189-596A: Modular forms
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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.
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.
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
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
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
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
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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