2023/2024

Introduction to Algebriac Numbers and Class Field Theory

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

When: 1, 2 module

Open to: students of all HSE University campuses

Instructors: Vladimir Zhgoon

Language: English

ECTS credits: 6

Contact hours: 60

Course Syllabus

Abstract

Algebraic number theory is a classical area of mathematics, appeared during the study of solutions to Diophantine equations, and was developing during attempts to prove Fermat's theorem. It is now a vast classical field of knowledge underlying Arithmetic geometry. This course is a continuation of the basic course in number theory. We will study the filtration of the Galois group: namely, the decomposition subgroup, inertia, and higher ramification groups and their norm maps. We learn about Galois cohomology of fields, as well as local and global class field theory. Time permitting, we will discuss an algebra-geometric analogue of this theory that allows us to describe Abelian coverings of curves in terms of their Jacobians. We will also talk about Arakelov's geometry, which allows us to construct a "compactification" of a curve over a ring of algebraic integers.