Categoria: seminari ALGANT

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.

STUDENT SEMINAR, NUMBER THEORY COURSE

Date and time: Thursday, May 5th, from 14.30

Room: Aula 4

1) Daniele Casazza

Title: Analytic and geometric tecniques for Dirichlet series.

Abstract: First we find analytical properties of Dirichlet series as region of convergence and residue in a simple pole, then we define zeta function for algebraic number fields and for ray classes of a modulus. Using geometric tools as lattices and the geometric view of the action of automorphism group we estimate the number of integral ideals of given norm in a ray class.

2) Fabio Malanchini

Title: Residue formula for zeta functions on number fields.

Abstract: Calculating the volume of a particular solid using lattice point counting we find the residue of a zeta function of a ray class and an explicit formula relating class number, regulator and discriminant of an algebraic number fields to the residue at s=1 of its zeta function. Using these informations we can obtain a slightly larger region of convergence for the zeta function of a number field.

3) Gao Ziyang (Jerry).

Title: An Introduction to Arakelov Geometry: Slope Theory (PDF note of the seminar)

Abstract:
In this lecture, I aim to introduce the slope theory, which is an important tool in Arakelov Geometry. I will start from the basic definitions of Arakelov Geometry and ends in the statement and proof of the slope inequality. At last, a proof of Siegel’s Lemma, which is one of the fundamental theorems of Arithmetic Algebraic Geometry, in flavor of the slope theory will be presented as an application.

Federico Binda and Ziyang Gao
will present two seminars on Lie Algebras – Representation theory

Here is a small note of the seminar.

Binda’s abstract:
Root systems provide a relatively uncomplicated way of completely characterizing simple and semisimple Lie algebras. We follow the axiomatic approach (as in Serre [2], Humphreys [1]). We introduce bases, the Weyl group and we explain its action on the set of bases (or, equivalently, on the Weyl chambers). Finally we introduce the classification theorem, using the Cartan matrix, Coxeter graphs and Dynkin diagrams.
References:
[1] J. Humphreys (1972), Introduction to Lie Algebras and Representation Theory, GTM, Springer-Verlag, NY.
[2] J. P. Serre (1966), Algébres de Lie semi-simples complexes, W.A. Benjamin, NY.

Gao’s abstract:
In this lecture, our main subject is the root space decomposition, a useful method to describe the representations of a Lie algebra. We will mainly focus on the case of . We will first characterize all irreducible representations of in terms of highest weight, then study the general root space decomposition. The notion of root system will be introduced here. Finally, we will generalize this notion and give the definition and a few basic properties of abstract root system.
References:
[1] J.Humphreys(1972), Introduction to Lie Algebras and Representation Theory, GTM-9, Springer-Verlag, NY.
[2] Jinpeng An, Lecture Notes, on-line version: ftp://162.105.69.120/teachers/anjp/李群与齐性空间