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).

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http://dx.doi.org/10.1090/cbms/102/01