Chapter 1

Modular Forms

This chapter introduces modular forms and congruence subgroups, which

are central objects in this book. We first introduce the upper half plane and

the group SL2(Z) then recall some definitions from complex analysis. Next

we define modular forms of level 1 followed by modular forms of general

level. In Section 1.4 we discuss congruence subgroups and explain a simple

way to compute generators for them and determine element membership.

Section 1.5 lists applications of modular forms.

We assume familiarity with basic number theory, group theory, and com-

plex analysis. For a deeper understanding of modular forms, the reader is

urged to consult the standard books in the field, e.g., [Lan95, Ser73, DI95,

Miy89, Shi94, Kob84]. See also [DS05], which is an excellent first intro-

duction to the theoretical foundations of modular forms.

1.1. Basic Definitions

The group

SL2(R) =

a b

c d

: ad − bc = 1 and a, b, c, d ∈ R

acts on the complex upper half plane

h = {z ∈ C : Im(z) 0}

by linear fractional transformations, as follows. If γ =

(

a b

c d

)

∈ SL2(R), then

for any z ∈ h we let

(1.1.1) γ(z) =

az + b

cz + d

∈ h.

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http://dx.doi.org/10.1090/gsm/079/01