Math 726: L-functions and modular forms Fall 2011

Week 3, lecture 10: Dimension of spaces of modular forms

Instructor: Henri Darmon Notes written by: Luiz Kazuo Takei

Recall

We are working with cusp forms f ∈ S_{k}(SL_{2}(ℤ)). For each such f, we defined

Λ(f, s) = (2π)^{−s}Γ(s)L(f, s),

which satisfies

Λ(f, s) = (−1)^{k−2}Λ(f, k − s).

We also defined, for each k ∈ ℤ_{>2} even, the Eisenstein series of weight k

E_{k}(q) ∈ ℚ ⊗(ℤ[[q]]).

Finally we defined the Ramanujan Δ-function by

Δ = \frac{E_{4}^{3} − E_{6}^{2}}{1728} = q + ⋯ ∈ S_{12}(SL_{2}(ℤ)).

Dimension of some spaces of modular forms

Before we start proving the theorem stated last time, we will present the following fact. Fact 1. The Ramanujan Δ-function satisfies

Δ(z) ≠ 0 for allz ∈ H .

There are at least two ways to prove this fact. We will present the idea behind them. Proof. (1. Using a product expansion for Δ) This proof uses the fact that

Δ(q) = q ∏_{n=1}^{∞}(1 − q^{n})^{24}.

Call the right-hand side Δ_{2}. To show that Δ_{2} = Δ, one first shows that Δ_{2} ∈ S_{12}(SL_{2}(ℤ)). By the definition of Δ_{2}, it is easy to see it satisfies the growth condition at ∞. It suffices then to show that (Δ_{2})|_{k}γ = Δ_{2} for all γ ∈ SL_{2}(ℤ). One further reduces to the case of showing this equality for γ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ), which is proved by taking the logarithmic derivative of Δ_{2} and comparing it with the Eisenstein series E_{2}:

2πi (1 − 24 ∑_{n=1}^{∞} σ(n)q^{n}) = ‘E_{2}(q)’.

where σ(n) = ∑_{d>0 , d|n} d.

One then realizes that Δ/Δ_{2} is an element of M_{0}(SL_{2}(ℤ)), which, as we shall see in the next theorem, is 1-dimensional.

(Cf. section VII.4.4 in ‘A Course in Arithmetic’ by J.-P. Serre)

Proof. (2. Using a relation between Δ and the theory of elliptic curves) This proof uses the fact that

Δ(τ) = disc(ℂ / < 1, τ > , 2πidz).

As a consequence, since ℂ /⟨1, τ⟩ is non-singular, Δ(τ) ≠ 0.

(Cf. corollary 1.4.2 in ‘A First Course in Modular Forms’ by Diamond & Shurman.)

Let us now restate the theorem we mentioned last class. In what follows, we will denote Γ := SL_{2}(ℤ).

Theorem 1. The spaces S_{k} := S_{k}(SL_{2}(ℤ)) and M_{k} := M_{k}(SL_{2}(ℤ)) are finite dimensional. More precisely:

(1) When k = 0 we have

dim S_{0} = 0 and dim M_{0} = 1.

(2) If k < 0, then

dim S_{k} = dim M_{k} = 0.

(3) If k is odd, then

dim S_{k} = dim M_{k} = 0.

(4) When k = 2 we have

dim M_{2} = 0.

(5) For any k,

dim S_{k+12} = dim M_{k}.

(6) For k > 2 even,

dim M_{k} = 1 + dim S_{k}.

Proof. (1) The quotient SL_{2}(ℤ)\ H has a complex structure which makes it a non-compact Riemann surface.

We also have

q ⟷ complex structure of SL_{2}(ℤ)\(H ∪ ℙ_{1}(ℚ)) =: X.

Using this complex structure, we can identify M_{0} with the space of holomorphic functions on X(ℂ). This implies that

M_{0} = ℂ .

Under the same identification, S_{0} correspond to the space of holomorphic functions on X(ℂ) which vanish at the cusp. This implies that

S_{0} = {0}.

(2) Suppose g ∈ M_{−r} for some r > 0. Then g^{12}Δ^{r} ∈ S_{0} = {0}. So g = 0.

(3) This follows from the fact that

(\begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix}) ∈ SL_{2}(ℤ).

(4) The space M_{2} can be identified with

M_{2} = Ω_{X−{∞}}^{1}(log(∞)),

which is the space of differentials on X − {∞} which are holomorphic on X − {∞} and have at worse a simple pole at ∞. This identification is given by

f ⟷ 2πif(z)dz = f(q)\frac{dq}{q} .

(Notice that \frac{dq}{q} = 2πidz.)

But, by the residue theorem, there can be no residue. So

M_{2} = Ω_{X}^{1}.

Now, since

SL_{2}(ℤ)\ H ≅ 𝔸_{1}(ℂ),

we obtain that

SL_{2}(ℤ)\(H ∪ ℙ_{1}(ℚ)) ≅ ℙ_{1}(ℂ),

which has genus 0 and, hence, Ω_{X}^{1} = {0}.

Exercise 1. Prove this in a more ‘hands-on’ way.

(5) Note that multiplication by Δ gives an injective linear transformation

M_{k} ⟶ S_{k+12}.

This map is also surjective because if f ∈ S_{k+12}, then f/Δ is still holomorphic on H and satisfies the growth condition at ∞.

(6) We have an injective map

M_{k}/S_{k} ⟶ ℂ

defined by

f ⟼ a_{0}(f),

where f = ∑_{n=0}^{∞} a_{n}q^{n} is its Fourier expansion. This implies that

dim (M_{k}/S_{k}) ≤ 1.

This inequality, together with the existence of the Eisenstein series (which are not cusp forms), yields

dim M_{k} = 1 + dim S_{k}.

As a consequence of this theorem, we may write the following table.

\begin{matrix}k & dim S_{k} & dim M_{k} \\ < 0 & 0 & 0 \\ 0 & 0 & 1 \\ 2 & 0 & 0 \\ 4 & 0 & 1 \\ 6 & 0 & 1 \\ 8 & 0 & 1 \\ 10 & 0 & 1 \\ 12 & 1 & 2 \\ 14 & 0 & 1 \\ 16 & 1 & 2 \end{matrix}

Note that combining items (5) and (6) we obtain

dim M_{k} ∼ \frac{k}{12}.

Corollary 1. Consider the graded ring (under multiplication) ⨁_{k=0}^{∞} M_{k}(SL_{2}(ℤ)). It is gen-

erated by E_{4} and E_{6}. In other words,

⨁_{k=0}^{∞} M_{k}(SL_{2}(ℤ)) = ℂ[E_{4}, E_{6}].

Corollary 2. The complex vector space M_{k}(SL_{2}(ℤ)) has a basis consisting of modular forms with rational (even integers) Fourier coefficients.

Hecke Operators

A modular form can be viewed in many different, \underline{more structured} ways.

Definition 2. A lattice in ℂ is a free ℤ-module Λ in ℂ of rank two such that ℂ /Λ is compact (equivalently, Λ = ℤ μ_{1} ⊕ ℤ μ_{2}, where μ_{1}, μ_{2} ∈ ℂ are linearly independent over ℝ).

Let Π denote the space of lattices in ℂ.

Definition 3. A complex-valued function F on Π is said to be homogeneous of weight k if

F (λΛ) = λ^{−k}F (Λ)

for all λ ∈ ℂ^{×} and Λ ∈ Π. Such an F is called holomorphic if

τ ∈ H ⟼ F (ℤ ⊕ ℤ τ)

is a holomorphic function.