Companion forms and weight one forms

Kevin Buzzard^{1} Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, U.S.A.

Richard Taylor ^{2} Department of Mathematics, Harvard University, Cambridge, MA 02138, U.S.A.

July 2, 1999

1KB was supported by a Miller Fellowship. He would also like to thank the Harvard University Clay Fund for funding a short visit to Harvard, during which time some of this work was completed.

2RT was partially supported by NSF Grant DMS-9702885

Introduction

In this paper we prove the following theorem. Let L/ℚ_{p} be a finite extension with ring of integers O_{L} and maximal ideal λ.

Theorem 1 Suppose that p ≥ 5. Suppose also that ρ : G_{ℚ} → GL_{2}(O_{L}) is a continuous representation satisfying the following conditions.

1. ρ ramifies at only finitely many primes.

2. (ρ mod λ) is modular and absolutely irreducible.

3. ρ is unramified at p and ρ(Frob_{p}) has eigenvalues α and β with distinct reductions modulo λ.

Then there exists a classical weight one eigenform f = ∑_{n=1}^{∞} a_{m}(f)q^{m} and an embedding of ℚ(a_{m}(f)) into L such that for almost all primes q we have a_{q}(f) = tr ρ(Frob_{q}). In particular ρ has finite image and for any embedding i : L ↪ ℂ the Artin L-function L(i ∘ ρ, s) is entire.

We have three motivations for looking at this theorem. Firstly it can be seen as partial confirmation of a conjecture of Fontaine and Mazur which asserts that if ρ : G_{ℚ} → GL_{n}(\overline{ℚ}_{p}) is a continuous irreducible representation ramified at only finitely many primes and such that the image of the inertia group at p is finite, then the image of ρ is finite (see [FM]). Our theorem verifies this conjecture in the case that n = 2, p ≥ 5 and the reduction (ρ mod λ) is modular, irreducible and takes Frob_{p} to an element with distinct eigenvalues. In our opinion the most serious assumption here is that (ρ mod λ) should be modular, but we remind the reader that if (ρ mod λ) is odd then a conjecture of Serre (see [S2]) predicts that it is necessarily modular.

Our second motivation is that our theorem gives a partial answer to the question: when is the weight one specialisation of a Hida family of ordinary eigenforms a classical weight one form. This question was raised by Mazur and Wiles (see [MW]), who showed that the answer is not “always”. We expect the answer to be whenever the image of inertia at p in the corresponding specialisation of the associated Galois representation is finite. Our theorem gives partial confirmation of this expectation (see corollary 2).

Our third motivation was that this work forms part of a program, outlined by one of us (RT) in 1992 (first to Wiles and later to other people), to prove the Artin conjecture for certain odd two dimensional icosahedral representations. If our main theorem could be extended to include the case p = 2, subject to the additional hypothesis that ρ is odd, then combining it with the main theorem of [ST] we would obtain the Artin conjecture in this case, subject to some local conditions (certainly in the case the representation is unramified at 2, 3 and 5 and the inertial degree of 2 in the A_{5} extension is > 2). The basic arguments of this paper seem to work for p = 2, but unfortunately some of our references exclude this prime. We believe the only place this poses a serious problem is the reference to the work of Wiles [W], Taylor-Wiles [TW] and Diamond [D]. Extensions of these results in a suitable way to the case p = 2 are currently being investigated by Mark Dickinson for his Harvard PhD.

Since writing the original draft of this paper we have discovered that our methods also give partial, but significant, answers both to a question of Gouvea and a question of Serre. The former concerns for which eigenvalues of U_{p} a p-adic modular form can be overconvergent (see [Go] page 53). The latter concerns when a mod p weight one eigenform of level prime to p can be lifted into characteristic zero. These applications are described at the end of sections 2 and 3 respectively.

We now outline the proof of our main theorem. We first use results of Gross [Gr] to find two weight 2 mod p eigenforms \overline{f}_{α} and \overline{f}_{β} of level Np (where p |̸N) whose associated Galois representations are (ρ mod λ). They differ however in that the eigenvalues of U_{p} on \overline{f}_{α} and \overline{f}_{β} are α and β. Then we use the results of Wiles [W] as completed by Taylor and Wiles [TW] and extended by Diamond [D] to show that there are Λ-adic eigenforms (in the sense of Hida [H2]) F_{α} and F_{β} lifting \overline{f}_{α} and \overline{f}_{β} respectively, whose associated Galois representations specialise in weight one to ρ. Specialising F_{α} and F_{β} in weight one gives two overconvergent p-adic weight one modular forms f_{α} and f_{β}. We let f = (αf_{α} −βf_{β})/(α−β) and f′ = (f_{α} −f_{β})/(α−β). There are two natural projections π_{1}, π_{2} : X_{0}(p) → X. We show that restricted to a certain rigid subspace π_{2}^{∗}f′ = π_{1}^{∗}h for some h defined on a rigid subspace of X = X_{1}(N) including the central parts of the supersingular discs. Moreover we show that h and f glue to give a weight one form defined on the whole of X. This is the weight one form we are looking for.

It is a pleasure to acknowledge the debt this work owes to the work of Gross [Gr] and to the various works of Coleman on p-adic modular forms. We would also like to thank Brian Conrad, Naomi Jochnowitz, Barry Mazur and particularly Robert Coleman for useful conversations.

1 Λ-adic companion forms.

Throughout this paper we will fix a prime p ≥ 5 and an integer N ≥ 5 which is not divisible by p.

We will let S_{k}(Γ_{1}(M)) denote the space of weight k cusp forms on Γ_{1}(M) with rational integer q-expansion at infinity. This space comes with an action of (ℤ/Mℤ)^{×} (via the diamond operators x ↦ ⟨x⟩), the Hecke operators T_{q} for any prime q |̸M, and the Hecke operators U_{q} for any prime q|M. For any prime q |̸M we define S_{q} = q^{k−1}⟨q⟩ (note that we have not followed our normal convention for the normalisation of S_{q}), and for any n ∈ ℤ_{≥1} we define T (n) by the following formulae:

• T (n_{1}n_{2}) = T (n_{1})T (n_{2}) if n_{1} and n_{2} are coprime;

• ∑_{r=0}^{∞} T (q^{r})X^{r} = (1 − T_{q}X + S_{q}X^{2})^{−1} if q |̸M;

• T (q^{r}) = U_{q}^{r} if q|M.

Let h_{k}(M) denote the ℤ-algebra generated by all these Hecke operators acting on S_{k}(Γ_{1}(M)). There is a perfect duality

\begin{matrix}S_{k}(Γ_{1}(M)) × h_{k}(M) & ⟶ & ℤ \\ (f, T) & ⟼ & a_{1}(f|T), \end{matrix}

where a_{m}(g) denotes the coefficient of q^{m} in the q-expansion at infinity of g.

Whenever it makes sense we will let e = lim_{r→∞} U_{p}^{r!}, the Hida idempotent. Following Hida we set

h^{0}(N) = lim_{←} e(h_{2}(Np^{r}) ⊗_{ℤ} ℤ_{p}).

Note that the operators T (n) for n ≥ 1 and S_{q} for q not dividing Np are compatible with the projection maps and commute with e, so give rise to operators T (n) and S_{q} in h^{0}(N). Moreover we have a natural map

⟨ ⟩ : ℤ_{p}^{×} × (ℤ/Nℤ)^{×} = lim_{←}(ℤ/Np^{r}ℤ)^{×} → h^{0}(N)^{×}.

We let Λ = ℤ_{p}[[(1 + pℤ_{p})^{×}]] and u = (1 + p) ∈ (1 + pℤ_{p})^{×} ⊂ Λ^{×}. Thus Λ ≅ ℤ_{p}[[(u−1)]]. If ζ is a p-power root of unity we let ψ_{ζ} : (1+pℤ_{p})^{×} → ℤ_{p}[ζ]^{×} denote the continuous homomorphism taking u to ζ. We also let it denote the corresponding homomorphism Λ → ℤ_{p}[ζ]. Note that h^{0}(N) is a Λ-module via (1 + pℤ_{p})^{×} → h^{0}(N)^{×} by u ↦ ⟨u⟩. Hida (see [H2]) proves that h^{0}(N) is a finite free Λ-module and that for any k ≥ 2 and ζ a primitive p^{r} root of unity

(h^{0}(N) ⊗_{ℤ_{p}} ℤ_{p}[ζ])/(u − (1 + p)^{k−2}ζ)(h^{0}(N) ⊗_{ℤ_{p}} ℤ_{p}[ζ])

is isomorphic to the maximal quotient of e(h_{k}(Np^{r+1}) ⊗_{ℤ} ℤ_{p}[ζ]) where ⟨x⟩ = ψ_{ζ}(x) for all x ∈ (ℤ/Np^{r+1}ℤ)^{×} with x ≡ 1 mod Np. This isomorphism further takes the Hecke operators T (n), T_{q}, S_{q} and U_{q} to themselves. For k = 1 it is known that in general such an isomorphism does not exist, see for instance [MW].

We set S^{0}(N) = Hom_{Λ}(h^{0}(N), Λ). Elements F ∈ S^{0}(N) are called Λ-adic forms. There is an injection S^{0}(N) ↪ Λ[[q]] which takes F to ∑_{n=1}^{∞} F (T (n))q^{n} and which is called the q-expansion map. If L is a finite field extension of the field of fractions of Λ then we call F ∈ S^{0}(N) ⊗_{Λ} L = Hom_{Λ}(h^{0}(N), L) a Λ-adic eigenform if it is a Λ-algebra homomorphism h^{0}(N) → L. We call two Λ-adic eigenforms F_{1} ∈ S^{0}(N) ⊗_{Λ} L_{1} and F_{2} ∈ S^{0}(N) ⊗_{Λ} L_{2} equivalent if there is a finite extension L_{3} of the field of fractions of Λ and embeddings of Λ-algebras L_{1} ↪ L_{3} and L_{2} ↪ L_{3} which send F_{1} and F_{2} to the same element of S^{0}(N) ⊗_{Λ} L_{3}. Equivalence classes of Λ-adic eigenforms are in bijection with height zero primes of h^{0}(N) via the map which sends an eigenform to its kernel. If k ∈ ℤ_{≥2} and ζ is a primitive p^{r}th root of unity then the map

F ⟼ ∑_{n=1}^{∞}(F (T (n)) mod (u − (1 + p)^{k−2}ζ))q^{n}

gives rise to an isomorphism of (S^{0}(N) ⊗_{ℤ_{p}} ℤ_{p}[ζ])/(u − (1 + p)^{k−2}ζ)(S^{0}(N) ⊗_{ℤ_{p}} ℤ_{p}[ζ]) with the maximal subspace of e(S_{k}(Γ_{1}(Np^{r+1})) ⊗_{ℤ} ℤ_{p}[ζ]) where ⟨x⟩ = ψ_{ζ}(x) for all x ∈ (ℤ/Np^{r+1}ℤ)^{×} with x ≡ 1 mod Np.

We will let χ^{cyclo} denote the usual character

\begin{matrix}G_{ℚ} → → Gal (ℚ(ζ_{Np^{∞}})/ℚ) ≅ ℤ_{p}^{×} × (ℤ/Nℤ)^{×} ↪ & ℤ_{p}[[ℤ_{p}^{×} × (ℤ/Nℤ)^{×}]]^{×} \\ = & Λ[(ℤ/Npℤ)^{×}]^{×}. \end{matrix}

Also note that the map q ↦ S_{q} extends (embedding the primes not dividing Np diagonally in ℤ_{p}^{×} × (ℤ/Nℤ)^{×}) to a homomorphism S : ℤ_{p}^{×} × (ℤ/Nℤ)^{×} → h^{0}(N)^{×} which sends (x_{p}, x^{p}) to x_{p}⟨(x_{p}, x^{p})⟩ for x_{p} ∈ ℤ_{p}^{×} and x^{p} ∈ (ℤ/Nℤ)^{×}. Thus S = S_{p} × S^{p} where S_{p} : ℤ_{p}^{×} → h^{0}(N)^{×} and S^{p} : (ℤ/Nℤ)^{×} → h^{0}(N)^{×}. Also S extends to a homomorphism Λ[(ℤ/Npℤ)^{×}] → h^{0}(N).

If 𝔪 is a maximal ideal of h^{0}(N) we will let k(𝔪) denote its residue field. There is a unique (up to conjugation) continuous semisimple representation \overline{ρ}_{𝔪} : G_{ℚ} → GL_{2}(k(𝔪)) such that for all primes q |̸Np the representation is unramified and tr \overline{ρ}_{𝔪}(Frob_{q}) = T_{q}. We call 𝔪 Eisenstein (respectively non-Eisenstein) if \overline{ρ}_{𝔪} is absolutely reducible (resp. absolutely irreducible). If 𝔪 is a non-Eisenstein maximal ideal of h^{0}(N) then there is a unique (up to conjugation) continuous representation

ρ_{𝔪} : G_{ℚ} ⟶ GL_{2}(h^{0}(N)_{𝔪})

such that for primes q |̸Np, ρ_{𝔪} is unramified at q and tr ρ_{𝔪}(Frob_{q}) = T_{q}. (See [H2].) Moreover det ρ_{𝔪} = S ∘ χ^{cyclo} and

ρ_{𝔪}|_{G_{p}} ∼ ( \begin{matrix}ϕ^{−1}(S ∘ χ^{cyclo}) & ∗ \\ 0 & ϕ \end{matrix} ) ,

where ϕ is the unramified character with ϕ(Frob_{p}) = U_{p}. We call a Λ-adic eigenform F ∈ S^{0}(N) ⊗_{Λ} L Eisenstein (resp. non-Eisenstein) if the unique maximal ideal of h^{0}(N) above ker F is Eisenstein (resp. non-Eisenstein). If F is non-Eisenstein we obtain a continuous representation

ρ_{F} : G_{ℚ} ⟶ GL_{2}(O_{L})

which for all primes q |̸Np is unramified and satisfies tr ρ_{F} (Frob_{q}) = F (T_{q}). Here O_{L} denotes the integral closure of Λ in L.

We call two Λ-adic eigenforms F and G ∈ S^{0}(N) ⊗ L companion forms with respect to height one primes ℘ and ℘′ of O_{L}, which do not divide p, if we can find embeddings O_{L}/℘ ↪ ℚ_{p}^{ac} and O_{L}/℘′ ↪ ℚ_{p}^{ac} such that

1. for all m ∈ ℤ_{≥1} not divisible by p we have

G(T (m)) mod ℘′ = F (T (m)S_{p}(m)^{−1}) mod ℘,

2. G(U_{p}) mod ℘′ = F (U_{p}^{−1}S^{p}(p)) mod ℘.

Note that this is equivalent to

1. (ρ_{G} mod ℘′) ∼ (ρ_{F} ⊗ (F ∘ S_{p} ∘ χ^{cyclo})^{−1} mod ℘);

2. if q|N then G(U_{q}) mod ℘′ = F (U_{q}S_{p}(q)^{−1}) mod ℘;

3. and in the case F ∘ S_{p} ≡ 1 mod ℘ we also require G(U_{p}) mod ℘′ = F (U_{p}^{−1}S^{p}(p)) mod ℘.

Note that we will only use this definition in the case where \overline{ρ}_{𝔪} is irreducible and (\overline{ρ}_{𝔪}|_{G_{p}})^{ss} does not act as scalars, where 𝔪 is the maximal ideal of h^{0}(N) containing ker F. It may be that one should modify this definition in other cases. From now on we suppose that L is fixed as a Galois extension of the fraction field of Λ large enough that all Λ-adic eigenforms of level N are equivalent to ones with values in L.

Theorem 2 Suppose F is a Λ-adic eigenform in S^{0}(N) ⊗_{Λ} L (L as above) and let 𝔪 be the maximal ideal of h^{0}(N) containing ker F. Suppose that for all q|N we have F (U_{q}) = 0. Suppose also that \overline{ρ}_{𝔪}|_{G_{ℚ(\sqrt{(−1)^{(p−1)/2}p})}} is absolutely

irreducible and (\overline{ρ}_{𝔪}|_{G_{p}})^{ss} does not consist of scalar matrices. Let ℘ be a height one prime of O_{L} not dividing p. Then the following are equivalent.

1. (ρ_{F} mod ℘)|_{G_{p}} ∼ ( \begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix} ).

2. F has a companion form G with respect to ℘ and some second height one prime ℘′ of O_{L} which does not divide p.

Before proving this theorem we make two remarks. First, the theorem is the Λ-adic analogue of a conjecture of Serre which was proved by Gross in his beautiful paper [Gr]. Second the condition that F (U_{q}) = 0 for q|N is mostly for simplicity. All we really need assume is that F (U_{q}) = 0 if q|N and \overline{ρ}_{𝔪} is unramified at q. In any case if F is an eigenform of level N and if N′ = N ∏_{q|N} q then there is an eigenform F′ of level N′ with ρ_{F′} = ρ_{F}, F′(U_{q}) = 0 for all q|N and F′(T (n)) = F (T (n)) for all n coprime to N.

We now turn to the proof of the theorem. We first show that condition two implies condition one. We know that

(ρ_{G} mod ℘′) ∼ (ρ_{F} ⊗ (F ∘ S_{p} ∘ χ^{cyclo})^{−1} mod ℘).

But we also know that

(ρ_{G} mod ℘′)|_{G_{p}} ∼ ( \begin{matrix}ϕ(F ∘ S_{p} ∘ χ^{cyclo})^{−1} & ∗ \\ 0 & ϕ^{−1}(F ∘ S^{p} ∘ χ^{cyclo}) \end{matrix} ) mod ℘

where ϕ is unramified with ϕ(Frob_{p}) = F (U_{p}). On the other hand

\begin{matrix}(ρ_{F} ⊗ (F ∘ S_{p} ∘ χ^{cyclo})^{−1} mod ℘)|_{G_{p}} \\ ∼ ( \begin{matrix}ϕ^{−1}(F ∘ S^{p} ∘ χ^{cyclo}) & ∗ \\ 0 & ϕ(F ∘ S_{p} ∘ χ^{cyclo})^{−1} \end{matrix} ) mod ℘. \end{matrix}

Thus we must have

(ρ_{G} mod ℘)|_{G_{p}} ∼ ( \begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix} ) ,

as desired.

Much deeper is the reverse implication. First note that by [Gr] (see also [CV] where the unproved hypotheses of [Gr] are removed) there is a maximal ideal 𝔫 of h^{0}(N) such that

\overline{ρ}_{𝔫} ∼ \overline{ρ}_{𝔪} ⊗ (S_{p} ∘ χ^{cyclo} mod 𝔪)^{−1};

(U_{p} mod 𝔫) = (U_{p}^{−1}S^{p}(p) mod 𝔪); and U_{q} ≡ 0 mod 𝔫 for q|N (after some embedding of k(𝔫) into k(𝔪)). Consider deformations ρ of \overline{ρ}_{𝔫} to complete noetherian local W (k(𝔪))-algebras with residue field k(𝔪) satisfying • ρ is unramified outside Np,

• ρ|_{G_{p}} ∼ ( \begin{matrix}∗ & ∗ \\ 0 & ϕ \end{matrix} ) with ϕ unramified.

Let ρ^{univ} : G_{ℚ} → GL_{2}(R) be the universal such deformation. Then by theorem 1.1 of [D] there is a surjection h^{0}(N) → → R taking

• T_{q} to tr ρ^{univ}(Frob_{q}) for q |̸Np,

• S_{q} to det ρ^{univ}(Frob_{q}) for q |̸Np,

• U_{q} to 0 for q|N,

• U_{p} to ϕ^{univ}(Frob_{p}). (Note that if q|N then q^{a+b}|N where a denotes the conductor of \overline{ρ}_{𝔫} at q and b = dim \overline{ρ}_{𝔫}^{I_{q}}. This is because U_{q} ∈ 𝔫.) However (ρ_{F} ⊗ (F ∘ S_{p} ∘ χ^{cyclo})^{−1} mod ℘) is an example of such a lifting and so we get a map h^{0}(N) → R → O_{L}/℘ taking

• T_{q} to (F ∘ S_{p} ∘ χ^{cyclo}(Frob_{q}))^{−1}tr ρ_{F} (Frob_{q}) for q |̸Np,

• S_{q} to (F ∘ S_{p} ∘ χ^{cyclo}(Frob_{q}))^{−2} det ρ_{F} (Frob_{q}) for q |̸Np,

• U_{q} to 0 for q|N,

• U_{p} to ϕ^{−1}(Frob_{p})(F ∘ S^{p} ∘ χ^{cyclo})(Frob_{p}), where ϕ is unramified and

ϕ(Frob_{p}) = F (U_{p}). We can take G to be any lifting of this map to a homomorphism h^{0}(N) → O_{L}.

The same method of proof gives the following result. Theorem 3 Let L/ℚ_{p} be a finite extension with ring of integers O_{L} and maximal ideal λ. Suppose that ρ : G_{ℚ} → GL_{2}(O_{L}) is a continuous representation satisfying

1. ρ is ramified at only finitely many primes;

2. (ρ mod λ) is irreducible;

3. (ρ mod λ) is modular;

4. ρ is unramified at p and ρ(Frob_{p}) has eigenvalues α and β with distinct reductions modulo λ.

Then there is an integer N coprime to p and two homomorphisms f_{α}, f_{β} : h^{0}(N) → O_{L} satisfying:

1. f_{α}(T_{q}) = f_{β}(T_{q}) = tr ρ(Frob_{q}) for all q |̸Np;

2. f_{α}(S_{q}) = f_{β}(S_{q}) = det ρ(Frob_{q}) for all q |̸Np;

3. f_{α}(U_{q}) = f_{β}(U_{q}) = 0 for all q|N;

4. f_{α}(U_{p}) = α and f_{β}(U_{p}) = β.

Gross’s result [Gr] (completed by [CV]) ensures the existence of f_{α} mod λ and f_{β} mod λ. Then arguing as above and invoking Diamond’s result [D] gives the theorem.

2 p-adic modular forms.

Let us first recall some facts about p-adic modular forms. Recall that X_{1}(N) has a natural model over ℤ_{p} (in fact over ℤ[1/N]). It is a natural completion of the moduli space of elliptic curves E together with an embedding i : μ_{N} ↪ E[N]. We will let X denote the pull back of X_{1}(N) to ℂ_{p}. We will let X_{0}(p)/ℂ_{p} (resp. X(p)/ℂ_{p}) denote the natural completion of the moduli space for elliptic curves E, an embedding i : μ_{N} ↪ E[N] and an isogeny E ⟶^{α} E′ of degree p (resp. and two points Q_{1}, Q_{2} in E[p] whose Weil pairing is ζ_{p} ∈ ℂ_{p} a fixed primitive p^{th} root of 1). Note that our use of the terminology X_{0}(p) and X(p) is non-standard, but not mentioning the level N structure keeps the notation less cluttered.

There are natural maps π_{3} : X(p) → X_{0}(p) (resp. π_{1} : X_{0}(p) → X, resp. π_{2} : X_{0}(p) → X) which take (E, i, Q_{1}, Q_{2}) to (E, i, E → E/⟨Q_{1}⟩) (resp. (E, i, E ⟶^{α} E′) to (E, i), resp. (E, i, E ⟶^{α} E′) to (E′, α ∘ i)). These maps are all etale away from the cusps (as long as N ≥ 5). The map π_{1} ∘ π_{3} is also Galois with group SL_{2}(𝔽_{p}), and π_{3} is thus also Galois and has group B(𝔽_{p}) ⊂ SL_{2}(𝔽_{p}), the subgroup of upper triangular matrices. We will let ω_{X} (resp. ω_{X_{0}(p)}, ω_{X(p)},…) denote the canonical extension to the cusps of the pull back by the identity section of the sheaf of relative differentials of the universal elliptic curve over the non-cuspidal locus of X (resp. X_{0}(p), X(p),…). Then π_{1}^{∗}ω_{X} = ω_{X_{0}(p)}, π_{3}^{∗}ω_{X_{0}(p)} = ω_{X(p)} and there is a natural map j = (α^{∨})^{∗} : ω_{X_{0}(p)} → π_{2}^{∗}ω_{X}. After one inverts p, j becomes an isomorphism. When it will not cause confusion, we shall refer to any of these sheaves as simply ω.

There is a natural identification of S_{k}(Γ_{1}(N))⊗_{ℤ}ℤ_{p} with Γ(X_{1}(N)/ℤ_{p}, ω^{⊗k}). There is a map from ℤ_{p}((q)) to X_{1}(N) corresponding to (𝔾_{m}/q^{ℤ}, i^{can})/ℤ_{p}((q)). Pulling back f ∈ Γ(X_{1}(N)/ℤ_{p}, ω^{⊗k}) to ℤ_{p}((q)) and dividing by the generator (dt/t)^{⊗k} of ω^{⊗k}/ℤ_{p}((q)) corresponds to taking the q-expansion of f at infinity. (Here t is the natural parameter on 𝔾_{m} and i^{can} is induced by the tautological inclusion μ_{N} ⊂ 𝔾_{m}.)

The curve X_{1}(N)/\overline{𝔽}_{p} has a finite number, s, of supersingular points. We let SS denote the union of their residue discs. This is a rigid analytic space isomorphic to the union of s open discs each of radius 1. Choose parameters T_{1}, ..., T_{s} such that the completed local ring of X_{1}(N)/ℤ_{p}^{nr} at the i^{th} supersingular point is ℤ_{p}^{nr}[[T_{i}]]. Here ℤ_{p}^{nr} denotes the Witt vectors of \overline{𝔽}_{p}. If I ⊆ [0, 1) is a closed, open, or half-open interval with endpoints in p^{ℚ}, then we define a rigid subspace SSI ⊆ SS to be the union over all supersingular residue discs of the points x such that if x is in the ith disc then |T_{i}(x)|_{p} ∈ I. Here the p-adic absolute value is normalised so that |p|_{p} = p^{−1}. So for example, SS[0, 1) = SS. If I = [0, r] (resp. [0, r), [r_{1}, r_{2}], etc.) then SSI is a disjoint union of closed discs of radius r (resp. open discs of radius r, closed annuli of radii r_{1} and r_{2}, etc.) In general, SSI will depend on the choices of the T_{i}, but if I ⊆ (1/p, 1) or [0, 1/p] ⊆ I then SSI is independent of these choices because we have an integral model over ℤ_{p}^{nr}. Note that we shall only ever consider I such that SSI is independent of choices in what follows. If r ∈ p^{ℚ} and 1 > r ≥ 1/p we let X_{>r} = X − SS[0, r]. If r ∈ p^{ℚ} and 1 ≥ r > 1/p we let X_{≥r} = X − SS[0, r).

We will let E denote the section of ω_{X}^{⊗(p−1)} over X with q-expansion at infinity

1 − (2(p − 1)/B_{p−1}) ∑_{n=1}^{∞} σ_{p−2}(n)q^{n}

where B_{p−1} denotes the Bernoulli number, and σ_{t}(n) = ∑_{0<d|n} d^{t}. Then E has a single simple zero in each connected component of SS[0, 1/p] and no other zeros on X.

The theory of the canonical subgroup (see [Ka], particularly theorem 3.10.7) provides rigid sections s_{1}, s_{2} : X_{>p^{−p/(1+p)}} → X_{0}(p) (corresponding to (E, i) ↦

(E, i, E → E/C) and (E, i) ↦ (E/C, p^{−1}i, E/C ⟶^{p} E) respectively, where

C denotes the canonical subgroup). These sections are isomorphisms onto their images. The induced map π_{1} : (s_{2}X_{>p^{−1/(1+p)}}) → X_{>p^{−p/(p+1)}} is finite and surjective of degree p. It restricts to a finite surjective degree p map s_{2}X_{≥r} → X_{≥r^{p}} for any r with 1 ≥ r > p^{−1/(p+1)}. The induced map π_{1} : s_{2}SS[p^{−1/(1+p)}, p^{−1/(1+p)}] → SS[0, p^{−p/(1+p)}] is finite surjective of degree p + 1. The induced map π_{1} : s_{2}SS(p^{−p/(1+p)}, p^{−1/(1+p)}) → SS(p^{−p/(1+p)}, p^{−1/(1+p)}) is an isomorphism. Thus π_{1}^{−1}SS[0, p^{−1/(1+p)}) = s_{2}SS(p^{−p/(1+p)}, p^{−1/(p(1+p))}). In fact, if p^{−p/(1+p)} < r < p^{−1/(1+p)} then

s_{2} : SS[1/(pr), r^{1/p}] →^{∼} π_{1}^{−1}SS[0, r].

We let S_{k}(N) denote the space of sections of ω_{X}^{⊗k} over X_{≥1} which vanish at each cusp. This is a p-adic Banach space and one of the norms in the natural equivalence class is given by ||f|| = sup_{n>0} |a_{n}(f)|, where f has q-expansion ∑_{n=1}^{∞} a_{n}(f)q^{n} at infinity. The operators T (n) for p |̸n and the diamond operators act naturally on these spaces. One can also define an operator U_{p} satisfying a_{n}(f|U_{p}) = a_{np}(f). Hida shows that his idempotent e is defined on this space and that eS_{k}(N) is finite dimensional, U_{p} is an isomorphism on eS_{k}(N) and that U_{p} is topologically nilpotent on (1 − e)S_{k}(N). (For this and the references to Hida in the next paragraph see [H1] and [Go].)

We also let S_{k}^{>r}(N) (resp. S_{k}^{≥r}(N)) denote the space of sections of ω_{X}^{⊗k} over X_{>r} (resp. X_{≥r}). These spaces are preserved by the Hecke operators T (n) for p |̸n and by the diamond operators. For 1 ≥ r > p^{−p/(1+p)} we define a continuous operator V : S_{k}^{≥r}(N) → S_{k}^{≥r^{1/p}}(N) as the composite

Γ(X_{≥r}, ω_{X}^{⊗k}) ⟶^{π_{1}^{∗}} Γ(s_{2}(X_{≥r^{1/p}}), ω_{X}^{⊗k}) ⟶^{p^{−k}(s_{2}^{∗}∘j)} Γ(X_{≥r^{1/p}}, ω_{X}^{⊗k}).

We have that

f|V = ∑_{n=1}^{∞} a_{n}(f)q^{np}.

For 1 ≥ r > p^{−p/(1+p)}, the Hecke operator U_{p} gives a continuous map U_{p} : S_{k}^{≥r^{1/p}}(N) → S_{k}^{≥r}(N). For k > 1 this is corollary II.3.7 of [Go]. As Coleman explained to us, the case k ≤ 1 can be reduced to the case k > 1 because the map U_{p} is the composite:

S_{k}^{≥r^{1/p}}(N) ⟶^{(E^{w}|V)} S_{k+w(p−1)}^{r^{1/p}}(N) ⟶^{U_{p}} S_{k+w(p−1)}^{≥r}(N) ⟶^{E^{−w}} S_{k}^{≥r}(N).

Let S^{0}(N)^{(k)} denote the set of F ∈ S^{0}(N) such that F |S_{p}(x) = x^{k−1}F for x ∈ μ_{p−1} ⊂ ℤ_{p}^{×}. Then Hida showed that

(S^{0}(N)^{(k)}/(u − (1 + p)^{k−2})S^{0}(N)^{(k)}) ⊗ ℂ_{p} →^{∼} eS_{k}(N)

via a map taking F to ∑_{n=1}^{∞}(F (T (n)) mod (u − (1 + p)^{k−2}))q^{n}. If k ≥ 2 then Hida also showed that

eS_{k}(N) = e(S_{k}(Γ_{1}(N) ∩ Γ_{0}(p)) ⊗ ℂ_{p})

and hence we may deduce that

eS_{k}(N) ⊂ S_{k}^{>p^{−p/(p+1)}}(N).

The following lemma seems to be well known to experts, but for lack of a reference we sketch a proof.

Lemma 1 eS_{k}(N) ⊂ S_{k}^{>p^{−p/(p+1)}}(N).

Let 1 > r > p^{−p/(1+p)}. The main point is to see that U_{p} : S_{k}^{≥r}(N) → S_{k}^{≥r}(N) is completely continuous. This follows because S_{k}^{≥r}(N) ↪ S_{k}^{≥r^{1/p}}(N) is completely continuous being the composite

S_{k}^{≥r}(N) ⟶^{E^{w}} S_{k+(p−1)w}^{≥r}(N) ↪ S_{k+w(p−1)}^{≥r^{1/p}}(N) ⟶^{E^{−w}} S_{k}^{≥r^{1/p}}(N)

where the middle map is known to be completely continuous for w sufficiently large (see [Go] corollary I.2.9). Moreover all eigenvalues of U_{p} are integral, because the same is true in S_{k}(N) as follows from the q-expansion. Thus by Serre’s theory [S1] (particularly propositions 7 and 12) we may write S_{k}^{≥r}(N) = e′S_{k}^{≥r}(N)⊕(1−e′)S_{k}^{≥r}(N), where e′ is an idempotent commuting with U_{p},where e′S_{k}^{≥r}(N) is finite dimensional and is spanned by generalised eigenvectors of U_{p} with unit eigenvalues, and where U_{p} is topologically nilpotent on (1 − e′)S_{k}^{≥r}(N). Thus if f ∈ S_{k}^{≥r}(N) we see that e′f = lim_{r→∞} U_{p}^{r!}f and so e′ = e|_{S_{k}^{≥r}(N)}.

Let f ∈ eS_{k}(N). We can find a sequence f_{n} ∈ eS_{k+p^{n}(p−1)}(N) such that f_{n} → f (in terms of their q-expansions at infinity). (For example f_{n} = e(fE^{p^{n+1}}).) Then by Hida’s result, for n sufficiently large f_{n} ∈ S_{k+p^{n}(p−1)}^{≥r}(N). We consider e(f_{n}/E^{p^{n+1}}) ∈ eS_{k}^{≥r}(N). We see that e(f_{n}/E^{p^{n+1}}) → f in eS_{k}(N), but as each e(f_{n}/E^{p^{n+1}}) lies in the finite dimensional subspace eS_{k}^{≥r}(N) ⊂ S_{k}^{≥r}(N) we also see that f ∈ eS_{k}^{≥r}(N), as desired.

We will now prove our key technical result.

Theorem 4 Suppose k ∈ ℤ, N ∈ ℤ_{≥5} and p ≥ 5 is a prime not dividing N. Suppose α and β are distinct non-zero elements of ℂ_{p}. Suppose that f_{α}, f_{β} ∈ S_{k}^{>t}(N) for some t < 1 are eigenvectors for U_{p} with eigenvalues α and β. Suppose finally that all positive integers n not divisible by p we have

a_{n}(f_{α}) = a_{n}(f_{β}).

Then f = (αf_{α} − βf_{β})/(α − β) is classical, i.e. lies in S_{k}(Γ_{1}(N)) ⊗ ℂ_{p}.

We first note that we may take t = p^{−p/(1+p)} (because α and β are nonzero). Set f′ = (f_{α} − f_{β})/(α − β) so that f′ = f|V ∈ S_{k}^{>p^{−1/(1+p)}}(N).

Choose r, r′ ∈ p^{ℚ} with

p^{−p/(1+p)} < r′ < r < p^{−1/(1+p)}.

Let S denote (π_{1} ∘ π_{3})^{−1}SS[0, r] ⊂ X(p). Define a section g of ω_{X(p)}^{⊗k} over S by g = π_{3}^{∗} ∘ j^{−1} ∘ π_{2}^{∗}(p^{k}f′). We will show below that g is invariant for the action of SL_{2}(𝔽_{p}). Because S → SS[0, r] is finite it will then follow that g = (π_{3} ∘ π_{1})^{∗}(h) for some section h of ω_{X} on SS[0, r]. On the other hand on s_{2}(SS(p^{−1/(1+p)}, r^{1/p}]) we have that π_{2}^{∗}(p^{k}f′) = j ∘ π_{1}^{∗}(f) and hence on π_{3}^{−1}s_{2}(SS(p^{−1/(1+p)}, r^{1/p}]) we have g = (π_{3} ∘ π_{1})^{∗}(f). Thus h|_{SS(p^{−p/(1+p)},r]} = f. Then we glue the sections h on SS[0, r] and f on X_{≥r′} to give the desired section f ∈ S_{k}(Γ_{1}(N)) ⊗_{ℤ} ℂ_{p} (by rigid GAGA, [Ko]).

We now turn to the proof that g is invariant under SL_{2}(𝔽_{p}). If C is a connected component of S let G_{C} denote the subgroup of SL_{2}(𝔽_{p}) stabilising C. Note that B(𝔽_{p}) acts transitively on the connected components of S lying over any given connected component of SS[0, r] and so SL_{2}(𝔽_{p}) = B(𝔽_{p}).G_{C}. On the other hand B(𝔽_{p}) fixes g. Thus it will suffice to show that G_{C} fixes g|_{C} for any connected component C. Let σ ∈ G_{C} and let C′ = π_{3}^{−1}s_{2}(SS(p^{−1/(1+p)}, r^{1/p}])∩σ^{−1}π_{3}^{−1}s_{2}(SS(p^{−1/(1+p)}, r^{1/p}]). On C′ we have σ^{∗}(g) = σ^{∗}π_{3}^{∗}j^{−1}π_{2}^{∗}(p^{k}f′) = σ^{∗}π_{3}^{∗}π_{1}^{∗}(f) = π_{3}^{∗}π_{1}^{∗}(f) = π_{3}^{∗}j^{−1}π_{2}^{∗}(p^{k}f′) = g. Thus it suffices to show that C′ is infinite. Suppose that π_{3} : C → π_{1}^{−1}SS[0, r] has degree d. Then π_{1}π_{3} : π_{3}^{−1}s_{2}(SS(p^{−1/(1+p)}, r^{1/p}]) → SS(p^{−p/(1+p)}, r] is of degree dp; as is π_{1}π_{3} : σ^{−1}π_{3}^{−1}s_{2}(SS(p^{−1/(1+p)}, r^{1/p}]) → SS(p^{−p/(1+p)}, r]. Hence, since the degree of C → SS[0, r] is d(p + 1), it follows be an elementary argument that the fibres of π_{1}π_{3} : C′ → SS(p^{−p/(1+p)}, r] have cardinality at least d(p−1) and C′ surjects onto SS(p^{−p/(1+p)}, r]. The theorem follows.

Before proceeding to the proof of our main theorem we note that this theorem gives a partial answer to a question of Gouvea. We will call two normalised eigenforms f_{1}, f_{2} ∈ S_{k}(N) equivalent away from p if a_{n}(f_{1}) = a_{n}(f_{2}) for all n not divisible by p. Gouvea notes ([Go], section II.3.3) that if f is any normalised eigenform in S_{k}(N) and if α ∈ ℂ_{p} satisfies |α|_{p} < 1 then there is an eigenform f_{α} ∈ S_{k}(N) equivalent away from p to f with f_{α}|U_{p} = αf_{α}. Gouvea asks how many of these eigenforms f_{α} can be overconvergent, i.e. lie in S_{k}^{>r}(N) for some r < 1. Our theorem implies the following result.

Corollary 1 With the notation as above, and for fixed f, f_{α} can be overconvergent for at most 2 non-zero values of α. If f_{α} and f_{β} are overconvergent for two distinct and non-zero α and β then (αf_{α}−βf_{β})/(α−β) ∈ S_{k}(Γ_{1}(N))⊗ℂ_{p}.

In particular k ≥ 1; αβ = p^{k−1}χ(p) where f|⟨p⟩ = χ(p)f; and the Galois representation ρ_{f} associated to f is crystalline at p.

3 Weight one forms

Putting the results of the last two sections together we are now in a position to conclude our main theorem. Let L/ℚ_{p} be a finite extension with ring of integers O_{L} and maximal ideal λ.

Theorem 5 Suppose that ρ : G_{ℚ} → GL_{2}(O_{L}) is a continuous representation satisfying the following conditions.

1. ρ ramifies at only finitely many primes.

2. (ρ mod λ) is modular and irreducible.

3. ρ is unramified at p and ρ(Frob_{p}) has eigenvalues α and β with distinct reductions modulo λ.

Then there exists an integer N coprime to p and an eigenform f ∈ S_{1}(Γ_{1}(N))⊗_{ℤ} O_{L} such that for almost all primes q we have a_{q}(f) = tr ρ(Frob_{q}). In particular ρ has finite image and for any embedding i : L ↪ ℂ the Artin L-function L(i ∘ ρ, s) is entire.

Combining theorem 3 and lemma 1 we can find such an integer N and two sections f_{α} and f_{β} in eS_{1}^{>−p/(p+1)}(N) which are eigenvectors for the Hecke operators T (n) for p |̸n and for U_{p} and have the following eigenvalues.

• f_{α}|T_{q} = (tr ρ(Frob_{q}))f_{α} and f_{β}|T_{q} = (tr ρ(Frob_{q}))f_{β} if q |̸Np.

• f_{α}|⟨q⟩ = (det ρ(Frob_{q}))f_{α} and f_{β}|⟨q⟩ = (det ρ(Frob_{q}))f_{β} if q |̸Np.

• f_{α}|U_{q} = 0 and f_{β}|U_{q} = 0 if q|N.

• f_{α}|U_{p} = αf_{α} and f_{β}|U_{p} = βf_{β}.

Then by theorem 4 we see that

f = (αf_{α} − βf_{β})/(α − β) ∈ S_{1}(Γ_{1}(N)) ⊗_{ℤ} O_{L}

is the desired form.

Corollary 2 Let L be a finite extension of the fraction field of Λ and let ℘ be a height one prime of O_{L} above ((1 + p)u − 1). Suppose that F ∈ S^{0}(N) ⊗ L is a non-Eisenstein Λ-adic eigenform for which F |⟨x⟩ = x^{−1}F for x ∈ μ_{p−1} ⊂ ℤ_{p}^{×}. Suppose also that (ρ_{F} mod ℘) is unramified at p and the eigenvalues of (ρ_{F} mod ℘)(Frob_{p}) are distinct modulo the maximal ideal of O_{L}/℘. Then ∑_{n=1}^{∞}(F (T (n)) mod ℘)q^{n} ∈ S_{1}(Γ_{1}(Np)) ⊗_{ℤ} (O_{L}/℘).

Gross has pointed out to us the following consequence of our main theorem. It partially answers a question posed to Gross by Serre.

Corollary 3 Suppose that p > 5 is a prime and N ≥ 5 is an integer not divisible by p. Suppose that \overline{f} ∈ H^{0}(X_{1}(N) ×_{ℤ_{p}} \overline{𝔽}_{p}, ω) is a normalised eigenform such that \overline{f}|U_{r} = 0 for all primes r|N and such that X^{2} − a_{p}(\overline{f})X + χ(p) has distinct roots, where \overline{f}|⟨p⟩ = χ(p)\overline{f}. Suppose also that the Galois representation \overline{ρ}\overline{f} : G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) associated to \overline{f} is irreducible. Then \overline{f} is in the image of

S_{1}(Γ_{1}(N)) ⊗_{ℤ} \overline{𝔽}_{p} ↪ H^{0}(X_{1}(N) ×_{ℤ_{p}} \overline{𝔽}_{p}, ω)

if and only if p does not divide the order of \overline{ρ}\overline{f}(G_{ℚ}).

(Before explaining the proof we remark that the condition that \overline{f}|U_{r} = 0 for r|N can almost certainly be suppressed, but we have not been through the slightly tedious details. We also remark that we could prove a similar theorem when p = 5, but that in that case the conclusion would have to be slightly modified. We leave the details to the reader.)

First suppose that \overline{f} is in the image of S_{1}(Γ_{1}(N)) ⊗_{ℤ} \overline{𝔽}_{p} ↪ H^{0}(X_{1}(N) ×_{ℤ_{p}} \overline{𝔽}_{p}, ω). Then we can find a finite extension L/ℚ_{p} and a normalised eigenform f ∈ S_{1}(Γ_{1}(N)) ⊗ O_{L} which reduces to \overline{f} (by the going up theorem applied to the Hecke algebra). The Galois representation ρ_{f} : G_{ℚ} → GL_{2}(O_{L}) associated to f by Deligne and Serre [DS] has finite image and reduces to \overline{ρ}\overline{f}. By the classification of finite subgroups of GL_{2}(ℂ) we see that the image of ρ_{f} can not have P SL_{2}(𝔽_{p}) as a subquotient (as p > 5), and hence the image of \overline{ρ}\overline{f} can not contain any conjugate of SL_{2}(𝔽_{p}). The classification of finite subgroups of GL_{2}(\overline{𝔽}_{p}) then shows that either \overline{ρ}\overline{f} is reducible or p does not divide the order of \overline{ρ}\overline{f}(G_{ℚ}).

Conversely suppose that p does not divide the order of \overline{ρ}\overline{f}(G_{ℚ}). Then a simple cohomological calculation shows that we can find a finite extension L/ℚ_{p} and a lifting ρ : G_{ℚ} → GL_{2}(O_{L}) of \overline{ρ}\overline{f} such that the image of ρ maps isomorphically to the image of \overline{ρ}\overline{f}. (If \overline{ρ}\overline{f} is defined over 𝔽_{q} then any L containing the Witt vectors of 𝔽_{q} will do.) By proposition 2.7 of [E], \overline{ρ}\overline{f} is unramified at p and \overline{ρ}(Frob_{p}) has two distinct eigenvalues, namely the two roots of X^{2} − a_{p}(\overline{f})X + χ(p). Thus ρ is also unramified at p, and so by our main theorem we can find a normalised eigenform f ∈ S_{1}(Γ_{1}(M))⊗O_{L} for some positive integer M, such that a_{r}(f) = tr ρ(Frob_{f}) for all but finitely many primes r. By the theory of newforms we may in fact take M equal to the conductor of ρ and ensure that a_{r}(f) = tr ρ^{I_{r}}(Frob_{r}) for all primes r. Alternatively we may take M to be the product of the conductor of ρ and the product over all primes r|N of r^{dim ρ^{Ir}}; and ensure that a_{r}(f) = tr ρ^{I_{r}}(Frob_{r}) if r |̸N, while a_{r}(f) = 0 if r|N. On the other hand because f|U_{r} = 0 if r|N we have that the product of

the conductor of \overline{ρ}\overline{f} and ∏_{r|N} r^{dim \overline{ρ}Ir ; \overline{f}} divides N. Also ρ and \overline{ρ}\overline{f} have the same conductor (as p does not divide the order of ρ(G_{ℚ}) and ρ is unramified at p). Thus this last form f has level dividing N and reduces modulo p to \overline{f}.

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