On icosahedral Artin representations

Kevin Buzzard, Department of Mathematics,

Imperial College, London, U.K.

Mark Dickinson, Department of Mathematics, Harvard University, Cambridge, MA 02138, U.S.A.

Nick Shepherd-Barron,

D.P.M.M.S., Cambridge University,

Cambridge, U.K.

Richard Taylor, ^{1} Department of Mathematics, Harvard University, Cambridge, MA 02138, U.S.A.

September 20, 1999

1Partially supported by NSF Grant DMS-9702885 and by the Miller Institute at the University of California at Berkeley

Introduction

Artin [A] conjectured that the L-series L(r, s) of any continuous representation r : Gal (ℚ^{ac}/ℚ) ⟶ GL_{n}(ℂ) is entire except possibly for a pole at s = 1 when r contains the trvial representation. The case n = 1 is simply a restatement of the Kronecker-Weber theorem and standard results on the analytic continuation of Dirichlet L-series. Artin proved his conjecture when r is induced from a 1–dimensional representation of an open subgroup of Gal (ℚ^{ac}/ℚ). Moreover, Brauer [Br] was able to show in general that L(r, s) is meromorphic on the whole complex plane. Since then the only real progress has been for n = 2.

When n = 2 such representations can be classified according to the image of the projectivised representation proj r : Gal (ℚ^{ac}/ℚ) ⟶ P GL_{2}(ℂ). This image is either cyclic, dihedral, the alternating group A_{4} (the tetrahedral case), the symmetric group S_{4} (the octahedral case) or the alternating group A_{5} (the icosahedral case). When the image of proj r is cyclic then r is reducible and Artin’s conjecture follows from the n = 1 case. When the image of proj r is dihedral then r is induced from a character of an open subgroup of index 2, and so Artin himself proved the conjecture in this case. (Note however that the result is implicit in earlier work of Hecke (see [He]). Langlands [Langl] proved Artin’s conjecture for tetrahedral and some octahedral representations. Tunnell [Tu] extended this to all octahedral representations. These results are based on Langlands’ theory of cyclic base change for automorphic representations of GL_{2}, and so the method seems to be restricted (at best) to cases where the image of r is soluble. A number of people, including Buhler [Buh] and Frey et al. [F], used computer calculations to check Artin’s conjecture for a few icosahedral examples.

The contribution here to the problem is to treat some (infinite families of) icosahedral cases. More precisely we prove the following theorem.

Theorem A. Suppose that r : Gal (ℚ^{ac}/ℚ) ⟶ GL_{2}(ℂ) is a continuous irreducible representation and that r is odd, i.e., the determinant of complex conjugation is −1. If r is icosahedral suppose that

• proj r is unramified at 2 and that the image of a Frobenius element at 2 under proj r has order 3,

• and proj r is unramified at 5.

Then there is a weight one newform f such that for all prime numbers p the p^{th} Fourier coefficient of f equals the trace of Frobenius at p on the inertia at p coinvariants of r. In particular the Artin L-series for r is the Mellin transform of a weight one newform and is an entire function.

The proof follows a strategy outlined by one of us (R.T.) to Wiles in 1992 (see [Ta2]), which has now been carried out by the four of us in three main steps (see [ST], [Di] and [BT]). The purpose of this article is simply to pull these results together and document some technical results which we require, but do not seem to be available in the literature. The result is that this paper is rather technical. The reader who simply desires to get an overview of the main ideas of the proof should consult [Ta2], perhaps followed by [ST], [BT] and [Di], rather than this paper.

One might hope that extensions of our method may treat all odd twodimensional icosahedral representations of Gal (ℚ^{ac}/ℚ), although considerable work remains to be done. On the other hand our method seems to offer no prospect of treating the general Artin conjecture.

1 Mod 2 icosahedral representations.

In this section we will give a slight extension of results in [ST]. This could be avoided by appealing to the results of [G]. However his results depend on certain “unchecked compatibilities”, and so we prefer to make our result unconditional by using this more ad hoc argument. We remark that the hypotheses in our main theorem could be weakened if one could make Gross’ theorem unconditional.

We start with a strengthening of theorem 3.4 of [ST].

Theorem 1.1. Fix a continuous homomorphism

\overline{ρ} : Gal (ℚ^{ac}/ℚ) ⟶ SL_{2}(𝔽_{4}).

Suppose that \overline{ρ} is unramified at 2 and that \overline{ρ}(Frob_{2}) has distinct eigenvalues α, β ∈ 𝔽_{4}^{×}. Then there is an abelian surface A/ℚ together with a principal polarisation λ : A ⟶ A^{∨} and an embedding i : ℤ[(1 + \sqrt{5})/2] ↪ End (A) (both defined over ℚ) such that

1. λ ∘ i(a) = i(a)^{∨} ∘ λ for all a ∈ ℤ[(1 + \sqrt{5})/2];

2. the action of Gal (ℚ^{ac}/ℚ) on A[2] ≅ 𝔽_{4}^{2} is equivalent to \overline{ρ};

3. A has good ordinary reduction at 2 and Frob_{2} = α on A[2]^{et} (the generic fibre of the maximal etale quotient of the two torsion on the Néron model of A over ℤ);

4. and the action of Gal (ℚ^{ac}/ℚ) on the \sqrt{5}-division points, A[\sqrt{5}] is via a surjection Gal (ℚ^{ac}/ℚ) → → GL_{2}(𝔽_{5}).

Proof. With the third condition removed this is the main result of [ST]. The proof of this strengthening is a slight variant of the argument of that paper. We start by recalling some of the constructions there.

We fix an identification of 𝔽_{4} with ℤ[(1 + \sqrt{5})/2]/(2) and of SL_{2}(𝔽_{4}) with A_{5}. We let Y/ℚ denote the smooth cubic surface given in ℙ^{4} by

∑_{i=1}^{5} y_{i} = ∑_{i=1}^{5} y_{i}^{3} = 0.

The group A_{5} acts on Y by permuting the variables. We let Y^{0} ⊂ Y (resp. Y^{1} ⊂ Y) denote the complement of the 15 lines conjugate to (s : −s : t : −t : 0) (resp. the complement of the 10 points conjugate to (1 : −1 : 0 : 0 : 0)). We let Y\overline{ρ} (resp. Y 0 ; \overline{ρ}, resp. Y 1 ; \overline{ρ}) denote the twist of Y (resp. Y^{0}, resp. Y^{1}) by \overline{ρ} : Gal (ℚ^{ac}/ℚ) → A_{5}. There is an etale ℙ^{1}-bundle C\overline{ρ} → Y 1 ; \overline{ρ} together with 6 distinguished sections s_{1}, ..., s_{6} : Y 1 ; \overline{ρ} × ℚ^{ac} → C\overline{ρ} × ℚ^{ac} such that the set \{s_{1}, ..., s_{6}\} is Gal (ℚ^{ac}/ℚ)-invariant. Over Y 0 ; \overline{ρ} the sections are distinct. (This is not explicitly proved in [ST], but one may argue as follows. We will use without comment some notation from section 2 of [ST]. By the formulae on pages 15-17 of [DO] the locus in P_{1}^{6} where s_{1}, ..., s_{6} are distinct is identified with the complement, Z^{0}, in ∑_{i=1}^{6} z_{i} = ∑_{i=1}^{6} z_{i}^{3} = 0 of the 15 S_{6}-conjugates of the plane (s : −s : t : −t : u : −u). Then using lemma 2.4 of [ST] it is easy to see that j^{−1}Z^{0} = Y^{0}, and the claim follows.)

We let W\overline{ρ}/ℚ denote the 𝔽_{4}-vector space scheme corresponding to \overline{ρ} : Gal (ℚ^{ac}/ℚ) → GL_{2}(𝔽_{4}). It comes with a standard pairing

W\overline{ρ} × W\overline{ρ} ⟶ μ_{2}

which on ℚ^{ac}-points sends

(a, b) × (c, d) ⟼ (−1)^{tr_{𝔽_{4}/𝔽_{2}}(ad−bc)}.

Then there is a coarse moduli space H\overline{ρ}/ℚ parametrising quadruples (A, λ, i, α) where (A, λ) is a principally polarised abelian surface, i : ℤ[(1 + \sqrt{5})/2] ↪ End (A) has image fixed by the λ-Rosati involution and where α : W\overline{ρ} →^{∼} A[2] is an isomorphism of 𝔽_{4}-vector space schemes taking the standard pairing to the λ-Weil pairing. There is a Zariski open subset H0 ; \overline{ρ} ⊂ H\overline{ρ} consisting of those geometric points for which the corresponding (A, λ) is a Jacobian. Then there is an isomorphism Y 0 ; \overline{ρ} ≅ H0 ; \overline{ρ} so that a geometric point y of Y 0 ; \overline{ρ} maps to the point parametrising a quadruple (A, λ, i, α) such that (A, λ) is the Jacobian of the curve which maps 2 : 1 to C_{\overline{ρ},y} ramified exactly at s_{1}(y), ..., s_{6}(y). (Again this is not explicitly stated in [ST]. To prove it one may assume that \overline{ρ} = 1. Recall from [ST] we have maps

Y − → H_{2}^{∗} ⟶ A_{2}^{∗}− → P_{1}^{6}.

(We keep the notation of [ST], so in particular H_{2}^{∗} is a compactification of what we are now calling H_{1}.) The locus of Jacobians in A_{2}^{∗} is the locus of points where A_{2}^{∗}− → P_{1}^{6} is regular and map to Z^{0} ⊂ P_{1}^{6}. Thus Y^{0} maps to H_{1}^{0} ⊂ H_{2}^{∗}. On the other hand H_{2}^{∗} is the disjoint union of the image of Y^{0} and some ℙ^{1}’s which get contracted to the points of P_{1}^{6} − (P_{1}^{6})^{s} (see section 2 of [ST]). Thus, if y is a point of H_{2}^{∗} not in the image of Y^{0} then either H_{2}^{∗}− → P_{1}^{6} is not regular at y or y gets mapped outside Z^{0}. In either case y does not lie in H_{1}^{0}.)

If X′ ; \overline{ρ} denotes the blow up of Y\overline{ρ} × Y\overline{ρ} along the diagonal, then X′ ; \overline{ρ} has an involution t which exchanges the two factors. We let X\overline{ρ} denote the twist of

X′ ; \overline{ρ} by

Gal (ℚ^{ac}/ℚ) → → Gal (ℚ(\sqrt{5})/ℚ) →^{∼} \{1, t\},

and X2 ; \overline{ρ} the complement in X\overline{ρ} of the strict transforms of L × L as L runs over lines on Y\overline{ρ}. Then there is a morphism

θ : X2 ; \overline{ρ} ⟶ Y\overline{ρ}

which (loosely speaking) sends (P, Q) to the third point of intersection of the line through P and Q with Y\overline{ρ}(see [ST] for details). We will let X0 ; \overline{ρ} (resp. X1 ; \overline{ρ}, resp. D\overline{ρ}/X1 ; \overline{ρ}) denote the preimage of Y 0 ; \overline{ρ} (resp. the preimage of Y 1 ; \overline{ρ}, resp. the pull back of C\overline{ρ}) under θ. Then it is proved in [ST] (lemma 3.1 and proposition 3.2) that X\overline{ρ}/ℚ is rational and that D\overline{ρ}/X1 ; \overline{ρ} is a Zariski ℙ^{1}-bundle.

The argument preceding lemma 2.7 of [ST] shows that given x ∈ X0 ; \overline{ρ} we can find a Zariski open subset U ⊂ X0 ; \overline{ρ} containing x and a principally polarised abelian surface (A_{U}, λ_{U})/U such that

1. for all x_{1} ∈ U the fibre (A_{U}, λ_{U})_{x_{1}} is the Jacobian of a curve which maps 2 : 1 to D_{\overline{ρ},x_{1}} ramified exactly at s_{1}(x_{1}), ..., s_{6}(x_{1});

2. there is an isomorphism α_{U} : W\overline{ρ} →^{∼} A_{U}[2] of finite flat group schemes over U with alternating pairings;

3. and there exists i_{U} : ℤ[(1 + \sqrt{5})/2] ↪ End (A_{U}) which is compatible with α_{U} and the action of 𝔽_{4} on W\overline{ρ}.

(In [ST] the existence of i_{U} is only explained over a non-empty open subset of U. That it extends to the whole of U follows from remark 1.10 (a) of chapter I of [CF].) We remind the reader that A_{U} is not canonical. Suppose that \overline{x} is a geometric point of U. If f is an automorphism of (A_{U}, λ_{U}, i_{U}, α_{U})\overline{x} then T_{2}(f) ≡ 1 mod 2 and so T_{2}(f^{2}) ≡ 1 mod 4. As f has finite order this implies that f^{2} = 1. If f ≠ ±1 then A_{U,\overline{x}} ≅ (1+f)/2A_{U,\overline{x}}⊕(1−f)/2A_{U,\overline{x}} and λ_{U} correspondingly decomposes as the direct sum of two polarisations. This contradicts the fact that θ(\overline{x}) ∈ Y 0 ; \overline{ρ} ≅ H0 ; \overline{ρ}. Thus we must have Aut ((A_{U}, λ_{U}, i_{U}, α_{U})\overline{x}) = {±1}. In particular if we set

Ũ = \{(a, b) ∈ (A_{U} × A_{U})[\sqrt{5}] | ⟨a, b⟩ ≠ 1\}/ ∼,

where (a, b) ∼ (a′, b′) if and only if (a, b) = ±(μa′, b′) for some μ ∈ 𝔽_{5}^{×}, then the construction of Ũ is canonical and so we can glue the Ũ/U to give an etale cover X̃0 ; \overline{ρ}/X0 ; \overline{ρ} of degree 60. The argument of lemma 2.7 of [ST] shows that X̃0 ; \overline{ρ} is geometrically irreducible.

Suppose for the moment that we can find a point x_{2} ∈ X0 ; \overline{ρ}(ℚ_{2}), a Zariski open U_{2} ⊂ X0 ; \overline{ρ} × ℚ_{2} as above and a continuous character χ_{2} : Gal (ℚ_{2}^{ac}/ℚ_{2}) → {±1} such that

• the twist A_{U_{2},x_{2}}(χ_{2}) of A_{U_{2},x_{2}} by χ_{2} has good reduction

• and, if \overline{A}_{U_{2},x_{2}}(χ_{2}) denotes the mod 2 reduction of the Néron model of A_{U_{2},x_{2}}(χ_{2}) over ℤ_{2}, then \overline{A}_{U_{2},x_{2}}(χ_{2})[2]^{et} ≠ (0) and Frob_{2} acts on \overline{A}_{U_{2},x_{2}}(χ_{2})[2]^{et} by α.

Then we can find a neighbourhood (for the 2-adic topology) U ⊂ X0 ; \overline{ρ}(ℚ_{2}) of x_{2} such that if x ∈ U then

• x ∈ U_{2},

• A_{U_{2},x}(χ_{2}) has good reduction at 2

• and \overline{A}_{U_{2},x}(χ_{2})[2] ≅ \overline{A}_{U_{2},x_{2}}(χ_{2})[2].

Because X\overline{ρ} is rational, it follows from Ekedahl’s version of the Hilbert irreducibility theorem (see theorem 1.3 of [E]) that we can find a point x ∈ X0 ; \overline{ρ}(ℚ) such that

• x ∈ U

• and if x̃ is a point of X̃0 ; \overline{ρ} above x then [ℚ(x̃) : ℚ] = 60.

Suppose that U is a Zariski neighbourhood of x in X0 ; \overline{ρ} as above. Then (A_{U}, λ_{U}, i_{U}, α_{U})_{x} × ℚ_{2} is a twist by some character χ′_{2} : Gal (ℚ_{2}^{ac}/ℚ_{2}) → {±1} of (A_{U_{2}}, λ_{U_{2}}, i_{U_{2}}, α_{U_{2}})_{x}. Choose a character χ : Gal (ℚ^{ac}/ℚ) ⟶ {±1} which restricts to χ_{2}χ′_{2} on Gal (ℚ_{2}^{ac}/ℚ_{2}). Then A_{U,x}(χ) has the following properties.

• (A_{U,x}(χ), λ_{U,x})/ℚ is a principally polarised abelian surface.

• i_{U,x} : ℤ[(1 + \sqrt{5})/2] ↪ End (A_{U,x}(χ)) and the image is fixed by the λ_{U,x}-Rosati involution.

• As an 𝔽_{4}[Gal (ℚ^{ac}/ℚ)]-module, A_{U,x}(χ)[2](ℚ^{ac}) is equivalent to ρ.

• A_{U,x}(χ) × ℚ_{2} ≅ A_{U_{2},x}(χ_{2}) and so A_{U,x}(χ) has good reduction at 2.

• \overline{A}_{U,x}(χ)[2] ≅ \overline{A}_{U_{2},x_{2}}(χ_{2})[2] and so \overline{A}_{U,x}(χ)[2]^{et} ≠ (0) and Frob_{2} acts on \overline{A}_{U,x}(χ)[2]^{et} by α.

• If G denote the image Gal (ℚ^{ac}/ℚ) in Aut_{𝔽_{4}}(A_{U,x}(χ)[\sqrt{5}]) ≅ GL_{2}(𝔽_{5}) then det G = 𝔽_{5}^{×} (because of the λ-Weil pairing) and

#G/G ∩ \{( \begin{matrix}μ & 0 \\ 0 & ν \end{matrix} )| ν = ±1, μ ∈ 𝔽_{5}^{×}\} = 60.

Then it is elementary to check that G = GL_{2}(𝔽_{5}).

It remains to explain the construction of x_{2}. This we will do in two steps. More precisely we will show the following two results.

1. There is a quadruple (A, λ, i, α) (as above) defined over K such that A has good reduction and, if \overline{A} denotes the reduction of its Néron model, then \overline{A}[2]^{et} ≠ (0) and Frob_{2} acts on \overline{A}[2]^{et} by α.

2. If y ∈ Y 0 ; \overline{ρ}(ℚ_{2}) then there is a point of X0 ; \overline{ρ}(ℚ_{2}) mapping to y under θ.

The first assertion gives a point y_{2} ∈ H0 ; \overline{ρ}(ℚ_{2}) = Y 0 ; \overline{ρ}(ℚ_{2}) and the second a point x_{2} ∈ X0 ; \overline{ρ}(ℚ_{2}) mapping to y_{2} under θ. This point x_{2} will suffice.

We turn first to the second assertion. Suppose y ∈ Y 0 ; \overline{ρ}(ℚ_{2}) and let Y\overline{ρ}(y)′ denote the complement in Y\overline{ρ} of the intersection of Y\overline{ρ} with the tangent plane to Y\overline{ρ} at y. Thus Y\overline{ρ}(y)′ is a smooth affine cubic surface. There is an involution ι_{y} of Y\overline{ρ}(y) which sends any point z to the third point of intersection of the line through y and z with the cubic surface Y\overline{ρ}. We will let Y\overline{ρ}(y) denote the twist of Y\overline{ρ}(y)′ by ι_{y} over Gal (ℚ_{2}(\sqrt{5})/ℚ_{2}). We may identify Y\overline{ρ}(y) as a Zariski open subset of the fibre of θ : X0 ; \overline{ρ} → Y 0 ; \overline{ρ} above y, and so it suffices to show that Y\overline{ρ}(y)(ℚ_{2}) ≠ ∅.

Note that the equations defining Y also define a smooth projective surface over ℤ_{2}, which we will also denote by Y. The construction of Y\overline{ρ}, Y\overline{ρ}(y)′ and Y\overline{ρ}(y) from Y all make sense over ℤ_{2} and give rise to smooth relative surfaces over ℤ_{2}, which we will denote by the same symbols. (We are using the fact that \overline{ρ} is unramified. We are not asserting that these integral models have any moduli theoretic meaning.) By Hensel’s lemma it will suffice to show that Y\overline{ρ}(y)(𝔽_{2}) is non-empty.

Without loss of generality the surface Y\overline{ρ} ×𝔽_{2} is given in ℙ^{3} by the equation

X_{1}^{3} + X_{1}X_{2}^{2} + X_{2}^{3} + X_{3}^{2}X_{4} + X_{3}X_{4}^{2} = 0.

(If γ is a root of T^{3} + T + 1 = 0, then (X_{1} : X_{2} : X_{3} : X_{4}) corresponds to the point

\begin{matrix}((X_{3} + X_{4}) + X_{1}γ + X_{2}γ^{2} : (X_{3} + X_{4}) + X_{1}γ^{2} + X_{2}γ^{4} : \\ (X_{3} + X_{4}) + X_{1}γ^{4} + X_{2}γ : X_{3} : X_{4}) \end{matrix}

of Y × 𝔽_{2}.) Thus Y\overline{ρ}(𝔽_{2}) has three points P = (0 : 0 : 1 : 0), Q = (0 : 0 : 0 : 1) and R = (0 : 0 : 1 : 1).

First suppose that y reduces to P. Then Y\overline{ρ}(y) × 𝔽_{2} is the surface given in affine 3-space by the equation

x_{1}^{3} + x_{1}x_{2}^{2} + x_{2}^{3} + x_{3} + x_{3}^{2} = 0

and ι_{y} maps (x_{1}, x_{2}, x_{3}) to

(x_{1}, x_{2}, x_{3} + 1).

(Here we set x_{i} = X_{i}/X_{4}.) Thus Y\overline{ρ}(y) × 𝔽_{2} is given in affine 3-space by the equation

y_{1}^{3} + y_{1}y_{2}^{2} + y_{2}^{3} + 1 + y_{3} + y_{3}^{2} = 0.

(Here we let (y_{1}, y_{2}, y_{3}) correspond to the point (x_{1}, x_{2}, x_{3}) = (y_{1}, y_{2}, y_{3} + (1 + \sqrt{5})/2).) Thus Y\overline{ρ}(y)(𝔽_{2}) consists of 6 points.

The case that y reduces to Q is exactly analogous, and again we see that Y\overline{ρ}(y)(𝔽_{2}) consists of 6 points.

Thirdly suppose that y reduces to R. Introducing a new variable X′_{4} = X_{3} + X_{4} we see that Y\overline{ρ} × 𝔽_{2} can also be described in ℙ^{3} by the equation

X_{1}^{3} + X_{1}X_{2}^{2} + X_{2}^{3} + X_{3}^{2}X′_{4} + X_{3}(X′_{4})^{2} = 0,

and that in these new coordinates R becomes the point (0 : 0 : 1 : 0). Thus the analysis is the same again and we see that Y\overline{ρ}(y)(𝔽_{2}) again consists of 6 points.

Finally we turn to our first assertion. Let K denote the field ℚ(a), where a is a root of

T^{4} + 13T^{2} + 41 = 0.

Then 13 + 2a^{2} is a square root of 5, which we will denote \sqrt{5}. Moreover K is a CM field with totally real subfield ℚ(\sqrt{5}). The inverse different 𝔡_{K/ℚ}^{−1} is principal with generator ξ = (13a + 2a^{3})^{−1}. We have the prime factorisation

2O_{K} = (((1 + \sqrt{5})/2 + a)/2)(((1 + \sqrt{5})/2 − a)/2).

As ±1 are the only roots of unity in K, the only elements of K^{×} with norm down to ℚ(\sqrt{5}) equal to 2 are (±(1 + \sqrt{5})/2 ± a)/2.

The normal closure of K/ℚ is K(\sqrt{41})/ℚ and Gal (K(\sqrt{41})/ℚ) is generated by two elements σ and τ, where

\begin{matrix}σ(a) = \sqrt{41}/a & τ(a) = a \\ σ(\sqrt{41}) = −\sqrt{41} & τ(\sqrt{41}) = −\sqrt{41}. \end{matrix}

Thus σ^{4} = τ^{2} = 1, τστ = σ^{3} and σ^{2} = c. By the Cebotarev density theorem we may choose a prime ℘ of O_{K} which is split completely and lies above a rational prime p ≡ 3 mod 4. Let α_{0} denote the character

O_{K,p}^{×} → → O_{K,℘}^{×} → → {±1}.

Fix an embedding K(\sqrt{41}) ↪ ℂ such that a has negative imaginary part, 13 + 2a^{2} > 0 and \sqrt{41} > 0. Then Φ = \{1, σ\} is a CM-type with reflex (L, Φ′), where L = K(\sqrt{41})^{\{1,στ\}} and Φ′ = \{1, σ^{3}\}. The field L is also a CM field and has totally real subfield ℚ(\sqrt{41}). It is isomorphic to the field obtained by adjoining a root of T^{4} + 26T^{2} + 5 to ℚ. Then L has class number 1 and O_{L}^{×} is generated by −1 and 32 + 5\sqrt{41}. We have a prime factorisation 2O_{l} = II^{c}J with #O_{L}/I = 2 and #O_{L}/J = 4. We have a homomorphism

\begin{matrix}N_{Φ′} : L^{×} & ⟶ & K^{×} \\ x & ⟼ & xσ^{3}(x). \end{matrix}

Then N_{Φ′} extends to a map 𝔸_{L}^{×} → 𝔸_{K}^{×}. Define a continuous homomorphism

α : 𝔸_{L}^{×} ⟶ K^{×}

by setting • α|_{L^{×}} = N_{Φ′},

• α|_{O_{L,p}^{×}} = α_{0} ∘ N_{Φ′},

• α|_{O_{L,p′}^{×}} = 1 for any rational prime p′ ≠ p,

• and α|_{L_{v}^{×}} = 1 for any infinite place v of L.

(This makes sense because the class number of L is one and because (α_{0} ∘ N_{Φ′})|_{O_{L}^{×}} = N_{Φ′}|_{O_{L}^{×}}.)

Then there is a triple (A, λ, i)/L (a principally polarised simple abelian surface with an action i of O_{K}) which has type (K, Φ, O_{K}, ξ) and character α. (See [Lang], especially theorems 3.6 and 4.5 of chapter 1 and corollary 5.3 of chapter 5.) Because α is trivial on O_{L,I}^{×}, we see from the fundamental theorem of complex multiplication (see theorem 1.1 of chapter 4 of [Lang]) that, for a rational prime l > 2, inertia at I acts trivially on T_{l}A, the l-adic Tate module of A. Thus A has good reduction at I. Let \overline{A} denote the reduction mod I of the Néron model of A. Moreover, if I = (a) then Frob_{2} acts on T_{l}A via ±N_{Φ′}a. As N_{K/ℚ(\sqrt{5})}N_{Φ′}a = 2 we see that ±N_{Φ′}a = (±(1 + \sqrt{5})/2 ± a)/2 and so

±N_{Φ′}a ≡ (1 + \sqrt{5})/2 mod (N_{Φ′}I)^{c}.

Thus \overline{A}[N_{Φ′}I^{c}] is etale and Frob_{2} acts on it as (1 + \sqrt{5})/2.

If α = (1+\sqrt{5})/2 then (A, λ, i|_{ℤ[(1+\sqrt{5})/2]})/L_{I} will suffice to give the desired example. If on the other hand α = (1 − \sqrt{5})/5 then (A, λ, i|_{ℤ[(1+\sqrt{5})/2]} ∘ σ)/L_{I} will suffice to give the desired example.

We would now like to apply this theorem to deduce the modularity of certain mod 2 representations. If N, M and k are positive integers we will denote by h_{k}(N; M) the ℤ-algebra generated by the Hecke operators T_{p} and ⟨p⟩ for any prime p |̸NM, and by the Hecke operators U_{p} for any prime p|NM acting on the space of weight k cusp forms for Γ_{1}(N) ∩ Γ_{0}(M). If M|N we will drop it from the notation and write simply h_{k}(N). If p |̸NM set S(p) = p^{k−2}⟨p⟩. Also for every positive integer n define T (n) by the relations

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

• (1 − T_{p}X + pS(p)X^{2}) ∑_{r=1}^{∞} T (p^{r})X^{r} = 1 for any prime p |̸NM,

• and T (p^{r}) = U_{p}^{r} for every prime p|NM.

Corollary 1.2. Fix a continuous homomorphism

\overline{ρ} : Gal (ℚ^{ac}/ℚ) ⟶ SL_{2}(𝔽_{4}).

Suppose that \overline{ρ} is unramified at 2 and 5 and that \overline{ρ}(Frob_{2}) has distinct eigenvalues α, β ∈ 𝔽_{4}^{×}. Then there is an odd positive integer N divisible by all primes at which \overline{ρ} ramifies and a homomorphism

\overline{f}_{α} : h_{2}(N) ⟶ 𝔽_{4}

which takes

1. T_{p} to tr \overline{ρ}(Frob_{p}) for all primes p |̸2N;

2. T_{2} to α;

3. and U_{p} to 0 for all p|N.

Proof. First note that in [BCDT] theorem 4.1 of [ST] is improved to suppress the condition on \overline{ρ}(I_{3}). Thus theorem 4.2 of [ST] can be improved to suppress the condition that A has semi-stable reduction at 3. The proof of this corollary is then the same as the proof of theorem 4.3 in [ST] except that we replace references to theorem 4.2 by this improvement and references to theorem 3.6 by references to theorem 1.1 of this paper.

2 2-adic modular forms.

In this section we will recall some facts about 2-adic modular forms. (The most important for us is the assertion that a 2-adic limit of ordinary classical modular forms is overconvergent - see lemma 2.9.) Many of these assertions appear in the literature, but we have not been able to locate proofs for them. For primes l > 3 such results are due to Katz [K], but we will follow Coleman’s approach via rigid geometry. In this section we will work with an arbitrary rational prime l as it makes no difference to the arguments.

Fix an integer N ≥ 5 which is not divisible by l. Let X_{1}(N)/ℤ_{l} denote the usual compactification of the moduli scheme for pairs (E, i) where E is an elliptic curve and i is an embedding μ_{N} ↪ E[N]. Also let X_{1}(N; l)/ℤ_{l} denote the usual compactification of the moduli scheme for pairs (E, i, E →^{α} E′) where E is an elliptic curve, i is an embedding μ_{N} ↪ E[N] and α : E → E′ is an isogeny of degree l. There are two natural projections π_{1} and π_{2} : X_{1}(N; l) → X_{1}(N), which take (E, i, E →^{α} E′) to (E, i) and (E′, α ∘ i) respectively.

We will let ω_{X_{1}(N)} (resp. ω_{X_{1}(N;l)}) denote the canonical extension to the cusps of the pullback by the identity section of the sheaf of relative differentials of the universal elliptic curve over the non-cuspidal locus of X_{1}(N) (resp. X_{1}(N; l)). Then π_{1}^{∗}ω_{X_{1}(N)} = ω_{X_{1}(N;l)} and there is a natural map j = (α^{∨})^{∗} : ω_{X_{1}(N;l)} → π_{2}^{∗}ω_{X_{1}(N)}. After one inverts l, j becomes an isomorphism.

We will let SS denote the finite set of points in X_{1}(N)(𝔽_{l}^{ac}) corresponding to supersingular elliptic curves. For s ∈ SS choose T_{s} ∈ O_{X_{1}(N)×W (𝔽_{l}^{ac}),s} so that

(X_{1}(N) × W (𝔽_{l}^{ac}))_{s}^{∧} ≅ Spf W (𝔽_{l}^{ac})[[T_{s}]],

and so that if σ ∈ Gal (𝔽_{l}^{ac}/𝔽_{l}) and s ∈ SS then

(1 × σ^{∗})^{∗}(T_{(1×σ^{∗})(s)}) = T_{s}.

(Here W (k) denotes the Witt vectors of k.) We will let X_{1}(N)^{an} denote the rigid analytic space over ℂ_{l} (the completion of ℚ_{l}^{ac}) associated to X_{1}(N). It is connected. If r ∈ l^{ℚ} and 1 ≥ r ≥ 1/l we will let X_{1}(N)_{≥r} (if r ≠ 1/l) (resp. X_{1}(N)_{>r} (if r ≠ 1)) denote the rigid analytic subspace of X_{1}(N)^{an} where for each s ∈ SS we remove all points x in the residue disc of s with |T_{s}(x)|_{l} < r (resp. ≤ r). (Here | |_{l} is the l-adic absolute value normalised by |l|_{l} = 1/l.) We note that X_{1}(N)_{≥r} is connected. (Suppose that X_{1}(N)_{≥r} had an admissible open cover {U, V }, with U and V non-empty and disjoint. For each s ∈ SS the preimage of s in X_{1}(N)_{≥r} is an annulus and hence connected and contained in either U or V. Let Ũ (resp. Ṽ) denote the union of U (resp. V) with the residue disc of each s ∈ SS for which the preimage of s in X_{1}(N)_{≥r} is contained in U (resp. V). Then {Ũ, Ṽ} is an admissible open cover of X_{1}(N)^{an} by disjoint non-empty sets, a contradiction.)

We will let M_{k}^{≥r}(N) (resp. M_{k}^{>r}(N)) denote the space of sections of (ω_{X_{1}(N)}^{an})^{⊗k} over X_{1}(N)_{≥r} (resp. X_{1}(N)_{>r}). The spaces M_{k}^{≥r}(N) have natural norms making them Banach spaces. More precisely we set

|f|_{r} = sup_{x∈X_{1}(N)_{≥r}(ℂ_{l})}|f|_{x},

where we define |f|_{x} as follows. Let \overline{x} ∈ X_{1}(N)(𝔽_{l}^{ac}) denote the reduction of x and let f_{0} denote a local generator for ω_{X_{1}(N)}^{⊗k} near \overline{x}. Then we set

|f|_{x} = |(f/f_{0})(x)|_{l},

which is easily checked to be independent of the choice of f_{0}. Note that if r_{1} ≥ r_{2} and if f ∈ M_{k}^{≥r_{2}}(N) then

|f|_{r_{1}} ≤ |f|_{r_{2}}.

We will let X_{1}(N)^{0} denote the formal completion of X_{1}(N) along its locally closed subscheme X_{1}(N) × 𝔽_{l} − SS. It is a formal scheme over ℤ_{l}. The base change to ℂ_{l} of the rigid analytic space associated to X_{1}(N)^{0} is just X_{1}(N)_{≥1}. Thus we get an identification

Γ(X_{1}(N)^{0}, ω_{X_{1}(N)}^{⊗k})ˆ⊗_{ℤ_{l}}ℂ_{l} →^{∼} M_{k}^{≥1}(N),

under which Γ(X_{1}(N)^{0}, ω_{X_{1}(N)}^{⊗k})ˆ⊗_{ℤ_{l}}O_{ℂ_{l}} is identified to the unit ball in M_{k}^{≥1}(N). There is a map

Spec ℤ_{l}((q)) ⟶ X_{1}(N)

corresponding to the pair (𝔾_{m}/q^{ℤ}, i^{can}), where 𝔾_{m}/q^{ℤ} denotes the Tate curve (Tate(q) in the notation of section 8.8 of [KM]) and where i^{can} comes from the tautological embedding μ_{N} ↪ 𝔾_{m} (see proposition 8.11.7 of [KM]). This map extends to a map

Spec ℤ_{l}[[q]] ⟶ X_{1}(N),

(use theorem 8.11.10 of [KM]) and this gives rise to a map

Spf ℤ_{l}[[q]] ⟶ X_{1}(N)^{0}.

If f ∈ Γ(X_{1}(N)^{0}, ω_{X_{1}(N)}^{⊗k}) then its pullback to Spf ℤ_{l}[[q]] has the form

(∑_{n=0}^{∞} c_{n}(f)q^{n}) (dt/t)^{⊗k},

where t is the usual parameter on 𝔾_{m} and where we refer to ∑_{n=0}^{∞} c_{n}(f)q^{n} as the q-expansion at infinity of f. This extends to a map

\begin{matrix}M_{k}^{≥1}(N) & ⟶ & ℂ_{l}[[q]] \\ f & ⟼ & ∑_{n=0}^{∞} c_{n}(f)q^{n}. \end{matrix}

From the q-expansion principle (see section 1.6 of [K] and note that X_{1}(N)×𝔽_{l}^{ac} is irreducible) we deduce that for f ∈ M_{k}^{≥1}(N) we have

|f|_{1} = sup_{n} |c_{n}(f)|_{l}.

If l ≥ 5 we will let E denote the section of ω_{X_{1}(N)}^{⊗(l−1)} over X_{1}(N) with qexpansion at infinity

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

where B_{k} denotes the Bernoulli number, and σ_{t}(n) = ∑_{0<d|n} d^{t}. Then the qexpansion at infinity of E is congruent to 1 modulo l. For l = 3 we may choose a section E of ω_{X_{1}(N)}^{⊗2} over X_{1}(N) with the same property (because N ≥ 5 and 3 |̸N, see section 2.1 of [K]). If l = 2 and if 5|N we can again choose a section E of ω_{X_{1}(N)} over X_{1}(N) with the same property (again see section 2.1 of [K]). In all these cases the section E, when pulled back to X_{1}(N)×𝔽_{l}^{ac}, has a simple zero at each element of SS and no other zeroes (see for instance theorem 12.4.3 of [KM]). In particular if for each s ∈ SS we make a choice of local generator f_{s} of ω_{X_{1}(N)}^{⊗(l−1)} near s such that for all σ ∈ Gal (𝔽_{l}^{ac}/𝔽_{l}) we have

(1 × σ^{∗})^{∗}f_{(1×σ^{∗})(s)} = f_{s},

then the E/f_{s} for s ∈ SS form one possible choice for a collection of local parameters T_{s} at s ∈ SS satisfying

(1 × σ^{∗})^{∗}T_{(1×σ^{∗})(s)} = T_{s}

for all σ ∈ Gal (𝔽_{l}^{ac}/𝔽_{l}) and s ∈ SS. Hence E has no zero on X_{1}(N)_{>1/l}.

If l ≥ 3 we will set E′ = E. If l = 2 we will take E′ to be the section of ω_{X_{1}(N)}^{⊗4} over X_{1}(N) with q-expansion at infinity

1 + 240 ∑_{n=1}^{∞} σ_{3}(n)q^{n}.

In either case the q-expansion at infinity of E′ is congruent to 1 modulo l and E′ has no zeroes in X_{1}(N)_{>l^{−1/4}}.

We recall some elementary results about rigid analytic functions on annuli. The set of analytic functions on the annulus β ≤ |z|_{l} ≤ α is the set of functions

f(z) = ∑_{n=−∞}^{∞} a_{n}z^{n}

for which |a_{n}|_{l}β^{n} → 0 as n → −∞ and |a_{n}|_{l}α^{n} → 0 as n → ∞. If r ∈ l^{ℚ} and β ≤ r ≤ α then the supremum of |f(z)|_{l} on |z|_{l} = r equals

sup_{n} |a_{n}|_{l}r^{n}.

(To see this set A = sup_{n} |a_{n}|_{l}r^{n}. Then

sup_{|z|_{l}=r} |f(z)|_{l} = A sup_{|w|_{l}=1} | ∑_{|a_{n}|_{l}r^{n}=A} c_{n}w^{n}|_{l}

for some c_{n} with |c_{n}|_{l} = 1. However for |w|_{l} = 1 we see that

| ∑_{|a_{n}|_{l}r^{n}=A} c_{n}w^{n}|_{l} ≤ 1,

with equality for some such choice of w.) In particular we see that |f(z)|_{l} will always achieve its maximum on either |z|_{l} = α or |z|_{l} = β (or possibly on both). In the former case this maximum equals

sup_{n} |a_{n}|_{l}α^{n} = sup_{n≥0} |a_{n}|_{l}α^{n}

and in the latter case it equals

sup_{n} |a_{n}|_{l}β^{n} = sup_{n≤0} |a_{n}|_{l}β^{n}.

Suppose now that f is an analytic function on the annulus β ≤ |z|_{l} < α such that |f(z)|_{l} is bounded by A. Then we have

f(z) = ∑_{n=−∞}^{∞} a_{n}z^{n},

where |a_{n}|_{l}β^{n} → 0 as n → −∞ and where for all n we have

|a_{n}|_{l} ≤ Aα^{−n}

and

|a_{n}|_{l} ≤ Aβ^{−n}.

If |f(z)|_{l} achieves its supremum it does so on |z|_{l} = β and the supremum equals

sup_{n} |a_{n}|_{l}β^{n} = sup_{n≤0} |a_{n}|_{l}β^{n}.

Lemma 2.1. Suppose that 1 > r > 1/l, that r ∈ l^{ℚ} and that f is a rigid analytic function on X_{1}(N)_{≥r}. Then |f(x)|_{l} achieves its supremum and does so at some point y which reduces to an element s ∈ SS and satisfies |T_{s}(y)|_{l} = r.

Proof. Because X_{1}(N)_{≥r} is a finite union of affinoids the maximum modulus principle tells us that |f(x)|_{l} does achieve its supremum. Thus we may assume that this supremum equals 1. If |f(x)|_{l} does not achieve its supremum in X_{1}(N)_{≥1} then it does so in the inverse image under reduction of some s ∈ SS and the lemma follows from the facts about rigid analytic functions on annuli which we recalled above. Thus suppose that f achieves its maximum in X_{1}(N)_{≥1}.

As |f(x)|_{l} ≤ 1 on X_{1}(N)_{≥1}, f is a global section of the structure sheaf of the formal completion of X_{1}(N) × O_{ℂ_{l}} along X_{1}(N) × 𝔽_{l}^{ac} − SS and thus reduces to give a regular function \overline{f} on X_{1}(N)×𝔽_{l}^{ac}−SS. Thus we may choose s ∈ SS such that either \overline{f} has a pole at s or \overline{f} is constant. Choose also an affine neighbourhood U of s in X_{1}(N) × 𝔽_{l}^{ac} which contains no other element of SS and which admits a regular function \overline{g} which has a simple zero at s and no other zero on U. Let the formal completion of X_{1}(N) × W (𝔽_{l}^{ac}) along U equal Spf A and let g ∈ A be a lift of \overline{g}. Note that the formal completion of X_{1}(N) × W (𝔽_{l}^{ac}) at s is isomorphic to Spf W (𝔽_{l}^{ac})[[g]]. The formal completion of X_{1}(N) × O_{ℂ_{l}} along U − {s} is Spf (Aˆ⊗O_{ℂ_{l}})⟨⟨S⟩⟩/(gS − 1). Thus we may expand f as

∑_{i=0}^{∞} f_{i}S^{i}

with f_{i} ∈ (Aˆ⊗O_{ℂ_{l}}) and f_{i} → 0 as i → ∞. The same expansion holds on the rigid analytic subspace of X_{1}(N)_{≥r} consisting of points which reduce to U (as this space is connected, being the inverse image under reduction of a Zariski connected space). Moreover on U we see that

\overline{f} = ∑_{i=0}^{∞} \overline{f}_{i}g^{−i},

where \overline{f}_{i} denotes the reduction of f_{i} and where now the sum is finite. In the formal completion of X_{1}(N) × O_{ℂ_{l}} at s we may expand

f_{i} = ∑_{j=0}^{∞} a_{ij}g^{j}

with a_{ij} ∈ O_{ℂ_{l}}. Thus, on the rigid analytic subspace of X_{1}(N)_{≥r} consisting of points which reduce to s, we see that

f = ∑_{k=−∞}^{∞}(∑_{i} a_{i,i+k})g^{k}.

(The second sum is over i ∈ ℤ such that i ≥ 0 and i + k ≥ 0.) Similarly we see that in the formal completion of X_{1}(N) × 𝔽_{l}^{ac} at s we have

\overline{f} = ∑_{k=−∞}^{∞}(∑_{i} a_{i,i+k})\overline{g}^{k}.

Write b_{k} for ∑_{i} a_{i,i+k}. Then b_{k} ∈ O_{ℂ_{l}} and either

• for some k < 0, b_{k} is a unit;

• or b_{0} is a unit and b_{k} reduces to zero for all k ≠ 0.

In the either case we see that the supremum of |f(x)|_{l} on |g(x)|_{l} = r (i.e. on |T_{s}(x)|_{l} = r) is ≥ 1, as desired.

Lemma 2.2. If 1 > r > 1/l then there is a constant C (depending on k, N and r) such that for all f ∈ M_{k}^{≥r}(N) we have

|f|_{r} ≤ C sup_{s,x} |f|_{x},

where s runs over SS and where x runs over elements of the residue disc of s with |T_{s}(x)|_{l} = r.

Proof. If l = 2 reduce to the case 5|N by passing to a cover. By lemma 2.1 we see that |f^{l−1}/E^{k}|_{l} on X_{1}(N)_{≥r} achieves its supremum at some point x which reduces to some s ∈ SS and which satisfies |T_{s}(x)|_{l} = r. Thus for all y ∈ X_{1}(N)_{≥r} we have

|f|_{y}^{l−1}/|E|_{y}^{k} ≤ sup_{s,x} |f|_{x}^{l−1}/|E|_{x}^{k}

where s and x run over the sets described in the statement of the lemma. Hence

|f|_{r}^{l−1} ≤ |E|_{r}^{k} sup_{s,x}(|f|_{x}^{l−1}/r^{k}),

where again s and x run over the sets described in the statement of the lemma. The lemma follows with C = (|E|_{r}/r)^{k/(l−1)}.

For each s ∈ SS, choose a local generator f_{s} of ω_{X_{1}(N)}^{⊗k} near s. If f ∈ M_{k}^{≥r}(N) and s ∈ SS then restricting f to the annulus 1 > |T_{s}(x)|_{l} ≥ r in the residue disc of s we see that f/f_{s} can be expanded

f/f_{s} = ∑_{n=−∞}^{∞} a_{n}(s, f)T_{s}^{n},

where the a_{n}(s, f) are bounded for n > 0 and where

|a_{n}(s, f)|_{l}r^{n} ⟶ 0

as n → −∞. Choose a non-negative integer M such that r^{−M} > C (the constant from the lemma), and choose π_{r} ∈ ℂ_{l} with |π_{r}|_{l} = r. Now consider the map Θ from M_{k}^{≥r}(N) to the direct sum of #SS Tate algebras ℂ_{l}⟨T ⟩^{SS} which sends f to

(∑_{n=0}^{∞} a_{M−n}(s, f)π_{r}^{M−n}T^{n})_{s∈SS}.

One clearly has |Θ(f)| ≤ |f|_{r}. (Here, as usual, we set |(∑_{n} b_{n}(s)T^{n})_{s∈SS}| = sup_{s,n} |b_{n}(s)|_{l}.) On the other hand for all n ∈ ℤ and s ∈ SS we have |a_{n}(s, f)π_{r}^{n}|_{l} ≤ |Θ(f)|. (Suppose not. Choose s and n so that |a_{n}(s, f)π_{r}^{n}|_{l} is maximal. Then we must have n > M and we see that

|f|_{r} ≥ |a_{n}(s, f)|_{l} = r^{−n} sup_{s,x} |f|_{x} > C sup_{s,x} |f|_{x} ≥ |f|_{r},

a contradiction.) Thus

C|Θ(f)| ≥ C sup_{s,x} |f|_{x} ≥ |f|_{r}.

We deduce that Θ is a homeomorphism onto a closed subspace of ℂ_{l}⟨T ⟩^{SS}.

Lemma 2.3. Suppose that 1 > r_{1} > r_{2} > 1/l. Then the natural inclusion

M_{k}^{≥r_{2}}(N) ↪ M_{k}^{≥r_{1}}(N)

is completely continuous.

Proof. We have a commutative diagram

M_{k}^{≥r_{2}}(N) ↪ M_{k}^{≥r_{1}}(N)
↓  ↓
ℂ_{l}⟨T ⟩^{SS} ⟶ ℂ_{l}⟨T ⟩^{SS}
(∑_{n=0}^{∞} b_{n}(s)T^{n})_{s∈SS} ⟼ (∑_{n=0}^{∞} b_{n}(s)(π_{r_{2}}/π_{r_{1}})^{n}T^{n})_{s∈SS},

where the vertical arrows are homeomorphisms onto closed subspaces (and where we have made the same choice of M to define both vertical arrows). The lower horizontal arrow is a limit of continuous operators with finite range and hence completely continuous. It follows that the upper horizontal arrow is completely continuous.

The reduction X_{1}(N; l) × 𝔽_{l}^{ac} of X_{1}(N,; l) has two irreducible components which we will denote \overline{X}_{1}(N; l)_{∞} and \overline{X}_{1}(N; l)_{0}. We choose the labelling so that

• π_{1} : \overline{X}_{1}(N; l)_{∞} ⟶^{∼} X_{1}(N) × 𝔽_{l}^{ac},

• π_{2} : \overline{X}_{1}(N; l)_{∞} ⟶ X_{1}(N) × 𝔽_{l}^{ac} has degree l,

• π_{1} : \overline{X}_{1}(N; l)_{0} ⟶ X_{1}(N) × 𝔽_{l}^{ac} has degree l,

• and π_{2} : \overline{X}_{1}(N; l)_{0} ⟶^{∼} X_{1}(N) × 𝔽_{l}^{ac}.

The two curves \overline{X}_{1}(N; l)_{∞} and \overline{X}_{1}(N; l)_{0} intersect in a finite number of points which we shall denote SS_{l}. Then π_{1} : SS_{l} →^{∼} SS and π_{2} : SS_{l} →^{∼} SS are both bijections. (See for instance lemma 5.3.1 of [KM] for these assertions.) If s ∈ SS_{l} we will write T_{s,i} for π_{i}^{∗}T_{π_{i}s}.

Lemma 2.4. If s ∈ SS_{l} then (X_{1}(N; l) × W (𝔽_{l}^{ac}))_{s}^{∧} is isomorphic to

Spf W (𝔽_{l}^{ac})[[T_{s,1}, T_{s,2}]]/((T_{s,1} − T_{s,2}^{l})(T_{s,2} − T_{s,1}^{l}) − lu_{s}),

for some u_{s} ∈ W (𝔽_{l}^{ac})[[T_{s,1}, T_{s,2}]]^{×}.

Proof. Theorem 6.6.2 of [KM] tells us that (X_{1}(N; l) × W (𝔽_{l}^{ac}))_{s}^{∧} ≅ Spf R, for some 2-dimensional, regular complete local ring R, which is flat over W (𝔽_{l}^{ac}). Theorem 13.4.7 of [KM] tells us that

R/lR ≅ 𝔽_{l}^{ac}[[T_{s,1}, T_{s,2}]]/((T_{s,1} − T_{s,2}^{l})(T_{s,2} − T_{s,1}^{l})).

Thus we have a surjection W (𝔽_{l}^{ac})[[T_{s,1}, T_{s,2}]] → → R and the kernel must be generated by one element f with

• f ≡ (T_{s,1} − T_{s,2}^{l})(T_{s,2} − T_{s,1}^{l}) mod l,

• and f ∉ (l, T_{s,1}, T_{s,2})^{2}.

The lemma follows.

Corollary 2.5. If s ∈ SS_{l} then

(X_{1}(N; l) × W (𝔽_{l}^{ac}))_{s}^{∧} ≅ Spf W (𝔽_{l}^{ac})[[X_{1}, X_{2}]]/(X_{1}X_{2} − l).

Proof. Take for instance X_{1} = (T_{s,1} − T_{s,2}^{l}) and X_{2} = (T_{s,2} − T_{s,1}^{l})u_{s}^{−1}.

For r ∈ l^{ℚ} and 1 ≥ r > 1/l we will define X_{1}(N; l)_{≥r}^{∞} (resp. X_{1}(N; l)_{≥r}^{0}) to be the admissible open subset of X_{1}(N; l)^{an} consisting of • all points of X_{1}(N; l)^{an} which reduce to a point of \overline{X}_{1}(N; l)_{∞} −SS_{l} (resp.

\overline{X}_{1}(N; l)_{0} − SS_{l})

• and all points x ∈ X_{1}(N; l)^{an} which reduce to some s ∈ SS_{l} and for which

|T_{s,1}(x) − T_{s,2}(x)^{l}|_{l} ≥ r

(resp.

|T_{s,2}(x) − T_{s,1}(x)^{l}|_{l} ≥ r).

If in fact 1 > r^{2} > 1/l and s ∈ SS_{l} then we will let U_{s}(r) denote the admissible open subset of X_{1}(N; l)^{an} consisting of points which reduce to s and which satisfy

|T_{s,1}(x) − T_{s,2}(x)^{l}|_{l} ≤ r

and

|T_{s,2}(x) − T_{s,1}(x)^{l}|_{l} ≤ r.

It is easy to check that these sets do not depend on the choice of {T_{s}} as long as they satisfy

(1 × σ^{∗})^{∗}(T_{(1×σ^{∗})(s)}) = T_{s}.

for σ ∈ Gal (𝔽_{l}^{ac}/𝔽_{l}). If r_{1}, r_{2}, r_{3} ∈ l^{ℚ}, 1 > r_{1}^{2} > 1/l, r_{1} > r_{2} > 1/l and r_{1} > r_{3} > 1/l then the sets

• X_{1}(N; l)_{≥r_{2}}^{∞},

• X_{1}(N; l)_{≥r_{3}}^{0},

• and for each s ∈ SS_{l} the set U_{s}(r_{1})

form an admissible cover of X_{1}(N; l)^{an} by connected admissible open subsets.

(This seems to be very well known, but as we are unable to find a reference let us sketch the argument. Take an affine Zariski cover U^{0}, U^{∞} and U_{s} for s ∈ SS_{l} of X_{1}(N; l) × 𝔽_{l}^{ac}, where for s ∈ SS_{l} we have SS_{l} ∩ U_{s} = {s}, where U^{0} = X_{1}(N; l) × 𝔽_{l}^{ac} − \overline{X}_{1}(N; l)_{∞} and where U^{∞} = X_{1}(N; l) × 𝔽_{l}^{ac} − \overline{X}_{1}(N; l)_{0}. Shrinking U_{s} if necessary, choose a regular function x_{s}^{0} on U_{s} which is identically zero on \overline{X}_{1}(N; l)_{∞} ∩ U_{s} and non-zero on (\overline{X}_{1}(N; l)_{0} ∩ U_{s}) − \{s\} with a simple zero at s. We can lift x_{s}^{0} to some affine open subset of X_{1}(N; l)×W (𝔽_{l}^{ac}) which intersects the special fibre in U_{s}. Set x_{s}^{∞} = p/x_{s}^{0}. In (X_{1}(N; l) × W (𝔽_{l}^{ac}))_{s}^{∧} we have x_{s}^{0} = ∑_{i=1}^{∞} a_{i}X_{2}^{i} +lf = X_{2}(∑_{i=1}^{∞} a_{i}X_{2}^{i−1} +X_{1}f), i.e. x_{s}^{0} is X_{2} times a unit (the same X_{1}, X_{2} as in corollary 2.5). Thus again shrinking U_{s} if necessary we may assume that x_{s}^{∞} is regular on U_{s}, identically zero on \overline{X}_{1}(N; l)_{0} ∩ U_{s} and non-zero on (\overline{X}_{1}(N; l)_{∞} ∩ U_{s}) − \{s\}. Moreover in (X_{1}(N; l) × W (𝔽_{l}^{ac}))_{s}^{∧}, x_{s}^{∞} is a unit times X_{1}. We will let U_{∞} (resp. U_{0}, resp. U_{s}) denote the preimage in X_{1}(N; l)^{an} of U_{∞} (resp. U_{0}, resp. U_{s}). They form an admissible affinoid cover of X_{1}(N; l)^{an}. For r ∈ l^{ℚ} and 1 ≥ r > 1/l set U_{s,≥r}^{0} ⊂ U_{s} (resp. U_{s,≥r}^{∞} ⊂ U_{s}) to be the locus where |x_{s}^{0}|_{l} ≥ r (resp. |x_{s}^{0}|_{l} ≥ r). Note also that U_{s}(r) is the subspace of U_{s} where |x_{s}^{0}|_{l} ≤ r and |x_{s}^{0}|_{l} ≤ r. Note that X_{1}(N; l)_{≥r}^{0} (resp. X_{1}(N; l)_{≥r}^{∞}) is the union of U_{0} and U_{s,≥r}^{0} for s ∈ SS_{l} (resp. U_{∞} and U_{s,≥r}^{∞} for s ∈ SS_{l}). If r_{1}, r_{2}, r_{3} ∈ l^{ℚ}, 1 > r_{1}^{2} > 1/l, r_{1} > r_{2} > 1/l and r_{1} > r_{3} > 1/l, then U_{s,≥r_{2}}^{∞}, U_{s,≥r_{3}}^{0} and U_{s}(r_{1}) form an admissible affinoid cover of U_{s}. Thus X_{1}(N; l)_{≥r_{2}}^{∞}, X_{1}(N; l)_{≥r_{3}}^{0} and U_{s}(r_{1}) for s ∈ SS_{l} form an admissible open cover of X_{1}(N; l)^{an}.

It remains to show that for r ∈ l^{ℚ} and 1 ≥ r > 1/l the spaces X_{1}(N; l)_{≥r}^{0} and X_{1}(N; l)_{≥r}^{∞} are connected. To save on notation we will only explain the case of X_{1}(N; l)_{≥r}^{0}. It suffices to check that U_{0} and U_{s,≥r}^{0} for s ∈ SS_{l} are all connected. This follows because in each case the reduction map gives a continuous map with connected fibres to a connected (in the Zariski topology) space.)

If r ∈ l^{ℚ} and 1 ≥ r > l^{−l/(1+l)} then it is easy to check that

π_{1}^{−1}X_{1}(N)_{≥r} = X_{1}(N; l)_{≥r}^{∞} ⨿ X_{1}(N; l)_{≥r^{1/l}}^{0}

and

π_{2}^{−1}X_{1}(N)_{≥r} = X_{1}(N; l)_{≥r^{1/l}}^{∞} ⨿ X_{1}(N; l)_{≥r}^{0}.

Moreover X_{1}(N; l)_{≥r}^{∞} and X_{1}(N; l)_{≥r^{1/l}}^{0} (resp. X_{1}(N; l)_{≥r}^{0} and X_{1}(N; l)_{≥r^{1/l}}^{∞}) form an admissible open cover of π_{1}^{−1}X_{1}(N)_{≥r} (resp. π_{1}^{−1}X_{1}(N)_{≥r}). As X_{1}(N; l) → X_{1}(N) is finite flat of degree l + 1, the same is true of the analytifications. Thus

π_{1} : X_{1}(N; l)_{≥r}^{∞} ⨿ X_{1}(N; l)_{≥r^{1/l}}^{0} ⟶ X_{1}(N)_{≥r}

and

π_{2} : X_{1}(N; l)_{≥r^{1/l}}^{∞} ⨿ X_{1}(N; l)_{≥r}^{0} ⟶ X_{1}(N)_{≥r}

are both finite and flat of degree l + 1. Looking at the cardinality of the preimages of points we deduce the following lemma.

Lemma 2.6. 1. Suppose that r ∈ l^{ℚ} and 1 ≥ r > l^{−l/(1+l)} then

π_{1} : X_{1}(N; l)_{≥r}^{∞} ⟶^{∼} X_{1}(N)_{≥r}

and

π_{2} : X_{1}(N; l)_{≥r}^{0} ⟶^{∼} X_{1}(N)_{≥r}.

2. Suppose that r ∈ l^{ℚ} and 1 ≥ r > l^{−1/(1+l)} then

π_{2} : X_{1}(N; l)_{≥r}^{∞} ⟶ X_{1}(N)_{≥r^{l}}

and

π_{1} : X_{1}(N; l)_{≥r}^{0} ⟶ X_{1}(N)_{≥r^{l}}

are both finite flat of degree l.

We define a bounded linear map

U = (1/l)tr_{π_{2}} ∘ j ∘ π_{1}|_{X_{1}(N;l)_{≥r}^{∞}}^{−1} : M_{k}^{≥r}(N) ⟶ M_{k}^{≥r^{l}}(N).

One may check that U is compatible with the map on q-expansions which sends

∑_{n=0}^{∞} a_{n}q^{n} ⟼ ∑_{n=0}^{∞} a_{nl}q^{n}.

Note that for 1 ≥ r ≥ l^{−l/(1+l)} using π_{1} to identify X_{1}(N; l)_{≥r}^{∞} and X_{1}(N)_{≥r} we get a map

Hom (h_{k}(N; l), ℂ_{l}) ↪ M_{k}^{≥r}(N)

which sends f to the form with q-expansion at infinity

∑_{n=1}^{∞} f(T (n))q^{n}.

Under this map the Hecke operator U_{l} corresponds to the linear map U.

Suppose that 1 > r > l^{−1/(1+l)}. Combining U : M_{k}^{≥r}(N) → M_{k}^{≥r^{l}}(N) with the inclusion M_{k}^{≥r^{l}}(N) ↪ M_{k}^{≥r}(N) we get a continuous endomorphism of M_{k}^{≥r}(N), which we will also denote U. It follows from lemma 2.3 that U is completely continuous as an endomorphism of M_{k}^{≥r}(N). From the theory of completely continuous operators on p-adic Banach spaces (see [S1]) we see that we may write

M_{k}^{≥r}(N) = M_{k}^{≥r}(N)^{0} ⊕ M_{k}^{≥r}(N)^{1}

as a direct sum of U-invariant subspaces, where M_{k}^{≥r}(N)^{0} is finite dimensional, all the eigenvalues of U|_{M_{k}^{≥r}(N)^{0}} are l-adic units and U|_{M_{k}^{≥r}(N)^{1}} is topologically nilpotent (i.e. if f ∈ M_{k}^{≥r}(N)^{1} then U^{r}f → 0 as r → ∞). We will let e denote projection onto the summand M_{k}^{≥r}(N)^{0}, so that

ef = lim_{r→∞} U^{r!}f.

Lemma 2.7. If f ∈ M_{k}^{≥r}(N)^{0} for some 1 > r > l^{−1/(1+l)} then

f ∈ M_{k}^{>l^{−l/(1+l)}}(N).

Proof. Choose a minimal integer i such that r^{l^{i}} ≤ l^{−1/(1+l)} and write f = U^{i+1}f′ for some f′ ∈ M_{k}^{≥r}(N)^{0}. Then we see that

U^{i}f′ ∈ M_{k}^{≥r^{l^{i}}}(N) ⊂ M_{k}^{>l^{−1/(1+l)}}(N)

and hence that

f = U(U^{i}f′) ∈ M_{k}^{>l^{−l/(1+l)}}(N).

Lemma 2.8. Suppose that 1 > r ≥ l^{−1/(1+l)}, that f ∈ M_{k}^{≥r}(N), that a ∈ ℂ_{l} is an l-adic unit and that ϵ ∈ ℝ_{>0}. If

|Uf − af|_{1} ≤ ϵ,

then

|f − ef|_{1} ≤ ϵ.

Proof. For all positive integers t we see that

|U^{t!}f − a^{t!}f|_{1} ≤ ϵ.

Taking the limit as t → ∞ and noting that | |_{1} ≤ | |_{r} the lemma follows. (We remark that a^{t!} → 1 as t → ∞.)

Lemma 2.9. Suppose we are given an integer k and a formal q-expansion

∑_{n=1}^{∞} a_{n}q^{n} ∈ ℂ_{l}[[q]]

such that for all n we have a_{nl} = a_{l}a_{n} and such that a_{l} is an l-adic unit. Suppose we also have two series of positive integers t_{i} and k_{i} and a series of abelian group homomorphisms

f_{i} : h_{k_{i}}(N; l) → ℂ_{l}

such that

1. t_{i} → ∞ as i → ∞,

2. k_{i} ≡ k mod (l − 1)l^{t_{i}−1},

3. and for all positive integers n and for all i we have

f_{i}(T (n)) ≡ a_{n} mod l^{t_{i}}.

Then ∑ a_{n}q^{n} is the q-expansion at infinity of an element of M_{k}^{>l^{−l/(1+l)}}(N).

Proof. By the last lemma we only need show that ∑ a_{n}q^{n} is the q-expansion at infinity of an element of M_{k}^{≥r}(N) for some r < 1. Choose such an r with r > l^{−1/4} and r > l^{−1/(1+l)}. We may suppose that each t_{i} ≥ 3. Set h = 4 if l = 2 and h = l − 1 otherwise. Then f_{i} corresponds to an element of M_{k_{i}}^{≥r}(N) which we will also denote by f_{i}. Moreover f_{i}/(E′)^{(k_{i}−k)/h} ∈ M_{k}^{≥r}(N) and has q expansion at infinity congruent to ∑_{n} a_{n}q^{n} modulo l^{t_{i}}. (If l = 2 note that E′ is congruent to 1 modulo 2^{4}.) Thus e(f_{i}/(E′)^{(k_{i}−k)/h}) ∈ M_{k}^{≥r}(N)^{0} also has q expansion at infinity congruent to ∑_{n} a_{n}q^{n} modulo l^{t_{i}}. As M_{k}^{≥r}(N)^{0} is finite dimensional all l-adic norms are equivalent. The e(f_{i}/(E′)^{(k_{i}−k)/h}) form a Cauchy sequence for | |_{1} and hence also for | |_{r}. Let f ∈ M_{k}^{≥r}(N)^{0} denote the limit of the e(f_{i}/(E′)^{(k_{i}−k)/h}) in both of these norms. Then f has q-expansion at infinity ∑_{n} a_{n}q^{n}, as desired.

Finally we state the generalisation of theorem 4 of [BT] to l = 2 and 3. Although in [BT] there is a running hypothesis that l ≥ 5, the proof given there of this theorem makes no use of that hypothesis.

Theorem 2.10. Let N and k denote integers with N ≥ 5. Let l |̸N be a prime. Suppose α and β are distinct non-zero elements of ℂ_{l} and that f_{α}, f_{β} ∈ M_{k}^{>l/(1+l)}(N) are eigenvectors for U with eigenvalues α and β. Suppose also f_{α} (resp. f_{β}) have q-expansions at infinity ∑_{n≥1} a_{n}(f_{α})q^{n} (resp. ∑_{n≥1} a_{n}(f_{β})q^{n}) and that for all positive integers n not divisible by l we have

a_{n}(f_{α}) = a_{n}(f_{β}).

Then f = (αf_{α} − βf_{β})/(α − β) is classical, i.e. there is an abelian group homomorphism f′ : h_{k}(N) → ℂ_{l} such that for all n

f′(T (n)) = (αa_{n}(f_{α}) − βa_{n}(f_{β}))/(α − β).

3 2-adic Hida theory and deformation theory.

In this section we will draw together some results about 2-adic Hida theory which are not well documented in the literature and deduce some slight extensions of the results of [Di].

If N ≥ 5 is an odd positive integer we will let h^{0}(N) denote

lim_{←r} e(h_{2}(2^{r}N) ⊗_{ℤ} ℤ_{2}),

where e denotes Hida’s idempotent

e = lim_{t→∞} U_{2}^{t!}.

Taking the limit of the homomorphisms

⟨ ⟩ : (ℤ/2^{r}Nℤ)^{×} ⟶ e(h_{2}(2^{r}N) ⊗_{ℤ} ℤ_{2})^{×}

we get a continuous homomorphism

S = S^{2} × S_{2} : (ℤ/Nℤ)^{×} × ℤ_{2}^{×} ⟶ h^{0}(N)^{×}.

We will let Λ denote the completed group ring ℤ_{2}[[(1 + 4ℤ_{2})]] ≅ ℤ_{2}[[T]], where T = S_{2}(5) − 1. Then S_{2} induces a continuous homomorphism Λ → h^{0}(N).

According to theorems 3.3 and 3.4 of [Hi], h^{0}(N) is a finitely generated, torsion free Λ-module and for any integer k ≥ 2 we have a surjection

h^{0}(N)/(S_{2}(5) − 5^{k−2}) → → e(h_{k}(4N) ⊗_{ℤ} ℤ_{2})

which sends T (n) to T (n) for all n and which becomes an isomorphism after tensoring with ℚ_{2}.

Set e_{±} = (1 ± S_{2}(−1))/2 and h^{0}(N)_{±} = e_{±}h^{0}(N) ⊂ h^{0}(N) ⊗_{ℤ_{2}} ℚ_{2}. Then

h^{0}(N) ⊂ h^{0}(N)_{+} ⊕ h^{0}(N)_{−} ⊂ (1/2)h^{0}(N).

Thus we see that h^{0}(N)_{±} are finitely generated torsion free Λ-modules, and so from the structure theory of finitely generated Λ-modules we see that we have exact sequences of Λ-modules

(0) ⟶ h^{0}(N)_{±} ⟶ Λ^{r_{±}} ⟶ X_{±} ⟶ (0),

where r_{±} are non-negative integers and where X_{±} have finite cardinality 2^{a_{±}}. If k ≥ 2 is an integer with the same parity as (1 ∓ 1)/2 then we see that there is a surjection

h^{0}(N)_{±}/(S_{2}(5) − 5^{k−2}) → → e(h_{k}(2N) ⊗_{ℤ} ℤ_{2})

which sends T (n) to T (n) for all n and which becomes an isomorphism after tensoring with ℚ_{2}. In fact the kernel of this surjection has order divisible by 2^{a_{±}}. (The key point is that for k ≡ (1 ∓ 1)/2 mod 2 we have an equality

ee_{±}(h_{k}(4N) ⊗ ℚ_{2}) = e(h_{k}(2N) ⊗ ℚ_{2}).

This results from the fact that U_{2} maps the space of modular forms of weight k and level Γ_{1}(2N)∩Γ_{0}(4) to the space of modular forms of weight k and level Γ_{1}(2N) (compare for instance proposition 8.3 of [Hi]).)

Similarly set h_{2}(4N)_{−} = e_{−}h_{2}(4N) ⊂ h_{2}(4N) ⊗ ℚ.

Lemma 3.1. Suppose that f : h^{0}(N)_{−} → ℚ_{2}^{ac} is a continuous ℤ_{2}-algebra homomorphism such that f(S_{2}(5)) = 1/5. Then

∑_{n=1}^{∞} f(T (n))q^{n}

is the q-expansion at infinity of an element of M_{1}^{>2^{−2/3}}(N).

Proof. For each integer r ≥ 1 set k(r) = 1 + 2^{a_{−}+r}. Then we can find a continuous homomorphism of ℤ_{2}-modules

f_{r} : e(h_{k(r)}(2N) ⊗_{ℤ} ℤ_{2}) ⟶ ℚ_{2}^{ac}

such that f_{r}(T (n)) ≡ f(T (n)) mod 2^{r+2} for all n. The lemma follows from lemma 2.9.

Suppose that k ≥ 2 is an integer. If ℘ is a minimal prime ideal of h^{0}(N)_{±} containing S_{2}(5) − 5^{k−2} then h^{0}(N)_{±}/℘ is a 1-dimensional integral domain. Thus ℘ contains

ker(h^{0}(N)_{±} → → ee_{±}(h_{k}(4N) ⊗_{ℤ} ℤ_{2})).

Thus contraction gives a bijection between prime ideals of ee_{±}(h_{k}(4N) ⊗_{ℤ} ℤ_{2}) and prime ideals of h^{0}(N)_{±} containing S_{2}(5) − 5^{k−2}, hence also a bijection between maximal ideals of ee_{±}(h_{k}(4N) ⊗_{ℤ} ℤ_{2}) and maximal ideals of h^{0}(N)_{±}. Hence to any maximal ideal 𝔪 of h^{0}(N)_{±} we can associate a continuous semisimple representation

\overline{ρ}_{𝔪} : Gal (ℚ^{ac}/ℚ) ⟶ GL_{2}(h^{0}(N)_{±}/𝔪)

such that for all but finitely many primes p we have

• tr \overline{ρ}_{𝔪}(Frob_{p}) = T_{p}

• and det \overline{ρ}_{𝔪}(Frob_{p}) = pS(p).

(To see that \overline{ρ}_{𝔪} can be defined over h^{0}(N)_{±}/𝔪 use the facts that its trace is valued in h^{0}(N)_{±}/𝔪 (which follows from the Chebotarev density theorem) and that the Brauer group of any finite field extension of h^{0}(N)_{±}/𝔪 is trivial.) We will call 𝔪 Eisenstein if \overline{ρ}_{𝔪} is not absolutely irreducible.

Note that the intersection over all integers k ≥ 2 with k ≡ (1 ∓ 1)/2 of

ker(h^{0}(N)_{±} → → e(h_{2}(2N) ⊗ ℚ_{2}))

equals

h^{0}(N)_{±} ∩ ⋂_{k}(S_{2}(5) − 5^{k−2})Λ^{r_{±}} = (0).

Thus

h^{0}(N)_{±} = lim_{←}(h^{0}(N)_{±}/ ⋂_{k∈K} ker(h^{0}(N)_{±} → → e(h_{k}(2N) ⊗ ℤ_{2}))),

where the inverse limit is over finite sets K of integers k ≥ 2 with k ≡ (1 ∓ 1)/2 mod 2.

For each k ≥ 2 there is a continuous 2-dimensional pseudo-representation (see [Ta1] for the definition of pseudo-representation)

T : Gal (ℚ^{ac}/ℚ) ⟶ e(h_{k}(2N) ⊗ ℤ_{2})

such that for all primes p |̸2N the pseudo-representation T is trivial on I_{p} (the inertia group at p) (i.e. T (στ) = T (τ) for all σ ∈ I_{p} and τ ∈ Gal (ℚ^{ac}/ℚ)) and T (Frob_{p}) = T_{p} and T (Frob_{p}^{2}) = T_{p}^{2} − 2pS(p). By the Chebotarev density theorem we see that there is at most one such pseudo-representation T. Thus for any finite set K as in the last paragraph we get a continuous pseudorepresentation

\begin{aligned}T : Gal (ℚ^{ac}/ℚ) & ⟶ & h^{0}(N)_{±}/ ⋂_{k∈K} ker(h^{0}(N)_{±} → → e(h_{k}(2N) ⊗ ℤ_{2})) \\ & ⊂ & ⨁_{k∈K} e(h_{k}(2N) ⊗_{ℤ} ℤ_{2}), \end{aligned}

such that for all primes p |̸2N the pseudo-representation T is trivial on I_{p}, T (Frob_{p}) = T_{p} and T (Frob_{p}^{2}) = T_{p}^{2} − 2pS(p). Taking the limit over K we find a continuous pseudo-representation

T : Gal (ℚ^{ac}/ℚ) ⟶ h^{0}(N)_{±}

such that for all primes p |̸2N the pseudo-representation T is trivial on I_{p}, T (Frob_{p}) = T_{p} and T (Frob_{p}^{2}) = T_{p}^{2} − 2pS(p). By the main theorem of [N] (see also [R]) we see that if 𝔪 is a non-Eisenstein maximal ideal of h^{0}(N)_{±} then there is a continuous representation

ρ_{𝔪}^{ord} : Gal (ℚ^{ac}/ℚ) ⟶ GL_{2}(h^{0}(N)_{±,𝔪})

such that ρ_{𝔪}^{ord} is unramified at all primes p |̸2N and satisfies

• tr ρ_{𝔪}^{ord}(Frob_{p}) = T_{p}

• and det ρ_{𝔪}^{ord}(Frob_{p}) = pS(p).

It is known (by [De] or [W], theorem 2.1.4) that \overline{ρ}_{𝔪}|_{Gal (ℚ_{2}^{ac}/ℚ_{2})}^{ss} is unramified. We will suppose that \overline{ρ}_{𝔪}|_{Gal (ℚ_{2}^{ac}/ℚ_{2})}^{ss}(Frob_{2}) has two distinct eigenvalues α and β. Then it is also known that α, β ∈ h^{0}(N)_{±}/𝔪 and that either U_{p} − α ∈ 𝔪 or U_{p} − β ∈ 𝔪 ([De] or [W], theorem 2.1.4). We will suppose it is the former. Choose an element σ_{0} ∈ Gal (ℚ_{2}^{ac}/ℚ_{2}) above Frob_{2}. It follows from Hensel’s lemma that ρ_{𝔪}^{ord}(σ_{0}) has distinct eigenvalues A and B in h^{0}(N)_{±,𝔪} with A ≡ α mod 𝔪 and B ≡ β mod 𝔪. Choose a basis (e_{B}, e_{A}) of h^{0}(N)_{±,𝔪}^{2} consisting of eigenvectors of ρ_{𝔪}^{ord}(σ_{0}) with eigenvalues B and A respectively. With respect to this basis write

ρ_{𝔪}^{ord}(σ) = ( \begin{matrix}a(σ) & b(σ) \\ c(σ) & d(σ) \end{matrix} ) .

Also write • ψ_{a} for the unramified character of Gal (ℚ_{2}^{ac}/ℚ_{2}) which takes Frob_{2} to a,

• χ_{2} for the 2-adic cyclotomic character,

• and S for the composite

Gal (ℚ^{ac}/ℚ) ⟶ Gal (ℚ(μ_{2^{∞}N})/ℚ) →^{∼} ℤ_{2}^{×} × (ℤ/Nℤ)^{×} ⟶^{S} h^{0}(N)^{×}.

Then by theorem 2.1.4 of [W] we see that for any integer k ≥ 2 with k ≡ (1 ∓ 1)/2 mod 2 and for any σ ∈ Gal (ℚ_{2}^{ac}/ℚ_{2}) we have

• a(σ) ≡ (χ_{2}Sψ_{U_{2}}^{−1})(σ),

• c(σ) ≡ 0,

• and d(σ) ≡ ψ_{U_{2}}(σ), all modulo

ker(h^{0}(N)_{±,𝔪} → → e(h_{k}(2N) ⊗_{ℤ} ℤ_{2})_{𝔪}).

We conclude that

ρ_{𝔪}^{ord}|_{Gal (ℚ_{2}^{ac}/ℚ_{2})} ∼ ( \begin{matrix}χ_{2}Sψ_{U_{2}}^{−1} & ∗ \\ 0 & ψ_{U_{2}} \end{matrix} )

and that A = U_{2}.

Now suppose that

\overline{ρ} : Gal (ℚ^{ac}/ℚ) ⟶ GL_{2}(𝔽_{2}^{ac})

is a continuous representation such that • \overline{ρ}(c) ≠ 1,

• \overline{ρ}|_{Gal (ℚ_{2}^{ac}/ℚ_{2})}^{ss} is unramified and \overline{ρ}|_{Gal (ℚ_{2}^{ac}/ℚ_{2})}^{ss}(Frob_{2}) has distinct eigenvalues α and β,

• \overline{ρ}|_{Gal (ℚ(\sqrt{−1})^{ac}/ℚ(\sqrt{−1}))} is irreducible,

• and such that there exists an odd integer N ≥ 5 and a homomorphism

\overline{f} : h_{2}(N) ⟶ 𝔽_{2}^{ac} satisfying – \overline{f}(T_{2}) = α,

– \overline{f}(T_{p}) = tr \overline{ρ}(Frob_{p}) for all primes p |̸2N,

– and \overline{f}(pS(p)) = det \overline{ρ}(Frob_{p}) for all primes p |̸2N.

We will let N(\overline{ρ}) denote the conductor of \overline{ρ}.

Suppose also that S is a finite set of odd primes containing all where \overline{ρ} ramifies and some prime p ≥ 5. Then set

N_{S}(\overline{ρ}) = N(\overline{ρ}) ∏_{p∈S} p^{dim \overline{ρ}^{Ip}}.

It follows from theorem 3.1 of [Buz] that we can find a ring homomorphism

h_{2}(2N_{S}(\overline{ρ})) ⟶ 𝔽_{2}^{ac}

such that • U_{2} maps to α,

• U_{p} maps to 0 if p ∈ S,

• T_{p} maps to tr \overline{ρ}(Frob_{p}) if p |̸2N_{S}(\overline{ρ}),

• and pS(p) maps to det \overline{ρ}(Frob_{p}) if p |̸2N_{S}(\overline{ρ}). (It is here we use that \overline{ρ}|_{Gal (ℚ(\sqrt{−1})^{ac}/ℚ(\sqrt{−1}))} is irreducible, rather than the weaker assumption that \overline{ρ} is irreducible.) We will let 𝔪_{S}(\overline{ρ}, α)_{+} denote the kernel of this homomorphism. Lemma 3.2. Keep the above notation and assumptions.

1. There is a ring homomorphism h_{2}(4N_{S}(\overline{ρ}))_{−} → 𝔽_{2}^{ac} such that • U_{2} maps to α,

• U_{p} maps to 0 if p ∈ S,

• T_{p} maps to tr \overline{ρ}(Frob_{p}) if p |̸2N_{S}(\overline{ρ}),

• and pS(p) maps to det \overline{ρ}(Frob_{p}) if p |̸2N_{S}(\overline{ρ}). We will denote its kernel 𝔪_{S}(\overline{ρ}, α)_{−}.

2. There is a surjection

h_{2}(4N_{S}(\overline{ρ}))_{−,𝔪_{S}(\overline{ρ},α)_{−}}/(2) → → h_{2}(2N_{S}(\overline{ρ}))_{𝔪_{S}(\overline{ρ},α)_{+}}/(2)

which takes T (n) to T (n) for all n.

Proof. Let 𝕋 denote the polynomial algebra over ℤ_{2} generated by variables t_{p} and s_{p} for p |̸2N_{S}(\overline{ρ}) and u_{p} for p|2N_{S}(\overline{ρ}). Then there is a natural map 𝕋 → h_{2}(2N_{S}(\overline{ρ}))_{𝔪_{S}(\overline{ρ},α)_{+}}/(2) which sends t_{p} to T_{p} etc. Let 𝔪 denote the pullback of 𝔪_{S}(\overline{ρ}, α)_{+}. It is a maximal ideal of 𝕋. Let Y denote the open (i.e. with the cusps removed) modular curve of level Γ_{1}(2N_{S}(\overline{ρ})) ∩ Γ_{0}(4). Let ϵ denote the order two character of Γ_{1}(4N_{S}(\overline{ρ}))/(Γ_{1}(2N_{S}(\overline{ρ})) ∩ Γ_{0}(4)) thought of as a character of the fundamental group of Y. It is known that

H^{1}(Y, ℤ_{2})_{𝔪} ≅ h_{2}(2N_{S}(\overline{ρ}))_{𝔪_{S}(\overline{ρ},α)_{+}}^{2},

where 𝕋 acts on the cohomology by sending t_{p} to T_{p} etc. (See proposition 12.10 of [G].) Because H^{2}(Y, ℤ_{2}) = (0) (as Y is affine) we conclude that

H^{1}(Y, 𝔽_{2})_{𝔪} ≅ (h_{2}(2N_{S}(\overline{ρ}))_{𝔪_{S}(\overline{ρ},α)_{+}}/(2))^{2}.

Thus to prove the lemma it suffices to see that the action of 𝕋 on H^{1}(Y, 𝔽_{2})_{𝔪} factors through h_{2}(4N_{S}(\overline{ρ}))_{−,𝔪_{S}(\overline{ρ},α)_{−}}. However

H^{1}(Y, 𝔽_{2})_{𝔪} = H^{1}(Y, 𝔽_{2}(ϵ))_{𝔪} ≅ H^{1}(Y, ℤ_{2}(ϵ))_{𝔪} ⊗ 𝔽_{2},

because H^{2}(Y, ℤ_{2}(ϵ)) = (0) (as Y is affine). Finally the action of 𝕋 on H^{1}(Y, ℤ_{2}(ϵ))_{𝔪} factors through h_{2}(4N_{S}(\overline{ρ}))_{−,𝔪_{S}(\overline{ρ},α)_{−}}, because H^{1}(Y, ℤ_{2}(ϵ))_{𝔪} is torsion free (because in turn H^{0}(Y, 𝔽_{2}(ϵ))_{𝔪} = H^{0}(Y, 𝔽_{2})_{𝔪} = (0), as 𝔪 is non-Eisenstein).

We remark that by our choice of N_{S}(\overline{ρ}), for p ∈ S we have U_{p} = 0 in each of h_{2}(2N_{S}(\overline{ρ}))_{𝔪_{S}(\overline{ρ},α)_{+}}, h_{2}(4N_{S}(\overline{ρ}))_{−,𝔪_{S}(\overline{ρ},α)_{−}} and h^{0}(N_{S}(\overline{ρ}))_{±,𝔪_{S}(\overline{ρ},α)_{±}}. (In fact it suffices to check that for p ∈ S we have U_{p} = 0 in h_{k}(4N_{S}(\overline{ρ}))_{𝔪_{S}(\overline{ρ},α)_{±}} whenever k ≥ 2 and k ≡ (1 ∓ 1)/2 mod 2. This is standard, see for example corollary 4.2.3 and the proof of lemma 5.1.1 of [CDT].)

We will let

ρ_{S,α,±}^{(2)} : Gal (ℚ^{ac}/ℚ) ⟶ GL_{2}(R_{S,α,±}^{(2)})

denote the universal deformation of \overline{ρ} to a representation which is unramified outside S ∪ {2} and which when restricted to Gal (ℚ_{2}^{ac}/ℚ_{2}) is of the form

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

where ϕ_{2} is unramified and ϕ_{2}(Frob_{2}) ≡ α modulo the maximal ideal, and where, thinking of ϕ_{1} as a character of ℚ_{2}^{×} by local class field theory, we have ϕ_{1}(−1) = ∓1 and ϕ_{1}(x) = x for all x ∈ (1 + 4ℤ_{2}). Similarly we will let

ρ_{S,α,±}^{ord} : Gal (ℚ^{ac}/ℚ) ⟶ (R_{S,α,±}^{ord})

denote the universal deformation of \overline{ρ} to a representation which is unramified outside S ∪ {2} and which when restricted to Gal (ℚ_{2}^{ac}/ℚ_{2}) is of the form

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

where ϕ_{2} is unramified and ϕ_{2}(Frob_{2}) ≡ α modulo the maximal ideal, and where, thinking of ϕ_{1} as a character of ℚ_{2}^{×} by local class field theory, we have ϕ_{1}(−1) = ∓1. The character ϕ_{1}χ_{2}^{−1} gives a continuous homomorphism, which we will denote S_{2},

(1 + 4ℤ_{2}) ⟶ (R_{S,α,±}^{ord})^{×}

and so makes R_{S,α,±}^{ord} into a Λ-algebra. From the definitions one sees that

R_{S,α,±}^{(2)} = R_{S,α,±}^{ord}/(S_{2}(5) − 1).

From the universal properties we get maps

• R_{S,α,+}^{(2)} ⟶ h_{2}(2N_{S}(\overline{ρ}))_{𝔪(\overline{ρ},α)_{+}},

• R_{S,α,−}^{(2)} ⟶ h_{2}(4N_{S}(\overline{ρ}))_{−,𝔪(\overline{ρ},α)_{−}},

• and R_{S,α,±}^{ord} ⟶ h^{0}(N_{S}(\overline{ρ}))_{±,𝔪(\overline{ρ},α)_{±}},

which are in fact surjections. (To see that these maps are surjections note that U_{p} = 0 if p ∈ S, that T_{p} = tr ρ_{𝔪(\overline{ρ},α)_{±}}^{(2)}(Frob_{p}) or tr ρ_{𝔪(\overline{ρ},α)_{±}}^{ord}(Frob_{p}) is in the image for p |̸2N_{S}(\overline{ρ}), that S(p) is similarly in the image for p |̸2N_{S}(\overline{ρ}) and that U_{2} is in the image by Hensel’s lemma (as it is an eigenvalue for an element of Gal (ℚ_{2}^{ac}/ℚ_{2}) above Frob_{2} in one of these representations).)

Theorem 4 and proposition 6 of [Di] show that the map

R_{S,α,+}^{(2)} ⟶ h_{2}(2N_{S}(\overline{ρ}))_{𝔪(\overline{ρ},α)_{+}}

is an isomorphism.

Proposition 3.3. The natural maps

R_{S,α,−}^{(2)} → → h_{2}(4N_{S}(\overline{ρ}))_{−,𝔪(\overline{ρ},α)_{−}}

and

R_{S,α,±}^{ord} → → h^{0}(N_{S}(\overline{ρ}))_{±,𝔪(\overline{ρ},α)_{±}}

are isomorphisms.

Proof. Consider the first of these maps. We have a commutative diagram

R_{S,α,−}^{(2)}/(2) → → h_{2}(4N_{S}(\overline{ρ}))_{−,𝔪(\overline{ρ},α)_{−}}/(2)
↓  ↓
R_{S,α,+}^{(2)}/(2) ⟶^{∼} h_{2}(2N_{S}(\overline{ρ}))_{𝔪(\overline{ρ},α)_{+}}/(2),

where the left hand vertical arrow is an isomorphism from the definitions. Thus

R_{S,α,−}^{(2)}/(2) ⟶^{∼} h_{2}(4N_{S}(\overline{ρ}))_{−,𝔪(\overline{ρ},α)_{−}}/(2)

and, because h_{2}(4N_{S}(\overline{ρ}))_{−,𝔪(\overline{ρ},α)_{−}} is torsion free over ℤ_{2}, we deduce that

R_{S,α,−}^{(2)} ⟶^{∼} h_{2}(4N_{S}(\overline{ρ}))_{−,𝔪(\overline{ρ},α)_{−}}.

Now the composite

R_{S,α,−}^{ord}/(S_{2}(5) − 1) → → h^{0}(N_{S}(\overline{ρ}))_{−,𝔪(\overline{ρ},α)_{−}}/(S_{2}(5) − 1) → → h_{2}(4N_{S}(\overline{ρ}))_{−,𝔪(\overline{ρ},α)_{−}}

is an isomorphism and so

R_{S,α,−}^{ord}/(S_{2}(5) − 1) ⟶^{∼} h^{0}(N_{S}(\overline{ρ}))_{−,𝔪(\overline{ρ},α)_{−}}/(S_{2}(5) − 1).

Because h^{0}(N_{S}(\overline{ρ}))_{−,𝔪(\overline{ρ},α)_{−}} is Λ-torsion free, we deduce that

R_{S,α,−}^{ord} →^{∼} h^{0}(N_{S}(\overline{ρ}))_{−,𝔪(\overline{ρ},α)_{−}}.

The same argument also shows that

R_{S,α,+}^{ord} →^{∼} h^{0}(N_{S}(\overline{ρ}))_{+,𝔪(\overline{ρ},α)_{+}}.

Putting this proposition together with corollary 1.2 and lemma 3.1 we obtain the following corollary.

Corollary 3.4. Suppose that K/ℚ_{2} is a finite extension with ring of integers O_{K} with maximal ideal ℘_{K}. Suppose also that

ρ : Gal (ℚ^{ac}/ℚ) ⟶ GL_{2}(O_{K})

is a continuous representation such that

1. (ρ mod ℘_{K}) has image SL_{2}(𝔽_{4}), 2. (ρ mod ℘_{K})(c) ≠ 1,

3. (ρ mod ℘_{K}) is unramified at 5,

4. ρ is unramified at 2 and ρ(Frob_{2}) has eigenvalues α and β in O_{K} with

distinct reduction modulo ℘_{K}. Then there exists an odd integer N ≥ 5 divisible by all primes at which ρ ramifies and a normalised eigenform f_{α} ∈ M_{1}^{>2^{−2/3}}(N) such that

• T_{p}f_{α} = (tr ρ(Frob_{p}))f_{α} for all primes p |̸2N,

• pS(p)f_{α} = (det ρ(Frob_{p}))f_{α} for all primes p |̸2N,

• U_{2}f_{α} = αf_{α},

• and U_{p}f_{α} = 0 for all p|N.

(We remark that it is presumably not hard to weaken the fourth assumption to simply require that ρ|_{Gal (ℚ_{2}^{ac}/ℚ_{2})}^{ss} is unramified and that α is an eigenvalue of ρ_{I_{2}}(Frob_{2}). We do not do so as we shall not need this result.)

4 The main theorem.

We now turn to the proof of theorem A. By the previous work cited in the introduction, it suffices to check the following special case, which is our only contribution. Theorem 4.1. Suppose that K/ℚ is a Galois extension with Galois group A_{5}. Suppose also that

• 2 is unramified in K and Frob_{2} ∈ Gal (K/ℚ) has order 3,

• 5 is unramified in K

• and K is not totally real. If r : Gal (ℚ^{ac}/ℚ) ⟶ GL_{2}(ℂ) is a continuous representation such that the image of proj r is Gal (K/ℚ) then there is a weight one newform f such that for all prime numbers p the p^{th} Fourier coefficient of f equals the trace of Frobenius at p on the inertia at p coinvariants of r. In particular the Artin L-series for r is the Mellin transform of a weight one newform and is an entire function.

Proof. Twisting r by a character of finite order we may suppose that the image of det r has two-power order, that r is unramified at 2 and 5, and that r(Frob_{2}) has order 3. Choose an isomorphism of fields ℚ_{2}^{ac} ≅ ℂ, so that we may think of r as a representation

Gal (ℚ^{ac}/ℚ) ⟶ GL_{2}(O_{K})

for some finite extension K/ℚ_{2} inside ℚ_{2}^{ac}. By corollary 3.4 we see that we may find an odd integer N ≥ 5 divisible by all primes at which r ramifies and normalised eigenforms f_{α}, f_{β} ∈ M_{1}^{>2^{−2/3}}(N) such that

• T_{p}f_{α} = (tr r(Frob_{p}))f_{α} and T_{p}f_{β} = (tr r(Frob_{p}))f_{β} for all primes p |̸2N,

• pS(p)f_{α} = (det r(Frob_{p}))f_{α} and pS(p)f_{β} = (det r(Frob_{p}))f_{β} for all primes p |̸2N,

• U_{2}f_{α} = αf_{α} and U_{2}f_{β} = βf_{β},

• and U_{p}f_{α} = U_{p}f_{β} = 0 for all p|N.

Theorem 2.10 tells us that

f = (αf_{α} − βf_{β})/(α − β)

is classical and theorem A follows from this.

Lastly let us give some examples. To that end we will call a number field K suitable if

• K is Galois over ℚ with group A_{5} ,

• 2 is unramified in K and Frob_{2} ∈ Gal (K/ℚ) has order 3,

• 5 is unramified in K

• and K is totally complex.

If K is such a number field then we can find a continuous homomorphism

r : Gal (ℚ^{ac}/ℚ) ⟶ GL_{2}(ℂ)

such that the image of proj r is Gal (K/ℚ) (see for instance the corollary to theorem 4 of [S2]). For any such r we have just shown that L(r, s) has analytic continuation to the whole complex plane. Thus to give examples of our theorem, one need only give examples of suitable number fields K.

Suppose that S is a finite set of places of ℚ including 2, 5 and ∞. For v ∈ S let L_{v}/ℚ_{v} be a finite Galois extension such that Gal (L_{v}/ℚ_{v}) embeds into A_{5}. Suppose that • L_{2}/ℚ_{2} is unramified of degree 3,

• L_{5}/ℚ_{5} is unramified

• and L_{∞} = ℂ.

According to [M] the quotient of affine 5 space over ℚ by the action of A_{5} which simply permutes the variables, is rational. Hence, by for example the discussion on page xiv of [S3] (see in particular theorem 2 and the remark which follows), we see that there is a number field K which is Galois over ℚ with group A_{5} and such that for v ∈ S we have K_{v} ≅ L_{v}^{60/[L_{v}:ℚ_{v}]}. By varying S we see in particular that there are infinitely many suitable number fields.

More concrete examples can be found in the literature. For example, according to Buhler [Buh], the splitting fields of the following are suitable:

x^{5} + 4x^{4} + 25x^{3} + 17x^{2} + 5x + 2
x^{5} + 6x^{4} + 19x^{3} + 25x^{2} + 11x + 2
x^{5} + 3x^{4} + 7x^{3} + 6x^{2} − 11x − 24
x^{5} + 3x^{4} + x^{3} − 4x^{2} + 17x − 8
x^{5} + 2x^{4} + 37x^{3} − 7x^{2} + 25x − 4.

Corrigenda for [Ta2].

One of us (R.T.) would like to take the opportunity to record some corrections to [Ta2]. He would like to thank Kevin Buzzard, Henri Darmon and Nick Shepherd-Barron for pointing these out.

• page 3, line 8: the formula defining T_{p} should have a factor p^{k−1} multiplying the second sum.

• page 3, line -1/2: between “if and only if” and “ f (as an element...” insert “c_{1}(f) = 1 and”.

• page 4, line -7: the SS_{>r} should read SS_{<r} and the X_{1}(N)_{<r} should read X_{1}(N)_{>r}.

• page 7, line 5: in theorem 1 we need to assume that the image of G_{l} under the projective representation associated to \overline{ρ} has order divisible by a prime other than l.

• page 7, line -8: in conjecture 1 we should have assumed that the image of G_{l} under the projective representation associated to \overline{ρ} has order divisible by a prime other than l.

• page 8, line -14: “together with an embedding i : ℤ[(1 + \sqrt{5})/2 ↪ End ((A, ψ)/ℚ) such that the representation of G_{ℚ} on A[2] is equivalent to \overline{ρ}.” should read “together with an embedding i : ℤ[(1 + \sqrt{5})/2 ↪ End (A/ℚ) such that the image of i is fixed by the ψ-Rosati involution and the representation of G_{ℚ} on A[2] is equivalent to \overline{ρ}.”

• page 9, line 5: it should read “A[2]”, not“A[\sqrt{2}]”.

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