Number theory II.b - selected topics

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

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

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

• 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

Resources

Virtual desktop infrastructure (VDI)

No

Moodle Link

• https://go.epfl.ch/MATH-417

Prerequisite for

Current research in number theory