Some local (at p) properties of residual Galois representations

Johnson Jia, Krzysztof Klosin

March 5, 2006

1 Preliminary results

In this talk we are going to discuss some local properties of (mod p) Galois representations at the prime p.

Let p be a prime number, N a positive integer prime to p, k an integer and ϵ : (Z/NZ)^{×} → \overline{F}_{p}^{×} a character. Let f be a cusp form of type (N, k, ϵ). We also assume that f is an eigenform for all the Hecke operators T_{l} (any prime l) with eigenvalue a_{l} ∈ \overline{F}_{p}. For the definition of the Hecke algebra we refer to Kirsten’s talk. We will denote by G_{Q} the absolute Galois group of Q, and by χ the mod p cyclotomic character. We work with arithmetic Frobenius elements.

Theorem 1.1 (Deligne). There exists a unique (up to isomorphism) continuous semi-simple representation ρ_{f} : G_{Q} → GL(V), where V is a two-dimensional \overline{F}_{p}-vector space, such that for all primes l ∤ Np,

• ρ_{f} is unramified at l,

• tr ρ_{f}(Frob_{l}) = a_{l}

• det ρ_{f}(Frob_{l}) = ϵ(l)l^{k−1}.

Remark 1.2. By choosing a basis of V we can treat ρ_{f} as having image in GL_{2}(\overline{F}_{p}). It then follows from continuity of ρ_{f} that there exists a finite extension κ of F_{p} such that ρ_{f} factors through some map G_{Q} → GL_{2}(κ) ↪ GL_{2}(\overline{F}_{p}) taht we may also denote ρ_{f}.

We fix an embedding \overline{Q} ↪ \overline{Q}_{p}. This determines a prime 𝔭 of the integral closure \overline{Z} of Z in \overline{Q} over p. Let G_{p} ⊂ G_{Q} denote the decomposition group of 𝔭. Our goal is to study ρ_{f,p} := ρ_{f}|_{G_{p}}. We will denote by I_{p} the inertia group inside G_{p}, by I_{p}^{w} the wild inertia subgroup, and set I_{p}^{t} = I_{p}/I_{p}^{w} to be the tame inertia quotient. We will identify G_{p} with Gal(\overline{Q}_{p}/Q_{p}) and G_{p}/I_{p} with Gal(\overline{F}_{p}/F_{p}). Moreover, for any a ∈ \overline{F}_{p}^{×} we denote by λ_{a} the unramified character ϕ : G_{p} → \overline{F}_{p}^{×} such that ϕ(Frob_{p}) = a.

Theorem 1.3 (Deligne). Let f be a cusp form of type (N, k, ϵ) with 2 ≤ k ≤ p + 1. Assume f is a Hecke eigenform with eigenvalues a_{l}, and that a_{p} ≠ 0. Then ρ_{f,p} is reducible and

ρ_{f,p} ≅ \left[{χ^{k−1}λ_{ϵ(p)/a_{p}} ∗ \atop λ_{a_{p}}}\right] .

Remark 1.4. Theorem 1.3 was proved by Deligne in a letter to Serre using p-adic etale cohomology and mod p “vanishing cycles” calculations on modular curves in characteristic p; in particular his proof uses his construction of p-adic Galois representations associated to eigenforms.

Proposition 1.5. Let f be a cusp form of type (N, 1, ϵ). Assume that f is a Hecke eigenform and T_{p}f = a_{p}f. Suppose that the polynomial

P (X) := X^{2} − a_{p}X + ϵ(p)  (1)

has two distinct roots in \overline{F}_{p}, or that f is the reduction of a characteristic zero eigenform of weight 1 and level N. Then ρ_{f,p} is unramified.

Proof. We omit the proof of the case when f is the reduction of a characteristic zero eigenform (cf. Theorem 4.1 in [DS74]), so assume P (X) has distinct roots in \overline{F}_{p}. Let f(q) = ∑ b_{n}q^{n} be the q-expansion of f at some cusp. In his talk Hui discussed the action of the operator V_{p} on q-expansions of cusp forms. In particular V_{p}f is a cusp form of type (N, p, ϵ) and V_{p}f(q) = ∑ b_{n}q^{np}. On the other hand, Bryden defined the Hasse invariant A, which is a modular form of type (1, p − 1, 1) and has q-expansion 1 at all cusps. Hence Af is a modular form of type (N, p, ϵ) and Af has q-expansion ∑ b_{n}q^{n}. As discussed in Hui’s talk, both Af and V_{p}f are eigenforms for all T_{l} with l ≠ p and their T_{l}-eigenvalues are the same as those of f. On the other hand, since T_{p}f = a_{p}f and ⟨p⟩ f = ϵ(p)f, it follows from general identities that

T_{p}(Af) = a_{p}Af − ϵ(p)V_{p}f  (2)

and

T_{p}(V_{p}f) = Af.  (3)

Let W be the two-dimensional space spanned by Af and V_{p}f. It follows from formulas (2) and (3) that P (X) is the characteristic polynomial for the action of T_{p} on W. Let α ≠ α′ be the roots of P (X). Let g, g′ be the corresponding normalized eigenforms of type (N, p, ϵ). Since f, g and g′ have the same eigenvalues for T_{l}, l ≠ p, we have tr ρ_{f} = tr ρ_{g} = tr ρ_{g′} by the Tchebotarev density theorem. We also have det ρ_{f} = det ρ_{g} = det ρ_{g′} = ϵ. Hence the semi-simple representations ρ_{f}, ρ_{g} and ρ_{g′} are isomorphic by the Brauer-Nesbitt theorem. Since αα′ = ϵ(p) ≠ 0, both α and α′ are non-zero. Hence we can apply Theorem 1.3 and conclude that we have

ρ_{g,p} ≅ \left[{λ_{ϵ(p)/α} ∗ \atop λ_{α}}\right] = \left[{λ_{α′} ∗ \atop λ_{α}}\right]  (4)

and

ρ_{g′,p} ≅ \left[{λ_{ϵ(p)/α′} ∗ \atop λ_{α′}}\right] = \left[{λ_{α} ∗ \atop λ_{α′}}\right] .  (5)

Since α ≠ α′, the representation space ρ_{f,p} contains lines with distinct characters λ_{α} and λ_{α′}, so ρ_{f,p} ≅ λ_{α} ⊕ λ_{α′} is unramified.

Theorem 1.6 (Gross). Let f = ∑ a_{n}q^{n} be as in Theorem 1.3, and suppose that a_{p}^{2} ≠ ϵ(p) if k = p. Then ρ_{f,p}|_{I_{p}^{w}} is trivial if and only if there exists a cuspidal eigenform f′ = ∑ a′_{n}q^{n} of type (N, k′, ϵ), where k′ = p + 1 − k, such that

la′_{l} = l^{k′}a_{l}  (6)

for all l.

Proof. We are only going to prove that the existence of f′ implies that ρ_{f,p} is tamely ramified. For the converse (which is the hard part of this theorem), see Theorem 13.10 in [Gro90]. Let f′ be as in the statement of the theorem and let a′_{l} denote the eigenvalue of T_{l} corresponding to f′. Formula (6) implies that tr ρ_{f′}(Frob_{l}) = tr (ρ_{f} ⊗ χ^{k′−1})(Frob_{l}) for all l ∤ Np, where χ denotes the mod p cyclotomic character. Hence by the Tchebotarev density theorem together with the Brauer-Nesbitt theorem as before, we get

ρ_{f′} ≅ ρ_{f} ⊗ χ^{k′−1}.  (7)

First assume k ≠ p. We begin by show that a′_{p} ≠ 0. Note that for all primes l ≠ p, we have a′_{l} = l^{k′−1}a_{l} by assumption. If a′_{p} = 0, we can also write a′_{p} = p^{k′−1}a_{p} as k′ ≠ 1. Hence f′ of weight k′ and θ^{k′−1}f of weight pk′ have the same q-expansions. Here θ is the operator whose properties were discussed in Kirsten’s talk. Hence θ^{k′−1}f = A^{k′}f′, which means that θ^{k′−1}f has filtration k′. However, it was proved in Kirsten’s talk that θ^{k′−1}f has filtration pk′, since f has weight at most p − 1, which yields a contradiction. Hence a′_{p} ≠ 0 if k ≠ p.

Applying Theorem 1.3 to the representation ρ_{f′} (still assuming k ≠ p), we see that there is a line L_{f′} in the space V of ρ_{f′} such that G_{p} acts on L_{f′} via the character χ^{k′−1}λ_{ϵ(p)/a′_{p}}. On the other hand applying Theorem 1.3 to the representation ρ_{f}, we get a line L̃_{f} in its representation space on which G_{p} acts via the character χ^{k−1}λ_{ϵ(p)/a_{p}} and hence a line L_{f} ⊂ V on which G_{p} acts via the character χ^{k′−1}(χ^{k−1}λ_{ϵ(p)/a_{p}}).

In particular, the action of I_{p} on L_{f} is via χ^{k′−1+k−1} = χ^{p−1} = 1 and on L_{f′} via χ^{k′−1}. Since k ≠ p by assumption, so χ^{k′} ≠ 1, L_{f} ≠ L_{f′} and so we conclude that V = L_{f} ⊕ L_{f′}. Thus ρ_{f′,p} and hence also ρ_{f,p} is diagonalizable; i.e., ρ_{f}(σ) has order prime to p for every σ ∈ G_{p}. Thus ρ_{f}|_{I_{p}^{w}} is trivial.

Now assume k = p. Then f′ = ∑ a′_{n}q^{n} is of weight 1. Note that f is in the span W of V_{p}f′ and Af′. Indeed, by looking at q-expansions, we see that θ((a_{p} − a′_{p})V_{p}f′ + Af′ − f) = 0, and so there exists a form h = ∑ c_{n}q^{n} of weight 1 such that

(a_{p} − a′_{p})V_{p}f′ + Af′ − f = V_{p}h,

and h is an eigenform for T_{l} with eigenvalue a_{l} if l ≠ p since V_{p} commutes with such T_{l} in weights 1 and p (and V_{p} is injective). Moreover, the p-th coefficient of the q-expansion of V_{p}h is zero, so c_{1} = 0. Since c_{l} = a_{l}c_{1} = 0 for all l ≠ p, we have θ(h) = 0. As θ is injective on forms of weight prime to p, we get h = 0, and so we conclude that f ∈ W, as claimed. As a_{p} ≠ ϵ(p)/a_{p} by assumption, and the characteristic polynomial for T_{p} on W has constant term ϵ(p) (by matrix calculation) with a_{p} as a root (since f ∈ W), there is a unique Hecke eigenform g ∈ W whose eigenvalue for T_{p} is b_{p} := ϵ(p)/a_{p}. Then by applying Theorem 1.3 to ρ_{f} and ρ_{g} ≅ ρ_{f}, we get a line L_{f} in the space W of ρ_{f} on which G_{p} acts via the unramified character λ_{ϵ(p)/a_{p}} and a line L_{g} on which G_{p} acts via λ_{ϵ(p)/b_{p}}. Since a_{p}^{2} ≠ ϵ(p), we again get V = L_{f} ⊕ L_{g} and conclude that ρ_{f}|_{I_{p}^{w}} is trivial. Note that in this case ρ_{f,p} is in fact unramified.

2 A theorem of Mazur

Let p be an odd prime and ℓ a prime with ℓ ≢ 1 mod p. (We allow ℓ = p.) Let f be a normalized cuspidal eigenform of weight 2, level N, and trivial charater. We drop the assumption that p ∤ N. Suppose N = Mℓ with ℓ ∤ M. Here is a basic “level lowering” theorem.

Theorem 2.1 (Mazur). Suppose the Galois representation ρ : G_{Q} → GL_{2}(\overline{F}_{p}) attached to f is irreducible. If ℓ ≠ p, then assume that ρ|_{G_{ℓ}} is unramified, and if ℓ = p, then assume that ρ is finite at p. Then ρ is modular of type (M, 2, 1).

Proof. We will suppress some of the technical details. Let 𝔛_{0}(N) over Z_{ℓ} be the coarse moduli space for the Γ_{0}(N)-moduli problem on generalized elliptic curves over Z_{(ℓ)}-schemes. Let J_{0}(N) be the jacobian of X_{0}(N) over Q. Let 𝔍_{0}(N) be the Neron model of J_{0}(N) over Z_{(ℓ)}. It follows from the work of Deligne-Rapoport that 𝔛_{0}(N)_{F_{ℓ}} is the union of two copies of 𝔛_{0}(M)_{F_{ℓ}}, which intersect transversally at geometric points corresponding to precisely supersingular elliptic curves. Their theory shows that 𝔛_{0}(M)_{F_{ℓ}} is smooth, as ℓ ∤ M.

Lemma 2.2. The normalization map 𝔛_{0}(M)_{F_{ℓ}} ⊔ 𝔛_{0}(M)_{F_{ℓ}} → 𝔛_{0}(N)_{F_{ℓ}} induces a short exact sequence

0 → T_{0}(N)_{F_{ℓ}} → 𝔍_{0}(N)_{F_{ℓ}}^{0} → 𝔍_{0}(M)_{F_{ℓ}} × 𝔍_{0}(M)_{F_{ℓ}} → 0,  (8)

where T_{0}(N)_{F_{ℓ}} is the torus whose character group X := Hom_{\overline{F}_{ℓ}}(T_{0}(N)_{\overline{F}_{ℓ}}, (G_{m})_{\overline{F}_{ℓ}}) as a Z[Gal(\overline{F}_{ℓ}/F_{ℓ})]- module is the group of degree zero divisors on 𝔛_{0}(M)_{\overline{F}_{ℓ}} with support in the supersingular points.

Proof. The existence of exact sequence (8) is a general fact about semi-stable curves, cf. Example 8 in section 9.2 of [BLR90]. To any semi-stable curve Y over an algebraically closed field one associates the graph Γ(Y) whose vertices are the irreducible components of Y and whose edges are the singular points of Y: an edge corresponding to a point P joins vertices V_{1} and V_{2} if and only if the irreducible components corresponding to V_{1} and V_{2} intersect at P. (We get a “loop” at a vertex if it lies on only one irreducible component, corresponding to “formal self-crossing”.) The character group X can be canonically identified with H_{1}(Γ(𝔛_{0}(N)_{\overline{F}_{ℓ}}), Z) as a Z[Gal(\overline{F}_{ℓ}/F_{ℓ})]-module. Using this identification, one proves that X has the alternative description in terms of the supersingular geometric points.

There is yet another tautological short exact sequence

0 → 𝔍_{0}(N)_{F_{ℓ}}^{0} → 𝔍_{0}(N)_{F_{ℓ}} → Φ_{0}(N)_{F_{ℓ}} → 0,  (9)

where Φ_{0}(N)_{F_{ℓ}} is the finite etale group scheme of components of 𝔍_{0}(N)_{F_{ℓ}}.

The Hecke algebra T of level N acts on J_{0}(N) and by the Neron mapping property also on 𝔍_{0}(N). This action respects the exact sequences (8) and (9) due to functoriality considerations with identity components and maximal toric parts.

Since ρ is absolutely irreducible and modular of type (N, 2, 1), by the Brauer-Nesbitt theorem and the triviality of Brauer groups of finite fields there exists a maximal ideal 𝔪 of T with an embedding κ := T/𝔪 ↪ \overline{F}_{p} and a two-dimensional κ-vector space V with G_{Q}-action such that the induced action on \overline{F}_{p} ⊗_{κ} V is via ρ and such that J_{0}(N)(\overline{Q})[𝔪] is isomorphic to the direct sum of a finite number of copies of V as κ[G_{Q}]-modules.

First assume that ℓ = p, as this is the case that will be of interest to us. Let W be the finite etale κ-vector space scheme over Q such that W (\overline{Q}) = V as κ[G_{Q}]-modules. By assumption ρ is finite at p, so (by oddness of p and Raynaud’s theorems when “e < p − 1”) there exists a finite flat κ-vector space scheme 𝔚 over Z_{(ℓ)} extending W. By the remarks above we can choose an injection of V into J_{0}(N)(\overline{Q}) which gives rise to a closed immersion of W into J_{0}(N).

Lemma 2.3. There exists a closed immersion 𝔚 ↪ 𝔍_{0}(N) prolonging the closed immersion W ↪ J_{0}(N).

Proof. We are going to prove the following more general statement:

Claim 2.4. Let R denote a complete dvr of mixed characteristic (0, p) with absolute ramification index e < p − 1. Denote by K the fraction field of R. Let 𝔊 be a finite flat commutative p-group over R, and A an abelian variety over K with semi-stable reduction. Let 𝔄 be the Neron model (over R) of A (so 𝔄 has closed fiber with semi-abelian identity component, whence 𝔄[n] is quasi-finite, flat, and separated over R for every nonzero n in Z). Suppose there exists a closed immersion ι_{K} : G ↪ A, where G = 𝔊_{K}. Furthermore, assume that the Galois module (𝔄[p]/𝔄[p]_{f})(\overline{K}) is unramified, where 𝔄[p]_{f} denotes the “finite” part of the quasi-finite flat separated R-group 𝔄[p]. Then ι_{K} extends to a closed immersion ι : 𝔊 ↪ 𝔄.

We remark here that the hypotheses of semi-stable reduction and unramifiedness are satisfied by 𝔄 = 𝔍_{0}(N) over Z_{(ℓ)}. This is a consequence of [Gro72] and uses the fact that 𝔛_{0}(N) has semi-stable reduction over Z_{(ℓ)} (as ℓ ∤ M).

Proof of the Claim. We first show that ι_{K}|_{𝔊_{K}^{0}} extends. The composite

𝔊_{K}^{0} ↪ G ↪ 𝔄[p^{n}]_{K} → (𝔄[p^{n}]/𝔄[p^{n}]_{f})_{K}

is the zero map since, as discussed by Nick in his talk, Raynaud’s results on finite flat group schemes imply that a finite flat connected R-group has no nonzero unramified quotients of its K-fiber. Hence we have a commutative diagram

\xymatrix{G \ar[r] & 𝔄[p]_{K} \\ 𝔊_{K}^{0} \ar[u] \ar[r] & (𝔄[p]_{f})_{K} \ar[u]}

where all the maps are closed immersions. Now, using the fact that 𝔄[p]_{f} is finite flat (as opposed to 𝔄[p], which is generally only quasi-finite and flat), the lower horizontal arrow extends to a closed immersion 𝔊^{0} ↪ 𝔄[p]_{f} by Raynaud’s Theorem ([Ray74], Corollary 3.3.6). As 𝔄[p]_{f}^{0} ⊂ 𝔄^{0}, we have proved that ι_{K}|_{𝔊_{K}^{0}} extends.

We will now show that ι extends. Since 𝔊^{0} is a finite flat commutative group, it makes sense to talk about the quotient (𝔊 × 𝔄^{0})/𝔊^{0} with 𝔊^{0} embedded by the “twisted diagonal” x ↦ (x, −x). We have a commutative diagram

\xymatrix{𝔊 \ar[r] & (𝔊 × 𝔄^{0})/𝔊^{0} \\ 𝔊^{0} \ar[u] \ar[r] & 𝔄^{0} \ar[u]} .  (10)

Our goal is to show that the closed embedding 𝔊^{0} ↪ 𝔄 factors through the left vertical arrow in diagram (10). First note that it suffices to show that (𝔊 × 𝔄^{0})/𝔊^{0} is smooth. Indeed, the natural addition map G × A → A factors through (G × A)/(𝔊^{0})_{K} → A, so if (𝔊 × 𝔄^{0})/𝔊^{0} is smooth, then (G × A)/(𝔊^{0})_{K} → A extends to ϕ : (𝔊 × 𝔄^{0})/𝔊^{0} → 𝔄 by the Neron mapping property. Diagram (10) gets amalgamated to a diagram

\xymatrix{𝔊 \ar[r] & (𝔊 × 𝔄^{0})/𝔊^{0} \ar[dr]^{ϕ} \\ 𝔊^{0} \ar[u] \ar[r] & 𝔄^{0} \ar[u] \ar[r] & 𝔄} .  (11)

To establish commutativity of diagram (11) it is enough (by separatedness and flatness over R) to check commutativity of the right triangle on generic fibers. Since (𝔄^{0})_{K} = A, the corresponding triangle on generic fibers clearly commutes, so we get the desired result concerning (10).

It remains to prove that (𝔊 × 𝔄^{0})/𝔊^{0} is smooth over R. Note that 𝔊 × 𝔄^{0} is of finite type and flat as are 𝔊 and 𝔄^{0}. As 𝔊^{0} is finite and flat over R, the map 𝔊 × 𝔄^{0} → (𝔊 × 𝔄^{0})/𝔊^{0} is finite, flat and surjective. Hence (𝔊 × 𝔄^{0})/𝔊^{0} is of finite type and flat over R. Thus to verify smoothness of (𝔊 × 𝔄^{0})/𝔊^{0}, it is enough to check smoothness of fibers. As the geometric generic fiber is clearly smooth (it is just a product of copies of A), it remains to check smoothness of the closed fiber. This follows from the following Fact.

Fact 2.5. Let k be a field, G a finite commutative k-group and H a smooth commutative k-group. Suppose we are given a closed immersion of k groups G^{0} ↪ H. Then (G × H)/G^{0} is k-smooth, where G^{0} ↪ G × H is g^{0} ↦ (g^{0}, (g^{0})^{−1}).

To prove Fact 2.5, we may assume that k is algebraically closed. Then G = G^{0}×G^{et}, hence (G×H)/G^{0} = G^{et} × H, which is smooth.

Lemma 2.6. The image of 𝔚_{F_{p}} in Φ_{0}(N)_{F_{p}} is zero.

Proof. It is a theorem of Ribet (cf. Theorem 3.12 in [Rib90]) that Φ_{0}(N)_{F_{p}} is Eisenstein in the sense that for all q prime to p, the operator T_{q} acts on it via multiplication by q + 1. Thus the action of T on Φ_{0}(N)_{F_{p}} factors through T/I, where I is the ideal of T generated by the elements T_{q} − q − 1 for q prime to p. Note that since ρ is irreducible, I is relatively prime to 𝔪. Indeed, if 𝔪 and I were not relatively prime, then T_{q} ≡ q + 1 mod 𝔪 for almost all q. By Tchebotarev Denisty Theorem we have then tr ρ = 1 + χ and det ρ = χ. Brauer-Nesbitt Theorem implies then that ρ^{ss} ≅ [ \begin{matrix}1 \\ & χ \end{matrix} ], which contradicts the irreducibility of ρ. The Lemma follows.

Lemma 2.6 implies that 𝔚_{F_{p}} lands inside 𝔍_{0}(N)_{F_{p}}^{0}. From now on assume that ρ is not modular of type (M, 2, 1). Then it follows (since ρ is irreducible) that the image of 𝔚_{F_{p}} inside 𝔍_{0}(M)_{F_{p}} × 𝔍_{0}(M)_{F_{p}} must be trivial ([Rib90], Thoerem 3.11). Hence, 𝔚_{F_{p}} lands inside the torus T_{0}(N)_{F_{p}}.

The Hecke algebra acts on the character group X, and Ribet showed in [Rib90] (Lemma 6.3) that V ⊂ Hom(X/𝔪X, μ_{p}), where μ_{p} denotes the Galois modules of p-th roots of 1 over Q.

Lemma 2.7. The action of Frob_{p} on X/𝔪X coincides with the action of T_{p} and of −w_{p}, where w_{p} denotes the level p Atkin-Lehner involution.

Proof. For the proof that the actions of T_{p} and of −w_{p} on T_{0}(N)_{\overline{F}_{p}} (and hence on X/𝔪X) coincide, see [Rib90], Proposition 3.7. We will show that the action of Frob_{p} and of −w_{p} coincide on X/𝔪X. Since X = H_{1}(Γ(𝔛_{0}(N)_{\overline{F}_{p}}), Z), it is enough to consider the action of Frob_{p} and of −w_{p} on Γ(𝔛_{0}(N)_{\overline{F}_{p}}). By using the moduli interpretation of 𝔛_{0}(N)_{F_{p}} one sees that Frob_{p} and w_{p} are both involutions and have the same effect on the edges of Γ(𝔛_{0}(N)_{\overline{F}_{p}}), but Frob_{p} fixes the vertices of Γ(𝔛_{0}(N)_{\overline{F}_{p}}), while w_{p} swaps them. Hence Frob_{p} ∘w_{p} swaps the vertices of Γ(𝔛_{0}(N)_{\overline{F}_{p}}) and fixes its edges, which on the homology has the same effect as multiplication by -1.

Since w_{p} is an involution and κ is a field, Lemma 2.7 implies that G_{p} acts on Hom(X/𝔪X, μ_{p}) via ψχ, where ψ is an unramified quadratic character. In particular the action of G_{p} on det_{κ} V is via ψ^{2}χ^{2} = χ^{2}. On the other hand, G_{p} acts on det_{κ} V by det ρ = χ, hence χ = χ^{2}, and thus χ must be trivial, which is an obvious contradiction since p is odd. This finishes the proof of Theorem 2.1 in the case when ℓ = p.

The case ℓ ≠ p proceeds along the same lines, but is simpler as one does not need an analogue of Lemma 2.3. Instead we note that when ℓ ≠ p, the Galois module V is unramified by assumption. Thus there exists a finite etale κ-vector space scheme 𝔚 over Z_{(ℓ)} such that V ≅ 𝔚(\overline{Q}) as κ[G_{Q}]-modules. Hence the injection V ↪ J_{0}(N)(\overline{Q})[𝔪] gives rise to the a closed immersion 𝔚_{Q} ↪ J_{0}(N). As ℓ ≠ p, 𝔚 is smooth over Z_{(ℓ)}, hence the extension 𝔚 → 𝔍_{0}(N) exists by the Neron mapping property. We claim that it is a closed immerion. Indeed, let \overline{𝔚} denote the schematic closure of the image of 𝔚 in 𝔍_{0}(N). Then \overline{𝔚} is finite and flat over R, but since it is an ℓ-group with ℓ ≠ p, it is also etale. Hence the fact that \overline{𝔚}_{Q} ⟶^{∼} 𝔚_{Q} implies that \overline{𝔚} ⟶^{∼} 𝔚. In that case the final conclusion that χ = χ^{2} implies that ℓ ≡ 1 mod p, which was assumed not to be the case.

References

[BLR90] Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud. Néron models, volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1990.

[DS74] Pierre Deligne and Jean-Pierre Serre. Formes modulaires de poids 1. Ann. Sci. École Norm. Sup. (4), 7:507–530 (1975), 1974.

[Gro72] Alexandre Grothendieck. Sga7 i, expose ix. Lecture Notes in Mathematics, 288(2):313–523, 1972.

[Gro90] Benedict H. Gross. A tameness criterion for Galois representations associated to modular forms (mod p). Duke Math. J., 61(2):445–517, 1990.

[Ray74] Michel Raynaud. Schémas en groupes de type (p, …, p). Bull. Soc. Math. France, 102:241–280, 1974.

[Rib90] K. A. Ribet. On modular representations of Gal(\overline{Q}/Q) arising from modular forms. Invent. Math., 100(2):431–476, 1990.