Modular curve

In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curves X(Γ) which are compactifications obtained by adding finitely many points (called the cusps of Γ) to this quotient (via an action on the extended complex upper-half plane). The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either over the field of rational numbers Q or a cyclotomic field Q(ζn). The latter fact and its generalizations are of fundamental importance in number theory.

Analytic definition

The modular group SL(2, Z) acts on the upper half-plane by fractional linear transformations. The analytic definition of a modular curve involves a choice of a congruence subgroup Γ of SL(2, Z), i.e. a subgroup containing the principal congruence subgroup of level N for some positive integer N, which is defined to be

Γ

(

N

)

=

{

(

a

b

c

d

)

:

a

≡

d

≡

1

mod

N

and

b

,

c

≡

0

mod

N

}

.

{\displaystyle \Gamma (N)=\left\{{\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}:\ a\equiv d\equiv 1\mod N{\text{ and }}b,c\equiv 0\mod N\right\}.}

The minimal such N is called the level of Γ. A complex structure can be put on the quotient Γ\H to obtain a noncompact Riemann surface called a modular curve, and commonly denoted Y(Γ).

Compactified modular curves

A common compactification of Y(Γ) is obtained by adding finitely many points called the cusps of Γ. Specifically, this is done by considering the action of Γ on the extended complex upper-half plane H* = H ∪ Q ∪ {∞}. We introduce a topology on H* by taking as a basis:

any open subset of H,

for all r > 0, the set

{

∞

}

∪

{

τ

∈

H

∣

Im

(

τ

)

>

r

}

{\displaystyle \{\infty \}\cup \{\tau \in \mathbf {H} \mid {\text{Im}}(\tau )>r\}}

for all coprime integers a, c and all r > 0, the image of

{

∞

}

∪

{

τ

∈

H

∣

Im

(

τ

)

>

r

}

{\displaystyle \{\infty \}\cup \{\tau \in \mathbf {H} \mid {\text{Im}}(\tau )>r\}}

under the action of

(

a

−

m

c

n

)

{\displaystyle {\begin{pmatrix}a&-m\\c&n\end{pmatrix}}}

where m, n are integers such that an + cm = 1.