ON MODULAR MOD ℓ GALOIS REPRESENTATIONS WITH EXCEPTIONAL IMAGES.

IAN KIMING, HELENA A. VERRILL

Abstract. We give a parametrization of the possible Serre invariants (N, k, ν) of modular mod ℓ Galois representations of the exceptional types A_{4}, S_{4}, A_{5}, in terms of local data attached to the fields cut out by the associated projective representations. We show how this result combined with certain global considerations leads to an effective procedure that will determine for a given modular form f and prime ℓ whether the mod ℓ representation attached to f is exceptional. We illustrate with numerical examples.

1. Introduction.

Suppose that f is a modular form of weight ≥ 2 on some Γ_{1}(M). If ℓ is a prime number we have attached to f a mod ℓ representation ρ̄: G_{ℚ} ⟶ GL_{2}(\overline{𝔽}_{ℓ}) of the absolute Galois group of ℚ. If ρ̄ is irreducible, the classification [13] of finite subgroups of GL_{2}(\overline{𝔽}_{ℓ}) implies that the image of the projectivisation ˜ ρ̄: G_{ℚ} ⟶ PGL_{2}(\overline{𝔽}_{ℓ}) of ρ̄ is either a (i) dihedral group, (ii) is isomorphic to one of the groups A_{4}, S_{4}, A_{5}, or (iii) is isomorphic to PGL_{2}(𝔽_{ℓ^{n}}) or PSL_{2}(𝔽_{ℓ^{n}}) for some n. Thanks to work by Swinnerton-Dyer, Serre, Ribet and Momose, cf. [36], [37], [31], [27], [28], [25], one knows that the cases (i) and (ii) occur for only a finite number of primes ℓ. Also, if we are in case (i) or (ii) for our particular prime ℓ, then this implies the existence of certain exceptional mod ℓ congruences for the Fourier coefficients of f (here, and in the following we use ‘mod ℓ’ as short-hand for ‘mod λ where λ is a prime over ℓ of a sufficiently large number field’). The link to exceptional mod ℓ congruences is one the the main reasons for interest in determining for the given pair (f, ℓ) whether ρ̄ has ‘small image’, i.e. whether we are in one of the cases (i) or (ii).

It is thus natural to ask whether we can devise a method that will determine by a finite amount of computation the image of ρ̄, or at least that of ˜ ρ̄. In principle the explicit version of Chebotarev’s theorem [23] provides us with such a method, but it seems impractical to actually use this since the bounds involved are generally astronomical. Of course, if one expects to be in case (iii), one can hope to quickly notice this by finding – via the Fourier coefficients of f – elements in the image of ρ̄ whose presence imply that case (iii) prevails. This seems to work remarkably well, see for example [14]. On the other hand, since case (i) means that ρ̄ is induced from a 1-dimensional character on on G_{M} for some quadratic M/ℚ, it is not hard to see how one can obtain through algebraic number theory combined with [35] a method that will confirm or deny that case (i) prevails. These considerations may convince us that the case(s) (ii) – which we refer to as the exceptional cases – are especially problematic to confirm for a given (f, ℓ). For example, in the classical works [36], [37], [31] that deal with classical forms f of level 1, it was precisely the case where f = QΔ is the normalized cusp form of weight 16 and level 1, ℓ = 59 and where Im˜ρ̄ turns out to be of S_{4}-type, that caused the most trouble. This case was finally settled by Haberland in [18] who had to work hard with the Fourier coefficients of QΔ.

The contribution of the present article is the systematic exploitation of the following simple idea: If ρ̄ is of the exceptional type (ii), then ˜ ρ̄ cuts out an extension K/ℚ with Galois group isomorphic to one of the groups A_{4}, S_{4}, A_{5}. Then K/ℚ gives rise to a complex projective Galois representation ρ̃_{0} (by choosing an embedding Gal(K/ℚ) ↪ PGL_{2}(ℂ)) which – in an appropriate sense – may be assumed to yield ˜ ρ̄ upon reducing mod ℓ, cf. the discussion in section 2 below. Then, if we lift ρ̃_{0} to a linear representation ρ_{0} : G_{ℚ} → GL_{2}(ℂ) and make this ℓ-adically integral, then ρ̄ is a twist of the mod ℓ reduction of ρ_{0}. Utilizing then the extensive information – as given in [3], [40], [21] – about the behavior of complex lifts upon twisting, we are able to determine purely in terms of local data attached to the extension K/ℚ the (infinitely many) possible Serre invariants (N, k, ν) of representations ρ̄ that cut out K/ℚ projectively. Here, and in the following, by ‘Serre invariant’ (N, k, ν) we means the quantities attached by Serre in [33] to an irreducible mod ℓ representation ρ̄, with the small modification in connection with the weight k as given in [16]. Combining this with results on Serre’s refined conjecture as given in [29], [11], [16], [7], we can predict the minimal types (N, k, ν) of modular forms whose attached projective mod ℓ representations cut out a given K/ℚ. This is what Theorem 1 below does: It gives a parametrization of these (infinitely many) minimal types (N, k, ν) purely in terms of local data attached to K/ℚ, albeit under an appropriate modularity assumption and a minor indeterminacy in the A_{5}-case (caused essentially by the existence of 2 inequivalent embeddings A_{5} ↪ PGL_{2}(ℂ)). One remark about the theorem: We discuss only minimal types (N, k, ν) where every prime divisor of N is a ramification point for the given K/ℚ; this is because, as one immediately checks, any other minimal type will then have the shape (N · m^{2}, k, ν · ϕ^{2}) where ϕ is a character of conductor m whose prime divisors are all different from ℓ and unramified in K.

The information obtained from Theorem 1 is sufficient to actually prove exceptionality for some particular examples of modular mod ℓ representations. We give some of these in section 4 below. For instance, we are able to quickly reprove Haberland’s result in connection with QΔ and ℓ = 59 without knowing a single Fourier coefficient of the form. Another and somewhat more complicated example concerns the unique normalized cusp form of weight 4 on Γ_{0}(8) and ℓ = 11.

Finally, in section 5 we show how Theorem 1 can be complemented by certain global considerations to yield an effective general procedure that will decide whether the exceptional case prevails for any explicitly given pair (f, ℓ).

Notation. Everywhere in this article ℓ denotes a fixed odd prime number but p may be any prime. The Galois group of a separable closure of a field k will be denoted G_{k}. Generally, K/ℚ will denote a not totally real extension with Galois group isomorphic to one of the groups A_{4}, S_{4}, A_{5}; in this situation, 𝔩 and 𝔭 will denote primes of K over ℓ and p respectively. More generally, if p is any prime it will be convenient to assume that a prime of \overline{ℚ} over p has been fixed. Corresponding to this prime over p we have a decomposition subgroup and an inertia subgroup of G_{ℚ}; the inertia subgroup will be denoted by I_{p}.

All Galois representations occurring are assumed to cut out a finite extension of the base field in question. If ρ is a linear or projective representation of G_{ℚ}, we shall denote by ρ_{p} the restriction of ρ to the decomposition group in G_{ℚ} corresponding to the fixed prime of ℚ over p. We then view ρ_{p} as a representation of G_{ℚ_{p}}. By the projective kernel field of ρ we mean the field fixed by the kernel of ρ if ρ is projective, or the kernel of the projectivisation of ρ if ρ is linear.

We denote by U_{p} the group of units in ℤ_{p}. If ϵ_{p} is a character of ℚ_{p}, we denote by c_{p}(ϵ_{p}) the exponent of the conductor of ϵ_{p}.

We use (local) class field theory freely and without special notice. Thus for example, if ϵ_{p} is a character of G_{ℚ_{p}} we may view the restriction of ϵ_{p} to the inertia subgroup of G_{ℚ_{p}} as a character on U_{p}.

The symbol χ always denotes the mod ℓ cyclotomic character χ: G_{ℚ} ⟶ 𝔽_{ℓ}.

Theorem 1. Let ℓ be an odd prime, and let K/ℚ be a not totally real Galois extension with Galois group G isomorphic to one of the groups A_{4},S_{4},A_{5}. If G ≅ A_{5}, assume weight 1 modularity of a complex representation of G_{ℚ} with projective kernel field K.

Let S denote the product of the finite primes different from ℓ that ramify in K. For p | S and ϕ_{p}: U_{p} → ℂ^{×} a homomorphism, define as in section 3.1 below the

natural number n_{p}(ϕ_{p}) = n_{p}(K_{𝔭}/ℚ_{p}, ϕ_{p}) and the homomorphism

ϵ_{p} = ϵ(K_{𝔭}/ℚ_{p}): U_{p} ⟶ \overline{𝔽}_{ℓ}^{×}

depending on the extension K_{𝔭}/ℚ_{p}. Also, as in sections 3.3–3.4, define the set 𝔚_{ℓ} = 𝔚(K_{𝔩}/ℚ_{ℓ}) of natural numbers in dependence on the extension K_{𝔩}/ℚ_{ℓ} and equipped with the partition 𝔚_{ℓ} = 𝔚_{ℓ}^{+} ∪ 𝔚_{ℓ}^{−} if ℓ ≠ 5 and K_{𝔩}/ℚ_{ℓ} has degree divisible by 5.

Suppose that N is a natural number whose prime divisors all divide S, that

ν: (ℤ/ℤN)^{×} ⟶ \overline{𝔽}_{ℓ}^{×}

is a homomorphism, and that k ∈ {2, …, ℓ − 1}.

(a) If there exists a mod ℓ modular form of type (N, k, ν) whose attached mod ℓ representation has Serre invariants (N, k, ν) and projective kernel field K, then k ∈ 𝔚_{ℓ} and there exists a global character ϕ: G_{ℚ} → ℂ^{×} unramified outside S · ∞ and of order prime to ℓ, such that

N = ∏_{p|S} p^{n_{p}(ϕ_{|Ip})} ,

and such that

ν(r) = ∏_{p|S} ϵ_{p}(r)^{−1} · (ϕ_{p}(r)^{−2} mod ℓ) , for (r, N) = 1.

(b) Conversely, suppose that N and ν have the above forms for some global character ϕ of order prime to ℓ. Then if either K_{𝔩}/ℚ_{ℓ} has degree not divisible by 5 or if ℓ = 5, there is for any k ∈ 𝔚_{ℓ} a modular form of type (N, k, ν) and with K the projective kernel field of the attached mod ℓ representation. If on the other hand, K_{𝔩}/ℚ_{ℓ} does have degree divisible by 5 and ℓ ≠ 5, we only state this conclusion for all k ∈ 𝔚_{ℓ}^{μ} for some choice of sign μ ∈ {+, −}.

Corollary 1. Suppose that ρ̄: G_{ℚ} ⟶ GL_{2}(𝔽̄_{ℓ}) is a modular representation of weight k ≥ 2 whose projective kernel field K/ℚ has Galois group isomorphic to one of the groups A_{4}, S_{4}, A_{5}. Again, if this Galois group is A_{5}, assume the weight 1 modularity of a complex representation of G_{ℚ} with projective kernel field K.

Then

ℓ ≤ \{ \begin{matrix}3k − 2 & if G ≅ A_{4} \\ 4k − 3 & if G ≅ S_{4} \\ 5k − 4 & if G ≅ A_{5} . \end{matrix}

Remarks: (1) In the theorem, the numbers n_{p}(ϕ_{p}) actually depend only on (K_{𝔭}/ℚ_{p} and) the conductor of ϕ_{p} except possibly when K_{𝔭}/ℚ_{p} is cyclic.

(2) The small indeterminacy concerning the weight k in part (b) of the theorem stems from the from the fact that there are 2 inequivalent embeddings A_{5} ↪ PGL_{2}(\overline{𝔽}_{ℓ}) if ℓ ≠ 5. It is true – in an appropriate sense – that swapping between these 2 embeddings will change the sign μ. The point is however that our proof of the theorem is by purely local means and that the simultaneous specification of all of the 3 quantities N, k and ν really requires some sort of global information. An example of this type of phenomenon can be found in [3], Chap. 6. We shall indicate in section 5 below how the question can be resolved by certain global information in any explicitly given case.

(3) In view of recent progress on the Artin conjecture for odd icosahedral representations of G_{ℚ}, cf. [5], [38], the modularity assumption in Theorem 1 is not an extremely strong one. It should be noted that it is actually only needed in part (b) of the theorem.

(4) It would have been possible to include a discussion of non-optimal levels ([12] and [4]). Also, we could have discussed weights outside the interval 2 ≤ k ≤ ℓ − 1; the main purpose for restricting to this interval is to exclude the case where the weight of the mod ℓ representation is 1. This case would occur if the given extension K/ℚ was unramified over ℓ which is a rather uninteresting situation in the context of the present paper.

Finally, one should notice that the proof of the theorem obviously enables one to actually count the number of modular mod ℓ with a given triple of Serre invariants (N, k, ν) (of course under the appropriate modularity assumption in the A_{5}-case and with the minor indeterminacy of part (b)).

These possible extensions of Theorem 1 have been discarded mainly in order to prevent overloading in the statement of the theorem.

2. Lifting exceptional mod l representations to complex representations.

In this section we prove some simple statements about lifting mod ℓ representations to complex ones. Everything in this section is probably fairly standard knowledge but we have included it for lack of appropriate references.

In the first part we work more generally with n-dimensional representations since the arguments will be the same in this more general setting.

Given an irreducible Galois representation ρ̃: G_{ℚ} → PGL_{n}(ℂ) with finite image, by the reduction mod ℓ of ρ̃ we mean the representation ˜ ρ̄: G_{ℚ} → PGL_{n}(\overline{𝔽}_{ℓ}) constructed as follows:

Let ρ: G_{ℚ} → GL_{n}(ℂ) be a lift of ρ̃, i.e., a representation which has projectivisation equal to ρ̃. By a theorem of Tate (cf. [32]) such a lift ρ does exist. We may consider ρ as a representation ρ: G_{ℚ} → GL_{n}(ℚ̄). Denote by ℤ̄ the ring of integers of ℚ̄. Since every finitely generated ideal of ℤ̄ is principal, we can conjugate ρ to a representation G_{ℚ} → GL_{n}(ℤ̄) which by abuse of notation we still denote by ρ.

Now choose a prime 𝔏 of ℤ̄ over ℓ, let ρ̄ be the mod 𝔏 reduction of ρ, and consider the projectivisation ˜ ρ̄: G_{ℚ} ⟶ PGL_{n}(\overline{𝔽}_{ℓ}) of ρ̄. One easily checks that ˜ ρ̄ is up to equivalence independent of the various choices made. We refer then to ˜ ρ̄ as the mod ℓ reduction of ρ̃. For clarity, the four representations ρ, ρ̄, ρ̃ and ˜ ρ̄ are shown in the following diagram:

\xymatrix{ & G_{ℚ} \ar[dl]_{ρ} \ar[dr]^{ρ̃} \ar[ddl]^{ρ̄} \ar[ddr]_{˜ρ̄} \\ GL_{n}(ℤ̄) \ar[d] &  & PGL_{n}(ℤ̄) \ar[d] \\ GL_{n}(𝔽̄_{ℓ}) \ar[rr] &  & PGL_{n}(𝔽̄_{ℓ})}

Reversing the roles of ρ̃ and ˜ ρ̄, if ˜ ρ̄: G_{ℚ} → PGL_{n}(𝔽̄_{ℓ}) is given and assumed to be irreducible, we may ask for a representation ρ̃: G_{ℚ} → PGL_{n}(ℂ) whose mod ℓ reduction is the given ˜ ρ̄. If such a ρ̃ exists we may and will then refer to it as a complex lift of ˜ ρ̄.

Proposition 1. Let ˜ ρ̄: G_{ℚ} → PGL_{2}(𝔽̄_{ℓ}) be an irreducible representation such that Im˜ρ̄ embeds in PGL_{2}(ℂ). Then ˜ ρ̄ has a complex lift.

In what follows we actually only use this proposition in case Im˜ρ̄ is of one of the exceptional types A_{4}, S_{4} or A_{5}, but for the sake of completeness, we also discuss the dihedral case. We begin the proof of this proposition by discussing the following partially more general but easy lemma.

Lemma 1. Let ˜ ρ̄: G_{ℚ} → PGL_{n}(𝔽̄_{ℓ}) be an irreducible representation such that Im˜ρ̄ embeds in PGL_{n}(ℂ), and such that the order of the image is not divisible by ℓ. Then ˜ ρ̄ has a complex lift.

Proof. : Let us first notice that we have

H^{2}(G_{ℚ}, 𝔽̄_{ℓ}^{×}) = 0 ,

where the action of G_{ℚ} on 𝔽̄_{ℓ}^{×} is trivial. This is because we have

𝔽̄_{ℓ}^{×} ≅ ∏_{p≠ℓ} ℚ_{p}/ℤ_{p} ,

and because we know that

H^{2}(G_{ℚ}, ℚ_{p}/ℤ_{p}) = 0 ,

cf. [32], §6.5. Consequently, ˜ ρ̄ has some lift ρ̄: G_{ℚ} → GL_{2}(𝔽̄_{ℓ}). The image of ρ̄ also has order prime to ℓ. Then a corollary of a theorem of Brauer and Nesbitt, see for instance [19], V Satz 12.9, shows that ρ̄ is in fact in the above sense the mod ℓ reduction of a complex representation ρ. The projectivisation of any such ρ is then a complex lift of ˜ ρ̄.

Proof. of Proposition 1: By lemma 1 we may assume that ℓ divides the order of the image G := Im˜ρ̄.

Notice first that G must be isomorphic to one of the groups A_{4}, S_{4}, A_{5}, or D_{m}, as it embeds into PGL_{2}(ℂ) and ˜ ρ̄ is irreducible. Now, ˜ ρ̄ can be realized over 𝔽_{ℓ^{n}} for some n, so we continue the discussion assuming that ˜ ρ̄ lands in PGL_{2}(𝔽_{ℓ^{n}}).

Enlarging n if necessary, in case ℓ ≠ 5, we may assume that 𝔽_{ℓ^{n}} contains a primitive 5th root of unity. Since we are assuming ℓ divides the order of G, in the dihedral case m must be a multiple of ℓ, but then by the classification of subgroups of PGL_{2}(𝔽_{ℓ^{n}}) (cf. [13]) we must have m = ℓ. We consider each case individually.

Suppose first that G ≅ D_{ℓ} = ⟨σ, τ | σ^{2} = τ^{ℓ} = 1, στσ^{−1} = τ^{−1}⟩. It is straightforward to see (cf. for example Dixon [13], §241 and §250–§253) that ˜ ρ̄ must be equivalent to a representation:

G_{ℚ} ⟶ G ≅ D_{ℓ} ↪ PGL_{2}(𝔽_{ℓ^{n}}) ,

of the form

σ ↦ ( \begin{matrix}−1 & a \\ 0 & 1 \end{matrix} ) , τ ↦ ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ,

with a ∈ 𝔽_{ℓ^{n}}. A complex lift of ˜ ρ̄ is then given by:

σ ↦ ( \begin{matrix}−1 & α+β \\ β & 1 \end{matrix} ) , τ ↦ ( \begin{matrix}ε & 1 \\ 0 & ε^{−1} \end{matrix} ) ,

where ε is a primitive ℓ’th root of unity, β := ε − ε^{−1} and α ∈ \overline{ℤ} is any element reducing mod ℓ to a.

Suppose then that G is isomorphic to one of A_{4}, S_{4} or A_{5}. In any of these cases, we first recall the well-known fact that there is at most one conjugacy class of subgroups isomorphic to G in PGL_{2}(𝔽_{ℓ^{n}}): For G ≅ A_{4} or S_{4} this follows from the results in [13], §257; if G ≅ A_{5}, the claim is easily derived from the discussions in [13], §§258–259. Consequently, any representation G_{ℚ} ⟶ PGL_{2}(𝔽_{ℓ^{n}}) with the same kernel field as ˜ ρ̄ will – up to equivalence – have the shape α∘˜ρ̄ where α is some automorphism of G = Im˜ρ̄ ≤ PGL_{2}(𝔽_{ℓ^{n}}). Now let us notice that α must in fact be conjugation by some element of PGL_{2}(𝔽_{ℓ^{n}}) except in the case where G ≅ A_{5} and ℓ ≠ 5: If G ≅ S_{4} this is clear as S_{4} has trivial outer automorphism group. If G ≅ A_{4} we are assuming that ℓ = 3; then the claim follows as the automorphism group of G ≅ PSL_{2}(𝔽_{3}) is isomorphic to PGL_{2}(𝔽_{3}) ≅ S_{4} acting via conjugation (cf. [34]). An analogous reasoning applies to the case where ℓ = 5 and G ≅ A_{5} ≅ PSL_{2}(𝔽_{5}), which has automorphism group PGL_{2}(𝔽_{5}) again acting via conjugation ([34]). Finally notice that if G ≅ A_{5} but ℓ ≠ 5, there are precisely 2 inequivalent embeddings G ↪ PGL_{2}(𝔽_{ℓ^{n}}), as PGL_{2}(𝔽_{ℓ^{n}}) will not in this case contain a subgroup isomorphic to PGL_{2}(𝔽_{5}) (and as PSL_{2}(𝔽_{5}) has index 2 in PGL_{2}(𝔽_{5})).

We can now clearly finish the proof by displaying explicitly in each of the cases we are considering a complex lift (in the above sense) of a given embedding G ↪ PGL_{2}(𝔽_{ℓ^{n}}).

If G ≅ S_{4} = ⟨x, y | x^{2} = y^{3} = (xy)^{4} = 1⟩ and ℓ = 3, the given embedding G ↪ PGL_{2}(𝔽_{ℓ^{n}}) is equivalent to the one given by

x ↦ ( \begin{matrix}1 & 1 \\ 0 & −1 \end{matrix} ) , y ↦ ( \begin{matrix}1 & 0 \\ −1 & 1 \end{matrix} ) ,

and we have a (well-known) complex lift given by

x ↦ ( \begin{matrix}−\sqrt{−2} & −1+\sqrt{−2} \\ −1−\sqrt{−2} & \sqrt{−2} \end{matrix} ) , y ↦ ( \begin{matrix}1 & −1−\sqrt{−2} \\ 1−\sqrt{−2} & −2 \end{matrix} ) .

This also finishes the case G ≅ A_{4}, ℓ = 3.

Suppose then that G ≅ A_{5} = ⟨x, y | x^{2} = y^{5} = (xy)^{3} = 1⟩. If ℓ ≠ 5 we have assumed n large enough so that 𝔽_{ℓ^{n}} contains a primitive 5th root of unity. Now, the following assignments define an embedding G ↪ PGL_{2}(𝔽_{ℓ^{n}}):

(♯)  x ↦ ( \begin{matrix}−c & ω \\ ω & c \end{matrix}) , y ↦ ( \begin{matrix}ε^{2} & −ω \\ 0 & ε^{−2} \end{matrix} ) ,

where ω := ε + ε^{−1}, c := ε^{2} − ε^{−2}, and where ε is a primitive 5th root of unity if ℓ ≠ 5, but ε = 1 if ℓ = 5. For ℓ ≠ 5 the substitution ε ↦ ε^{2} brings one embedding to the other inequivalent one. Viewing the matrices in (♯) as elements of GL_{2}(ℂ) by interpreting ε as a primitive complex 5th root of unity, we have explicit complex lifts in all cases.

Remark: The authors do not know to what extent Proposition 1 generalizes to higher-dimensional situations. The argument given for n = 2 in case Im˜ρ̄ has order divisible by ℓ is admittedly awkward, relying as it does on the classification of subgroups of PGL_{2}(𝔽_{ℓ^{n}}). But we have not been able to find a more conceptual approach in this case.

3. Proof of Theorem 1.

In this section we prove Theorem 1 and the Corollary. So, let K/ℚ, G and S be as in the statement of the Theorem. Definitions of the quantities n_{p}(K_{𝔭}/ℚ_{p}, ϕ_{p}) and ϵ(K_{𝔭}/ℚ_{p}) will be given in section 3.1 below, and the definition of the set 𝔚_{ℓ}(K_{𝔩}/ℚ_{ℓ}) is stated in section 3.3.

We start with the proof of part (a). Let

ρ̄: G_{ℚ} ⟶ GL_{2}(\overline{𝔽}_{ℓ})

be an irreducible modular mod ℓ representation with Serre invariants (N, k, ν) where every prime divisor of N divides S, has 2 ≤ k ≤ ℓ−1, and is such that the extension cut out by the projectivisation

˜ρ̄: G_{ℚ} ⟶ PGL_{2}(\overline{𝔽}_{ℓ})

of ρ̄ is the given field K.

Let ρ̃_{0} : G_{ℚ} ⟶ PGL_{2}(ℂ) denote a complex lift of ˜ ρ̄ in the sense of section 2 (which exists according to Proposition 1).

Now, if ρ_{0} : G_{ℚ} ⟶ GL_{2}(ℂ) is any lift of ρ̃_{0} to a linear representation unramified outside Sℓ·∞, and ρ̄_{0} : G_{ℚ} ⟶ GL_{2}(𝔽̄_{ℓ}) is its mod ℓ reduction, the definition of the notion ‘complex lift’ implies that ρ̄ is a twist of ρ̄_{0} by some character G_{ℚ} ⟶ \overline{𝔽}_{ℓ}^{×}; we may write this character as ϕ̄ · χ^{i} for some i ∈ ℤ and with ϕ̄ a character unramified at ℓ. We may and will consider ϕ̄ as the mod ℓ reduction of a complex character ϕ: G_{ℚ} ⟶ ℂ^{×} of order prime to ℓ, and view the restriction ϕ_{p} := ϕ_{|I_{p}} for p | S as a homomorphism U_{p} → ℂ^{×}.

In the following subsection we utilize the results of [21] to show the existence of a special linear lift ρ_{0} which is unramified outside Sℓ · ∞ and has the property that each of the quantities

c_{p}(ρ_{0} ⊗ ϕ) , for p | S,

and

det(ρ_{0})_{|U_{p}} , for p | S,

can (and will) be explicitly determined. The first of these quantities depends on the data K_{𝔭}/ℚ_{p} and ϕ_{p}, whereas the second depends only on K_{𝔭}/ℚ_{p}. Call the first of these quantities ñ_{p}(ϕ_{p}) = ñ_{p}(K_{𝔭}/ℚ_{p}, ϕ_{p}), and let ϵ_{p} = ϵ_{p}(K_{𝔭}/ℚ_{p}) denote the mod ℓ reduction of the second.

In section 3.2 we compute the number

δ_{p}(ϕ_{p}) = δ_{p}(K_{𝔭}/ℚ_{p}, ϕ_{p}) := ñ_{p}(ϕ_{p}) − c_{p}(ρ̄_{0} ⊗ ϕ̄)

for p | S, which depends only on the datum (K_{𝔭}/ℚ_{p}, ϕ_{p}). Defining then

n_{p}(ϕ_{p}) = n_{p}(K_{𝔭}/ℚ_{p}, ϕ_{p}) := ñ_{p}(ϕ_{p}) − δ_{p}(ϕ_{p})

the conductor of ρ̄ = ρ̄_{0} ⊗ ϕ̄χ^{i} is

∏_{p|S} p^{n_{p}(ϕ_{p})} ,

which by the assumption on ρ̄ coincides with N. Further, for p dividing S, the restriction of the determinant to I_{p} of ρ̄ is the character ∏_{p|S} ϵ_{p} ·(ϕ_{p}^{2} mod ℓ) which by definition coincides with ν_{|I_{p}} when, via global class field theory ν is viewed as a character of G_{ℚ}. As ν is unramified outside S · ∞ we have – again by global class field theory – that ν is given by

r ↦ ∏_{p|S} ϵ_{p}(r)^{−1} · (ϕ_{p}(r)^{−2} mod ℓ) , for (r, N) = 1 ,

as a character on (ℤ/ℤN)^{×}.

In subsection 3.3 we study the local behavior of ρ̄_{0} ⊗ χ^{i} at ℓ and define the set 𝔚_{ℓ} = 𝔚(K_{𝔩}/ℚ_{ℓ}) such that k ∈ 𝔚_{ℓ} may be inferred. This will then complete the proof of part (a) of Theorem 1.

3.1. Local lifts: Conductors and determinants. First we use a theorem of Tate, cf. [32], to reduce to a purely local question the problem of specifying a convenient lift ρ_{0} of ρ̃_{0} to a linear representation: Define the representation ρ̃_{0,p}: G_{ℚ_{p}} ⟶ PGL_{2}(ℂ) as the restriction of ρ̃_{0} to G_{ℚ_{p}} ≤ G_{ℚ}. Thus ρ̃_{0,p} factors as

ρ̃_{0,p}: G_{ℚ_{p}} ↠ Gal(K_{𝔭}/ℚ_{p}) ↪ PGL_{2}(ℂ) .

Now, if we choose for each p | S · ℓ a lift ρ_{0,p} of ρ̃_{0,p} to a linear representation, there is a uniquely determined lift ρ_{0} of ρ̃_{0} which is unramified outside Sℓ · ∞ and satisfies:

(ρ_{0})_{|I_{p}} ≅ (ρ_{0,p})_{|I_{p}} for p | Sℓ .

We shall now address the problem of specifying for each p a particular choice of ρ_{0,p} with the property that the quantities

c_{p}(ρ_{0,p} ⊗ ϕ_{p}) and det(ρ_{0,p})_{|U_{p}} ,

can be explicitly given for any character ϕ_{p}: U_{p} → ℂ^{×}. Then

c_{p}(ρ_{0} ⊗ ϕ) = c_{p}(ρ_{0,p} ⊗ ϕ_{|I_{p}}) ,

and

det(ρ_{0})_{|U_{p}} = det(ρ_{0,p})_{|U_{p}}

are also explicitly known for any global character ϕ: G_{ℚ} → ℂ^{×}. This then completes the first step of the proof as described above.

The theory of local lifts ρ_{0,p} was initiated by Weil in [39], further developed by Buhler and Zink, cf. [3], [40], and complemented by the results in [21] where a full solution to the problem was given in all cases. Thus, we can simply quote from the latter article and briefly review the results given there. There are 3 subcases corresponding to whether the image of ρ̃_{0,p} is cyclic, dihedral or isomorphic to A_{4} or S_{4} respectively. Surprisingly, the most difficult of these subcases is the dihedral one. For the following review, let ϕ_{p} denote an arbitrary character G_{ℚ_{p}} → ℂ^{×}.

3.1.1. Cyclic cases. This case is trivial: If Imρ̃_{0,p} is cyclic then ρ̃_{0,p} is equivalent to

ρ̃_{0,p} ∼ ( \begin{matrix}α & 0 \\ 0 & 1 \end{matrix} ) : G_{ℚ_{p}} ⟶ PGL_{2}(ℂ) ,

for some character α: G_{ℚ_{p}} → ℂ^{×} which we may and will view as a character on ℚ_{p}^{×}. We can then choose ρ_{0,p} ≅ α ⊕ 1 which gives

ñ_{p}(ϕ_{p}) := c_{p}(ρ_{0,p} ⊗ ϕ_{p}) = c_{p}(α_{|U_{p}}ϕ_{p}) + c_{p}(ϕ_{p}) ,

and

ϵ_{p} := (α_{|U_{p}} mod ℓ) .

3.1.2. Dihedral cases. Suppose that Imρ̃_{0,p} is a dihedral group. Then K_{𝔭}/ℚ_{p} contains a quadratic extension M/ℚ_{p} such that K_{𝔭}/M is cyclic and ramified. If the degree [K_{𝔭} : ℚ_{p}] is greater than 4, the field M is uniquely determined. If [K_{𝔭} : ℚ_{p}] = 4 we let M be any quadratic subfield of K_{𝔭} such that K_{𝔭}/M is ramified. Let α: ℚ_{p}^{×} → ℂ^{×} be the quadratic character corresponding to M via local class field theory. Now, the restriction of ρ̃_{0,p} to G_{M} has the form

( \begin{matrix}β & 0 \\ 0 & 1 \end{matrix} )

for some character β on G_{M} which by local class field theory may and will be viewed as a character on M^{×}. In Theorem 1 of [21] a special lift ρ_{0,p} of ρ̃_{0,p} is given, together with explicit formulas depending on the data (M, β) for det(ρ_{0,p})_{|U_{p}} and for c_{p}(ρ_{0,p} ⊗ ϕ_{p}). So, we let

ϵ_{p} := (det(ρ_{0,p})_{|U_{p}} mod ℓ)

and

ñ_{p}(ϕ_{p}) := c_{p}(ρ_{0,p} ⊗ ϕ_{p})

where ρ_{0,p} is the special lift given loc. cit.; we have then explicit formulas for these quantities, but as these are somewhat involved in the general case we shall not restate them in all detail but will be content to review their general shape and give a few special cases:

Let π be a uniformizer of M, let b ≥ 1 be the exponent of the conductor of β (i.e. β has conductor π^{b}), and let t denote the break in the sequence of ramification groups of 𝔊 := Gal(M/ℚ_{p}) in the lower numbering:

𝔊 = 𝔊_{0} = … = 𝔊_{t} ≠ 𝔊_{t+1} = 0 .

Then

p^{ñ_{p}(ϕ_{p})} = p^{c_{p}(ρ_{0,p}⊗ϕ_{p})} = D(M/ℚ_{p}) · N_{M/ℚ_{p}}(π^{max{t+b,γ(ϕ_{p})}}) ,

where D(·) and N_{M/ℚ_{p}}(·) denote discriminant and norm respectively, and where

γ(ϕ_{p}) \{ \begin{matrix}= c_{p}(ϕ_{p}) & if M/ℚ_{p} unramified \\ ≤ t + 1 & if M/ℚ_{p} ramified and c_{p}(ϕ_{p}) ≤ t + 1 \\ = 2 · c_{p}(ϕ_{p}) + 1 − t & if M/ℚ_{p} ramified and c_{p}(ϕ_{p}) ≥ t + 2. \end{matrix}

Thus, as is immediately seen, the number ñ_{p}(ϕ_{p}) actually only depends on the conductor of ϕ_{p}. The character det(ρ_{0,p})_{|U_{p}} has the general shape

det(ρ_{0,p})_{|U_{p}} = α_{|U_{p}} · ψ_{1} · ψ_{2}

where ψ_{1}, ψ_{2} are the characters given loc. cit., which are both of 2-power order. The character ψ_{1} is at most tamely ramified, whereas ψ_{2} is wildly ramified if it nontrivial, which happens only for p = 2. The definitions of ψ_{1} and ψ_{2} are complicated in general and will not be restated, but we give a few special cases:

If the degree m := [K_{𝔭} : M] is odd then ψ_{1} = ψ_{2} = 1 so that ϵ_{p} is the trivial or quadratic character

ϵ_{p} = (α_{|U_{p}} mod ℓ) .

If additionally we have p ∤ m then one finds ñ_{p}(1) = 2.

On the other hand, if K_{𝔭}/ℚ_{p} is dihedral of order 8, but p ≠ 2, then p ≡ 3 (4) and K_{𝔭} is unique. One has in this case that M is the unramified quadratic extension ℚ_{p}(\sqrt{−1})/ℚ_{p} so that α is unramified. Further, ψ_{2} = 1, but ψ_{1} is the non-trivial quadratic character on {±1} ≤ ℚ_{p}^{×}. With the above notation one has b = 1 and t = 0 so that

ñ_{p}(ϕ_{p}) = 2 · max{1, c_{p}(ϕ_{p})} .

3.1.3. The ‘primitive’ cases for p = 2. Finally, Imρ̃_{0,p} may be isomorphic to A_{4} or S_{4} in which case we must have p = 2. The study of these cases was initiated by Weil in [39]. In [3] Buhler computed the minimal possible conductor of a lift of ρ̃_{0,2} and subsequently Zink determined in [40] all possible conductors of lifts of ρ̃_{0,2}. In [21] these results were complemented by a discussion of the associated determinants. We can refer to Theorem 2 of [21] where a special lift ρ_{0,2} of ρ̃_{0,2} was given together with formulas for the restriction of its determinant to U_{2} and for the conductor of ρ_{0,2} ⊗ ϕ_{2}. We define then

ñ_{2}(ϕ_{2}) := c_{2}(ρ_{0,2} ⊗ ϕ_{2})

and

ϵ_{2} := (det(ρ_{0,2})_{|U_{2}} mod ℓ) ,

where ρ_{0,2} is a lift of ρ̃_{0,2} as in Theorem 2 of [21]. Thus the formulas of that theorem give – depending on the extension K_{𝔭_{2}}/ℚ_{2} (𝔭_{2} a prime of K over 2) – explicit formulas for the quantities ñ_{2}(ϕ_{2}) and ϵ_{2}. Then ϵ_{2} is at most quadratic, and the number ñ_{2}(ϕ_{2}) actually only depends on c_{2}(ϕ_{2}) in every case. We shall not restate the formulas in all cases but be content to quote them in a particular instance: There exist 3 distinct extensions of ℚ_{2} with Galois group isomorphic to S_{4}, cf. [39]. If K_{𝔭_{2}} is the particular one denoted by M_{4} in [21], i.e. if

K_{𝔭_{2}} = ℚ_{2}(ζ, π, \sqrt{1 + π^{2}}, \sqrt{1 + ζπ^{2}}, \sqrt{1 + ζ^{2}π^{2}})

with ζ a primitive 3rd root of unity and π^{3} = 2, then the formulas of Theorem 2 in [21] give ϵ_{2} = 1 and

ñ_{2}(ϕ_{2}) = \{ \begin{matrix}3 & for c_{2}(ϕ_{2}) ≤ 1 \\ 2 · c_{2}(ϕ_{2}) & for c_{2}(ϕ_{2}) ≥ 2. \end{matrix}

3.2. Degeneration of local conductors. We now determine how the conductor of ρ_{0,p} ⊗ ϕ_{p} degenerates upon reduction mod ℓ, i.e. we determine the number

δ_{p}(ϕ_{p}) = δ_{p}(K_{𝔭}/ℚ_{p}, ϕ_{p}) := c_{p}(ρ_{0,p} ⊗ ϕ_{p}) − c_{p}(ρ̄_{0,p} ⊗ ϕ̄)

for characters ϕ_{p}: U_{p} → ℂ^{×}. There is a well-developed understanding of how local conductors may degenerate, cf. [24], [6], but in our special situation it will be more convenient and in fact basically trivial to determine δ_{p}(ϕ_{p}) directly by ‘inspection’. Recall that if p and ℓ are distinct primes, and if r: G_{ℚ_{p}} → GL_{n}(\overline{ℚ}_{ℓ}) is a continuous ℓ-adic representation with r̄: G_{ℚ_{p}} → GL_{n}(\overline{𝔽}_{ℓ}) a mod ℓ reduction, then

c_{p}(r) − c_{p}(r̄) = dim r̄^{I_{p}} − dim r^{I_{p}}

where r̄^{I_{p}} and r^{I_{p}} denote the space of fixed points of r̄(I_{p}) and r(I_{p}) respectively (in the associated underlying vector space). Cf. for example [24], 1.

We split up into 3 main subcases corresponding to the behavior of ρ̄_{0,p} on the inertia group I_{p}.

(1) Suppose that ρ̄_{0,p} is irreducible on I_{p}. This case will occur if p = 2 and ρ_{0,2} is of A_{4}- or S_{4}-type, or if Imρ̃_{0,p} is totally ramified dihedral of order 2m with ℓ ∤ m.

Then ρ̄_{0,p} ⊗ ϕ̄ and ρ_{0,p} ⊗ ϕ_{p} are also both irreducible on I_{p}. Thus,

δ_{p}(ϕ_{p}) = 0

in these cases.

(2) Suppose that ρ̄_{0,p} is reducible but semisimple on I_{p}. This case occurs if Imρ̃_{0,p} is either cyclic of order m not divisible by ℓ, or is ramified, but not totally ramified, dihedral of order 2m with m not divisible by ℓ.

(2a) In the first case we have

(ρ_{0,p})_{|I_{p}} ∼ ( \begin{matrix}α & 0 \\ 0 & 1 \end{matrix} ) , (ρ̄_{0,p})_{|I_{p}} ∼ ( \begin{matrix}ᾱ & 0 \\ 0 & 1 \end{matrix} ) ,

with α a ramified character of order m. We obtain

δ_{p}(ϕ_{p}) = 0

since – as α and ϕ have order prime to ℓ – the characters αϕ and ᾱϕ̄ are seen to have the same conductors, and similarly for ϕ and ϕ̄.

(2b) In the second case we have ρ_{0,p} = Ind_{M/ℚ_{p}}(β) where M/ℚ_{p} is the unramified quadratic extension, and β is a ramified character on G_{M} that may be – and in fact has been – chosen to have order prime to ℓ. Then if σ denotes the non-trivial automorphism of M/ℚ_{p} we have

(ρ_{0,p} ⊗ ϕ_{p})_{I_{p}} ∼ ( \begin{matrix}β·(ϕ_{p})_{G_{M}} & 0 \\ 0 & β^{σ}·(ϕ_{p})_{G_{M}} \end{matrix} ) ,

(ρ̄_{0,p} ⊗ ϕ̄_{p})_{I_{p}} ∼ ( \begin{matrix}β̄·(ϕ̄_{p})_{G_{M}} & 0 \\ 0 & β̄^{σ}·(ϕ̄_{p})_{G_{M}} \end{matrix} )

and thus again

δ_{p}(ϕ_{p}) = 0 .

(3) Suppose finally that ρ̄_{0,p} is reducible but not semisimple on I_{p}. Then Imρ̃_{0,p} is either cyclic of order m divisible by ℓ, or is dihedral of order 2m with m divisible by ℓ. Hence these cases may only occur for ℓ ∈ {3, 5}, and we have m = ℓ.

(3a) In the first case we have

ρ_{0,p} ∼ ( \begin{matrix}ϵ_{p} & 0 \\ 0 & 1 \end{matrix} )

whence a priori

dim(ρ_{0,p} ⊗ ϕ_{p})^{I_{p}} = \{ \begin{matrix}1 & if ϕ_{p} ∈ \{1, ϵ_{p}^{−1}\} \\ 0 & otherwise, \end{matrix}

where however ϕ_{p} = ϵ_{p}^{−1} does not occur because ϕ_{p} has order prime to ℓ whereas ϵ_{p} has order ℓ.

On the other hand,

ρ̄_{0,p} ∼ ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} )

with ∗ ≠ 0 and so

dim(ρ̄_{0,p} ⊗ ϕ̄_{p})^{I_{p}} = \{ \begin{matrix}1 & if ϕ̄_{p} = 1 \\ 0 & otherwise. \end{matrix}

We obtain

δ_{p}(ϕ_{p}) = 0 .

(3b) In the second case we have as in (2) above that ρ_{0,p} = Ind_{M/ℚ_{p}}(β) with M/ℚ_{p} a quadratic extension and β a ramified character on G_{M}, and

(ρ_{0,p} ⊗ ϕ_{p})_{I_{p}} ∼ ( \begin{matrix}β·(ϕ_{p})_{G_{M}} & 0 \\ 0 & β^{σ}·(ϕ_{p})_{G_{M}} \end{matrix} )

with σ the non-trivial automorphism of M/ℚ_{p}. Noticing that (ϕ_{p})_{G_{M}} is fixed under conjugation with σ whereas β is not (β^{−1}β^{σ} is a character or order m), we obtain

dim(ρ_{0,p} ⊗ ϕ_{p})^{I_{p}} = 0 .

On the other hand, denoting by α the restriction to I_{p} of the quadratic character corresponding to M/ℚ_{p}, we have

(ρ̄_{0,p} ⊗ ϕ̄_{p})_{I_{p}} ∼ ( \begin{matrix}ᾱϕ̄_{p} & ∗ \\ 0 & ϕ̄_{p} \end{matrix} )

so that

dim(ρ̄_{0,p} ⊗ ϕ̄_{p})^{I_{p}} = \{ \begin{matrix}1 & if ϕ̄_{p} = ᾱ \\ 0 & otherwise. \end{matrix}

Consequently,

δ_{p}(ϕ_{p}) = \{ \begin{matrix}1 & if ϕ̄_{p} = ᾱ \\ 0 & otherwise. \end{matrix}

3.3. Local lifts: The weight. In this section we define in dependence on the extension K_{𝔩}/ℚ_{ℓ} a set 𝔚_{ℓ} of natural numbers so that k ∈ 𝔚_{ℓ} may be deduced.

We start by quoting two theorems about the structure of the restriction ρ̄_{ℓ} of our given modular mod ℓ representation ρ̄ to the decomposition group G_{ℚ_{ℓ}} ≤ G_{ℚ}. Let the corresponding mod ℓ cusp form be denoted by f. Thus f is a form of weight k ∈ \{2, …, ℓ − 1\}. Denote by a_{n} ∈ \overline{𝔽}_{ℓ} the Fourier coefficients of f. The first theorem is due to Deligne; a proof can be found in [17]. The statement is that if a_{ℓ} ≠ 0 then ρ̄_{ℓ} is reducible and more precisely has the shape

ρ̄_{ℓ} ∼ (\begin{matrix}u · χ^{k−1} & ∗ \\ 0 & v \end{matrix})

where u and v are unramified characters. The second theorem – due to Fontaine with a proof to be found in [16] – states that if a_{ℓ} = 0 then ρ̄_{ℓ} is irreducible and

(ρ̄_{ℓ})_{|I_{ℓ}} ∼ (\begin{matrix}(ψ′)^{k−1} & 0 \\ 0 & ψ^{k−1} \end{matrix})

where ψ, ψ′: I_{ℓ} → \overline{𝔽}_{ℓ} are the 2 fundamental characters of level 2.

As usual, we refer to these two cases as the ordinary and the supersingular case respectively.

Now on the other hand we have defined above in section 3.1 a certain lift ρ_{0} of ρ̃_{0}, and we know that

ρ̄_{ℓ} ≅ ρ̄_{0,ℓ} ⊗ χ^{i} · (unramified character) ,

for some i, where ρ̄_{0,ℓ} denoted the restriction of the mod ℓ reduction of ρ_{0} to the decomposition group G_{ℚ_{ℓ}} ≤ G_{ℚ}. We proceed now to compare these two descriptions of ρ̄ in order to make statements about k.

The first observation is that if K_{𝔩}/ℚ_{ℓ} were unramified, we would obviously be in the ordinary case whence the assumption 2 ≤ k ≤ ℓ − 1 would imply the contradiction k = 1. So we may start by defining

𝔚_{ℓ} := ∅ if K_{𝔩}/ℚ_{ℓ} unramified.

Secondly, as ℓ is odd the extension K_{𝔩}/ℚ_{ℓ} is either cyclic or dihedral. We split up the discussion into 5 subcases.

3.3.1. Tamely ramified cyclic case. Suppose that K_{𝔩}/ℚ_{ℓ} is tamely ramified and cyclic of order m ∈ {2, 3, 4, 5}. Thus, ℓ ≡ 1 mod m. As ℓ ∤ m we have that ρ̄_{0,ℓ} is semisimple and reducible. We are then in the ordinary case and necessarily χ^{k−1} has order m. On the other hand, this order is \frac{ℓ−1}{(ℓ−1,k−1)} so we see that

k − 1 = a · \frac{ℓ − 1}{m}

for some natural number a which must be less than m, as k ≤ ℓ − 1.

So, if m = 2, 3, 5 we have a = 1, a ∈ {1, 2} and a ∈ {1, 2, 3, 4} respectively. If m = 4 we must have a ∈ {1, 3} as a = 2 would imply O(χ^{k−1}) = 2. So we may define

𝔚_{ℓ} = 𝔚_{ℓ}(K_{𝔩}/ℚ_{ℓ}) := \{ \begin{matrix}\{\frac{ℓ+1}{2}\} & for m = 2 \\ \{\frac{ℓ+2}{3}, \frac{2ℓ+1}{3}\} & for m = 3 \\ \{\frac{ℓ+3}{4}, \frac{3ℓ+1}{4}\} & for m = 4 \\ \{\frac{ℓ+4}{5}, \frac{2ℓ+3}{5}, \frac{3ℓ+2}{5}, \frac{4ℓ+1}{5}\} & for m = 5. \end{matrix}

3.3.2. Wildly ramified cyclic case. If K_{𝔩}/ℚ_{ℓ} is wildly ramified cyclic of order m, we have that ρ̄_{0,ℓ} is twist-equivalent to

(\begin{matrix}u & ∗ \\ 0 & v \end{matrix})

with unramified characters u, v and ∗ ≠ 0. We deduce the contradiction k = 1 and may thus define 𝔚_{ℓ} := ∅ in this case.

3.3.3. Wildly ramified dihedral cases. If K_{𝔩}/ℚ_{ℓ} is wildly ramified dihedral of order 2m then as ℓ is odd we must have ℓ | m and thus m = ℓ. Consequently, ℓ ∈ {3, 5}. Now, K_{𝔩}/ℚ_{ℓ} has a unique quadratic subextension M/ℚ_{ℓ} and we may choose our lift ρ_{0} in such a way that ρ̄_{0,ℓ} modulo a twist by an unramified character appears in the form

ρ̄_{0,ℓ} ∼ (\begin{matrix}α & ∗ \\ 0 & 1 \end{matrix})

where α is the quadratic character corresponding to M/ℚ_{ℓ}. So, if M/ℚ_{ℓ} is unramified, that is, if K_{𝔩}/ℚ_{ℓ} is wildly but not totally ramified, we deduce the contradiction k = 1. On the other hand, if M/ℚ_{ℓ} is ramified, i.e. K_{𝔩}/ℚ_{ℓ} is totally and wildly ramified, we must have α = χ^{k−1} and consequently k = \frac{ℓ+1}{2} as α has order 2. So, we define

𝔚_{ℓ} := \{ \begin{matrix}∅ & for K_{𝔩}/ℚ_{ℓ} wildly but not totally ramified \\ \{\frac{ℓ+1}{2}\} & for K_{𝔩}/ℚ_{ℓ} totally and wildly ramified. \end{matrix}

3.3.4. Tamely ramified dihedral case. Suppose finally that K_{𝔩}/ℚ_{ℓ} is tamely ramified dihedral of order 2m, m ∈ {2, 3, 4, 5}. Then K_{𝔩} is cyclic of degree m over the unramified quadratic extension M/ℚ_{ℓ} and ℓ ≡ −1 mod m. The representation has the form ρ̄_{0,ℓ} = Ind_{M/ℚ_{ℓ}}(β) for a tamely ramified character β on G_{M}. Thus, we are in the supersingular case and

(ρ̄_{0,ℓ})_{|I_{ℓ}} ∼ (\begin{matrix}β & 0 \\ 0 & β^{σ} \end{matrix})

where σ denotes the non-trivial automorphism of M/ℚ_{ℓ}. The characters β, β^{σ} are of level 2, we have

β^{σ} = β^{ℓ}

and m is the order of the character β^{σ}β^{−1} = β^{ℓ−1}. Denote as above by ψ, ψ′ the two fundamental characters of level 2. Interchanging ψ and ψ′ if necessary we may write

β = ψ^{r}(ψ′)^{s} , β^{σ} = ψ^{s}(ψ′)^{r}

for integers r, s with 0 ≤ r < s ≤ ℓ − 1. Then

(ρ̄_{0,ℓ} ⊗ χ^{−r})_{|I_{ℓ}} ∼ (\begin{matrix}(ψ′)^{a} & 0 \\ 0 & ψ^{a} \end{matrix})

with a := s − r which must satisfy 1 ≤ a ≤ ℓ − 2 because ρ̄_{0,ℓ} is irreducible. Now,

ψ^{a}(ψ′)^{−a} = ψ^{a·(ℓ−1)}

must still have order m and this allows us to determine the possibilities for a: As ψ has order ℓ^{2} − 1 the number a must have the form

a = a_{0} · \frac{ℓ + 1}{m}

with 1 ≤ a_{0} ≤ m. If m = 4 then a_{0} ≠ 2 because ψ^{a·(ℓ−1)} would otherwise have order 2.

On the other hand we know that the restriction (ρ̄_{ℓ})_{|I_{ℓ}} has the form

(ρ̄_{ℓ})_{|I_{ℓ}} ∼ (\begin{matrix}(ψ′)^{k−1} & 0 \\ 0 & ψ^{k−1} \end{matrix})

and is the twist by a power of χ = ψ^{ℓ+1} of the above (ρ̄_{0,ℓ})_{|I_{ℓ}}. We deduce that k − 1 has form a + i(ℓ + 1) or ℓa + i(ℓ + 1) for some i, and since k − 1 is a number in {1, …, ℓ − 2} we find k = a + 1 or k = ℓ + 2 − a where a is one of the numbers given above. We can then check that k ∈ 𝔚_{ℓ} if we define the set 𝔚_{ℓ} thus:

𝔚_{ℓ} = 𝔚_{ℓ}(K_{𝔩}/ℚ_{ℓ}) := \{ \begin{matrix}\{\frac{ℓ+3}{2}\} & for m = 2 \\ \{\frac{ℓ+4}{3}, \frac{2ℓ+5}{3}\} & for m = 3 \\ \{\frac{ℓ+5}{4}, \frac{3ℓ+7}{4}\} & for m = 4 \\ \{\frac{ℓ+6}{5}, \frac{2ℓ+7}{5}, \frac{3ℓ+8}{5}, \frac{4ℓ+9}{5}\} & for m = 5. \end{matrix}

3.4. End of proof. We finish the proof of Theorem 1 by proving part (b). So, let ϕ, N, k and ν be given with the properties and relations stated in the theorem. In particular we have that k is a number ≤ ℓ − 1 in the set 𝔚_{ℓ} = 𝔚_{ℓ}(K_{𝔩}/ℚ_{ℓ}), and because of our definition of 𝔚_{ℓ} we have that K is ramified over ℓ. Choose an embedding G ↪ PGL_{2}(ℂ); this gives us a projective Galois representation

ρ̃_{0} : G_{ℚ} ⟶ PGL_{2}(ℂ) .

Denote by

ρ_{0} : G_{ℚ} ⟶ GL_{2}(ℂ)

the lift of ρ̃_{0} that we specified in section 3.1 above. The hypothesis that K not be totally real implies that ρ_{0} is an odd representation. So, ρ_{0} is modular of weight 1 (by Langlands-Tunnell if G ≅ A̸_{5}, and by assumption if G ≅ A_{5}). Accordingly, the mod ℓ reduction ρ̄_{0} of ρ_{0} is an irreducible modular mod ℓ representation. The same is then true of the representation

ρ̄:= ρ̄_{0} ⊗ ϕ̄

with ϕ̄ the mod ℓ reduction of ϕ. By construction ρ̄ has conductor N and character ν. Now it follows from Ribet’s fundamental work [29] on ‘level-lowering’ as complemented by Diamond in [11], Corollary 1.2, that ρ̄ is the mod ℓ representation attached to a mod ℓ cusp form of type (N, k′, ν) where k′ is the weight attached to ρ̄ by Serre (as modified in [16]). Tensoring with some power of χ if necessary, we may further assume that k′ ≤ ℓ + 1. Now, it is a simple matter to check from the definition of k′ as given by Serre-Edixhoven, that in any of our cases we must actually have k′ ≤ ℓ.

We claim now that k′ ∈ 𝔚_{ℓ}(K_{𝔩}/ℚ_{ℓ}), though possibly only after tensoring with some power of χ. First notice that if k′ = 1 then ρ̄ is unramified at ℓ by [7], Corollary 0.2; this is contrary to our assumptions, so k′ ≥ 2.

Suppose then that k′ = ℓ and that we are in the ordinary case. By the theorem of Deligne quoted above in section 3.3 we have then

ρ̄_{ℓ} := (ρ̄)_{G_{ℚ_{ℓ}}} ∼ (\begin{matrix}u & ∗ \\ 0 & v \end{matrix})

for some unramified characters u, v. Then ∗ ≠ 0 again because ρ̄ is ramified at ℓ. We deduce that ℓ ∈ {3, 5} and that K_{𝔩}/ℚ_{ℓ} is either wildly ramified cyclic of order ℓ, or wildly but not totally ramified dihedral of order 2ℓ. Again, this is in contradiction to our assumption k ∈ 𝔚_{ℓ} and the definition of 𝔚_{ℓ} in these cases.

Assume then k′ = ℓ but that we are in the supersingular case so that ρ̄_{ℓ} has a projective kernel field which is tamely ramified of order 2m, m ∈ {2, 3, 4, 5}. Now, the theorem of Fontaine quoted above implies that we have

(ρ̄_{ℓ})_{I_{ℓ}} ∼ (\begin{matrix}(ψ′)^{ℓ−1} & 0 \\ 0 & ψ^{ℓ−1} \end{matrix}) ,

and we deduce as above in section 3.3 that ψ^{(ℓ−1)^{2}} has order m. On the other hand, this order is \frac{ℓ+1}{2}, so (m, ℓ) is one of the following: (2,3), (3,5), (4,7). Now, it follows from [16], Proposition 3.3, that for a suitable power χ^{j} of χ, the representation ρ̄ ⊗ χ^{j} is modular of weight 3. As one checks that in fact 3 ∈ 𝔚_{ℓ} in all cases, we are done.

We have now seen that our desired conclusion, i.e. the existence of a mod ℓ cusp form of type (N, k, ν) with associated projective kernel field K is true for at least one number k in the set 𝔚_{ℓ}(K_{𝔩}/ℚ_{ℓ}). In order to finish the proof we must now show the following two statements:

(1) In case either ℓ = 5 or K_{𝔩}/ℚ_{ℓ} has degree not divisible by 5, we must show that (N, k̃, ν) ‘occurs’ in the above sense for every number k̃ ∈ 𝔚_{ℓ} which is ≤ ℓ − 1.

(2) In case ℓ ≠ 5 and K_{𝔩}/ℚ_{ℓ} has degree divisible by 5, we must define a partition 𝔚_{ℓ} = 𝔚_{ℓ}^{+} ∪ 𝔚_{ℓ}^{−} and prove the existence of a sign μ ∈ {+, −} such that (N, k̃, ν) occurs for every number k̃ ∈ 𝔚_{ℓ}^{μ} which is ≤ ℓ − 1.

This is essentially accomplished via the theory of companion forms [17], [7], and the theory of θ-cycles [20], [16], applied to the ordinary and supersingular cases respectively: Suppose that |𝔚_{ℓ}| > 1, and that f is a mod ℓ cusp form of type (N, k, ν) where k ∈ 𝔚_{ℓ}. Let ρ̄_{f} denote the mod ℓ representation attached to f.

Suppose first that K_{𝔩}/ℚ_{ℓ} is tamely ramified cyclic of degree m ∈ {3, 4, 5}. According to the main results of [17] and [7], the form f has a companion form of weight k_{1} := ℓ + 1 − k. One checks that if m is 3 or 4 then 𝔚_{ℓ} consists precisely of the numbers k and k_{1}. If m = 5, we define

𝔚_{ℓ}^{+} := \{\frac{ℓ + 4}{5}, \frac{4ℓ + 1}{5}\} , 𝔚_{ℓ}^{−} := \{\frac{2ℓ + 3}{5}, \frac{3ℓ + 2}{5}\}

so that

𝔚_{ℓ} = 𝔚_{ℓ}^{+} ∪ 𝔚_{ℓ}^{−} ,

and one checks that if a subset 𝔚_{ℓ}^{μ} contains k, then it also contains k_{1}.

Suppose then that K_{𝔩}/ℚ_{ℓ} is tamely ramified dihedral of order 2m, m ∈ {3, 4, 5}, so that we are in the supersingular case. Then we first observe that any number in 𝔚_{ℓ} is ≥ 3 except if ℓ = 3 and m = 4; however, in the latter case we have 𝔚_{ℓ} = {2, 4} and there is nothing to show, as 4 > ℓ − 1 in this case. So, we may assume k ≥ 3. Now the theory of θ-cycles [20], [16], see specifically Proposition 3.3 in [16], implies that ρ̄_{f} ⊗ χ^{ℓ+1−k} is modular of type (N, ℓ + 3 − k, ν) (and obviously with K its projective kernel field). One checks that for m = 3 or 4, the sum of the 2 numbers in 𝔚_{ℓ} is indeed ℓ + 3, so we are done in those cases. If m = 5 we can split 𝔚_{ℓ} into 2 subsets:

𝔚_{ℓ} = 𝔚_{ℓ}^{+} ∪ 𝔚_{ℓ}^{−}

with

𝔚_{ℓ}^{+} := \{\frac{ℓ + 6}{5}, \frac{4ℓ + 9}{5}\} , 𝔚_{ℓ}^{−} := \{\frac{2ℓ + 7}{5}, \frac{3ℓ + 8}{5}\} .

As the sum of the 2 numbers in each 𝔚_{ℓ}^{μ} is ℓ + 3, we are done. Proof of the Corollary: If ℓ ≤ k the conclusion of the corollary is true. Otherwise, we are in the situation of Theorem 1, and the proof is finished by reviewing the definitions of the sets 𝔚_{ℓ}. Thus for example, if Gal(K/ℚ) ≅ A_{5} one finds that Theorem 1 implies

k ≥ \frac{ℓ + 4}{5}

and thus the desired conclusion. Similarly, the conclusion is checked in the 2 other cases.

4. Examples.

In this section we give a few simple examples of the use of Theorem 1 in connection with questions about the image of modular mod ℓ representations.

4.1. An interesting class of S_{4}-type representations arises in connecting with nonreal S_{4}-extensions K/ℚ where K_{𝔩}/ℚ_{ℓ} is dihedral of order 8. In that case, we must have ℓ ≡ 3 (4) as there are no dihedral extensions of ℚ_{ℓ} of order 8 if ℓ ≡ 1 (4). Theorem 1 informs us that there are exceptional modular mod ℓ representations of weights

\frac{ℓ + 5}{4} and \frac{3ℓ + 7}{4}

associated with such a field K. Let us give a few concrete examples.

4.1.1. Suppose additionally that K/ℚ is unramified outside ℓ · ∞. In [15] such fields were studied using other methods than ours and under an additional condition on ℓ made to ensure the solvability of a certain embedding problem associated with lifting G_{ℚ} ↠ Gal(K/ℚ) ↪ PGL_{2}(ℂ) to a linear representation. By solving explicitly this embedding problem and studying the behavior of the solution locally at ℓ the author of that paper was able to find the connection to cusp forms of weight \frac{ℓ+5}{4}. Here we proceed without any further assumptions than those already made on K and conclude from Theorem 1 that there are modular mod ℓ representations of level 1 and the above weights with K the associated projective kernel field. The classical example of this phenomenon concerns the unique cusp form QΔ of level 1 and weight 16 in connection with ℓ = 59. This case was first considered by Swinnerton-Dyer in [36] who found strong reasons to believe that the mod 59 representation attached to QΔ is exceptional of type S_{4}. This was then subsequently proved in [18] by Haberland who – having less powerful theorems at his disposal than we do – had to work hard numerically with the Fourier expansion of QΔ. We can reprove the statement easily and without knowing any Fourier coefficients: One knows (cf. [18]) that there is a unique S_{4}-extension of ℚ unramified outside 59 and ℓ. In fact, the splitting field K of the polynomial

x^{4} − x^{3} − 7x^{2} + 11x + 3

is seen to be such a field. It is non-real, and dihedral of order 8 over 59. Theorem 1 implies the existence of modular mod 59 representations of level 1 and weights \frac{59+5}{4} = 16, \frac{3·59+7}{4} = 46 with K as projective kernel field. Since QΔ is the unique cusp form of level 1 and weight 16, the claim follows.

4.1.2. A somewhat more complicated example occurs in connection with the mod 11 representation ρ̄ attached to the unique cusp form η(2z)^{4} · η(4z)^{4} of weight 4 on Γ_{0}(8). Numerical experiments performed by the second author suggested strongly that ρ̄ is exceptional of S_{4}-type; verifying this was one of the initial starting points of the present article. If we wish to verify this via Theorem 1 we must begin by looking for a candidate K/ℚ for the corresponding projective kernel field. Thus, K/ℚ should a non-real S_{4}-extension unramified outside 2 · 11 · ∞. By class field theory one finds that the unique S_{3}-subextention L/ℚ contained in K/ℚ would have to be the splitting field of

x^{3} − x^{2} − x − 1 ,

which is in fact an S_{3}-extension of ℚ unramified outside 2 · 11 · ∞. Now we have to look for an embedding of this L/ℚ in an S_{4}-extension K/ℚ of the desired type. It turns out that there is more than one such K/ℚ but we shall focus on the one that is relevant for our example. We seek to find the candidate K/ℚ by utilizing the method of [2]:

Consider the elliptic curve −22y^{2} = x^{3}−x^{2}−x−1 which has minimal Weierstrass model

E: y^{2} = x^{3} + x^{2} − 645x + 14771 .

The curve E has good reduction outside 2 · 11, and has the rational point

P = (−26, 121) ∈ E(ℚ)\2 · E(ℚ) .

According to [2], the field generated over ℚ by the x-coordinates of points Q ∈ E(\overline{ℚ}) with 2·Q = P is a (non-real) S_{4}-extension K/ℚ unramified outside 2·11·∞. Using the 2-division polynomial for E we find that this K is the splitting field of the polynomial

x^{4} + 104x^{3} + 1394x^{2} − 185248x + 1893125 .

Applying to this polynomial the function ‘polred’ in PARI we find that K can also be given as the splitting field of

x^{4} − 2x^{3} − 4x^{2} + 16x − 13 .

One can then verify directly that K/ℚ is indeed a non-real S_{4}-extension unramified outside 2 · 11 · ∞, that the local extension over 11 is the unique D_{4}-extension of ℚ_{11}, and finally that the local extension over 2 is the S_{4}-extension that was denoted M_{4} in [21] and in section 3.1 above. Thus, n_{2}(1) = 3, ϵ_{2} = 1, and 𝔚_{11} = {4, 10}. Theorem 1 then implies the existence of an exceptional S_{4}-type modular mod 11 representation of level 8, weight 4 and trivial nebentypus, which must then be our representation ρ̄.

Similarly, one can show that the mod 19 representation attached to the unique cusp form η(2z)^{12} of weight 6 on Γ_{0}(4) is exceptional of S_{4}-type with projective kernel field the splitting field of

x^{4} − x^{3} − 2x^{2} − 6x − 2 .

4.2. An interesting A_{5}-type example occurs in Ribet’s paper [30], p. 284: There are exactly 2 newforms of weight 2 on Γ_{0}(23); call them f_{1}, f_{2}. The example concerns the mod 3 representations ρ̄_{i} attached to f_{i}. Arguments given loc. cit. suggested that each of the representations ρ̄_{i} is of A_{5}-type, and in fact with projective kernel field the A_{5}-extension of ℚ given as the splitting field K of the polynomial

(∗)  x^{5} + 3x^{3} + 6x^{2} + 9 .

It was further stated loc. cit. that computations performed by Mestre had confirmed this. We shall show here how Theorem 1 immediately implies this statement under the assumption of modularity a 2-dimensional complex representation associated with K.

The polynomial (∗) occurs as the second entry of the table in [1], and if we choose an embedding A_{5} ↪ PGL_{2}(ℂ) we obtain a complex, odd A_{5}-type representation ρ of conductor 3^{3} · 23^{2}. It should be fairly clear that a rigorous verification of the above statement about the ρ̄_{i} must involve either a proof of the Artin conjecture of this complex ρ or else the application of some form of explicit Chebotarev. The authors have not considered how to practically implement the second option. Concerning the first, let us remark that the results of the papers [5] and [38] do not suffice to prove the Artin conjecture for ρ (in K, a Frobenius over 2 has order 5, and 3 is totally ramified of degree 6). Furthermore, an attempt at a direct computational verification of Artin’s conjecture for this ρ utilizing methods similar to those employed in [22], can be estimated to lead to an intractably large computation. Thus, we shall be content with giving a relative statement: Assuming the weight 1 modularity for this particular ρ, the ρ̄_{i} are both of A_{5}-type with projective kernel field the above K. This is readily done: Denoting by 𝔭_{23} and 𝔭_{3} primes of K over 23 and 3 respectively, one finds that K_{𝔭_{23}}/ℚ_{23} is the unique dihedral extension of degree 6 of ℚ_{23}, and that K_{𝔭_{3}}/ℚ_{3} is totally ramified dihedral of degree 6. Consequently we have n_{23}(1) = 1, ϵ_{23} = 1, and 𝔚_{3} = {2}. Theorem 1 thus implies the existence of a modular mod 3 representation ρ̄ of level 23, weight 2 and trivial character. On the other hand, choosing the other embedding A_{5} ↪ PGL_{2}(ℂ), we obtain similarly another modular mod 3 representation which is not equivalent to ρ̄ but has the same level, weight and character. As dim S_{2}(Γ_{0}(23)) = 2, the desired conclusion follows.

5. Additional remarks.

5.1. The purpose of Theorem 1 is to give the possible Serre invariants (N, k, ν) of mod ℓ representations of one of the exceptional types A_{4}, S_{4}, or A_{5} in terms of local data attached to a fixed projective kernel field K. One may of course reverse the question, i.e. start with a triple (N, k, ν) and ask whether there are any cusp forms with these data and an attached exceptional mod ℓ representation. In connection with this question, the Theorem will immediately give information about whether there are any local Galois theoretic obstructions for this to be the case, and if there are not, will give detailed information about the local structure of possible corresponding projective kernel fields. This information will suffice to determine all possible candidates for these projective kernel fields, either via class field theory in the A_{4}- and S_{4}-cases, or via geometry of numbers (or tables) in the A_{5}-case. This gives an algorithm which will decide by a finite amount of computation the above question, – again of course under the assumption of the Artin conjecture in the A_{5} case. Examples of this were given above in section 4. In these examples the dimension of the space of cusp forms in question was sufficiently low that we could actually ‘point’ to the exceptional form. In general however, the theorem will only inform us about the existence of an exceptional mod ℓ cusp form with the given data (N, k, ν), though – as was pointed out above – we may in fact count the number of such forms. If one wishes additionally to ‘point’ to these forms one must first realize that all this can mean in general is to give an algorithm that computes their Fourier expansions. We shall now briefly indicate how this can be done.

Starting with an extension K/ℚ of A_{4}, S_{4} or A_{5} type, we choose an embedding Gal(K/ℚ) ↪ PGL_{2}(ℂ) and obtain thus a complex projective Galois representation ρ̃. The Fourier expansion that we desire is obtained via reduction mod ℓ from the L-series of a twist of a lift of ρ̃ to a linear representation G_{ℚ} ⟶ GL_{2}(ℂ). If one uses the results the papers [26], [8], [9], and [10], one has an algorithm that computes the L-series of some lift ρ_{1} of ρ̃ (corresponding to a central embedding problem with cyclic kernel of 2-power order). The desired Fourier expansions can be computed from twists of the mod ℓ reduction ρ̄_{1} of ρ_{1}. Now the problem is that one does not a priori know how the conductor of ρ̄_{1} behaves under twists. However, this can easily be determined by Theorem 1 or rather the proof if it: The representation ρ̄_{1} is a twist of the mod ℓ reduction ρ̄_{0} of the special linear lift ρ_{0} occurring in the proof of Theorem 1 (section 3 above), say

ρ̄_{1} ≅ ρ̄_{0} ⊗ ξ .

If we can determine ξ then because we have complete information about the behavior of ρ̄_{0} with respect to twists, we have determined the Serre invariants of all twists of ρ̄_{1} (essentially parametrized by the conductor of the character by which we are twisting). But ξ is easily determined: Let Σ denote the product of ℓ and the finite primes at which ρ̄_{0} or ρ̄_{1} ramifies. As ρ_{1} is given via an explicitly solved embedding problem, one can determine (ρ̄_{1})_{|I_{p}} for each of the finitely many primes p | Σ. On the other hand (ρ̄_{0})_{|I_{p}} is known by construction. Hence, the quantity ξ_{p} := ξ_{|I_{p}} can be determined for p | Σ. Identifying ξ_{p} with a character on U_{p}, we have ξ explicitly determined as a mod ℓ Dirichlet character by

ξ: r ↦ ∏_{p|Σ} ξ_{p}(r)^{−1} , for (r, Σ) = 1.

We can also use the global information obtained by solving an embedding problem to resolve the indeterminacy in part (b) of Theorem 1: In case that ℓ ≠ 5 and K_{𝔩}/ℚ_{ℓ} has degree divisible by 5, the theorem informs us – under the appropriate modularity assumption – that ρ̄_{0} is mod ℓ modular of type (N_{0}, k, ν_{0}) for any of the 2 numbers in 𝔚_{ℓ}^{μ} for some choice of sign μ ∈ {+, −}. However, since we now know

(ρ̄_{0})_{I_{ℓ}} ≅ (ρ̄_{1} ⊗ ξ^{−1})_{I_{ℓ}}

explicitly, we can – as is easily checked – determine μ by looking at this restriction.

5.2. Continuing the above analysis, if we have given a mod ℓ cusp form f̄ of minimal level and weight k ∈ {2, … ℓ−1}, and if we know that the Galois representation ρ̄_{f} attached to f̄ is irreducible, then we have – under the assumption of the Artin conjecture – an effective procedure which will determine whether ρ̄_{f} is of one of the exceptional types A_{4}, S_{4}, A_{5}. If it is not, the corresponding projective kernel field has Galois group either dihedral or isomorphic to PSL_{2}(𝔽_{ℓ^{m}}) or PGL_{2}(𝔽_{ℓ^{m}}) for some m. Since the dihedral case means that ρ̄_{f} is induced from a global 1dimensional character, it is easy to see – as was also noted in the introduction – that it can be effectively decided whether this case prevails. If it does not, we are in the PSL_{2} or PGL_{2} case and the question arises how we can effectively distinguish between these, and in particular how to determine m computationally. In general, these questions lead to certain not completely trivial considerations that we shall be addressing elsewhere.

kiming@math.ku.dk verrill@math.ku.dk

References

[1] J. Basmaji, I. Kiming: ‘A table of A_{5}-fields.’ On Artin’s conjecture for odd 2-dimensional representations, Lecture Notes in Math., Vol. 1585, Springer, 1994, pp. 37–46 and 122–141.

[2] P. Bayer, G. Frey: ‘Galois representations of octahedral type and 2-coverings of elliptic curves.’ Math. Z. 207 (1991), 395–408.

[3] J. P. Buhler: Icosahedral Galois representations, Lecture Notes in Mathematics 654, (Springer, [4] K. M. Buzzard: ‘The levels of modular representations.’ Thesis, Cambridge 1995.

[5] K. M. Buzzard, M. Dickinson, N. Shepherd-Barron, R. Taylor: ‘On icosahedral Artin representations.’ Preprint, 1999.

[6] H. Carayol: ‘Sur les représentations galoisiennes modulo ℓ attachées aux formes modulaires.’ Duke Math. J. 59 (1989), 785–801.

[7] R. F. Coleman, J. F. Voloch: ‘Companion forms and Kodaira-Spencer theory.’ Invent. Math. 110 (1992), 263–281.

[8] T. Crespo: ‘Central extensions of the alternating group as Galois groups.’ Acta Arith. 66 (1994), 229–236.

[9] T. Crespo: ‘Galois realization of central extensions of the symmetric group with kernel a cyclic 2-group.’ Acta Arith. 70 (1995), 183–192.

[10] T. Crespo: ‘Construction of 2^{m}S_{n}-fields containing a C_{2^{m}}-field.’ J. Algebra 201 (1998), 233–242.

[11] F. Diamond: ‘The refined conjecture of Serre.’ Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), 22–37, Ser. Number Theory, I, Internat. Press, Cambridge, MA, 1995.

[12] F. Diamond, R. Taylor: ‘Non-optimal levels of mod l modular representations.’ Invent. Math. 115 (1994), 435–462.

[13] L. E. Dickson: Linear groups with an exposition of the Galois field theory, (Teubner, Leipzig 1901).

[14] L. Dieulefait, N. Vila: ‘Projective Linear Groups as Galois groups over ℚ via Modular Representations.’ To appear in J. Symbolic Comput. 2000.

[15] D. Doud: ‘S_{4} and S̃_{4} extensions of ℚ ramified at only one prime.’ J. Number Theory 75 (1999), 185–197.

[16] B. Edixhoven: ‘The weight in Serre’s conjectures on modular forms.’ Invent. Math. 109 (1992), 563–594.

[17] B. Gross: ‘A tameness criterion for Galois representations associated to modular forms (mod p).’ Duke Math. J. 61 (1990), 445–517.

[18] K. Haberland: ‘Perioden von Modulformen einer Variabler und Gruppenkohomologie, III.’ Math. Nachr. 112 (1983), 297–315.

[19] B. Huppert: Endliche Gruppen I, (Springer, 1983).

[20] N. Jochnowitz: ‘A study of the local components of the Hecke algebra mod l.’ Trans. Amer. Math. Soc. 270 (1982), 253–267.

[21] I. Kiming: ‘On the liftings of 2-dimensional projective Galois representations over Q.’ J. Number Theory 56 (1996), 12–35.

[22] I. Kiming, X. Wang: ‘Examples of 2-dimensional, odd Galois representations of A_{5}-type over ℚ satisfying the Artin conjecture.’ On Artin’s conjecture for odd 2-dimensional representations, Lecture Notes in Math., Vol. 1585, Springer, 1994, pp. 109–121.

[23] J. C. Lagarias, H. L. Montgomery, A. M. Odlyzko: ‘A bound for the least prime ideal in the Chebotarev density theorem.’ Invent. Math. 54 (1979), 271–296.

[24] R. Livné: ‘On the conductors of mod l Galois representations coming from modular forms.’ J. Number Theory 31 (1989), 133–141.

[25] F. Momose: ‘On the l-adic representations attached to modular forms.’ J. Fac. Sci. Univ. Tokyo 28 (1981), 89–109.

[26] J. Quer: ‘Liftings of projective 2-dimensional Galois representations and embedding problems.’ J. Algebra 171 (1995), 541–566.

[27] K. A. Ribet: ‘On ℓ-adic representations attached to modular forms.’ Invent. Math. 28 (1975), 245–275.

[28] K. A. Ribet: ‘On l-adic representations attached to modular forms. II. Glasgow Math. J. 27 (1985), 185–194.

[29] K. A. Ribet: ‘On modular representations of Gal(Q̄/Q) arising from modular forms.’ Invent. Math. 100 (1990), 431–476.

[30] K. A. Ribet: ‘Images of semistable Galois representations.’ Pacific J. Math. 181 (1997), 277–297.

[31] J.-P. Serre: ‘Congruences et formes modulaires [d’après H. P. F. Swinnerton-Dyer].’ Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, 319–338. Oevres III, pp. 74–88.

[32] J.-P. Serre: ‘Modular forms of weight one and Galois representations.’ Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 193–268. Academic Press, London, 1977.

[33] J.-P. Serre: ‘Sur les représentations modulaires de degré 2 de Gal(Q̄/Q).’ Duke Math. J. 54 (1987), 179–230.

[34] O. Schreier, B. L. van der Waerden: ‘Die Automorphismen der Projektiven Gruppen.’ Abh. Math. Sem. Univ. Hamburg 6 (1928), 303–322.

[35] J. Sturm: ‘On the congruence of modular forms.’ Number Theory, New York, 1984-85, Lecture Notes in Math., Vol. 1240, Springer, 1987, pp. 275–280.

[36] H. P. F. Swinnerton-Dyer: ‘On l-adic representations and congruences for coefficients of modular forms.’ Modular functions of one variable, III, Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 1–55.

[37] H. P. F. Swinnerton-Dyer: ‘On l-adic representations and congruences for coefficients of modular forms. II.’ Modular functions of one variable, V, Lecture Notes in Math., Vol. 601, Springer, Berlin, 1977, pp. 63–90.

[38] R. Taylor: ‘On icosahedral Artin representations II.’ Preprint, 2000.

[39] A. Weil: ‘Exercices dyadiques.’ Invent. Math. 27 (1974), 1–22.

[40] E.-W. Zink: ‘Ergänzungen zu Weils Exercises dyadiques.’ Math. Nachr. 92 (1979), 163–183.

Dept. of Mathematics, University of Copenhagen, Universitetsparken 5,, 2100 Copenhagen Ø, Denmark