9Hecke theory for Γ_{0}(N)

III Modular Forms and L-functions

9 Hecke theory for Γ

0

(N)

Note that it is possible to do this for other congruence subgroups. The key case

is

Γ

1

(N) =

a b

c d

∈ SL

2

(Z) : c ≡ 0 (mod N), d, a ≡ 1 (mod N)

What is special about this? There are two things

–

The map

a b

c d

7→ d mod N

is a homomorphism Γ

0

(

N

)

→

(

Z/nZ

)

×

, and

the kernel is Γ

1

(N).

So we can describe

S

k

(Γ

1

(N)) =

M

χ∈

\

(Z/NZ)

×

S

k

(Γ

1

(N), χ),

where f ∈ S

k

(Γ

1

(n), χ) if

f |

k

a b

c d

= χ(d)f for all

a b

c d

∈ Γ

0

(N).

Of course,

S

k

(Γ

1

(

N

)

, χ

trivial

) =

S

k

(Γ

0

(

N

)). In general, everything we can do for

Γ

0

(N) can be done for S

k

(Γ

1

(N), χ).

But why not study Γ(N) itself? We can check that

1 0

0 N

Γ(N)

1 0

0 N

−1

⊇ Γ

1

(N

2

).

So we can go from modular forms on Γ(N ) to ones on Γ

1

(N

0

).

For various representation-theoretic reasons, things work much better on

Γ

1

(N).

Last time, we used in various places the matrix

W

N

=

0 −1

N 0

.

Then we decomposed

S

k

(Γ

0

(N)) = S

k

(Γ

2

(N))

+

⊕ S

k

(Γ

0

(N))

−

,

according to the

±

-eigenspaces of the operator

W

N

. A while ago, we proved

some theorem about the function equation of L-functions.

Theorem. Let f ∈ S

k

(Γ

0

(N))

ε

, where ε = ±1. Then define

L(f, s) =

X

n≥1

a

n

n

−s

.

Then L(f, s) is am entire function, and satisfies the functional equation

Λ(f, s) = (2π)

−s

Γ(s)L(f, s) = ε(−N)

k/2

Λ(f, k −s).

Proof. We have f |

k

W

N

= εf, and then we can apply our earlier result.

This is a rather remarkable thing. We didn’t really use much about the

properties of f .

Now we want to talk about Hecke operators on Γ

0

(

N

). These are a bit more

complicate. It is much better to understand these in terms of representation

theory instead, but that is outside the scope of the course. So we will just state

the relevant definitions and results.

Recall that a modular form of level 1 is defined by the

q

-expansion, and if

what we have is in fact a Hecke eigenform, then it suffices to know the Hecke

eigenvalues, i.e. the values of

a

p

. We describe this as having “multiplicity one”.

Theorem

(Strong multiplicity one for

SL

2

(

Z

))

.

Let

f, g ∈ S

k

(Γ(1)) be normal-

ized Hecke eigenforms, i.e.

f|T

p

= λ

p

f λ

p

= a

p

(f)

g|T

p

= µ

p

g µ

p

= a

p

(g).

Suppose there exists a finite set of primes

S

such that such that for all

p 6∈ S

,

then λ

p

= µ

p

. Then f = g.

Note that since the space of modular forms is finite dimensional, we know

that the modular forms can only depend on finitely many of the coefficients. But

this alone does not give the above result. For example, it might be that

a

2

(

f

) is

absolutely crucial for determining which modular form we are, and we cannot

miss it out. The strong multiplicity one theorem says this does not happen.

Idea of proof. We use the functional equations

Λ(f, k −s) = (−1)

k/2

Λ(f, s)

Λ(g, k − s) = (−1)

k/2

Λ(g, s)

So we know

L(f, k −s)

L(f, s)

=

L(g, k − s)

L(g, s)

.

Since these are eigenforms, we have an Euler product

L(f, s) =

Y

p

(1 − λ

p

p

−s

+ p

k−1−2s

)

−1

,

and likewise for g. So we obtain

Y

p

1 − λ

p

p

s−k

+ p

2s−k−1

1 − λ

p

p

−s

+ p

k−1−2s

=

Y

p

1 − µ

p

p

s−k

+ p

2s−k−1

1 − µ

p

p

−s

+ p

k−1−2s

.

Now we can replace this

Q

p

with

Q

p∈S

. Then we have some very explicit

rational functions, and then by looking at the appropriate zeroes and poles, we

can actually get λ

p

= µ

p

for all p.

This uses L-functions in an essential way.

The reason we mention this here is that a naive generalization of this theorem

does not hold for, say, Γ

0

(

N

). To even make sense of this statement, we need to

say what the Hecke operators are for Γ

0

(

N

). We are going to write the definition

in a way as short as possible.

Definition (Hecke operators on Γ

0

(N)). If p - N , we define

T

p

f = p

k

2

−1

f |

k

p 0

0 1

+

p−1

X

k=0

f |

k

1 b

0 p

!

which is the same as the case with Γ(1).

When p | N , then we define

U

p

f = p

k

2

−1

p−1

X

n=0

f |

k

1 b

0 p

.

Some people call this T

p

instead, and this is very confusing.

We can compute the effect on

q

-expansions as follows — when

p - N

, then

we have

a

n

(T

p

f) = a

np

(f) + p

k−1

a

n/p

(f),

where the second term is set to 0 if p - n. If p | N, then we have

a

n

(U

p

f) = a

np

(f).

Proposition. T

p

, U

p

send S

k

(Γ

0

(N)) to S

k

(Γ

0

(N)), and they all commute.

Proof. T

p

, U

p

do correspond to double coset actions

Γ

0

(N)

1 0

0 p

Γ

0

(N) =

(

Γ

0

(N)

p 0

0 1

q

`

b

Γ

0

(N)

1 b

0 p

p - N

`

b

Γ

0

(N)

1 b

0 p

p | N

.

Commutativity is checked by carefully checking the effect on the

q

-expansions.

However, these do not generate all the Hecke operators. For example, we

have W

N

!

Example. Consider S

12

(Γ

0

(2)). This contains f = ∆(z) and

g = f |

12

(

2 0

0 1

) = 2

6

∆(2z) = ∆ |

12

W

2

,

using the fact that

∆|

k

0 −1

1 0

= ∆.

So the matrix of W

2

on span{f, g} is

0 1

1 0

.

We can write out

f =

X

τ(n)q

n

= q −24q

2

+ 252q

3

− 1472q

4

+ ···

g = 2

6

X

τ(n)q

2n

= 2

6

(q

2

+ 24q

4

+ ···)

So we find that

U

2

g = 2

6

f.

It takes a bit more work to see what this does on f. We in fact have

U

2

f =

X

τ(2n)q

n

= −24q −1472q

4

+ ··· = −24f − 32g.

So in fact we have

U

2

=

24 64

−32 0

.

Now

U

2

and

W

2

certainly do not commute. So the Hecke algebra is not com-

mutative. In fact, ∆ generates a two-dimensional representation of the Hecke

algebra.

This makes life much worse. When we did Hecke algebras for Γ(1), all our

representations are 1-dimensional, and we can just work with linear spans. Now

everything has higher dimensional, and things go rather wrong. Similarly, we

can consider ∆(

dz

)

∈ S

12

(Γ

0

(

N

)) for any

d | N

, and this gives a whole bunch of

things like this.

This turns out to be the only obstruction to the commutativity of the action

of the Hecke algebra. We know S

k

(Γ

0

(N)) contains

{f(dz) : f ∈ S

k

(Γ

0

(M)), dM | N, M 6= N}.

We let

S

k

(Γ

0

(

N

))

old

be the span of these. These are all the forms that come

from a smaller level.

Now

S

k

(Γ

0

(

N

)) has an inner product! So the natural thing to do is to

consider the orthogonal complement of

S

k

(Γ

0

(

N

))

old

, and call it

S

k

(Γ

0

(

N

))

new

.

Theorem

(Atkin–Lehner)

.

The Hecke algebra

H

(

G,

Γ

0

(

N

)) fixes

S

k

(Γ

0

(

N

))

new

and

S

k

(Γ

0

(

N

))

old

, and on

S

k

(Γ

0

(

N

))

new

, it acts as a commutative subalgebra

of the endomorphism ring, is closed under adjoint, and hence is diagonalizable.

Moreover, strong multiplicity one holds, i.e. if

S

is a finite set of primes, and we

have

{λ

p

:

p 6∈ S}

given, then there exists at most one

N ≥

1 and at most one

f ∈ S

k

(Γ

0

(N), 1)

new

(up to scaling, obviously) for which

T

p

f = λ

p

f for all p - N, p 6∈ S.