**189-726B:**Modular Forms II

**Professor:** Henri Darmon

**Classes: ** MW 10:30-12:00.

**Room: ** McGill University, Burnside Hall 1214

**Office hours: ** MW 12:00-1:00, in my office (BH 1111).

This course will be a continuation of "Modular Forms I" which was
taught by Eyal Goren in the Fall. The guiding
theme of the course will be modular forms
and their relations to L-functions and their special values.
Topics to be covered will include:

* *L-functions as Mellin transforms of modular forms.*
The
Hecke L-functions attached to modular forms, and their
functional equations (Hecke's theory).

* *L-functions as constant terms of modular forms. *
Serre's theory of p-adic modular forms; application
to the construction of p-adic L-functions of totally real fields.

* *L-functions as values of modular forms at CM points.*
Katz's geometric point of view on the theory of p-adic modular forms.
The Shimura-Mass differential operator acting on nearly holomorphic modular
forms, and its relation with the theta operator acting
on p-adic modular forms.
Rationality and integrality properties of values of
nearly holomorphic or p-adic modular forms at CM points.
Application to Katz's
two-variable p-adic L-function attached to imaginary quadratic or CM fields.

* (Time permitting).
The Rankin-Selberg method. p-adic L-functions attached to Rankin
convolutions.

**Detailed syllabus**:

** Grading Scheme **:
Your grade will be based on your participation in class,
on your work in the assignments, and on the
final in-class exam to be given at the end of the term.