Modular Forms, Elliptic Curves, and Modular Curves

Abstract

This chapter introduces three central objects of the book.

Modular forms are functions on the complex upper half plane. A matrix group called the modular group acts on the upper half plane, and modular forms are the functions that transform in a nearly invariant way under the action and satisfy a holomorphy condition. Restricting the action to subgroups of the modular group called congruence subgroups gives rise to more modular forms.