Topics in number theory
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 (poisson summation formula).
-Adelic points of algebraic groups.
-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.
Keywords
Local and Global Fields
Archimedean and non-archimedean absolute values
Topological Fields and Rings
Groups of Matrices
L-functions
Learning Prerequisites
Required courses
Analysis III & IV
Introduction to Analytic Number Theory.
Recommended courses
Not strictly required but certainly useful
- Introduction to Analytic Number Theory.
- Introduction to Algebraic Number Theory.
- Some knowledge of modular forms (such as MATH-511 "Modular forms and applications" ) will be usefull since at the end of the course to present modular forms from the adelic viewpoint.
Important concepts to start the course
Analysis III & IV
Rings and Modules
Galois Theory
Measure an Integration
Learning Outcomes
- Synthesize the theory of adeles and ideles
- Solve basic problems involving adeles and ideles
- Interpret classical problems in the adelic language
- Solve advanced problems in analytic number theory
- Synthesize the analytic aspects of the theory of numbers
Transversal skills
- Access and evaluate appropriate sources of information.
- Make an oral presentation.
- Demonstrate the capacity for critical thinking
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
Oral Exam
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
In the programs
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Topics in number theory
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Topics in number theory
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Topics in number theory
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks