On icosahedral Artin representations II

Richard Taylor ^{1} Department of Mathematics,

Harvard University, Cambridge, MA 02138,

U.S.A.

April 4, 2000

1Partially supported by NSF Grant DMS-9702885

Introduction

This paper is a sequel to [BDST]. In that paper we proved the Artin conjecture for certain odd icosahedral representations of Gal (ℚ^{ac}/ℚ) by proving that they were modular. In this paper we will prove a variant of this result with different local hypotheses. Neither set of hypotheses is strictly weaker/stronger than the other. The key innovation in this paper is to work with the prime 5 rather than the prime 2. To do this we have to prove Serre’s conjecture for many continuous representations \overline{ρ} : Gal (ℚ^{ac}/ℚ) → GL_{2}(𝔽_{5}^{ac}) which have projective image A_{5}. (Note that such representations do not have cyclotomic determinant.) The key to doing this is to combine base change arguments with a beautiful method of Ramakrishna (see [R]) and with an extension to totally real fields of results of Wiles and of the author and Wiles ([W],[TW]). (Such a generalisation can be found in [SW2], however we make no use of the main innovation of [SW2]. There are two key ingredients in the result we do use. One is due independently to Diamond [Dia] and Fujiwara [F], the other is due to Skinner and Wiles [SW1]. Results along the lines of the one we use have been previously announced by Fujiwara.) We no longer have to make appeal to the main results of [SBT] and [Dic], but we do still make essential use of [BT].

More precisely an example of our main result is the following.

Theorem A Let ρ : 𝔾_{ℚ} → GL_{2}(ℂ) be an irreducible continuous representation with det ρ(c) = −1 (where c denotes a complex conjugation). Suppose moreover that if the projective image of ρ is isomorphic to A_{5} then the projective image of the inertia group at 3 has odd order and the projective image of the decomposition group at 5 is unramified of order 2. Then ρ is modular and its Artin L-function, L(ρ, s), is entire.

This paper is organised as follows. In the first section we prove a slight extension of one of the main results of [R]. We emphasise that the method is entirely Ramakrishna’s, we simply make some minor technical improvements. In the second section we apply this result to prove our main theorems.

Notation

If K is a perfect field we will let K^{ac} denote its algebraic closure and G_{K} denote its absolute Galois group Gal (K^{ac}/K). If moreover p is a prime number different from the characteristic of K then we will let ϵ_{p} : G_{K} → ℤ_{p}^{×} denote the p-adic cyclotomic character and ω_{p} the Teichmuller lift of ϵ_{p} mod p. If V is a ℤ_{p}[G_{K}] module we will write V (n) for V ⊗_{ℤ_{l}} ℤ_{l}(ϵ_{p}^{n}). If K is a local field we will let W_{K} denote the Weil group of K. If K is a number field and x is a finite place of K we will write G_{x} for a decomposition group above x, I_{x} for the inertia subgroup of G_{x} and Frob_{x} for an arithmetic Frobenius element in G_{x}/I_{x}. We will also let O_{K} denote the integers of K and k(x) denote the residue field of O_{K} at x. We will let c denote complex conjugation on ℂ.

We will write μ_{N} for the group scheme of N^{th} roots of unity. We will write W (k) for the Witt vectors of k. We will write ad^{0}ρ for the trace zero submodule of the adjoint Hom (ρ, ρ) of ρ.

Suppose that E/K is an elliptic curve. If m is a positive integer prime to the characteristic of K we will write \overline{ρ}_{E,m} for the representation of G_{K} on E[m](K^{ac}). If l is rational prime coprime to the characteristic of K, we will write T_{l}E for the l-adic Tate module of E, V_{l}E for T_{l}E ⊗_{ℤ} ℚ and ρ_{E,l} for the representation of G_{K} on V_{l}E.

Suppose that F is a totally real number field and that π is an algebraic cuspidal automorphic representation of GL_{2}(𝔸_{F}) with field of definition (or coefficients) M ⊂ ℂ. In some cases, including the cases that π_{∞} is regular and the case π_{∞} is weight (1, ..., 1), then it is known that M is a CM number field and that for each prime λ of O_{M} there is a continuous irreducible representation

ρ_{π,λ} : G_{F} → GL_{2}(M_{λ})

canonically associated to π. (See [Ta] for details.) We may always conjugate ρ_{π,λ} so that it is valued in GL_{2}(O_{M,λ}) and then reduce it to get a continuous representation G_{F} → GL_{2}(O_{M}/λ). If for one such choice of conjugate the resulting representation is irreducible then it is independent of the choice of conjugate and we will denote it \overline{ρ}_{π,λ}.

1 Generalisation of a result of Ramakrishna.

In this section we give a slight generalisation of a result of Ramakrishna [R]. We stress that both the result and the arguments we use are essentially his.

In this section we will let l denote an odd rational prime, ϵ denote ϵ_{l} and ω denote ω_{l}. We will also let k denote a finite extension of 𝔽_{l} and \overline{ρ} : G_{ℚ} → GL_{2}(k) a continuous representation such that \overline{ρ}(G_{ℚ}) is insoluble. Define a positive integer n as follows. If \overline{ρ}|_{G_{l}} is absolutely reducible set n = 1. Otherwise choose 1 ≤ n ≤ l − 1 such that

\overline{ρ}|_{I_{l}} ∼ ψ^{n+(l+1)m} ⊕ ψ^{nl+(l+1)m}

(over k^{ac}) where ψ is a fundamental character of level 2 (see [S]).

Let S denote a finite set of rational primes which contains l and all primes where \overline{ρ} is ramified. We will let G_{S} denote the Galois group of the maximal extension of ℚ which is unramified outside S. Thus \overline{ρ} factors through G_{S}. By a deformation of \overline{ρ} (resp. \overline{ρ}|_{G_{v}}) we shall mean a complete noetherian local ring (R, 𝔪) with residue field k and a continuous representation ρ : G_{S} → GL_{2}(R) (resp. ρ : G_{v} → GL_{2}(R)) such that (ρ mod 𝔪) = \overline{ρ} and ϵ^{−n} det ρ has finite order prime to l. In the global case we will write S-deformation if we wish to emphasise the choice of set S.

Now suppose that for each v ∈ S we are given a pair (C_{v}, L_{v}) where C_{v} is a collection of deformations of \overline{ρ}|_{G_{v}} and L_{v} is a subspace of H^{1}(G_{v}, ad^{0}\overline{ρ}) satisfying the following properties.

P1. (k, \overline{ρ}|_{G_{v}}) ∈ C_{v}.

P2. C_{v} is closed under conjugation by elements of 1 + M_{2}(𝔪).

P3. If (R, ρ) ∈ C_{v} and f : R → S is a morphism of complete local noetherian rings which induces an isomorphism on residue fields then (S, f ∘ρ) ∈ C_{v}.

P4. Suppose that (R_{1}, ρ_{1}) and (R_{2}, ρ_{2}) ∈ C_{v}, that I_{1} (resp. I_{2}) is an ideal of R_{1} (resp. R_{2}) and that ϕ : R_{1}/I_{1} →^{∼} R_{2}/I_{2} is an isomorphism such that α(ρ_{1} mod I_{1}) = ρ_{2} mod I_{2}. Let R_{3} be the subring of R_{1} ⊕ R_{2} consisting of pairs with the same image in R_{1}/I_{1} →^{∼} R_{2}/I_{2}. Then (R_{3}, ρ_{1} ⊕ρ_{2}) ∈ C_{v}.

P5. If (R, ρ) is a deformation of \overline{ρ}|_{G_{v}} and if \{I_{i}\} is a nested sequence of ideals in R with intersection (0) such that each (R/I_{i}, ρ) ∈ C_{v} then (R, ρ) ∈ C_{v}.

P6. Suppose (R, 𝔪) is a complete noetherian local ring with residue field k and suppose that I is an ideal of R with 𝔪I = (0). If (R/I, ρ) ∈ C_{v} then there is a deformation ρ̃ of \overline{ρ}|_{G_{v}} to R such that (R, ρ̃) ∈ C_{v} and (ρ̃ mod I) = ρ.

P7. Suppose that ((R, 𝔪), ρ_{1}) and ((R, 𝔪), ρ_{2}) are deformations of \overline{ρ} with ((R, 𝔪), ρ_{1}) ∈ C_{v}, and that I is an ideal of R with 𝔪I = (0) and (ρ_{1} mod I) = (ρ_{2} mod I). Thus σ ↦ ρ_{2}(σ)ρ_{1}(σ)^{−1} − 1 defines an element of H^{1}(G_{v}, ad^{0}\overline{ρ})⊗_{k}I which we shall denote [ρ_{2}−ρ_{1}]. Then [ρ_{2}−ρ_{1}] ∈ L_{v}⊗_{k}I if and only if (R, ρ_{2}) ∈ C_{v}.

Let us next give some examples of such pairs (C_{v}, L_{v}).

E1. Suppose that v ≠ l and that l |̸#\overline{ρ}(I_{v}). Take C_{v} to be the class of lifts ρ of \overline{ρ}|_{G_{v}} which factor through G_{v}/(I_{v} ∩ ker \overline{ρ}) and take L_{v} to be H^{1}(G_{v}/I_{v}, (ad^{0}\overline{ρ})^{I_{v}}). Note that • H^{2}(G_{v}/(I_{v} ∩ ker \overline{ρ}), ad^{0}\overline{ρ}) ≅ H^{2}(G_{v}/I_{v}, (ad^{0}\overline{ρ})^{I_{v}}) = (0),

• H^{1}(G_{v}/(I_{v} ∩ ker \overline{ρ}), ad^{0}\overline{ρ}) = L_{v} ⊂ H^{1}(G_{v}, ad^{0}\overline{ρ})

• and dim L_{v} = dim H^{0}(G_{v}, ad^{0}\overline{ρ}). E2. Suppose that v = 2, l = 3 and that (ad^{0}\overline{ρ})(G_{v}) ≅ A_{4}. Take C_{v} to be the

class of lifts ρ of \overline{ρ}|_{G_{v}} which factor through G_{v}/(I_{v} ∩ ker \overline{ρ}) and take L_{v} to be H^{1}(G_{v}/I_{v}, (ad^{0}\overline{ρ})^{I_{v}}). Note that

• H^{i}(\overline{ρ}(I_{v}), ad^{0}\overline{ρ}) = (0) for all i ≥ 0,

• H^{2}(G_{v}/(I_{v} ∩ ker \overline{ρ}), ad^{0}\overline{ρ}) = (0),

• H^{1}(G_{v}/(I_{v} ∩ ker \overline{ρ}), ad^{0}\overline{ρ}) = (0) = L_{v} ⊂ H^{1}(G_{v}, ad^{0}\overline{ρ})

• and dim L_{v} = dim H^{0}(G_{v}, ad^{0}\overline{ρ}) = 0. E3. Suppose that with respect to some basis e_{1}, e_{2} of k^{2} the restriction \overline{ρ}|_{G_{v}}

has the form

( \begin{matrix}ϵ\overline{χ} & ∗ \\ 0 & \overline{χ} \end{matrix} )

and that either v ≢ 1 mod l or l|#\overline{ρ}(G_{v}). Take C_{v} to be the class of deformations ρ of \overline{ρ}|_{G_{v}} of the form

( \begin{matrix}ϵχ & ∗ \\ 0 & χ \end{matrix} )

where χ lifts \overline{χ} and take L_{v} to be the image of

H^{1}(G_{v}, Hom (ke_{2}, ke_{1})) ⟶ H^{1}(G_{v}, (ad^{0}\overline{ρ})).

If I and R are as in the statement of property P6 then

H^{0}(G_{v}, R^{∨}) ⟶ H^{0}(G_{v}, I^{∨})

is surjective, so that

H^{2}(G_{v}, I(1)) ⟶ H^{2}(G_{v}, R(1))

is injective and

H^{1}(G_{v}, R(1)) ⟶ H^{1}(G_{v}, (R/I)(1))

is surjective. Thus C_{v} has property P6. Note also that

dim L_{v}
= dim H^{1}(G_{v}, Hom (ke_{2}, ke_{1})) − dim H^{0}(G_{v}, (ad^{0}\overline{ρ})/Hom (ke_{2}, ke_{1}))
+ dim H^{0}(G_{v}, ad^{0}\overline{ρ}) − dim H^{0}(G_{v}, Hom (ke_{2}, ke_{1}))
= δ_{vl} + 1 + dim H^{0}(G_{v}, Hom (ke_{2}, ke_{1})) − 1 + H^{0}(G_{v}, ad^{0}\overline{ρ})
−H^{0}(G_{v}, Hom (ke_{2}, ke_{1}))
= δ_{vl} + dim H^{0}(G_{v}, ad^{0}\overline{ρ}),

where δ_{vl} = 1 if v = l and 0 otherwise.

E4. Suppose v = l and that with respect to some basis e_{1}, e_{2} of k^{2} \overline{ρ}|_{G_{l}} has

the form

( \begin{matrix}ϵ\overline{χ}_{1} & ∗ \\ 0 & \overline{χ}_{2} \end{matrix} ) .

Suppose also that \overline{χ}_{1} ≠ \overline{χ}_{2} and that if ϵ\overline{χ}_{1} = \overline{χ}_{2} then \overline{ρ}_{G_{l}} is wildly ramified. Take C_{v} to consist of all deformations of the form

( \begin{matrix}ϵχ_{1} & ∗ \\ 0 & χ_{2} \end{matrix} ) ,

where χ_{1} and χ_{2} are tamely ramified and χ_{2} lifts \overline{χ}_{2}. Also take L_{v} to be the image in H^{1}(G_{v}, (ad^{0}\overline{ρ})) of the kernel of the natural map

H^{1}(G_{v}, Hom (ke_{2}, ke_{1}) ⊕ ((Hom (ke_{1}, ke_{1}) ⊕ Hom (ke_{2}, ke_{2})) ∩ ad^{0}\overline{ρ}))

↓

H^{1}(I_{v}, ((Hom (ke_{1}, ke_{1}) ⊕ Hom (ke_{2}, ke_{2})) ∩ ad^{0}\overline{ρ})).

If I and R are as in the statement of property P6 then

H^{0}(G_{v}, I^{∨}(χ_{2}/χ_{1}) = (0),

so that H^{2}(G_{v}, I(ϵχ_{1}/χ_{2})) = (0) and

H^{1}(G_{v}, R(ϵχ_{1}/χ_{2})) ⟶ H^{1}(G_{v}, (R/I)(ϵχ_{1}/χ_{2}))

is surjective. Thus C_{l} has property P6. One can also check that (see for instance tables 2 and 3 in [R])

dim L_{l} = 1 + dim H^{0}(G_{l}, ad^{0}\overline{ρ}).

E5. Suppose that v = l and that for some 1 ≤ n ≤ l − 1 we have

\overline{ρ}|_{I_{l}} ∼ ψ^{n+(l+1)m} ⊕ ψ^{nl+(l+1)m}

where ψ is a fundamental character of level 2. Take C_{l} to be the collection of lifts ρ : G_{l} → GL_{2}(R) such that ρ ⊗ ω^{−n} is crystalline (in the sense that it is the inverse limit of crystalline representations over Artinian quotients of R). A calculation using the theory of Fontaine and Lafaille shows that C_{l} has property P6 and that there is a suitable L_{l} with dim L_{l} = 1. (This calculation basicly goes back to Ramakrishna’s thesis, see for instance the paragraph before proposition 2 of [R].)

Recall that the trace gives a perfect pairing ad^{0}\overline{ρ} × ad^{0}\overline{ρ} → k. By Tate local duality this induces a perfect pairing

H^{1}(G_{v}, ad^{0}\overline{ρ}) × H^{1}(G_{v}, (ad^{0}\overline{ρ})(1)) ⟶ k.

We will let L_{v}^{⊥} denote the annihilator of L_{v} under this pairing. We will let H_{\{L_{v}\}}^{1}(G_{S}, ad^{0}\overline{ρ}) denote the preimage under the restriction map

H^{1}(G_{S}, ad^{0}\overline{ρ}) ⟶ ⨁_{v∈S} H^{1}(G_{v}, ad^{0}\overline{ρ})

of ⨁_{v∈S} L_{v}. Similarly we will let H_{\{L_{v}^{⊥}\}}^{1}(G_{S}, (ad^{0}\overline{ρ})(1)) denote the preimage under the restriction map

H^{1}(G_{S}, (ad^{0}\overline{ρ})(1)) ⟶ ⨁_{v∈S} H^{1}(G_{v}, (ad^{0}\overline{ρ})(1))

of ⨁_{v∈S} L_{v}^{⊥}. Ramakrishna’s first observes the following.

Lemma 1.1 Keep the above notation and assumptions. If

H_{\{L_{v}^{⊥}\}}^{1}(G_{S}, (ad^{0}\overline{ρ})(1)) = (0)

then we can find an S-deformation (W (k), ρ) of \overline{ρ} such that for all v ∈ S the restriction (W (k), ρ|_{G_{v}}) ∈ C_{v}.

Proof: The Poitou-Tate exact sequence gives us an exact sequence

H^{1}(G_{s}, ad^{0}\overline{ρ}) ⟶ ⨁_{v∈S} H^{1}(G_{v}, ad^{0}\overline{ρ})/L_{v} ⟶ H_{\{L_{v}^{⊥}\}}^{1}(G_{S}, (ad^{0}\overline{ρ})(1))^{∨}
↓
⨁_{v∈S} H^{2}(G_{v}, ad^{0}\overline{ρ}) ⟵  H^{2}(G_{S}, ad^{0}\overline{ρ})

(see for instance the proof of theorem 2.18 of [DDT]). Thus we see that

H^{1}(G_{S}, ad^{0}\overline{ρ}) → → ⨁_{v∈S} H^{1}(G_{v}, ad^{0}\overline{ρ})/L_{v}

and

H^{2}(G_{S}, ad^{0}\overline{ρ}) ↪ ⨁_{v∈S} H^{2}(G_{v}, ad^{0}\overline{ρ}).

Now we recursively look for S-deformations (W (k)/l^{m}, ρ_{m}) of \overline{ρ} such that for all v ∈ S we have (W (k)/l^{m}, ρ_{m}|_{G_{v}}) ∈ C_{v}. For m = 1 there is nothing to prove. In general, for all v ∈ S we can lift ρ_{m−1}|_{G_{v}} to a continuous homomorphism ρ_{v} : G_{v} → GL_{2}(W (k)/l^{m}). By injectivity of the restriction map on H^{2}’s this means that we can lift ρ_{m−1} to a continuous homomorphism ρ : G_{S} → GL_{2}(W (k)/l^{m}). By surjectivity of the map on H^{1}’s we may find a class ϕ ∈ H^{1}(G_{S}, ad^{0}\overline{ρ}) mapping to

([ρ|_{G_{v}} − ρ_{v}])_{v∈S} ∈ ⨁_{v∈S} H^{1}(G_{v}, ad^{0}\overline{ρ})/L_{v}.

Thus we may find a second lifting ρ_{m} of ρ_{m−1} to W (k)/l^{m} such that for all v ∈ S we have (W (k)/l^{m}, ρ_{m}|_{G_{v}}) ∈ C_{v}. The lemma follows. □

In fact under these conditions essentially the same argument shows that the universal S-deformation of \overline{ρ} of type C_{v} for all v ∈ S is a power series ring over W (k) in dim H_{\{L_{v}\}}^{1}(G_{S}, ad^{0}\overline{ρ}) variables.

Ramakrishna’s main innovation is the following result which gives conditions under which the hypotheses of the last lemma can be achieved.

Lemma 1.2 Let \overline{ρ}, S, \{(C_{v}, L_{v})\} be as above and suppose that

∑_{v∈S} dim L_{v} ≥ ∑_{v∈S∪\{∞\}} dim H^{0}(G_{v}, ad^{0}\overline{ρ}).

Then we can find a finite set of rational primes T ⊃ S and data (C_{v}, L_{v}) for v ∈ T − S satisfying the above conditions P1-P7 and such that

H_{\{L_{v}^{⊥}\}}^{1}(G_{T} , (ad^{0}\overline{ρ})(1)) = (0).

Proof: Suppose first that l = 5 and ad^{0}\overline{ρ}(G_{ℚ}) ≅ A_{5}, because in this case we will require a little extra argument. Choose w ∉ S such that w ≡ 1 mod 5 and ad^{0}\overline{ρ}(Frob_{w}) has order 5. Adding w to S with the pair (C_{w}, L_{w}) as in example E3, we see that in this case we may assume that

H_{\{L_{v}\}}^{1}(G_{S}, ad^{0}(\overline{ρ})) ∩ H^{1}(ad \overline{ρ}(G_{ℚ}), ad \overline{ρ}) = (0).

(For if ϕ lies in this intersection and is non-zero then ϕ restricts to a non-zero element of H^{1}(G_{w}/I_{w}, ad^{0}\overline{ρ}), while

H^{1}(G_{w}/I_{w}, ad^{0}\overline{ρ}) ∩ L_{w} = (0).)

Now return to the general case. Suppose that

0 ≠ ϕ ∈ H_{\{L_{v}^{⊥}\}}^{1}(G_{S}, (ad^{0}\overline{ρ})(1)).

We will show below that we can find a prime w ∉ S and (C_{w}, L_{w}) satisfying properties P1-P7 and such that

• dim L_{w} = dim H^{1}(G_{w}/I_{w}, ad^{0}\overline{ρ}),

• H_{\{L_{v}\}}^{1}(G_{s}, ad^{0}\overline{ρ}) → → H^{1}(G_{w}/I_{w}, ad^{0}\overline{ρ})

• and ϕ does not map to zero in H^{1}(G_{w}, (ad^{0}\overline{ρ})(1))/L_{w}^{⊥}.

Suppose for a moment that we have done this. We have an injection

H_{\{L_{v}^{⊥}\}}^{1}(G_{S}, (ad^{0}\overline{ρ})(1)) ↪ H_{\{L_{v}^{⊥}\}∪\{H^{1}(G_{w},(ad^{0}\overline{ρ})(1))\}}^{1}(G_{S∪\{w\}}, (ad^{0}\overline{ρ})(1))

and a formula of Wiles (see for instance theorem 2.18 of [DDT]) tells us that

#H_{\{L_{v}^{⊥}\}∪\{H^{1}(G_{w},(ad^{0}\overline{ρ})(1))\}}^{1}(G_{S∪\{w\}}, (ad^{0}\overline{ρ})(1)) =

#H_{\{L_{v}^{⊥}\}}^{1}(G_{S}, (ad^{0}\overline{ρ})(1))#coker (H_{\{L_{v}\}}^{1}(G_{S}, ad^{0}\overline{ρ}) → H^{1}(G_{w}/I_{w}, ad^{0}\overline{ρ})).

Thus in our case

H_{\{L_{v}^{⊥}\}}^{1}(G_{S}, (ad^{0}\overline{ρ})(1)) = H_{\{L_{v}^{⊥}\}∪\{H^{1}(G_{w},(ad^{0}\overline{ρ})(1))\}}^{1}(G_{S∪\{w\}}, (ad^{0}\overline{ρ})(1))

and so we get a left exact sequence

(0) ⟶ H_{\{L_{v}^{⊥}\}}^{1}(G_{S∪\{w\}}, (ad^{0}\overline{ρ})(1)) ⟶ H_{\{L_{v}^{⊥}\}}^{1}(G_{S}, (ad^{0}\overline{ρ})(1))
↓
H^{1}(G_{w}, (ad^{0}\overline{ρ})(1))/L_{w}^{⊥}.

Thus

ϕ ∉ H_{\{L_{v}^{⊥}\}}^{1}(G_{S∪\{w\}}, (ad^{0}\overline{ρ})(1)) ⊂ H_{\{L_{v}^{⊥}\}}^{1}(G_{S}, (ad^{0}\overline{ρ})(1))

and arguing recursively the lemma will follow.

We now turn to the proof of the existence of a prime w ∉ S and a pair (C_{w}, L_{w}) with the above properties. In fact it suffices to show that if 0 ≠ ϕ ∈ H_{\{L_{v}^{⊥}\}}^{1}(G_{S}, (ad^{0}\overline{ρ})(1)) and 0 ≠ ψ ∈ H_{\{L_{v}\}}^{1}(G_{S}, (ad^{0}\overline{ρ})(1)), then we can find a prime w ∉ S and (C_{w}, L_{w}) satisfying properties P1-P7 and such that • dim H^{1}(G_{w}/I_{w}, ad^{0}\overline{ρ}) = dim L_{w} = 1,

• ψ does not map to zero in H^{1}(G_{w}/I_{w}, ad^{0}\overline{ρ})

• and ϕ does not map to zero in H^{1}(G_{w}, (ad^{0}\overline{ρ})(1))/L_{w}^{⊥}. (To see this note that by the assumption of the lemma and by Wiles’ formula (see for instance theorem 2.18 of [DDT]) we have

dim H_{\{L_{v}\}}^{1}(G_{s}, ad^{0}\overline{ρ}) ≥ dim H_{\{L_{v}^{⊥}\}}^{1}(G_{s}, (ad^{0}\overline{ρ})(1)),

so that if H_{\{L_{v}^{⊥}\}}^{1}(G_{s}, (ad^{0}\overline{ρ})(1)) ≠ (0) then we can find

0 ≠ ψ ∈ H_{\{L_{v}\}}^{1}(G_{S}, (ad^{0}\overline{ρ})(1)).)

Let K/ℚ be the field generated by a primitive l^{th} root of unity and by the fixed field of ker(ad^{0}\overline{ρ}). Note that

• H^{1}(Gal (K/ℚ), ad^{0}\overline{ρ}) = (0),

• H^{1}(Gal (K/ℚ), (ad^{0}\overline{ρ})(1)) = (0)

• and there is an element σ ∈ Gal (K/ℚ) such that ad^{0}\overline{ρ}(σ) has an eigen-

value ϵ(σ) ≢ 1 mod l. (This is a straightforward exercise using the following facts.

• ad^{0}\overline{ρ}(G_{ℚ}) ≅ A_{5}, P SL_{2}(𝔽_{l^{r}}) or P GL_{2}(𝔽_{l^{r}}) for some r ∈ ℤ_{>0}.

• If l is an odd prime, r ∈ ℤ_{>0} and l^{r} ≠ 5 then H^{1}(P SL_{2}(𝔽_{l^{r}}), ad^{0}) = (0).

• H^{1}(P GL_{2}(𝔽_{5}), ad^{0}) = (0).

• In the case l = 5 and ad^{0}\overline{ρ}(G_{ℚ}) ≅ A_{5}, we have seen that we may assume

that

H_{\{L_{v}\}}^{1}(G_{S}, ad^{0}(\overline{ρ})) ∩ H^{1}(ad \overline{ρ}(G_{ℚ}), ad \overline{ρ}) = (0).)

Let σ̃ be a lift of σ to G_{ℚ}. Then \overline{ρ}(σ̃) has two distinct eigenevalues α, β ∈ k with α/β = ϵ(σ). Let e_{α} and e_{β} denote corresponding eigenvectors. We get a decomposition

ad^{0}\overline{ρ} = Hom (ke_{β}, ke_{α})⊕

((Hom (ke_{α}, ke_{α}) ⊕ Hom (ke_{β}, ke_{β})) ∩ ad^{0}\overline{ρ}) ⊕ Hom (ke_{α}, ke_{β}).

Let ϕ and ψ be cohomology classes as above. We will use the same symbols to denote some choice of cocycles representing these cohomology classes. The restrictions of ϕ and ψ are non-zero homomorphisms ϕ : G_{K} → (ad^{0}\overline{ρ})(1) and ψ : G_{K} → ad^{0}\overline{ρ}. Let K_{ϕ} and K_{ψ} denote the fixed field of their kernels. Because ad^{0}\overline{ρ} is absolutely irreducible and not isomorphic to its twist by ϵ we see that K_{ψ} and K_{ϕ} are disjoint over K. Moreover the images ψ(G_{K}) and ϕ(G_{K}) are not contained in any proper subspaces of ad^{0}\overline{ρ} and (ad^{0}\overline{ρ})(1), respectively. Thus we can find a τ ∈ Gal (K_{ϕ}K_{ψ}/K) such that

ϕ(τσ̃) = ϕ(τ) + ϕ(σ̃) ∉
Hom (ke_{β}, ke_{α})(1) ⊕ ((Hom (ke_{α}, ke_{α}) ⊕ Hom (ke_{β}, ke_{β})) ∩ ad^{0}\overline{ρ})(1)

and

ψ(τσ̃) = ψ(τ) + ψ(σ̃) ∉ Hom (ke_{β}, ke_{α}) ⊕ Hom (ke_{α}, ke_{β}).

Now choose a prime w ∉ S which is unramified in K_{ϕ}K_{ψ}/ℚ and such that Frob_{w} = τσ̃. Take C_{w} as in example E3 so that L_{w} = H^{1}(G_{w}, Hom (ke_{β}, ke_{α})). It follows easily that (w, C_{w}, L_{w}) has the desired properties. □

Theorem 1.3 Let k be a finite extension of 𝔽_{l} and \overline{ρ} : G_{ℚ} → GL_{2}(k) a continuous representation such that det \overline{ρ}(c) = −1 and \overline{ρ}(G_{ℚ}) is insoluble.

Suppose first that

\overline{ρ}|_{G_{l}} ∼ ( \begin{matrix}ϵ\overline{χ}_{1} & ∗ \\ 0 & \overline{χ}_{2} \end{matrix} )

and that if ϵ\overline{χ}_{1} = \overline{χ}_{2} then \overline{ρ}|_{G_{l}} is wildly ramified. Then there is a continuous representation ρ : G_{ℚ} → GL_{2}(W (k)) such that

• (ρ mod l) = \overline{ρ},

• ρ|_{G_{l}} ∼ ( \begin{matrix}ϵχ_{1} & ∗ \\ 0 & χ_{2} \end{matrix} ) with χ_{i} a tamely ramified lift of \overline{χ}_{i} for i = 1, 2

• and for some prime p ≠ l we have ρ|_{G_{p}} ∼ ( \begin{matrix}ϵχ & ∗ \\ 0 & χ \end{matrix} ) for some character χ.

Now suppose that for some 1 ≤ n ≤ l − 1 we have

\overline{ρ}|_{I_{l}} ∼ ψ^{n+(l+1)m} ⊕ ψ^{nl+(l+1)m}

where ψ is a fundamental character of level 2, then there is a continuous representation ρ : G_{ℚ} → GL_{2}(W (k)) such that • (ρ mod l) = \overline{ρ},

• (ρ ⊗ ω^{−m})|_{G_{l}} is crytsalline with Hodge-Tate numbers 0 and n

• and for some prime p ≠ l we have ρ|_{G_{p}} ∼ ( \begin{matrix}ϵχ & ∗ \\ 0 & χ \end{matrix} ) for some character χ.

2 Icosahedral Galois representations

We begin with some elementary lemmas on number fields. They are presumably well known, but it is easier to prove them than find a reference.

Lemma 2.1 Let K be a number field (finite extension of ℚ) and S a finite set of places of K. We will let K_{S}^{×} denote the subgroup of K^{×} consisting of elements which are units at all finite places v ∉ S and positive at all real places v ∉ S. Then for any positive integer n we can find an open subgroup U ⊂ ∏_{v∉S, v |̸∞} O_{K,v}^{×} such that

K_{S}^{×} ∩ U ⊂ (K_{S}^{×})^{n}.

Proof: We may suppose that S contains all infinite places and that \sqrt{−1} ∈ K (as K^{×} ⊂ K(\sqrt{−1})^{×}). We may also suppose that n is a prime power, say n = p^{r}. Thus if ζ denotes a primitive n^{th} root of unity then Gal (K(ζ)/K) is cyclic. Because K_{S}^{×} is finitely generated, it suffices to prove that if a ∈ K^{×} and for all y ∉ S we have a ∈ (K_{y}^{×})^{n} then a ∈ (K^{×})^{n}. Thus suppose a ∈ K^{×} and for all y ∉ S we have a ∈ (K_{y}^{×})^{n}. Then all but finitely many places split completely in K(\sqrt[n]{a}, ζ)/K(ζ) so that K(ζ) contains every n^{th} root of a. We may choose y ∉ S which is unramified in K(ζ) and such that Frob_{y} generates the cyclic group Gal (K(ζ)/K). We conclude that K contains an n^{th} root of a, as desired. □

Lemma 2.2 Let K be a number field and S a finite set of places of K. For each v ∈ S let L_{v} be a finite Galois extension of K_{v}. Then we can find a finite, soluble, Galois extension M of K, such that for each place w of M above a place v ∈ S we have L_{v} ≅ M_{w} as K_{v}-algebras.

Proof: We need only find a finite extension M/K such that M embeds in a soluble Galois extension of K and such that for each place w of M above a place v ∈ S we have L_{v} ≅ M_{w} as K_{v}-algebras. (Then replace M by its normal closure over K.) We may also suppose that S contains all infinite places of K. Then, by a simple induction argument, we may reduce to the case that each L_{v}/K_{v} is an abelian Galois extension. Thus suppose that L_{v}/K_{v} corresponds by class field theory to an open finite index subgroup W_{v} ⊂ K_{v}^{×}. Set W = ∏_{v∈S} W_{v} and choose an open subgroup U ⊂ ∏_{v∉S} O_{K,v}^{×} such that

K_{S}^{×} ∩ U ⊂ K_{S}^{×} ∩ W.

Finally take M to be the abelian extension of K corresponding to

K^{×}(K_{∞}^{×})^{0}W U.

□

Lemma 2.3 Suppose that \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{5}^{ac}) is a continuous representation which satisfies the following conditions.

• det \overline{ρ}(c) = −1.

• \overline{ρ} has projective image A_{5}.

• The projective image of I_{3} has odd order.

• The projective image of G_{5} has order 2 and the corresponding map ℚ_{5}^{×} → {±1} sends 5 to −1.

Then there is a finite, soluble, totally real extension F/ℚ and an elliptic curve E/F satisfying the following conditions.

• F ⊃ ℚ(\sqrt{5}) and \sqrt{5} splits completely in F.

• \overline{ρ}_{E,5} is equivalent to a twist of \overline{ρ}|_{G_{F}} by some character.

• \overline{ρ}_{E,3} : G_{F} → → GL_{2}(𝔽_{3}).

• E has good ordinary reduction at 3 and potentially good ordinary reduction at 5.

• For all primes v of F above 3 we have

\overline{ρ}_{E,3}|_{G_{Fv}} ∼ ( \begin{matrix}χ_{1,v} & ∗ \\ 0 & χ_{2,v} \end{matrix} )

with χ_{1,v} ≠ χ_{2,v}.

Proof: The obstruction to lifting the continuous homomorphism

G_{ℚ(\sqrt{5})} ⟶^{\overline{ρ}} A_{5} ≅ P SL_{2}(𝔽_{5})

to a continuous homomorphism G_{ℚ(\sqrt{5})} → SL_{2}(𝔽_{5}) lies in

H^{2}(G_{ℚ(\sqrt{5})}, \{±1\}) ↪ ⨁_{v} H^{2}(G_{ℚ(\sqrt{5})_{v}}, \{±1\}).

Also the local component at (3) is trivial as 3 is inert in ℚ(\sqrt{5}). Thus we can find a totally real, biquadratic field F_{1}

• such that F_{1} contains ℚ(\sqrt{5}) and both (3) and (\sqrt{5}) splits in F_{1},

• and such that the image of this obstruction vanishes at all finite places of F_{1}.

As det \overline{ρ}(c) = −1 the image of this obstruction is non-trivial at all infinite places of F_{1}. Similarly the obstruction to lifting the mod 5 cyclotomic character

G_{ℚ(\sqrt{5})} ⟶ \{±1\} ⊂ 𝔽_{5}^{×}

to a character G_{ℚ(\sqrt{5})} → μ_{4} with square the mod 5 cyclotomic character lies in

H^{2}(G_{ℚ(\sqrt{5})}, \{±1\}) ↪ ⨁_{v} H^{2}(G_{ℚ(\sqrt{5})_{v}}, \{±1\})

and has trivial image at all finite places and non-trivial image at infinity. Thus the sum of the two obstructions vanishes in H^{2}(G_{F_{1}}, {±1}) and so we can lift

G_{F_{1}} ⟶^{\overline{ρ}} A_{5} ≅ P SL_{2}(𝔽_{5})

to a continuous representation

ρ̃: G_{F_{1}} ⟶ GL_{2}(𝔽_{5})

with det ρ̃ = ϵ_{5}.

Choose a finite, soluble, totally real extension F_{2}/F_{1} such that (\sqrt{5}) splits completely in F_{2}, such that ρ̃ is trivial on the decomposition group of every prime of F_{2} above 3, but such that the ramification index of any prime above 3 in F_{2} is odd. Finally let F be the Galois closure of F_{2}/ℚ.

Let Xρ̃/F be the twist of X_{5} defined in section 1 of [SBT]. By lemma 1.1 of [SBT] we see that Xρ̃ is isomorphic over F to a Zariski open subset of the projective line. Also let Yρ̃/Xρ̃ be the cover defined in the proof of theorem 1.2 of [SBT]. Thus Yρ̃ is geometrically irreducible and Yρ̃/Xρ̃ has degree 24.

Suppose that v is a prime of F above 3. Then ϵ_{5}(Frob_{v}) ≡ 1 mod 5 so that the residue field of v contains 𝔽_{81}. Thus the elliptic curve y^{2} = x^{3} + x^{2} − x − 1 defines an element of Xρ̃(F_{v}) with good ordinary reduction at v. The same will be true of any point of Xρ̃(F_{v}) sufficiently close to this one in the 3-adic topology. Let U_{v} ⊂ Xρ̃(F_{v}) be a non-empty open set (for the 3adic topology) consisting of points corresponding to elliptic curves with good ordinary reduction.

Suppose now that v is a prime of F above 5. We claim that we can find a non-empty open subset (for the 5-adic topology) U_{v} ⊂ Xρ̃(F_{v}) consisting of points corresponding to elliptic curves with good ordinary reduction. It suffices to find one such point (and then take U_{v} to be a sufficiently small open neighbourhood of that point). Note that up to twist by quadratic characters

ρ̃|_{G_{Fv}} ∼ χδ ⊕ χ

where δ is a quadratic character corresponding to a character of ℚ_{5}(\sqrt{5})^{×} taking \sqrt{5} to −1, and χ is a tamely ramified character of order 4 corresponding to a character of ℚ_{5}(\sqrt{5})^{×} taking \sqrt{5} to 2. Moreover if δ is ramified then we may take χ unramified, while if δ is unramified the restriction of χ to inertia also has order 4. In the first case the elliptic curve y^{2} = x^{3} + x provides a point in Xρ̃(F_{v}). This elliptic curve has CM by ℤ[\sqrt{−1}] over F_{v} and a suitable quartic twist provides a point on Xρ̃(F_{v}) in the second case.

By Ekedahl’s version of the Hilbert irreducibility theorem [E] we may find a point P ∈ Xρ̃(F) which lies in U_{v} for all v|15 and such that any point of Yρ̃ above F cuts out an extension of F of degree 24. Let E/F be the elliptic curve corresponding to P. □

Theorem 2.4 Let \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{5}^{ac}) be a continuous representation which satisfied the following conditions.

• det \overline{ρ}(c) = −1.

• \overline{ρ} has projective image A_{5}.

• The projective image of I_{3} has odd order.

• The projective image of G_{5} has order 2 and the corresponding map ℚ_{5}^{×} → {±1} sends 5 to a non-trivial element.

Then \overline{ρ} is modular.

Proof: Choose F and E as in the previous lemma. By the Langlands-Tunnell theorem there is a cuspidal automorphic representation π′′′ of GL_{2}(𝔸_{F}) and a place μ of the field of coefficients of π′′′ above 3 such that the following conditions are satisfied.

• For each infinite place v the component π′′′_{v} is lowest discrete series.

• \overline{ρ}_{π′′′,μ} ∼ \overline{ρ}_{E,3}.

• For any place v of F above 3 we have

ρ_{π′′′,μ}|_{G_{v}} ∼ ( \begin{matrix}χ_{1,v}ϵ_{3} & ∗ \\ 0 & χ_{2,v} \end{matrix} ) ,

where χ_{1,v} and χ_{2,v} are finitely tamely ramified, and χ_{1,v}ϵ_{3} ≢ χ_{2,v} mod μ. (See [L] and [Tu], as well as [RT] for the method for arranging the conditions on \overline{ρ}_{π′′′,v} for v infinite.)

Applying theorem 5.1 of [SW2] to the 3-adic Tate module of E, we see that there is a cuspidal automorphic representation π′′ of GL_{2}(𝔸_{F}) satisfying the following conditions.

• For each infinite place v the component π′′_{v} of π′′ is lowest discrete series.

• π′′ has field of coefficients ℚ.

• For every rational prime l, the representation ρ_{π′′,l} ∼ ρ_{E,l}. Twisting π′′ we see that there is a cuspidal automorphic representation π′ of GL_{2}(𝔸_{F}) and a place λ′ of the field of coefficients of π′ above 5 such that the following conditions are satisfied.

• For each infinite place v the component π′_{v} is lowest discrete series.

• There is an embedding of the residue field of λ′ in 𝔽_{5}^{ac} such that \overline{ρ}_{π′,λ′} ∼ \overline{ρ}.

• For any place v of F above 5 we have

ρ_{π′,λ′}|_{G_{v}} ∼ ( \begin{matrix}χ_{1,v}ϵ_{5} & ∗ \\ 0 & χ_{2,v} \end{matrix} ) ,

where χ_{1,v} and χ_{2,v} are finitely tamely ramified.

Note that by our assumptions on \overline{ρ}, χ_{1,v}ϵ_{5} ≢ χ_{2,v} mod λ′.

We will next explain how to descend (in a mod l sense) π′ to ℚ while maintaining ρ_{π′,λ′} ∼ \overline{ρ}. We learned this argument from C.Khare.

By theorem 1.3 we may choose a continuous representation ρ : G_{ℚ} → GL_{2}(ℚ_{5}^{ac}) satisfying the following conditions.

• ρ is a lift of \overline{ρ}.

• ρ|_{G_{5}} ∼ ( \begin{matrix}χ_{1}ϵ_{5} & ∗ \\ 0 & χ_{2} \end{matrix} ) where χ_{1} and χ_{2} are finitely tamely ramified.

The existence of π′ above and theorem 5.1 of [SW2] tell us that there is a cuspidal automorphic representation π of GL_{2}(𝔸_{F}) and a place λ of the field of coefficients of π above 5 such that the following conditions are satisfied.

• For each infinite place v the component π_{v} is lowest discrete series.

• There is an embedding of the λ-adic completion of the field of coefficients of π into ℚ_{5}^{ac} such that ρ_{π,λ} ∼ ρ|_{G_{F}}.

Let F = F_{1} ⊃ F_{2} ⊃ ... ⊃ F_{n} = ℚ with F_{i}/F_{i+1} Galois and cyclic of prime degree for all i. We will show by induction on i that there is a cuspidal automorphic representation π_{i} of GL_{2}(𝔸_{F_{i}}) and a place λ_{i} of the field of coefficients of π_{i} above 5 such that the following conditions are satisfied.

• For each infinite place v the component π_{i,v} is lowest discrete series.

• There is an embedding of the λ_{i}-adic completion of the field of coefficients of π_{i} into ℚ_{5}^{ac} such that ρ_{π_{i},λ_{i}} ∼ ρ|_{G_{F_{i}}}.

We have treated the case i = 1 above. Suppose we have treated the case of i. Let σ be a generator of Gal (F_{i}/F_{i+1}). Then we see that π_{i}^{σ} = π_{i} and so, by Langlands base change theorem [L], π_{i} descends to a cuspidal automorphic representation π′_{i+1} of GL_{2}(𝔸_{F_{i+1}}) with π′_{i+1,v} lowest discrete series for each infinite place v of F′_{i+1}. Then there is an embedding of the field of coefficients of π′_{i+1} into ℚ_{5}^{ac}, which gives rise to a place λ′_{i+1}, such that ρ_{π′_{i+1},λ′_{i+1}}|_{G_{F_{i}}} ∼ ρ|_{G_{F_{i}}}. As ρ|_{G_{F_{i}}} is irreducible we see that ρ_{π′_{i+1},λ′_{i+1}} is the twist of ρ|_{G_{F_{i+1}}} by a character of Gal (F_{i}/F_{i+1}). Thus replacing π′_{i+1} by a twist the claim follows for i + 1. The case i = n of the claim implies the theorem. □

Corollary 2.5 Let ρ : 𝔾_{ℚ} → GL_{2}(ℂ) be a continuous representation satisfying the following conditions.

• det ρ(c) = −1.

• ρ has projective image A_{5}.

• The projective image of I_{3} has odd order.

• The projective image of G_{5} has order 2 and the corresponding map ℚ_{5}^{×} → {±1} sends 5 to −1.

Then ρ is modular.

Proof: This follows from the previous theorem and the main theorem of [Buz]. (In the case that the projective representation associated to \overline{ρ} is unramified at 5, one may appeal instead to the main theorem of [BT].) □

We will finish by giving some concrete examples where this corollary can be applied. We list quintic polynomials whose splitting fields are A_{5} extensions of ℚ. In each case this A_{5} extension can be lifted to a Galois representation ρ satisfying the conditions of the above corollary. None of these examples satisfy the conditions of the main theorem of [BDST]. They are all taken from the tables in [Buh].

x^{5} + 2x^{4} + 6x^{3} + 8x^{2} + 10x + 8
x^{5} + 6x^{4} + x^{3} + 4x^{2} − 24x + 32
x^{5} − 2x^{3} + 2x^{2} + 5x + 6
x^{5} + 5x^{4} + 8x^{3} − 20x^{2} − 21x − 5.

Corrigendum to [Ta].

I would like to thank Fred Diamond for pointing out an error in [Ta]. More precisely, with the definition of the inner product given on page 271 of [Ta], the calculation of the adjoint of a Hecke operator is in general wrong. This may be corrected as follows.

• Change the definition of ⟨f, g⟩ on page 271 to read

⟨f, g⟩ = ∑_{[x]∈X(U)}[D^{×} ∩ xUx^{−1} : F^{×} ∩ U]^{−1}⟨f(x), g(x)⟩(Nνx)^{μ}.

• Make the corresponding changes to the calculation on page 271 of the adjoint of [UxU′]. The final formula remains unchanged.

• At the start of line 4 on page 274 of [Ta] and the following sentence. “Note that [D^{×} ∩ t_{j}u_{l}Uu_{l}^{−1}t_{j}^{−1} : F^{×} ∩ U]|[D^{×} ∩ t_{j}U_{0}t_{j}^{−1} : F^{×} ∩ U_{0}] and so there are only finitely many possibilities (independent of U ⊂ U_{0}) for [D^{×} ∩ t_{j}u_{l}Uu_{l}^{−1}t_{j}^{−1} : F^{×} ∩ U].”

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