189-726A: L-functions and Modular Forms Assignment 5

Due: Monday, November 21

Note: You may hand in this last assignment one week late, on November 28, but no later than that.

1. Let f = ∑_{n} a_{n}q^{n} be a normalised Hecke eigenform of weight k on SL_{2}(Z). a) Show that

∑_{n} a_{n}^{2}n^{−s} = ζ(s − k + 1) ∑_{n} a_{n^{2}}n^{−s}.

b) Letting

⟨f, g⟩ := ∫_{SL_{2}(Z)\H} y^{k}f(z)\overline{g(z)}\frac{dxdy}{y^{2}}

denote the Petersson scalar product on forms of weight k, show that

⟨f, f⟩ = \frac{π(k − 1)!}{3(4π)^{k}} ∑_{n=1}^{∞} a_{n^{2}}n^{−s}|_{s=k}.

2. Let K be a finite extension of Q_{p} and let R denote its ring of integers. Let

ρ : G_{Q} ⟶ GL_{2}(K̄)

be a continuous odd representation, satisfying

char.poly(ρ(frob_{p})) belongs to R[x],

for all primes p at which ρ is unramified.

Write

ρ(σ) = ( \begin{matrix}a(σ) & b(σ) \\ c(σ) & d(σ) \end{matrix} ) , σ ∈ G_{Q}.

a) Show that ρ can be conjugated in such a way that a(σ), d(σ), and b(σ)c(τ) belong to R for all σ, τ ∈ G_{Q}.

b) After normalising ρ as in a), let B be the ideal of R generated by the expressions b(σ)c(τ) as σ, τ run over G_{Q}. Show that ρ has a conjugate taking values in GL_{2}(R) when B ≠ 0. Show by an example that the conclusion need not hold when B = 0.

c) Assume that B ≠ 0 and that ρ is conjugated as in b). Show that the reduction

ρ̄: G_{Q} ⟶ GL_{2}(R/m_{R}R)

of ρ modulo the maximal ideal m_{R} of R is irreducible if and only if B = R.

d) Let f = ∑_{n=1}^{∞} a_{n}q^{n} be a normalised Hecke eigenform of weight k, level D and character ϵ, with fourier coefficients in a field K_{f}. Choose a prime ideal λ of K_{f} and assume that the function

p ↦ a_{p} (mod λ)

is not a sum of two characters. Let K_{f,λ} be the completion of K_{f} at λ and let O_{f,λ} be its ring of integers. Show that there is a continuous Galois representation

ρ_{f} : G_{Q} ⟶ GL_{2}(O_{f,λ})

satisfying

char.poly(ρ_{f}(frob_{p})) = x^{2} − a_{p}x + ϵ(p)p^{k−1},

for all primes p not dividing DNorm(λ). (You may of course use any of the theorems on the existence of ℓ-adic representations attached to cusp forms that was stated in class.)