Math 726: L-functions and modular forms Fall 2011

Lecture 13 : Hecke Operators and Hecke theory

Instructor: Henri Darmon Notes written by: Celine Maistret

We previously proved the following statements about modular forms and Hecke operators in the context of Γ = SL_{2}(ℤ)

1. M(Γ) = ⨁_{k} M_{k}(Γ) = ℂ[E_{4}, E_{6}] which implies in particular that each M_{k} is finite dimensional and :

M(Γ, ℚ) = ⨁_{k} M_{k}(Γ, ℚ) = ℚ[E_{4}, E_{6}]

where M_{k}(Γ, ℚ) is the space of modular forms with rational Fourier coefficients.

2. M_{k}(Γ) has extra structures :

• Hermitian inner product

• Hecke Operators (T_{n})_{n≥1}

3.

M_{k}(Γ) = ⨁_{ϕ∈Hom_{Alg}(𝕋,ℂ)}M_{k}(Γ)^{ϕ}

where 𝕋 = ℂ[T_{1}, T_{2}, ...] ⊂ End_{ℂ}(M_{k}(Γ)) and dim_{ℂ}(M_{k}(Γ))^{ϕ} = 1

= ⨁_{ϕ} ℂf_{ϕ}

where f_{ϕ} is the normalized eigenform attached to ϕ, completely characterized by :

- T_{n}(f_{ϕ}) = ϕ(T_{n})f_{ϕ}

- a_{1}(f_{ϕ}) = 1

4. If f ∈ S_{k}(Γ), f = ∑_{n=1}^{∞} a_{n}q^{n}, L(f, s) = ∑_{n=1}^{∞} a_{n}n^{−s} then L(f, s) has a functional equation relating s and k − s. Moreover, if f is a normalized eigenform then :

L(f, s) = ∏_{p}(1 − a_{p}p^{−s} + p^{k−1−2s})^{−1}

In particular, L(f, s) ≠ 0 when ℜe(s) > 1 + k/2

What about other congruence groups ? We will discuss the four statements above in the case of Γ = Γ_{0}(N) or Γ_{1}(N)

1. We still have M(Γ) = ⨁_{k} M_{k}(Γ) and the M_{k}(Γ)’s are still finite dimensional but they need not be generated by Eisenstein Series.

In particular, this is still the case that

M(Γ, ℚ) = ⨁_{k} M_{k}(Γ, ℚ)

and

M_{k}(Γ) = M_{k}(Γ, ℚ) ⊗_{ℚ} ℂ

One can produce elements of M_{k}(Γ) using the following tricks :

• Any f ∈ M_{k}(SL_{2}(ℤ)) belongs to M_{k}(Γ)

• If f ∈ M_{k}(SL_{2}(ℤ)) then f(dz) ∈ M_{k}(Γ_{0}(d)) (Exercise)

• E_{l}(d_{1}, z)E_{m}(d_{2}, z) ∈ M_{l+m}(Γ_{0}(lcm(d_{1}, d_{2})))

• Hecke translates

In fact, one can prove that if Γ = Γ_{0}(N) and k is large enough, then these basic tricks are enough to generate all of M_{k}(Γ_{0}(N))

2. M_{k}(Γ) is still a Hilbert space with Hecke operators but we have to consider different cases :

• If n is prime to N,

T_{n}f(z) = n^{k−1} ∑_{γ∈Γ_{1}(N)\M_{n}(N)} f|_{γ}(z)

where M_{n}(N) are upper triangular unipotent matrices modulo N of determinant n and

T_{n}f(z) = n^{k−1} ∑_{γ∈Γ_{0}(N)\M_{n}(N)} f|_{γ}(z)

where M_{n}(N) are upper triangular matrices modulo N of determinant n

• If l is prime and l ∤ N,

T_{l}f(q) = ∑_{n=1}^{∞} a_{n}lq^{n} + l^{k−1} ∑_{n=1}^{∞} a_{n} < l > fq^{nl}

where f ∈ M_{k}(Γ_{0}(N)) and < l > is the Diamond Operator defined below.

These are called good Hecke Operators.

• If l|N, we still define some kind of Hecke Operators :

T_{l}f(q) = ∑_{n=1}^{∞} a_{nl}q^{n}

These are called bad Hecke Operators.

• Diamond Operators < a > for a ∈ (ℤ/Nℤ)^{∗} :

< a > f(z) := f|_{k}γ_{a}(z)

where γ_{a} ∈ Γ_{0}(N) and γ_{a} ≡ (\begin{matrix}a & ∗ \\ 0 & a^{−1} \end{matrix}) mod(N) Remark 1. T_{l} is not self adjoint in general

T_{l}^{∗} =< l > T_{l} ( if l ∤ N)

Remark 2. The bad Hecke operators T_{l} do not commute with their adjoints. Let 𝕋 = ℂ(T_{l}, < a >) where l ∤ N and (a, N) = 1. Then 𝕋 ⊂ End_{ℂ}(M_{k}(Γ)) and 𝕋 is an algebra of commuting operators. We have :

M_{k}(Γ) = ⨁_{ϕ∈Hom_{Alg}(𝕋,ℂ)}M_{k}(Γ)^{ϕ}

but M_{k}(Γ)^{ϕ} is not necessary 1-dimensional. Note that if f_{1}, f_{2} ∈ M_{k}(Γ)^{ϕ} are both normalized then

ϕ(T_{n}) = a_{n}(f_{1}) = a_{n}(f_{2}) (∀(n, N) = 1)

Example 1. Construction of 2 such functions : Let f_{1} ∈ S_{k}(Γ_{1}(M)) be normalized, with M|N, M ≠ N and define :

f_{2}(z) = f_{1}(z) + λf_{1}(dz)

= ∑_{n=1}^{∞} a_{n}q^{n} + λ ∑_{n=1}^{∞} a_{n}q^{nd}

where d|\frac{M}{N} This example motivates the following definition :

Definition 2. A modular form in M_{k}(Γ_{1}(N)) which is a linear combination of forms of type

g(dz)

where g ∈ M_{k}(Γ_{1}(M)) with M|N, M ≠ N and d|\frac{M}{N} is called an old form.

Considering the space generated by old forms, we have :

Definition 3. S_{k}(Γ_{1}(N))^{old} = Space of old forms S_{k}(Γ_{1}(N))^{new} = (S_{k}(Γ_{1}(N))^{old})^{⊥} = Space of new forms where the orthogonality is relative to the Petersson inner product.

Theorem 4. (Atkin-Lehner) The space S_{k}(Γ_{1}(N))^{new} decomposes as :

S_{k}(Γ_{1}(N))^{new} = ⨁_{ϕ∈Hom_{Alg}(𝕋^{new},ℂ)}S_{k}(Γ_{1}(N))^{ϕ}

where dim_{ℂ}S_{k}(Γ_{1}(N))^{ϕ} = 1 and 𝕋^{new} = 𝕋|_{S_{k}(Γ_{1}(N))^{new}}

Definition 5. A normalized eigenform f ∈ S_{k}(Γ_{1}(N))^{new} is also called a new form of weight k and level N

It remains to discuss L(f, s) = ∑_{n=1}^{∞} a_{n}n^{−s} in this context. Can we find a functional equation ? We note that there is an extra symmetry on S_{k}(Γ_{1}(N)) :

F ricke or Atkin − Lehner involution : w_{N} ↔ (\begin{matrix}0 & −1 \\ N & 0 \end{matrix})

Fact : w_{N} normalizes Γ_{0}(N) and Γ_{1}(N) :

w_{N} (\begin{matrix}a & b \\ Nc & d \end{matrix}) w_{N}^{−1} = (\begin{matrix}d & −c \\ −Nb & a \end{matrix})

Remark 3. w_{N} does not commute in general with the action of 𝕋 :

w_{N}T_{l} =< l > T_{l}w_{N} (∀l ∤ N)

But if Γ = Γ_{0}(N) then w_{N} does commute. In particular, w_{N}f = wf for w ∈ {±1} Theorem 6. Let f ∈ S_{k}(Γ_{0}(N)) and w_{N}f = wf with w ∈ {±1} then :

Λ(f, s) : = (2π)^{−1}N^{s/2}Γ(s)L(f, s)
= (−1)^{k/2}wΛ(f, k − s)

proof : Assignment 2 □