Syllabus
Completions, adeles and ideles
Main theorems of class field theory, Hilbert class field
Artin map
Proof tools: L-functions and group cohomology
Tchebotareff density theorem
Hasse local-global principle for quadratic forms
Higher reciprocity laws, Hilbert symbol
Local class fields: Lubin-Tate theory
Annotation
Class field theory, which provides a substantial generalization of quadratic reciprocity law, forms the foundation for many advanced areas of number theory including Langlands program. In principle, it is concerned with the description of abelian extensions of number fields and p-adic numbers.
In the course we will explain the main statements of this theory for global or local fields including the main tools for their proofs and various applications, mostly on the structure of number fields and quadratic forms. Specific choice of topics will depend on the interests of participants.