Modular forms from the representation theoretic point of view | |
Student No.： | 50 |
Time： | Mon 13:00-14:50, 2017-02-20~ 2017-06-07 (No classes on public holidays) |
Instructor： | Zongbin Chen [Tsinghua University] |
Place： | Conference Room 1, Floor 1, Jin Chun Yuan West Building |
Starting Date： | 2017-2-20 |
Ending Date： | 2017-6-7 |
Description:
This is the first part of two lecture courses on modular forms, our goal is to introduce the students to the circle of ideas in the Langlands program by examining carefully the case of GL_2. For this first part, we will look at the modular forms from the representation theoretic point of view. This point of view has the advantage that the symmetries of modular forms become transparent, and is hence much more structural. In this framework, the modular forms are special cases of automorphic forms/representations. The main concern of Langlands program is to classify the automorphic representations. The problem is of global nature and hard to attack directly. However, the automorphic representations are tensor products of local representations, which are much better understood. In fact, they have been classified for the group GL_n, and this is the local Langlands program. Unfortunately, we still don't know when the local representations can be patched to automorphic representations.
In this course, we will not be able to go as far, we are only able to explain the basics. Topics that will be covered includes: Modular forms and functional equation of L-function (Converse theorem); Harmonic analysis on SL_2(R) (Plancherel formula); Satake isomorphism; Selberg trace formula and Jacquet-Langlands correspondence.
The second part should be at Autumn semester 2017/18, in which we will discuss modular forms from the algebraic geometric point of view.
Prerequisite:
I will try to make the course available to a large audience, including advanced undergraduates. But a familiarity with basic notions from differential manifolds (eg. What is a differential form), and classical groups (eg. What is SL_n, Sp_2n) will be appreciated.
Reference:
Gelbart, Automorphic forms on adele groups.
Lang, SL_2(R)
Bump, Automorphic forms and automorphic representation
Deligne, Forme modulaire et representation de GL_2
Jacquet-Langlands, Automorphic forms on GL_2