Math 726: L-functions and modular forms Fall 2011

Lecture 27 : Artin Representations Attached to Forms of Weight One

Instructor: Henri Darmon Notes written by: Victoria de Quehen

Recall what we had this morning: Let f ∈ S_{1}(Γ_{0}(D), ε) with Fourier coefficients in O_{K_{f}} ⊆ K_{f}. Given λ ◁ O_{K_{f}} with λ|ℓ and ℓ > 3 we considered:

F_{λ} = fE_{ℓ−1} ∈ S_{ℓ}(Γ_{0}(N), ε).

Here F_{λ} is an eigenform modulo λ^{n}, ie, T_{n}(F_{λ}) − a_{n}(F_{λ}) ∈ λO_{K_{f,λ}}[[q]].

Theorem 1 (Delign-Serre Lifting Lemma). Given F_{λ} there exists

1. a finite extension K′ of K = K_{f},

2. an ideal λ′ ∈ O_{K′} such that λ′ ∩ O_{K} = λ, and

3. a normalized eigenform F̃_{λ′} ∈ S_{ℓ}(Γ_{0}(D), ε) with Fourier coefficients in O_{K} such that

a_{n}(F̃_{λ′}) ≅ a_{n}(F_{λ}) (mod λ′) for all (n, D) = 1.

Proof. Let 𝕋 be the ring generated by the Hecke operators T_{n} with (n, D) = 1 inside End(S_{l}(Γ_{0}(D), ε)).

Structure of 𝕋

Basic facts we will need. (For this we will assume ε is quadratic.)

1. 𝕋 is a maximal commutative subring of M_{d}(ℤ),. This implies 𝕋 is isomorphic to ℤ^{d} as a ℤ-module (not as a ring).

If L is any ring then 𝕋_{L} = 𝕋 ⊗_{ℤ} L ≅ L^{d} as a ℤ-module.

2. 𝕋_{O_{K,λ}} is equipped with a homomorphism to O_{K,λ} arising from the mod λ eigenform F_{λ}.

φ_{F_{λ}} : 𝕋_{O_{K,λ}} → (O_{K,λ}/λ) = F_{ℓ^{s}}
T_{n} ↦ a_{n} (mod λ).

\underline{Goal}: We want to lift φ_{F_{λ}} to a homomorphism φ_{f,λ} in characteristic 0.

\xymatrix{ & O_{K′,λ′} \ar[d] \\ 𝕋_{O_{K,λ}} \ar@{.>}[ur] \ar[r] & O_{K,λ}}

(We can do this for a larger field K′ and λ′.)

Let M = ker(φ_{F_{λ}}). M is a maximal ideal of 𝕋_{O_{K,λ}}.

Lemma 1. The ideal M properly contains a prime ideal I of 𝕋_{O_{K,λ}}.

Proof. Consider the injective map

𝕋 ⊗_{ℤ} O_{L} ↪^{(φ_{i})_{i}} O_{L}^{d}

induced by the eigenvalues of the mod λ eigenform F_{λ} Let I_{j} := ker(φ_{j}) be the various kernels.

We claim that there exists an I_{j} ⊂ M. Indeed, suppose I_{j} ⊈ M for all j. Then for each j there exists x_{j} ∈ I_{j} \ M. We then have that the product x_{1} … x_{d} ∈ I_{1} ∩ ⋯ ∩ I_{d}. But this intersection is trivial as the above map is injective. Thus, x_{1} ⋯ x_{d} = 0 which contradicts x_{1}, …, x_{d} ∉ M.

Take F̃_{λ′} ∈ S_{ℓ}(D, ε)[I] normalized. Then T_{n} − a_{n}(F̃_{λ′}) ∈ I, hence F̃_{λ′} is an eigenform and

a_{n}(F̃_{λ′}) ≡ a_{n}(F_{λ}) (mod λ′).

Theorem 2. If f is a normalized eigenform in S_{1}(Γ_{0}(D), ε), then for each λ ◁ O_{K_{f}} there exists

\overline{ρ_{f,λ}} : G_{ℚ} → GL_{2} (O_{K_{f}} /λ)

satisfying

char(\overline{ρ_{f,λ}}(Frob_{p})) = x^{2} − a_{p}x + ε(p).

Proof. Let F_{λ} = E_{ℓ−1}f and let F̃_{λ′} be the eigenform obtained from F_{λ} be the Deligne-Serre lifting lemma. Recall that F̃_{λ′} ∈ S_{ℓ}(Γ_{0}(D), ε) has Fourier coefficients in O_{K′} where K′ ⊇ K.

Deligne’s construction of λ′-adic representations:

There exists a ρ_{F̃_{λ′},λ′} : G_{ℚ} → GL_{2}(OF̃_{λ′}) satisfying

char(ρ_{F̃_{λ′},λ′}(Frob_{p})) = x^{2} − a_{p}(F̃_{λ′})x + p^{ℓ−1}ε(p)
≡ x^{2} − a_{p}(F_{λ})x + ε(p) (mod λ′)
≡ x^{2} − a_{p}(f)x + ε(p).

Hence ρ_{F̃_{λ′},λ′} reduced mod λ′ has the desired properties; that is,

ρ̄_{F̃_{λ′},λ′} : G_{ℚ} → GL_{2} (O_{K′,λ′}/λ′) .

This reduced representation can be conjugated to take values in GL_{2}(O_{K,λ}/λ) following an argument similar to the one of this morning.

f ⇝ ρ_{f,λ} : G_{ℚ} → GL_{2}(O_{K_{f}} /λ)

We have now constructed a mod λ representation that has all the properties we want, thus the only remaining problem is:

Problem 1. Lift these mod λ representations to a common representation in characteristic 0.