Number Theory II: Introduction to Modular Forms Spring 2023

Lecturer
Prof. Dr. Özlem Imamoglu | ozlem.imamoglu@math.ethz.ch
Coordinator
Dr. Sebastián Herrero | sebastian.herrero.m@gmail.com

Lectures

Tuesdays 10:15-12:00 HG E 22 (updated!)
Fridays 10:15-12:00 HG F 5

Exercise classes

Fridays 10:15-12:00 ML J 37.1 (every two weeks)

Course catalogue

401-3112-23L Number Theory II: Introduction to Modular Forms

Content

\( \bullet \) Basic definitions and properties of \( \mathrm{SL}(2,\mathbb{Z}) \), subgroups and modular forms for \( \mathrm{SL}(2,\mathbb{Z}) \).
\( \bullet \) Eisenstein and Poincaré series.
\( \bullet \) L-functions of modular forms.
\( \bullet \) Hecke operators.
\( \bullet \) Theta functions.
\( \bullet \) Possibly Maass forms.
\( \bullet \) Possibly automorphic forms for more general groups.

Schedule

Date Room Content Notes
21st February 2023 HG G 5 Motivation and background Lecture 1
24th February 2023 HG F 5 The modular group and its action on \( \mathbb{H} \) Lecture 2
28th February 2023 HG E 22 Fundamental domain and generators for the modular group Lecture 3
3rd March 2023 ML J 37.1 (Exercise class #1) Review of Lectures 1 to 3 Exercise class 1
7th March 2023 HG E 22 Modular forms for \( \mathrm{SL}_2(\mathbb{Z}) \) Lecture 4
10th March 2023 HG F 5 Eisenstein series for \( \mathrm{SL}_2(\mathbb{Z}) \) Lecture 5
14th March 2023 HG E 22 The valence formula and the structure of \( M_k \) Lecture 6
17th March 2023 ML J 37.1 (Exercise class #2) The cotangent identity, the weight 2 Eisenstein series and relations among modular forms Exercise class 2
21st March 2023 HG E 22 Klein's \( j \)-invariant and the discriminant modular function Lecture 7
24th March 2023 HG F 5 Growth of Fourier coefficients and \( L \)-functions of modular forms Lecture 8
28th March 2023 HG E 22 Analytic continuation and functional equation of Riemann's zeta function Lecture 9
31st March 2023 ML J 37.1 (Exercise class #3) Algebraic independence of \( E_4 \) and \(E_6 \), infinite product expansion of \( \Delta \) and values of \( j\) Exercise class 3
4th April 2023 HG E 22 Modular \(L\)-functions and Poincaré series Lecture 10
Easter break No classes
18th April 2023 HG E 22 Properties of Poincaré series Lecture 11
21st April 2023 HG F 5 Hecke operators (part I) Lecture 12
25th April 2023 HG E 22 Hecke operators (part II) Lecture 13
28th April 2023 ML J 37.1 (Exercise class #4) Duplication formula for \( \Gamma \), modular \( L \)-functions, Hecke eigenforms Exercise class 4
2nd May 2023 HG E 22 Hecke operators (part III) Lecture 14
5th May 2023 HG F 5 Review of sections 1 to 6 Lecture 15
9th May 2023 HG E 22 Class number formula and binary quadratic forms Lecture 16
12th May 2023 ML J 37.1 (Exercise class #5) Petersson inner product, Hecke operators are hermitian on cusp forms, Miller basis Exercise class 5
16th May 2023 HG E 22 Reduced quadratic forms and class numbers Lecture 17
19th May 2023 HG F 5 Number of representations of an integer by a quadratic form Lecture 18
23rd May 2023 HG E 22 Number of primitive representations and class number formula Lecture 19
Additional material: Fourier expansion of the non-holomorphic Eisenstein series and Maass forms Additional notes
26th May 2023 ML J 37.1 (Exercise class #6)
30th May 2023 HG E 22 Applications of modular forms Lecture 20
2nd June 2023 HG F 5 Review of section 7 Lecture 21

Exercises

The new problem sheets will be posted here every other Tuesday. We expect you to look at the problems and to prepare questions for the exercise class. Please hand in to the course coordinator the problems that you want to receive feedback on, either at HG J 16.5 or via e-mail to sebastian.herreromiranda@math.ethz.ch.

Exercise sheet Hand-in by Solutions
Exercise sheet 1 10th March 2023 Solutions 1
Exercise sheet 2 24th March 2023 Solutions 2 (updated!)
Exercise sheet 3 6th April 2023 Solutions 3
Exercise sheet 4 5th May 2023 Solutions 4
Exercise sheet 5 19th May 2023 Solutions 5 (updated!)
Exercise sheet 6 2nd June 2023 Solutions 6

Literature