**Teaching in winter semester 2021/2022 :**

**Algebraic number theory**.

This lecture (4 hrs) will be taught in presence in the room

**A 027**(Tuesday, Thursday 10:00-12:00).

We will give an introduction to classical algebraic number theory, which means the study of the ring of algebraic integers OK in a number field K (a finite extension of the field of rational number Q ). We will start with some general facts on Dedekind rings, and will then specialize to the rings of the form OK . We will give the proof of classical facts, like the Dirichlet theorem on the structure of the group of units of OK , as well as the finiteness of the class group also due to Dirichlet. We will then introduce and study the ramification in a finite extension K < L , and local phenomena. We will prove the fundamental result due to Hermite-Minkowski that any non trivial number field admits nontrivial ramification with respect to Q. We will finish by studying the behaviour of rings of integers in Ga- lois extensions, and aim to prove the famous Theorem of Kronecker-Weber classifying abelian extensions of Q.

**References**: P. Samuel, Theorie algebrique des nombres (french, also available in english version under "algebraic number theory").

J.-P. Serre, Corps locaux (french, also available in english version under "local fields").

S. Lang, algebraic number theory.

Exercises classes (Tom Bachmann).

**Trees, groups and amalgams**.

This lecture (2 hrs) will be taught in presence in the room

**B 040**(Tuesday 16:00-18:00).

The aim of this lecture is to give an introduction to combinatorial groups theory in the following sense: given a group G acting on a combinatorial graph, what can we say on the structure of G if we know for instance the structure of each of the isotropy subgroups of the action of G at each vertex of the graph ? We will emphasize the case of actions of groups on ``trees" and will give several applications. For instance any subgroup of a free group is a free group. Other arithmetical applications will involve the group SL2 acting on ``its" tree that will be introduced in the second part of the lecture.

**Reference**: Arbres, Amalgames, SL2, by J.-P. Serre, Astérisque 1977, also available in english:

Trees, by J.-P. Serre, Springer.

**Previous semesters .**