Introduction to Modular Forms

Boston College, Fall 2011

Modular forms are classical analytic objects which were the center of much attention early in the last century. For some time their interest appeared to have diminished, but then remarkable connections with a huge range of other areas in pure mathematics were discovered. Currently, modular forms and their close relatives, the automorphic forms, are present in almost every area of modern number theory. The most celebrated is perhaps the role they played in the proof of Fermat's last theorem, through the conjecture of Shimura-Taniyama-Weil that elliptic curves are modular. Modular forms also play an important role in the Langlands program and in recent advancement towards the quantum unique ergodicity conjecture.

The aim of this course is to cover the classical theory of modular forms and point out some classical applications to number theory. In particular, some of the topics that will be covered are

• Elliptic functions
• Eisenstein series
• The modular group and its subgroups
• Basic properties of modular forms
• Hecke operators
• Dirichlet series attached to modular forms
• Theta functions and quadratic forms
• Maass waveforms

Prerequisites

The course is intended for the second year graduate students; first year graduate students are also welcome.

I will assume knowledge of basic courses in complex analysis, linear algebra, and group theory.

Bibliography

• H. Iwaniec, Topics in classical automorphic forms
• N. Koblitz, Introduction to elliptic curves and modular forms
• J.-P. Serre, A Course in Arithmetic

Schedule

MWF 9-10

Room: Gasson 208

Homework

There will be periodic homework assignments posted here.

Assignments

1. Assignment 1, due date: Sep. 28, 2011.
2. Assignment 2, due date: Oct. 12, 2011.
3. Assignment 3, due date: Nov. 09, 2011.
4. Assignment 4, due date: Dec. 14, 2011.