Modular Forms

First lecture: Thursday, February 21, 2013.

Contents

The main goal of the course is to give an introduction to the theory of automorphic forms from the analytic point of view. There will be an emphasis on theta-functions and their applications in analytic number theory, especially to representations by quadratic forms.

Prerequisites

Complex Analysis (Funktionentheorie), Algebra I, Measure and Integration.

First exercise session: Monday, February 25, 2013

Requirements for the certificate (Testat): Reasonable attempt to solve 50% of all tasks.

Exercises

Exercises will be available here (approximately one exercise every second week).
Please hand in your written solutions in class on the due date given below.
In general, you will have your marked exercises returned at the beginning of the subsequent exercise class (on monday).
If there are any questions and/or corrections please send an email to Jonas Jermann.

Lecture Notes

The congruent numer elliptic curve y^2=x^3-x (not part of the exam).

Exam

It is a 20 minutes oral exam at the end of the semester (Semesterendprüfung), i.e. not during the exam session. The exercices are also part of the exam.

Literature

• Iwaniec, "Topics in classical automorphic forms"
• Zagier: Elliptic modular froms and their applications (in "The 1-2-3 of modular forms")
• Neal Koblitz, "Introduction to Elliptic Curves and Modular Forms"

Additional Literature:

• Tom M. Apostol, "Modular functions and Dirichlet Series in Number Theory"
• Daniel Bump, "Automorphic Forms and Representations"
• Fred Diamond and Jerry Shurman, "A First Course in Modular Forms"
• Eberhard Freitag, Rolf Busam, "Funktionentheorie I" (in german)
• Toshitsune Miyake, "Modular Forms"
• Jean-Pierre Serre, "A Course in Arithmetic"
• Goro Shimura, "Introduction to the Arithmetic Theory of Automorphic Functions"