Subconvexity, Periods and Equidistribution


Lecturer(s) :

Michel Philippe




Only this year


Spring 2021 - Fridays


This course is a modern exposition of "Duke's Theorems" which describe the distribution of representations of large integers by a fixed ternary quadratic form. It will be the occasion to introduce the students to the adelic language, the theory of automorphic forms and their associated L-functions


Duke's theorems establish that the set of representations of an integer by a fixed ternary quadratic form are usually equidistributed along the level sets hypersurface of the form as the integer becomes large. It has alternative formulations in terms of distribution of Heegner points and their reductions as well as closed geodesics on modular and Shimura curves.


This course will present a modern and unifed proof of these theorems by recasting them in the adelic language and by using harmonic analysis on the corresponding adelic locally homogeneous spaces. It will be the occasion to discuss  from the adelic viewpoint, the theory of automorphic forms (on PGL_2 and its forms), their periods, their L-functions as well as the subconvexity problem for L-functions.


Course content:


Duke's theorems. Representations by quadratic forms in the classical language.

The local-global principle. The duality principle. Alternative formulations of Duke's Theorems in terms of distribution of Heegner points and closed geodesics.


Recollections on adeles; Adelization of Duke's theorems.

Automorphic forms on adelic quotients, especially for quaternion algebras.

Automorphic Periods and L-functions: Hecke's and Waldspurger's formula.

The subconvexity problem and its applications to Duke's theorems.




Automorphic forms

Quadratic forms

Learning Prerequisites

Required courses

Basic course  on classical modular forms (cf Iwaniec Book)

Basic course in algebraic number theory (like MATH-482)

Recommended courses

A basic course on the theory of quadratic forms (over Q)

Introduction to analytic number theory (like MATH-313)

MATH-417 (2019) Adelic Number Theory

Teaching methods

Course ex-cathedra

Expected student activities

Participate actively to the course via questions and comments.

To prepare for the lectures by reading in advance the indicated references


Assessment methods

Oral Examination based on the course content



Bump, Daniel Automorphic forms and representations. Cambridge Studies in Advanced Mathematics, 55. Cambridge University Press, Cambridge, 1997.

Iwaniec, Henryk Topics in classical automorphic forms.
Graduate Studies in Mathematics, 17. American Mathematical Society, Providence, RI, 1997.

O'Meara, O. Timothy Introduction to quadratic forms. Reprint of the 1973 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2000.

Ressources en bibliothèque

Lecture notes will be distributed as the course progress

In the programs

    • Semester
    • Exam form
    • Credits
    • Subject examined
      Subconvexity, Periods and Equidistribution
    • Number of places
    • Lecture
      28 Hour(s)
    • Practical work
      28 Hour(s)

Reference week

      Exercise, TP
      Project, other


  • Autumn semester
  • Winter sessions
  • Spring semester
  • Summer sessions
  • Lecture in French
  • Lecture in English
  • Lecture in German