Class Field Theory

Content

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Description

Time: Mondays, 09.00 - 11.00

Location for Maxwell Institute students: Bayes Centre (room 5.46)

Module Leader: James Rawson (University of Glasgow)

Module Summary

Class field theory is the study of abelian extensions of fields. These naturally arise in number theory as they represent studying the "easiest" part of the (absolute) Galois group, its abelianization. Such questions can also be of interest to algebraic geometers, where these extensions describe coverings of varieties with abelian automorphism group. 

The main focus of this course will be local class field theory, which studies the class field theory of local fields - a class of fields equipped with a topology, which includes the p-adic numbers and the field of Laurent series over a finite field.

The goal will be to describe the structure of these local fields and to give an explicit description of their abelian extensions via Lubin—Tate theory. This entails the study of formal groups, groups defined via certain power series, which appear not just in number theory, but also in other branches of mathematics such as algebraic topology (via homotopy theory).

The course will end with a discussion of the main results for number fields ("global class field theory") and for function fields of varieties ("geometric class field theory").

Assessment

TBC