ALGANT Student seminar – Number theory course
Date and time: Thursday, May 19th, at 2 pm
Room: Aula 4
Lecture 1: Riccardo Ghezzi
Title: Topics on profinite groups, infinite Galois theory and properties of norm groups
This lecture divides into two parts. In the first part, we will give an account of the background material that is required to study local class field theory: we will introduce profinite groups and we will show some properties of their topology. Then we will present some results from infinite Galois theory. In the second part will show some properties of the norm groups and some consequences of the main theorems of local class field theory.
Lecture 2: Mattia Righetti
Title: the cohomology of groups
Abstract: The aim is to give an introduction to cohomolgy of a group G with coefficients in a G-module. We also introduce the Tate groups of G, which come from both homology and cohomology, and are a valuable tool. As examples we consider particular groups G and G-modules, relevant to Number Theory.
Lecture 3: Federico Binda
Title: the local reciprocity law
In this lecture we will develop the tools for a cohomological approach to local class field theory. We first study the cohomology of unramified extensions of a nonarchimedean local field K and then we extend the results obtained in this setting to the more general case of ramified extensions: we will see how the invariant map is defined and introduce the fundamental class of a Galois extension L/K. Using Tate’s theorem, we will prove the existence of the local Artin map for any finite Galois extension of local fields L/K, which yields to the local reciprocity law.
Lecture 4: Francesco Monopoli
Title: the existence theorem
In this lecture we will present a proof of the so-called “local existence theorem”, which gives a characterization of the norm groups in K* for a local field K, under the assumption that K has characteristic zero. The definition of the Hilbert symbol and the cohomological tools developed in the previous lectures will be used to show that, if K is a local field containing a primitive n-th root of 1, any element of K* that is a norm from every cyclic extension of K is an n-th power. We will use this result as a tool in our proof of the existence theorem.