189-596A: Modular forms

-------------- Course blog ----------- Exercises -----------

Professor: Henri Darmon
Classes: MW 10:30-12:00 AM, in BH 920
Problem sessions: F 10:30-12:00 AM, in BH 920
Office Hours: MW 12:00-1:00 or by appointment, in Burnside Hall 1111.

Optional Textbooks:

The long article by Diamond-Im and the textbook by Diamond and Shurman are highly recommended as supplementary references for this course.

Modular forms are objects that arise naturally in describing the moduli space of elliptic curves endowed with various extra structures, and their study is therefore grounded in the theory of elliptic curves. For the basics of elliptic curves I can do no better than to recommend the eponymous textbook of J. Silverman which belongs on the shelf of any aspiring number theorist. It is based on the no less classic article by Tate which is a wonderful, insightful and efficient presentation of the important ideas and results in the subject.

For the more advanced students interested in learning more about the finer issues concerning various moduli problems attached to elliptic curves over rings, the classic articles by Katz-Mazur and Deligne-Rapoport are the standard references.

Syllabus:
This course is meant to be an introduction to modular forms and will assume no previous exposure to the subject. For the students who already took last year's course with Stephan Ehlen, (or the venerable old-timers who can remember taking my previous course on modular forms) and still want to attend the lectures, I will try to present the material in a different way stressing the geometric over the analytic aspects of the theory (although analysis, both complex and p-adic, will inevitably enter into our discussion). We will begin by describing modular forms from the point of view of the moduli of elliptic curves, and then discuss such topics as:

Modular forms over general rings (with an emphasis on rings in which 6 is invertible);
Modular curves as curves over Q and even Z;
Weierstrass theory and modular forms as holomorphic functions;
The theory of the Tate curve and q-expansions;
Cusps, modular units, and the Manin-Drinfeld theorem;
Differential operators on modular forms;
Hecke theory and L-functions attached to modular forms;
p-adic L-functions attached to modular forms.
p-adic modular forms in the style of Katz.

Problem sessions : Fridays will generally be reserved for problem sessions where students will be expected work out questions that are assigned during the lectures and consigned to the course blog.

Grading Scheme : The grade will be based on your written work and your participation in the problem sessions.

The obligatory statements

McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/integrity for more information).

In accord with McGill University's Charter of Students' Rights, students in this course have the right to submit in English or in French any written work that is to be graded.

In the event of extraordinary circumstances beyond the University's control, the content and/or evaluation scheme in this course is subject to change.