On Galois Representations of Weight One

Gabor Wiese

Received: September 13, 2013 Revised: November 22, 2013

Communicated by Don Blasius

Abstract. A two-dimensional Galois representation into the Hecke algebra of Katz modular forms of weight one over a finite field of characteristic p is constructed and is shown to be unramified at p in most cases.

2010 Mathematics Subject Classification: Primary 11F80M Secondary 11F33, 11F25 Keywords and Phrases: Galois representations, modular forms of weight one, Hecke algebras

1 Introduction

Let g = ∑_{n=1}^{∞} a_{n}q^{n} be a holomorphic cuspidal Hecke eigenform of weight 1 on Γ_{1}(N) with Dirichlet character ϵ. Deligne and Serre [DS74] constructed a Galois representation

ρ_{g} : Gal(\overline{ℚ}/ℚ) → GL_{2}(ℂ),

which is irreducible, unramified outside N and characterised by ρ_{g}(Frob_{ℓ}) having characteristic polynomial X^{2} − a_{ℓ}X + ϵ(ℓ) for all primes ℓ ∤ N. Here, and throughout, Frob_{ℓ} denotes an arithmetic Frobenius element. Let now p ∤ N be a prime number. Reducing ρ_{g} modulo (a prime above) p and semisimplifying yields a Galois representation

\overline{ρ}_{g} : Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝔽}_{p}),

which is still unramified outside N (in particular, at p) and still satisfies the respective formula for the characteristic polynomials at all unramified primes. In fact, \overline{ρ}_{g} only depends on the reduction of (the coefficients of) g modulo (a prime above) p.

In this article, we shall work more generally and study normalised cuspidal Katz eigenforms g over \overline{𝔽}_{p} of weight 1 on Γ_{1}(N) with Dirichlet character ϵ and q-expansion (at ∞) ∑_{n=1}^{∞} a_{n}q^{n} for p ∤ N (see Section 2). Unlike when the weight is at least 2, not all such g can be obtained by reducing holomorphic weight 1 forms. The first such nonliftable example was found by Mestre (see Appendix A of [Edi06]). Nevertheless, g also has an attached Galois representation \overline{ρ}_{g} which is unramified outside Np and such that the characteristic polynomials at unramified primes look as before. In their study of companion forms, Gross, Coleman and Voloch proved that \overline{ρ}_{g} is also unramified at p in almost all cases.

Proposition 1.1 (Gross, Coleman, Voloch). If p = 2, assume that a_{p}^{2} ≠ 4ϵ(p) (i.e. a_{2} ≠ 0). Then the residual representation \overline{ρ}_{g} is unramified at p.

Proof. The case p > 2 is treated by [CV92] without any assumption on a_{p}.

[Gro90] proves the result for all p (i.e. including p = 2), but with the extra condition, subject to some unchecked compatibilities, which have now been settled by Bryden Cais in [Cai07], Chapter 10.

In this article we give a somewhat different proof and remove the condition in the case p = 2. The main objective, however, is to extend the representation \overline{ρ}_{g} to a 2-dimensional Galois representation with coefficients in the weight 1 Hecke algebra and to show, in most cases, that it is also unramified outside N, in particular at p, with the natural characteristic polynomials at all unramified primes. We now introduce the notation necessary to state the main result. Let S_{1}(Γ_{1}(N); \overline{𝔽}_{p})_{Katz} be the \overline{𝔽}_{p}-vector space of cuspidal Katz modular forms of weight 1 on Γ_{1}(N) over \overline{𝔽}_{p} and let 𝕋_{1} be the Hecke algebra acting on it, i.e. the 𝔽_{p}-subalgebra inside End_{\overline{𝔽}_{p}}(S_{1}(Γ_{1}(N); \overline{𝔽}_{p})_{Katz}) generated by all Hecke operators T_{n} for n ∈ ℕ (see also Section 2). Let 𝔪 be the maximal ideal of 𝕋_{1} defined as the kernel of the ring homomorphism 𝕋_{1} ⟶^{T_{n}↦a_{n}} \overline{𝔽}_{p}. Let 𝕋_{1,𝔪} denote the localisation of 𝕋_{1} at 𝔪. For the representation \overline{ρ}_{g} we shall also write \overline{ρ}_{𝔪}.

Theorem 1.2. Assume that \overline{ρ}_{𝔪} is irreducible and that, if \overline{ρ}_{𝔪} is unramified at p, then \overline{ρ}_{𝔪}(Frob_{p}) is not scalar. Then there is a Galois representation

ρ_{𝔪} : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝕋_{1,𝔪})

which is unramified outside N and such that for all primes ℓ ∤ N (including ℓ = p) the characteristic polynomial of ρ_{𝔪}(Frob_{ℓ}) is X^{2} − T_{ℓ}X + ⟨ℓ⟩ ∈ 𝕋_{1,𝔪}[X].

Note that we are not assuming that \overline{ρ}_{𝔪} is unramified at p, but, that this can be deduced from the theorem. This removes the condition in the case p = 2 from Proposition 1.1.

Corollary 1.3. The representation \overline{ρ}_{g} is unramified outside N and the characteristic polynomial of \overline{ρ}_{g}(Frob_{ℓ}) equals X^{2} − a_{ℓ}X + ϵ(ℓ) for all primes ℓ ∤ N, including ℓ = p.

Proof. If \overline{ρ}_{𝔪} = \overline{ρ}_{g} is reducible, then it is automatically unramified at p as it is semisimple. If \overline{ρ}_{𝔪} is irreducible, the result follows by reducing ρ_{𝔪} (from Theorem 1.2) modulo 𝔪.

The proof of Theorem 1.2 will be given in Section 4 and we will illustrate the theorem with examples in Section 5. The essential point that makes the proof work is that cusp forms of weight 1 over \overline{𝔽}_{p} sit in weight p in two different ways; on q-expansions the situation is precisely the same as in the theory of oldforms, when passing from level N to level Np. Let us call this ‘doubling’. We shall see that it leads to a ‘doubling of Hecke algebras’ and finally to a ‘doubling of Galois representations’. It is from the latter that we deduce the main statement. In the proof of Theorem 1.2 we use essentially that \overline{ρ}_{𝔪} satisfies multiplicity one (see Section 3); hence, the case when \overline{ρ}_{𝔪} is unramified at p with scalar \overline{ρ}_{𝔪}(Frob_{p}), where multiplicity one fails by Corollary 4.5 of [Wie07], has to remain open here. Since the present article was finished and first put on arXiv (arXiv:1102.2302), I made some unsuccessful efforts to remove the multiplicity one-assumption. Since then, Frank Calegari and David Geraghty released an impressive preprint [CG12], in which they manage to extend Theorem 1.2 to all cases (for odd primes p) and, moreover, achieve an R = 𝕋-theorem, using a detailed analysis of the local deformation rings. They also prove that the relevant multiplicity is 2 if it is not 1, completing the main result of [Wie07]. We finish this introduction by expressing our optimism that the methods of the present paper might generalise to some extent to Hilbert modular forms. We intend to investigate this in future work.

Acknowledgements

I would like to thank Mladen Dimitrov for clarifying discussions and Jean-Pierre Serre for comments and hints on analysing the examples. I also thank Kevin Buzzard for some comments on the first version of this article. Thanks are also due to the referee for a careful reading, pointing out a notational ambiguity and some helpful remarks. I acknowledge partial support during the writing of this work by the SFB/TRR 45 and the SPP 1489 of the Deutsche Forschungsgemeinschaft.

2 Modular forms and Hecke algebras of weight one

In this section we provide the statements on modular forms and Hecke algebras that are needed for the sequel. In particular, we deduce a ‘doubling of Hecke algebras’ from a ‘doubling of modular forms’. We shall use the following notation and assumptions throughout the article.

Notation 2.1. • Let p be a prime number and N ≥ 5 an integer not divisible by p.

• Frob_{ℓ} always denotes an arithmetic Frobenius element at ℓ.

• ζ_{n} always denotes a primitive n-th root of unity (for n ∈ ℕ).

• If R is a ring, M, N are R-modules and S ⊆ M is a subset, then we put

Hom_{R}(M, N)^{S=0} := {f ∈ Hom_{R}(M, N) | f(s) = 0 ∀ s ∈ S}.

Katz modular forms

For the treatment in this article, it is essential to dispose of the geometric definition of modular forms given by Katz. Since the tools we need are nicely exposed in [Edi06], we follow this article, and, in particular, we work with Katz modular cusp forms for the moduli problem [Γ_{1}(N)]′_{𝔽_{p}} of elliptic curves with an embedding of the group scheme μ_{N}. We use the notation S_{k}(Γ_{1}(N); 𝔽_{p})_{Katz} for these. Replacing 𝔽_{p} by a field extension 𝔽 of 𝔽_{p}, one also defines S_{k}(Γ_{1}(N); 𝔽)_{Katz}. By flatness, S_{k}(Γ_{1}(N); 𝔽)_{Katz} = 𝔽 ⊗_{𝔽_{p}} S_{k}(Γ_{1}(N); 𝔽_{p})_{Katz}. Let 𝕋_{k}(Γ_{1}(N); 𝔽_{p})_{Katz} be the 𝔽_{p}-subalgebra of End_{\overline{𝔽}_{p}}(S_{k}(Γ_{1}(N); \overline{𝔽}_{p})_{Katz}) generated by all Hecke operators T_{n} and let 𝕋′_{k}(Γ_{1}(N); 𝔽_{p})_{Katz} be its subalgebra generated only by those T_{n} with p ∤ n. Note that both contain the diamond operators due to the formula ℓ^{k−1}⟨ℓ⟩ = T_{ℓ}^{2} −T_{ℓ^{2}} for a prime ℓ. The q-expansion principle (see e.g. [DI95], Theorem 12.3.4) and the formula a_{1}(T_{n}f) = a_{n}(f) show that the pairing 𝕋_{k}(Γ_{1}(N); 𝔽_{p})_{Katz} × S_{k}(Γ_{1}(N); 𝔽_{p})_{Katz} → 𝔽_{p}, given by ⟨T, f⟩ = a_{1}(T f) is nondegenerate and, thus, provides the identification

Hom_{𝔽_{p}}(𝕋_{k}(Γ_{1}(N); 𝔽_{p})_{Katz}, 𝔽) \underrightarrow{φ↦∑_{n=1}^{∞} φ(T_{n})q^{n}} S_{k}(Γ_{1}(N); 𝔽)_{Katz}  (2.1)

for 𝔽/𝔽_{p}.

Classical modular forms

It is useful to point out the relation with classical holomorphic cusp forms, for which we use the notation S_{k}(Γ_{1}(N)) and S_{k}(Γ_{1}(N))_{cl}. The corresponding Hecke algebra 𝕋_{k}(Γ_{1}(N))_{cl} is defined as the ℤ-subalgebra of End_{ℂ}(S_{k}(Γ_{1}(N))_{cl}) generated by all Hecke operators T_{n}. By the existence of an integral structure and the q-expansion principle, the map Hom_{ℤ}(𝕋_{k}(Γ_{1}(N))_{cl}, ℂ) → S_{k}(Γ_{1}(N))_{cl} which sends a map φ to the Fourier series ∑_{n=1}^{∞} φ(T_{n})q^{n} with q = q(z) = e^{2πiz} is an isomorphism. We let S_{k}(Γ_{1}(N); R)_{cl} = Hom_{ℤ}(𝕋_{k}(Γ_{1}(N))_{cl}, R) for any ℤ-algebra R. Note S_{k}(Γ_{1}(N); ℂ)_{cl} = S_{k}(Γ_{1}(N))_{cl}. Note also that due to the freeness and the finite generation of 𝕋_{k}(Γ_{1}(N))_{cl} as a ℤ-module

R_{2} ⊗_{R_{1}} S_{k}(Γ_{1}(N); R_{1})_{cl} ≅ S_{k}(Γ_{1}(N); R_{2})_{cl}  (2.2)

for any R_{1}-algebra R_{2}. For a subring R ⊆ ℂ the R-module S_{k}(Γ_{1}(N); R)_{cl} agrees with the subset of S_{k}(Γ_{1}(N)) consisting of those forms with q-expansion having coefficients in R, as e.g. in [DI95], Section 12.3. The following proposition states that for weights at least 2, Katz cusp forms over \overline{𝔽}_{p} coincide with reductions of classical ones of the same level Γ_{1}(N).

Proposition 2.2. Let k ≥ 2. Assume N ≠ 1 or p ≥ 5.

(a) There is an isomorphism S_{k}(Γ_{1}(N); 𝔽)_{cl} ≅ S_{k}(Γ_{1}(N); 𝔽)_{Katz} which is compatible with the Hecke operators and q-expansions for any 𝔽/𝔽_{p}.

(b) The map 𝔽_{p} ⊗_{ℤ} 𝕋_{k}(Γ_{1}(N))_{cl} ⟶^{1⊗T_{n}↦T_{n}} 𝕋_{k}(Γ_{1}(N); 𝔽_{p})_{Katz} is an isomorphism of rings.

Proof. (a) By Theorem 12.3.2 of [DI95] (see also Lemma 1.9 of [Edi97] for the cases N < 5) one has

𝔽_{p} ⊗_{ℤ[ \frac{1}{N}]} S_{k}(Γ_{1}(N); ℤ[\frac{1}{N}])_{Katz} ≅ S_{k}(Γ_{1}(N); 𝔽_{p})_{Katz}

compatible with the Hecke operators. By [Edi06], 4.7, one also has

S_{k}(Γ_{1}(N); ℤ[\frac{1}{N}])_{Katz} ≅ S_{k}(Γ_{1}(N); ℤ[\frac{1}{N}])_{cl}.

Both identifications respect q-expansions. Invoking them together with Equation (2.2) gives the statement.

(b) From (a) it follows that 𝔽_{p} ⊗_{ℤ} 𝕋_{k}(Γ_{1}(N))_{cl} ⟶^{1⊗T_{n}↦T_{n}} 𝕋_{k}(Γ_{1}(N); 𝔽_{p})_{Katz} is a surjection of rings. To see it is an isomorphism it suffices to invoke Equations (2.1) and (2.2) to give:

Hom_{𝔽_{p}}(𝕋_{k}(Γ_{1}(N); 𝔽_{p})_{Katz}, 𝔽_{p}) ≅ S_{k}(Γ_{1}(N); 𝔽_{p})_{Katz} ≅^{(a)} S_{k}(Γ_{1}(N); 𝔽_{p})_{cl} ≅ Hom_{ℤ}(𝕋_{k}(Γ_{1}(N))_{cl}, 𝔽_{p}) ≅ Hom_{𝔽_{p}}(𝔽_{p} ⊗_{ℤ} 𝕋_{k}(Γ_{1}(N))_{cl}, 𝔽_{p}),

which is the map induced from 1 ⊗ T_{n} ↦ T_{n} due to the compatibility of qexpansions.

Note that the corresponding statement for weight k = 1 is false. We shall explain examples in Section 5. That failure is actually la raison d’être of this article.

Doubling of weight one forms

Towards the goal of this article, the construction and study of a Galois representation into the weight 1 Hecke algebra, it is necessary to increase the weight, since weight 1 is not a cohomological weight. The increased weight will enable us to see the Galois representation on the Jacobian of a modular curve, thus, permitting the use of geometric tools. We shall map weight 1 forms into weight p. This can be done in two different ways: multiplying by the Hasse invariant A (a modular form over 𝔽_{p} of weight p − 1 with q-expansion 1); the Frobenius F (f) = f^{p}. The former does not change the q-expansion and the latter maps ∑_{n=1}^{∞} a_{n}q^{n} to ∑_{n=1}^{∞} a_{n}q^{np}. The formula for F is clear for modular forms over 𝔽_{p}; if we work with coefficients in 𝔽/𝔽_{p}, then as in [Edi06] we take the 𝔽-linear extension of F. Note that on the level of q-expansions, the two maps A and F correspond precisely to the two degeneracy maps from level N to Np. Hence, weight one forms in weight p are very analogous to oldforms. That is the price to pay for the use of geometric tools. Let 𝔽/𝔽_{p} and consider the map

Ψ : (S_{1}(Γ_{1}(N); 𝔽)_{Katz})^{⊕2} → S_{p}(Γ_{1}(N); 𝔽)_{Katz},
(f, g) ↦ A(f) + F (g) = Af + g^{p}. (2.3)

By Proposition 4.4 of [Edi06] this is an injection. We shall write T_{p} for the Hecke operator in weight 1 and U_{p} for the one in weight p. According to Equation (4.2) of loc. cit. one has

⟨a⟩∘Ψ = Ψ∘( \begin{matrix}⟨a⟩ & 0 \\ 0 & ⟨a⟩ \end{matrix} ) , T_{n} ∘Ψ = Ψ∘( \begin{matrix}T_{n} & 0 \\ 0 & T_{n} \end{matrix} ) , U_{p} ∘Ψ = Ψ∘( \begin{matrix}T_{p} & id \\ −⟨p⟩ & 0 \end{matrix} ) (2.4)

for p ∤ n and a ∈ ℤ/Nℤ^{×}.

The weight one Hecke algebra and doubling of Hecke algebras

>From now on we use the abbreviations 𝕋_{k} and 𝕋′_{k} for 𝕋_{k}(Γ_{1}(N); 𝔽_{p})_{Katz} and 𝕋′_{k}(Γ_{1}(N); 𝔽_{p})_{Katz}, respectively. Note that Equation (2.4) implies that 𝕋′_{p} acts on S_{1}(Γ_{1}(N); 𝔽)_{Katz} via the embedding given by A. In particular, mapping T_{n} ↦ T_{n} for p ∤ n defines a ring surjection 𝕋′_{p} ↠ 𝕋′_{1}. Define

I := 𝕋′_{p} ∩ U_{p}𝕋′_{p},

where the intersection is taken inside 𝕋_{p}. We shall see in Corollary 2.6 (c) that I is the kernel of the previous surjection.

Lemma 2.3. (a) Inside S_{p}(Γ_{1}(N); 𝔽)_{Katz} the equality

Hom_{𝔽_{p}}(𝕋_{p}, 𝔽)^{𝕋′_{p}=0} = F S_{1}(Γ_{1}(N); 𝔽)_{Katz}

holds (via Equation (2.1)).

(b) Inside S_{p}(Γ_{1}(N); 𝔽)_{Katz} the equality

U_{p}Hom_{𝔽_{p}}(𝕋_{p}, 𝔽)^{𝕋′_{p}=0} = AS_{1}(Γ_{1}(N); 𝔽)_{Katz}

holds (via Equation (2.1)).

Proof. (a) As 𝕋′_{p} is generated as 𝔽_{p}-vector space by the Hecke operators T_{n} for p ∤ n, the left hand side is equal to {f ∈ S_{p}(Γ_{1}(N); 𝔽)_{Katz} | a_{n}(f) = 0 ∀n s.t. p ∤ n}. As this is precisely the kernel of Θ defined in [Kat77], part (3) of the main theorem of loc. cit. implies that it is equal to F S_{1}(Γ_{1}(N); 𝔽)_{Katz}. (b) follows from Equation (2.4), namely one has U_{p}F = A.

Proposition 2.4. (a) Let f ∈ S_{1}(Γ_{1}(N); 𝔽)_{Katz} satisfy a_{n}(f) = 0 whenever p ∤ n. Then f = 0.

(b) 𝕋′_{1} = 𝕋_{1}.

(c) 𝕋_{p} = 𝕋′_{p} + U_{p}𝕋′_{p} as 𝕋′_{p}-modules.

(d) There are T, D ∈ 𝕋′_{p} such that U_{p}^{2} − T U_{p} + D = 0 in 𝕋_{p}.

(e) I is an ideal of 𝕋_{p}.

Proof. (a) The theorem of [Kat77] already used in the previous proof gives a contradiction for f ≠ 0, since that f is in the kernel of Θ and has weight 1, so that it would have to come from an even smaller weight under Frobenius, which is impossible.

(b) If 𝕋_{1}/𝕋′_{1} were nonzero, then S_{1}(Γ_{1}(N); 𝔽_{p})^{𝕋′_{1}=0} = Hom_{𝔽_{p}}(𝕋_{1}, 𝔽_{p})^{𝕋′_{1}=0} would be nonzero and, hence, there would be a nonzero f ∈ S_{1}(Γ_{1}(N); 𝔽_{p})_{Katz} such that a_{n}(f) = 0 whenever p ∤ n. This is, however, excluded by (a).

(c) Let g ∈ S_{p}(Γ_{1}(N); 𝔽_{p})^{𝕋′_{p}+U_{p}𝕋′_{p}=0} = Hom_{𝔽_{p}}(𝕋_{p}, 𝔽_{p})^{𝕋′_{p}+U_{p}𝕋′_{p}=0}. Now g satisfies a_{n}(g) = 0 whenever p^{2} ∤ n. Thus, there is f ∈ S_{1}(Γ_{1}(N); 𝔽_{p})_{Katz} such that F f = g (again by [Kat77]) satisfying a_{n}(f) = 0 whenever p ∤ n, so that by (a) it is zero, implying the claim.

(d) This is immediate from (c).

(e) Let x ∈ 𝕋′_{p} ∩ U_{p}𝕋′_{p}. Thus, there is y ∈ 𝕋′_{p} such that x = U_{p}y. We have

U_{p}x = U_{p}^{2}y = T U_{p}y − Dy = T x − Dy ∈ 𝕋′_{p},

whence U_{p}x ∈ I.

Let 𝔪′ be a maximal ideal of 𝕋′_{p}. By 𝕋_{p,𝔪′} and 𝕋′_{p,𝔪′} we denote localisation at 𝔪′. We also use similar notation in similar circumstances.

Lemma 2.5. Let 𝔪′ be a maximal ideal of 𝕋′_{p}.

(a) The following statements are equivalent:

(i) 𝕋′_{p,𝔪′} ≠ 𝕋_{p,𝔪′}.

(ii) There is a normalised eigenform g ∈ S_{1}(Γ_{1}(N); \overline{𝔽}_{p})_{Katz} with qexpansion ∑_{n=1}^{∞} a_{n}q^{n} such that the map 𝕋′_{p} ⟶^{T_{n}↦a_{n}} \overline{𝔽}_{p} defines a ring homomorphism with kernel 𝔪′.

If the equivalent statements hold, then we say that 𝔪′ comes from weight 1.

(b) We have 𝕋_{p,𝔪′} ≅ ∏_{i=1}^{n} 𝕋_{p,𝔪̃_{i}} with n ∈ {1, 2}, where the 𝔪̃_{i} are the maximal ideals of 𝕋_{p} containing 𝔪′. If one of the 𝔪̃_{i} is ordinary (meaning that U_{p} ∈ 𝕋_{p,𝔪̃_{i}}^{×}), then all are and we say that 𝔪′ is ordinary. Suppose now that 𝔪′ comes from weight one with g ∈ S_{1}(Γ_{1}(N); \overline{𝔽}_{p})_{Katz} as in (a). Then 𝔪′ is ordinary. Furthermore, n = 2 if and only if the polynomial X^{2} − a_{p}(g)X + ϵ(p) has two distinct roots in 𝕋′_{p}/𝔪′.

Proof. (a) Statement (ai) means that U_{p} ∈ 𝕋_{p,𝔪′} is not contained in

𝕋′_{p,𝔪′}, i.e. that 𝕋_{p,𝔪′}/𝕋′_{p,𝔪′} ≠ 0 and, equivalently, S_{p}(Γ_{1}(N); 𝔽_{p})_{Katz,𝔪′}^{𝕋′_{p,𝔪′} =0} =

Hom_{𝔽_{p}}(𝕋_{p,𝔪′}, 𝔽_{p})^{𝕋′_{p,𝔪′} =0} ≠ 0. This, however, is equivalent to the existence of a cusp form f ∈ S_{p}(Γ_{1}(N); \overline{𝔽}_{p})_{Katz} such that a_{n}(f) = 0 for all p ∤ n and such that it is an eigenfunction for all T_{n} with p ∤ n with eigenvalues corresponding to 𝔪′. By the theorem of [Kat77] used already in the proof of Lemma 2.3, any such is of the form f = F h with h ∈ S_{1}(Γ_{1}(N); \overline{𝔽}_{p})_{Katz}. Hence, there is a normalised eigenform g ∈ S_{1}(Γ_{1}(N); \overline{𝔽}_{p})_{Katz} such that the a_{n}(g) are the eigenvalues of T_{n} on f for p ∤ n, whence (aii).

Conversely, the existence of g implies that S_{p}(Γ_{1}(N); 𝔽_{p})_{Katz,𝔪′}^{𝕋′_{p,𝔪′} =0} ≠ 0, as it contains F g, so that 𝕋′_{p,𝔪′} ≠ 𝕋_{p,𝔪′}.

(b) The product decomposition into its localisations is a general fact of Artinian rings. If 𝕋′_{p,𝔪′} = 𝕋_{p,𝔪′}, there is nothing to show. So we assume now that this equality does not hold. >From Proposition 2.4 (d) we have the surjection of rings

𝕋′_{p,𝔪′}[X]/(X^{2} − T X + D) ⟶^{X↦U_{p}} 𝕋_{p,𝔪′}.

Taking it mod 𝔪′ yields 𝔽[X]/(X^{2} − T̄X + D̄) on the left hand side with 𝔽 = 𝕋′_{p}/𝔪′, which has at most two local factors, depending on whether the quadratic equation has two distinct roots or a double one. Thus there can at most be two local factors on the right hand side. Modulo 𝔪′, the quadratic polynomial is in fact X^{2} − a_{p}(g)X + ϵ(p), which follows from the explicit shape of U_{p} given in Equation (2.4); see also Corollary 2.6 (a). The ordinarity is now also clear since ϵ(p) is non-zero in 𝔽.

We remark that it can happen that a_{p}(g)^{2} ≠ 4ϵ(p) (this is the so-called pdistinguished case), but that nevertheless the algebra 𝕋_{p,𝔪′} is local because the distinct roots of X^{2} − a_{p}(g)X + ϵ(p) might only lie in a quadratic extension of 𝔽. (In a previous version of this article we had referred to the case when 𝕋_{p,𝔪′} is non-local as ‘p-distinguished’, which was very misleading.) We assume henceforth that 𝔪′ comes from weight 1 and is hence ordinary. We write Ψ_{𝔪′} for the localisation of Ψ (from Equation (2.3)) at 𝔪′ and similarly I_{𝔪′} = 𝕋′_{p,𝔪′} ∩ U_{p}𝕋′_{p,𝔪′}.

Corollary 2.6. Let 𝔪′ be a maximal ideal of 𝕋′_{p} which comes from weight 1.

(a) We have (S_{1}(Γ_{1}(N); 𝔽_{p})_{Katz,𝔪′})^{⊕2} ≅^{Ψ_{𝔪′}} S_{p}(Γ_{1}(N); 𝔽_{p})_{Katz,𝔪′}^{I_{𝔪′} =0}. The operator

U_{p} acts on the left hand side as ( \begin{matrix}T_{p} & 1 \\ −⟨p⟩ & 0 \end{matrix} ). The operator T_{n} for p ∤ n acts

as ( \begin{matrix}T_{n} & 0 \\ 0 & T_{n} \end{matrix} ).

(b) There is a natural isomorphism 𝕋_{p,𝔪′}/I_{𝔪′} ≅ 𝕋_{1,𝔪′} ⊕𝕋_{1,𝔪′} of 𝕋′_{p,𝔪′}-modules. The operator U_{p} acts on the right hand side as ( \begin{matrix}T_{p} & −⟨p⟩ \\ 1 & 0 \end{matrix} ). The operator T_{n}

for p ∤ n acts as ( \begin{matrix}T_{n} & 0 \\ 0 & T_{n} \end{matrix} ).

(c) The ring homomorphism 𝕋′_{p,𝔪′} ⟶^{T_{n}↦T_{n}} 𝕋_{1,𝔪′} is surjective with kernel I_{𝔪′}.

Proof. (a) By Lemma 2.5, U_{p} is invertible and by Proposition 2.4 (e), I_{𝔪′} is an ideal of 𝕋_{p,𝔪′}. Thus, we have I_{𝔪′} = U_{p}^{−1}I_{𝔪′} = 𝕋′_{p,𝔪′} ∩ U_{p}^{−1}𝕋′_{p,𝔪′}. Since by Proposition 2.4 (c) 𝕋′_{p,𝔪′} + U_{p}𝕋′_{p,𝔪′} = 𝕋_{p,𝔪′}, we have the natural isomorphism 𝕋_{p,𝔪′}/I_{𝔪′} ≅ 𝕋_{p,𝔪′}/𝕋′_{p,𝔪′} ⊕ 𝕋_{p,𝔪′}/(U_{p}𝕋′_{p,𝔪′}) of 𝕋′-modules. It follows that

Hom_{𝔽_{p}}(𝕋_{p,𝔪′}, 𝔽)^{I_{𝔪′} =0} ≅ Hom_{𝔽_{p}}(𝕋_{p,𝔪′}, 𝔽)^{𝕋′_{p,𝔪′} =0} ⊕ Hom_{𝔽_{p}}(𝕋_{p,𝔪′}, 𝔽)^{U_{p}^{−1}𝕋′_{p,𝔪′} =0}

= Hom_{𝔽_{p}}(𝕋_{p,𝔪′}, 𝔽)^{𝕋′_{p,𝔪′} =0} ⊕ U_{p}Hom_{𝔽_{p}}(𝕋_{p,𝔪′}, 𝔽)^{𝕋′_{p,𝔪′} =0}

≅ A(S_{1}(Γ_{1}(N))_{Katz})_{𝔪′} ⊕ F (S_{1}(Γ_{1}(N))_{Katz})_{𝔪′}
= im(Ψ_{𝔪′}),

using Lemma 2.3. Moreover, Equation (2.1) gives an isomorphism

S_{p}(Γ_{1}(N); 𝔽_{p})_{Katz,𝔪′}^{I_{𝔪′} =0} ≅ Hom_{𝔽_{p}}(𝕋_{p,𝔪′}, 𝔽)^{I_{𝔪′} =0}.

The shapes of U_{p} and T_{n} are taken from Equation (2.4).

(b) Using Equation (2.1), (a) can be reformulated as an isomorphism

Hom_{𝔽_{p}}(𝕋_{p,𝔪′}/I_{𝔪′}, 𝔽_{p}) ≅ Hom_{𝔽_{p}}(𝕋_{p,𝔪′}, 𝔽_{p})^{I_{𝔪′} =0} ≅ Hom_{𝔽_{p}}(𝕋_{1,𝔪′}, 𝔽_{p})^{⊕2}.

Dualising it yields the statement, and the matrices are just the transposes of the matrices in the previous part.

(c) The algebra generated by the T_{n} with p ∤ n on the left hand side of (b) is 𝕋′_{p,𝔪′}/I_{𝔪′} and on the right hand side 𝕋′_{1,𝔪′}, which equals 𝕋_{1,𝔪′} by Proposition 2.4 (b).

We refer to (b) as ‘doubling of Hecke algebras’. Part (c) is the key for the definition of the Galois representation with coefficients in 𝕋_{1,𝔪′}.

Passage to weight two

In order to work on the Jacobian of a modular curve, we pass from weight p to weight 2, which is only necessary if p > 2. We assume this for this subsection.

Proposition 2.7. Let N ≥ 5, p ∤ N, p > 2 and 𝔪̃ be an ordinary maximal ideal of the Hecke algebra 𝕋_{p}(Γ_{1}(N); 𝔽_{p})_{Katz}. Then there is a unique maximal ideal 𝔪_{2} of 𝔽_{p} ⊗_{ℤ} 𝕋_{2}(Γ_{1}(Np))_{cl} such that T_{n} ↦ T_{n} for all n defines a ring isomorphism 𝕋_{p}(Γ_{1}(N); 𝔽_{p})_{Katz,𝔪̃} ≅ (𝔽_{p} ⊗_{ℤ} 𝕋_{2}(Γ_{1}(Np))_{cl})_{𝔪_{2}}.

Proof. This is due to Hida and follows, for instance, from combining Proposition 2.2 and [KW08], Proposition 2.3.

Remembering 𝕋_{p,𝔪′} = ∏_{i=1}^{n} 𝕋_{p,𝔪̃_{i}} (see Lemma 2.5), we obtain that after localisation at ordinary 𝔪′, the Hecke algebra 𝕋_{p,𝔪′} acts on the p-torsion of the Jacobian of X_{1}(Np). We shall henceforth use this action without specifying the isomorphism from Proposition 2.7 explicitly.

3 The Galois representation of weight one

In this section we shall construct the Galois representation ρ_{𝔪}, identify it on the Jacobian of a suitable modular curve and derive that it ‘doubles’ from the ‘doubling of Hecke algebras’. We collect some statements and pieces of notation which are in place for the whole of this section.

Notation 3.1. Next to Notation 2.1 we use the following pieces of notation and the following assumptions.

• 𝕋_{p} = 𝕋_{p}(Γ_{1}(N); 𝔽_{p})_{Katz} denotes the full Hecke algebra acting on the space S_{p}(Γ_{1}(N); 𝔽_{p})_{Katz} and 𝕋′_{p} is its subalgebra generated by those T_{n} with p ∤ n. The p-th Hecke operator is denoted U_{p}.

• 𝕋_{1} is the Hecke algebra on S_{1}(Γ_{1}(N); 𝔽_{p})_{Katz} and it is equal to 𝕋′_{1} (see Proposition 2.4). The p-th Hecke operator is denoted T_{p}.

• The map 𝕋′_{p} ⟶^{T_{n}↦T_{n}} 𝕋_{1} defines a ring surjection with kernel I = 𝕋′_{p} ∩ U_{p}𝕋′_{p} (see Corollary 2.6 (c)).

• Let 𝔪′ be a maximal ideal of 𝕋′_{p} which comes from weight 1 and corresponds to a normalised eigenform g ∈ S_{1}(Γ_{1}(N); \overline{𝔽}_{p})_{Katz} (see Lemma 2.5). Let ϵ be the Dirichlet character of g. Then 𝔪′ is ordinary (see Lemma 2.5). Denote by 𝔪 the maximal ideal of 𝕋_{1} the preimage of which in 𝕋′_{p} is 𝔪′, whence 𝕋′_{p}/𝔪′ = 𝕋_{1}/𝔪. Then 𝔪 corresponds to the Gal(\overline{𝔽}_{p}/𝔽_{p})-conjugacy class of g, i.e. it is the kernel of the ring homo-

morphism 𝕋_{1} ⟶^{T_{n}↦a_{n}(g)} \overline{𝔽}_{p}.

• Either 𝕋_{p,𝔪′} ≅ 𝕋_{p,𝔪̃_{1}} × 𝕋_{p,𝔪̃_{2}} (the non-local case), or 𝕋_{p,𝔪′} is local (see Lemma 2.5).

Existence

By work of Shimura and Deligne there is a Galois representation

\overline{ρ}_{𝔪} = \overline{ρ}_{𝔪′} : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝕋_{1}/𝔪) = GL_{2}(𝕋′_{p}/𝔪′)

characterised by the property that it is unramified outside Np and

charpoly(\overline{ρ}_{𝔪})(Frob_{ℓ}) = X^{2} − T_{ℓ}X + ⟨ℓ⟩ ∈ 𝕋_{1}/𝔪[X] ≅ 𝕋′_{p}/𝔪′[X]

for all primes ℓ ∤ Np. Under the assumption that \overline{ρ}_{𝔪} is absolutely irreducible, Carayol in [Car94], Théorème 3, shows the existence of a Galois representation

ρ_{𝔪′} : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝕋′_{p,𝔪′})

characterised by the property that it is unramified outside Np and

charpoly(ρ_{𝔪′}(Frob_{ℓ})) = X^{2} − T_{ℓ}X + ⟨ℓ⟩ ∈ 𝕋′_{p,𝔪′}[X]

for all primes ℓ ∤ Np. In fact, the reference gives a twist of this representation. Later on, we are going to be more precise about which twist it is. As a general rule, we denote \overline{ρ} a representation with coefficients in a finite field or \overline{𝔽}_{p} and ρ when the coefficients are in a Hecke algebra.

Proposition 3.2. Let 𝔪 be a maximal ideal of 𝕋_{1} such that \overline{ρ}_{𝔪} is absolutely irreducible. Then there is a Galois representation

ρ_{𝔪} : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝕋_{1,𝔪})

characterised by the property that it is unramified outside Np and

charpoly(ρ_{𝔪}(Frob_{ℓ})) = X^{2} − T_{ℓ}X + ⟨ℓ⟩ ∈ 𝕋_{1,𝔪}[X]

for all primes ℓ ∤ Np.

Proof. It suffices to compose ρ_{𝔪′} with GL_{2}(𝕋′_{p,𝔪′}) → GL_{2}(𝕋_{1,𝔪}) coming from Corollary 2.6 (c).

The p-divisible group for p = 2

Assume for the moment that p = 2. Let J be the Jacobian J_{1}(N) of X_{1}(N), which is defined over ℚ. Let G be the 𝔪′-component of the p-divisible group J[p^{∞}]_{ℚ} attached to J. A word of explanation is necessary (see also [Gro90], Section 12). The maximal ideals 𝔪̃ of 𝕋_{p} containing 𝔪′ correspond under pull-back to unique maximal ideals of the Hecke algebra ℤ_{p} ⊗_{ℤ} 𝕋_{p}(Γ_{1}(N))_{cl}, using Proposition 2.2. This Hecke algebra acts on the Tate module of J and localisation at each 𝔪̃ gives a direct factor of it. Then G is the direct product of the (at most two by Lemma 2.5) corresponding p-divisible groups. If 𝕋_{p,𝔪′} is non-local, then we shall denote by G_{1} and G_{2} the two p-divisible groups such that G = G_{1} × G_{2}.

The p-divisible group for p > 2

Assume now p > 2. We proceed very similarly to the above: Let J be the Jacobian J_{1}(Np) of X_{1}(Np), which is defined over ℚ. Let G be the 𝔪′-component of the p-divisible group J[p^{∞}]_{ℚ} attached to J. Here we use that the ideals 𝔪̃ of 𝕋_{p} containing 𝔪′ correspond to unique maximal ideals of the Hecke algebra 𝔽_{p} ⊗_{ℤ} 𝕋_{2}(Γ_{1}(Np))_{cl} by Proposition 2.7. In turn they give rise, by taking preimages, to unique maximal ideals of ℤ_{p} ⊗_{ℤ} 𝕋_{2}(Γ_{1}(Np))_{cl}. For each of these (at most two, by Lemma 2.5) maximal ideals we take the pdivisible group of the corresponding factor of the Tate module of J. Thus, if 𝕋_{p,𝔪′} is non-local, then G is of the form G_{1} × G_{2}.

Properties of the p-divisible group

We assume that G (and G_{1} and G_{2} in the non-local case) is as defined above (for either p = 2 or p > 2).

Proposition 3.3. The p-divisible group G acquires good reduction over ℤ_{p}[ζ_{p}]. Let G^{0} and G^{e} be the connected component and the étale quotient of G over ℤ_{p}[ζ_{p}], respectively.

(a) The module G^{0}[p](\overline{ℚ}_{p}) is unramified over ℚ_{p}(ζ_{p}) and there is an isomorphism G^{0}[p](\overline{ℚ}_{p}) ≅ 𝕋_{p,𝔪′} of 𝕋_{p,𝔪′}-modules, under which any arithmetic Frobenius Frob_{p} ∈ Gal(\overline{ℚ}_{p}/ℚ_{p}(ζ_{p})) acts as U_{p}^{−1}.

(b) The exact sequence 0 → G^{0} → G → G^{e} → 0 gives rise to the exact sequence

0 → 𝕋_{p,𝔪′} → G[p](\overline{ℚ}_{p}) → Hom_{𝔽_{p}}(𝕋_{p,𝔪′}, 𝔽_{p}) → 0

of 𝕋_{p,𝔪′}-modules, under the identification of (a) and its dual.

Proof. This follows immediately from applying [Wie07], Proposition 2.2, Corollary 2.3 and Theorem 3.1 for all maximal ideals 𝔪̃ ⊂ 𝕋_{p} containing 𝔪′. We stress that those results were all derived from [Gro90].

Since in this article we are using arithmetic Frobenius elements, and on modular curves (with level structure of the type μ_{N} ↪ E[N]) geometric ones are more natural, we have to twist our representations at various places. It is well-known that \overline{ρ}_{𝔪} ⊗ ϵ^{−1} is contained in the 𝔪′-kernel G[p](\overline{ℚ})[𝔪′] of G[p](\overline{ℚ}) (possibly more than once, see e.g. [Wie07], Proposition 4.1). The following theorem is the result of the work of many authors. We do not intend to give all the original references, but, content ourselves by referring to a place in the literature where the statements appear as we need them.

Theorem 3.4 (Mazur, Wiles, Gross, Ribet, Buzzard, Tilouine, Edixhoven, W.). Assume that \overline{ρ}_{𝔪} is absolutely irreducible. Then the following statements are equivalent:

(i) \overline{ρ}_{𝔪} is unramified at p and \overline{ρ}_{𝔪}(Frob_{p}) is non-scalar or \overline{ρ}_{𝔪} is ramified at p.

(ii) G[p](\overline{ℚ})[𝔪̃] ≅ \overline{ρ}_{𝔪} for any maximal ideal 𝔪̃ ⊂ 𝕋_{p} containing 𝔪′, i.e. \overline{ρ}_{𝔪} satisfies multiplicity one on the Jacobian.

(iii) 𝕋_{p,𝔪′} ≅ Hom_{𝔽_{p}}(𝕋_{p,𝔪′}, 𝔽_{p}), i.e. 𝕋_{p,𝔪′} is Gorenstein.

If the equivalent statements hold, then G[p](\overline{ℚ}) ≅ 𝕋_{p,𝔪′} ⊕ 𝕋_{p,𝔪′}.

Proof. For the implication (i) ⇒ (ii) we refer, for instance, to [KW08], Theorem 1.2. The implication (ii) ⇒ (i) is the content of [Wie07], Corollary 4.5. The equivalence of (ii) and (iii) is proved, for instance, in [KW08], Proposition 2.2 (b). That (ii) implies the final statement is, for example, proved in [KW08], Proposition 2.1.

Note that by Proposition 1.1 the case that \overline{ρ}_{𝔪} is ramified at p is known not to occur in almost all cases. We are proving in Corollary 1.3 that it actually never occurs.

The Galois representation on the Jacobian

We proceed under the following assumptions:

Assumption 3.5. We continue to use Notation 2.1 and 3.1. Moreover:

• \overline{ρ}_{𝔪} denotes the residual Galois representation introduced above. It is the residual representation attached to g. We assume that \overline{ρ}_{𝔪} is absolutely irreducible.

• Let G (and G_{1}, G_{2} in the non-local case) be the p-divisible group introduced earlier.

• We assume that \overline{ρ}_{𝔪} satisfies multiplicity one on the Jacobian so that G[p](\overline{ℚ}) ≅ 𝕋_{p,𝔪′} ⊕ 𝕋_{p,𝔪′} (see Theorem 3.4).

• Let ϵ̃ : Gal(\overline{ℚ}/ℚ) ↠ Gal(ℚ(ζ_{N})/ℚ) ≅ ℤ/(N)^{×} ⟶^{a↦⟨a⟩} 𝕋_{p}^{′×}. Note that composing ϵ̃ with 𝕋_{p,𝔪′}^{′×} → (𝕋′_{p}/𝔪′)^{×} equals ϵ, the Dirichlet character of g, seen as a character of Gal(\overline{ℚ}/ℚ).

The next proposition can be considered as a ‘doubling of Galois representations’.

Proposition 3.6. We use Assumption 3.5. Let ρ_{I_{𝔪′}} := G[p](\overline{ℚ}) ⊗_{𝕋′_{p,𝔪′}} 𝕋_{1,𝔪}. Then there is an isomorphism

ρ_{I_{𝔪′}} ≅ (ρ_{𝔪} ⊗ ϵ̃^{−1}) ⊕ (ρ_{𝔪} ⊗ ϵ̃^{−1}).

of 𝕋_{1,𝔪}[Gal(\overline{ℚ}/ℚ)]-modules.

Proof. >From [Car94], 3.3.2, and Theorem 3.4 it follows that H := G[p](\overline{ℚ}) ≅ 𝕋_{p,𝔪′} ⊕ 𝕋_{p,𝔪′} as 𝕋_{p,𝔪′}[Gal(\overline{ℚ}/ℚ)]-modules and that it is characterised by the property that it is unramified outside Np and that the characteristic polynomial of Frob_{ℓ} is X^{2} − T_{ℓ}/⟨ℓ⟩X + 1/⟨ℓ⟩ ∈ 𝕋′_{p,𝔪′}[X] for all primes ℓ ∤ Np. We recall that [Car94] works with geometric Frobenius elements, whereas we are using arithmetic ones, accounting for the differences in the formulae. By Théorème 2 of loc. cit., H is obtained by scalar extension of some 𝕋′_{p,𝔪′}[Gal(\overline{ℚ}/ℚ)]-module H′ ≅ 𝕋′_{p,𝔪′} ⊕ 𝕋′_{p,𝔪′}, i.e. H = H′ ⊗_{𝕋′_{p,𝔪′}} 𝕋_{p,𝔪′}. Note

that in this description H is a Gal(\overline{ℚ}/ℚ)-module via an action on H′ and a 𝕋_{p,𝔪′}-module via the natural action on 𝕋_{p,𝔪′} in the tensor product. Next we have the following isomorphisms of 𝕋′_{p,𝔪′}[Gal(\overline{ℚ}/ℚ)]-modules:

H ⊗_{𝕋_{p,𝔪′}} 𝕋_{p,𝔪′}/I_{𝔪′} ≅ (H′⊗_{𝕋′_{p,𝔪′}} 𝕋_{p,𝔪′})⊗_{𝕋_{p,𝔪′}} 𝕋_{p,𝔪′}/I_{𝔪′} ≅ H′⊗_{𝕋′_{p,𝔪′}} 𝕋_{p,𝔪′}/I_{𝔪′}

≅ H′ ⊗_{𝕋′_{p,𝔪′}} (𝕋_{1,𝔪} ⊕ 𝕋_{1,𝔪}) ≅ (H′ ⊗_{𝕋′_{p,𝔪′}} 𝕋_{1,𝔪}) ⊕ (H′ ⊗_{𝕋′_{p,𝔪′}} 𝕋_{1,𝔪}), where we used Corollary 2.6 (b). Note that the 𝕋′_{p,𝔪′}[Gal(\overline{ℚ}/ℚ)]-action factors through to give a 𝕋_{1,𝔪}[Gal(\overline{ℚ}/ℚ)]-action. Recall that ρ_{𝔪} : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝕋′_{p,𝔪′}/I_{𝔪′}) ≅ GL_{2}(𝕋_{1,𝔪}) is characterised by it being unramified outside Np and the characteristic polynomial of Frob_{ℓ} for ℓ ∤ Np being equal to X^{2} − T_{ℓ}X + ⟨ℓ⟩. Hence, the characteristic polynomial of (ρ_{𝔪} ⊗ ϵ̃^{−1})(Frob_{ℓ}) is X^{2} − T_{ℓ}/⟨ℓ⟩X + 1/⟨ℓ⟩. Since H′ ⊗_{𝕋′_{p,𝔪′}} 𝕋_{1,𝔪} satisfies the same properties, it agrees with ρ_{𝔪} ⊗ ϵ̃^{−1} by [Car94], Théorème 1.

4 Proof of Theorem 1.2

In this section we prove Theorem 1.2. We deal with the cases when 𝕋_{p,𝔪′} is local or not separately. Let us first remark that N ≥ 5 can be assumed without loss of generality as follows. One can increase the level at some unramified auxiliary prime q ≥ 5, q ≠ p and apply the theorem in level Nq, yielding a Galois representation with coefficients in the weight 1 Hecke algebra for Γ_{1}(Nq) which is unramified outside Nq. Since the Hecke algebra for Γ_{1}(N) is a quotient of the one for Γ_{1}(Nq), one obtains the desired Galois representation, which is hence also unramified outside Nq. Choosing a different auxiliary q, one sees that the Galois representation for Γ_{1}(N) is unramified at the auxiliary prime.

No tame ramification

We first show that there cannot be any tame ramification.

Lemma 4.1. Let 𝕋 be a finite dimensional local 𝔽-algebra with maximal ideal 𝔪 for a finite extension 𝔽/𝔽_{p}. If A ∈ ker(GL_{n}(𝕋) → GL_{n}(𝕋/𝔪)) is a matrix such that A^{p−1} = 1, then A = 1.

Proof. There is a matrix M all of whose entries are in 𝔪 such that A = 1 + M. Thus A = A^{p^{r}} = (1 + M)^{p^{r}} = 1 + M^{p^{r}} for all r ∈ ℕ. As 𝔪 is a nilpotent ideal and all entries of M^{p^{r}} lie in 𝔪^{p^{r}}, it follows that M = 0.

Proposition 4.2. Let 𝕋 be a finite dimensional local 𝔽-algebra with maximal ideal 𝔪 for a finite extension 𝔽/𝔽_{p}. Let C be a subgroup of ℤ/(p − 1) and ρ : C → GL_{n}(𝕋) a representation such that the residual representation \overline{ρ} : C → GL_{n}(𝕋/𝔪) is trivial. Then ρ is trivial.

Proof. As \overline{ρ} is trivial, ρ takes its values in ker(GL_{n}(𝕋) → GL_{n}(𝕋/𝔪)). But, this group does not have any nontrivial element of order dividing p − 1 by Lemma 4.1, whence ρ is the trivial representation.

Corollary 4.3. Let 𝕋 be a finite dimensional local 𝔽-algebra with maximal ideal 𝔪 for a finite extension 𝔽/𝔽_{p}. Let ρ : Gal(\overline{ℚ}_{p}/ℚ_{p}) → GL_{2}(𝕋) be a representation and let \overline{ρ} : Gal(\overline{ℚ}_{p}/ℚ_{p}) → GL_{2}(𝕋/𝔪) be its residual representation. Assume that the semisimplification of \overline{ρ} is unramified and that the restriction of ρ to Gal(\overline{ℚ}_{p}/ℚ_{p}(ζ_{p})) is unramified.

Then ρ is unramified.

Proof. As the semisimplification of \overline{ρ} is unramified and the restriction of ρ to Gal(\overline{ℚ}_{p}/ℚ_{p}(ζ_{p})) is also unramified, it follows that \overline{ρ} is unramified. Moreover, the image of the inertia group has to be a subgroup of ℤ/(p − 1), whence it acts trivially by Proposition 4.2.

The non-local case

Proof of Theorem 1.2 – the non-local case. Note that the assumptions of Theorem 1.2 imply that Assumption 3.5 is satisfied due to Theorem 3.4. We now assume that we are in the non-local case, i.e. 𝕋_{p,𝔪′} ≅ 𝕋_{p,𝔪̃_{1}} × 𝕋_{p,𝔪̃_{2}}. Let m_{i} ∈ 𝔽_{p}[X] be the minimal polynomial of U_{p}^{−1} acting on 𝕋_{p,𝔪̃_{i}}. Then m_{1} and m_{2} are powers of coprime irreducible polynomials. We obtain

G[p](\overline{ℚ})/I_{𝔪′} = G_{1}[p](\overline{ℚ})/I_{𝔪′} ⊕ G_{2}[p](\overline{ℚ})/I_{𝔪′}

and G_{i}[p](\overline{ℚ})/I_{𝔪′} is characterised by the fact that m_{i}(U_{p}^{−1}) annihilates it. >From Proposition 3.6 it follows that G[p](\overline{ℚ})/I_{𝔪′} is isomorphic to (ρ_{𝔪} ⊗ ϵ̃^{−1}) ⊕ (ρ_{𝔪} ⊗ ϵ̃^{−1}) as 𝕋_{1,𝔪}[Gal(\overline{ℚ}/ℚ)]-modules. But, as such G_{1}[p](\overline{ℚ})/I_{𝔪′} ≅ G_{2}[p](\overline{ℚ})/I_{𝔪′}, thus for both i = 1, 2 we have ρ_{𝔪} ⊗ ϵ̃^{−1} ≅ G_{i}[p](\overline{ℚ})/I_{𝔪′}. We are now going to work locally and let G = Gal(\overline{ℚ}_{p}/ℚ_{p}(ζ_{p})) and I its inertia group. By Proposition 3.3 (a) applied to G_{i} we obtain for i = 1, 2 that

G_{i}^{0}[p](\overline{ℚ}_{p})/I_{𝔪′} ↪ (G_{i}[p](\overline{ℚ}_{p})/I_{𝔪′})^{I} ≅ (ρ_{𝔪} ⊗ ϵ̃^{−1})^{I}

and that Frob_{p} on the left hand side acts through U_{p}^{−1}, whence the image of the map is annihilated by m_{i}(Frob_{p}). As the polynomials m_{1} and m_{2} are coprime, G^{0}[p](\overline{ℚ}_{p})/I_{𝔪′} ≅ G_{1}^{0}[p](\overline{ℚ}_{p})/I_{𝔪′} ⊕G_{2}^{0}[p](\overline{ℚ}_{p})/I_{𝔪′} is a subrepresentation of (ρ_{𝔪}⊗ ϵ̃^{−1})^{I}. Counting 𝔽_{p}-dimensions, it follows that

G^{0}[p](\overline{ℚ}_{p})/I_{𝔪′} ≅ (ρ_{𝔪} ⊗ ϵ̃^{−1})^{I} ≅ ρ_{𝔪} ⊗ ϵ̃^{−1}.

Consequently, ρ_{𝔪} is unramified at p, using Corollary 4.3 and taking into account that the semisimplification of \overline{ρ}_{𝔪} restricted to Gal(\overline{ℚ}_{p}/ℚ_{p}) is unramified at p. Moreover, again due to Proposition 3.3 (a) the characteristic polynomial of Frob_{p} on ρ_{𝔪} ⊗ ϵ̃^{−1} is the one of U_{p}^{−1}, which is X^{2} − T_{p}/⟨p⟩X + 1/⟨p⟩ (see Corollary 2.6), so that the one of ρ_{𝔪}(Frob_{p}) is as claimed.

The local case

Proposition 4.4. Let R be a local 𝔽_{p}-algebra which is finite dimensional as an 𝔽_{p}-vector space and let 𝔪 be its maximal ideal. Put 𝔽 = R/𝔪. Let G be a group. Let M, N, Q be R[G]-modules which are free of rank 2 as R-modules and suppose we have an exact sequence

0 → N ⟶^{α} M ⊕ M → Q → 0

of R[G]-modules. Suppose further that \overline{N} := R/𝔪 ⊗_{R} N is indecomposable as an 𝔽[G]-module. Then N ≅ M ≅ Q as R[G]-modules.

Proof. Write \overline{M} := R/𝔪 ⊗_{R} M. Counting dimensions as 𝔽-vector spaces it follows that the sequence

0 → \overline{N} ⟶^{\overline{α}} \overline{M} ⊕ \overline{M} → \overline{Q} → 0

is an exact sequence of 𝔽[G]-modules. Consider the composite map

ϕ_{i} : N ↪^{α} M ⊕ M ⟶^{pr_{i}} M

for i = 1, 2, where pr_{i} is the projection on the i-th summand. Note that the ϕ_{i} are homomorphisms of R[G]-modules. Tensor ϕ_{i} with R/𝔪 to obtain

\overline{ϕ_{i}} : \overline{N} ↪^{\overline{α}} \overline{M} ⊕ \overline{M} ⟶^{\overline{pr_{i}}} \overline{M}.

Note that the cases dim_{𝔽} im(\overline{ϕ_{i}}) ≤ 1 for i = 1, 2 cannot occur: If one of the dimensions is 1 and the other 0 or if both are 0, then one has a contradiction to the injectivity of \overline{α}. If both are 1, then \overline{N} ≅ im(\overline{ϕ_{1}}) ⊕ im(\overline{ϕ_{2}}) as 𝔽[G]- modules, which contradicts the assumed indecomposability of \overline{N}. Hence, there is i ∈ \{1, 2\} such that dim_{𝔽} im(\overline{ϕ_{i}}) = 2. Hence, \overline{ϕ_{i}} is an isomorphism \overline{N} → \overline{M}. It follows that ϕ_{i} : N → M is surjective. Indeed, tensoring the exact sequence N ⟶^{ϕ_{i}} M → S → 0 over R with R/𝔪, shows that \overline{S} = R/𝔪 ⊗_{R} S = 0, whence S = 0 by Nakayama’s lemma. As N and M are finite sets, ϕ_{i} is an isomorphism of R[G]-modules.

Proof of Theorem 1.2 – the local case. Note that the assumptions of Theorem 1.2 imply that Assumption 3.5 is satisfied due to Theorem 3.4. We now assume that 𝕋_{p,𝔪′} is local. We are going to deduce the statement from Proposition 4.4. For R we take 𝕋_{1,𝔪} and we let G := Gal(\overline{ℚ}_{p}/ℚ_{p}(ζ_{p})). In Proposition 3.6 we have seen that ρ_{I_{𝔪′}} is isomorphic to (ρ_{𝔪} ⊗ϵ̃^{−1})⊕(ρ_{𝔪} ⊗ϵ̃^{−1}) as R[G]-modules and we take M to be ρ_{𝔪} ⊗ ϵ̃^{−1}. Next we reduce the exact sequence of Proposition 3.3 (b) modulo I_{𝔪′}. Due to multiplicity one, it remains exact (since it is split as a sequence of 𝕋_{p,𝔪′}- modules), whence we obtain an exact sequence of R[G]-modules

0 → ρ̃ → ρ_{I_{𝔪′}} → G^{e}[p](\overline{ℚ}_{p})/I_{𝔪′} → 0,

where ρ̃ = G^{0}[p](\overline{ℚ}_{p})/I_{𝔪′}. By Proposition 3.3 (a) we know that ρ̃ is unramified as a G-module and it is free of rank 2 over R. Moreover, any arithmetic Frobenius at p acts through multiplication by U_{p}^{−1}. Also G^{e}[p](\overline{ℚ}_{p})/I_{𝔪′} is free of rank 2 as an R-module. Taking ρ̃ modulo 𝔪 we obtain an indecomposable R/𝔪[G]-module, where the indecomposability is due to the formula for U_{p} (see Corollary 2.6). Hence, we take N to be ρ̃, restricted to G.

Thus, from Proposition 4.4 we obtain ρ_{𝔪} ⊗ ϵ̃^{−1} ≅ ρ̃ as 𝕋_{1,𝔪}[Gal(\overline{ℚ}_{p}/ℚ_{p}(ζ_{p}))]- modules, and, in particular, that ρ_{𝔪} is unramified at p, using Corollary 4.3 and taking into account that the semisimplification of \overline{ρ}_{𝔪} restricted to Gal(\overline{ℚ}_{p}/ℚ_{p}) is unramified at p. Moreover, again due to Proposition 3.3 (a) the characteristic polynomial of Frob_{p} on ρ_{𝔪}⊗ϵ̃^{−1} is the one of U_{p}^{−1}, which is X^{2}−T_{p}/⟨p⟩X+1/⟨p⟩ (see Corollary 2.6), so that the one of ρ_{𝔪}(Frob_{p}) is as claimed.

5 Examples

We illustrate Theorem 1.2 by two examples. They both appeared first in Mestre’s appendix to [Edi06] and we work them out in our context. Both examples are of the following shape. Let 𝔽 be a finite field of characteristic p and 𝕋 := 𝔽[ϵ] := 𝔽[X]/(X^{2}). Then we have the split exact sequence of groups:

0 → Mat_{2}(𝔽)^{0} ⟶^{A↦1+ϵA} SL_{2}(𝕋) ⟶^{ϵ↦0} SL_{2}(𝔽) → 1,

where Mat_{2}(𝔽)^{0} denotes the 2 × 2-matrices of trace zero (considered here as an abelian group with respect to addition), on which SL_{2}(𝔽) acts by conjugation (i.e. it is the adjoint representation). If p > 2, then this representation is irreducible, if p = 2 it has non-trivial submodules.

Example p = 2, N = 229

In this case there is only one normalised eigenform g ∈ S_{1}(Γ_{1}(N); \overline{𝔽}_{2})_{Katz} and thus only a unique maximal ideal 𝔪 ⊂ 𝕋_{1}. For example using Magma ([BCP97]) and a package developed by the author (see Appendix B of [Edi06] for an old version), one computes that 𝕋_{1} ≅ 𝔽_{2}[ϵ] and that \overline{ρ}_{𝔪} = \overline{ρ}_{g} : Gal(\overline{ℚ}/ℚ) ↠ SL_{2}(𝔽_{2}) ≅ S_{3}. If ker(\overline{ρ}_{g}) = Gal(\overline{ℚ}/K), then K is the Hilbert class field of ℚ(\sqrt{229}). Let us call G the image of ρ_{𝔪} : Gal(\overline{ℚ}/ℚ) → SL_{2}(𝕋) coming from Theorem 1.2. It turns out that G ∩ Mat_{2}(𝔽_{2})^{0} = \{( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} )\} (with the intersection being taken with respect to the map A ↦ 1 + ϵA) and that G ≅ S_{4}. In fact, this example can be obtained by reducing ρ_{f}, where f is a holomorphic weight 1 cuspidal eigenform with ρ_{f} having projective image S_{4}. Hence, the fact that ρ_{𝔪} is unramified at 2, which follows from Theorem 1.2, can already be deduced from the theorem of Deligne and Serre.

Example p = 2, N = 1429

In this case there is a normalised eigenform g ∈ S_{1}(Γ_{1}(N); \overline{𝔽}_{2})_{Katz} such that the image of \overline{ρ}_{g} is isomorphic to SL_{2}(𝔽_{8}). As SL_{2}(𝔽_{8}) is not a quotient of any finite subgroup of GL_{2}(ℂ), this implies, as noted by Mestre, that g is not the reduction of any holomorphic weight 1 eigenform. Let 𝔪 be the maximal ideal of 𝕋_{1} corresponding to g. One computes that 𝕋_{1,𝔪} ≅ 𝔽_{8}[ϵ]. If ker(\overline{ρ}_{g}) = Gal(\overline{ℚ}/K), then K is a Galois extension of ℚ with Galois group SL_{2}(𝔽_{8}) which is unramified outside the prime 1429. Now consider ρ_{𝔪} : Gal(\overline{ℚ}/ℚ) → SL_{2}(𝕋_{1,𝔪}) from Theorem 1.2 and let L be the Galois extension of ℚ such that Gal(\overline{ℚ}/L) ≅ ker(ρ_{𝔪}), which is unramified outside 1429. After checking many Frobenius traces it seems very likely that L is K(\sqrt{1429}) and, hence, that G := im(ρ_{𝔪}) ≅ SL_{2}(𝔽_{8}) × ℤ/(2). Explicitly,

G ∩ Mat_{2}(𝔽_{2})^{0} = \{( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )\}.

In this case it is clear that L is unramified at 2 without appealing to Theorem 1.2. However, the remarkable phenomenon is that this extension L/ℚ is detected by weight one Katz forms through their Hecke algebras. This points in the direction that one should ask if 𝕋_{1,𝔪} is in fact a universal deformation ring of \overline{ρ}_{𝔪} in the category of local 𝔽_{p}-algebras with residue field 𝕋_{1}/𝔪 for the local conditions of being unramified at p and minimally ramified elsewhere.

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Gabor Wiese

Université du Luxembourg

Faculté des Sciences, de la Technologie et de la Communication 6, rue Richard Coudenhove-Kalergi L-1359 Luxembourg Luxembourg gabor.wiese@uni.lu 708

Documenta Mathematica 19 (2014)