**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 1. It is aimed to 4th-grade undergraduates and for master course students.

For any question please

-- More details will follow here before the start of the lectures --

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

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 report of the problems assigned in the lectures.

The deadline is not decided yet.

For any question please

**feel free to visit my office**(Math. Building Room 457) at any time or**write an email**(henrik.bachmann (at) math.nagoya-u.ac.jp).-- More details will follow here before the start of the lectures --

Schedule:Schedule:

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

**Lecture 1 (10/30)**: (details will follow)**Lecture 2 (11/06)**: (details will follow)**Lecture 3 (11/13)**: (details will follow)**Lecture 4 (11/20)**: (details will follow)**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
- The space of modular and its dimension
- Cusp forms and Ramanujan's Delta-function
- Application: Relations among Fourier coefficients of modular forms

**Grading:**Based on the report of the problems assigned in the lectures.

The deadline is not decided yet.