Number Theory II: Introduction to Modular Forms Spring 2025

Lecturer
Emmanuel Kowalski
Coordinator
Ana Marija Vego
Lectures
Wednesdays, 14 to 16 in HG E 1.1
Fridays, 10 to 12 in HG E 1.1.
Exercise classes will be held on average once every two weeks, usually during the Wednesday lecture. They will be announced in class and below.
The lectures should be automatically recorded; the video recordings will then be available on the ETH Video Portal.

Important note: the schedule of the classes is altered between Wednesday 12.3 and Wednesday 26.3, included. See below for the precise information and links to the video recordings and lecture notes which replace the regular lectures during this time.

Summary

The course is an introduction to modular forms and their applications in number theory.

Lecture notes

The course will be mostly based on the books of Serre and Iwaniec listed below. I will also post the scans of my handwritten lecture notes after each class with the summary of the content below.
For the basic background material in number theory, I will use mostly the lecture notes from the Number Theory I class of Fall 2024.

Exercises

The lecture will be accompanied by roughly biweekly exercise classes, usually during the Wednesday class. We will announce the precise dates in the lecture as well as here. You should submit your exercise sheets as a PDF upload to the SamUpTool.

Dates of exercise classes (subject to changes!)
February 26
March 19
March 26
April 9
April 30
May 14
May 28
Exercise sheet Due by Solutions
Exercise sheet 1 March 4 Solutions 1
Exercise sheet 2 (corrected version) March 18

Summary of the lectures

We will summarize here briefly the content of each lecture, and provide links to scans of the corresponding (handwritten) lecture notes.
DayContentLecture notes
19.2.2025 Introduction to the class: some examples of results that can be proved with modular forms, although the statements do not mention them at all. Basic definitions of modular forms. Basic definition of modular forms of level one. Transitivity of the action of SL2(R) on the upper-half plane, stabilizer of i.
References for the results mentioned during the lecture:
  • Fermat's Great Theorem: a full account of the proof written by H. Darmon, F. Diamond and R. Taylor.
  • Optimal sphere packings: a survey paper by H. Cohn.
  • The Banach-Ruziewicz problem: see chapters 1 and 2 of 'Some applications of modular forms' by P. Sarnak (which also contains details of other surprising results proved using modular forms).
  • Equidistribution of roots of quadratic congruences: the original paper of Duke, Friedlander and Iwaniec.
Lecture notes
21.2.2025 A fundamental domain for the action of the modular group; generators of the modular group.
First examples of modular forms: Poincaré series and Eisenstein series. 		Lecture notes
26.2.2025 Exercises Exercise notes
28.2.2025 Proof of convergence of Poincaré and Eisenstein series. Definition of the Fourier coefficients of modular forms at infinity. Definition of spaces of modular forms which are holomorphic including at infinity and of the subspace of cusp forms. Statement of the fact that their dimensions are finite. Sketch of applications resulting from this fact. Lecture notes
5.3.2025 The number of zeros of a holomorphic modular form, counted with multiplicity. Dimension of the space of modular forms of given weight, examples of bases in low weights. Definition of the Δ function. Lecture notes
7.3.2025 Proof of the dimension formula. Corollaries concerning the structure of the algebra of modular forms. Definition of the j-function and properties. Lecture notes
12.3, 14.3 and 21.3.205 Recorded lectures (follow links to access the Zoom video recording)
19.3.2025 Exercises
26.3.2025 Exercises

Literature