The seminar will meet weekly at the Student Lounge or Thesis Room, from 4:00pm-5:00pm on Friday. The official homepage for the seminar is here.

I will be talking about various aspect of classical modular forms, especially its application in number theory. My first goal is to give you a fundamental treatment of modular forms, a first course if you will (which mean that it comes with little pre-requisite and could be viewed as a typical graduate course like algebraic topology).

The (incomplete list of) topics I want to talk about are:

- Definition of modular forms; most importantly, precise definition of various notions such as cusps, Fourier expansion and holomorphic at cusps; work out examples of modular forms notes
- Interpretation of modular forms as differential on modular curves notes
- Dimensions of the vector space of modular forms notes (for a Review of Riemann surfaces)
- Hecke operators and the theory of new forms
- L-functions and converse theorems
- Algebraic theory: algebraicity of eigenvalues of Hecke eigenforms, p-adic modular forms
- Applications of modular forms: Galois representation and non-abelian class field theory

Along the way, I shall review some topic such as Riemann surfaces, sheaf and algebraic geometry.

Unlike last semester, I shall upload the combined note when I think they are ready.

Here is Kindle version of William Stein's book *Modular Forms: A Computational Approach* that I recompiled from his TeX source code.

This time, I make all my notes in Kindle size.

### Modular Forms I

#### September 7th, 2018

*Abstract*: Definition of modular forms, examples, classification of linear fractional transformations

### Modular Forms II

#### September 14th, 2018

*Abstract*: Absolute convergence of Eisenstein series (missing from last time), cusps, elliptic points, Fourier expansion

### Modular Forms III

#### September 21st, 2018

*Abstract*: Review of Riemann surfaces: the category of Riemann surfaces, uniformization theorem, compact Riemann surfaces, differentials

### Modular Forms IV

#### September 28st, 2018

*Abstract*: Modular curves as Riemann surfaces, interpretation of modular forms as differential on modular curves

### Modular Forms V

#### October 5th, 2018

*Abstract*: Dimension formulas for the space of modular forms

### Modular Forms VI

#### October 12th, 2018

*Abstract*: Dimension formulas for the space of modular forms (Continued)

### Modular Forms VII

#### October 19th, 2018

**NO SEMINAR**

### Modular Forms VIII

#### October 26th, 2018

*Abstract*: Hecke operators

### Modular Forms IX

#### October 26th, 2018

*Abstract*: Primitive forms

### Modular Forms X

#### November 2nd, 2018

**NO SEMINAR. I WILL SPEAK AT JNTD.**