Course Archives Theoretical Statistics and Mathematics Unit
Course: Special Topics - Class Field Theory
Level: Postgraduate
Time: Currently not offered
Past Exams


The aim of this course is to introduce class field theory and related topics. The syllabus and how much we cover will depend on time and motivation.
Below is an outline of the topics which are most likely to be treated, not necessarily in the order given below.
1. Review of absolute values, completions, local fields, Ostrowskis theorem (from Algebraic Number Theory course).
2. Global fields, ring of adeles, ideles; compactness of idele class group.
3. Applications: finiteness of class group, Dirichlet unit theorem.
4. Cohomology of finite groups.
5. Local Class Field Theory.
6. Global Class Field Theory.
7. Pontryagin Duality, Fourier series, Poisson summation formula.
8. Tates Thesis.
9. Artin L-functions; introduction to automorphic forms and automorphic representations on GL2.

Suggested Texts :
(a) J. Neukirch, Class Field Theory, the Bonn Lectures, edited by Alexander Schmidt (Springer-Verlag, 2013)
(b) E. Artin & J. Tate, Class Field Theory (American Mathematical Society, 2009).
(c) J.S. Milne, Class Field Theory, lecture notes available at (2020).

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