SoSe 2021

Algebraic Number Theory II

Lecture notes

Weekly material

Week 1
Week 2
Week 3

Content:

L-functions are of central importance for modern number theory. In this course, we will treat various different aspects of L-functions.

In the first part of this lecture, we will discuss the Riemann zeta function and its basic properties. Among other things, we will deal with the prime number theorem, prove Euler's formula for the values of the Riemann zeta function at the positive even integers and show the irrationality of zeta(3).

In the second part, we turn our attention to cyclotomic fields. First, we will prove the Kronecker-Weber theorem, which gives an explicit description of all Abelian extensions of the field of rational numbers. In this context, we will introduce Dirichlet L-functions and relate them to Dedekind zeta functions of cyclotomic fields. As a nice application, we will outline the proof of the Kummer criterion. This gives a surprising connection between class groups of cyclotomic extensions and the values of the Riemann zeta function; it can be seen as the starting point of Iwasawa theory.

In the last part of the lecture, we will investigate the question of how to generalize the concept of Dirichlet L-functions to arbitrary number fields. This will naturally lead to Hecke L-functions.

At the end of each section, we will provide an outlook on interesting follow-up questions. Here, we will briefly address the topics of the Riemann Hypothesis, the question of odd zeta values, global class field theory, Iwasawa theory and Tate’s thesis.

Format of the lecture

The lecture will take place online in a 'flipped classroom' format, i.e., I will provide material (detailed lecture notes, videos) for each lecture and we will meet once or twice a week to discuss the content and to answer questions. Additionally, there will be a weekly exercise session.

Weekly meeting

The weekly meeting/question hour will be on Wednesday at 8:30 via zoom.

Exercises

The exercise class will be on Tuesday at 16:15 via zoom.

Lecture notes

Lecture notes.

First meeting

The first meeting was on Wednesday, April 14th, at 8:15 am, via zoom. If you are interested in the lecture but missed the first meeting, just write me an email. The first meeting was just about organizational aspects. You can find the slides of the first meeting here: Slides of the first meeting.

Video: Overview

Moodle

You will find more information about this lecture in Moodle. The password for the Moodle course is name for the imaginative animal which appears at minute 3:40 in the introductory video.

Questions

Please do not hesitate to write me an email if you have any further questions.