Math 726: L-functions and modular forms Fall 2011

Lecture 28 : Artin Representations Attached to Forms of Weight One

Instructor: Henri Darmon Notes written by: Victoria de Quehen

We are trying to associate to f a homomorphism ρ_{f} : G_{ℚ} → GL_{2}(ℂ). The steps we have taken are:

1. Given λ ⊲ O_{K_{p}} we get F_{λ} = fE_{ℓ−1} ∈ S_{ℓ}(D, ϵ), an eigenform mod λ.

2. Prove the Delign-Serre lifting lemma: that there exists F̃_{λ′} ∈ S_{ℓ}(D, ϵ) which is an eigenform with coefficients in K′_{λ}/K_{f} with a_{p}(F̃_{λ′}) ≅ a_{p}(f) (mod λ′) for all p ∤ D.

3. Now consider the mod λ′ reduction of the λ′-adic representation attached to F̃_{λ′}:

ρ_{f,λ′} : G_{ℚ} → GL_{2}(O_{K′}/λ′).

The characteristic polynomial of Frobenius mod λ′ is:

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

Let ρ̄_{f,λ} denote the semisimplification of ρ_{λ}. We will apply the following lemma to ρ̄_{f,λ}.

Lemma 1. If ρ : G_{ℚ} → GL_{2}(L) is a semisimple odd representation and the characteristic polynomial of every element is in K[x], then ρ has a conjugate whose image is in GL_{2}(K).

At this stage we have

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

with the desired property.

Let Σ = {λ ⊲ O_{K_{f}} : λ|ℓ and ℓ splits completely in K_{f}/ℚ}. By Chebotarev’s density theorem Σ is infinite. For each λ ∈ Σ let ℓ be the rational prime lying below it. Furthermore let

G_{λ} := \overline{ρ_{f,λ}}(G_{ℚ}) ⊆ GL_{2}(𝔽_{ℓ}).

We wish to bound |G_{λ}| independently of ℓ.

Definition 1. Fix an integer X > 0. A subgroup G ⊂ GL_{2}(𝔽_{ℓ}) is \underline{X-sparse} if there is a subset H ⊂ G such that

1. |H| ≥ \frac{3}{4} |G| , and

2. the elements of H have at most X distinct characteristic polynomials.

Lemma 2. There exists an X > 0 such that all the groups G_{λ} (λ ∈ Σ) are X-sparse.

Proof. Recall from last week that for all η > 0 there exists finite set X_{η} ⊂ ℂ such that a_{p} ∈ X_{η} for all p outside a set Y_{η} of density η. Take η < \frac{1}{4} and set X = |X_{η}| ord(ε). We claim that G_{λ} is X-sparse. Let

H = ⋃_{p∈Y_{η}} ρ_{f}(Frob_{p}).

1. The density of Y_{η} > \frac{3}{4} implies |H| ≥ \frac{3}{4} |G| (by Chebotarev density theorem).

2. The number of distinct characteristic polynomials of elements of H is less that |X_{η}| ord(ε).

Definition 2. A subgroup G of GL_{2}(𝔽_{ℓ}) is semisimple if the underlying 2 dimensional representation of G is semisimple, (that is, either irreducible or a direct sum of 1 dimensional representations).

Example 3. The groups GL_{2}(𝔽_{ℓ}) and SL_{2}(𝔽_{ℓ}) are irreducible, hence semisimple.

Example 4. If G = \{(\begin{matrix}a & 0 \\ 0 & b \end{matrix}) |a, b ∈ 𝔽_{ℓ}^{×}\} (the “split Cartan subgroup”), then G is semisim-

ple and reducible.

Example 5. Let 𝔽_{ℓ^{2}}^{×} act by left multiplication on 𝔽_{ℓ^{2}} viewed as a 𝔽_{l} vector space with any choice of basis. Let G be the image of 𝔽_{ℓ^{2}}^{×} in Aut_{𝔽_{ℓ}}(𝔽_{ℓ^{2}}) ≅ GL_{2}(𝔽_{ℓ}) (a “non-split Cartan subgroup”). Then G is semisimple.

Example 6. If G is the normalizer of a Cartan subgroup, then it is semisimple. (Note that the Cartan subgroup then has index 2 in G).

Example 7. The group G = \{(\begin{matrix}a & b \\ 0 & d \end{matrix})\} is reducible but not totally reducible, hence G is

not semisimple.

Remark 1. Note that all the G_{λ} are semisimple by construction.

Theorem 8. Fix X, there exists a constant A (depending on X but not ℓ) such that |G| < A for all semisimple X-sparse subgroups of GL_{2}(𝔽_{ℓ}).

The proof relies on the following group theory fact. Group Theory Fact: If G ⊂ GL_{2}(𝔽_{ℓ}) is semisimple, then one of the following cases is true:

1. G ⊇ SL_{2}(𝔽_{ℓ}),

2. G ⊆ T for some Cartan subgroup T,

3. G ⊆ N(T) the normalizer of a Cartan subgroup but G ⊈ T, or

4. the image of G in P GL_{2}(𝔽_{ℓ}) is isomorphic to one of A_{4}, S_{4}, A_{5}.