Seminar Modular Forms

ETH Zürich - Spring Semester 2021

In the seminar we will learn about the basic theory of classical elliptic modular forms.

We start with the action of the modular group on the complex upper half-plane by Moebius transformations and describe its fundamental domain. As first examples of modular forms, we will investigate Eisenstein series, Ramanujan's Delta function, the Dedekind eta function, and the modular j-invariant. We will show that the space of modular forms of a fixed weight is finite dimensional, and determine its dimension. We will also study Hecke operators and the Petersson inner product on spaces of modular forms, and the L-functions associated with modular forms. If time permits, we will discuss some more advanced topics, such as the general theory of theta series, the CM values of the j-function, and the periods of modular forms.

General informations

The seminar takes place Fridays from 12-14 via Zoom, starting on 26.02. until 04.06. (13 talks). The Zoom link will be sent by email.

The talks should take about 100 minutes. Two students share a talk. For your presentation you can use slides (e.g. Beamer LaTex) or write on a tablet, for example. It will not be required to prepare extended lecture notes of your talk, but your slides or your handwritten notes should be made available to the participants on this website after your talk, so keep this in mind during preparation.

For a large part of the seminar we will follow my lecture notes on modular forms.

Here is a list of topics.