Math 726: L-functions and modular forms Fall 2011

Lecture 12 : Hecke Operators and Hecke theory

Instructor: Henri Darmon Notes written by: Celine Maistret

Recall that we found a formula for multiplication of operators :

T_{n}T_{m} = T_{nm} when gcd(m, n) = 1
T_{p^{r}} = T_{p}T_{p^{r−1}} − p^{k−1}T_{p^{r−2}}

In particular, we note that all Hecke Operator commute.

Let 𝕋_{k} be the algebra generated over ℂ by the Hecke Operator (T_{1}, T_{2}, ....) in End_{ℂ}(M_{k}(SL_{2}(ℤ))), ie 𝕋_{k} ⊆ M_{d}(ℂ) where d = dim_{ℂ}(M_{k}(SL_{2}(ℤ)))

Definition 1. A modular form f ∈ M_{k}(SL_{2}(ℤ)) is called an eigenform or Hecke eigenform if it is a simultaneous eigenvector for all the Hecke Operators:

T_{n}(f) = λ_{n}f (∀λ_{n} ∈ ℂ withf ≠ 0)

Also, the eigenform f is said to be normalized if a_{1}(f) = 1

Remark 1. If f is a normalized eigenform we have :

T_{n}(f) = λ_{n}f for some λ_{n}

but

a_{n}(f) = a_{1}(T_{n}(f)) = a_{1}(λ_{n}f) = λ_{n}a_{1}(f) = λ_{n}

since a_{1}(f) = 1. Hence we can recover the Fourier coefficients of a normalized eigenform by considering the associated eigenvalues of Hecke Operators : λ_{n} = a_{n}(f).

This leads to the following property : Multiplicity one property : An eigenspace for any collection {λ_{n}} of eigenvalues for T_{n} is exactly 1-dimensional, spanned by the normalized eigenform.

Theorem 2. (Hecke) If f is a normalized eigenform on SL_{2}(ℤ) and if L(f, s) is its associated L-function then for ℜe(s) > k/2 + 1

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

proof : Relations among the T_{n}’s and T_{p^{r}}’s imply that if f is a normalized eigenform :

a(mn) = a(m)a(n) if (m, n) = 1

and

a(p^{r}) = a(p)a(p^{r−1}) − p^{k−1}a(p^{r−2})

thus

L(f, s) = ∑_{n=1}^{∞} a_{n}n^{−s}

= ∏_{p} ( ∑_{r=0}^{∞} a_{p}rp^{−rs})

= ∏_{p}(1 − a(p)p^{−s} + p^{k−1−2s})^{−1}

□ We now see some examples of eigenforms and Dirichlet Series :

• Eisenstein Series : G_{k} ∈ M_{k}(SL_{2}(ℤ)) is an eigenform. Normalizing appropriately, we get G̃_{k} with a_{1}(G̃_{k}) = 1 Now recall that the Fourier expansion of G_{k} involves δ_{k−1}(n) = ∑_{d|n} d^{k−1}, we have :

a_{n}(G̃_{k}) = δ_{k−1}(n) = ∑_{d|n} d^{k−1}

where δ_{k−1} is multiplicative and satisfy :

δ_{k−1}(p^{r}) = δ_{k−1}(p)δ_{k−1}(p^{r−1}) − p^{k−1}δ_{k−1}(p^{r−2})

• A cuspidal example : Δ ∈ S_{12}(SL_{2}(ℤ)) is also an eigenform.

Writing the q-exansion of Δ:

Δ(q) = ∑_{n=1}^{∞} τ(n)q^{n}

where τ(n) is the n^{th} Fourier coefficient, also called the Ramanujan τ-function, is in particular multiplicative.

= q ∏_{n}(1 − q^{n})^{24}

Then

L(Δ, s) = ∑_{n} τ(n)n^{−s} = ∏_{p}(1 − τ(p)p^{−s} + p^{11−2s})^{−1}

Now, we can ask how often do we find eigenforms if we look at S_{k}(SL_{2}(ℤ)) when k > 2 is even.

Theorem 3. (Hecke) The space M_{k}(SL_{2}(ℤ)) has a basis consisting of normalized eigenforms

M_{k}(SL_{2}(ℤ)) = ⨁_{ϕ∈Hom_{Alg}(𝕋_{k},ℂ)}M_{k}(SL_{2}(ℤ))^{T_{n}=ϕ(T_{n})}

where M_{k}(SL_{2}(ℤ))^{T_{n}=ϕ(T_{n})} := {f ∈ M_{k}(SL_{2}(ℤ))|T_{n}(f) = ϕ(T_{n})(f)}

proof : S_{k}(SL_{2}(ℤ)) is naturally equipped with a hermitian inner product, the Petersson scalar product

< f, g >= ∫_{SL_{2}(ℤ)/H} y^{k}f(z)\overline{g(z)} \frac{dz\overline{dz}}{y^{2}}

which is non-degenerate and positive.

Key fact : The operator T_{n} is self adjoint with respect to this pairing:

< T_{n}(f), g >=< f, T_{n}(g) >

By diagonalisibility of self adjoint operators we have :

S_{k}(SL_{2}(ℤ)) = ⨁_{ϕ∈Hom_{Alg}(𝕋_{k},ℂ)}S_{k}(SL_{2}(ℤ))^{T_{n}=ϕ(T_{n})}

which is an orthogonal direct sum. But since we know that M_{k}(SL_{2}(ℤ)) is a direct sum of the line spanned by the Eisenstein Series E_{k} with the space of cusp forms, we also have such a decomposition for M_{k}(SL_{2}(ℤ)):

M_{k}(SL_{2}(ℤ)) = ℂ.E_{k} ⨁ S_{k}(SL_{2}(ℤ))

Note however that < f, g > does not converge for f, g ∈ M_{k}(SL_{2}(ℤ)), hence is it not a orthogonal direct product. □

Proposition 1. If f ∈ S_{k}(SL_{2}(ℤ)) is a normalized eigenform then the Fourier coefficients a_{n}(f) are algebraic numbers. More precisely, ℚ(a_{1}(f), ..., a_{n}(f), ...) is a totally real extension of ℚ of degree at most the dimension of S_{k}(SL_{2}(ℤ)).

proof :

As we saw before, S_{k}(SL_{2}(ℤ)) has a basis consisting of modular forms with rational Fourier coefficients. Let (f_{1}, ...., f_{d}) be such a basis. T_{n} preserves the rational vector space ℚf_{1} ⨁ ℚf_{2} ⨁ .... ⨁ ℚf_{d} and therefore can be viewed as a d by d matrix with rational entries relative to this basis. But the eigenvalues of such a matrix are algebraic, and moreover, if we let 𝕋_{k}(ℚ) be the ℚ-algebra generated by T_{2}, T_{3}, T_{4}, ... acting on S_{k}(SL_{2}(ℤ)), then 𝕋_{k}(ℚ) is isomorphic to a commutative subring of M_{d}(ℚ). Therefore, any ϕ ∈ Hom(𝕋_{k}(ℚ), \overline{ℚ}) has its image contained in an extension of ℚ of degree less or equal to d = dim S_{k}(SL_{2}(ℤ)) Note : Hom(𝕋_{k}(ℚ)) is totally real by self adjointness.□

Remark 2. The first interesting case, where dim(S_{k}(SL_{2}(ℤ))) > 1 is k = 24

As a possible exercise, one can compute the field of definition of the Fourier coefficients of eigenforms in S_{24}(SL_{2}(ℤ)))

This complete the study of Hecke Theory and L-functions of modular forms in case of level 1. It remains to study the theory when the level is arbitrary, and discuss the features that generalize and those that do not.