A Brief Introduction to Modular Forms

In this chapter, we give an overview on the theory of “classical” modular forms. What we mean by classical here are holomorphic modular forms on the full modular group or on its congruence subgroups. We give the definition of modular forms for real weights and give properties of the multiplier system. A few explicit examples of modular forms (Eisenstein series, discriminant function, holomorphic Poincare series, and theta series) are also presented. Then, we introduce Hecke operators and L-functions and give their properties that will be needed in later chapters.