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
- Interpretation of modular forms as differential on modular curves; computation of dimensions of the vector space of modular forms
- 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.

### 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*: Modular forms as differential on modular curves, review of Riemann surfaces