**Modular forms and their combinatorial variants**

(Nagoya University, Spring 2023)

(Nagoya University, Spring 2023)

**Please make sure that you are in the TACT group for this course. If you can not enter this group please contact me.**

**Materials**

- Course information
- Lecture notes (ver. 12, 16th July 2023)
- Homework: Homework 1, Homework 2, Homework 3, Homework 4

**Content:**

Modular forms are functions appearing in several areas of mathematics as well as mathematical physics. There are two cardinal points about them which explain why they are interesting. First of all, the space of modular forms of a given weight is finite dimensional and algorithmically computable. Secondly, modular forms occur naturally in connection with problems arising in many areas of mathematics. Together, these two facts imply that modular forms have a huge number of applications and the purpose of this lecture is to demonstrate this on examples coming from classical number theory, such as identities among divisor sums. We will cover the following topics (tentative):

- The action of the modular group on the complex upper half-plane and modular forms
- Eisenstein series and their Fourier expansion
- Cusp forms and Ramanujan's Delta function
- The space of modular and its dimension
- Modular forms of higher level
- Derivatives of modular forms and quasi-modular forms
- Hecke-Petersson theory
- L-functions and periods of modular forms
- Combinatorial modular forms

**Place & Time**

**Lecture:****Friday 1st Period (8:45 - 10:30) @ Room****多-453****(Math building)**

**Grading & Homework submission**

**:**

Based on homework assignments which you will submit at TACT.

**Course schedule (Tentative)**

**:**

The following gives a tentative overview of the topics we will cover each week.

Week 01 (04/10-04/16): Introduction & Motivation: Four square theorem

Week 02 (04/17-04/23): Divisor sums, Hurwitz Identity, Modular group, Fundamental domain

Week 03 (04/24-04/30): Modular functions, modular forms, Eisenstein series

**Week 3.5 (05/01-05/07): Golden week (No lecture)**

Week 04 (05/08-05/14): Cusp forms and the Ramanujan Delta function

Week 05 (05/15-05/21): Structure of the space of modular forms of level 1

Week 06 (05/22-05/28): Structure of the space of modular forms of level 1 II & Modular forms for congruence subgroups I

Week 07 (05/29-06/04): Modular forms for congruence subgroups II & Quasimodular forms I

Week 08 (06/05-06/11): Meidai-sai (No lecture)

Week 09 (06/12-06/18): Quasimodular forms II, Derivations, sl2-action, Hecke operators I

Week 10 (06/19-06/25): Hecke operators II

Week 11 (06/26-07/02): - No lecture -

Week 12 (07/03-07/09): L-series, Periodpolynomials

Week 13 (07/10-07/16): Periodpolynomials II

Week 14 (07/17-07/23): Combinatorial modular forms I

Week 15 (07/24 - 07/30): Combinatorial modular forms II

****Last update: 21st July 2023.