## McGill University

# Department of Mathematics & Statistics

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

# 189-726A

## Detailed Syllabus

**Note-taking**: One of the components of your final grade will be the notes that you will have written up for the course,
in the week of your choice. If you have not done so yet, you should volunteer for the week in which you prefer to take
the notes (otherwise I will assign you arbitrarily.)

In order to keep the appearance of the notes uniform, you should use the following nice tex files which Luca has provided:

template.tex,

preamble.tex.

Once you have finished typing your notes, please send me the files by email *in both pdf and tex formats.*

**September 7**. 10:30-12:00: Organisational meeting. Overview of the course.

Luca's notes of lecture 1.

**Week 1 (Sept 12 and 14)**. Note-taker: * Luca Candelori*

The Riemann zeta function, its analytic continuation, and functional equation.
Transformation property of the theta series.

Luca's notes of lecture 2.

Luca's notes of lecture 3.

Luca's notes of lecture 4.

**Week 2 (Sept 19 and 21)**. Note-taker: * Juan Restrepo*

The functional equation for Dirichlet *L*-functions. Remarks on Artin *L*-functions
attached to higher-dimensional representations. Connection with modular forms of
weight one, and Hecke theory.

Juan's notes of lectures 5 and 6.

Juan's notes of lecture 7.

**Week 3 (Sept 26 and 28)**. Note-taker: * Luiz Takei *

Modular forms and Hecke theory, continued.

Luiz's notes of lecture 8.

Luiz's notes of lecture 9.

Luiz's notes of lecture 10.

Assignment 1 is due on September 28.

**Week 4 (Oct 3 and 5)**. Note-taker: * Céline Maistret *

Hecke operators and Hecke theory.

Céline's notes of lecture 11.

Céline's notes of lecture 12.

Céline's notes of lecture 13.

**Week 5 (Oct 12 and 13)**. Note-taker: * Maxime Turgeon *

Galois representations attached to modular forms.
Algebraic interpretation of modular forms. Modular curves.

Maxime's notes of lecture 14.

Maxime's notes of lecture 15.

Maxime's notes of lecture 16.

Assignment 2 is due on October 12.

**Week 6 (Oct 17 and 19)**. Note-taker: * Dylan Attwell-Duval *

Algebraic interpretation of modular forms, cont'd.
Eichler-Shimura theory.

Dylan's notes of lecture 17.

Dylan's notes of lecture 18.

Dylan's notes of lecture 19.

Assignment 3 is due on October 19.

**Week 7 (Oct 24 and 26)**. Note-taker: * Bahare Mirza *

The Rankin-Selberg Method.

Bahare's notes of lecture 20.

Bahare's notes of lecture 21.

Bahare's notes of lecture 22.

**Week 8 (Oct 31 and Nov 2)**. Note-taker: * Jason Polak*

*I will be away this week and Fu-Tsun Wei will be lecturing in my place.*

The Rankin-Selberg method, cont'd. Application to the fourier coefficients
of modular forms of weight one, following Chapter 5 of Deligne-Serre.

Jason's notes of lecture 23.

Jason's notes of lecture 24.

Jason's notes of lecture 25.

**Week 9 (Nov 7 and Nov 9)**. Note-taker: * Victoria de Quehen*

The construction of Artin representations attached to forms of weight one,
following Deligne-Serre.

Victoria's notes of lecture 26.

Victoria's notes of lecture 27.

Victoria's notes of lecture 28.

Assignment 4 is due on November 7.

**Week 10 (Nov 14 and Nov 16)**. Note-taker: * Francesca Gala*

End of proof of the Serre-Deligne theorem.
Special values of zeta functions (the case of the Riemann zeta-function).

Francesca's notes of lecture 29.

Francesca's notes of lecture 30.

Francesca's notes of lecture 31.

**Week 11 (Nov 21 and Nov 23)**. Note-taker: * Clément Gomez*

Values of zeta functions. p-adic banach spaces, p-adic measures, and integration.
Kummer-Clausen-Von Staudt congruences.

Clément's notes of lectures 32 and 33.

Cléments's notes of lecture 34.

Assignment 5 is due on
Monday, November 21.

*
However, it is OK to hand in this assignment
one week late, on November 28 (but *__no later__ than that.)

**Week 12 (Nov 28 and Nov 30)**. Note-taker: *Francesc Castella *

Construction of the Kubota Leopoldt p-adic zeta function.
Proof of the *p*-adic class number formula.

Francesc's notes of lecture 35.

Francesc's notes of lecture 36.

Francesc's notes of lecture 37.

** Monday, December 5. ** 10:00-1:00. Final exam
will be held in BH-1214.