top of the page

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