Content of the Lecture:
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.
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.
For further information, please consult the course page on Moodle. Passwort: "L-ephant".