Automorphic Forms and Representation Theory

6 EC

Semester 1, period 1, 2

5334AFRT6Y

Owner Master Mathematics
Coordinator prof. dr. Jasper Stokman
Part of Master Mathematics, specialization Algebra and Geometry, year 1Master Mathematics, specialization Analysis and Dynamical Systems, year 1Master Mathematics, specialisation Discrete Mathematics and Quantum Information, year 1Master Mathematics, specialization Stochastics , year 1

Course manual 2023/2024

Course content

This is an advanced master course aimed at revealing some of the deep connections between analytic number theory and representation theory. The course will first focus on the number theory side, with an introduction to modular forms and their L-functions, and a discussion of the classical approach for obtaining functional equations and Euler products. Then attention is turned to representation theory of reductive groups, especially GL(2) over the real numbers. The two topics are brought together via the theory of automorphic forms. 

This course will give a first glimpse on the broad modern mathematical program connecting representation theory, number theory and geometry, known as the Langlands program.

Study materials

Literature

  • D. Bump, "Automorphic forms and representations", Cambridge Studies in Math 55, Cambridge Univ. Press, 1998.

     

Objectives

  • has learned basic techniques from the theory of modular forms and representation theory of reductive groups.
  • can apply these techniques to the theory of automorphic forms.

Teaching methods

  • Lecture
  • Self-study

Learning activities

Activity

Hours

Hoorcollege

28

Tentamen

3

Self study

137

Total

168

(6 EC x 28 uur)

Attendance

This programme does not have requirements concerning attendance (TER-B).

Assessment

Item and weight Details

Final grade

1 (100%)

Tentamen

The final exam will be a written exam, in which you are allowed to use the Bump's book. The reexam will be an oral exam.

On a regular basis, we will give homework exercises, which will be corrected and marked. The average mark of the homework exercises counts for 30% of the final grade. 

The homework marks will be discarded in 3 cases:

  • The mark of the final written exam is higher than the average mark of the homework.
  • The mark of the final written exam is less than 5.
  • In case of a reexamination.

Inspection of assessed work

Via canvas

Assignments

Biweekly homework exercises.

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Weeknummer Onderwerpen Studiestof
1 The modular group Bump, sect. 1.2
2 Modular forms Bump, sect. 1.3
3 Hecke operators Bump, sect. 1.4
4 L-functions Bump, sect. 1.4
5 GL(2,R) rep. theory Bump, chpt. 2
6 GL(2,R) rep. theory  Bump, chpt. 2
7 Maass forms Bump, chpt. 2
8 Autumn break  
9 Automorphic forms Bump, chpt. 2
10 Automorphic forms Bump, chpt. 2
11 Dirichlet L-functions, p-adic numbers Bump, sect. 1.1 and chpt. 3
12 Automorphic representations of GL(1,A) Bump, chpt. 3
13 Automorphic representations of GL(1,A) Bump, chpt. 3
14 Automorphic representations of GL(1,A) Bump, chpt. 3
15 GL(2,A) and automorphic forms Bump, chpt. 3
16    

Timetable

The schedule for this course is published on DataNose.

Contact information

Coordinator

  • prof. dr. Jasper Stokman