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
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