MODULAR FORMS

Problem sheet 3 KMB, Michaelmas 1997

This sheet is mostly about Hecke operators. Apologies for not handing it out earlier.

1. This question gives some corollaries of the theorem (not proved in lectures) that h_{0}(N) is a finitely generated abelian group and the theorem (proved in lectures) that h_{0}(N; R) and S_{k}(Γ_{0}(N); R) are dual to one another.

(a) Deduce that S_{k}(Γ_{0}(N); ℤ) is a finitely generated abelian group, and moreover that S_{k}(Γ_{0}(N); ℤ) ⊗_{ℤ} ℂ = S_{k}(Γ_{0}(N)).

(b) Deduce that for all subrings R of ℂ, S_{k}(Γ_{0}(N); R) is isomorphic to Hom_{ℤ}(h_{0}(N), R) = Hom_{R}(h_{0}(N; R), R). Moreover, show that f ∈ S_{k}(Γ_{0}(N)) is a normalised eigenform if and only if the corresponding map h_{0}(N) → ℂ is a homomorphism of rings.

(c) Deduce that if f ∈ S_{k}(Γ_{0}(N); ℚ) then the denominators of c_{n}(f) are bounded, independently of n.

(d) Deduce that if σ is a field automorphism of ℂ and f = ∑_{n>0} a_{n}q^{n} is in S_{k}(Γ_{0}(N)) then f^{σ} := ∑_{n>0} σ(a_{n})q^{n} is also in S_{k}(Γ_{0}(N)).

2.

In lectures, I proved that

T (m)T (n) = T (nm), if m and n are coprime

and

T (p)T (p^{e}) = T (p^{e+1}) ( +pS_{p}T (p^{e−1}) if p is prime to N ).

Furthermore, I claimed that (at least formally) these identities implied that the Lfunction ∑_{n≥1} a_{n}n^{−s} of a normalised eigenform f = ∑_{n≥1} a_{n}q^{n} had an Euler product

∑_{n≥1} a_{n}n^{−s} = ∏_{p|N}(1 − a_{p}p^{−s})^{−1} ∏_{p∤N}(1 − a_{p}p^{−s} + p^{k−1−2s})^{−1}.

Verify this claim.

3. L-functions are ubiquitous in number theory. Generally they are formed by taking some series c_{n}, n > 0, and forming the sum ∑_{n>0} c_{n}n^{−s}. In many cases of interest, these L-functions have an Euler product (e.g. the zeta function, and the L-function associated to a normalised eigenform). Here is another example.

(a) Show that if T is an indeterminate, and α, β, γ, δ are complex numbers, then

\begin{matrix}∑_{r≥0} T^{r}(α^{r} + α^{r−1}β + … + β^{r})(γ^{r} + γ^{r−1}δ + … + δ^{r}) \\ = (1 − αβγδT^{2}) ∑_{r≥0} T^{r} ∑_{{r=r_{1}+r_{2}+r_{3}+r_{4} \atop r_{i}≥0}} α^{r_{1}+r_{2}}β^{r_{3}+r_{4}}γ^{r_{1}+r_{3}}δ^{r_{2}+r_{4}}. \end{matrix}

(b) If (1 − αT)^{−1}(1 − βT)^{−1} = ∑_{r≥0} c_{r}T^{r} and (1 − γT)^{−1}(1 − δT)^{−1} = ∑_{r≥0} d_{r}T^{r}, then

∑_{r≥0} c_{r}d_{r}T^{r} = (1 − αβγδT^{2})(1 − αγT)^{−1}(1 − αδT)^{−1}(1 − βγT)^{−1}(1 − βδT)^{−1}.

(c) Suppose f = ∑ a_{n}q^{n} ∈ S_{k}(Γ_{0}(N)) and g = ∑ b_{n}q^{n} ∈ S_{l}(Γ_{0}(N)) are normalised eigenforms. Show that

\begin{matrix}ζ(2s + 2 − k − l) ∑_{n≥1} a_{n}b_{n}n^{−s} \\ = ∏_{p}((1 − α_{p}γ_{p}q^{−s})(1 − α_{p}δ_{p}p^{−s})(1 − β_{p}γ_{p}p^{−s})(1 − β_{p}δ_{p}p^{−s}))^{−1}, \end{matrix}

where, for p prime to N, α_{p} and β_{p} are the roots of T^{2} − a_{p}T + p^{k−1} and γ_{p}, δ_{p} are the roots of T^{2} − b_{p}T + p^{l−1}, and for p|N, α_{p} = a_{p}, γ_{p} = c_{p}, β_{p} = δ_{p} = 0.

4. This question shows how the action of the Hecke operators can change if the level changes.

(a) If f : ℌ → ℂ and d > 0 is an integer then define η_{d}f by (η_{d}f)(Z) = f(dz). Show that if d divides m then η_{d} defines an injection S_{k}(Γ_{0}(N)) → S_{k}(Γ_{0}(Nm)).

(b) Show that T (p) commutes with η_{d} if p is prime to m.

(c) Say p is a prime, prime to N, and f ∈ S_{k}(Γ_{0}(N)) is an eigenvector for T (p), with eigenvalue λ. Show that T (p) acting on S_{k}(Γ_{0}(Np)) preserves the 2-dimensional subspace generated by f and η_{p}f. Show furthermore that if λ^{2} ≠ 4p^{k−1} then T (p) is diagonalisable on this 2-dimensional space. What are the eigenvalues of T (p) on this space? Remark: I believe there is no example known where λ^{2} = 4p^{k−1} and it’s a theorem that it can never happen if k = 2.

(d) The same, but for level Np^{2}. Show that f, η_{p}f and η_{p^{2}}f span a 3-dimensional subspace of S_{k}(Γ_{0}(Np^{2})) and work out the action of T (p) on this space. Show that if λ^{2} ≠ 4p^{k−1} then T (p) is diagonalisable, and work out the eigenvalues.

(e) Show that for level Np^{r}, for any r > 2, that f, η_{p}f, …, η_{p^{r}}f span a subspace of S_{k}(Γ_{0}(Np^{r})) which is preserved by T (p) and furthermore show that T (p) is never diagonalisable on this space.

5.

(a) Show that m(X(SL_{2}(ℤ))) = π/3.

(b) And that reminds me: also do all the other exercises that I mentioned during the lectures.