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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.