Fiches de cours

Topics in number theory

MATH-417

Enseignant(s) :

Michel Philippe

Langue:

English

Summary

This year's topic is "Adelic Number Theory" or how the language of adeles and ideles and harmonic analysis on the corresponding spaces can be used to revisit classical questions in algebraic number theory

Content

This year we will discuss the theory of adéles and idèles.

Idèles were invented by Claude Chevalley to provide an compact reformulation of Class Field Theory and Artin's reciprocity law.

However the group of idèles together with its associated ring of adèles are powerful tools to encode all sorts of local-global principles in number theory and arithmetic geometry; the associated terminology is by now the lingua franca of the Langlands program.

The course will introduce the language and we will use it to revisit various aspects of classical algebraic number theory. For instance we will give new proofs of classical results like the finiteness of the class group, Dirichlet's units theorem or the class number formula.

-completions in number fields. Ostrowski's Theorem.

-Local-global principles : the case of the space of lattices.

- Topology and harmonic analysis on adeles and ideles.

-The ring of adeles and the group of ideles associated to a number field. Finitness of the class group and Dirichlet's unit theorem all in one.

-Tate's thesis. Analytic properties of Dedekind and Dirichlet L-functions.

- The adelic formulation of class field theory (without proofs)

- Modular forms in the adelic language.

Learning Prerequisites

Required courses

Anbalysis III & IV

Galois Theory.

Rings and Modules.

Introduction to Algebraic Number Theory.

Recommended courses

Not required but possibly useful

Introduction to Analytic Number Theory.

It will be good to have had some exposure to the theory of Riemann's zeta function.

Automorphic forms and L-functions.

Some exposure to the classical theory of modular forms migh be used at the end of the course to compare with the adelic viewpoint.

Important concepts to start the course

-Good knowledge of Galois theory, Algebraic Number Theory and the associated commutative algebra.

Learning Outcomes

By the end of the course, the student must be able to:

Transversal skills

Teaching methods

Ex-Cathedra Course

Expected student activities

We expect a proactive attitude during the courses and the exercises sessions (possibly with individual presentation of the solution of various problems).

Assessment methods

Exam Oral

Dans le cas de l¿art. 3 al. 5 du Règlement de section, l¿enseignant décide de la forme de l¿examen qu¿il communique aux étudiants concernés.

Supervision

Office hours No
Assistants Yes
Forum No
Others a moodle with ressources for the course will be maintained

Prerequisite for

Current research in number theory

Dans les plans d'études

  • Mathématiques - master, 2019-2020, Master semestre 1
    • Semestre
      Automne
    • Forme de l'examen
      Oral
    • Crédits
      5
    • Matière examinée
      Topics in number theory
    • Cours
      2 Heure(s) hebdo x 14 semaines
    • Exercices
      2 Heure(s) hebdo x 14 semaines
  • Mathématiques - master, 2019-2020, Master semestre 3
    • Semestre
      Automne
    • Forme de l'examen
      Oral
    • Crédits
      5
    • Matière examinée
      Topics in number theory
    • Cours
      2 Heure(s) hebdo x 14 semaines
    • Exercices
      2 Heure(s) hebdo x 14 semaines
  • Ingénierie mathématique, 2019-2020, Master semestre 1
    • Semestre
      Automne
    • Forme de l'examen
      Oral
    • Crédits
      5
    • Matière examinée
      Topics in number theory
    • Cours
      2 Heure(s) hebdo x 14 semaines
    • Exercices
      2 Heure(s) hebdo x 14 semaines
  • Ingénierie mathématique, 2019-2020, Master semestre 3
    • Semestre
      Automne
    • Forme de l'examen
      Oral
    • Crédits
      5
    • Matière examinée
      Topics in number theory
    • Cours
      2 Heure(s) hebdo x 14 semaines
    • Exercices
      2 Heure(s) hebdo x 14 semaines

Semaine de référence

LuMaMeJeVe
8-9
9-10
10-11 MAA110
11-12
12-13
13-14 MAA112
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
Cours
Exercice, TP
Projet, autre

légende

  • Semestre d'automne
  • Session d'hiver
  • Semestre de printemps
  • Session d'été
  • Cours en français
  • Cours en anglais
  • Cours en allemand