CHAPTER 1
Basic facts
Modular forms and functions play many roles in number theory. Before pro-
ceeding to the main topics of this monograph, we begin by recalling (without proof)
the fundamental objects and the most basic facts about modular forms. More com-
plete details of this introductory material may be found in many texts such as
[Apo, Hi1, Iw2, Kna, Kno, Kob2, La2, Mi2, Ogg1, Ran, Sch, Se2, Shi1].
1.1. Congruence subgroups
Let A =
a b
c d
SL2(Z) act on H, the upper half of the complex plane, by
the linear fractional transformation
Az =
az + b
cz + d
.
The fundamental domain for the action of SL2(Z) on H, which we denote by F, is
given by
(1.1) F :=
1
2
(z)
1
2
and |z| 1
1
2
(z) 0 and |z| = 1 .
The end points of the lower boundary of F are the roots of unity i and
(1.2) ω :=
−1 +

−3
2
.
Modular forms are meromorphic functions which transform in a suitable way
with respect to groups of such transformations. For our purposes, we are mostly
concerned with certain congruence subgroups of SL2(Z).
Definition 1.1. If N is a positive integer, then define the level N congruence
subgroups Γ0(N), Γ1(N), and Γ(N) by
Γ0(N) :=
a b
c d
SL2(Z) : c 0 mod N ,
Γ1(N) :=
a b
c d
SL2(Z) : a d 1 mod N, and c 0 mod N ,
Γ(N) :=
a b
c d
SL2(Z) : a d 1 mod N, and b c 0 mod N .
It is straightforward to verify that these groups are indeed subgroups of SL2(Z).
1
http://dx.doi.org/10.1090/cbms/102/01
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