**189-726A:**Topics in Number Theory: L-functions and modular forms

**Professor:** Henri Darmon

**Classes: ** Mondays 9:00-10:00 and 2:30-3:30, and Wednedsay 2:30-3:30.

**Room: ** BH1205 from 9:00-10:00, and
MAASS 328 from 2:30-3:30. (In spite of this,
Maass forms will not be covered in the afternoon lectures.)

**Office hours: ** By appointment, in my office (BH 1111).

The unifying theme of this course will be the notion of
L-functions attached to analytic data (Dirichlet characters,
modular forms) or to algebro-geometric data (such as Artin characters,
or
elliptic curves; these are all special cases of the
far-reaching notion of *motives*, a framework which will
be discussed, lightly, in the course.)
The class will focus (more or less chronologically)
on the following basic examples of L-functions.

1. The Riemann zeta-function.

2. Dirichlet *L*-functions.

3. Artin *L*-functions.

4. The Hasse-Weil *L*-function of an elliptic curve.

5. *L*-functions attached to modular forms.

6. *L*-functions attached to (double and triple)
Rankin convolutions.

Aspects of the theory of *L*-functions to be covered
will include

a) *Analytic continuation and functional equation. *
*L*-functions as Mellin transforms of modular forms.
Riemann's and Hecke's proofs of the functional equations.
The Rankin-Selberg method.

b) *Special values.*
The notion of a *critical value* in the sense of Deligne.
Introduction to the Deligne conjectures. Algebraicity results of
Euler, Dirichlet, and Shimura.

c) *p-adic interpolation of special values, and p-adic L-functions.*
The Kubota-Leopoldt *p*-adic *L*-functions.
The *L*-functions of Mazur-Swinnerton-Dyer and Mazur-Kitagawa.
Hida's *p*-adic *L*-function attached to Rankin convolutions of
modular forms.

**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.

** The usual official disclaimers**

1. * 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.*

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

3. * 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/students/srr/honest/
for more information).*