**Introduction to modular forms (2018 Fall)**

Perspectives in Mathematical Science IV (Part II)

This course will give a basic introduction to modular forms of level one. It is aimed to 4th-grade undergraduates and for master course students.

For any question please

All lectures will take place at: Tuesday, 3rd period (13:00 - 14:30) at Room 109, Math. Building.

The dates for Part III by H. Furusho will be 12/11, 12/18, 01/15 and 01/22. So in particular there are no lectures on 11/27 and 12/4.

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 discuss on the following topics:

Based on the exercises given above.

The deadline is:

Please put your solutions into the report box for the lecture series in front of the secretary office on the first floor, hand them in personally or send them via email.

Last update: 20th November 2018.

For any question please

**feel free to visit my office**(Math. Building Room 457) during my**office hours (Thursday 12:00 - 13:00)**or**write an email**(henrik.bachmann (at) math.nagoya-u.ac.jp).**Materials****Schedule****:**All lectures will take place at: Tuesday, 3rd period (13:00 - 14:30) at Room 109, Math. Building.

**Lecture 1 (10/30)**: Motivation, modular group, fundamental domain. (until Definition 2.7)**Lecture 2 (11/06)**: Modular functions & modular forms, Eisenstein series. (until Proposition 3.3)**Lecture 3 (11/13)**: Cusp forms, Delta function, the space of modular forms, valence formula. (until Theorem 5.6)**Lecture 4 (11/20)**: Dimension formula, derivatives of modular forms, applications, modular forms of higher level.

The dates for Part III by H. Furusho will be 12/11, 12/18, 01/15 and 01/22. So in particular there are no lectures on 11/27 and 12/4.

**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 discuss on the following topics:

- 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
- Application: Relations among Fourier coefficients of modular forms

**Grading:**Based on the exercises given above.

The deadline is:

**24th December 2018**.Please put your solutions into the report box for the lecture series in front of the secretary office on the first floor, hand them in personally or send them via email.

Last update: 20th November 2018.