## McGill University

# Department of Mathematics & Statistics

# Topics in Number Theory: L-functions and modular forms

# 189-726A

## Detailed Syllabus

**September 9**. 11:00-12:00: Organisational meeting.

Assignment 1 is Due Wednesday, January 16.

*References*. The material treated this week is fairly standard. I
have followed the treatment in
Lecture 1 of

James Cogdell. *Lectures on $L$-functions, converse theorems,
and functoriality for ${\rm GL}\sb n$.*
Lectures on automorphic $L$-functions, 1-96, Fields Inst. Monogr.,
20, Amer. Math. Soc., Providence, RI, 2004.

**Week 2 (January 14-16)**.
Hecke theory, continued. The petersson scalar product,
diagonalisability of Hecke operators.

Assignment 2 is Due Wednesday, January 23.

*References*.
See for example Ogg's book "Modular forms and Dirichlet series"
or the treatment in Knapp's book on elliptic curves.

**Week 3 (January 21-23) **.
Twisting by Dirichlet characters.
Functional equation for twisted L-series.

Assignment 3 is Due Wednesday, January 30.

**Week 4 (January 28-30) **.
Modular symbols. The Mazur-Swinnerton-Dyer p-adic
L-function of an elliptic curve.

Assignment 4 is Due Wednesday, February 6.

**Week 5 (February 4-6) **.
The Mazur-Swinnerton-Dyer p-adic
L-function, continued.

Assignment 5 is Due Wednesday, February 13.

**Week 6 (February 11-13) **.
The theory of p-adic modular forms (following Serre).

Assignment 6 is Due Wednesday, February 20.

**Week 7 (February 18-20) **.
The theory of p-adic modular forms following Serre (cont'd).

**Spring Break: ** February 25-29.

* Note that there will be no classes in the week of March 3-7, as I
will be out of town that week. I've posted an assignment that is
somewhat longer than the usual ones. Allow enough time
to do it well, and to go over the material we've covered so far.
*

Assignment 7 is Due Wednesday, March 12.

**Week 8 (March 10-12) **.
Zeta-functions attached to imaginary quadratic fields.
Zeta values as special values of modular forms at CM points.

Assignment 8 is Due Wednesday, March 19.

**Week 9 (March 17-19) **.
Algebraic modular forms, and algebraicity properties of modular forms evaluated
at CM points.

A good reference for the material this week and next week
is the following
article by Hida.

Assignment 9 is Due Friday, March 28.

**Week 10 (Wednesday March 26 and Thursday, March 27) **.

**Note**: *There will be no class on Monday because of Easter.
To compensate, there will be an extra class on Thursday, March 27
at 2:00 PM in the McGill Lounge.*

*p*-adic properties of the special values of modular forms at CM points.
Application to the
construction of the Katz two-variable *p*-adic *L*-function.

Assignment 10 is Due Wednesday, April 2.

**Week 11 (March 31-April 2) **.

More on algebraic modular forms, and the Tate curve.
The supersingular and ordinary locus of the modular curve.
p-adic modular forms as ``rigid analytic functions" on the ordinary
locus.
Application to p-adic interpolation for (certain)
values of the Katz two-variable L-function.

**References**. The following article by Matthew Emerton is a well-written
overview of some of the concepts we're covering this week (and much more).
It also contains useful pointers to the literature.

Assignment 11 is Due Wednesday, April 9.

**Week 12 (April 7-9) **.

De Rham cohomology and the Gauss-Manin connection: the case of elliptic curves.

Assignment 12 is Due Wednesday, April 16.

**Week 13 (April 14-16) **.

Interpretation of the differential operators
$\delta_k$ and $d$ in terms of the
Gauss-Manin connection.
Application to the Katz p-adic L-function.

** April 18**: Final exam.