Serre’s conjecture.

Bas Edixhoven

April 1996

The aim of the first section is to state Serre’s conjecture and to tell what is presently known about it, without proof. We start by recalling what modular forms are. Then we recall the result, due to Deligne, that to a mod p modular form one can associate a mod p Galois representation. After that we state Serre’s conjecture and what we know about it. In Section 2 we will see which cases of it are actually needed in order to prove, following Wiles, that all semi-stable elliptic curves over ℚ are modular. In the last two sections we will sketch the proofs in those cases.

1 Serre’s conjecture: statement and results

Let N ≥ 1 and k be integers and let R be a ℤ[1/N]-algebra. We will first recall Katz’s definition [22] of modular forms of level N and weight k over R. This definition may seem more complicated than necessary, but it is very convenient in order to deal with modular forms over fields of positive characteristic p. For example, one has the Hasse invariant (see [24, §12.4]), which cannot be lifted to characteristic zero as a form of level 1 and weight p − 1 for p equal to 2 or 3, and one has the derivation q d/dq sending modular forms to modular forms, increasing the weight by p + 1 (p being the characteristic of the finite field). A good reference for more details concerning modular forms in various settings is [14].

Let [Γ_{1}(N)]_{R} denote the category whose objects are pairs (E/S/R, α), with S an R-scheme, E/S a generalized elliptic curve in the sense of [10] and α: (ℤ/Nℤ)_{S} → E[N] an embedding of group schemes such that the image of α meets all irreducible components of all geometric fibres of E/S. The morphisms from (E′/S′/R, α′) to (E/S/R, α) in [Γ_{1}(N)]_{R} are Cartesian diagrams:

(1.1)  \begin{aligned}E′ & → & E \\ ↓ & & ↓ \\ S′ & → & S \end{aligned}

which are compatible with α and α′. For a generalized elliptic curve E over a scheme S we have the invertible O_{S}-module \underline{ω}_{E/S} := 0^{∗}Ω_{E/S}^{1}, obtained by pulling back the sheaf of Kähler differentials of E over S by the zero section 0 in E(S). A modular form f of level N and weight k over R is then a rule, that assigns to every object (E/S/R, α) of [Γ_{1}(N)]_{R} an element f(E/S/R, α) of \underline{ω}_{E/S}^{⊗k}(S), compatible with morphisms in [Γ_{1}(N)]_{R}. The R-module of such modular forms will be denoted by M(N, k)_{R}. Let us give one example: for p a prime, the Hasse invariant is an element of M(1, p − 1)_{𝔽_{p}}.

There are more down to earth ways to describe M(N, k)_{R}. For N ≥ 5 the category [Γ_{1}(N)]_{R} has a final object, called (E_{univ}/X_{1}(N)_{R}, α_{univ}), and in that case one has simply M(N, k) = H^{0}(X_{1}(N)_{R}, \underline{ω}^{⊗k}). Recall that X_{1}(N)_{R} is a smooth projective curve over R whose fibres are geometrically irreducible. For N ≥ 1 and n ≥ 3 invertible in R one has the description:

(1.2)  M(N, k)_{R} = H^{0}(\overline{M}([Γ_{1}(N), Γ(n)]_{R}), \underline{ω}^{⊗k})^{G},

where \overline{M}([Γ_{1}(N), Γ(n)]_{R}) is the moduli scheme parametrizing triples (E/S/R, α, β) of generalized elliptic curves with α: (ℤ/Nℤ)_{S} → E[N] and β: (ℤ/nℤ)_{S}^{2} → E[n] embeddings of group schemes such that the image of α + β meets all irreducible components of all geometric fibres, and where G is the group GL_{2}(ℤ/nℤ). This \overline{M}([Γ_{1}(N), Γ(n)]_{R}) is a smooth projective curve over R, but its fibres are not geometrically irreducible.

In the special case R = ℂ one easily shows that M(N, k)_{ℂ} is naturally isomorphic to the space of modular forms defined as certain holomorphic functions on the upper half plane ℍ (see Rohrlich’s chapter in this book).

One can show, for example by using the modular form Δ in M(1, 12)_{R}, that M(N, k)_{R} = 0 for all k < 0.

Let R → R′ be a morphism of ℤ[1/N]-algebras. Then we have a morphism of R′-modules M(N, k)_{R} ⊗_{R} R′ → M(N, k)_{R′}. Such a morphism is not always an isomorphism (consider for example ℤ → 𝔽_{2}, N = 1 and k = 1). However, it is an isomorphism when R → R′ is flat (use (1.2)).

Over ℤ[[q]] one has the Tate curve Tate(q), which is equipped with a basis dt/t of \underline{ω}_{Tate(q)/ℤ[[q]]}. One has an isomorphism of groups:

(1.3)  (ℤ/Nℤ)^{2} ⟶ Tate(q)[N](\overline{ℚ((q))}), (a, b) ↦ ζ_{N}^{a}(q^{1/N})^{b},

where ζ_{N} ∈ \overline{ℚ}^{∗} is a fixed root of unity of order N. For R a ℤ[1/N, ζ_{N}]-algebra one has the standard q-expansion map M(N, k)_{R} → R[[q]] sending f to the series ∑_{n≥0} a_{n}(f)q^{n} defined by:

(1.4)  f(Tate(q), 1 ↦ ζ_{N}) = (∑_{n≥0} a_{n}(f)q^{n}) (dt/t)^{k}.

This map is injective. The R-modules M(N, k)_{R} can also be defined as follows. One replaces “generalized elliptic curve” in the definition we gave by “elliptic curve” and one demands that the q-expansions obtained from all points of order N of the Tate curve over ℤ[ζ_{N}]((q^{1/N})) are power series in q^{1/N} (a priori they are Laurent series). The R-module of cusp forms of level N and weight k is defined to be the submodule M^{0}(N, k)_{R} of M(N, k)_{R} of those f all of whose q-expansions have zero as constant coefficient. Equivalently, they are the forms that vanish on degenerate elliptic curves.

Suppose that k ≥ 1. The R-modules M(N, k)_{R} and M^{0}(N, k)_{R} are equipped with certain endomorphisms. For n ≥ 1 one has the Hecke operator T_{n}, defined in terms of isogenies of degree n. For a in (ℤ/Nℤ)^{∗} one has the diamond operator ⟨a⟩, induced by the automorphism of [Γ_{1}(N)_{R}] that sends (E/S/R, α) to (E/S/R, aα). The action of these operators is given by the usual formulas in terms of q-expansions (see Rohrlich’s chapter).

In particular, one has a_{1}(T_{n}(f)) = a_{n}(f). The construction of the T_{n} in this generality is a bit complicated, especially for k = 1. See for example [17, §4] for a construction of T_{p} on M(N, 1)_{𝔽_{p}}. In general, one can construct the T_{n} as follows. It suffices to construct the T_{p} with p prime. If p divides N then p is invertible in R and the construction is easy. So suppose that p does not divide N. In this case one proceeds as in [17, §4]. Take f in M(N, k)_{R}, view it as a G-invariant section of \underline{ω}^{⊗k} on \overline{M}([Γ_{1}(N), Γ(n)]_{R}), as in (1.2). Then restrict it to M([Γ_{1}(N), Γ(n)]_{R}), the complement of the cusps. This is affine, hence this restriction of f is a linear combination, with coefficients in R, of sections of \underline{ω}^{⊗k} over M([Γ_{1}(N), Γ(n)]_{ℤ[1/Nn]}) and one knows what T_{p} does with those. One verifies that the T_{p}(f) obtained in this way is again G-invariant and regular at the cusps.

The endomorphisms T_{n} and ⟨a⟩ of M(N, k)_{R} all commute with each other. For ε a character of (ℤ/Nℤ)^{∗} with values in R^{∗} let M(N, k, ε)_{R} denote the R-submodule of M(N, k)_{R} of elements f such that ⟨a⟩(f) = ε(a)f for all a; such f will be called forms of type (N, k, ε). If R is a field and f a non-zero element of M(N, k)_{R} which is an eigenform for all T_{n}, then the formula a_{1}(T_{n}(f)) = a_{n}(f) implies that the corresponding eigenspace for the T_{n} has dimension one, and that there is a unique character ε such that f is of type (N, k, ε).

1.5 Theorem. (Deligne) Let p be a prime, N an integer prime to p, and k ≥ 1. Let f be a non-zero eigenform in M(N, k, ε)_{\overline{𝔽}_{p}}, and let a_{n} be its eigenvalue for T_{n}. Then there is a unique semi-simple continuous representation ρ_{f}: G_{ℚ} := Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝔽}_{p}), unramified outside pN, such that for all primes l not dividing pN the (arithmetic) Frobenius element ρ_{f}(Frob_{l}) has trace a_{l} and determinant ε(l)l^{k−1}.

(The topology on GL_{2}(\overline{𝔽}_{p}) in this statement is the discrete one.) In fact, Deligne [8] showed that for an eigenform f of weight k ≥ 2 and with coefficients in \overline{ℚ}_{p} one even has a p-adic representation ρ_{f} of G_{ℚ} with values in GL_{2}(\overline{ℚ}_{p}). The case k = 2 had already been treated by Shimura in [40]. A detailed and reasonably elementary proof of Theorem 1.5 can be found in [17].

Let f be as in Theorem 1.5. The fact that every elliptic curve has the automorphism −1 implies that ε(−1)f = ⟨−1⟩(f) = (−1)^{k}f, hence that ε(−1) = (−1)^{k}. It follows that, for σ in G_{ℚ} any complex conjugation, det(ρ_{f}(σ)) = ε(−1)(−1)^{k−1} = −1. A representation of G_{ℚ} with this property will be called odd. Serre’s conjecture [37, (3.2.4)_{?}] states that any continuous odd representation ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) is isomorphic to some ρ_{f}, and for irreducible ρ it claims that such a f exists of a certain “minimal” type (N(ρ), k(ρ), ε(ρ)). Before we can state Serre’s conjecture, we need some terminology.

Let p be a prime and ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) a continuous representation. Then the image G of ρ is finite, let’s say equal to Gal(K/ℚ) with K a finite Galois extension of ℚ in \overline{ℚ}. It follows that ρ is unramified at all but finitely many primes. The number N(ρ) is by definition the Artin conductor of ρ, except that one doesn’t take the prime p into account. More precisely, N(ρ) will be a positive integer prime to p. Let l be a prime number different from p. Let I_{l} denote the inertia subgroup at l of G corresponding to a place of K above l, and let I_{l} = I_{l,0} ⊃ I_{l,1} ⊃ ⋯ be the higher ramification subgroups: I_{l,i} is the subgroup of the decomposition group at the chosen place whose elements act trivially on the ring of integers modulo the (i + 1)th power of the maximal ideal. The valuation of N(ρ) at l, i.e., the number of factors l in it, is then given by the formula:

(1.6)  v_{l}(N(ρ)) = ∑_{i≥0} \frac{1}{[I_{l,0} : I_{l,i}]} dim(V/V^{I_{l,i}}),

where V is a two-dimensional \overline{𝔽}_{p}-vector space with a G-action giving ρ, and where V^{I_{l,i}} denotes its subspace of invariants under I_{l,i}. For a motivation for formula (1.6), see [38], pp. 171–174. Let us note, by the way, that replacing V/V^{I_{l,i}} by the kernel of the map to the coinvariants V → V_{I_{l,i}} would give the same result. For i > 0 this is clear since then I_{l,i} is an l-group, for i = 0 one uses that V_{I_{l,0}} is obtained by first taking the coinvariants for the l-group I_{l,1} and then the coinvariants for the cyclic group I_{l,0}/I_{l,1}.

We can now define the character ε(ρ) and the image of k(ρ) in ℤ/(p−1)ℤ. For each positive integer n let χ_{n}: G_{ℚ} → (ℤ/nℤ)^{∗} denote the cyclotomic character of the nth roots of unity, i.e., the character such that for all σ in G_{ℚ} and all x in \overline{ℚ} with x^{n} = 1 we have σ(x) = x^{χ(σ)}. We will often view a character of the group (ℤ/nℤ)^{∗} as a character of G_{ℚ}, by composing it with χ_{n}. With these conventions, Theorem 1.5 implies that for f an eigenform in M(N, k, ε)_{\overline{𝔽}_{p}} one has det^{∘}ρ_{f} = εχ_{p}^{k−1}. This equality shows that det^{∘}ρ_{f} is a character of the group (ℤ/pNℤ)^{∗}. This group is canonically isomorphic to (ℤ/Nℤ)^{∗} × 𝔽_{p}^{∗} via the corresponding isomorphism of rings. Under this decomposition, the restriction of det^{∘}ρ_{f} to (ℤ/Nℤ)^{∗} is ε, and the restriction to 𝔽_{p}^{∗} is χ_{p}^{k−1}. Let us now go back to the representation ρ. By the definition of N(ρ), it is unramified away from pN(ρ). Since the maximal abelian extension of ℚ is the cyclotomic extension, det^{∘}ρ can be considered as a character of (ℤ/p^{n}ℤ)^{∗} × (ℤ/N(ρ)^{m})^{∗}, for some n, m ≫ 0. Since the character has values in \overline{𝔽}_{p}^{∗}, one can take n = 1. Comparing formula (1.6) for ρ and det^{∘}ρ, and using some class field theory, one sees that one can take m = 1. Then one defines ε(ρ) to be the restriction of det^{∘}ρ to the factor (ℤ/N(ρ)ℤ)^{∗}, and the image of k(ρ) in 𝔽_{p}^{∗} such that the restriction of det^{∘}ρ to 𝔽_{p}^{∗} is χ_{p}^{k(ρ)−1}.

Now we get to the exact definition of k(ρ). The first thing to note is that k(ρ) depends only on the restriction of ρ to some inertia subgroup I_{p} ⊂ G_{ℚ} at p. Hence the level N(ρ) reflects the ramification away from p and the weight k(ρ) the ramification at p. In order to state the definition of k(ρ) we first have to classify the two-dimensional representations over \overline{𝔽}_{p} of a decomposition subgroup G_{p} ⊂ G_{ℚ} at p (this subgroup G_{p} depends on the choice of a maximal ideal containing p, but since these are permuted transitively by G_{ℚ} the choice will not matter). For this classification, we need some terminology.

Let \overline{ℤ} be the integral closure of ℤ in \overline{ℚ}. Let us choose a morphism of rings \overline{ℤ} → \overline{𝔽}_{p}. This gives us a decomposition group G_{p} (the stabilizer of the kernel). The action of G_{p} on \overline{𝔽}_{p} gives a surjection G_{p} → G_{𝔽_{p}}; its kernel I_{p} is the inertia group. We can identify G_{p} with Gal(\overline{ℚ}_{p}/ℚ_{p}) and I_{p} with Gal(\overline{ℚ}_{p}/ℚ_{p}^{unr}), where ℚ_{p}^{unr} is the maximal unramified extension of ℚ_{p}. Let ℚ_{p}^{tr} be the maximal tamely ramified extension of ℚ_{p}. Then the quotient I_{p,t} := Gal(ℚ_{p}^{tr}/ℚ_{p}^{unr}) of I_{p} is called the tame ramification group, and the subgroup I_{p,w} := Gal(\overline{ℚ}_{p}/ℚ_{p}^{tr}) of I_{p} the wild ramification group. The tame extensions of ℚ_{p}^{unr} are obtained by taking nth roots of elements p, with n prime to p. It follows that I_{p,t} can be identified with lim_{⟵} 𝔽_{p^{n}}^{∗}, where the limit is taken over the n ≥ 1 and where the transition morphisms are norm maps. A character ϕ: I_{p,t} → \overline{𝔽}_{p}^{∗} is called of level n, if n is the smallest integer such that ϕ factors through 𝔽_{p^{n}}^{∗}. The n characters I_{p,t} → 𝔽_{p^{n}}^{∗} → \overline{𝔽}_{p}^{∗} that are induced by embeddings of fields 𝔽_{p^{n}} → \overline{𝔽}_{p} are called the fundamental characters of level n.

Suppose now that ρ_{p}: G_{p} → GL(V) is a continuous 2-dimensional representation on a \overline{𝔽}_{p}- vector space. Let V^{ss} be the semi-simplification of V for the action of G_{p}. we claim that I_{p,w} acts trivially on V^{ss}. To see that, let W be an irreducible factor of V^{ss}. Then the subspace W^{I_{p,w}} of W is stable under G_{p} because I_{p,w} is a normal subgroup of it, so either W^{I_{p,w}} is trivial or it is W. Since the image of G_{p} in GL(W) is finite, the representation of G_{p} on W can be realized over a finite extension of 𝔽_{p}. For such a realization W′, the number of fixed points for I_{p,w} is a multiple of p since the image of I_{p,w} is a p-group, hence W′ has non-trivial I_{p,w}-invariants. This proves that the action of I_{p} on V^{ss} is given by two characters ϕ, ϕ′: I_{p,t} → \overline{𝔽}_{p}^{∗}. Since Gal(\overline{𝔽}_{p}/𝔽_{p}) acts by conjugation on I_{p,t}, it follows that {ϕ^{p}, ϕ^{′p}} = {ϕ, ϕ′}. This means that there are two cases: either ϕ and ϕ′ are both of level one, or ϕ and ϕ′ are both of level two and ϕ^{p} = ϕ′, ϕ′^{p} = ϕ. In the first case ρ_{p} is reducible, whereas in the second case ρ_{p} is absolutely irreducible.

We need one more ingredient before we can define k(ρ_{p}): the notion of “finiteness at p”. Let 𝔽 be a finite extension of 𝔽_{p} such that ρ_{p} can be realized over 𝔽. An 𝔽-vector space scheme V over a scheme S is then an S-scheme V, together with a structure of 𝔽-vector space on all the sets of points V (T) for S-schemes T, functorially in T. Equivalently, an 𝔽-vector space scheme over S is a commutative group scheme over S given with an action of 𝔽 on it by endomorphisms. An 𝔽-vector space scheme V over ℚ_{p} gives rise to the representation of G_{p} on the 𝔽-vector space V (\overline{ℚ}_{p}). This construction induces an equivalence between the category of 𝔽-vector space schemes finite over ℚ_{p} and the category of representations of G_{p} on finite dimensional 𝔽-vector spaces. Concretely, the 𝔽-vector space scheme V_{ℚ_{p}} corresponding to ρ_{p} can be obtained as the quotient of the scheme 𝔽_{ℚ_{p}}^{2} ×_{Spec(ℚ_{p})} Spec(\overline{ℚ}_{p}) by the group G_{p}, with σ in G_{p} acting as (ρ_{p}(σ)^{−1}, Spec(σ)). One says that ρ_{p} is finite at p if V_{ℚ_{p}} can be extended to a finite flat 𝔽-vector space scheme over ℤ_{p}. It is equivalent to demand that V_{ℚ_{p}} can be extended to a finite flat 𝔽-vector space scheme over the ring of integers ℤ_{p}^{unr} in ℚ_{p}^{unr}, or to demand that it can be extended over ℤ_{p} or ℤ_{p}^{unr} as a finite flat group scheme. Finally, one can formulate the condition that ρ_{p} be finite at p in Galois theoretic terms, without using group schemes (see [15, Proposition 8.1], and its proof, and [37, (2.4.7)]).

We will now give the definition of k(ρ_{p}). Let ψ and ψ′ be the two fundamental characters of level 2.

1.7 Definition. Let ρ_{p}: G_{p} → GL(V), V^{ss}, ϕ and ϕ′ be as above. We associate an integer k(ρ_{p}) to ρ_{p} as follows.

1. Suppose that ϕ and ϕ′ are of level 2. we have:

ρ_{p}|_{I_{p}} ≅ ( \begin{matrix}ϕ & 0 \\ 0 & ϕ′ \end{matrix} ) .

After interchanging ϕ and ϕ′ if necessary, we have (uniquely) ϕ = ψ^{a+pb} = ψ^{a}ψ′^{b} and ϕ′ = ψ′^{a}ψ^{b} with 0 ≤ a < b ≤ p − 1. We set k(ρ_{p}) = 1 + pa + b.

2. Suppose that ϕ and ϕ′ are of level 1.

(a) If ρ_{p}|_{I_{p,w}} is trivial then we have:

ρ_{p}|_{I_{p}} ≅ ( \begin{matrix}χ^{a} & 0 \\ 0 & χ^{b} \end{matrix} )

with 0 ≤ a ≤ b ≤ p − 2. We set k(ρ_{p}) = 1 + pa + b.

(b) Suppose that ρ_{p}|_{I_{p,w}} is not trivial. We have:

ρ_{p}|_{I_{p}} ≅ ( \begin{matrix}χ^{β} & ∗ \\ 0 & χ^{α} \end{matrix} )

for unique α and β with 0 ≤ α ≤ p − 2 and 1 ≤ β ≤ p − 1. We set a = min(α, β), b = max(α, β). If χ^{β−α} = χ and ρ_{p} ⊗ χ^{−α} is not finite at p then we set k(ρ_{p}) =

1 + pa + b + p − 1; otherwise we set k(ρ_{p}) = 1 + pa + b.
□

For ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) continuous, we define k(ρ) to be k(ρ_{p}), where ρ_{p} is the restriction of ρ to a decomposition group at p.

1.8 Conjecture. (Serre) Let p be a prime number and ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) a continuous irreducible odd representation. Then there exists an eigenform f in M^{0}(N(ρ), k(ρ), ε(ρ))_{\overline{𝔽}_{p}} such that ρ is isomorphic to ρ_{f}.

Several remarks should be made, at this point. First of all, it is easy to see that for f an eigenform which is not cuspidal, the representation ρ_{f} is reducible (restrict f to the cusps, and study the action of the Hecke operators on the cusps). This explains that one can demand in the conjecture f to be a cuspform.

The second remark is more important. The conjecture, as we state it here (i.e., as suggested in [15, 4.3]), is not equivalent the one stated by Serre in [37, (3.2.4)_{?}]. The difference comes from the fact that the modular forms over \overline{𝔽}_{p} considered in [37] are defined to be those obtained by reduction mod p of forms over \overline{ℚ}. More precisely, in [37], a cuspidal modular form over \overline{𝔽}_{p} of some type (N, k, ε), with N prime to p and k ≥ 2, is an element in the image of the map M^{0}(N, k, ε̃)_{\overline{ℤ}_{p}} → M^{0}(N, k, ε)_{\overline{𝔽}_{p}}, where \overline{ℤ}_{p} is the integral closure of ℤ_{p} in \overline{ℚ}_{p} and ε̃: (ℤ/Nℤ)^{∗} → \overline{ℤ}_{p}^{∗} is the Teichmüller lift of ε, i.e., ε̃ induces ε and they have the same order. The problem is that these reduction maps are not all surjective. Before we discuss what is known today about Serre’s conjecture, we will discuss the differences between Conjecture 1.8 and [37, (3.2.4)_{?}]. It was suggested by Serre in his 1987–1988 course at the Collège de France to replace the mod p modular forms in [37] by those defined by Katz, i.e., the ones we are using here.

Let us first consider the problem of lifting modular forms from \overline{𝔽}_{p} to \overline{ℤ}_{p}, without paying attention to the character. Then we have the following result.

1.9 Lemma. Let p be a prime, N ≥ 1 prime to p.

1. Suppose that k ≥ 2. Then the map M^{0}(N, k)_{\overline{ℤ}_{p}} → M^{0}(N, k)_{\overline{𝔽}_{p}} is surjective if N ≠ 1 or if p > 3.

2. The map M^{0}(1, k)_{\overline{ℤ}_{2}} → M^{0}(1, k)_{\overline{𝔽}_{2}} is not surjective if and only if k ≥ 12 and (k ≡ 1 mod 2 or k ≡ 2 mod 12).

3. The map M^{0}(1, k)_{\overline{ℤ}_{3}} → M^{0}(1, k)_{\overline{𝔽}_{3}} is not surjective if and only if k ≥ 12 and k ≡ 2 mod 12.

Proof. Let us prove the first statement; the other two can be proved using the explicit descriptions of the rings of modular forms of level one over ℤ, 𝔽_{2} and 𝔽_{3} found in [9, Proposition 6.2]. Suppose first that N ≥ 5. Because \overline{ℤ}_{p} is flat over ℤ_{p} and \overline{𝔽}_{p} is flat over 𝔽_{p}, it suffices to prove that the reduction map induced by ℤ_{p} → 𝔽_{p} is surjective. Consider the long exact cohomology sequence arising from the short exact sequence of sheaves on X_{1}(N)_{ℤ_{p}}:

(1.9.1)  0 → \underline{ω}^{⊗k}(−cusps) → \underline{ω}^{⊗k}(−cusps) → i_{∗}\underline{ω}^{⊗k}(−cusps) → 0,

where the map \underline{ω}^{⊗k}(−cusps) → \underline{ω}^{⊗k}(−cusps) is multiplication by p and i: X_{1}(N)_{𝔽_{p}} → X_{1}(N)_{ℤ_{p}} denotes the closed immersion. To get the surjectivity we want, it is sufficient to show that H^{1}(X_{1}(N)_{𝔽_{p}}, \underline{ω}^{⊗k}(−cusps)) = 0, since by Nakayama’s lemma and the long exact sequence this implies that H^{1}(X_{1}(N)_{ℤ_{p}}, \underline{ω}^{⊗k}(−cusps)) = 0. The Kodaira-Spencer isomorphism (see [22, A1.4]) and Serre duality give isomorphisms:

H^{1}(X_{1}(N)_{𝔽_{p}}, \underline{ω}^{⊗k}(−cusps)) = H^{1}(X_{1}(N)_{𝔽_{p}}, Ω ⊗ \underline{ω}^{⊗k−2}) = H^{0}(X_{1}(N)_{𝔽_{p}}, \underline{ω}^{⊗2−k})^{∨}.

So if k > 2, this shows what we want, since the degree of \underline{ω} is positive. The case k = 2 is in fact easy, since the Kodaira-Spencer isomorphism identifies weight 2 cuspforms with differential forms, and those can be lifted. Another way to phrase the argument is to say that the dimensions of M^{0}(N, k)_{𝔽_{p}} and M^{0}(N, k)_{ℚ_{p}} are given by the Riemann–Roch formula since the H^{1}-term vanishes, and that hence the reduction map is surjective.

Suppose now that p > 3. Then M^{0}(N, k)_{𝔽_{p}} = H^{0}(\overline{M}([Γ_{1}(N), Γ(3)]_{𝔽_{p}}), \underline{ω}^{⊗k}(−cusps))^{G}, with G = GL_{2}(𝔽_{3}). Since p does not divide the order of G, the functor M ↦ M^{G} from ℤ_{p}[G]-modules to ℤ_{p}-modules is exact. Combining this with the long exact sequence arising from the short exact sequence 0 → \underline{ω}^{⊗k}(−cusps) → \underline{ω}^{⊗k}(−cusps) → i_{∗}\underline{ω}^{⊗k}(−cusps) → 0 on \overline{M}([Γ_{1}(N), Γ(3)])_{ℤ_{p}} gives the result.

Suppose now that N = 2 or N = 4. Then p ≠ 2. In these cases, the category [Γ_{1}(N)]_{ℤ_{p}} is the quotient, in the sense of algebraic stacks, for the action of a subgroup G of GL_{2}(ℤ/4ℤ) acting on [Γ(4)]_{ℤ_{p}}. This gives a formula analogous to (1.2). The group G is a 2-group, hence or order prime to p. One can then apply an argument which is similar to the one used in the case p > 3 above.

Suppose now that N = 3. Then p ≠ 3. Let ℤ_{p}[ζ_{3}] be the subring of \overline{ℤ}_{p} generated by ℤ_{p} and a third root of unity ζ_{3}. Let [Γ(3)^{ζ_{3}−can}]_{ℤ_{p}[ζ_{3}]} be the category of generalized elliptic curves over schemes over ℤ_{p}[ζ_{3}] with an embedding α of the constant group scheme (ℤ/3ℤ)^{2}, such that the Weil pairing of α(1, 0) and α(0, 1) equals ζ_{3}. Then [Γ_{1}(N)]_{ℤ_{p}[ζ_{3}]} is the quotient of [Γ(3)^{ζ_{3}−can}]_{ℤ_{p}[ζ_{3}]} for the action of a group of order 3. This means that one can again use the same argument. □

The proof of this lemma indicates that the case k = 1 is very different, since the degree of \underline{ω} is, as one sees from the Kodaira-Spencer isomorphism, too small to make the H^{1}-term in the Riemann–Roch theorem vanish. Mestre has indeed found examples with p > 3 where the map M^{0}(N, 1)_{ℤ_{p}} → M^{0}(N, 1)_{𝔽_{p}} is not surjective. In these examples one has an eigenform f in M^{0}(N, 1)_{\overline{𝔽}_{p}} such that the image of the representation ρ_{f} is too big to be embeddable in GL_{2}(ℂ); if ρ could be lifted to characteristic zero this would contradict the theorem of Deligne–Serre (Theorem 4.1 of [11]). Let us note that for a representation ρ as in Conjecture 1.8 it can very well happen that k(ρ) = 1; one can check that this is equivalent to ρ being unramified at p. This explains that the weight k_{ρ} for f that one finds in [37, (3.2.4)_{?}], is not in all cases the same as k(ρ) defined above. The difference between k_{ρ} and k(ρ) can be summarized as follows, in the notation of Definition 1.7. There are only two cases where they are different; in both cases the characters ϕ and ϕ′ are of level 1. In the first case, the restriction of ρ to the wild inertia group I_{p,w} is trivial and a = 0 = b; then k(ρ) = 1 and k_{ρ} = p. In the second case p = 2, ρ is wildly ramified at 2, α = 0, β = 1 and ρ is not finite at 2; then k(ρ) = 3 and k_{ρ} = 4.

Other problems arise if we take the character into account. In his course at the Collège de France, 1987–1988, Serre gave some counter examples against his conjecture [37, (3.2.4)_{?}]. These examples are found by considering the genus two curve X_{1}(13). On this curve there are two eigenforms and of weight 2 over ℤ[ζ_{3}], and the two corresponding characters are of order 6. The reductions mod 2 and 3 of these eigenforms have characters of order 3 and 2, respectively. One verifies that the Galois representations corresponding to these mod 2 and mod 3 forms are irreducible, and that the weights associated to them equal two (Definition 1.7 and [37, §2] coincide in these cases). In fact, these representations are dihedral, induced from G_{ℚ(\sqrt{−1})} and G_{ℚ(\sqrt{−3})}, respectively. According to [37, (3.2.4)_{?}], the mod 2 and mod 3 reductions of the two eigenforms should have lifts to weight two eigenforms in characteristic zero on X_{1}(13) with a character of the same order as the reduction. But since the genus is two, there are no such forms.

In the same course at the Collège de France, Serre showed that the only mod p eigenforms f of weight at least 2 that cannot be lifted to an eigenform with the same level, weight and order of character are among those in characteristic 2 or 3, whose representation ρ_{f} is induced from G_{ℚ(\sqrt{−1})} or G_{ℚ(\sqrt{−3})}, respectively. This result was obtained independently by Carayol, see [4, §4.4], and is usually called Carayol’s Lemma. Serre’s proof uses a result of Nakajima implying that for n ≥ 3 prime to p and k ≥ 2 the ℤ_{p}[GL_{2}(ℤ/nℤ)]-module H^{0}(X(n)_{ℤ_{p}}, \underline{ω}^{⊗k}) is projective, if all stabilizers are of order prime to p.

Carayol’s proof uses the realization of the Galois representation associated to modular forms in the first cohomology group of certain p-adic sheaves on modular curves over \overline{ℚ}. His arguments can be adapted to the sheaves \underline{ω}^{⊗k}. This gives the following result, that we state without proof, and which can also be found in Serre’s notes. Yet another version of it can be found in [12, §2].

1.10 Proposition. (Carayol’s Lemma) Let N ≥ 1, p a prime not dividing N, and k ≥ 2. Let ε: (ℤ/Nℤ)^{∗} → \overline{ℤ}_{p}^{∗} be a character with ε(−1) = (−1)^{k}, and let ε̄: (ℤ/Nℤ)^{∗} → \overline{𝔽}_{p}^{∗} be its reduction. Consider the map ϕ: M^{0}(N, k, ε)_{\overline{ℤ}_{p}} → M^{0}(N, k, ε̄)_{\overline{𝔽}_{p}}. If p ≥ 5, then ϕ is surjective. If p = 3 (resp., p = 2) and f ∈ M^{0}(N, k, ε̄)_{\overline{𝔽}_{p}} is an eigenform with ρ_{f} irreducible and f not in the image of ϕ, then ρ_{f} is induced from ℚ(\sqrt{−3}) (resp., ℚ(\sqrt{−1})). □

In both proofs of this result it is quite clear where the ℚ(\sqrt{−3}) and ℚ(\sqrt{−1}) come from. Suppose for simplicity that N ≥ 5. In Carayol’s proof, it comes from the fact that the points of X_{0}(N)\overline{ℚ} with an automorphism (i.e., an automorphism of the pair (E/\overline{ℚ}, G) corresponding to it) of order 3 (resp., order 4) are defined over abelian extensions of ℚ(\sqrt{−3}) (resp., ℚ(\sqrt{−1})). In Serre’s proof, it comes from the fact that for primes l ≡ −1 mod 3 (resp., mod 4) there is no elliptic curve with an automorphism of order 6 (resp., 4) fixing a subgroup of order l, implying that if an eigenform is not liftable (in the sense of Proposition 1.10) then it is annihilated by T_{l} for such l; this implies that the character of ρ_{f} vanishes on Frob_{l} for such l; then it follows that ρ_{f} is induced as stated.

The statement of Carayol’s Lemma in [4] is actually different from Proposition 1.10 above. It says that if an irreducible representation ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) arises from some eigenform f in M(N, k, ε)_{\overline{ℚ}_{p}} with k ≥ 2, then for every character ε′: (ℤ/Nℤ)^{∗} → \overline{ℚ}_{p}^{∗} inducing the same character (ℤ/Nℤ)^{∗} → \overline{𝔽}_{p}^{∗} as ε, there exists an eigenform f′ inducing ρ. In this statement, which does not speak of modular forms mod p, one does not suppose that N is prime to p.

Let us now discuss what is known about Conjecture 1.8 and about its relation to [37, (3.2.4)_{?}].

1.11 Proposition. Let p be a prime, and let ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) be continuous, irreducible and odd.

1. If p = 2 (resp., p = 3), suppose that ρ is not induced from ℚ(\sqrt{−1}) (resp., ℚ(\sqrt{−3})). Then if ρ satisfies Conjecture 1.8, it satisfies [37, (3.2.4)_{?}].

2. If p = 2, suppose that the restriction of ρ to a decomposition group at 2 is not an extension of some character by itself. Then if ρ satisfies [37, (3.2.4)_{?}], it satisfies Conjecture 1.8.

1.12 Theorem. Let p > 2 be a prime, and let ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) be continuous, irreducible and odd. Suppose that ρ comes from a modular form of some type. Then ρ satisfies Conjecture 1.8.

Moreover, if ρ comes from a mod p modular form of some type (N, k, ε) with N prime to p, then N is a multiple of N(ρ), k ≥ k(ρ) and ε is obtained from ε(ρ) via composition by ℤ/Nℤ → ℤ/N(ρ)ℤ.

The proof of these results is quite long and many people have contributed to it. A complete proof can be found by reading Diamond’s article [12], and the references therein. A very good overview of the strategy of the whole proof is given in Ribet’s report [32]. In the next section we will see which parts of these results are used in the proof of the conjecture of Shimura– Taniyama–Weil for semi-stable elliptic curves over ℚ and the proof of Fermat’s Last Theorem. In Sections 3 and 4, we will then describe the proofs of those cases. To finish this section, we will briefly recall the history of the proofs of Proposition 1.11 and Theorem 1.12.

For the rest of this section, let p be prime and ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) be continuous, irreducible and odd. We will say that ρ is modular of type (N, k, ε)_{\overline{𝔽}_{p}} (resp., (N, k, ε)_{\overline{ℚ}_{p}}) if there is an eigenform f in M^{0}(N, k, ε)_{\overline{𝔽}_{p}} (resp., M^{0}(N, k, ε)_{\overline{ℚ}_{p}}) such that f gives the representation ρ. Serre formulated, in a letter to Mestre dated August 13, 1985, a part of his conjectures that, together with the Shimura–Taniyama–Weil conjecture, implies Fermat’s Last Theorem. Mazur proved, in a letter to Mestre dated August 16, 1985, the following result: suppose moreover that p > 2, that ρ is modular of some type (N, 2, 1)_{\overline{ℚ}_{p}}, that l is a prime not congruent to 1 mod p, that l divides N but l^{2} does not, that ρ is unramified at l if l ≠ p and that ρ is finite at p if l = p; then ρ is modular of type (N/l, 2, 1)_{\overline{ℚ}_{p}}. In 1987, Ribet removed the condition “l ≢ 1 mod p” from Mazur’s result, under the assumption that p does not divide N (see [33]). These two results together imply already that Fermat’s Last Theorem is a consequence of the Shimura–Taniyama–Weil conjecture. Together with Mazur [29], Ribet extended his result to the case where p divides N, but where p^{2} does not.

Langlands, Deligne and Carayol have proved [5] that for f a newform the conductor of the system of l-adic representations associated to f is equal to the level of f. From this it follows easily that for f an eigenform in some M^{0}(N, k, ε)_{\overline{ℚ}_{p}}, and ρ_{f} the mod p Galois representation that it gives, N(ρ_{f}) divides N. Carayol [4] and Livné [26] classified, independently, in terms of the admissible irreducible representation of GL_{2}(ℚ_{l}) associated to a newform f in some M^{0}(N, k, ε)_{\overline{ℚ}_{p}}, the cases where the l-adic valuations of N(ρ_{f}) and N are different (here ρ_{f} denotes the mod p representation associated to f, ρ_{f} is supposed to be irreducible and l is a prime different from p). Carayol [4] showed that if ρ is modular of type (N, k, ε)_{\overline{ℚ}_{p}} and not induced from ℚ(\sqrt{−1}) (resp., ℚ(\sqrt{−3})) if p = 2 (resp., p = 3), then it is modular of type (N, k, ε′)_{\overline{ℚ}_{p}} for all ε′ whose mod p reduction equals that of ε and such that ε′(−1) = (−1)^{k} (this last condition is implied by the first if p ≠ 2). This result shows that in order to prove Proposition 1.11 and Theorem 1.12 one need not pay attention anymore to the character, so we will drop it from the notation in what follows.

Suppose now that ρ is modular of some type (N, k)_{\overline{ℚ}_{p}}. Then one wants to prove that ρ is modular of type (N(ρ), k). In [4] Carayol reduces the proof of this, for p ≥ 5 and k ≥ 2, to the following two statements:

(A) There exists a prime number q not dividing Nl and a newform of type (N′q, k, ε′)_{\overline{ℚ}_{p}} with N′ dividing N and ε′ trivial mod q, whose associated mod p Galois representation is isomorphic to ρ.

(B) If l ≠ p divides N, l^{2} does not divide N and l does not divide N(ρ) (i.e., ρ is unramified at l), then ρ is modular of type (N/l, k)_{\overline{ℚ}_{p}}.

The first of these two statements is used to switch in certain cases from modular curves to Shimura curves associated to indefinite quaternion algebras over ℚ, via the Jacquet–Langlands correspondence. The main part of Ribet’s article [32] is about establishing some geometric integral version of this correspondence in the case of weight two and trivial character. Statement (A) for weight two and trivial character was proved first by Ribet in [34] and more generally by Diamond for 2 ≤ k ≤ p + 1 and arbitrary character in [13]. Note that statement (B) for weight two, trivial character and p^{2} not dividing N, is the result of Mazur and Ribet above. A crucial point in their proof is that ρ has multiplicity one in the p-torsion of the jacobian J_{0}(N), in some sense (see Section 3.3). A semi-simplicity result in [2] made it possible for Ribet to prove statement (B) for weight two and trivial character, but without the condition that p^{2} does not divide N (see [35] and [32]). In [15] it was shown, using work of Gross [17] and of Coleman and Voloch [7], that one can always adapt the weight, in the following sense: if p ≠ 2 and ρ is modular of some type (N, k, ε)_{\overline{𝔽}_{p}} with N prime to p, then ρ is modular of type (N, k(ρ), ε)_{\overline{𝔽}_{p}}. The definition of k(ρ) makes it clear that the mechanism behind the proof of this result was known to Serre; in Section 4 we will discuss a part of it. This mechanism includes the fact that if ρ is modular of some type (N, k, ε)_{\overline{𝔽}_{p}}, with N prime to p, then for some integer a, ρ ⊗ χ_{p}^{a} is modular of type (N, k′, ε)_{\overline{𝔽}_{p}} with 2 ≤ k′ ≤ p + 1, and ρ is modular of type (Np^{2}, 2)_{\overline{ℚ}_{p}}. It follows from this that in order to prove statement (B), one may assume that the weight is two. This is used in [32] to show statement (B) for p ≥ 3 and ρ with det(ρ) = χ_{p}; Ribet also remarks that he expects his proof to extend without difficulty to det(ρ) arbitrary. Finally, Diamond [12] proved statement (B) for p ≥ 3, following [32]. Another proof of statement (B), not using the reduction to weight two, but extending the arguments of [33] to weights k between 2 and p + 1, was suggested by Jordan and Livné in [21]. The multiplicity one result needed for that is proved in [16].

2 The cases we need

Special cases of Theorem 1.12 are used at three different places in the proof of the Shimura– Taniyama–Weil conjecture and of Fermat’s Last Theorem. First of all, Ribet’s proof that the Shimura–Taniyama–Weil conjecture implies Fermat’s Last Theorem is a special case of Theorem 1.12. We briefly recall the situation. One supposes that Fermat’s Last Theorem is not true. Then there exist a prime p > 3 and non-zero integers a, b and c that are pairwise relatively prime and satisfy a^{p} + b^{p} + c^{p} = 0. This leads, via a construction of Hellegouarch (see [19] and [20]), to a semi-stable elliptic curve E over ℚ that is usually called the Frey curve associated to (a^{p}, b^{p}, c^{p}). Let ρ_{p} be the representation G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) given by the p-torsion of E. It follows from Mazur’s work on isogenies between elliptic curves over ℚ (see [27] and [28]) that ρ_{p} is irreducible. Moreover, E has the miraculous property that ρ_{p} is unramified away from 2p and that its ramification at 2 and p is very well-behaved: one has N(ρ_{p}) = 2, k(ρ_{p}) = 2 and ε(ρ_{p}) = 1; see [37, §4]. The conductor N of E is the product of all primes dividing abc; note that it is square free. If E is modular, i.e., if the Shimura–Taniyama–Weil conjecture is true for E, then ρ_{p} is modular of type (N, 2, 1)_{ℚ_{p}}. So in this case it suffices to have Theorem 1.12 for ρ_{p} that are modular of some type (N, 2, 1)_{\overline{ℚ}_{p}} with N square free and with p > 3.

Let us now look where Theorem 1.12 intervenes in Wiles’s proof of the Shimura–Taniyama– Weil conjecture for semi-stable elliptic curves. One has the following proposition, that was given by Oesterlé in the seminar held in Paris on the work of Wiles.

2.1 Proposition. Let E be a semi-stable elliptic curve over ℚ, p be a prime number and ρ_{p}: G_{ℚ} → GL_{2}(𝔽_{p}) the representation obtained by choosing some basis of the p-torsion of E. Then either ρ_{p} is surjective, or it is reducible and its semi-simplification is isomorphic to the direct sum 1 ⊕ χ_{p} of the trivial character with the cyclotomic character.

Proof. If p > 5 or if the image G of ρ_{p} has order divisible by p, this is part of Proposition 21 of [39]. Suppose that we are not in this case. We use the same arguments as Serre does. If ρ_{p} is reducible Serre’s argument shows that its semi-simplification is 1 ⊕ χ_{p}. So we suppose that ρ_{p} is irreducible and we have to derive a contradiction. The morphism det: G → 𝔽_{p}^{∗} is surjective because det^{∘}ρ_{p} = χ_{p}. We have p = 2, 3 or 5 and ρ_{p} is unramified outside p. The image under ρ_{p} of inertia at p is a non-split Cartan subgroup (i.e., cyclic of order p^{2} − 1) or “half” of a split Cartan (i.e., a conjugate of the subgroup of elements of the form ({∗ \atop 0} {0 \atop 1})). The first case arises if and only if E has good, supersingular reduction at p.

Suppose that p = 2. Then G has order 3 since ρ_{p} is irreducible. It follows that ℚ has a Galois extension of degree 3 which is ramified only at 2; there is no such extension. This finishes the proof for p = 2.

Suppose that p = 3. Then G has order dividing 16. The absolute irreducibility of ρ_{p} implies that G has order 8 or 16. But then G has a quotient of order 4, leading to a Galois extension of ℚ of degree 4 which is unramified outside 3; again, such an extension does not exist.

Suppose that p = 5. By §2.6 of [39], G is contained in the normalizer of a Cartan subgroup or the image \overline{G} of G in GL_{2}(𝔽_{p})/𝔽_{p}^{∗} is isomorphic to A_{4}, S_{4} or A_{5}. Since G is of order prime to 5, we cannot have \overline{G} isomorphic to A_{5}. If \overline{G} is isomorphic to A_{4} or S_{4} we get a Galois extension of degree 3 of ℚ or of ℚ(\sqrt{5}) which is unramified outside 5, and such extensions do not exist. Hence G is contained in the normalizer N of a Cartan subgroup C. The absolute irreducibility of ρ_{p} implies that G is not contained in C. This gives us a degree 2 extension K of ℚ. We claim that K is unramified, which gives us the desired contradiction. The extension K is clearly unramified outside p, so we have to show that ρ_{p}(I_{p}) is contained in C. If ρ_{p}(I_{p}) is a non-split Cartan subgroup then this Cartan subgroup has to be C (look at the action on the projective line), hence we get what we need. If ρ_{p}(I_{p}) is “half” of a split Cartan subgroup, then again ρ_{p}(I_{p}) has to be contained in C because of its action on the projective line. □

Let E be a semi-stable elliptic curve over ℚ. If for a prime number p the representation ρ_{p} is not surjective, then E admits a ℚ-rational isogeny of degree p. Hence, if ρ_{3} and ρ_{5} are both not sujective, E admits a ℚ-rational isogeny of degree 15, hence defines a ℚ-rational non-cuspidal point of X_{0}(15). According to the tables in [1], X_{0}(15) is an elliptic curve with exactly eight rational points, four of which are cusps. The four non-cuspidal points correspond (up to twist) to elliptic curves of conductor 50 of which one can easily see that no twist of them is semi-stable. So we conclude that at least one among ρ_{3} and ρ_{5} is surjective.

Suppose that ρ_{3} is surjective. Results of Langlands [25] and Tunnell [43] show that ρ_{3} is modular of type (3^{m}N(ρ_{3})^{2}, 1)_{ℚ_{3}}, for some m ≥ 0. This will be discussed in Section 4. In that same section, we will show that ρ_{3} is then modular of type (N(ρ_{3}), 2, 1)_{\overline{ℚ}_{3}} if ρ_{3} is finite at 3, and of type (3N(ρ_{3}), 2, 1)_{\overline{ℚ}_{3}} if ρ_{3} is not finite at 3. We will not need the results on companion forms, and we can use an older version of Carayol’s Lemma dating back to Mazur’s [28]. Carayol’s reductions in [4] imply that ρ_{3} is modular of type (N(ρ_{3}), 2, 1)_{\overline{ℚ}_{3}} if ρ_{3} is finite at 3, and of type (3N(ρ_{3}), 2, 1)_{\overline{ℚ}_{3}} if ρ_{3} is not finite at 3. Starting at this point, Wiles and Taylor prove (see [44] and [42]) that all deformations of ρ_{3} that are ramified at only finitely many primes, that are “semi-stable” at 3 and whose determinant is the cyclotomic character are modular. Hence E is modular. (Let us note that the restriction of ρ_{3} to Gal(\overline{ℚ}/ℚ(\sqrt{−3})) is absolutely irreducible since its image is SL_{2}(𝔽_{3}).) So in this case we don’t need Ribet’s part of the proof of Theorem 1.12.

Suppose that ρ_{3} is not surjective. Then ρ_{5} is surjective. In this case, Wiles [44] shows that there exists an elliptic curve E′ over ℚ such that its representation ρ′_{3} is irreducible and with ρ′_{5} isomorphic to ρ_{5} (this follows from the fact that X(5)\overline{ℚ} is a disjoint union of four projective lines). The curve E′ is semi-stable since this can be read off from ρ′_{5}, even at the prime 5 itself. The previous argument shows that E′ is modular, hence ρ_{5} is modular of type (N′, 2, 1)_{ℚ_{5}}, where N′ is the conductor of E′. Note that N′ is square free. Ribet’s part of the proof of Theorem 1.12 shows that ρ_{5} is modular of type (N(ρ_{5}), 2, 1)_{\overline{ℚ}_{5}} if ρ_{5} is finite at 5, and of type (5N(ρ_{5}), 2, 1)_{\overline{ℚ}_{5}} if ρ_{5} is not finite at 5. Wiles and Taylor ([44] and [42]) show that all deformations of ρ_{5} that are ramified at only finitely many primes, that are “semi-stable” at 5 and whose determinant is the cyclotomic character are modular. Hence E is modular.

The final conclusion of this section is the following. We need Ribet’s part of the proof of Theorem 1.12 in the case where p > 3 and ρ is modular of some type (N, 2, 1)_{\overline{ℚ}_{p}} with N square free. We also need Carayol’s reductions for p = 3 and ρ modular of type (N^{2}, 2, 1)_{\overline{ℚ}_{3}} with N square free. It should be noted that Carayol’s reductions use Mazur’s result, so in the next section we will explain that result for p ≥ 3.

3 Weight two, trivial character and square free level

The aim of this section is to describe a proof of the following theorem.

3.1 Theorem. Let p ≥ 3 be prime. Let ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) be irreducible and modular of some type (N, 2, 1)_{\overline{ℚ}_{p}} with N square free. If ρ is finite at p it is modular of type (N(ρ), 2, 1)_{\overline{ℚ}_{p}}. If ρ is not finite at p it is modular of type (pN(ρ), 2, 1)_{\overline{ℚ}_{p}}.

The proof of this theorem has two parts. The first part, due to Mazur, deals with the primes l dividing N/N(ρ) that are not congruent to 1 modulo p. The second part, due to Ribet, deals with an arbitrary l ≠ p at the cost of introducing a prime q in the level that one can get rid of by the first part. These two results will be described in the following two sections. For a detailed proof of in fact more general results we refer to [32, §6–§8]; our aim is just to give a good idea of what happens in [32].

3.2 Mazur’s result

3.2.1 Theorem. Let p ≥ 3 be prime. Let ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) be irreducible and modular of some type (N, 2, 1)_{\overline{ℚ}_{p}}. Suppose that l is a prime not congruent to 1 mod p, that l divides N but l^{2} does not, that ρ is unramified at l if l ≠ p and that ρ is finite at p if l = p. Then ρ is modular of type (N/l, 2, 1)_{\overline{ℚ}_{p}}.

The proof of this result is by contradiction: let M := N/l and suppose that ρ is not modular of type (M, 2, 1)_{\overline{ℚ}_{p}}. Let X_{0}(N) denote the model over the localization ℤ_{(l)} of ℤ at l which is constructed in [10] as a coarse moduli space (see also [24]). Let J_{0}(N)_{ℚ} be the jacobian of X_{0}(N)_{ℚ}, and let J_{0}(N) be its Néron model over ℤ_{(l)}. Then J_{0}(N) represents in fact the degree zero part of the relative Picard functor of X_{0}(N) over ℤ_{(l)} by [30]; this implies that one can describe J_{0}(N) in terms of X_{0}(N). By [10, V–VI], the fibre X_{0}(N)_{𝔽_{l}} of X_{0}(N) over 𝔽_{l} is the union of two copies of the smooth curve X_{0}(M)_{𝔽_{l}}, which intersect transversally at the supersingular points. This gives the following “devissage” of J_{0}(N)_{𝔽_{l}}. One has the connected component of the identity element J_{0}(N)_{𝔽_{l}}^{0} of J_{0}(N)_{𝔽_{l}}. The quotient J_{0}(N)_{𝔽_{l}}/J_{0}(N)_{𝔽_{l}}^{0} is a finite etale group scheme over 𝔽_{l}, which is in fact a constant group scheme that we will denote by Φ_{0}(N)_{𝔽_{l}}. By construction we have an exact sequence:

(3.2.2)  0 → J_{0}(N)_{𝔽_{l}}^{0} → J_{0}(N)_{𝔽_{l}} → Φ_{0}(N)_{𝔽_{l}} → 0.

The normalization map X_{0}(M)_{𝔽_{l}} ∐ X_{0}(M)_{𝔽_{l}} → X_{0}(N)_{𝔽_{l}} induces another short exact sequence:

(3.2.3)  0 → T_{0}(N)_{𝔽_{l}} → J_{0}(N)_{𝔽_{l}}^{0} → J_{0}(M)_{𝔽_{l}}^{2} → 0,

where T_{0}(N)_{𝔽_{l}} is a torus whose character group Hom_{\overline{𝔽}_{l}}(T_{0}(N)_{\overline{𝔽}_{l}}, G_{m}_{\overline{𝔽}_{l}}) can be identified with the group of degree zero divisors on X_{0}(M)_{\overline{𝔽}_{l}} with support in the supersingular points. Let 𝕋_{0}(N) be the Hecke algebra of level N: it is the subring of End(J_{0}(N)_{ℚ}) generated by the Hecke operators T_{n}, n ≥ 1. By the Néron property of J_{0}(N), 𝕋_{0}(N) acts on it. This action can be described explicitly by correspondences over ℤ_{(l)}. The short exact sequences (3.2.2) and (3.2.3) are compatible with the action of 𝕋_{0}(N).

Let us now look what all this has to do with our representation ρ. The fact that ρ is modular of type (N, 2, 1)_{\overline{ℚ}_{p}} means that there exist a maximal ideal m of 𝕋_{0}(N), an embedding of k := 𝕋_{0}(N)/m into \overline{𝔽}_{p}, an integer d ≥ 1 and a 2-dimensional k-vector space V with an action by G_{ℚ} with \overline{𝔽}_{p} ⊗_{k} V giving ρ, such that J_{0}(N)(\overline{ℚ})[m] is isomorphic to the direct sum V^{d} of d copies of V as k[G_{ℚ}]-modules. (Here J_{0}(N)(\overline{ℚ})[m] denotes the kernel of m, i.e., the elements x such that tx = 0 for all t in m.) The fact that J_{0}(N)(\overline{ℚ})[m] is semi-simple is proved in [2].

We will now first treat the case where l ≠ p, which is technically simpler than the case l = p. So we suppose that l ≠ p. Then ρ is unramified at l by hypothesis, and there is a unique finite etale k-vector space scheme W over ℤ_{(l)} such that V ≅ W (\overline{ℚ}) as k[G_{ℚ}]-modules. We choose an injection of V into J_{0}(N)(\overline{ℚ})[m]. This gives us an injection of W_{ℚ} into J_{0}(N)_{ℚ}. The Néron property of J_{0}(N) implies that this injection extends to a morphism W → J_{0}(N), which must be injective since J_{0}(N)[p] is etale. Consider the image of W_{𝔽_{l}} in Φ_{0}(N)_{𝔽_{l}} under (3.2.2). It was proved in [33] that the action of 𝕋_{0}(N) on Φ_{0}(N)_{𝔽_{l}} is “Eisenstein”, in the sense that for q prime to N the operator T_{q} acts as multiplication by q + 1. Since ρ is irreducible, it follows that the image of W_{𝔽_{l}} in Φ_{0}(N)_{𝔽_{l}} is zero. Hence W_{𝔽_{l}} lands in J_{0}(N)_{𝔽_{l}}^{0}. Since we suppose that ρ is not modular of type (M, 2, 1)_{\overline{ℚ}_{p}}, the image of W_{𝔽_{l}} in J_{0}(M)_{𝔽_{l}}^{2} is zero. So W_{𝔽_{l}} lands in the torus T_{0}(N)_{𝔽_{l}}. This has strong consequences for the Frobenius element ρ(Frob_{l}). Namely, Frob_{l} acts on T_{0}(N)_{𝔽_{l}}(\overline{𝔽}_{l}) simultaneously as lT_{l} and as −lw_{l}, with w_{l} the Atkin–Lehner involution of level l. This implies that ρ(Frob_{l}) is in k^{∗} and it follows that l = det(ρ(Frob_{l})) = ρ(Frob_{l}^{2}) = l^{2}. This contradicts the assumption that l is not congruent to 1 modulo p.

Let us now assume that l = p. We have the information that ρ is finite at p. Let W_{ℚ} be finite k-vector space scheme over ℚ such that W_{ℚ}(\overline{ℚ}) gives V. Then W can be extended to a finite flat k-vector space scheme over ℤ_{(p)}. Such an extension is unique by results of Raynaud in [31] (here we use that p ≠ 2). We choose an injection of V into J_{0}(N)(\overline{ℚ})[m]. This gives an injection of W_{ℚ} into J_{0}(N)_{ℚ}. Ribet showed in [33, Lemma 6.2] that this injection extends to an injection of W into J_{0}(N). Note that this is not just a consequence of the Néron property of J_{0}(N), since W is not smooth as det(ρ) = χ_{p}. Once one has this result the rest of the proof of Theorem 3.2.1 proceeds as in the case l ≠ p. So it remains to explain the proof of [33, Lemma 6.2]. We may replace ℤ_{(p)} by its completion ℤ_{p}. Then W sits in a short exact sequence:

(3.2.4)  0 → W^{0} → W → W^{et} → 0,

where W^{0} denotes the connected component of the zero section of W, and where W^{et} is the largest finite etale quotient of W. Grothendieck proved, see [18, 7, IX, §11.6], that the action of G_{ℚ_{p}} on J_{0}(N)(\overline{ℚ}_{p})/J_{0}(N)^{0}(\overline{ℤ}_{p}) is unramified. This implies that the injection of W_{ℚ_{p}}^{0} into J_{0}(N)_{ℚ_{p}} extends to a morphism of W^{0} into J_{0}(N)^{0}, which is a closed immersion by [31]. Then one can finish by applying the Néron property to the amalgamated sum (W ⊕ J_{0}(N)^{0})/W^{0} (this trick is due to Grothendieck, see [18, 7, IX, 5.9.2]).

3.3 Ribet’s result

3.3.1 Theorem. Let p ≥ 3 be prime. Let ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) be irreducible and modular of some type (N, 2, 1)_{\overline{ℚ}_{p}}. Suppose that l ≠ p is a prime, that l divides N but l^{2} does not, and that ρ is unramified at l. Then there exists a prime number q, not dividing N and congruent to −1 mod p, such that ρ is modular of type (qN/l, 2, 1)_{\overline{ℚ}_{p}}.

We define M := N/l. Let q be a prime number not dividing Np with the property that ρ(Frob_{q}) = ρ(c), with c a complex conjugation. This implies that q is congruent to −1 modulo p.

In what follows we will consider the modular curves X_{0}(Mlq)_{ℚ}, X_{0}(Ml)_{ℚ}, X_{0}(Mq)_{ℚ}, and a certain Shimura curve C_{ℚ} that we will now define (see [33, §4] for more details). Let B_{ℚ} be a quaternion algebra over ℚ of discrminant lq and let B be a maximal order in it. Then C_{ℚ} is the Shimura curve associated to B_{ℚ} of level Γ_{0}(M). More precisely, this means that C_{ℚ} is the coarse moduli scheme for objects (A/S, α, G), with S a ℚ-scheme, A/S an abelian scheme of relative dimension two, with α: B → End_{S}(A) a morphism of rings and G ⊂ A a finite flat closed subgroup scheme of rank M^{2}, which is killed by M and which is stable for the action by B. We define J_{ℚ} to be the jacobian of C_{ℚ}. From the moduli interpretation of C_{ℚ} it is clear that one can define Hecke correspondences on C_{ℚ}, inducing endomorphisms T_{n}, n ≥ 1, of J_{ℚ}. Let 𝕋 be the subring of End_{ℚ}(J_{ℚ}) generated by these T_{n}.

We have the usual pairs of degeneracy morphisms from X_{0}(Mlq)_{ℚ} to X_{0}(Ml)_{ℚ} and to X_{0}(Mq)_{ℚ}, inducing a morphism J_{0}(Mlq)_{ℚ} → J_{0}(Ml)_{ℚ}^{2} × J_{0}(Mq)_{ℚ}^{2}. The lq-new subvariety J_{0}(Mlq)_{ℚ}^{l,q−new} of J_{0}(Mlq)_{ℚ} is defined as the connected component of the identity element of the kernel of this last morphism. One knows that J_{ℚ} is isogeneous to J_{0}(Mlq)_{ℚ}^{l,q−new} (this results from trace formula calculations by Eichler, Shimizu, Jacquet–Langlands and Faltings’s isogeny theorem). Ribet has given a more precise version of this in terms of the character groups of the torus parts of the reductions mod l and q of the jacobians of the curves under consideration. For G a commutative algebraic group over a field k, let X(G) := Hom\overline{k}(T\overline{k}, G_{m}\overline{k}) be the character group of the maximal torus T\overline{k} of G\overline{k}. Then Ribet constructed a short exact sequence:

(3.3.2)  0 → X(J_{𝔽_{q}}) → X(J_{0}(Mlq)_{𝔽_{l}}) → X(J_{0}(Ml)_{𝔽_{l}}^{2}) → 0

which is Hecke-equivariant in the sense that for each n ≥ 1, the element T_{n} in 𝕋_{0}(Mlq) induces the element T_{n} of 𝕋 on X(J_{𝔽_{q}}). The induced action of T_{n} on X(J_{0}(Ml)_{𝔽_{l}}^{2}) can be described in terms of a two by two matrix with coefficients in 𝕋_{0}(Ml). Since l and q play symmetric roles, we also have the following exact sequence:

(3.3.3)  0 → X(J_{𝔽_{l}}) → X(J_{0}(Mlq)_{𝔽_{q}}) → X(J_{0}(Mq)_{𝔽_{q}}^{2}) → 0.

To construct these sequences, Ribet relies heavily on work of Cerednik, Drinfeld and Jordan– Livné concerning the q-adic uniformization of C_{ℚ_{q}}. A detailed account of this uniformization can be found in [3]. An amazing feature of these sequences is that they compare character groups of tori over fields of distinct characteristics. Since J_{𝔽_{q}}^{0} is its maximal torus, it follows from (3.3.2) that 𝕋_{0}(Mlq) acts on J_{ℚ} via a (necessarily unique) morphism of rings 𝕋_{0}(Mlq) → 𝕋 that sends T_{n} to T_{n}.

Let η_{q} be the element T_{q}^{2} − 1 of 𝕋_{0}(Mlq), and let Φ_{𝔽_{q}} be the group of connected components of J_{𝔽_{q}}. For M a finite abelian group, let M^{∗} := Hom_{ℤ}(M, ℚ/ℤ) be its Pontrjagin dual. Theorem 4.3 of [33] asserts that there is a Hecke equivariant exact sequence:

(3.3.4)  0 → K_{l} → X(J_{0}(Ml)_{𝔽_{l}}^{2})/η_{q}X(J_{0}(Ml)_{𝔽_{l}}^{2}) → Φ_{𝔽_{q}}^{∗} → C_{q} → 0,

with K_{l} and C_{q} “Eisenstein” in the sense we saw in the previous section. Likewise, one has an exact sequence:

(3.3.5)  0 → K_{q} → X(J_{0}(Mq)_{𝔽_{q}}^{2})/η_{q}X(J_{0}(Mq)_{𝔽_{q}}^{2}) → Φ_{𝔽_{l}}^{∗} → C_{l} → 0

with K_{q} and C_{l} “Eisenstein”. Since ρ is modular of type (Ml, 2, 1)_{\overline{ℚ}_{p}}, ρ arises from a maximal ideal of 𝕋_{0}(Ml), in the way we have seen in Section 3.2. It is not hard to see that then ρ also arises from a maximal ideal m of 𝕋_{0}(Mlq) (note that we do not claim that ρ arises from a newform whose level is divisible by q). More precisely, we have a maximal ideal m of 𝕋_{0}(Mlq), an embedding of k := 𝕋_{0}(Mlq)/m into \overline{𝔽}_{p}, a two-dimensional k-vector space V with an action by G_{ℚ} with \overline{𝔽}_{p} ⊗_{k} V giving ρ, such that J_{0}(Mlq)(\overline{ℚ})[m] is isomorphic to V^{λ} for some positive integer λ (this λ is called the multiplicity at m of ρ in J_{0}(Mlq)). Let μ be the multiplicity at m of ρ in J_{ℚ}: J_{ℚ}(\overline{ℚ})[m] ≅ V^{μ} (it follows again from [2] that J_{ℚ}(\overline{ℚ})[m] is semi-simple). It is clear that μ ≥ 0.

From now on we suppose that ρ is not modular of type (Mq, 2, 1)_{\overline{ℚ}_{p}}. From this assumption and the exact sequences above Ribet then derives that 2μ ≤ λ and that 2λ ≤ 2μ, which gives a contradiction because we know that λ > 0. So let us describe the arguments of Ribet. One starts by localizing the exact sequence (3.3.5) at m; this shows that Φ_{𝔽_{l},m} is zero, because m is not in the support of X(J_{0}(Mq)_{𝔽_{q}}^{2}). This implies that J_{𝔽_{l}}^{0}(\overline{𝔽}_{l})[m] is isomorphic to V^{μ} (as k-vector spaces), so that:

(3.3.6)  dim_{k}(X(J_{𝔽_{l}}) ⊗_{𝕋_{0}(Mlq)} k) = 2μ

The exact sequence (3.3.3) shows that:

(3.3.7)  dim_{k}(X(J_{0}(Mlq)_{𝔽_{q}}) ⊗_{𝕋_{0}(Mlq)} k) = dim_{k}(X(J_{𝔽_{l}}) ⊗_{𝕋_{0}(Mlq)} k).

Next we have the following exact sequence, obtained by replacing N by Mlq in (3.2.2):

(3.3.8)  0 → J_{0}(Mlq)_{𝔽_{l}}^{0} → J_{0}(Mlq)_{𝔽_{l}} → Φ_{0}(Mlq)_{𝔽_{l}} → 0.

Since Φ_{0}(Mlq)_{𝔽_{l}} is “Eisenstein”, it follows that J_{0}(Mlq)(\overline{ℚ})[m] specializes into J_{0}(Mlq)_{𝔽_{l}}^{0}(\overline{𝔽}_{l}). As in (3.2.3), the normalization of X_{0}(Mlq)_{𝔽_{l}} induces a short exact sequence:

(3.3.9)  0 → T_{0}(Mlq)_{𝔽_{l}} → J_{0}(Mlq)_{𝔽_{l}}^{0} → J_{0}(Mq)_{𝔽_{l}}^{2} → 0.

Since ρ is not modular of type (Mq, 2, 1)_{\overline{ℚ}_{p}}, it follows that:

(3.3.10)  dim_{k}(X(J_{0}(Mlq)_{𝔽_{l}}) ⊗_{𝕋_{0}(Mlq)} k) = 2λ.

Ribet shows that the Frobenius endomorphism of J_{𝔽_{q}}^{0} is equal to qT_{q}. It follows from this that ρ(Frob_{q}) acts as a scalar (i.e., an element of k) on J_{𝔽_{q}}^{0}(\overline{𝔽}_{q})[m]. But by the choice of q, ρ(Frob_{q}) is in the conjugacy class of ({1 \atop 0} {0 \atop −1}). It follows that:

(3.3.11)  dim_{k}(X(J_{𝔽_{q}}) ⊗_{𝕋_{0}(Mlq)} k) ≤ μ.

The same argument applied to the maximal torus T_{0}(Mlq)_{𝔽_{q}} in J_{0}(Mlq)_{𝔽_{q}} gives:

(3.3.12)  dim_{k}(X(J_{0}(Mlq)_{𝔽_{q}}) ⊗_{𝕋_{0}(Mlq)} k) ≤ λ.

3.3.13 Lemma. We have dim_{k}(X(J_{0}(Ml)_{𝔽_{l}}^{2}) ⊗_{𝕋_{0}(Mlq)} k) ≤ μ.

Proof. First note that η_{q} is in m. The exact sequence (3.3.4) shows that:

dim_{k}(X(J_{0}(Ml)_{𝔽_{l}}^{2}) ⊗_{𝕋_{0}(Mlq)} k) = dim_{k}(Φ_{𝔽_{q}}^{∗}) ⊗_{𝕋_{0}(Mlq)} k = dim_{k}(Φ_{𝔽_{q}}[m]).

Grothendieck’s description [18, 7, IX, §11] of Φ_{𝔽_{q}} gives an exact sequence:

(3.3.14)  0 → X(J′_{𝔽_{q}}) → X(J_{𝔽_{q}})^{∨} → Φ_{𝔽_{q}} → 0,

where for M a ℤ-module M^{∨} denotes its ℤ-dual, and where J′_{𝔽_{q}} is just J_{𝔽_{q}}, but with the dual 𝕋_{0}(Mlq)-action: t in 𝕋_{0}(Mlq) acts as t^{∗}, the dual of the endomorphism given by t (this uses the natural autoduality of jacobians). This makes the sequence (3.3.14) Hecke equivariant. Consider multiplication by p on the exact sequence (3.3.14). Applying the snake Lemma and then taking kernels for m gives an injection of Φ_{𝔽_{q}}[m] into (X(J′_{𝔽_{q}})⊗_{ℤ}𝔽_{p})[m]. Ribet shows, using more results from [18, 7, IX, §11] and a description of the action of Frob_{q} on Hom(X(J_{𝔽_{q}}), μ_{p}) and on X(J′_{𝔽_{q}}), that one has an exact sequence:

(3.3.15)  0 → Hom(X(J_{𝔽_{q}}), μ_{p})[m] → J(\overline{ℚ})[m] → (X(J′_{𝔽_{q}}) ⊗ 𝔽_{p})[m] → 0.

The middle term in this sequence has dimension 2μ by the definition of μ. The fact that ρ(Frob_{q}) has the two distinct eigenvalues 1 and −1 then implies that the other two terms both have dimension μ. □

We can now finish our description of the proof of Theorem 3.3.1. The sequence (3.3.2) gives: (3.3.16) dim_{k}(X(J_{0}(Mlq)_{𝔽_{l}}) ⊗_{𝕋_{0}(Mlq)} k) ≤ dim_{k}(X(J_{𝔽_{q}}) ⊗_{𝕋_{0}(Mlq)} k) + dim_{k}(X(J_{0}(Ml)_{𝔽_{l}}^{2}) ⊗_{𝕋_{0}(Mlq)} k).

Combining (3.3.6), (3.3.7) and (3.3.12) gives 2μ ≤ λ. On the other hand, combining (3.3.10), (3.3.16), (3.3.11) and Lemma 3.3.13 gives 2λ ≤ 2μ.

4 Dealing with the Langlands–Tunnell form

Let E be a semi-stable elliptic curve over ℚ, with ρ_{3} irreducible. It follows, Proposition 2.1, that ρ_{3}: G_{ℚ} → GL_{2}(𝔽_{3}) is surjective. The aim of this section is to show that ρ_{3} is modular of type (N(ρ_{3}), 2, 1)_{\overline{ℚ}_{3}} if ρ_{3} is finite at 3 and of type (3N(ρ_{3}), 2, 1)_{\overline{ℚ}_{3}} if ρ_{3} is not finite at 3.

4.1 The theorem of Langlands and Tunnell

The first step is to show that ρ_{3} is modular, by lifting ρ_{3} to a continuous representation of G_{ℚ} with values in GL_{2}(ℂ), i.e., a continuous representation ρ: G_{ℚ} → GL_{2}(ℂ) (for the discrete topology on ℂ) such that ρ_{3} is a reduction modulo 3 of ρ, and applying a theorem of Langlands and Tunnell to ρ. To do this, we compose ρ_{3} with a two-dimensional complex representation of GL_{2}(𝔽_{3}). The irreducible representations of the groups GL_{2}(k), where k is a finite field, are well-known, see for example [6]. If q denotes the cardinality of k, then the irreducible representations have dimensions 1, q − 1, q or q + 1. Those of dimension one factor via the determinant. Hence for q > 3 there are no faithful two-dimensional representations. In the case of interest for us, i.e., q = 3, there are three irreducible two-dimensional representations. One of them can be realized in GL_{2}(ℤ), but it is not faithful (it factors through the quotient S_{3}). The other two are Galois conjugates and can be realized in GL_{2}(ℤ[\sqrt{−2}]). Let π: GL_{2}(𝔽_{3}) → GL_{2}(ℤ[\sqrt{−2}]) be one of these, and let ρ := π^{∘}ρ_{3}: G_{ℚ} → GL_{2}(ℤ[\sqrt{−2}]). From the character table given in [6], one sees immediately that π is faithful (our π is one of the π(Λ) with Λ of order eight), and that the restriction of π to the subgroup ({∗ \atop 0} {0 \atop 1}) is the regular representation. It follows that ρ is odd and that det^{∘}ρ is the composition of χ_{3}: G_{ℚ} → 𝔽_{3}^{∗} with the embedding of 𝔽_{3}^{∗} into ℂ^{∗}. Looking again at the character table shows that reduction of ρ modulo one of the two maximal ideals containing 3 gives ρ_{3}. According to [37, §5.3] the conductor N(ρ) of ρ is 3^{m}N(ρ_{3})^{2}, for some m ≥ 0. This is explained as follows. Let l ≠ 3 be a prime. If ρ_{3} is unramified at l then ρ is unramified too. Suppose that ρ_{3} is ramified at l. Since E is semi-stable, ρ_{3}(I_{l}) is a conjugate of the subgroup ({1 \atop 0} {∗ \atop 1}). From the character table one sees that π({1 \atop 0} {1 \atop 1}) has trace −1, hence its eigenvalues are the two roots of unity of order 3. This means that the dimension of the space of I_{l}-invariants for ρ_{3} is one, and that it is zero for ρ. Hence the exponent of l in N(ρ) is twice that of l in N(ρ_{3}). It will be of use for us to determine the exponent m of 3 in N(ρ) in the case where ρ_{3}|_{I_{3}} is isomorphic to the direct sum χ_{3} ⊕ 1. In this case, in which the ramification is tame, (1.6) shows that m = 1. One can see easily that in all other cases m > 1.

A deep result of Langlands and Tunnell ([25] and [43]) says that ρ is modular of type (N(ρ), 1, det(ρ))_{ℤ[\sqrt{−2}]}, i.e., there exists a cuspidal eigenform f of level N(ρ), of weight 1 and with character det(ρ) such that ρ ≅ ρ_{f}, where ρ_{f}: G_{ℚ} → GL_{2}(ℂ) is the representation associated to f by Deligne–Serre in [11] (see Gelbart’s chapter). At this point we know that ρ_{3} arises from the modular form f.

4.2 Some purely mod 3 arguments

The second step is now to show that ρ_{3} is modular of type (N(ρ_{3})^{2}, k(ρ_{3}), 1)_{\overline{𝔽}_{3}}. First we want to see that ρ is modular of type (N(ρ_{3})^{2}, k)_{\overline{𝔽}_{3}}, for some k ≥ 1. This is folklore. Ribet has given a proof of this in [32, §2] for p > 2. We remark that there are (unpublished) conceptual short proofs, valid for all p, using either the results of Katz and Mazur in [24] on the reduction mod p of modular curves (the idea is just to restrict the form to a suitable irreducible component of this reduction), or the representation theory of GL_{2}(𝔽_{p}) on 𝔽_{p}-vector spaces and some group cohomology or etale cohomology.

We can now assume that ρ_{3} is modular of type (N(ρ_{3})^{2}, k)_{\overline{𝔽}_{3}}, for some k ≥ 1. We want to show that it is modular of type (N(ρ_{3})^{2}, k(ρ_{3}))_{\overline{𝔽}_{3}}, i.e., we want to adjust the weight. Theorem 4.5 of [15] says that this adjustment of the weight can always be done; in [15] one can find a detailed proof. We will now describe only the main features of that proof, and indicate a simplification, due to Taylor, in the case of our ρ_{3}.

The proof of Theorem 4.5 of [15] runs as follows. Let p be a prime number and ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) be irreducible, continuous and modular of some level N which is prime to p. Then one knows that there exist a in ℤ/(p − 1)ℤ and k in {2, …, p + 1} such that ρ ≅ ρ_{f} ⊗ χ_{p}^{a} for some eigenform f of type (N, k)_{\overline{𝔽}_{p}}. Since f has low weight, i.e., k ≤ p + 1, one knows that ρ_{f}|_{I_{p}} is an extension of the trivial character by χ_{p}^{k−1} if f is ordinary (i.e., T_{p}f ≠ 0) and that it is the direct sum of ψ^{k−1} and ψ′^{k−1} if f is supersingular (i.e., T_{p}f = 0); here ψ and ψ′ are the two fundamental characters of level two that occur in Definition 1.7. The first part of this result is due to Deligne (see [17] for a proof). The second part was first proved by Fontaine (see [15] for a proof). It follows that a and k are completely determined by ρ|_{I_{p}}, except if ρ|_{I_{p}} is a direct sum of two distinct powers of χ_{p} or if it is a non-split extension of χ_{p}^{a} by χ_{p}^{a+1}. In the first case there are two candidates for a; in the second case one cannot decide between the values 2 and p + 1 for k. In the second case Serre conjectured, and Mazur showed under the assumption that p > 2, that one can take k = 2 if and only if ρ ⊗ χ_{p}^{−a} is finite at p. Recall that Mazur’s argument was described in §3.2. In the first case, Serre conjectured that both candidates for a should work and called the two forms corresponding to the two values of a companions of each other. The fact that such companion forms exist was proved in [17], [7] and [16], except for some cases when p = 2. With these results, it remains to construct a form f′ of type (N, k′)_{\overline{𝔽}_{p}}, with k′ as small as possible, such that ρ ≅ ρ_{f′}. This last equality is equivalent to: a′_{l} = l^{a}a_{l} for all primes l not dividing Np, where the a′_{l} and a_{l} are the eigenvalues of f′ and f for T_{l}. There is a operation θ on modular forms over \overline{𝔽}_{p} that has the effect of the derivation qd/dq on q-expansions. This derivation θ was first constructed, more or less by hand, for forms of level one by Swinnerton-Dyer and Serre in [41], and in general, in terms of the Gauss–Manin connection, by Katz in [23]. The properties proved about θ in [23] imply that one knows exactly how to construct f′: one should take f′ := A^{−n}θ^{a}(f) with n as large as possible (here A denotes the Hasse invariant), or the same expression with f replaced by its companion if it exists. The results in [23] imply that one knows the value of n.

Let us now look at our ρ_{3}. We want to show that we do not need the results on companion forms to show that ρ_{3} is modular of type (N(ρ_{3})^{2}, k(ρ_{3})) (this is an observation of Taylor). We know that ρ_{3} or ρ_{3} ⊗ χ_{3} arises from a form of weight k with 2 ≤ k ≤ 4. If E has bad or good and ordinary reduction at 3, then ρ_{3}|_{I_{3}} is an extension of the trivial character 1 by χ_{3}; k(ρ_{3}) = 2 if ρ_{3} is finite at 3 (this is the case if and only if the exponent of 3 in the discriminant of E is a multiple of 3), otherwise k(ρ_{3}) = 4. If E has good and supersingular reduction at 3, then ρ_{3}|_{I_{3}} is the direct sum of ψ and ψ′, the two fundamental characters of level two.

Suppose first that E is good and supersingular at 3. Then ρ_{3} ⊗ χ_{3} cannot arise from a form of weight between 2 and 4, because its restriction to I_{3} is not of the right form. Hence ρ_{3} itself arises from an eigenform f of weight k with 2 ≤ k ≤ 4. Because det(ρ_{3}) = χ_{3}, we must have k = 2 or k = 4. It remains to see that we can take k = 2. So suppose that f has weight 4. We claim that then f = Ag, where A is the Hasse invariant and g an eigenform of weight two. To prove this, it suffices to show that f vanishes at the supersingular points of the modular curve on which it lives. Now if f would not vanish at all the supersingular points, then the regular differential form ω(f) on X_{0}(3N(ρ_{3})^{2})_{\overline{𝔽}_{3}} that will be constructed in the next section has a non-zero residue at some supersingular point. A study of the action of T_{3} then shows that T_{3}f = ±f, but we have T_{3}f = 0. The eigenform g has the same eigenvalues as f, hence ρ_{3} ≅ ρ_{g}.

Suppose now that E has bad or good and ordinary reduction. Then ρ_{3}|_{I_{3}} is an extension of 1 by χ_{3}. Let us first deal with the case where this extension is not split. Then ρ_{3} ⊗ χ_{3} cannot arise from a form of weight between 2 and 4, hence there is a form f of weight 2 or 4 such that ρ_{3} ≅ ρ_{f}. If k(ρ_{3}) = 4 we have k = 4 and there is nothing to prove. So suppose that k(ρ_{3}) = 2. If k = 2 there is nothing to prove, so suppose moreover that k = 4. Let ω(f) be the differential form on X_{0}(3N(ρ_{3})^{2})_{\overline{𝔽}_{3}} as constructed in the next section. Since ω(f) can be lifted this shows that ρ_{3} is modular of type (3N(ρ_{3})^{2}, 2, 1)_{\overline{ℚ}_{3}}. Since ρ_{3} is finite at 3, Mazur’s result (Theorem 3.2.1) shows that ρ_{3} is modular of type (N(ρ_{3})^{2}, 2, 1)_{\overline{ℚ}_{3}}. It remains to deal with the split case. So suppose that ρ_{3}|_{I_{3}} ≅ χ_{3} ⊕ 1. Then we have seen that the weight one form f given by Langlands and Tunnell has level 3N(ρ_{3})^{2}. Then fE_{1,χ}, with χ: 𝔽_{3}^{∗} → ℂ^{∗} non-trivial, shows that ρ_{3} is modular of type (3N(ρ_{3})^{2}, 2, 1)_{\overline{ℚ}_{3}}. Theorem 3.2.1 shows that ρ_{3} is modular of type (N(ρ_{3})^{2}, 2, 1)_{\overline{ℚ}_{3}}.

4.3 Differential forms and a version of Carayol’s lemma

In the last section we have seen that ρ_{3} is modular of type (N(ρ_{3})^{2}, k(ρ_{3}), 1)_{\overline{𝔽}_{3}}. Now we want to deduce that ρ_{3} is modular of type (N(ρ_{3})^{2}, 2, 1)_{\overline{ℚ}_{3}} if ρ_{3} is finite at 3 and of type (3N(ρ_{3})^{2}, 2, 1)_{\overline{ℚ}_{3}} otherwise. It is well known that cuspidal modular forms of a given type (N, 2, 1)_{K}, where K is a field of characteristic zero, correspond via the Kodaira–Spencer isomorphism to differential forms on the modular curve X_{0}(N)_{K}, i.e, to sections of the sheaf Ω^{1} on X_{0}(N)_{K}. This correspondence is compatible with the action of Hecke operators. We define N := N(ρ_{3})^{2} if ρ_{3} is finite at 3 and N := 3N(ρ_{3})^{2} otherwise. We will show in fact that there exists a nonzero differential form ω on X_{0}(N)_{\overline{ℚ}_{3}} which is an eigenform for the Hecke algebra and whose associated Galois representation ρ_{ω}: G_{ℚ} → GL_{2}(\overline{𝔽}_{3}) is isomorphic to ρ_{3}. We will construct this ω by first constructing it over \overline{𝔽}_{3}, which is in fact the hardest part (it involves a version of Carayol’s lemma); a standard argument from commutative algebra shows that it can be lifted to an eigenform (see for example [17, §9]).

Let us first do this construction in the case where ρ_{3} is finite at 3. So then N is prime to 3 and we have a cuspidal eigenform f of type (N, 2, 1)_{\overline{𝔽}_{3}}. We want to construct a differential form ω on X_{0}(N)_{\overline{𝔽}_{3}} which is an eigenform for all Hecke operators, with the same eigenvalues as f. The problem is that in general we cannot view f as a section of a sheaf \underline{ω}^{⊗2} on X_{0}(N)_{\overline{𝔽}_{3}} and apply a Kodaira–Spencer isomorphism, because X_{0}(N)_{\overline{𝔽}_{3}} does not carry a universal family of generalized elliptic curves with a level structure of type Γ_{0}(N). We will see that this problem is not just artificial, in the sense that for certain eigenforms of type (N, 2, 1)_{\overline{𝔽}_{3}} there does not exist a differential form as we want; it will exist for our f because ρ_{3} is not induced from G_{ℚ(\sqrt{−3})}. Let n ≥ 3 be an integer prime to 3. According to (1.2) we can view f as a section of the sheaf \underline{ω}^{⊗2} on the smooth projective curve X := \overline{M}([Γ_{1}(N), Γ(n)]_{\overline{𝔽}_{3}}) over \overline{𝔽}_{3} that carries a universal family of generalized elliptic curves with a point of order N and a trivialization of the n-torsion. Let Y denote the curve X_{0}(N)_{\overline{𝔽}_{3}}. Then Y is the quotient of X by the action of the group G := (ℤ/Nℤ)^{∗} × GL_{2}(ℤ/nℤ) acting on X, and f is a G-invariant section of \underline{ω}^{⊗2}. Let ω(f) be the global differential form on X obtained by applying the Kodaira–Spencer isomorphism to f. The functorial properties of this isomorphism imply that ω(f) is G-invariant. We would like to conclude that ω(f) is the pullback of a unique differential form, that we will also call ω(f), on Y. The uniqueness is guaranteed by the fact that the morphism π: X → Y is separable, which is implies that the pullback morphism π^{∗} on differential forms is injective. The problem is the existence. The rest of this section is motivated by [28, II, Lemma 4.4], and its proof. Let V be the biggest open part of Y over which π is etale. Note that G acts on X via its quotient \overline{G} by the subgroup generated by (−1, −1). The group \overline{G} acts faithfully, hence V is the complement in Y of the image of under π of those points of X with non-trivial stabilizer in \overline{G}. Since π: π^{−1}V → V is etale, the restriction of ω(f) to π^{−1}V is the pullback of a unique differential form ω(f) on V. So we have to show that this ω(f) has no poles in the complement of V. The information we have is that the pullback of ω(f) to X has no poles. Let y be a point of Y over which π is ramified, and let x be in π^{−1}y. Let s and t be uniformizers at x and y, respectively. Let e be the ramification index at x, i.e., t = s^{e}u with u in O_{X,x}^{∗}. Let r be the valuation v_{x}(dt) at x of dt. Since dt = s^{e−1}(e+u′s)ds, where u′ = du/ds, one has r ≥ e−1 with equality if and only if π is tamely ramified at x. We have v_{x}(ω(f)) = ev_{s}(ω(f)) + r. It follows that ω(f) is regular at y if π is tamely ramified over y, so it remains to look at those y over which π is wildly ramified. Such points y all correspond to the elliptic curve E of j-invariant zero over \overline{𝔽}_{3}. The automorphism group of E is isomorphic to 𝔽_{3}^{∗} ×GL_{2}(𝔽_{2}); the projection to 𝔽_{3}^{∗} comes from the action of Aut(E) on the tangent space at zero of E, the other projection comes from the action on the 2-torsion of E. The invariants e and r are the same at all x at which π is wildly ramified; note that we have e = 3 or e = 6. A global calculation as in [28, II, §2], using the Hurwitz formula for π, shows that we have r = e, which means that the wild ramification is of the mildest form. (In fact, this can also be derived from Table 1 of [28, II, §2].) It follows that ω(f) has a pole of order at most one at y. Now the number of y over which π is wildly ramified can be easily computed: it is 1 if N = 1, it is 0 if N is divisible by a prime number congruent to −1 modulo 3, and otherwise it is 2^{ν−1} where ν is the number primes dividing N. In the situation of [27, II, Lemma 4.4] this number is zero or one, and it follows that in fact ω(f) is regular because the sum of its residues must be zero. But if all primes dividing N are 1 modulo 3 and ν > 1, there really exist eigenforms f such that ω(f) has poles, so we have to show why our ω(f) is regular. So suppose that ω(f) is not regular. Let D be the set of y at which π is wildly ramified, viewed as an effective divisor on Y, and let V be the \overline{𝔽}_{3}-vector space of \overline{𝔽}_{3}-valued functions on D. Consider the map R: H^{0}(Y, Ω^{1}(D)) → V which sends a form to its residues at the y in D. Then the image of ω(f) is not zero. There is a natural action of the Hecke operators on V which is compatible with R. For n ≥ 1 the Hecke operator T_{n} acts on V via isogenies of degree n between elements y of D, hence by endomorphisms of degree n of E. Let σ denote one of the two automorphisms of order 3 of E. An endomorphism ϕ of E of degree n that does not commute with σ contributes zero to the action of T_{n} on V, since the correspondence inducing T_{n} is wildly ramified at ϕ. Let l be prime and congruent to −1 mod 3. Then E has no endomorphism of degree l commuting with σ, hence T_{l} acts as zero on V. It follows that T_{l}f = 0 for all such l, which contradicts that ρ_{3} is not induced from G_{ℚ(\sqrt{−3})}.

Let us now consider the case where ρ_{3} is not finite at 3. Then N = 3N′, with N′ = N(ρ_{3})^{2} prime to 3, and k(ρ_{3}) = 4. In this case X_{0}(N)_{\overline{𝔽}_{3}} has two irreducible components, both isomorphic to X_{0}(N′)_{\overline{𝔽}_{3}}, which intersect transversally at the supersingular points. The sheaf of Kähler differentials on X_{0}(N)_{ℤ_{3}} is not locally free of rank one at the double points over \overline{𝔽}_{3}, and it is better to work with the dualizing sheaf Ω on it. This sheaf Ω can be obtained as follows: let X_{0}(N)_{ℤ_{3}}^{sm} be the smooth locus of X_{0}(N)_{ℤ_{3}} (i.e., the complement of the double points), let j: X_{0}(N)_{ℤ_{3}}^{sm} → X_{0}(N)_{ℤ_{3}} be the inclusion and let Ω^{1} be the sheaf of Kähler differentials on X_{0}(N)_{ℤ_{3}}^{sm}; then Ω = j_{∗}Ω^{1}. The dualizing sheaf Ω is locally free of rank one and it is dualizing in the sense of Serre duality. For a more detailed description of Ω in the context of modular curves see [29, §§6–7], [17, §§8–9] and references therein. Let Y := X_{0}(N)_{\overline{𝔽}_{3}}, and let Y_{0} (resp., Y_{∞}) be the irreducible component of Y containing the cusp 0 (resp., ∞). The restrictions of Ω to Y_{0} and Y_{∞} are the sheafs of Kähler differentials with poles of order at most one in the supersingular points. Let n ≥ 3 be prime to 3, let X := \overline{M}([Γ_{1}(N′), Γ(n)]_{\overline{𝔽}_{3}}) and let G := (ℤ/N′ℤ)^{∗} × GL_{2}(ℤ/nℤ). Then f is a G-invariant section of \underline{ω} on X. It follows that f/A (recall that A is the Hasse invariant) is a rational section of \underline{ω}^{⊗2} with poles of at most order one in the supersingular points. Applying the Kodaira–Spencer isomorphism to f/A gives a G-invariant rational differential form ω(f) on X which has poles of order at most one in the supersingular points. We identify the quotient of X by G with Y_{∞}. Since ω(f) is G-invariant, we can view ω(f) as a rational differential form on Y_{∞}, with poles only at the supersingular points and at the points where X → Y_{∞} is ramified. A calculation as above shows that there are only poles of order at most one in the supersingular points. The proof of [17, Prop. 9.3] shows that there exists a section of Ω on X_{0}(N)_{\overline{ℤ}_{3}} which us an eigenform for the Hecke algebra and whose restriction to Y_{∞} is ω(f).

We end this section with some remarks. First of all, one can show that the Hecke action on the \overline{𝔽}_{3}-vector space V that occurs in the arguments for ρ_{3} finite at 3, is “Eisenstein”: for l prime and not dividing 3N, T_{l} acts as l + 1 on V. Hence it would have been sufficient to use that ρ_{3} is irreducible. For more general quotients of X by subgroups of the form H × GL_{2}(ℤ/nℤ) of G, the vector space V is not necessarily “Eisenstein”. This is related to results on groups of connected components of Néron models in [36].

Secondly, instead of studying in detail the wild ramification in the morphism X → Y, we could have used Theorem 3.2.1 as follows. Let q be any prime number that is congruent to −1 mod 3 and that does not divide N. Then, replacing N by qN, one gets X → Y tamely ramified, hence a differential form on X_{0}(Nq)_{\overline{𝔽}_{3}}. This shows that ρ_{3} is modular of type (Nq, 2, 1)_{\overline{ℚ}_{3}}. Then Mazur’s result shows that ρ_{3} is modular of type (N, 2, 1)_{\overline{ℚ}_{3}}. One reason to give the argument above is to illustrate the problems one gets when interpreting modular forms as differential forms.

4.4 Carayol’s reductions

At this point we know that ρ_{3} is modular of type (N(ρ_{3})^{2}, 2, 1)_{\overline{ℚ}_{3}} if ρ_{3} is finite at 3 and of type (3N(ρ_{3})^{2}, 2, 1)_{\overline{ℚ}_{3}} otherwise. We want to show that ρ_{3} is modular of type (N(ρ_{3}), 2, 1)_{\overline{ℚ}_{3}} if ρ_{3} is finite at 3 and of type (3N(ρ_{3}), 2, 1)_{\overline{ℚ}_{3}} otherwise. Before explaining how this is done, it is good to recall some results of Langlands, Deligne and Carayol (see [5]). Let p be a prime number and let f be a newform of some type (N, k, ε)_{\overline{ℚ}_{p}} with k ≥ 2. Then this gives us a representation ρ_{f}: G_{ℚ} → GL_{2}(\overline{ℚ}_{p}), determined by the property that it is unramified outside Np and that for l not dividing Np the Frobenius element ρ_{f}(Frob_{l}) has trace a_{l}(f). On the other hand, there is also a representation π_{f}: GL_{2}(ℤ̂ ⊗ ℚ) → GL(V), with V an infinite dimensional \overline{ℚ}_{p}-vector space, associated to f in the following way. Let W be the direct limit, taken over all multiples n ≥ 1 of N, of the \overline{ℚ}_{p}-vector spaces H^{0}(\overline{M}([Γ(n)]_{\overline{ℚ}_{p}}), \underline{ω}^{⊗k}). It is clear that GL_{2}(ℤ̂) acts on W, and it is not hard to see that this action extends to one of GL_{2}(ℤ̂ ⊗ ℚ). Then V is the subspace of W that is generated by the g(f), for g in GL_{2}(ℤ̂ ⊗ ℚ). One knows that V is an irreducible representation of GL_{2}(ℤ̂ ⊗ ℚ) and that V = ⊗′_{l}V_{l} is the restricted tensor product, over all primes l, of irreducible admissible representations π_{f,l}: GL_{2}(ℚ_{l}) → GL(V_{l}). The result alluded to describes, for all l ≠ p, the restriction ρ_{f,l} of ρ_{f} to some decomposition group at l in terms of π_{f,l}. It follows from this result that the conductor N(\overline{ρ}_{f}) of the reduction \overline{ρ}_{f}: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) divides N (here we suppose that \overline{ρ}_{f} is irreducible, since otherwise it is not well defined). Carayol [4] and Livné [26] have classified, in terms of the π_{f,l}, the l ≠ p dividing N/N(\overline{ρ}_{f}).

The strategy for proving that ρ_{3} is modular of type (N(ρ_{3}), 2, 1)_{\overline{ℚ}_{3}} if ρ_{3} is finite at 3 and of type (3N(ρ_{3}), 2, 1)_{\overline{ℚ}_{3}} otherwise will be the following. Suppose that we know that ρ_{3} is modular of type (N, 2, 1)_{\overline{ℚ}_{3}} for some N dividing N(ρ_{3})^{2}. Let l ≠ 3 be a prime number and suppose that l^{2} divides N. Then we want to show that ρ_{3} is modular of type (N′, 2, 1)_{\overline{ℚ}_{3}} for some N′ dividing N with l^{2} not dividing N′. Of course, this suffices.

So suppose that f is a cuspidal eigenform of type (N, 2, 1)_{\overline{ℚ}_{3}} with N dividing N(ρ_{3})^{2} and that l ≠ 3 is a prime number such that l^{2} divides N. The newform associated to f has level dividing N, so we may in fact assume that f is a newform. The classification of Carayol and Livné then says that ρ_{f,l} is of one of the following two types:

1. ρ_{f,l} is a direct sum of two ramified characters α, β: G_{ℚ_{l}} → \overline{ℚ}_{3}^{∗} whose reductions \overline{α}, \overline{β}: G_{ℚ_{l}} → \overline{𝔽}_{3}^{∗} are unramified,

2. ρ_{f,l} = Ind_{G_{K}}^{G_{ℚ_{l}}}ψ with K the unique unramified quadratic extension of ℚ_{l} and ψ: G_{K} → \overline{ℚ}_{3}^{∗} a ramified character with unramified reduction \overline{ψ}: G_{K} → \overline{𝔽}_{3}^{∗}.

Let us deal with the first case first. Let χ̃_{3}: G_{ℚ} → ℤ_{3}^{∗} be the character giving the action on all roots of unity of 3-power order. Recall that since f has trivial character and weight two we have det(ρ_{f}) = χ̃_{3}. This implies that αβ is unramified. There is a unique character ϵ of 𝔽_{l}^{∗} = Gal(ℚ(ζ_{l})/ℚ) with values in the kernel of \overline{ℤ}_{3}^{∗} → \overline{𝔽}_{3}^{∗} such that αε is unramified. Let f′ be the newform corresponding to the twist f ⊗ ε of f by ε. One way to express this is to say that a_{n}(f′) = a_{n}(f)ε(n) for all n prime to l. Another way is to say that ρ_{f′} = ρ_{f} ⊗ ε. Anyway, f′ is a newform of type (N/l, 2, ε^{2})_{\overline{ℚ}_{3}} giving ρ_{3}. Since \overline{ε} = 1 and ρ_{3} is not induced from ℚ(\sqrt{−3}) Carayol’s Lemma implies that ρ_{3} is modular of type (N/l, 2, 1)_{\overline{ℚ}_{3}}.

Let us now say something about case 2. The analog of this case with 3 replaced by a prime p ≥ 5 is treated in [4, §5], and uses the Jacquet–Langlands correspondence to switch to a certain Shimura curve. The generalization to the case p = 3, using that ρ_{3} is not induced from ℚ(\sqrt{−3}), is explained in [12, §5]. One might also say that this generalization is done in [4], if one admits that the remarks in [4, §4.4] concerning modular curves also hold for the Shimura curves used in [4, §5]. We will now sketch the argument. Let q be a prime number not dividing 3N, such that ρ_{3}(Frob_{q}) is conjugated to ρ_{3}(c), with c a complex conjugation. Then a result of Ribet (see [34]) says that there exists a newform f′ of type (N′q, 2, 1)_{\overline{ℚ}_{3}}, with N′ dividing N, such that \overline{ρ}_{f′} ≅ ρ_{3} and with ρ_{f′,q} special, i.e., ρ_{f′,q} is a non-split extension of an unramified character α with α^{2} = 1 by αχ̃_{3} (see also [44, II, Lemma 2.3]). If ρ_{f′,l} is not in case 2, then one applies the method to deal with case 1 to get rid of the l^{2} in the level and then one applies Mazur’s Theorem 3.2.1 to get rid of q. So we may assume that ρ_{f′,l} is in case 2. Let B be the quaternion algebra over ℚ with discriminant pq. By the Jacquet–Langlands correspondence and the results of [5], ρ_{f′} can be constructed from the 3-adic Tate module of the jacobian of a Shimura curve of a certain level associated to B. On this Shimura curve one has an action by the group 𝔽_{l^{2}}^{∗} × 𝔽_{q^{2}}^{∗} which is analogous to the action of the diamond operators on modular curves. A version of Carayol’s Lemma then shows that ρ_{3} actually arises from the quotient of this Shimura curve by that group. Switching back to modular curves by the Jacquet–Langlands correspondence then shows that ρ_{3} is modular of type (N′′q, 2, 1)_{\overline{ℚ}_{3}}, with N′′ dividing N/l. Then one finishes by applying Theorem 3.2.1.

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