Math 726: L-functions and modular forms Fall 2011

Lecture 29 : End of proof of the Serre-Deligne theorem

Instructor: Henri Darmon Notes written by: Francesca Gala

The goal of this lecture is to conclude the proof of the Serre-Deligne theorem, which accompanied us along the last few lectures. Recall that last week we associated to our eigenform f ∈ S_{1}(D, ϵ) a family of representations:

\overline{ρ_{f,λ}} : G_{ℚ} ⟶ GL_{2}(𝔽_{λ}),

where λ ∈ Σ = {λ ◁ O_{K_{f}} |O_{K_{f}} /λ ≅ 𝔽_{ℓ}}. We let

G := G_{ℓ} = \overline{ρ_{f,λ}}(G_{ℚ}).

We would like to bound the cardinality of G_{ℓ} independently of ℓ. Recall that G is a subgroup of GL_{2}(𝔽_{ℓ}), it is semisimple by construction and it is X-sparse for a suitable integer X > 0, i.e. there exists a subgroup H ≤ G such that |H| ≥ \frac{3}{4} |G| and the elements in H have at most X distinct characteristic polynomials.

Theorem 1. If G is a semisimple, X-sparse subgroup of GL_{2}(𝔽_{l}), then ∃A independent of ℓ such that |G| ≤ A.

Proof. To prove this theorem we will use the following proposition:

Proposition 1. If G is a semisimple subgroup of GL_{2}(𝔽_{ℓ}) then only the following four cases can arise:

1. G ⊃ SL_{2}(𝔽_{ℓ})

2. G is contained in a Cartan subgroup T, either split or non-split, which means that T ≃ 𝔽_{ℓ}^{×} × 𝔽_{ℓ}^{×} or T ≃ 𝔽_{ℓ^{2}}^{×}.

3. G ⊂ N_{GL_{2}(𝔽_{ℓ})}(T), where N_{GL_{2}(𝔽_{ℓ})}(T) is the normaliser of a Cartan subgroup T Note that [N(T) : T] = 2 and there exists a split exact sequence: 1 → T → N(T) → ±1 → 1.

4. G is an ’exceptional subgroup’, namely its image in PGL_{2}(𝔽_{ℓ}) is A_{4}, S_{4} or S_{5}

Proof. Reference: J. P. Serre, Proprietes galoisiennes des points d’ordre fini des courbes elliptiques, chapter 2.

Remark 1. The semisimplicity assumption is crucial, for example if we consider

G = \{( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) |a ∈ 𝔽_{ℓ}\} ,

then G is 1-sparse but |G| = ℓ, so clearly we are not able to bound the cardinality of G independently of ℓ.

Now we can prove Theorem 1 by analysing separately the four cases of the proposition. Our strategy will be to bound the order of H by bounding the number of elements in GL_{2}(𝔽_{ℓ}) which have the same characteristic polynomial, i.e. by bounding the number of elements in a given conjugacy class.

1. We know that |GL_{2}(𝔽_{ℓ})| = (ℓ^{2} − 1)(ℓ^{2} − l) = ℓ(ℓ + 1)(ℓ − 1)^{2}. Let σ ∈ GL_{2}(𝔽_{ℓ}), then the cardinality of the set C(σ) := {τστ^{−1}, τ ∈ GL_{2}(𝔽_{ℓ})} is given by

|C(σ)| = \frac{|GL_{2}(𝔽_{ℓ})|}{|Z(σ)|},

where Z(σ) = {τ|τσ = στ} is the centraliser of σ in GL_{2}(𝔽_{ℓ}).

Let us suppose that char(σ) = (x − a)^{2}. This means that σ ∈ C(( \begin{matrix}a & 1 \\ 0 & a \end{matrix} )) ∪ C(( \begin{matrix}a & 0 \\ 0 & a \end{matrix} )).

Since

|Z(( \begin{matrix}a & 1 \\ 0 & a \end{matrix} )) = \{( \begin{matrix}u & v \\ 0 & u \end{matrix} ) |u ∈ 𝔽_{ℓ}^{×}, v ∈ 𝔽_{ℓ}\}| = (ℓ − 1)ℓ

we have that |C(( \begin{matrix}a & 1 \\ 0 & a \end{matrix} ))| = ℓ^{2} − 1 ,while clearly |C(( \begin{matrix}a & 0 \\ 0 & a \end{matrix} ))| = 1. So we have that:

|{σ| char(σ) = (x − a)^{2}}| = ℓ^{2}. Now clearly:

|\{σ| char(σ) = (x − a)(x − b), a, b ∈ 𝔽_{ℓ}^{×}a ≠ b\}| = |C(( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ))| = ℓ^{2} + ℓ

since |Z(( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ))| = (ℓ − 1)^{2}.

The last case to consider is the one of σ ∈ GL_{2}(𝔽_{ℓ}) with char(σ) equal to a polynomial p(x) = x^{2} + ax + b, which is irredubile over 𝔽_{ℓ}. In this case |Z(σ)| = ℓ^{2} − 1, so that:

|C(σ)| = (ℓ − 1)ℓ.

Therefore we can deduce the following bound:

\frac{3}{4} |SL_{2}(𝔽_{ℓ})| = \frac{3}{4}ℓ(ℓ + 1)(ℓ − 1) ≤ |H| ≤ X(ℓ^{2} + ℓ),

which is a bound on H, independent of ℓ since for the inequalities to hold we must have ℓ − 1 ≤ X \frac{4}{3}.

2. In T there are at most two elements with a given characteristic polynomial, in fact since char(σ) = x^{2} − tr(σ) + det(σ), then we have \{σ ∈ T | char(σ) = x^{2} − ax + b\} = \{σ, \overline{σ}\}. Hence in this case we have |H| ≤ 2X which implies:

|G| ≤ \frac{8}{3}X.

3. Let G_{0} = G ∩ T so that, since [N(T) : T] = 2, |G_{0}| = \frac{1}{2} |G| and let H_{0} = H ∩ T so that |H_{0}| ≥ \frac{1}{2} |G_{0}|. Now from case 2. we can deduce that |H_{0}| ≤ 2X, so that |G_{0}| ≤ 4X which implies |G| ≤ 8X.

4. Consider the map:

\begin{aligned}η : & G & → & PGL_{2}(𝔽_{ℓ}) × 𝔽_{ℓ}^{×} \\ & σ & ↦ & (\overline{σ}, det(σ)) \end{aligned}

Since we know that the image of G in PGL_{2}(𝔽_{ℓ}) is A_{4}, S_{4} or A_{5} and X is the number of different characteristic polynomials in H we have that: |η(H)| ≤ |A_{5}| X = 60X and also

ker(η) = \{( \begin{matrix}±1 & 0 \\ 0 & ±1 \end{matrix} )\} ≃ ℤ/2ℤ.

This implies that |H| ≤ 120X and |G| ≤ 160X. This concludes case 4. and the proof of the theorem.

We are now able to deduce a significant bound on the size of the Fourier coefficients of the eigenform f ∈ S_{1}(Γ_{0}(D), ϵ).

Theorem 2. For all primes p, the coefficient a_{p}(f) is a sum of roots of unity. In particular |a_{p}(f)| ≤ 2.

Proof. Let

ℙ\overline{K_{f}} = \{g(x) = (x − α)(x − β) ∈ O\overline{K_{f}} [x]|α and β are roots of unity of order ≤ A\}

and, for every λ ∈ Σ,

ℙ_{λ} = \{g(x) = (x − α)(x − β) ∈ O_{K_{f}} /λ[x]|α, β ∈ \overline{O_{K_{f} /λ}} and ord(α), ord(β) ≤ A\}.

We have that ℙ\overline{K_{f}} and ℙ_{λ} are finite and there exists a reduction map mod λ:

red_{λ} : ℙ\overline{K_{f}} → ℙ_{λ},

which is bijective if ℓ > A. If we let σ = Frob_{p}, then the characteristic polynomial char(\overline{ρ_{f,λ}}) = x^{2} − a_{p}x + ϵ(p) ∈ O_{K_{f}} /λ[x] belongs the set ℙ_{λ}. Since ℙ\overline{K_{f}} is finite, there exists a polynomial g ∈ ℙ\overline{K_{f}} such that red_{λ}(g) ≅ x^{2}−a_{p}x+ϵ(p) mod λ, for infinitely many λ, which implies that g = x^{2}−a_{p}x+ϵ(p). So we can conclude that

x^{2} − a_{p}x + ϵ(p) ∈ ℙ\overline{K_{f}} ,

and the roots of x^{2} − a_{p}x + ϵ(p) are roots of unity of order ≤ A.

End of the proof of Serre-Deligne’s theorem. The embedding of G_{ℓ} in GL_{2}(𝔽_{ℓ}) gives a two-dimensional representation ρ_{ℓ} of G_{ℓ} over the field 𝔽_{ℓ}. Because G_{ℓ} is of cardinality prime to ℓ, there is a complex two-dimensional representation ρ of G_{ℓ} satisfying

tr(ρ(σ)) = tr(ρ_{ℓ}(σ)) mod λ

for a suitable prime λ above ℓ in the field generated by the traces of ρ. This representation is the desired lift.