Modular forms 2_SS 2025

The course will introduce automorphic forms on GL_2(A_Q). We will first introduce adeles A_Q over the rational numbers Q and Fourier analysis on it. Then we will introduce the automorphic forms on GL_1(A_Q) (the ideles over Q), which is a generalisation of Dirichlet characters. After that we will move to GL_2(A_Q). We will generalise the notion of modular forms to automorphic functions and study Maass forms and their Fourier expansions. Hecke operators will then be defined in an adelic way. Then we will introduce automorphic forms on GL_2(A_Q) and automorphic representations of GL_2(A_Q).

Prerequisites: the course Modular Forms-1 (chapters 1-5 of Diamond-Shurman, A first course in modular forms; in particular the definitions of modular forms and Hecke operators).

Time and Room: Lectures: Tuesday 10h15-11h45 in WSC-S-U-3.02 and Friday 12h15-13h45 in WSC-S-U-3.01; Exercise Session: Friday 14h15-15h45 in WSC-S-U-3.01

References:

  1. Goldfeld, Hundley, Automorphic Representations and L-functions for the General Linear Group, vol. 1 (online access through university library here)
  2. Bump, Automorphic Forms and Representations (online access through university library here)
  3. Miyake, Modular Forms