Advanced analytic number theory

MATH-521 / 5 crédits

Enseignant: Michel Philippe

Langue: Anglais

Remark: Cours donné en alternance tous les deux ans

Summary

This year we will present some further applications of the theory of modular forms (compared to MATH-511). These may include the following: - Equidistribution of points on spheres - Construction of Ramanujan Graph - Invariant means on the spheres - Complex multiplication for elliptic curves

Content

The course will have some overlap with MATH-511 but the applications presented are somewhat different.

Harmonic analysis on the sphere

The upper half plane and the group of fractional linear transformations

Modular forms: définitions and examples (Eisenstein series, theta series)

Fourier coefficients of modular forms; Petersson inner product and Ptersson formula.

Hecke operators.

Equidistribution on the sphere.

Expander graphs/Ramanujan graphs

Modular forms and elliptic curves.

Keywords

Modular forms, theta series, Fourier coefficients

Equidistribution

Graphs

Elliptic curves

Learning Prerequisites

Required courses

Fourier Analysis

Harmonic analysis

Modular Forms

 

 

 

Recommended courses

Analytic Number Theory

Algebraic Number Theory

 

Learning Outcomes

  • Quote the main results of the course
  • Use the main results of the course
  • Prove the main results of the course

Teaching methods

Ex cathedra lecture and exercises in the classroom

Expected student activities

Attendence to the lectures and active participation to thhe exercise sessions

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 moodle

Resources

Virtual desktop infrastructure (VDI)

No

Bibliography

 

Fred Diamond: A first course on modular forms.

Henryk Iwaniec: Topics on Classical automorphic forms

Peter Sarnak: Some applications of modular forms

 

Ressources en bibliothèque

Moodle Link

Prerequisite for

Fields medal

Dans les plans d'études

  • Semestre: Printemps
  • Forme de l'examen: Oral (session d'été)
  • Matière examinée: Advanced analytic number theory
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel
  • Semestre: Printemps
  • Forme de l'examen: Oral (session d'été)
  • Matière examinée: Advanced analytic number theory
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel
  • Semestre: Printemps
  • Forme de l'examen: Oral (session d'été)
  • Matière examinée: Advanced analytic number theory
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel

Semaine de référence

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