Siegel modular forms and associated representations

Overview

Modular forms and automorphic representations have played a central role in the research in number theory in the last half century. Several important problems in mathematics, including Fermat's Last Theorem, have been solved using modular forms. Two of the seven million dollar millennium problems are related to modular forms and related topics.

In this course, we plan to introduce the audience to the topic of Siegel modular forms. The idea is that, via the study of Siegel modular forms, the students will get to know the current directions of research in automorphic forms. We will cover topics ranging from the basic definitions and properties of Siegel modular forms, to the recent research on Langlands transfer and Deligne's conjectures on special values of L-functions, and will also include several important open problems like the Bocherer's conjecture.

This course is intended for graduate students interested in research in number theory. Familiarity with elliptic modular forms and associated representations is preferable but not essential. We will provide a handout for the attendees to read to get them prepared for the lectures. The advanced topics of this course will also be of interested to young as well as established researchers in number theory.

Objectives

The main objectives of this course are:

• Classical theory of Siegel modular forms
• Automorphic representations associated Siegel modular forms
• Important results and open problems for Siegel modular forms coming from analytical number theory and automorphic representation theory
• In particular, discussion of Ramanujan conjecture, Langlands conjecture, Deligne's conjecture and Bocherer's conjecture.

Click here for the poster for this course.