modular form

modular form

Let

SL

2

⁢

(

ℝ

)

be the group of real

2

×

2

matrices with

determinant

1

(see entry on

special linear groups

). The group

SL

2

⁢

(

ℝ

)

acts on

H

, the

upper half plane

, through

fractional linear transformations

. That is, if

γ

=

(

a

b

c

d

)

,

and

τ

∈

H

, then we let

γ

⁢

τ

=

a

⁢

τ

+

b

c

⁢

τ

+

d

.

(1)

For any natural number

N

≥

1

, define the

congruence subgroup

Γ

0

⁢

(

N

)

of level

N

to be the following

subgroup

of the group

SL

2

⁢

(

ℤ

)

of

integer

coefficient matrices of determinant

1

:

Γ

0

⁢

(

N

)

:=

{

(

a

b

c

d

)

∈

SL

2

⁢

(

ℤ

)

|

c

≡

0

(

mod

N

)

}

.

Fix an integer

k

. For

γ

∈

SL

2

⁢

(

ℤ

)

and a

function

f

defined on

H

, we define

f

∣

γ

⁢

(

τ

)

=

f

⁢

(

γ

⁢

τ

)

(

c

⁢

τ

+

d

)

k

.

For a finite index subgroup

Γ

of

SL

2

⁢

(

ℤ

)

containing a congruence subgroup, a function

f

defined on

H

is said to be a weight

k

modular form

if:

1.

f

=

f

∣

γ

for

γ

∈

Γ

.

2.

f

is

holomorphic

on

H

.

3.

f

is holomorphic at the cusps.

This last condition requires some explanation. First observe that the element

μ

=

(

1

m

0

1

)

∈

Γ

0

⁢

(

N

)

,

and

μ

⁢

z

=

z

+

m

, while if

f

satisfies all the other conditions above,

f

∣

μ

=

f

. In other words,

f

is

periodic

with period

1

. Thus, convergence permitting,

f

admits a Fourier expansion. Therefore, we say that

f

is holomorphic at the cusps if, for all

γ

∈

Γ

,

f

∣

γ

admits a a Fourier expansion

f

∣

γ

⁢

(

τ

)

=

∑

n

=

0

∞

a

n

⁢

q

n

,

(2)

where

q

=

e

2

⁢

i

⁢

π

⁢

τ

.

If all the

a

n

are zero for

n

≤

0

, then a modular form

f

is said to be a

cusp form

. The set of modular forms for

Γ

(respectively cusp forms for

Γ

) is often denoted by

M

k

⁢

(

Γ

)

(respectively

S

k

⁢

(

Γ

)

). Both

M

k

⁢

(

Γ

)

and

S

k

⁢

(

Γ

)

are finite dimensional

vector spaces

.

The space of modular forms for

SL

2

⁢

(

ℤ

)

(respectively cusp forms) is non-trivial for any

k

even and greater than 4 (respectively greater than

12

and not

14

). Examples of modular forms for

SL

2

⁢

(

ℤ

)

are:

1.

The

Eisenstein series

E

m

, where

m

is even and greater than

4

, is a modular form of weight

m

. Here

B

m

denotes the

m

-th

Bernoulli number

and, as usual,

q

=

e

2

⁢

i

⁢

π

⁢

τ

:

E

m

⁢

(

τ

)

=

1

-

2

⁢

m

B

m

⁢

∑

n

=

1

∞

⁢

σ

m

-

1

⁢

(

n

)

⁢

q

n

.

(3)

For instance,

E

4

⁢

(

τ

)

=

1

+

240

⁢

∑

n

=

1

∞

⁢

σ

3

⁢

(

n

)

⁢

q

n

(4)

and

E

6

⁢

(

τ

)

=

1

-

504

⁢

∑

n

=

1

∞

⁢

σ

5

⁢

(

n

)

⁢

q

n

.

(5)

2.

The Weierstrass

Δ

function, also called the

modular discriminant

, is a modular form of weight

12

:

Δ

⁢

(

τ

)

=

q

⁢

∏

n

=

1

∞

⁢

(

1

-

q

n

)

24

.

(6)

Every modular form is expressible as

f

⁢

(

τ

)

=

∑

n

=

0

⌊

k

/

12

⌋

⁢

a

n

⁢

E

k

-

12

⁢

n

⁢

(

τ

)

⁢

(

Δ

⁢

(

τ

)

)

n

,

(7)

where the

a

n

are arbitrary constants,

E

0

⁢

(

τ

)

=

1

and

E

2

⁢

(

τ

)

=

0

. Cusp forms are the forms with

a

0

=

0

.

Title

modular form

Canonical name

ModularForm

Date of creation

2013-03-22 14:07:37

Last modified on

2013-03-22 14:07:37

Owner

olivierfouquetx (2421)

Last modified by

olivierfouquetx (2421)

Numerical id

31

Author

olivierfouquetx (2421)

Entry type

Definition

Classification

msc 11F11

Related topic

TaniyamaShimuraConecture

Related topic

HeckeAlgebra

Related topic

AlgebraicNumberTheory

Related topic

RamanujanTauFunction

Defines

cusp form

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