Control Theorems and Applications

Haruzo Hida^{∗}

February 15, 2000

Contents

1 Introduction 2

2 Vertical control theorem 3

3 Deformation theory of mod p representations 6 3.1 Galois representations . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Deformation rings . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Theorem 7: R = 𝕋 . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 Theorem 8: R^{φ} = T^{φ} . . . . . . . . . . . . . . . . . . . . . . . . 9 3.5 Theorem 8 + Vertical control ⟹ Theorem 7 . . . . . . . . . . 12

4 Horizontal control theorems 12 4.1 Galois side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Hecke side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Proof of Theorem 11 . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Taylor-Wiles systems 19

6 Proof of Theorem 8 25

7 Proof of Vertical control theorem 32 7.1 Modular definition of modular forms . . . . . . . . . . . . . . . . 32 7.2 False modular forms . . . . . . . . . . . . . . . . . . . . . . . . . 34 7.3 p–Adic modular forms . . . . . . . . . . . . . . . . . . . . . . . . 44 7.4 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.5 Family of ordinary p–adic modular forms . . . . . . . . . . . . . 51 8 Applications 54

∗The research of the author is partially supported by an NSF grant.

8.1 F –motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 8.2 Endomorphism algebras of F –motives . . . . . . . . . . . . . . . 58 8.3 Modularity of Galois representations of ℚ–motives . . . . . . . . 61 8.4 Abelian varieties with real multiplication . . . . . . . . . . . . . . 67 8.5 Cusp forms with quadratic “Neben” type. . . . . . . . . . . . . . 69 8.6 Base-change to totally real fields . . . . . . . . . . . . . . . . . . 71

1 Introduction

Sections 1-7 of this article are based on notes of lectures given by the author during the Special Year on Automorphic Forms at the Tata Institute of Fundamental Research. The purpose of the lectures was to introduce the audience to control theorems and to show how such theorems have been useful in establishing that certain Hecke algebras are universal deformation rings for certain mod p representations.

The universality of the Hecke algebra was first proved by A. Wiles [W], assuming a ring theoretic property of the Hecke algebra, which was in turn proved in [TW]. The appendix to [TW], contains an alternative method for proving the universality of the Hecke algebra, assuming that the ramification of deformations outside p is minimal. This latter method has been simplified by F. Diamond [Di1] and K. Fujiwara [Fu]. In these notes, we describe Fujiwara’s treatment, assuming that the deformations are unramified outside p.

We have added an extra section (Section 8) to the lecture notes. In it we descibe some arithmetic applications of the control theorems and of the universality of the Hecke algebra. These results are new, and were obtained after completing the lectures series at TIFR.

In particular, we give a sufficient condition for the modularity of a ℚ–motive M of rank 2 with coefficients in an anti-CM field E. Here, a number field is said to be anti-CM if it does not contain a CM field, and, a motive M defined over the algebraic closure \overline{ℚ} of ℚ is said to be a ℚ–motive if, for each automorphism σ of \overline{ℚ}, σ(M) is isomorphic (over \overline{ℚ}) to M as motives with coefficients in E. Following Ribet’s argument for abelian varieties [R4], we prove that any ℚ– motive as above has a model defined over a composite L of quadratic extensions of ℚ, and define an obstruction class Ob(M) ∈ H^{2}(ℚ, E^{×}) to finding a model of M over ℚ. We shall show that the 𝔭–adic Galois representation attached to Res_{L/ℚ}M is a deformation of an induced representation of a character if there is no 𝔭–adic unit valued cocycle representing Ob(M). Since two dimensional induced mod p representations are modular, if the deformation attached to M is odd, it must be modular under some mild conditions prescribed by Wiles’ theorem mentioned above.

A result similar to the above, for elliptic ℚ–curves, was discussed in [H00b] Chapter V. Recently, it has been announced by J. Ellenberg that every elliptic ℚ–curve, satisfying a certain local condition at the prime 3, appears as a factor, over the algebraic closure of ℚ, of the jacobian of a modular curve X_{0}(N).

Although our result has some restrictive assumptions, it does cover general ℚ– motives (and not just elliptic curves).

Let F be a totally real field. There is a generalization of Wiles’ theorem for Hilbert modular forms on GL(2)_{/F} due to Fujiwara [Fu]. Applying this to the motive Res_{LF/F} M, we show that the base-change of the Galois representation attached to Res_{L/ℚ}M is associated to a Hilbert modular form, under some conditions on M and F. At the end of Section 8, we also briefly describe a conjectural method for reducing the base-change problem for elliptic modular forms to the case of ℚ–motives, by the vertical control theorem (see [HM]; this was the subject of the colloquium talk at TIFR).

The author would like to take this opportunity to express his thanks to the audience of the lectures at TIFR for their patience, and to the Institute for its hospitality. The notes of the lectures were taken by Eknath Ghate, and this article has been prepared in collaboration with him.

2 Vertical control theorem

Let N ≥ 1 and k ≥ 1 be integers. Let χ : (ℤ/Nℤ)^{×} → ℂ^{×} denote a Dirichlet character. Let S_{k}(N, χ) (resp. G_{k}(N, χ)) be the complex vector space of elliptic cusp forms (resp. elliptic modular forms) of weight k for Γ_{0}(N) with Neben character χ.

There is an integral structure on S_{k}(N, χ) coming from Fourier expansion. Let ℤ[χ] denote the ℤ-algebra generated by the values of the character χ. For each ℤ[χ]-algebra A sitting inside ℂ we set

S_{k}(N, χ, A) = {f = ∑_{n=1}^{∞} a(n, f)q^{n} | a(n, f) ∈ A}

G_{k}(N, χ, A) = {f = ∑_{n=0}^{∞} a(n, f)q^{n} | a(n, f) ∈ A}.

It is a fact that for A as above, S_{k}(N, χ, A) = S_{k}(N, χ, ℤ[χ]) ⊗_{ℤ[χ]} A and G_{k}(N, χ, A) = G_{k}(N, χ, ℤ[χ]) ⊗_{ℤ[χ]} A (see [H00b] III.1 or [H93] Chapter 5 for different proofs). For a general algebra A, not necessarily in ℂ, we define F_{k}(n, χ, A) by F_{k}(N, χ, ℤ[χ]) ⊗_{ℤ[χ]} A, where F_{k} is S_{k} or G_{k}.

For each n = 1, 2, …, there is a Hecke operator T (n) which acts A-linearly on the space G_{k}(N, χ, A). It is given by the formula

a(m, f|T (n)) = ∑_{d|m,d|n} χ(d)d^{k−1}a(mn/d^{2}, f).  (1)

See Subsection 7.4 (and [H00a] 3.2) for a more intrinsic definition of the operators T (n). It is customary to write U(p) for T (p) if p is a prime and p|N, because they have different effect on Fourier expansion.

Let now p denote an odd prime (although we will include the case p = 2 later in Section 7). Fix a complete discrete valuation ring O lying over ℤ_{p}. Let T denote the maximal split torus in PGL_{2}. Thus T ⟶^{∼} 𝔾_{m}. We have

T(ℤ_{p}) = ℤ_{p}^{×} = μ_{p−1} × Γ,

with Γ ⟶^{∼} ℤ_{p}, via γ^{s} ↦ s, for γ = 1 + p, and s ∈ ℤ_{p}. Let

Λ := O[[Γ]] = lim_{←} O[Γ/Γ^{p^{n}}] ⟶^{∼} O[[t]]

denote the Iwasawa algebra. Then

O[[T(Z_{p})]] = Λ[μ_{p−1}].

If R is a p-profinite ring, then any continuous character φ : Γ → R^{×} extends to a character φ : Λ → R, which we shall again denote by φ. We may also consider φ as a character φ : O[[T(ℤ_{p})]] → O. Thus if we write ν for the canonical inclusions

ν : Γ ↪ T(ℤ_{p}) = ℤ_{p}^{×} ↪ O^{×},

then we may consider ν as a character ν : Λ → O. In particular we have ν^{k−1}(Φ(t)) = Φ(γ^{k−1} − 1) ∈ O, for Φ(t) ∈ Λ. We may also consider ν as a character ν : O[[T(ℤ_{p})]] → O.

Let us now assume that N is such that (p, N) = 1, and that k ≥ 2. For simplicity, we assume that the prime p ≥ 5 until Section 7. This does not cause much harm if N = 1 because the space of p–ordinary cusp forms (that we will be mainly interested in) vanishes if p ≤ 7 and N = 1. We fix a character χ : (ℤ/Nℤ)^{×} → O^{×}. Sometimes we will think of the mod N Dirichlet character χ as a mod Np Dirichlet character. If we need to indicate the modulus M defining χ, we will write χ_{M} instead of just χ.

There is a projective O[[T(ℤ_{p})]]-module S_{Γ_{0}(N),Λ}[χ] of finite type, the so called space of ‘ordinary Λ-adic cusp forms of level Np^{∞}’, which has an action of Hecke operators, again denoted by T (n), such that

S_{Γ_{0}(N),Λ}[χ] ⊗_{O[[T(ℤ_{p})]],ν^{k−1}} O ↪ S_{k}(Np, χ, O).

Tensoring over O[[T(ℤ_{p})]] through ν^{k−1} does not alter the ‘Neben’ character χ we have fixed, but tensoring over Λ produces all spaces with ‘Neben’ characters χω^{a} (for the Teichmüller character ω) as we will state later. Roughly, this map is realized as follows. Define the Hecke algebra h(N, χ) to be the algebra generated over Λ by the Hecke operators T (n) inside End_{Λ}(S_{Γ_{0}(N),Λ}[χ]). Let λ be a Λ-algebra homomorphism of h(N, χ) into an algebraic closure of the field of fractions of Λ. For simplicity let us assume that λ(T (n)) ∈ Λ, for n. Write

F (t, q) = ∑_{n=1}^{∞} λ(T (n))(t)q^{n} ∈ Λ[[q]]]

for the corresponding Λ-adic cusp form in S_{Γ_{0}(N),Λ}[χ]. Then F (γ^{k−1} − 1, q) ∈ O[[q]], and is a classical modular form for k ≥ 2. These classical forms interpolate the family of p-adic modular forms F (γ^{s} − 1, q) for s ∈ ℤ_{p}. In fact the following theorem is true:

Theorem 1. (Hida) Fix a mod N Dirichlet character χ with values in O^{×}. We have

• S_{Γ_{0}(N),Λ}[χ] ⊗_{O[[T(ℤ_{p})]],ν^{k−1}} O ⟶^{∼} S_{k}^{ord}(Np, χ, O) ⟶^{∼} S_{k}^{ord}(N, χ, O), for

all k ≥ 3,

• S_{Γ_{0}(N),Λ}[χ] ⊗_{Λ,ν^{k−1}} O ⟶^{∼} ⨁_{a=0}^{p−2} S_{k}^{ord}(Np, χω^{a}, O), for all k ≥ 2,

where ω : (ℤ/pℤ)^{×} → ℂ^{×} is the Teichmüller character.

A similar assertion holds for G_{k} in place of S_{k} (see Theorem 22 for details).

In the theorem, the superscript ord means the ordinary part of the corresponding space of cusp forms. Let us recall, for the reader’s convenience, what this means. For an O-module X of finite type, and an O-linear operator T : X → X, we may decompose X

X = X^{ord} ⊕ X^{nil}

where both X^{ord} and X^{nil} are T -stable, and moreover T : X^{ord} ⟶^{∼} X^{ord} and ⋂_{m=1}^{∞} T^{m}(X^{nil}) = 0. The idempotent corresponding to X^{ord} in the above decomposition is lim_{m→∞} T^{m!}.

Apply this to X = F_{k}(M, χ, O) for M = N and Np with the operator

T = \begin{cases}T (p) & if M = N, \\ U(p) & if M = Np. \end{cases}

Then F_{k}^{ord}(M, χ, O) is the ordinary part, X^{ord}, of X = F_{k}(M, χ, O), where F_{k} is S_{k} or G_{k}.

The Hecke operator T (p) acting on the space G_{k}(N, χ, O) and T (p) acting on the larger space G_{k}(Np, χ, O) are not equal; so, we have written U(p) for the p–th Hecke operator acting on G_{k}(Np, χ, O). These two operators T (p) and U(p) are congruent modulo p if k ≥ 2. If further k ≥ 3, the dimension of S_{k}^{ord}(Np, χ, O) ⊗_{ℤ} ℤ/pℤ coincides with the level N counterpart, because all p– new forms in the level Np space with trivial ‘Neben’ p–character has eigenvalue ±\sqrt{χ_{N}(p)}\sqrt{p}^{k−2} for U(p) (see [Mk] Theorem 4.6.17), which is divisible by \sqrt{p} if k ≥ 3. Therefore, if

e_{0} = lim_{m→∞} T (p)^{m!} and e = lim_{m→∞} U(p)^{m!},

are the ordinary projectors, we conclude that the p–ordinary projector e acting on S_{k}(Np, χ, O) induces an isomorphism of e_{0}S_{k}(N, χ, O) = S_{k}^{ord}(N, χ, O) onto S_{k}^{ord}(Np, χ, O) as long as k ≥ 3. Thus the isomorphism between the level N space and the level Np space in the theorem is given by e^{−1} when k ≥ 3. We emphasize that this isomorphism is induced by e and not by the natural inclusion; that is, S_{k}^{ord}(N, χ, O) is possibly distinct from S_{k}^{ord}(Np, χ, O) inside S_{k}(Np, χ, O). If k = 2, the image under e of S_{2}^{ord}(N, χ, O) could be smaller than S_{2}^{ord}(Np, χ, O).

Remark 1. When k = 1, the above theorem is false. We describe this phenomenon in more detail in Subsection 3.4, Remark 2.

Now define the Hecke algebra h_{k}^{ord}(M, χ, O) as the algebra generated by the Hecke operators T (n) inside End_{O}(S_{k}^{ord}(M, χ, O)). We have:

Theorem 2 (Vertical control theorem). (Hida) We have

• h(N, χ) ⊗_{O[[T(ℤ_{p})]],ν^{k−1}} O ⟶^{∼} h_{k}^{ord}(N, χ, O), for all k ≥ 3,

• h(N, χ) ⊗_{Λ,ν^{k−1}} O ⟶^{∼} ⨁_{a=0}^{p−2} h_{k}^{ord}(Np, χω^{a}, O), for all k ≥ 2.

We will restate the vertical control theorems in Section 7 in a more general setting which will include p = 2 and 3, and will give a proof of it there.

3 Deformation theory of mod p representations

3.1 Galois representations

Assume as before that p ≥ 5, that (N, p) = 1 and that k ≥ 2. However after this subsection, we allow a Dirichlet character modulo Np, since χω^{a} appears as the ‘Neben’ character of the specialization of the Hecke algebra h(N, χ). Thus we take a Dirichlet character ψ modulo Np with values in O^{×}.

Fix an O-algebra homomorphism λ : h_{k}(Np, ψ, O) → O. The following theorem is a consequence of a general theorem due to Eichler, Shimura, and Deligne:

Theorem 3. There exists a continuous Galois representation

ρ_{λ} : Gal(\overline{ℚ}/ℚ) → GL_{2}(O),

satisfying

• ρ_{λ} is unramified outside Np, and,

• det(1 − ρ_{λ}(Frob_{ℓ})X) = 1 − λ(T (ℓ))X + ψ(ℓ)ℓ^{k−1}X^{2}, for ℓ | N̸p.

Let D_{p} denote a decomposition group at p. If λ factors through the p– ordinary part h_{k}^{ord}(Np, ψ, O), the following further information at p is due to Deligne, Mazur and Wiles:

Theorem 4. Suppose that λ factors through h_{k}^{ord}(Np, ψ, O). Then the Galois representation ρ_{λ} is p-ordinary. That is, there are characters ϵ_{ρ_{λ}} : D_{p} → O^{×}, and δ_{ρ_{λ}} : D_{p} → O^{×} with δ_{ρ_{λ}} unramified, such that

ρ_{λ}|_{D_{p}} ∼ (\begin{matrix}ϵ_{ρ_{λ}} & ∗ \\ 0 & δ_{ρ_{λ}} \end{matrix}) ,

and δ_{ρ_{λ}}(Frob_{p}) = λ(U(p)).

For a detailed proof of the above two theorems, see, for example, [H00b] IV.2. Let \overline{ρ}_{λ} be the mod p Galois representation defined by reduction:

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

where 𝔽 is the residue field of O. Let us now impose the conditions (ai_{p}(ℚ)) and (reg_{p}) on \overline{ρ}_{λ}, where:

(ai_{p}(F)): If F/ℚ is a number field, then the restriction of to \overline{ρ}_{λ} to F:

\overline{ρ}_{λ|F} : Gal(\overline{ℚ}/F) ⊂ Gal(\overline{ℚ}/ℚ) ⟶^{\overline{ρ}_{λ}} GL_{2}(𝔽),

is absolutely irreducible.

(reg_{p}): Let I_{p} denote the inertia subgroup at p. Then

\overline{ρ}_{λ}|_{I_{p}} ∼ (\begin{matrix}ϵ_{\overline{ρ}_{λ}} & ∗ \\ 0 & δ_{\overline{ρ}_{λ}} \end{matrix})

with ϵ_{\overline{ρ}_{λ}} ≠ δ_{\overline{ρ}_{λ}} on I_{p}. In view of Theorem 4 above, this happens exactly when ϵ_{\overline{ρ}_{λ}} is ramified at p.

Note that there is a decomposition of the finite Λ-module h = h(N, χ):

h = ⊕h_{𝔪}

as 𝔪 varies through the maximal ideals of h. Let 𝕋 = h_{𝔪} be the local ring through which λ : h_{k}^{ord}(Np, ψ, O) → O factors. We write 𝕋^{red} for the quotient of 𝕋 by its nilradical. The two algebras 𝕋 and 𝕋^{ord} are equal if the character χ is primitive modulo N. The following theorem is due to Hida and Wiles:

Theorem 5. There exists a Galois representation ρ_{𝕋} : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝕋^{red}) satisfying

• ρ_{𝕋} is unramified outside Np,

• det(1 − ρ_{𝕋}(Frob_{ℓ})X) = 1 − T (ℓ)|_{𝕋^{red}}X + χ(ℓ)⟨ℓ⟩X^{2}, for ℓ | N̸p, where ⟨ℓ⟩ denotes the image of ℓ under the natural map

O[[T(ℤ_{p})]] → h(N, χ) → 𝕋^{red},

and,

• ρ_{𝕋} is p-ordinary, that is

ρ_{𝕋}|_{D_{p}} ∼ (\begin{matrix}ϵ_{ρ_{𝕋}} & ∗ \\ 0 & δ_{ρ_{𝕋}} \end{matrix}) .

with δ_{ρ_{𝕋}} unramified and δ_{ρ_{𝕋}}(Frob_{p}) = U(p)|_{𝕋^{red}}.

A construction of the representation as above with coefficients in the total quotient ring of 𝕋^{red} was first given in [H86b]. Later by the technique of pseudo-representations invented by Wiles, we found ρ_{𝕋} with coefficients in 𝕋 in the isomorphism class of the representations constructed by [H86b]. The local behavior of ρ_{𝕋} described above then follows from Theorem 4. See [H00a] 2.2 for generalities on pseudo-representations, and see [H93] Chapter 7 for the construction of ρ_{𝕋} by means of pseudo-representations.

3.2 Deformation rings

We keep the assumptions and notation of the previous section. In particular \overline{ρ}_{λ} is the reduction of the Galois representation ρ_{λ} attached to λ : h_{k}^{ord}(Np, ψ, O) → O. However, to make our exposition simple, from now on (until Section 7) we will assume that

N = 1.

As before, we write ψ for the Dirichlet character modulo p of λ. Thus the local ring 𝕋 = 𝕋^{red} is the direct factor of h(1, 1) (through which λ factors) for the identity character χ = 1 (modulo 1).

Theorem 6. (Mazur) If (ai_{p}(ℚ)) and (reg_{p}) hold, then the universal deformation ring R and universal Galois representation ρ_{R} : Gal(\overline{ℚ}/ℚ) → GL_{2}(R) exist.

Here R is a p-profinite local O-algebra with R/𝔪_{R} = 𝔽. Furthermore ρ_{R} : Gal(\overline{ℚ}/ℚ) → GL_{2}(R) satisfies:

• ρ_{R} is unramified outside p,

• ρ_{R} is p-ordinary, that is

ρ_{R}|_{D_{p}} ∼ (\begin{matrix}ϵ_{ρ_{R}} & ∗ \\ 0 & δ_{ρ_{R}} \end{matrix}) ,

with δ_{ρ_{R}} : D_{p} → R^{×} unramified, and,

• ρ_{R} mod 𝔪_{R} ∼ \overline{ρ}_{λ}.

Moreover the couple (R, ρ_{R}) is universal in the sense that for any pair (A, ρ_{A}) with

• A a p-profinite local O-algebra with A/𝔪_{A} = 𝔽,

• ρ_{A} : Gal(\overline{ℚ}/ℚ) → GL_{2}(A) an unramified outside p, p-ordinary representation in the sense above, and,

• ρ_{A} mod 𝔪_{A} ∼ \overline{ρ}_{λ},

there is a unique map of local O-algebras ϕ_{A} : R → A, such that

ϕ_{A} ∘ ρ_{R} ∼ ρ_{A}.

See [H00a] 2.3 and 3.2.3 for a proof of the existence of the universal couple (R, ρ_{R}).

3.3 Theorem 7: R = 𝕋

By Theorem 5, we see that there exists a map of local O-algebras

ϕ_{𝕋} : R ⟶ 𝕋.

The following theorem, was conjectured by Mazur:

Theorem 7. (Wiles) Assume that \overline{ρ}_{λ} satisfies (reg_{p}) and (ai_{p}(κ)) , for κ = ℚ(\sqrt{(−1)^{(p−1)/2}p}). (In particular it satisfies (ai_{p}(ℚ)).) Then the map ϕ_{𝕋} : R → 𝕋 is an isomorphism of local O-algebras.

Note that the surjectivity of ϕ_{𝕋} is obvious:

Lemma 1. The map ϕ_{𝕋} : R → 𝕋 is surjective.

Proof. 𝕋 is generated by the operators T (ℓ), for ℓ ≠ p. But

T (ℓ) = Tr(ρ_{𝕋}(Frob_{ℓ})) = Tr(ϕ_{𝕋} ∘ ρ_{R}(Frob_{ℓ})) = ϕ_{𝕋}(Tr(ρ_{R}(Frob_{ℓ})).

So we are done.

3.4 Theorem 8: R^{φ} = T^{φ}

We now investigate the deformation problem when the determinant is fixed.

Fix a character φ : Gal(\overline{ℚ}/ℚ) → O^{×}, unramified outside p and ∞ and let us assume that det \overline{ρ}_{λ} = φ mod 𝔪_{O}. Note that for each local O-algebra A as above, we have the algebra homomorphism ι_{A} : O^{×} → A^{×} giving the O–algebra structure on A, so that it makes sense to impose the further condition

det ρ_{A} = ι_{A} ∘ φ

on the deformation problem considered above. This new deformation problem also has a solution; call the universal couple (R_{φ}, ρ_{R^{φ}}).

On the other hand φ factors through the Galois group of the maximal abelian extension ℚ(μ_{p^{∞}}) of ℚ unramified outside p (and ∞). Since

Gal(ℚ(μ_{p^{∞}})/ℚ) = ℤ_{p}^{×} = T(ℤ_{p}),

we can think of φ as a character

φ : O[[T(ℤ_{p})]] → O.  (2)

Now set

T^{φ} = 𝕋 ⊗_{O[[T(ℤ_{p})]],φ} O = 𝕋/P_{φ}𝕋,

where P_{φ} is the kernel of φ in (2). Also set ρ_{T^{φ}} = ρ_{𝕋} mod P_{φ}. We now have the following key theorem:

Theorem 8. (Wiles-Taylor) Assume that \overline{ρ}_{λ} satisfies (reg_{p}) and (ai_{p}(κ)), where κ = ℚ(\sqrt{(−1)^{(p−1)/2}p}). Then

(R^{φ}, ρ_{R^{φ}}) =^{∼} (T^{φ}, ρ_{T^{φ}}).

The above theorem was first proved by Wiles in [W] as Theorem 3.3, under the assumption that T^{φ} is a local complete intersection. The local complete intersection property was then proved by Taylor and Wiles in [TW], where they also give another direct proof of the theorem. As mentioned in the Introduction, we wish to describe this latter method of Taylor-Wiles [TW] incorporating improvements due to Diamond and Fujiwara. We will follow Fujiwara’s treatment [Fu]. Note that [Fu] also contains a generalization of the above theorem to totally real fields.

Let us now describe the content of the above theorem in more down-to-earth terms. Let A be a valuation ring finite flat over O. We pick any Galois representation ρ : Gal(\overline{ℚ}/ℚ) → GL_{2}(A) which is a p–ordinary deformation with determinant φ = ψν^{k−1} for the p–adic cyclotomic character ν and a positive integer k ≥ 2, where ψ : Gal(ℚ(μ_{p})/ℚ) → ℤ_{p}^{×} is a character (which could be trivial). Then by the above theorem, we have a unique O–algebra homomorphism π : R^{φ} = T^{φ} → A such that π ∘ ρ_{T^{φ}} is equivalent to ρ. Under the condition k ≥ 2, the ring T^{φ} is a direct factor of the Hecke algebra h_{k}(p, ψ, O); so, we may regard π as an O–algebra homomorphism π : h_{k}(p, ψ, O) → A. Since h_{k}(p, ψ, O) = h_{k}(p, ψ, ℤ[ψ]) ⊗_{ℤ[ψ]} O, π induces a ℂ–algebra homomorphism π_{ℂ} of h_{k}(p, ψ, ℂ) = h_{k}(p, ψ, ℤ[ψ]) ⊗_{ℤ[ψ]} ℂ into ℂ, which is associated to a Hecke eigenform f = ∑_{n=1}^{∞} π_{ℂ}(T (n))q^{n}. Thus ρ is isomorphic to the p–adic Galois representation ρ_{f} associated to the Hecke eigenform f.

Remark 2. When k = 1, we have a natural homomorphism

π : T^{φ} → h_{1}^{ord}(p, φ, O)

if φ is a finite order character. The homomorphism often has a non-trivial kernel ([H98] and [ChV]) and could have trivial image. A result of Deligne and Serre says that to each weight 1 classical (or equivalently ‘true’) Hecke eigenform one may associate a two dimensional Artin Galois representation. Thus the image of π is expected to be non-trivial if ρ_{𝕋} specializes to a Galois representation with finite image at weight 1. This expectation was proved by Langlands [L] (and Tunnel [Tu]) for Artin representations whose image in PGL_{2}(ℂ) is a tetrahedral or octahedral group (for example, see [H00b] V.1.3). On the other hand, as was shown by Mazur-Wiles [MW], often the image of the specialization is infinite (so the image of π would have to be trivial in this case). However, recently, Taylor and Buzzard [BT] found that if the ramification of \overline{ρ}_{λ} at p is tame, ρ_{𝕋} specializes to an Artin representation and that the Artin representation is associated to a Hecke eigenform of weight 1. Combining this result with ideas of Wiles (the same ideas that have led to a proof of the Shimura-Taniyama conjecture), Taylor et al have shown that certain icosahedral Artin representations are modular (see [BuDST]).

Remark 3. The argument of Taylor-Wiles (and also the original argument of Wiles in [W] proving Theorem 3.3 there) actually gives a result covering general N ≥ 1 when the ramification of the deformations is minimal (see [W] page 455-8 for a description of the term minimal). Then Wiles analyzed carefully the difference of the Hecke algebra in the minimal case and those without the minimality conditions and reached the following more general (and more convenient) result. To describe it, we need some more notation and conditions: Let Σ be a finite set of primes including the fixed odd prime p and ℚ^{Σ} be the maximal extension unramified outside Σ and ∞. Let 𝔊 = Gal(ℚ^{Σ}/ℚ). We pick an absolutely irreducible odd representation \overline{ρ} : 𝔊 → GL_{2}(𝔽) for a finite field 𝔽 of characteristic p. Hereafter all valuation rings will be finite flat over ℤ_{p}. We consider the following conditions:

(Ordinarity) \overline{ρ}|_{D_{p}} ≅ ( \begin{matrix}\overline{ε} & ∗ \\ 0 & \overline{δ} \end{matrix} ) with \overline{δ} unramified and \overline{δ} ≠ \overline{ε} on the decomposition group D_{p} at p;

(Flatness) \overline{ρ} restricted to the decomposition group at p is isomorphic to a Galois module associated to a locally free group scheme over ℤ_{p} of rank |𝔽|^{2};

(Irreducibility) \overline{ρ} restricted to Gal(ℚ^{Σ}/ℚ(μ_{p})) remains absolutely irreducible;

(Modularity) \overline{ρ} is associated to an elliptic cusp form, that is, \overline{ρ} = \overline{ρ}_{λ} for λ : h_{k}(p, ψ, O) → O.

Theorem 9. (Wiles-Taylor-Diamond) Suppose modularity and irreducibility and either ordinarity or flatness of \overline{ρ}. Let ρ : 𝔊_{Σ} → GL_{2}(O) be a Galois representation for a DVR O such that

• ρ ≡ \overline{ρ} mod 𝔪_{O};

• det ρ = ν_{p}^{k−1} up to finite order characters for k ≥ 2, where ν_{p} is the p–adic cyclotomic character;

• ρ|_{D_{p}} ≅ ( \begin{matrix}ε & ∗ \\ 0 & δ \end{matrix}) for an unramified character δ ≡ \overline{δ} mod 𝔪_{O} when \overline{ρ} is ordinary;

• When \overline{ρ} is flat, k = 2 and det ρ|_{I_{p}} = ν_{p}|_{I_{p}} and ρ is associated to a p– divisible group over ℤ_{p} in the sense of Tate.

Then there exist a positive integer N and a Hecke eigenform f ∈ S_{k}(Γ_{1}(N)) such that ρ ≅ ρ_{f}.

The above theorem was proven in [W] as Theorem 0.2 under an additional condition on auxiliary ramification outside p, which was later removed by Diamond in [Di]. Wiles proved the result assuming the complete intersection property of a certain Hecke algebra in the minimal ramification case, which was in turn proved in [TW]. The flatness condition has now been eased by Conrad-Diamond-Taylor to potential flatness over an extension of ℤ_{p} with ramification index ≤ p−1 [CoDT], and Breuil-Conrad-Diamond-Taylor have treated the case of arbitrary ramification index.

3.5 Theorem 8 + Vertical control ⟹ Theorem 7

It turns out that Theorem 8 and the vertical control theorem (Theorem 2) imply Wiles’ main theorem (Theorem 7). To see this we first prove the following lemma:

Lemma 2. 𝕋 is a free Λ-module of finite rank.

Proof. By Theorem 2 (for χ = 1), applied with φ = ν^{k−1}, we see that 𝕋/P_{φ}𝕋 sits inside a Hecke algebra of weight k, and so in particular it is O-free, of finite rank, say r. By Nakayama’s lemma the minimal number of generators of 𝕋 over Λ is r as well. Let x_{1}, …, x_{r} be generators of 𝕋, and let

π : Λ^{r} ↠ 𝕋
(λ_{1}, …, λ_{r}) ↦ ∑_{i=1}^{r} λ_{i}x_{i}.

Now, by Nakayama’s lemma again, 𝕋/P_{ν^{j−1}}𝕋 must be O-free of finite rank r for each j ≥ 2. This forces ker(π) ⊂ P_{ν^{j−1}}^{r} for each j, and so

ker(π) ⊂ ∩_{j≥2}P_{ν^{j−1}}^{r} = 0,

as desired.

Proposition 1. Theorem 8 + Vertical control ⟹ Theorem 7.

Proof. Choose φ = ν^{k−1} for some k ≥ 2. Since R/P_{φ}R = 𝕋/P_{φ}𝕋 by Theorem 8, we know that R/P_{φ}R is O-free of finite rank r, by Lemma 2. By Nakayama’s lemma there is a surjective map Λ^{r} → R. On the other hand, by Lemma 1, the map ϕ_{𝕋} : R → 𝕋 is surjective. Since 𝕋 ⟶^{∼} Λ^{r} (by Lemma 2), ϕ_{𝕋} : R → 𝕋 is forced to be an isomorphism.

4 Horizontal control theorems

In this section we will establish horizontal control theorems on both the Galois and Hecke sides. These will be useful in proving Theorem 8 above.

4.1 Galois side

Recall that \overline{ρ}_{λ} was the reduction of ρ_{λ}, for fixed λ : h_{k}^{ord}(p, ψ, O) → O. This representation was unramified outside p and was p-ordinary. In addition we had imposed the conditions (ai_{p}(κ)) and (reg_{p}) on \overline{ρ}_{λ}.

Let Q = {q_{1}, … q_{r}} denote a finite set of primes, with q_{i} ≡ 1 mod p, for q_{i} ∈ Q. Let us consider the deformation problem, that allows, in addition, ramification at primes in Q (we also fix a determinant φ). Let (R_{Q}^{φ}, ρ_{R_{Q}^{φ}}) be the universal couple for this deformation problem.

We now impose the following condition: for each q ∈ Q, assume (reg_{q}): If

\overline{ρ}_{λ}(Frob_{q}) = (\begin{matrix}ᾱ_{q} & ∗ \\ 0 & β̄_{q} \end{matrix}) ,

then ᾱ_{q} ≠ β̄_{q}.

Let D_{q} denote a decomposition group at q and let I_{q} denote the corresponding inertia group at q. We have the following theorem on the local behavior of ρ_{R_{Q}^{φ}} at q:

Theorem 10. (Faltings) Let q ∈ Q and assume (reg_{q}) above. Then

ρ_{R_{Q}^{φ}}|_{I_{q}} ∼ (\begin{matrix}φδ_{q}^{−1} & 0 \\ 0 & δ_{q} \end{matrix}) ,

for some character δ_{q} : D_{q} → R_{Q}^{φ}^{×}.

Recall that local class field theory gives us the following commutative diagram:

Gal(ℚ_{q}^{ab}/ℚ_{q}) =^{∼} D_{q}^{ab} {dense \atop ↩} ℚ_{q}^{×}
∪  ∪
I_{q}^{ab}  ←^{∼} ℤ_{q}^{×} = Δ_{q} × Δ′_{q} × 1 + qℤ_{q},

where Δ_{q}, respectively Δ′_{q}, is the p-Sylow subgroup, respectively prime-to-p part, of 𝔽_{q}^{×}.

Now since \overline{ρ}_{λ} is unramified at q, we have

δ_{q}(I_{q}^{ab}) ⊂ 1 + 𝔪_{R_{Q}^{φ}}.

Since 1 + 𝔪_{R_{Q}^{φ}} is p-profinite, δ_{q}|_{I_{q}^{ab}} factors through Δ_{q}. Thus we get a map

Δ_{Q} := ∏_{q∈Q} Δ_{q} \underrightarrow{∏_{q∈Q} δ_{q}} R_{Q}^{φ}^{×},

via which we may consider R_{Q}^{φ} as an O[Δ_{Q}]-algebra.

We note in passing that O[Δ_{Q}] is a p-profinite local O-algebra with residue field 𝔽.

We denote by 𝔄_{Q} the augmentation ideal of the group algebra O[Δ_{Q}], that is, 𝔄_{Q} is the kernel of the augmentation map O[Δ_{Q}] → O, which sends σ ∈ Δ_{q} to 1.

The following result establishes horizontal control for the rings R_{Q}^{φ}:

Proposition 2 (Horizontal control: Galois side). The natural map

R_{Q}^{φ} ⟶ R^{φ}

induces an isomorphism

R_{Q}^{φ}/𝔄_{Q}R_{Q}^{φ} ⟶^{∼} R^{φ}.

Proof. By the universal property of R^{φ} it suffices to show that the couple (R_{Q}^{φ}/𝔄_{Q}R_{Q}^{φ}, ρ_{R_{Q}^{φ}} mod 𝔄_{Q}) is universal for the ‘unramified outside p, p-ordinary, det = φ’ deformation problem. So say that ρ_{A} : Gal(\overline{ℚ}/ℚ) → GL_{2}(A) is such a deformation. Since, vacuously, ρ_{A} is unramified outside Q ∪ {p}, there is a unique morphism ϕ_{A} : R_{Q}^{φ} → A such that

ϕ_{A} ∘ ρ_{R_{Q}^{φ}} = ρ_{A}.  (3)

Restricting (3) to I_{q} we see that (cf. Theorem 10):

(\begin{matrix}φδ_{q}^{−1} & 0 \\ 0 & δ_{q} \end{matrix}) = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) ,

on I_{q}. Thus ϕ_{A}(δ_{q}(σ) − 1) = 0, for σ ∈ Δ_{q}. This shows that ϕ_{A}(𝔄_{Q}) = 0, and we may consider ϕ_{A} as a map

ϕ_{A} : R_{Q}^{φ}/𝔄_{Q}R_{Q}^{φ} → A.

If there is another ϕ : R_{Q}^{φ}/𝔄_{Q}R_{Q}^{φ} → A giving rise to ρ_{A}, the pull back of ϕ has to coincide with ϕ_{A} by the universality of R_{Q}^{φ}. Thus this ϕ_{A} is unique; so, (R_{Q}^{φ}/𝔄_{Q}R_{Q}^{φ}, ρ_{R_{Q}^{φ}} mod 𝔄_{Q}) is universal for the ‘unramified outside p, pordinary, det = φ’ deformation problem.

4.2 Hecke side

We now establish horizontal control on the Hecke side. This is a much more delicate matter (compared to the Galois side), and the proof will use some algebraic geometry. Since we can choose φ congruent to det \overline{ρ}_{λ} modulo p, we may assume that φ = ν^{k−1} with k ≥ 3 and ψ = χ = 1.

Let N_{Q} = ∏_{q∈Q} q, and consider the space of cusp forms on Γ_{1}(N_{Q}) ∩ Γ_{0}(p). We can define Hecke algebras h_{k}^{ord}(p, Γ_{1}(N_{Q}), ψ, O) as the algebra generated by all the Hecke operators inside End_{O}(S_{k}(Γ_{1}(N_{Q})∩Γ_{0}(p)). We have the following commutative diagram

h_{k}^{ord}(p, Γ_{1}(N_{Q}), ψ, O) ⊃ T_{Q}^{φ}
↓  ↓
h_{k}^{ord}(p, ψ, O)  ⊃ T^{φ},

where the first vertical maps is the natural projection, and T_{Q}^{φ} is defined as a local factor of the pre-image of T^{φ} under this projection. The choice of T_{Q}^{φ} depends on the choice of the \overline{α}_{q} out of the two eigenvalues of \overline{ρ}_{λ}(Frob_{q}) for q ∈ Q as we will see later.

Since the new Hecke algebra is defined with respect to Γ_{1}(N_{Q}) ∩ Γ_{0}(p), the action of ⟨ℓ⟩ factors through T(ℤ_{p} ×(ℤ/N_{Q}ℤ)) instead of T(ℤ_{p}). We decompose

T(ℤ_{p} × (ℤ/N_{Q}ℤ)) = Γ × μ_{p−1} × Δ′_{Q} × Δ_{Q}

so that Δ_{Q} is the p–Sylow subgroup of (ℤ/N_{Q}ℤ)^{×}. The action of the quotient group Γ_{0}(N_{Q})/Γ_{1}(N_{Q}) = (ℤ/N_{Q}ℤ)^{×} gives rise to an action of

Δ_{Q} ⊂ T(ℤ_{p} × (ℤ/N_{Q}ℤ))

on the Hecke algebra h_{k}^{ord}(p, Γ_{1}(N_{Q}), ψ, O). This action induces a canonical O– algebra homomorphism of the completed group algebra O[[T(ℤ_{p} × (ℤ/N_{Q}ℤ))]] into T_{Q}^{φ}. Since T_{Q}^{φ} is local, this algebra homomorphism factors through a local ring of O[[T(ℤ_{p}×(ℤ/N_{Q}ℤ))]], which is canonically isomorphic to Λ[Δ_{Q}] (that is, the quotient of O[[T(ℤ_{p} × (ℤ/N_{Q}ℤ))]] by the augmentation ideal of the group algebra O[μ_{p−1} × Δ′_{Q}]). For primes ℓ ∤ pN_{Q}, we write ⟨ℓ⟩ for the image of ℓ ∈ T(ℤ_{p} × (ℤ/N_{Q}ℤ))]] in T_{Q}^{φ}.

As in Theorem 5 we can construct a continuous Galois representation:

ρ′ = ρ′_{T_{Q}^{φ}} : Gal(\overline{ℚ}/ℚ) → GL_{2}(T_{Q}^{φ}),

with det ρ′_{T_{Q}^{φ}}(Frob_{ℓ}) = ℓ^{k−1}⟨ℓ⟩ and U(q)|_{T_{Q}^{φ}} ≡ \overline{α}_{q} mod 𝔪 for the maximal ideal

𝔪 = 𝔪_{T_{Q}^{φ}} of T_{Q}^{φ}. Thus the choice of T_{Q}^{φ} depends on the choice of the root \overline{α}_{q} for each q ∈ Q, but the isomorphism class of the O–algebra T_{Q}^{φ} and that of the Galois representation ρ′ are independent of the choice, because the algebra is generated by T (ℓ) for primes ℓ with ℓ ∤ pN_{Q} and the Galois representation is determined by Tr(ρ′(Frob_{ℓ})) = T (ℓ). Since det ρ′ ≡ φ mod 𝔪, the ratio φ/ det ρ′ has a p–power order. Since p is odd, we have a unique \sqrt{φ/ det ρ′}. We define

ρ_{T_{Q}^{φ}} = \sqrt{φ/ det ρ′} ⊗ ρ′_{T_{Q}^{φ}},

which satisfies det ρ_{T_{Q}^{φ}} = φ and ρ_{T_{Q}^{φ}} ≡ \overline{ρ}_{λ} mod 𝔪.

We can now state the horizontal control on the Hecke side:

Theorem 11 (Horizontal control: Hecke side). Fix φ = ν^{k−1} for k ≥ 2. Then

• T_{Q}^{φ}/𝔄_{Q}T_{Q}^{φ} ⟶^{∼} T^{φ}, where 𝔄_{Q} is the augmentation ideal inside O[Δ_{Q}],

• T_{Q}^{φ} is O[Δ_{Q}]-free of rank d_{Q}, say, and,

• d_{Q} = d is independent of Q.

4.3 Proof of Theorem 11

In this section, we shall give a brief sketch of the proof of Theorem 11, referring the reader to [H00b] III.1-2 for more details. For simplicity, we assume an extra condition: k ≥ 3, in order not to worry about the reduction modulo p of modular curves of level divisible by p.

Let W be a complete d.v.r. over ℤ_{p} with p ≥ 5. Let S be a scheme over W. Let E ⟶^{π} S be an elliptic curve over S. Assume that π_{∗}Ω_{E/S} = O_{S}ω. Consider the functor

W − Schemes ⟶ Sets
S ↦ [(E, ω)_{/S}],

where [ ] means the set of isomorphism classes. This functor is represented by the affine scheme M = Spec(A) where

A = W[g_{2}, g_{3}, \frac{1}{Δ}],

with Δ = 27g_{2}^{3} − g_{3}^{2}. Each W–algebra homomorphism ϕ : A → A gives rise to the elliptic curve E_{/A} : y^{2} = 4x^{3} − ϕ(g_{2})x − ϕ(g_{3}) with differential ω = \frac{dx}{y}.

Now say S = Spec(A) is affine. Then λ ∈ G_{m}(S) = A^{×} acts on the above functor via

(E, ω) ↦ (E, λω).

It therefore acts on A. Let A_{j} denote the corresponding eigenspace under the action of G_{m} corresponding to λ · x = λ^{−j}x. Then we may decompose (e.g.

[H00b] I.6.5)

A = ⊕_{j∈ℤ}A_{j}.

Let \overline{A} = W[g_{2}, g_{3}] ⊂ A, and let \overline{A}_{j} be the integral closure of A_{j} in A. We have

\overline{A} = ⊕_{j≥0}\overline{A}_{j}

Note that g_{2} ∈ \overline{A}_{4} and g_{3} ∈ \overline{A}_{6}.

Definition 1. For any W–algebra A, a modular form of weight k integral over A is an element f ∈ \overline{A}_{k} ⊗_{W} A.

This definition agrees with the earlier definition of modular forms of level one, since G_{k}(1, 1, W) = \overline{A}_{k} is classically known to be spanned over W by the monomials g_{2}^{a}g_{3}^{b} for 4a+6b = k, where g_{2} (resp. g_{3}) is the weight 4 (resp. weight 6) Eisenstein series in G_{k}(SL_{2}(ℤ)) normalized so that its constant term is equal to 1. Polynomials of g_{2} and g_{3} with this skewed homogeneity condition: 4a+6b = k are called isobaric polynomials of weight k. The above definition includes A = W/p^{n}W, giving a geometric definition of modular forms (independent of its analytic behavior). We write G_{k}(1, 1, A) for \overline{A}_{k} ⊗_{W} A, which is consistent with our earlier definition.

Let (𝔼, ω) → M denote the universal elliptic curve, defined over A by the equation Y^{2} = 4X^{3} − g_{2}X − g_{3} with ω = \frac{dX}{Y}. For each elliptic curve (E, ω)_{/S}, with S = Spec(A), there is a unique morphism ι : S → M such that

(E, ω) = (S ×_{ι,M} 𝔼, ι^{∗}ω).

We define f(E, ω) = i^{∗}(f) ∈ A, where ι^{∗} : A → A is the pullback map of ι. Note that this makes f a function on pairs (E, ω)_{/S=Spec(A)} such that

• f(E, ω) ∈ A, • we have ϕ(f(E, ω)) = f((E, ω) ⊗_{A,ϕ} B) if ϕ : A → B is a W–algebra

homomorphism,

• the values of f(E, ω) only depend on the isomorphism class of (E, ω), and,

• f(E, λω) = λ^{−k}f(E, ω), for λ ∈ A^{×}.

By Fourier expansion, we can embed A into W((q)) = W[[q]][\frac{1}{q}]. Then we define T ate(q) = 𝔼⊗_{A}W((q)) ([T]). We write ω_{∞} for the image of ω on T ate(q). The curve T ate(q) is canonically embedded into the projective space P_{/W((q))}^{2} using the equation Y^{2}Z = 4X^{3} − g_{2}XZ^{2} − g_{3}Z^{3}. Since the coefficients of the above equation is contained in W[[q]], the curve T ate(q)_{/W((q))} canonically extends to a proper flat curve T ate′(q)_{/W[[q]]} defined by the same equation. Removing singular locus of T ate′(q) concentrated at q = 0, we get a flat group scheme (T ate(q), ω_{∞})_{/W[[q]]}, which is called the Tate curve. Its special fiber at q = 0 is isomorphic to G_{m/W} as easily seen (by manipulation of the equation; see [H00b] II.5); so, T ate(q)_{/W[[q]]} is no longer an elliptic curve, but, removing the fiber G_{m} at q = 0, T ate(q)_{/W((q))} is an elliptic curve. Writing the special fiber G_{m} at q = 0 as Spec(W[t, t^{−1}]) (so, t is the local parameter of G_{m} at the origin), we find ω_{∞} = \frac{dt}{t}. By evaluating f ∈ G_{k}(1, 1, A) at (T ate(q), ω_{∞})_{/W((q))}, we get the q–expansion f(q) = f(T ate(q), ω_{∞}) ∈ W((q)). Since G_{k}(1, 1, A) is made up of isobaric polynomials of g_{2} and g_{3} of weight k, the expansion coincides with the analytic Fourier expansion by substituting exp(2πiz) for q if A is embedded into ℂ, and the q–expansion f(q) actually falls in A[[q]]. If it falls further in qA[[q]], we call f a cusp form. We write S_{k}(1, 1, A) for the subspace of cusp forms.

Now let N be a positive integer prime to p. We introduce a level N structure. Consider the functors

℘_{N} : W − Schemes ⟶ Sets
S ↦ [(E, ϕ_{Γ(N)} : (ℤ/Nℤ)^{2} ⟶^{∼} E[N], ω)_{/S}],
℘_{Γ_{1}(N)} : W − Schemes ⟶ Sets,
S ↦ [(E, ϕ_{Γ_{1}(N)} : ℤ/Nℤ ↪ E[N], ω)_{/S}], and,
℘_{Γ_{0}(N)} : W − Schemes ⟶ Sets
S ↦ [(E, C, ω)_{/S}],

where [ ] = { }/ ≅ denotes isomorphims class and C is a subgroup scheme of E[N] defined over S, cyclic of order N (that is, C ≅ ℤ/Nℤ after changing base to a finite étale extension of S). We often use the symbol: ϕ_{Γ_{0}(N)} to indicate the level structure “C” of Γ_{0}(N)–type (to make our notation consistent with the other level structures of Γ_{1}(N) and Γ(N) type).

These functors are representable by the schemes M_{N} = M_{Γ(N)} = Spec(A_{N}), respectively M_{Γ_{1}(N)} = Spec(A_{N}^{Γ_{1}(N)}), M_{Γ_{0}(N)} = Spec(A_{N}^{Γ_{0}(N)}). Here the superscript “Γ” of A_{N}^{Γ} indicates symbolically the subring fixed by the modular group Γ. Define the ‘bars’ of these algebras by the integral closure of \overline{A} in each algebra, and put M_{Γ} = Spec(A_{N,0}) and \overline{M}_{Γ} = Proj(\overline{A}_{N}^{Γ}) for Γ = Γ_{0}(N) and Γ_{1}(N). Then \overline{M}_{Γ} − M_{Γ} is the set of cusps of \overline{M}_{Γ}.

Now let Q be a finite set of primes as in Section 4.1. Recall that N_{Q} = ∏_{q∈Q} q. To simplify the notation, we write M_{Q}^{?} and A_{Q}^{?}, etc. for M_{Γ_{?}(N_{Q})} and

A_{N_{Q}}^{Γ_{?}(N_{Q})} in this section. Note that M_{Q}^{1} → M_{Q}^{0} is étale, and moreover that

\overline{M}_{Q}^{1} = Proj(\overline{A}_{Q}^{1}) → Proj(\overline{A}_{Q}^{0}) = \overline{M}_{Q}^{0}

is an étale Galois covering with Galois group (ℤ/N_{Q}ℤ)^{×}, the Galois action of a ∈ (ℤ/N_{Q}ℤ)^{×} being given by (E, ϕ_{N}^{1}, ω) ↦ (E, ϕ_{N}^{1} ∘ a, ω). We remark that étaleness at cusps holds because N_{Q} is square free. This can be easily checked by using the fact that the monodromy group at infinity is given by upper unipotent subgroup of GL_{2}(ℤ/N_{Q}ℤ) (and that Bruhat-Tits decomposition holds for GL_{2}(𝔽_{q}) for primes q ∈ Q). This shows that

H^{0}((ℤ/N_{Q}ℤ)^{×}, \overline{A}_{Q}^{1}/p^{n}\overline{A}_{Q}^{1}) = \overline{A}_{Q}^{0}/p^{n}\overline{A}_{Q}^{0}

for all 0 < n ∈ ℤ. We now make the following definition:

Definition 2. G_{k}(Γ_{?}(N_{Q}), A) = \overline{A}_{Q,k}^{?} ⊗_{W} A = H^{0}(\overline{M}_{Q}^{?}, \underline{ω}^{k}), where \underline{ω}^{k} is the

invertible sheaf over \overline{M}_{Q}^{?} generated by degree k homogeneous elements in \overline{A}_{Q}^{?}.

The subspace of cusp forms S_{k}(Γ_{?}(N_{Q}), A) in G_{k}(Γ_{?}(N_{Q}), A) is defined to be made up of modular forms f whose q–expansion f(T ate(q), ϕ_{N}^{?}, ω_{∞}) is without constant term for any choice of ϕ_{N}^{?}. This definition is again consistent with the earlier one. Note that we have written here S_{k}(Γ_{0}(N_{Q}), A) for S_{k}(N_{Q}, 1, A) defined in the previous section to emphasize two different level groups Γ_{1}(N_{Q}) and Γ_{0}(N_{Q}). Using our new notation, what we have shown is:

H^{0}((ℤ/N_{Q}ℤ)^{×}, G_{k}(Γ_{1}(N_{Q}), A)) = G_{k}(Γ_{0}(N_{Q}), A),
H^{0}((ℤ/N_{Q}ℤ)^{×}, S_{k}(Γ_{1}(N_{Q}), A)) = S_{k}(Γ_{0}(N_{Q}), A)

A variant of this is that if ξ : (ℤ/N_{Q}ℤ)^{×} → A is a character then

S_{k}(Γ_{1}(N_{Q}), A)[ξ] = S_{k}(N_{Q}, ξ, A).

This follows by a similar argument applied to the sheaf \underline{ω}^{k}(ξ) obtained from \underline{ω}_{/\overline{M}_{Q}^{1}}^{k} by twisting it by the character ξ of Gal(\overline{M}_{Q}^{1}/\overline{M}_{Q}^{0}). Applying this to

A = W/p^{n}W and taking injective limit with respect to n, we get

S_{k}(Γ_{1}(N_{Q}), W ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p})[ξ] = S_{k}(Γ_{0}(N_{Q}), ξ, W) ⊗_{ℤ_{p}} (ℚ_{p}/ℤ_{p}).  (4)

We now take W to be O to prove the horizontal control theorem on the Hecke side. Writing the q–expansion at ∞ of a cusp form f as f(q) = ∑_{n=1}^{∞} a(n, f)q^{n}, it is easy to deduce from (1) that the pairing (f, h) ↦ a(1, f|h) induces a perfect Pontryagin duality between h_{k}(Γ_{1}(N_{Q}), O) and S_{k}(Γ_{1}(N_{Q}), O ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p}).

Since k ≥ 3, we know that the projector e induces an isomorphism of the Hecke algebra h_{k}^{ord}(p, Γ_{1}(N_{Q}), ψ, O) onto h_{k}^{ord}(Γ_{1}(N_{Q}), O) for ψ = 1. By duality, (4) may be rephrased as

h_{k}(Γ_{1}(N_{Q}), O)/Σ_{σ∈(ℤ/N_{Q}ℤ)^{×}}(σ − ξ(σ))h_{k}(Γ_{1}(N_{Q}), O) = h_{k}(N_{Q}, ξ, O). (5)

Since T_{Q}^{φ} is local, the algebra homomorphism of O[Δ_{Q}] into T_{Q}^{φ} factors through its augmentation quotient; so, it induces the trivial character on Δ′_{Q}. Thus on T_{Q}^{φ}, what matters is only the action of Δ_{Q}; so, (5) yields

T_{Q}^{φ}/Σ_{σ∈Δ_{Q}}(σ − ξ(σ))T_{Q}^{φ} is a local ring of h_{k}^{ord}(N_{Q}, ξ, O)  (6)

Taking ξ = 1 and applying the fact that the local ring T_{Q}^{φ} of h_{k}(N_{Q}, 1, O) with given eigenvalues \overline{α}_{q} of ρ′_{T_{Q}^{φ}}(Frob_{q}) (q ∈ Q) is isomorphic to T^{φ} in h_{k}(1, 1, O), we conclude

T_{Q}^{φ}/Σ_{σ∈Δ_{Q}}(σ − 1)T_{Q}^{φ} ≅ T^{φ}.

Since O[Δ_{Q}] is a local ring, by Nakayama’s lemma, this shows that the minimal number of generators of T_{Q}^{φ} over O[Δ_{Q}] is d = rank_{O} T^{φ}; so, we have an O[Δ_{Q}]– linear surjection π : O[Δ_{Q}]^{d} → T_{Q}^{φ}. By tensoring

O = O[Δ_{Q}]/Σ_{σ∈Δ_{Q}}(σ − ξ(σ))O[Δ_{Q}]

over O[Δ_{q}] with π, the linear map π induces a surjection

π_{ξ} : O^{d} → T_{Q}^{φ}/Σ_{σ∈Δ_{Q}}(σ − ξ(σ))T_{Q}^{φ}.

Since the right-hand-side is O–free by (6) with minimal number of generators d by Nakayama’s lemma, it has to be O–free of rank d. Thus π_{ξ} is an isomorphism. Since Ker π ⊂ ⋂_{ξ} Ker π_{ξ} = 0 regarding π_{ξ} as a linear map: O[Δ_{Q}]^{d} → T_{Q}^{φ}/Σ_{σ∈Δ_{Q}}(σ − ξ(σ))T_{Q}^{φ}, we know that π is an isomorphism. This finishes the proof.

5 Taylor-Wiles systems

In this section we introduce the commutative algebra machine invented by Taylor and Wiles [TW] (with simplifications due to Faltings, Diamond and Fujiwara) which allows one to deduce the equality of the Hecke algebra and universal deformation ring in Theorem 8. We prefer to work axiomatically and a bit more generally here (over arbitrary number fields rather than ℚ); this section can in fact be read independently of the previous ones. Our treatment follows that of Fujiwara [Fu].

Fix an odd prime p. Let O be an extension of ℤ_{p}. Let F be a number field, and let O_{F} denote the ring of integers of F.

Consider the set Q defined via

Q = {𝔮 ⊂ O_{F} | 𝔮 prime, N𝔮 ≡ 1 (mod p)}.

For each finite subset Q ⊂ Q, let • Δ_{Q} = the p-Sylow subgroup of ∏_{𝔮∈Q}(O_{F} /𝔮)^{×},

• O[Δ_{Q}] = the corresponding group algebra over O, and,

• 𝔄_{Q} the augmentation ideal of O[Δ_{Q}]. Definition 3 (Taylor-Wiles system). For each m = 1, 2, …, fix a finite subset Q_{m} ⊂ Q. A Taylor-Wiles system is a local O-algebra R, and a collection of tuples (R_{m}, T_{m}, M_{m}), indexed by m = 1, 2, …, where

• R_{m} is a local O-algebras, with a map R_{m} → R,

• T_{m} is a local O-algebras, with a map R_{m} → T_{m},

• M_{m} is a T_{m}-module, satisfying:

• R_{m} (so T_{m}) is an O[Δ_{Q_{m}}]-algebra,

• R_{m}/𝔄_{Q_{m}}R_{m} ⟶^{∼} R,

• As an O[Δ_{Q_{m}}]-module, M_{m} is free of finite rank d, which is independent

of m. Theorem 12 (Isomorphism Criterion). Let R, (R_{m}, T_{m}, M_{m}) be a Taylor-Wiles system. Assume that the following conditions hold:

(TW1): If 𝔮 ∈ Q_{m} then N𝔮 ≡ 1 (mod p^{m}), (TW2): |Q_{m}| is independent of m, say this cardinality is r, (TW3): R_{m} is generated by at most r elements as an O-algebra, (TW4): Assume that the kernel of

R → End_{O}(M_{m}/𝔄_{Q_{m}}M_{m})

is independent of m, and let T = R/kernel. Then R is a complete intersection, free of finite rank over O. Moreover, the natural map

R ↠ T,

is an isomorphism. Proof. Fix m. For each 𝔮 ∈ Q_{m}, let Δ_{𝔮}, be the p-Sylow subgroup of (O_{F} /𝔮)^{×}. By (TW1) we know that this is cyclic of order at least p^{m}. Let δ_{𝔮} be a generator of Δ_{𝔮}. Suppose that 1 ≤ n ≤ m. Note that Δ_{𝔮}/ < δ_{𝔮}^{p^{n}} > is cyclic of order p^{n}. Thus,

\frac{O[Δ_{𝔮}]}{(δ_{𝔮}^{p^{n}} − 1)} ⟶^{∼} \frac{O[[S]]}{(1 + S)^{p^{n}} − 1},

via δ_{𝔮} ↦ 1 + S.

By (TW2) we know that the cardinality of Q_{m} is r (and this number is independent of m). Suppose that Q_{m} = {𝔮_{1}, …, 𝔮_{r}}. Let I_{n} ⊂ O[Δ_{Q_{m}}] be the ideal generated by p^{n}, δ_{𝔮_{1}}^{p^{n}} − 1, …, δ_{𝔮_{r}}^{p^{n}} − 1. Then we have

\frac{O[Δ_{Q_{m}}]}{I_{n}} ⟶^{∼} \frac{O[[S_{1}, …, S_{r}]]}{(p^{n}, (1 + S_{1})^{p^{n}} − 1, …, (1 + S_{r})^{p^{n}} − 1)},

via δ_{𝔮_{i}} ↦ 1 + S_{i}, for i = 1, …, r.

Let us set A_{n} = O[Δ_{Q_{m}}]/I_{n}. It is an easy computation to check that |A_{n}| = p^{ntp^{nr}}, where t = rank_{ℤ_{p}}(O). In particular the cardinality of A_{n} is independent of m. Now define

R_{n,m} = image (\frac{R_{m}}{I_{n}R_{m}} → End_{A_{n}} (\frac{M_{m}}{I_{n}M_{m}} )) ,
R̃_{n,m} = \frac{R_{n,m}}{𝔄_{Q_{m}}} .

Since M_{m} is free over O[Δ_{Q_{m}}] of rank d (independent of m) there are natural maps

A_{n} →^{α} R_{n,m} ↪^{β} End_{A_{n}} (\frac{M_{m}}{I_{n}M_{m}} ) = M_{d}(A_{n}).

Since β is injective we see that |R_{n,m}| ≤ |A_{n}|^{d^{2}} is bounded independently of m. Also β ∘ α is injective, so that α is injective as well. This will be used below.

By (TW3), R_{m} is generated by r elements over O, and thus, so is R_{n,m}. Pick generators f_{1}, …, f_{r} of R_{n,m}.

Now fix n, and suppose m, m′ ≥ n. We say two triples

((R_{n,m}, α, β), R̃_{n,m}, (f_{1}, …, f_{r})) and ((R_{n,m′}, α′, β′), R̃_{n,m′}, (f′_{1}, …, f′_{r}))

are isomorphic if there is an isomorphism of A_{n}-algebras

ι : R_{n,m} ⟶^{∼} R_{n,m′}

inducing

ι : R̃_{n,m} ⟶^{∼} R̃_{n,m′},

such that ι(f_{j}) = f′_{j} and such that the following two diagrams commute:

A_{n} →^{α} R_{n,m} ↪^{β} M_{d}(A_{n})
∥  ↓ ι  ∥
A_{n} →^{α} R_{n,m′}  ↪^{β′} M_{d}(A_{n})

and

R_{n,m} ⟶ R̃_{n,m}
↓ ι  ↓ ι
R_{n,m′} ⟶ R̃_{n,m′}.

Since the cardinality of R_{n,m} is bounded independently of m, we see that for n = 1, there exists infinitely m ≥ 1 (call this set ℕ_{1} ⊂ ℕ) such that the

((R_{1,m}, α, β), R̃_{1,m}, (f_{1}, …, f_{r})) ,

are mutually isomorphic.

Suppose, for purposes of induction, that we have constructed ℕ_{n} ⊂ ℕ_{n−1} ⊂ ⋯ ⊂ ℕ_{1} ⊂ ℕ such that

((R_{n,m}, α, β), R̃_{n,m}, (f_{1}, …, f_{r})) ,

are mutually isomorphic for all n ∈ ℕ_{n}. Then, again, by varying m ≥ n + 1 in ℕ_{n} we may pick an infinite subset ℕ_{n+1} ⊂ ℕ_{n}, such that

((R_{n+1,m}, α, β), R̃_{n+1,m}, (f_{1}^{(n)}, …, f_{r}^{(n)})) ,

are mutually isomorphic for m ∈ ℕ_{n+1}.

Now, for each n, let m(n) denote the minimal element in ℕ_{n}, and let the corresponding triple be

((R_{n,m(n)}, α, β), R̃_{n,m(n)}, (f_{1}, …, f_{r})) .

Note that the following diagram commutes:

\frac{R_{m(n+1)}}{I_{n+1}R_{m}(n+1)} ⟶ End_{A_{n+1}} (\frac{M_{m(n+1)}}{I_{n+1}M_{m(n+1)}} )
↓ ^{mod I_{n}}  ↓
\frac{R_{m(n+1)}}{I_{n}R_{m}(n+1)} ⟶ End_{A_{n}} (\frac{M_{m(n+1)}}{I_{n}M_{m(n+1)}} ) .

Moreover R_{n,m(n+1)} (and its tuple) is isomorphic to R_{n,m(n)} (and its tuple) since both m(n + 1) and m(n) ∈ ℕ_{n}. This shows that the process of going modulo I_{n} yields a projective system of tuples

((R_{n,m(n)}, α, β), R̃_{n,m(n)}, (f_{1}^{(n)}, …, f_{r}^{(n)})) ,

as n varies. Denote the projective limit by

((R_{∞}, α_{∞}, β_{∞}), R̃_{∞}, (f_{1}^{(∞)}, …, f_{r}^{(∞)})) .

Let us now set

A := lim_{←} A_{n} = O[[S_{1}, … S_{r}]], and B := O[[T_{1}, … T_{r}]].

By the injectivity of the α and β at each level, we have maps

A ↪^{α_{∞}} R_{∞} ↪^{β_{∞}} M_{d}(A).  (7)

Also we have a map

B ↠ R_{∞}  (8)
T_{i} ↦ f_{i}^{(∞)}.

The maps (7) show that R_{∞} is a torsion free A-module of finite type, and so by the going-up and going-down theorems, the Krull dimension of R_{∞} is given by dim(R_{∞}) = r + 1. This shows that the map (8) must be injective. For if it were not, we could always choose some 0 ≠ f in its kernel K, and a height 1 prime P of B containing f, and we would get the absurdity:

r + 1 = dim R_{∞} = dim B/K ≤ dim B/(f) = dim B/P = r.

This shows that R_{∞} is isomorphic to the power series ring B = O[[T_{1}, … T_{r}]].

We now claim that R̃_{∞} is a complete intersection, free of finite rank over O. To see this note that we have an exact sequence

R_{∞}^{r} ⟶ R_{∞} ⟶ R̃_{∞}
(x_{i}) ↦ ∑ S_{i}x_{i}

This shows that R̃_{∞} is the homomorphic image of a power series ring over O in r variables, with kernel generated by r elements, i.e. that R̃_{∞} is a complete intersection. The fact that R̃_{∞} is free of finite rank over O follows from the following lemma:

Lemma 3. Let A ↪ B be power series rings over O in the same number of variables. Assume that as an A-module B is of finite type. Then B is A-free.

Indeed, in our situation, since A ↪ B ↪ M_{d}(A), B is indeed an A-module of finite type. Thus R_{∞} is A-free (of finite rank), and so R̃_{∞} is O-free (of the same rank). This proves the claim.

We now claim that R̃_{∞} = T. For this we need one more general lemma, (which follows the Auslander-Buchsbaum formula):

Lemma 4. Say A ↪ B are power series rings over O in the same number of variables. Say L is a B-module. Then L is A-free ⟺ L is B-free.

Let L = lim_{←n} M_{m(n)}/I_{n}M_{m(n)} = lim_{←n} A_{n}^{d} = A^{d}. By Lemma 4 L is a free B-module (of finite rank), and so consequently L/I_{n}L = M_{m(n)}/I_{n}M_{m(n)} is R_{∞}/I_{n}R_{∞} = R_{n,m(n)}-free. Now note that generally, for a C-module X, which is free of finite rank, and an idea 𝔄 ⊂ C, C/𝔄 acts faithfully on X/𝔄X, and so we may consider C/𝔄 ⊂ End(X/𝔄X). Applying this to our situation we see

R̃_{n,m(n)} ⊂ End_{O} (M_{m(n)}/(I_{n} + 𝔄_{Q_{m(n)}})M_{m(n)}) .

Now note that the following diagram is commutative (see (TW4)):

R_{m(n)} ↠ R_{n,m(n)} ↠ R̃_{n,m(n)} ⊂ End_{O} (M_{m(n)}/(I_{n} + 𝔄_{Q_{m(n)}})M_{m(n)})
∥  ↑
R_{m(n)} ↠  T  ⊂  End_{O} (M_{m(n)}/𝔄_{Q_{m(n)}}M_{m(n)})

Letting n → ∞, we see that the map R_{∞} ↠ R̃_{∞} factors through T:

R_{∞} ↠ T ↠ R̃_{∞}.  (9)

On the other hand we see that under the map R_{∞} ↠ T, the ideal (T_{1}, …, T_{r}) maps to 0, so that we have a map from R̃_{∞} = R_{∞}/(T_{1}, …, T_{r}) ↠ T. Along with (9) this says that R̃_{∞} ⟶^{∼} T.

We now claim that in fact R ⟶^{∼} T. Let

K_{n} = ker(R_{∞} ↠ R_{n,m(n)}),

and 𝔪_{∞} the maximal ideal of R_{∞}. Then for each N > 0, there exists n(N) >> 0, such that K_{n(N)} ⊂ 𝔪_{∞}^{N}. Consider the following commutative diagram with exact rows:

R_{m(n(N))}^{r}  ⟶  R_{m(n(N))}  ⟶  R  ⟶ 0
↓  ↓  ↓
R_{n(N),m(n(N))}^{r} ⟶ R_{n(N),m(n(N))} ⟶ R̃_{n(N),m(n(N))} ⟶ 0
∥
T/p^{n(N)}T

Tensoring this with R_{∞}/𝔪_{∞}^{N} (over R_{∞}) we get another commutative diagram with exact rows:

(R_{m(n(N))}^{r}/𝔪_{∞}^{N})^{r}  ⟶  R_{m(n(N))}/𝔪_{∞}^{N}  ⟶ R/𝔪_{∞}^{N} ⟶ 0
↓  ↓  ↓
(R_{n(N),m(n(N))}^{r}/𝔪_{∞}^{N})^{r} ⟶ (R_{n(N),m(n(N))}/𝔪_{∞}^{N})^{r} ⟶ T/𝔪_{T}^{N} ⟶ 0,

where 𝔪_{T} is the maximal ideal of T. By our choice of n(N) the first two vertical maps are isomorphisms, so that

R/𝔪_{R}^{N} ⟶^{∼} T/𝔪_{T}^{N},

for all N > 0. Taking limits, we see that R ⟶^{∼} T, and this finishes the proof of the Theorem 12.

Theorem 13 (Freeness Criterion). Let the hypothesis and assumptions be as in Theorem 12. Assume in addition that

(TW5): There is an R-module M, such that M_{m}/𝔄_{Q_{m}}M_{m} =^{∼} M, as R-modules for each m = 1, 2, . . ..

Then M is a free T -module.

Proof. We have

M = lim_{←} M/p^{n}M = lim_{←n} M_{m(n)}/(I_{n} + 𝔄_{Q_{m(n)}})M_{m(n)} = L/(S_{1}, … S_{r})L.

But L is a free R_{∞}-module (of finite rank) and R̃_{∞} = R_{∞}/(S_{1}, … S_{r})R_{∞} = T. So M is a free T -module (of finite rank).

6 Proof of Theorem 8

We use the isomorphism criterion of Theorem 12 to give a proof of Theorem 8.

In our proof, the Galois cohomological argument is a version of the argument of Wiles in [W] Chapters 1 and 3. In [TW], this type of argument is applied to the Taylor-Wiles system obtained from the topological cohomology groups M_{Q} = H^{1}(X_{1}(pN_{Q}), ℤ_{p}); so, a new aspect of the proof here is the use of the Hecke algebra T_{Q}^{φ} in place of the cohomology groups.

We refer the reader to [H00a] 3.2.7-8 for further details of the argument and to [H00a] 4 for the Tate duality theorem, Poitou-Tate exact sequence, local and global Euler characteristic formulas etc. that we use below.

Let F = ℚ. Let p ≥ 5 and O/ℤ_{p} be as in previous sections. Let

Q_{m} = \{q ∈ Q | q ≡ 1 (mod p^{m}) and \overline{ρ}_{λ}(Frob_{q}) has distinct eigenvalues\}.

It is a fact (shown by Wiles) that under the condition (ai_{p}(κ)), there are infinitely many primes in Q_{m}.

We now show that there exist sets Q_{m} ⊂ Q_{m} for m = 1, 2, 3, …, satisfying (TW1), (TW2) and (TW3), such that, if, in the notation of Sections 3.4 and 4,

R := R^{φ}, R_{m} := R_{Q_{m}}^{φ} and T_{m} := M_{m} := T_{Q_{m}}^{φ},

with φ = ν^{k−1}, then, {R, (R_{m}, T_{m}, M_{m})_{m∈ℕ}} is a Taylor-Wiles system in the sense of Definition 3. In fact many of the requirements of the definition have already been checked (see in particular Proposition 2 and Theorem 11). Indeed Theorem 11 shows that (TW5), and therefore (TW4), holds.

As a result we will get, by Theorem 12, that:

R^{φ} ⟶^{∼} T^{φ}

which is the desired conclusion of Theorem 8.

Fix m, and fix a finite set Q ⊂ Q_{m} of arbitrary cardinality. Denote by ℚ^{Q∪{p}} the maximal extension of ℚ unramified outside p, ∞ and the primes in Q. Let 𝔊_{Q} = Gal(ℚ^{Q∪p}/ℚ). Let W = Ad^{0}(\overline{ρ}_{λ}) be the three dimensional 𝔊_{Q}- module over 𝔽, arising from the adjoint representation on the 2 by 2 matrices over 𝔽 of trace 0.

For each q ∈ Q ∪ {p}, let B_{q} be an 𝔽-subspace of H^{1}(D_{q}, W). We define the Selmer group attached to the data

• the 𝔊_{Q}-module W, and,

• the local conditions B_{q}, for q ∈ Q ∪ {p},

as follows. Set

Sel_{Q}(W) := β_{Q}^{−1} ( ∏_{q∈Q∪{p}} B_{q}) ,

where β_{Q} is the natural restriction map:

β_{Q} : H^{1}(𝔊_{Q}, W) ⟶ ∏_{q∈Q∪{p}} H^{1}(D_{q}, W).

Eventually we will choose the following local conditions in the definition of the Selmer group. We will assume that

• B_{q} = H^{1}(D_{q}, W), for each q ∈ Q, and,

• B_{p} is given by

B_{p} = ker (H^{1}(D_{p}, W) → H^{1}(I_{p}, W/W_{0})) ,

where W_{0} is the sub-representation of W consisting of matrices of the form

( \begin{matrix}0 & ∗ \\ 0 & 0 \end{matrix} ).

This is because of the following lemma.

Lemma 5. Let R_{Q} denote the universal deformation ring (with fixed determinant) as above. Let t_{Q} = 𝔪_{R_{Q}}/𝔪_{R_{Q}}^{2} + 𝔪_{O} be the cotangent space of R_{Q}. Then with the choice of local conditions B_{q} as above, we have

Hom_{𝔽}(t_{Q}, 𝔽) ⟶^{∼} Sel_{Q}(W).

In particular, the number of generators of R_{Q} as an O-algebra is equal to dim Sel_{Q}(W).

This is a standard formula for the tangent space of a universal deformation ring. One can find a proof of this fact and related topics in [H00a] 2.3.4, 3.2.4 and 5.2.4.

Now let us consider the Selmer group attached to the dual representation of W. Set W^{∗} = Hom_{𝔽}(W, 𝔽) and set W^{∗}(1) = W^{∗} ⊗_{𝔽} μ_{p}. (In our case since W ⊂ M_{2}(𝔽), the non-degenerate pairing (X, Y) ↦ Tr(XY) shows that W^{∗} ⟶^{∼} W. Thus we have W^{∗}(1) ⟶^{∼} W (1)).

Theorem 14 (Local Tate duality). For r = 0, 1, 2, there is a natural duality between H^{2−r}(D_{q}, W) and H^{r}(D_{q}, W^{∗}(1)) = H^{r}(D_{q}, W (1)).

For an arbitrary subspace B_{q} ⊂ H^{1}(D_{q}, W), with q ∈ Q ∪ {p}, let B_{q}^{⊥} ⊂ H^{1}(D_{q}, W^{∗}(1)) denote the orthogonal compliment of B_{q} under the pairing of Theorem 14. Define the Selmer group Sel_{Q}(W^{∗}(1)) with respect to the subspaces B_{q}^{⊥} as follows. For the restriction map

β_{Q}^{∗} : H^{1}(𝔊_{Q}, W^{∗}(1)) ⟶ ∏_{q∈Q∪{p}} H^{1}(D_{q}, W^{∗}(1)),

Sel_{Q}(W^{∗}(1)) := β_{Q}^{∗}^{−1} ( ∏_{q∈Q∪{p}} B_{q}^{⊥}) .

Note that for the eventual choice of B_{q} that we have mentioned above, we have B_{q}^{⊥} = 0 for q ∈ Q.

Theorem 15 (Poitou-Tate exact sequence). Let B_{q} be arbitrary subspaces of H^{1}(D_{q}, W), for q ∈ Q ∪ {p}. Then, there is an exact sequence

0 → Sel_{Q}(W) → H^{1}(𝔊_{Q}, W) → ∏_{q∈Q∪\{p\}} \frac{H^{1}(D_{q}, W)}{B_{q}} → Sel_{Q}(W^{∗}(1))

→ H^{2}(𝔊_{Q}, W) → ∏_{q∈Q∪{p}} H^{2}(D_{q}, W) → H^{0}(𝔊_{Q}, W^{∗}(1))^{∗} → 0.

Note that the condition (ai_{p}(κ)) holds ⟺ W is an absolutely irreducible 𝔊_{Q}-module. Consequently

H^{0}(𝔊_{Q}, W) = 0 = H^{0}(𝔊_{Q}, W^{∗}(1))^{∗}.  (10)

We need another formula:

Theorem 16 (Global Euler characteristic formula).

dim H^{0}(𝔊_{Q}, W) − dim H^{1}(𝔊_{Q}, W) + dim H^{2}(𝔊_{Q}, W)
= dim H^{0}(Gal(ℂ/ℝ), W) − dim W.

Let c denote complex conjugation. Since det \overline{ρ}_{λ}(c) = −1, we have \overline{ρ}_{λ}(c) ∼ ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), so that the eigenvalues of c on W are −1, +1, −1. This means that the value of the expression on the RHS of Theorem 16 is = 1 − 3 = −2.

Now let us assume that we have made the choice of the local conditions at q ∈ Q ∪ {p}, as mentioned above. We set

d_{Q} = dim Sel_{Q}(W) − dim Sel_{Q}(W^{∗}(1)).

By the Poitou-Tate exact sequence, the global Euler characteristic formula, local Tate duality, and (10), we have:

d_{Q} = h_{p} + 2 + ∑_{q∈Q} dim H^{0}(D_{q}, W^{∗}(1)),  (11)

where

h_{p} = dim H^{0}(D_{p}, W^{∗}(1)) − dim \frac{H^{1}(D_{p}, W)}{B_{p}}

depends only on p. We now impose an additional condition: (Sel) Assume that φ = ν^{k−1} with k ≢ 2 (mod p − 1). Proposition 3. Recall that Q ⊂ Q_{m}. Assume that (Sel) holds. We have

1. h_{p} + 2 ≤ 0,

2. dim H^{0}(D_{q}, W^{∗}(1)) = 1, for all q ∈ Q. Proof. Since the cohomological dimension of D_{p} is 2, the exact sequence

0 → W_{0} → W → W/W_{0} → 0,

yields the following long exact sequence in cohomology:

0 → image(u) → H^{1}(D_{p}, W/W_{0}) → H^{2}(D_{p}, W_{0})

→ H^{2}(D_{p}, W) → H^{2}(D_{p}, W/W_{0}) → 0

where u is the map defined by

u : H^{1}(D_{p}, W) → H^{1}(D_{p}, W/W_{0}).

Thus we have, dim image(u) =

dim H^{1}(D_{p}, \frac{W}{W_{0}} ) − dim H^{2}(D_{p}, W_{0}) + dim H^{2}(D_{p}, W) − dim H^{2}(D_{p}, \frac{W}{W_{0}} ).

Now consider the following commutative diagram

H^{1}(D_{p}, W)

↓ u  ↘ δ
0 → H^{1}(D_{p}/I_{p}, \frac{W}{W_{0}} )^{I_{p}} → H^{1}(D_{p}, \frac{W}{W_{0}} ) → H^{1}(I_{p}, \frac{W}{W_{0}} )^{D_{p}} → 0,

where the bottom row comes from the inflation-restriction sequence, and the map δ is defined by the triangle above. Note that by the definition of the Selmer group,

h_{p} = dim H^{0}(D_{p}, W^{∗}(1)) − dim image(δ).

In any case we see that dim image(δ) ≥ dim image(u) − dim H^{1}(D_{p}/I_{p}, W/W_{0})^{I_{p}}.

But

H^{1}(D_{p}/I_{p}, W/W_{0})^{I_{p}} ⟶^{∼} (W/W_{0})^{I_{p}}/(Frob_{p} − 1)(W/W_{0})^{I_{p}}.

Moreover there is an exact sequence

0 → H^{0}(D_{p}/I_{p}, W/W_{0})^{I_{p}} → (W/W_{0})^{I_{p}} ⟶^{Frob_{p}−1}

(W/W_{0})^{I_{p}} → H^{1}(D_{p}/I_{p}, W/W_{0})^{I_{p}} → 0.

This shows that

dim H^{1}(D_{p}/I_{p}, W/W_{0})^{I_{p}} = dim H^{0}(D_{p}/I_{p}, W/W_{0})^{I_{p}} = H^{0}(D_{p}, W/W_{0}).

Putting things together we get:

dim image(δ) ≥ dim H^{1}(D_{p}, W/W_{0}) − dim H^{2}(D_{p}, W_{0})
+ dim H^{2}(D_{p}, W) − dim H^{2}(D_{p}, W/W_{0}) − dim H^{0}(D_{p}, W/W_{0}).

Proposition 4 (Local Euler characteristic formula).

dim H^{0}(D_{p}, \frac{W}{W_{0}} ) − dim H^{1}(D_{p}, \frac{W}{W_{0}} ) + dim H^{2}(D_{p}, \frac{W}{W_{0}} ) = dim(\frac{W}{W_{0}} ) = 2.

Consequently we have:

h_{p} = dim H^{0}(D_{p}, W^{∗}(1)) − dim image(δ)
≤ dim H^{0}(D_{p}, W^{∗}(1)) − 2 + dim H^{2}(D_{p}, W_{0}) − dim H^{2}(D_{p}, W)
= −2 + dim H^{0}(D_{p}, W^{∗}(1)),

where the last equality follows by local Tate-duality (applied twice!)

We now claim that when (Sel) holds, H^{0}(D_{p}, W^{∗}(1)) = 0. Note that part (1) of the proposition then follows immediately. To see this note that by (Sel) we may write

\overline{ρ}_{λ|I_{p}} = (\begin{matrix}ψ & ∗ \\ 0 & 1 \end{matrix}) ,

where ψ is a non-trivial power of the Teichmüller character. Thus we see that H^{0}(D_{p}, W^{∗}(1)) = 0.

As for part (2), note that q ∈ Q ⟹ q ≡ 1 (mod p), so W^{∗}(1) = W (1) = W. Consequently since the eigenvalues of Frob_{q} on W are ᾱ_{q}/β̄_{q}, 1, and β̄_{q}/ᾱ_{q}, we see that dim H^{0}(D_{q}, W^{∗}(1)) = 1 by (reg_{q}).

Remark 4. The condition (Sel) is not necessary in the sense that the Proposition also holds under other hypothesis.

Corollary 1.

d_{Q} ≤ |Q|.

Proof. This follows immediately from (11) and Proposition 3.

Let

r := dim Sel_{∅}(W^{∗}(1))

be the dimension of the Selmer group when Q = ∅. Our next objective is to show that for each m, we may choose Q ∈ Q_{m} above such that |Q| = r, and so that Sel_{Q}(W^{∗}(1)) = 0. Lemma 5 and Corollary 1 will then show that these choices of Q will satisfy (TW1 - 3). Indeed, since Sel_{Q}(W^{∗}(1)) = 0, we see that d_{Q} = dim t_{Q}, and so each R_{Q} will be generated by at most r elements as an O-algebra.

Proposition 5. There are infinitely many sets Q ∈ Q_{m} such that

1. |Q| = r = dim Sel_{∅}(W^{∗}(1)), and,

2. Sel_{Q}(W^{∗}(1)) = 0.

Proof. let Q ∈ Q_{m} be arbitrary. Since B_{q}^{⊥} = 0 for all q ∈ Q, we have

Sel_{Q}(W^{∗}(1)) ⊂ ker β′_{Q}

where

βQ^{′} : H^{1}(𝔊_{Q}, W^{∗}(1)) ⟶ ∏_{q∈Q} H^{1}(D_{q}, W^{∗}(1))

is the map obtained from β_{Q}^{∗} by ignoring the restriction at p. In fact one may check that ker β_{Q} ⊂ ker β_{Q}^{†}, where

β_{Q}^{†} : H^{1}(𝔊_{∅}, W^{∗}(1)) ⟶ ∏_{q∈Q} H^{1}(D_{q}, W^{∗}(1))

We now show that when the cardinality of |Q| is sufficiently large, β_{Q}^{†} is injective. This will force the vanishing of Sel_{Q}(W^{∗}(1)) when |Q| is sufficiently large.

First let us introduce some notation. Let 𝔥_{Q} = Gal(ℚ^{Q∪p}/K(μ_{p})) ⊂ 𝔊_{Q}. When Q = ∅, we write 𝔊 (respectively 𝔥) for 𝔊_{Q} (respectively 𝔥_{Q}). Let K be the fixed field of ker \overline{ρ}_{λ}, and let G = Gal(K(μ_{p})/ℚ).

It is a fact that

H^{1}(G, W (1)) = 0.

Consequently, the inflation-restriction sequence yields:

0 = H^{1}(G, W (1)) → H^{1}(𝔊, W (1)) →^{ι} H^{1}(𝔥, W (1))^{G} = Hom_{G}(𝔥, W (1)).

Now fix 0 ≠ x ∈ ker β_{Q}^{†}, and let f = f_{x} : 𝔥 → W (1) be the image of x under the map ι.

Since \overline{ρ}_{λ} satisfies (ai_{p}(κ)), we see that \overline{ρ}_{λ} is not induced from κ. Consequently the image of \overline{ρ}_{λ} in PGL_{2}(𝔽) has cardinality divisible by p or is isomorphic to one of the groups A_{4}, S_{4} or A_{5}. A lemma of Wiles then shows that there is an element σ ∈ 𝔊 such that (1) \overline{ρ}_{λ}(σ) has order ℓ, where ℓ ≥ 3 is a prime different from p, and,

(2) σ fixes ℚ(μ_{p^{m}}) (so det \overline{ρ}_{λ}(σ) = 1).

We claim that we may additionally assume that

(3) f(σ^{ℓ}) ≠ 0.

To see this, let L be the fixed field of the kernel of f. Then X = Gal(L/K(μ_{p})) is an abelian extension of exponent p. We have the exact sequence

0 → X → Gal(L/ℚ) → G → 0.

Note that G acts on X via conjugation, and f : X → W (1) = W is an isomorphism of G modules (note that f ≠ 0, f is injective, and W is irreducible, so f is surjective as well). Now let σ′ satisfy the conditions (1) and (2) above. Then Ad(\overline{ρ}_{λ})(σ′) has three distinct eigenvalues on W and therefore on X, and one of them is equal to 1. Write

X = X[1] ⊕ X′,

where X[1], X′ ≠ 0, σ′ = 1 on X[1] and σ′ − 1 is an automorphism of X′. Since f is an isomorphism, we see that f(X[1]) ≠ 0. Thus we may choose an element τ ∈ X[1] such that f(τ) ≠ 0.

Since K(μ_{p^{m}})/K(μ_{p}) is a prime to p extension, we see that L and K(μ_{p^{m}}) are linearly disjoint over K(μ_{p}). We now set

τ′ := 1 × τ ∈ Gal(K(μ_{p^{m}})/K(μ_{p})) × Gal(L/K(μ_{p})) = Gal(L(μ_{p^{m}})/K(μ_{p})).

Since τ ∈ X[1], τ′ commutes with σ′ in Gal(L(μ_{p^{m}})/ℚ). Then, noting that σ′^{ℓ} ∈ X, we have

f((τ′σ′)^{ℓ}) = f(τ′^{ℓ}σ′^{ℓ}) = ℓf(τ′) + f(σ′^{ℓ}).

Since ℓ ≠ p, ℓf(τ′) ≠ 0, and so one of τ′σ′ or σ′ satisfies (1), (2) and (3) above. This proves the claim.

Now choose a prime q ∉ Q such that Frob_{q} = σ in Gal(L(μ_{p^{m}})/ℚ). Then the fact that f(Frob_{q}^{ℓ}) ≠ 0 implies that β_{Q∪{q}}(x) ≠ 0. By (2) we see that q ≡ 1 mod p^{m}. Further (1) and (2) imply that the characteristic roots α and β of \overline{ρ}_{λ}(σ) satisfy αβ = 1 and hence are primitive ℓ^{th} roots of unity. Since ℓ ≥ 3, we have α ≠ β. This shows that

Q ∪ {q} ∈ Q_{m}.

We have shown that for 0 ≠ x ∈ ker β_{Q}, there is a prime q ∉ Q, such that Q ∪ {q} ∈ Q_{m}, and β_{Q∪{q}}(x) ≠ 0. Iterating this statement, we may assume that Q ∈ Q_{m} is such that β_{Q}^{†} is injective.

Now, by the local Euler characteristic formula, dim H^{1}(D_{q}, W^{∗}(1)) = 2. So we may now remove primes from Q, one at a time, preserving the injectivity of β_{Q}^{†}, at least until we reach |Q| = r. This finishes the proof of the proposition.

Since we have proved Proposition 5 we have in fact completed the proof of Theorem 8 as well.

7 Proof of Vertical control theorem

Three different methods of proof are known:

1. Hida Inventiones 1986 [H86b]

2. Hida Ann. E. N. S. 1986 [H86a]

3. Wiles 1989 (see for example, [H93] Chapter 7)

We will follow the second approach since it seems the most susceptible to generalization to modular forms on Shimura varieties (see [H99a]). This approach is via the moduli of elliptic curves. In this section, we do not assume N = 1 and p ≥ 5. The theory of p–adic modular forms was initiated by Serre in the early 70’s, and Deligne and Katz later gave a geometric interpretation (in terms of moduli of elliptic curves), which we adopt for our proof of vertical control theorems. Let us start with some background.

7.1 Modular definition of modular forms

We use the notation used in Section 4.3. Say that A is a local W-algebra, and E is an elliptic curve over S = Spec(A). Let

E[p^{n}] = {x ∈ E|p^{n}x = 0}

be the group scheme of p^{n}-division points of E. This is a finite flat group scheme over A. Now there are two possibilities:

E[p^{n}](\overline{A/𝔪_{A}}) ≅ \begin{cases}0 & the supersingular case, or, \\ ℤ/p^{n}ℤ & the ordinary case, \end{cases}

where \overline{A/𝔪_{A}} is an algebraic closure of A/𝔪_{A} and E[p^{n}](\overline{A/𝔪_{A}}) is the set of points of E[p^{n}] with coordinates in \overline{A/𝔪_{A}}. We now remove the super-singular elliptic curves from (𝔼, ω) → M. This is done via the Hasse invariant which is defined as follows. Fix (E, ω)_{/A}.

1. Assume char(A) = p. We have H^{0}(E, Ω_{E/A}) = A · ω. By Serre duality H^{1}(E, O_{E}) = A · η(ω). One checks that

η(λω) = λ^{−1}η(ω).

Now let Frob : A → A be the map x ↦ x^{p}. Then Frob^{∗}η(ω) = H(E, ω) · η(ω). Then H(E, ω) ∈ A and is called the Hasse invariant of (E, ω).

Lemma 6. H is a modular form of weight p − 1 defined over 𝔽_{p}.

Proof: We have

H(E, λω)η(λω) = Frob^{∗}(η(λω))
= Frob^{∗}(λ^{−1}η(λ))
= λ^{−p}Frob^{∗}η(ω)
= λ^{−p}H(E, ω)η(ω)
= λ^{1−p}H(E, ω)η(λω).

Thus we have shown that H ∈ A_{p−1}. In fact H(E, ω) ∈ \overline{A}_{p−1}. Moreover H(E, ω) = 0 ⟺ E is super-singular. We set

M^{ord} = Spec ((\overline{A} ⊗_{W} 𝔽)[1/H])

which is a scheme over 𝔽 = W/𝔪_{W}.

2. Assume char(A) = 0. Now ℙ(j)_{/W} = Proj(\overline{A}). We have

H^{0}(ℙ(j)_{/W}, \underline{ω}_{/W}^{k}) ⊗_{W} 𝔽 = H^{0}(ℙ(j)_{/A_{N,k}𝔽}, \underline{ω}_{/𝔽}^{k})

for k ≥ 4. Let E be the lift of H^{j} (j = 4, 2, 1 according to p = 2, 3 or ≥ 5), and we can choose k = p − 1 if p ≥ 5. Then E is a weight p − 1 Eisenstein series with constant term 1, and E ≡ 1 mod p. When p = 2, 3, we can choose E to be g_{2} of weight 4. In this case we set

M_{/W}^{ord} = Spec (\overline{A}[1/E])

Thus M_{/W}^{ord} ⊗_{W} 𝔽 = M_{/𝔽}^{ord}.

Now let us introduce a level. Let N ≥ 3 be such that (N, p) = 1. For the universal elliptic curve (𝔼, ω) → M, set 𝔼[N] be the groups scheme of N-division points. This is an étale group scheme over M. Now set

M_{N} = Isom_{M}(𝔼[N], (ℤ/Nℤ)_{/M}^{2})

which is a scheme over W, and is the moduli for tuples

(E, ϕ_{N} : E[N] =^{∼} (ℤ/Nℤ)^{2}, ω).

We write M_{N} = Spec(A_{N}). Note that M_{N} → M is étale and finite. Note that λ ∈ G_{m} acts via

(E, ϕ_{N}, ω) ↦ (E, ϕ_{N}, λω).

We have A_{N} = ⊕_{k∈ℤ}A_{N,k}, where A_{N,k} are the eigenspaces for the G_{m} action. We let \overline{A}_{N}, respectively \overline{A}_{N,k}, be the integral closure of \overline{A}, respectively of \overline{A}_{k}, in A_{N}.

Let us set M_{N} = Spec(A_{N,0}), and \overline{M}_{N} = Proj(\overline{A}_{N}). Then \overline{M}_{N} − M = set of cusps. The universal curve 𝔼 does not depend on the choice of ω; so, it is actually defined over M_{N}.

We now introduce level p^{n} structure. Set M_{N}^{ord} = Spec(A_{N}[1/E]_{0}) = M_{N}[\frac{1}{E}]

and \overline{M}_{N}^{ord} = Spec(\overline{A}_{N}[1/E]_{0}) = \overline{M}_{N}[\frac{1}{E}], where the subscript “0” indicate the

degree 0 component. Now set S_{m} = \overline{M}_{N}^{ord} ⊗_{W} W_{m} where W_{m} = W/p^{m}. Recall that there is a connected-étale exact sequence of group schemes:

0 → 𝔼^{ord,conn}[p^{n}] → 𝔼^{ord}[p^{n}] → 𝔼^{ord,et}[p^{n}] → 0.

The universal curve 𝔼 extends to \overline{M}^{ord} so that it is isomorphic to a Tate curve on the formal infinitesimal neighborhood of the cusps. Since the Tate curve is a group scheme with singular fibers given by G_{m}, the connected component 𝔼^{ord,conn}[p^{n}] of 𝔼[p^{n}]_{/\overline{M}^{ord}} is a well defined finite flat group scheme. Take the Cartier dual of 𝔼^{ord,conn}[p^{n}], which extends the étale finite group scheme 𝔼^{ord,et}[p^{n}] to \overline{M}^{ord}, for which we use again the symbol 𝔼^{ord,et}[p^{n}]_{/\overline{M}^{ord}}. We now set

T_{m,n} = Isom_{S_{m}}(𝔼^{ord,et}[p^{n}], (ℤ/p^{n}ℤ)),

and write T_{m,n} = Spec(V_{m,n}). Let us set S_{∞} = lim_{←} S_{m}. Finally set V = lim_{←} V_{m,∞}.

Definition 4. We call

• elements in H^{0}(\overline{M}_{N}, \underline{ω}^{k}) (true) modular forms of weight k,

• elements in H^{0}(S_{∞}, \underline{ω}^{k}) false modular forms of weight k, and,

• elements of V, p-adic modular forms

Note that we have the inclusions:

{true mod. forms} ⊂ {false mod. forms} ⊂ {p − adic mod. forms}

As we will see later, true modular forms are dense in the space of p–adic modular forms (see Theorem 20).

7.2 False modular forms

In this subsection, we explore false modular forms in a more abstract setting, since the first half of the proof of the vertical control theorem is formal and is best understood in a formal setting. The theory of false modular forms was invented by Deligne and Katz in the early 70’s ([K] and [K1]), which has been generalized to modular forms on Shimura varieties in [H99a]. We follow the treatment of [K] Appendix.

In this subsection, we fix the level subgroup Γ which is one of Γ_{0}(N), Γ_{1}(N) and Γ(N). We write \overline{M} for \overline{M}_{Γ/W}. We set \overline{M}_{m} to be \overline{M} ⊗_{W} W_{m} for W_{m} = W/p^{m}W and define S_{m} by \overline{M}_{m}[\frac{1}{E}] for the lift E of the Hasse invariant H. Thus S_{m} is an affine scheme.

Let us give some background on locally constant sheaves (or étale sheaves as they are often referred to in other places; see [Mi] II.1 for details). Let Schemes_{/S_{m}} (resp. AB) be the category of schemes over S_{m} (resp. the category of abelian groups). For U ∈ Schemes_{/S_{m}}, an étale covering {U_{i} → U}_{i} of U is a collection of étale S_{m}–morphisms U_{i} → U such that the image of U_{i} in aggregate covers S_{m} (⇔ ⨆_{i} U_{i} is faithfully flat over S_{m}). A locally constant sheaf F over S_{m} is a contravariant functor: Schemes_{/S_{m}} → AB satisfying the

sheaf condition, that is, if {U_{i} ⟶^{f_{i}} U} is an étale covering, then the sequence

F(U) → P (U_{i}) ⇉ P (U_{i} ×_{S_{m}} U_{j})

is exact. Here the first arrow is F(f_{i}) and the second two arrows indicate F(U_{i} × U_{j}) → F(U_{i}) and F(U_{i} × U_{j}) → F(U_{j}). The sequence is called exact, if the following two conditions are satisfied:

1. F(f_{i})(ϕ) = 0 for all i, then ϕ = 0;

2. For each i, ϕ_{i} ∈ F(U_{i}) is given. If the image of ϕ_{i} and ϕ_{j} in F(U_{i} × U_{j}) coincides for all i, j, there exists ϕ ∈ F(U) such that the image of ϕ in F(U_{i}) is ϕ_{i} for all i.

Obviously, for a finite étale group scheme G_{/S_{m}},

P_{G}(U) = Hom_{SCH_{/U}} (U, G_{/U}) = Hom_{SCH_{/Sm}} (U, G)

for G_{/U} = G ×_{S_{m}} U is a locally constant sheaf. When G is a constant group scheme, the corresponding locally constant sheaf is called constant.

Let S be a connected scheme over ℤ_{p} or ℤ/p^{k}ℤ. Let ℘ be a sheaf of sets on the small étale site over S. We say that the sheaf ℘ is locally constant, if there is a faithfully flat étale cover T → S, such that for all étale covers T′ → T, with T′ connected, there is a fixed set V such that ℘(T′) = V.

Lemma 7. Say ℘ is a locally constant sheaf on the small étale site over S. There is a unique faithfully flat étale cover P → S such that

℘(X) ⟶^{∼} Hom_{S}(X, P),

for all étale covers X → S.

We shall give a proof of the lemma shortly. Now suppose that ℘ is as above, with V = ℤ/p^{n}ℤ. Let P denote the scheme over S arising from the Lemma.

Consider another functor

S − Schemes ⟶ Sets,
X ↦ Isom_{X}(P ×_{S} X, (ℤ/p_{n}ℤ)_{/X}),

This is a sheaf on the small étale site over S which is locally constant. By the Lemma above there is a unique faithfully flat cover I_{P} → S such that

Isom_{X}(P ×_{S} X, (ℤ/p^{n}ℤ)_{/X}) = Hom_{S}(X, I_{P}).

Let P be a ℤ_{p}–rank 1 p–adic locally constant sheaf on the S_{m}’s. This means that P is the projective limit of locally constant sheaves P_{n/S_{m}} associated to a projective system of finite étale group schemes G_{n/S_{m}} such that

G_{n/S_{m+1}} ×_{S_{m+1}} S_{m} = G_{n/S_{m}} and G_{n} ×_{S_{m}} S′_{m} ≅ ℤ/p^{n}ℤ_{/S′_{m}}

over a finite étale extension S′_{m} of S_{m}. For an étale map f : U → S_{m}, the value of the sheaf P_{n} is given by P_{n}(U) = Hom_{SCH_{/S}}(U, G_{n}). In particular, P_{n/S_{m+1}} ×_{S_{m+1}} S_{m} = P_{n/S_{m}}, and P is isomorphic to the constant sheaf ℤ_{p} over a possibly infinite étale extension of S_{m}.

We define

π_{m,n} : T_{m,n} = Isom_{S_{m}}(P_{n}, (ℤ/p^{n}ℤ)) → S_{m}

to be a finite étale S_{m}–scheme which represents the following functor defined on the category Schemes_{/S_{m}}:

(π : X → S_{m}) ↦ {ℤ_{p}–linear isomorphisms ψ_{n} : P_{n/X} ≅ (ℤ/p^{n}ℤ)_{/X}} .

Lemma 8. Let Y′/Y be a general étale Galois covering. A scheme X′/Y′ descends to X/Y such that X ×_{Y} Y′ ≅ X′ if and only if X′_{/Y′} has an action of Gal(Y′/Y) compatible with that on Y′.

For a proof of this fact, see [H00b] Example I.11.1.

Let us prove the existence of T_{m,n}, which at the same time gives a proof of Lemma 7. We may assume that S_{m} is connected. When P_{n} is a constant sheaf, T_{m,n} = ⨆_{u∈GL_{1}(ℤ/p^{n}ℤ)} S_{m} represents the functor, because, identifying P_{n} with ℤ/p^{n}ℤ once and for all, the value of the functor on a connected scheme is canonically isomorphic (ℤ/p^{n}ℤ)^{×}. When P_{n} is not constant, we take an étale Galois covering S′_{m}/S_{m} with Galois group G such that P_{n} becomes constant over S′_{m}. For example, we could take as S′_{m} a connected component of P_{n} containing the generator over ℤ at one fiber. Then the functor is represented by T′_{m,n} over S′_{m}. Since P_{n} is defined over S_{m}, σ ∈ G induces an automorphism σ of P_{n/S′_{m}} = P_{n} ×_{S_{m}} S′_{m}. By composition of the automorphism σ of P_{n/S′_{m}}, T′_{m,n/S′_{m}} has an action of G, and we can descend T′_{m,n} to an étale finite scheme T_{m,n} over S_{m}, which represents the functor.

By construction, T_{m,n/S_{m}} is étale. Since each geometric fiber of T_{m,n} over S_{m} is isomorphic to (ℤ/p^{r}ℤ)^{×} everywhere, it is faithfully flat and finite. Therefore T_{m,n} is affine over S_{m}. We define V_{m,n} = H^{0}(T_{m,n}, O_{T_{m,n}}).

The finite group (ℤ/p^{n}ℤ)^{×} acts on T_{m,n} freely by ψ ↦ gψ for g ∈ (ℤ/p^{n}ℤ)^{×}, and we have T_{m,n}/Γ_{n,n′} ≅ T_{m,n′} for all n′ ≤ n, where

Γ_{n,n′} = {x ∈ (ℤ/p^{n}ℤ)^{×}|x ≡ 1 mod p^{n′}}.

We have a tower:

V_{m,0} ⊂ V_{m,1} ⊂ ⋯ ⊂ V_{m,n}

with V_{m,0} = H^{0}(S_{m}, O_{S_{m}}). We put V_{m,∞} = ⋃_{n} V_{m,n} and T_{m,∞} = \underleftarrow{lim}_{n}T_{m,n}. We define

V = V_{∞,∞} = \underleftarrow{lim}_{m}V_{m,∞}.  (12)

We call V the space of p–adic modular forms with respect to P. If necessary, we write V_{Γ,m,α} for V_{m,α} to indicate the dependence of this space on the choice of the level subgroup Γ. If we take P_{n} to be 𝔼^{ord,et}[p^{n}]_{/S_{m}} as in the previous subsection, we simply call V the space of p–adic modular forms.

Let T = 𝔾_{m/ℤ_{p}} = Spec(ℤ_{p}[t, t^{−1}]). Write X(T) for the group of homomorphisms: T → G_{m} of algebraic groups defined over W. We have X(T) ≅ ℤ, and if κ corresponds to an integer κ, then κ(x) = x^{κ} for x ∈ T(A), and T = Spec(ℤ_{p}[t^{ℤ}]) = Spec(ℤ_{p}[X(T)]). Following the tradition, we call each element κ ∈ X(T) a weight of T. We have the weight decomposition:

R_{A} = H^{0}(T_{/A}, O_{T/A}) = ⨁_{κ∈X(T)} R_{A}[κ]

for the space R_{A}[κ] of weight κ, and

R_{A}[κ] = {f : T → 𝔸^{1} ∈ Γ(T, O_{T})|f(ht) = κ(t)f(h)} = Aκ

for t, h ∈ T. Here T acts on R_{A}[κ] by the character κ (cf. [H00b] I.6.5).

The evaluation of ϕ ∈ R_{A}[κ] at the origin 1 ∈ T gives rise to a canonical isomorphism ℓ_{can} : R_{A}[κ] ≅ A sending κ to 1. We have a tautological embedding R_{A}[κ] ↪ Γ(T, O_{T}) given by

ϕ ↦ {h ↦ ℓ_{can}(κ(h^{−1})ϕ)}.

By definition, ϕ then induces a function on T(ℤ_{p}) = ℤ_{p}^{×}. In other words,

We have R_{A}[κ] ≅ C(T(ℤ_{p}), A)[κ] = Aκ.  (c)

Here C(T(ℤ_{p}), A) is the space of (p–adic) continuous functions with values in A on T(ℤ_{p}), and “[κ]” indicates the κ–eigenspace under the right action of T(ℤ_{p}). A function ϕ : T(ℤ_{p}) → A for a topological ring A is called “locally constant” if we have an open neighborhood for each point x ∈ T(ℤ_{p}) such that ϕ|_{U} is a constant. By definition, a locally constant function is continuous. When A is a finite ring (with discrete topology), the space of continuous functions C(T(ℤ_{p}), A) is equal to the space of locally constant functions

LC(T(ℤ_{p}), A),

and we use LC(T(ℤ_{p}), A) instead of C(T(ℤ_{p}), A) when A is finite.

Let \underline{ω}_{m} be the line bundle P ⊗_{ℤ_{p}} O_{S_{m}} = P_{m} ⊗_{ℤ/p^{m}ℤ} O_{S_{m}}. We use the symbol \underline{ω} to denote this invertible sheaf, because it coincides with the line bundle we have defined already when we apply our theory to modular curves (as we will see later). We also write

\underline{ω}_{m,n} = P ⊗_{ℤ_{p}} O_{T_{m,n}} = P_{m} ⊗_{ℤ/p^{n}ℤ} O_{T_{m,n}}.

Over T_{m,n} with n ≥ m, we have the universal isomorphism

I_{can} : π_{m,n}^{∗}P_{n} ≅ (ℤ/p^{m}ℤ),

so we have an action of Gal(T_{m,∞}/S_{m}) on π_{m,n}^{∗}P_{m}, and

ω_{can} = I_{can} ⊗ id : \underline{ω}_{m,n} ≅ O_{T_{m,n}}

is an isomorphism. We can identify T_{/T_{m,n}} = 𝔾_{m/T_{m,n}} with the constant covering

p_{Y} : Y = Y_{m,n} = T × T_{m,n} = GL(\underline{ω}_{m,n}) → T_{m,n}

on V_{m,n} (n ≥ m). Thus ω_{can} induces an isomorphism:

ω_{can}^{κ} : p_{Y,∗}(O_{Y} [κ]) ≅ R_{V_{m,n}}[−κ].

We write \underline{ω}_{m,n}^{κ} for the sheaf O_{Y} [−κ] on T_{m,n}. By definition, the Galois group Gal(T_{m,n}/S_{m}) = (ℤ/p^{n}ℤ)^{×} acts on Y on the left. The group Gal(T_{m,n}/S_{m}) acts on \underline{ω}_{m,n}^{κ} via the rational structure given from \underline{ω}_{m}, and we then descend the sheaf to \underline{ω}_{m}^{κ} on S_{m} (by Lemma 8). In other words, for an O_{T_{m,n}}–algebra A, f ∈ H^{0}(Spec_{T_{m,n}}(A), \underline{ω}_{m,n}^{κ}) is a functorial rule assigning f(X, ψ) ∈ R_{A}[−κ] to X_{/A} and ψ : P_{n/X} ≅ (ℤ_{p}/p^{n}ℤ_{p})_{/X}.

We let h ∈ (ℤ_{p}/p^{n}ℤ_{p})^{×} = Gal(T_{m,n}/S_{m}) act on f in the following way: f ↦ {(X, ψ) ↦ κ(h)f(X, hψ)}. Since the scheme S_{m} is the quotient of T_{m,n} by (ℤ_{p}/p^{n}ℤ_{p})^{×}, for any O_{S_{m}}–algebra A,

A ↦ H^{0}(Gal(T_{m,n}/S_{m}), H^{0}(Spec(A) ×_{S_{m}} T_{m,n}, \underline{ω}_{m,n}^{κ}))

defines a coherent sheaf on S_{m} (as in the proof of the horizontal control theorem), which we write \underline{ω}_{m}^{κ}. In other words, \underline{ω}_{m}^{κ} is the coherent sheaf associated to the following O_{S_{m}}–module:

H^{0}(S_{m}, \underline{ω}_{m}^{κ})

= {f ∈ H^{0}(T_{m,m}, R_{V_{m,m}}[−κ])|f(X, hψ) = κ^{−1}(h)f(X, ψ) ∀h ∈ (ℤ/p^{m}ℤ)^{×}} .

This shows that \underline{ω}^{κ} defined above is actually the invertible sheaf of modular forms of weight κ restricted to S_{m}; so, our use of the symbol \underline{ω}_{/S_{m}}^{κ} is justified.

There is another description of \underline{ω}_{m/S_{m}}^{κ}. Since P_{m} ≅ (ℤ/p^{m}ℤ) on T_{m,m}, the action of Gal(T_{m,n}/S_{m}) on P_{m} extends to an action of the Galois group on R_{ℤ/p^{m}ℤ}[κ], which determines an étale torsion sheaf P_{m}^{κ} over S_{m}. Then ˆ \underline{ω}_{m}^{κ} = O_{S_{m}} ⊗_{ℤ} P_{m}^{κ}, where ˆ \underline{ω}_{m} = Hom(\underline{ω}_{m}, O_{S_{m}}) is the dual sheaf of \underline{ω}_{m}.

By our definition, for each f ∈ H^{0}(S_{m}, \underline{ω}_{m}^{κ}),

φ(f) = ω_{can}^{κ}(f) ∈ H^{0}(T_{m,m}, R_{T_{m,m}}[−κ]),

which can be regarded as a functorial rule assigning to each test object

(X_{/S_{m}}, ψ : P_{m/X} ≅ (ℤ_{p}/p^{m}ℤ_{p})_{/X})

a value φ(f)(X, ψ) ∈ H^{0}(X, R_{O_{X}} [−κ]) such that

φ(f)(X, hψ) = κ^{−1}(h)φ(f)(X, ψ)

for all h ∈ (ℤ_{p}/p^{m}ℤ_{p})^{×} and φ(f)(Y, ϕ^{∗}ψ) = ϕ^{∗}(φ(f)(X, ψ)) for any morphism ϕ : Y → X of S_{m}–schemes. Similarly, φ ∈ V_{m,n} is a functorial rule assigning (X, ψ) a value φ(X, ψ_{n} : P_{n} ≅ (ℤ_{p}/p^{n}ℤ_{p})) ∈ H^{0}(X, O_{X}) such that φ(Y, ϕ^{∗}ψ_{n}) = ϕ^{∗}(φ(X, ψ_{n})) for any morphism ϕ : Y → X of S_{m}–schemes.

We let h ∈ T (ℤ_{p}) act on test objects by (X, ψ) ↦ (X, hψ). In this way, we identify T(ℤ_{p}) with Gal(T_{m,∞}/S_{m}). We can think of the image of R_{V_{m,m}}[−κ] inside V_{m,m}[−κ] over V_{m,0}[(ℤ_{p}/p^{m}ℤ_{p})^{×}], which is the homomorphic image of H^{0}(S_{m}, \underline{ω}_{m}^{κ}) under f ↦ ℓ_{can} ∘ φ(f). We have a natural map

i : H^{0}(S_{m}, \underline{ω}_{m}^{κ}) → V_{m,∞}[−κ],  (13)

where V_{m,m}[−κ] is the κ–eigenspace of the right action of T. The above map is injective if m = ∞. We define

R′_{m} = ⨁_{κ∈X(T)_{+},κ≫0} H^{0}(S_{m}, \underline{ω}_{m}^{κ}).  (14)

Here “≫” implies “sufficiently positive”.

We note that S_{m} is an affine scheme, because we inverted a positive degree element E in \overline{M}_{m} = Proj(\overline{A}_{N})⊗_{W} W/p^{m}W to get S_{m}. Since cohomology groups of affine scheme of coherent sheaves vanishes in positive degree, the short exact sequence:

0 ⟶ \underline{ω}^{κ} ⟶^{p^{m}} \underline{ω}^{κ} ⟶ \underline{ω}_{m}^{κ} ⟶ 0

gives rise to an exact sequence:

(F1) 0 ⟶ H^{0}(S, \underline{ω}^{κ}) ⟶^{p^{m}} H^{0}(S, \underline{ω}^{κ}) ⟶ H^{0}(S_{m}, \underline{ω}_{m}^{κ}) ⟶ 0; and

(F2) V_{m,∞} = V/p^{m}V.

Thus we have

R′_{∞}/p^{m}R′_{∞} ≅ R′_{m} and H^{0}(S, \underline{ω}^{κ})/p^{m}H^{0}(S, \underline{ω}^{κ}) ≅ H^{0}(S_{m}, \underline{ω}_{m}^{κ}).  (15)

We now define a homomorphism

β(m) : R′_{m} → V_{m,m}

in the following way. Over T_{m,m}, we have a universal isomorphism from I_{can} : P_{n} ≅ (ℤ/p^{m}ℤ). This gives rise to

ω_{can} = I_{can} ⊗ id ∈ H^{0}(Isom_{T_{m,m}}(\underline{ω}_{m,m}, O_{T_{m,m}})),

which is invertible. We have

β(m)(∑_{κ≫0} f_{κ}) = {(X_{/T_{m,m}}, ψ) ↦ ∑_{κ} ℓ_{can}(ω_{can}^{κ})(f_{κ}(X, ψ))}

for f_{κ} ∈ H^{0}(S_{m}, \underline{ω}_{m}^{κ}). By construction, β(n) mod p^{m} = β(m) for all n > m. Thus taking the projective limit, we have

β(∞) : R′_{∞} → V = \underleftarrow{lim}_{m}V_{m,∞}.

The κ–eigenspace V [−κ] of V. By definition, β = β(∞) induces

β = β_{κ} : H^{0}(S_{∞}, \underline{ω}_{∞}^{κ}) = \underleftarrow{lim}_{m}H^{0}(S_{m}, \underline{ω}_{m}^{κ}) → V [−κ].

Proposition 6. The above map β_{κ} is an injection.

Proof. Since T_{m,n} is faithfully flat and étale over S_{m}, we may make a basechange: T_{m,n/S_{m}} to T_{m,n/S_{m}} ×_{S_{m}} T_{m,m}, and hence we may suppose that P is constant. Then V_{m,∞} is made of locally constant function on T(ℤ_{p}) with values in V_{m,0}. Taking the limit, V is the space C(T(ℤ_{p}), V_{∞,0}) of continuous function on T(ℤ_{p}) with values in V_{∞,0} = \underleftarrow{lim}_{m}V_{m,0}. Then H^{0}(S_{∞}, \underline{ω}_{∞}^{κ}) is inside the limit of global sections of \underleftarrow{lim}_{m}R_{V_{m,0}}[−κ], which injects into C(T(ℤ_{p}), V_{∞,0})[−κ]. This shows the assertion.

We now put

D′ = β(R′_{∞}) [\frac{1}{p}] ⋂ V = β(⨁_{κ≫0} H^{0}(S, \underline{ω}^{κ})) [\frac{1}{p}] ⋂ V.

Theorem 17. The inclusion β = β(∞) : D′ → V induces an isomorphism

D′/p^{m}D′ ≅ V/p^{m}V for all m.

In other words, D′ is p–adically dense in V.

Proof. The injectivity of D′/p^{m}D′ → V/p^{m}V follows from the definition (see [K] Appendix III) (or as easily seen after faithfully flat extension to T_{m,m}).

We thus need to prove that D′/pD′ → V/pV = V_{1,∞} is surjective, which implies the surjectivity of D′/p^{m}D′ → V/p^{m}V for all m by Nakayama’s lemma. Since T_{m,n}/S_{m} is étale finite, replacing S_{m} by T_{m,∞}, we may assume that P is constant (see [K] Appendix III pages 364-5), because we can recover the global sections of \underline{ω}^{κ} over S_{m} as Galois invariants of that over T_{m,n}. We have

O_{T_{1,n}} = O_{S_{1}} ⊗_{W} W[(ℤ/p^{n}ℤ)^{×}] = O_{T_{1,0}}[(ℤ/p^{n}ℤ)^{×}].

This shows V_{1,∞} = LC(T(ℤ_{p}), V_{1,0}), where LC(T(ℤ_{p}), V_{1,0}) is the space of locally constant functions on the p–adic analytic manifold T(ℤ_{p}) with values in H^{0}(S_{1}, O_{S_{1}}) = V_{1,0}. Writing V_{∞,0} as a union of W–free module X of finite rank, we have LC(T(ℤ_{p}), V_{1,0}) = ⋃_{X} LC(T(ℤ_{p}), X). We need to prove that

LC(T(ℤ_{p}), X/pX) = D_{X}/pD_{X},

where D_{X} is the space of polynomial functions of homogeneous degree ≫ 0 (with coefficients in K = W [\frac{1}{p}] on the torus T) which has values in X over T(ℤ_{p}). This last fact follows from Mahler’s theorem of the density of binomial polynomials in the space of continuous functions on ℤ_{p} with values in ℤ_{p}.

By the properness of \overline{M} = \overline{M}_{N}, H^{1}(\overline{M}, \underline{ω}^{κ}) is a W–module of finite type. Taking the projective limit with respect to m of the exact sequences:

0 ⟶ H^{0}(\overline{M}, \underline{ω}^{κ}) ⊗_{W} W_{m} ⟶ H^{0}(\overline{M}_{m}, \underline{ω}_{m}^{κ}) ⟶ H^{1}(\overline{M}_{m}, \underline{ω}^{κ})[p^{m}] ⟶ 0,

we get \underleftarrow{lim}_{m}H^{0}(\overline{M}_{m}, \underline{ω}_{m}^{κ}) = H^{0}(\overline{M}, \underline{ω}_{m}^{κ}). Let R_{m} = ⨁_{κ≫0} H^{0}(\overline{M}_{m}, \underline{ω}_{m}^{κ}). We know R_{∞} is p–adically dense in \underleftarrow{lim}_{m}R_{m}. By definition, R_{∞} ⊂ R′_{∞}. Note that \underline{ω}^{p−1} is trivial on S_{1}. Let a ∈ H^{0}(S_{1}, \underline{ω}^{p−1}) be the section corresponding to 1 ∈ \underline{ω}_{1}^{p−1} ≅ O_{S_{1}}. The section a extends to the Hasse invariant H defined over \overline{M}_{1}, and H vanishes outside S_{1}. We also have a section E ∈ H^{0}(\overline{M}, \underline{ω}^{(p−1)}) such that E mod p = H. By definition,

H^{0}(S_{m}, \underline{ω}_{m}^{κ}) = \underrightarrow{lim}_{n} \frac{H^{0}(\overline{M}_{m}, \underline{ω}_{m}^{κ} ⊗ \underline{ω}_{m}^{n(p−1)})}{E^{n}}.

We would like to show that β(R_{∞}[\frac{1}{p}]) ∩ V is dense in V. Pick ∑_{κ} f_{κ} ∈ p^{m}V for f_{κ} ∈ H^{0}(S, \underline{ω}^{κ}). We need to approximate f = f_{κ} modulo p^{m+1}V by an element in H^{0}(\overline{M}, \underline{ω}^{κ} ⊗ \underline{ω}^{k}). This section f ∈ H^{0}(S, \underline{ω}^{κ}) can be written as f ≡ g_{ℓ}/E^{ℓ} mod p^{m+1} for g_{ℓ} ∈ H^{0}(\overline{M}, \underline{ω}^{κ} ⊗ \underline{ω}^{ℓ}). For k > ℓ, we have f ≡ g_{ℓ}E^{k−ℓ}/E^{k} mod p^{m}. Ee may assume that k = p^{m}(p − 1). As a function of (X, ψ), E^{k} mod p^{m+1} is a constant, and f ≡ g_{ℓ}E^{k−ℓ} mod p^{m+1}R_{∞}. This shows the density of β(R_{∞} [\frac{1}{p}]) ⋂ V in V.

Corollary 2. Put

D = β(⨁_{κ≫0} H^{0}(\overline{M}, \underline{ω}^{κ})) [\frac{1}{p}] ⋂ V.

Then D is p–adically dense in V.

We now suppose to have an idempotent e acting on V, which projects down D′ and V [−κ] into eD and H^{0}(S_{m}, \underline{ω}^{κ}), respectively (for κ ≫ 0). Here eD may not be sitting inside D. We have eV [−κ] = eH^{0}(\overline{M}, \underline{ω}^{κ}) for sufficiently positive κ. We put V = \underrightarrow{lim}_{n}V/p^{n}V = \underrightarrow{lim}_{n}V_{n,∞}. Since V is W–flat, V is p–divisible. We have V[−κ] = \underrightarrow{lim}_{m}V_{m,m}[−κ].

Now we apply e and get

eV[−κ] = eH^{0}(S_{/W}, \underline{ω}^{κ} ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p}) = eH^{0}(\overline{M}_{/W}, \underline{ω}^{κ} ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p})  (16)

if κ ≫ 0.

By (F1), H^{0}(S_{/W}, \underline{ω}^{κ} ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p}) is p–divisible, and eV[−κ] is p–divisible. Since H^{0}(\overline{M}_{/W}, \underline{ω}^{κ}) is a W–module of finite type, eV[−κ] is a p–divisible module of finite corank. Let V^{∗} be the Pontryagin dual module of V.

Since V is a discrete T(ℤ_{p})–module, V^{∗} is a compact W[[T(ℤ_{p})]]–module. Let Γ = 1 + pℤ_{p} be the p–profinite part of T(ℤ_{p}). Then T(ℤ_{p}) = Γ × μ_{p−1}. We fix a character \overline{χ} : μ_{p−1} → 𝔽^{×} for the residue field 𝔽 = W/𝔪_{W}. We write χ̃ : μ_{p−1} → W^{×} for the Teichmüller lift of \overline{χ}. We define Xχ̃ ⊂ X_{+}(T) by the set of algebraic characters κ : T → 𝔾_{m} such that κ ≡ χ̃ mod 𝔪_{W} and κ is sufficiently regular so that the above equation (16) holds. Then Xχ̃ is Zariskidense in Spec(W[[Γ]])(W). We write V_{ord}^{∗} for eV^{∗}.

Let us decompose

V_{ord}^{∗} = ⨁_{χ̃∈μ̂_{p−1}} V_{ord}^{∗}[χ̃]  (17)

into the direct sum of the χ̃–eigenspaces under the action of μ_{p−1}. Then V_{ord}^{∗}[χ̃] is a compact module over W[[Γ]]. By (16), V_{ord}^{∗}[χ̃]⊗_{W[[Γ]],κ} W is W–free of finite rank s(χ̃) for κ ∈ Xχ̃. By topological Nakayama’s lemma, V_{ord}^{∗}[χ̃] is W[[Γ]]– module of finite type with minimum number s(χ̃) of generators. Since Xχ̃ is Zariski-dense in Spec(W[[Γ]]), we see that V_{ord}^{∗}[χ̃] is W[[Γ]]–free of rank s(χ̃). Thus we have

rank_{W[[Γ]]} V_{ord}^{∗}[χ̃] = rank_{W} V_{ord}^{∗}[χ̃] ⊗_{W[[Γ]],κ} W

= rank_{W}(eH^{0}(\overline{M}, \underline{ω}^{κ}) ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p})^{∗} = rank_{W} eH^{0}(\overline{M}, \underline{ω}^{κ}) (18)

for all κ ∈ Xχ̃, and we get

Theorem 18. Suppose the existence of the idempotent e : V → V as above. Then V_{ord}^{∗} is a well controlled W[[T(ℤ_{p})]]–projective module of finite type. This means that

V_{ord}^{∗} ⊗_{W[[T (ℤ_{p})]],κ} W ≅ Hom_{W}(eH^{0}(\overline{M}, \underline{ω}^{κ}), W)

canonically if κ is sufficiently positive. For each χ̃–component V_{ord}^{∗}[χ̃] is free of finite rank over W[[Γ]] for the maximal p–profinite subgroup Γ of T(ℤ_{p}).

In Subsection 7.4, we construct the projector e as the limit lim_{n→∞} U(p)^{n!} acting on the space of p–adic modular forms out of the Hecke operator U(p). The following lemma shows us the convergence of the above limit:

Lemma 9. Let T be a linear operator acting on V. If we can write V = \underleftarrow{lim}_{i}V_{i} so that V_{i} are of finite type as W–modules for all i and T preserves Ker(V ↠ V_{i}) for all i, then lim_{n→∞} T^{n!} exists and gives an idempotent acting on V.

Proof. We need to show the existence of the limit for each V_{i}, which has been shown in Section 2 (just after Theorem 1).

We suppose the operator U(p) satisfies the assumption of the above lemma. Writing K for the field of fractions of W, we suppose the following condition:

dim_{K} eH^{0}(M_{/K}, \underline{ω}_{/K}^{k}) is bounded independent of k.  (F)

Under this, if \overline{f} ∈ eH^{0}(S_{m}, \underline{ω}^{κ}) ⊂ H^{0}(S_{/W}, \underline{ω}^{κ} ⊗ ℚ_{p}/ℤ_{p}) is outside the image of H^{0}(M_{m}, \underline{ω}^{κ}), we can lift \overline{f} to f ∈ H^{0}(S_{/W}, \underline{ω}^{κ}) (by (F1)), multiply f by E^{p^{n}(p−1)} for large n so that

(i) E^{p^{n}(p−1)}f extends to M and

(ii) E^{p^{n}(p−1)}f mod p^{m}, eE^{p^{n}(p−1)}f mod p^{m} and \overline{f} are all equal.

Hence multiplying \underline{ω}^{κ} by a sufficiently large power \underline{ω}^{p^{n}(p−1)}, we have \overline{f} in the image. In this way, we can create eE^{p^{n}(p−1)}f whenever we find \overline{f} = eE^{p^{n}(p−1)}f mod p^{m} outside the image. Then eE^{p^{n}(p−1)}f mod p^{N} for all N ≥ m is also outside the image. If we can find

\overline{f} ∈ H^{0}(S_{/W}, (\underline{ω}^{κ+p^{n}(p−1)}) ⊗ ℚ_{p}/ℤ_{p})

outside the image for infinitely many n, the dimension of

eH^{0}(M_{/K}, \underline{ω}_{/K}^{p^{n}(p−1)})

would grows as n grow, which is impossible by (F). Thus

eH^{0}(M_{m}, \underline{ω}^{κ}) ≅ eH^{0}(S_{m}, \underline{ω}^{κ}) if κ ≫ 0.

The condition (F) can be proven in our application by a group cohomology means using the Eichler-Shimura isomorphism combined with the p–adic density of D in D′ (see [H93] page 217).

If the condition (F) is replaced by much stronger

dim_{K} eH^{0}(M_{/K}, \underline{ω}_{/K}^{k}) depends only on k mod p − 1 if k ≥ κ,  (E)

then we can prove the assertion of Theorem 18 for all k ≥ κ in the following way: We know that V^{ord}[−k] is W–cofree of rank given by the above constant value d, since it is the Pontryagin dual of V_{ord}^{∗} ⊗_{W[[ℤ_{p}^{×}]],−k} W which is W–free of rank d. If k ≥ κ, (eH^{0}(M_{/W}, \underline{ω}_{/W}^{k}) ⊗ K/W coincides with V^{ord}[−k] by comparing the corank. This shows the assertion.

7.3 p–Adic modular forms

We fix a prime p and a positive integer N prime to p. We call a ℤ_{p}–algebra a p– adic algebra if A = \underleftarrow{lim}_{n}A/p^{n}A. We write p–Algebras for the category of p–adic algebras. Morphisms in this category are continuous algebra homomorphisms (taking identity to identity).

For simplicity, we assume that W contains a primitive N–th root of unity ζ_{N}. Let us consider the functor E_{Γ,α} : p–Algebras → Sets given by

E_{Γ,α}^{ord}(A) = [(E, ϕ_{p} : μ_{p^{α}} ↪ E, ϕ_{N})_{/A}] ,  (19)

where Γ = Γ(N), Γ_{0}(N), Γ_{1}(N) and ϕ_{Γ} is a level N–structure of type Γ. Here ϕ_{p} : μ_{p^{α}} ↪ E is an embedding of group schemes defined over A. For α > β, we define projection morphism of functors: π_{β,α} : E_{Γ,β}^{ord} → E_{Γ,α}^{ord} by

(E, ϕ_{p} : μ_{p^{β}} ↪ E, ϕ_{N})_{/A} ↦ (E, ϕ_{p}|_{μ_{p^{α}}} , ϕ_{N})_{/A}.

These functors are representable (if p^{α} > 3) by M_{Γ,α/ℤ_{p}}^{ord} = M_{Γ,α}[\frac{1}{E}]_{/ℤ_{p}} for the lift E of Hasse invariant. When Γ = Γ_{1}(N) or Γ_{0}(N), the moduli scheme M_{Γ,α}^{ord} is given by the quotient of M_{N}^{ord} by the constant group scheme G = Γ/Γ(N), that is, M_{N}^{ord} = Spec(A_{N,0}^{G}), because if p^{α} > 3, the triple (E, ϕ_{p}, ϕ_{N}) does not have non-trivial automorphism (see Theorem 19 and its proof).

By the representability, we have the morphism of schemes π_{β,α} : M_{Γ,β/ℤ_{p}}^{ord} → M_{Γ,α/ℤ_{p}}^{ord}, and E_{Γ,∞}^{ord} is pro-representable by \underleftarrow{lim}_{α}M_{Γ,β/ℤ_{p}}^{ord}.

Let S_{/W} = \overline{M}_{Γ,0}^{ord} and \overline{M}_{/W} = \overline{M}_{Γ}. The scheme S is an affine curve over W, because S is obtained by inverting the global section E of H^{0}(\overline{M}_{Γ}, \underline{ω}^{p−1}) and \underline{ω}^{p−1} is ample on the projective scheme \overline{M}_{Γ} = Proj(\overline{A}_{N}^{G}) for G = Γ/Γ(N).

Theorem 19. Let P_{n/W} be the Cartier dual of the connected component of the p^{n}–torsion points 𝔼[p^{n}] of the universal elliptic curve (completed at cusps by Tate curves) over S. We put S_{m} = S ×_{W} W/p^{m}W. Then Spec(V_{Γ,m,α}) represents the functor Isom_{S_{m}}(P_{α}, ℤ/p^{α}ℤ) over S_{m}. In particular, we have \overline{M}_{Γ,α}^{ord} ×_{W} W/p^{m}W = Spec(V_{Γ,m,α}).

Proof. We pick a homomorphism f ∈ Hom_{W_{m}–alg}(V_{Γ,m,α}, A) for a W_{m}– algebra A. We have a morphism: Spec(A) → \overline{M}_{Γ,α/W_{m}}^{ord}, for which we use

the same symbol f. Then f : Spec(A) → \overline{M}_{Γ,α/W_{m}}^{ord} induces a unique triple (E, ϕ_{p}^{f} : μ_{p^{α}} ↪ E, ϕ_{N})_{/A}. Since E = \overline{𝔼} ×_{S_{m},π∘ϕ} Spec(A) for the projection

π : \overline{M}_{Γ,α}^{ord} ↠ S_{m}, the level p^{α}–structure ϕ_{p}^{f} induces a unique isomorphism ϕ_{p}^{f} : μ_{p^{α}} ≅ C_{α} ×_{S_{m},π∘ϕ} Spec(A), where C_{α} ⊂ \overline{𝔼}[p^{α}] is the connected component of the identity, and \overline{𝔼} is the completed universal elliptic curve over S_{m}. We take the Cartier dual ϕ̂_{p}^{f} : P_{α} ×_{S_{m}} Spec(A) ≅ ℤ/p^{α}ℤ of ϕ_{p}^{f} (see [H00b] I.7), since P_{α} is by definition the Cartier dual of C_{α}. We get a morphism of functors: I : Hom_{W_{m}–alg}(V_{Γ,m,α}, A) → Isom_{S_{m}}(P_{α/A}, ℤ/p^{α}ℤ_{/A}) by the correspondence f ↦ ϕ̂_{p}^{f}. Since Cartier duality is perfect, this morphism is an isomorphism for each W_{m}–algebra A.

Corollary 3. Let the notation and the assumption be as in Theorem 19. Let f : \overline{𝔼} → S_{m} be the universal elliptic curve. We have the following canonical isomorphism of invertible sheaves:

\underline{ω} = 0^{∗}Ω_{𝔼/S_{m}} ≅ P_{m} ⊗_{ℤ_{p}} O_{S_{m}},

where 0 : S_{m} → \overline{𝔼} is the identity section. In particular, H^{0}(S_{m}, P_{m}^{k}⊗O_{S_{m}}) is the

space of false modular forms of weight k over S_{m}, where P_{m}^{k} = \overbrace{P_{m} ⊗ ⋯ ⊗ P_{m}}^{k}.

Proof. Let C_{m} ⊂ 𝔼[p^{m}] be the identity connected component. Over a finite faithfully flat étale extension (for example, \overline{M}_{Γ,m}/S_{m}), C_{m} is isomorphic to μ_{p^{m}/S_{m}}. The tangent space of μ_{p^{m}} at the origin can be computed easily by using the expression μ_{p^{m}} = Spec(ℤ[T]/((1 + T)^{p^{m}} − 1)), and the tangent sheaf at the origin of μ_{p^{m}/S_{m}} is free of rank 1 over O_{S_{m}}. This shows that

Hom_{O_{Sm}} (0^{∗}Ω_{𝔼/S_{m}}, O_{S_{m}}) ≅ Hom_{O_{Sm}} (0^{∗}Ω_{C_{m}/S_{m}}, O_{S_{m}}),

since the tangent space at the origin of 𝔼 is locally free of rank 1 over O_{S_{m}}. We get 0^{∗}Ω_{𝔼/S_{m}} ≅ 0^{∗}Ω_{C_{m}/S_{m}}, because O_{S_{m}}–duality is perfect for locally free O_{S_{m}}–modules. By definition, we have P_{m} ≅ Hom_{gpsch}(C_{m}, G_{m}). Writing G_{m} = Spec(ℤ[t, t^{−1}]), we associate to each ϕ ∈ P_{m} a section ϕ^{∗} \frac{dt}{t} of Ω_{C_{m}/S_{m}}, which induces a surjective sheaf morphism: P_{m} ⊗_{ℤ_{p}} O_{S_{m}} ↠ 0^{∗}Ω_{C_{m}/S_{m}}. This map is an isomorphism, since the two sides are locally free of rank 1.

By definition, 0^{∗}Ω_{𝔼/\overline{S}_{m}} is the invertible sheaf generated by the degree 1

graded component \overline{A}_{N}^{Γ} [\frac{1}{E} ]. The sheaf \underline{ω}^{k} is isomorphic to k–fold tensor product of P_{m} ⊗ O_{S_{m}} (that is, P_{m}^{k} ⊗ O_{S_{m}}), and on the other hand, it is isomorphic to the sheaf generated by degree k component of the graded algebra \overline{A}_{N}^{Γ} [\frac{1}{E} ]. This shows the last assertion.

Let V_{Γ,m,∞} = \underrightarrow{lim}_{α}V_{Γ,m,α} and V_{Γ} = \underleftarrow{lim}_{m}V_{Γ,m,∞}. We call each element in V_{Γ/W} a p–adic modular form with level N structure of type Γ. For each p–adic W–algebra A, we put V_{Γ/A} = \underleftarrow{lim}_{n}V_{Γ/W} ⊗_{W} A/p^{n}A. By Theorem 19, the formal spectrum Spf(V_{Γ/W}) removed cusps classifies the triples (E, ϕ_{p} : μ_{p^{∞}} ↪ E, ϕ_{N}) made of an elliptic curve E_{/A}, an embedding of (indo-)group schemes ϕ_{p} and a level N–structure ϕ_{N} over p–adic W–algebra A. The Tate curve T ate(q) has a canonical ϕ_{p}^{can} (e.g. [H00b] II.5.3).

By the above interpretation, we can regard an element f ∈ V_{Γ/A} as a rule assigning, to a triple (E, ϕ_{p} : μ_{p^{∞}} ↪ E, ϕ_{N}) defined over a p–adic A–algebra R (for a level Γ–structure ϕ_{N}), a value f(E, ϕ_{p} : μ_{p^{∞}} ↪ E, ϕ_{N}) ∈ R such that

(M_{p}1) (E, ϕ_{p}, ϕ_{N})_{/R} ≅ (E′, ϕ′_{p}, ϕ_{N})_{/R} ⇒ f(E, ϕ_{p}, ϕ_{N}) = f(E′, ϕ′_{p}, ϕ′_{N});

(M_{p}2) If ρ : R → R′ be a continuous A–algebra homomorphism, then

f((E, ϕ_{p}, ϕ_{N}) ×_{R,ρ} R′) = ρ(f(E, ϕ_{p}, ϕ_{N}));

(M_{p}3) For all level N–structure ϕ_{N} of type Γ on the Tate curve T ate(q^{1/N}),

f(T ate(q^{1/N}), ϕ_{p}^{can}, ϕ_{N}) ∈ A[[q^{1/N}]].

Replacing (M_{p}3) by the following condition:

(S_{p}3) For all level N–structure ϕ_{N} of type Γ on the Tate curve T ata(q^{1/N}),

f(T ate(q^{1/N}), ϕ_{p}^{can}, ϕ_{N}) ∈ q^{1/N}A[[q^{1/N}]],

we can define the subspace of the p–adic cusp forms of level N–structure Γ.

We can therefore apply the theory of false modular forms to the present circumstance. As a consequence of Corollary 2, we have

Theorem 20 (Density theorem). Let W be a discrete valuation ring finite flat over ℤ_{p} and Γ = Γ_{0}(N), Γ_{1}(N) and Γ(N) for a positive integer N prime to p. If Γ = Γ(N), we assume that W contains μ_{N}. Write \overline{A}_{Γ}(A) for the graded algebra ⨁_{k≥0} H^{0}(\overline{M}_{Γ/A}, \underline{ω}^{k}) for a W–algebra A. Then \overline{A}_{Γ}(K) ∩ V is dense in V_{Γ/W} under the p–adic topology. The p–adic topology is induced by the p–adic norm: |f| = Sup_{r}|a(r; f)|_{p}, where | |_{p} is the p–adic absolute value of W, and f(T ate(q^{1/N}), ϕ_{p}^{can}, ϕ_{N}) = ∑_{r} a(r; f)q^{r} for any fixed level N structure of type Γ on the Tate curve.

The algebra \overline{A}_{Γ}(A) may be slightly different from \overline{A}_{N}^{Γ} ⊗_{W} A in the low degree components, because \underline{ω}^{k} may not be ample if k is small.

We shall give a proof assuming that W contains a primitive N–th root of unity, although one can deduce from this case the general result with a little more effort (because ℤ_{p}[μ_{N}] is unramified over ℤ_{p}). By this assumption, the generic fiber of M_{Γ,α/W} is irreducible. Since this scheme is smooth over W, its special fiber over 𝔽 is also irreducible.

Proof. We only need to prove that the p–adic topology of V_{Γ} coincides with the norm topology induced by the q–expansion. Write f(q) for the q–expansion at the Tate curve (T ate(q^{1/N}), ϕ_{p}^{can}, ϕ_{N}). Since M_{Γ,α}^{ord} is irreducible over the residue field 𝔽 = W/𝔪_{W} of W (as explained above), for any p–adic algebras A ⊂ B, if f ∈ V_{Γ/B} and f(q) ∈ A[[q^{1/N}]], then f ∈ V_{Γ/A}. This implies that if f ∈ p^{n}W[[q^{1/N}]], then f mod p^{n} vanishes in V_{Γ/W_{m}}, and therefore, f ∈ p^{n}V_{Γ/W}. This shows that the norm induces the p–adic topology on V_{Γ/W}.

By the above proof of the theorem, we have verified the following q–expansion principle:

Corollary 4. Let f be a p–adic modular from in V_{Γ/A} for a p–adic algebra A. If the value f(T ate(q^{1/N}), ϕ_{p}^{can}, ϕ_{N}) vanishes for one level N–structure ϕ_{N} on the Tate curve, the modular form f vanishes identically.

For a given ϕ_{p} : μ_{p^{∞}} ↪ E, the invariant differential ω_{can} = \frac{dt}{t} on G_{m} = Spec(ℤ[t, t^{−1}]) induces an invariant differential ω_{can} on μ_{p^{∞}}. Since the tangent space at 0 of the connected component C_{m} of E[p^{m}] coincides with that of E if p^{m} = 0 in A, ω_{can} on C_{∞} = ⋃_{α} C_{α} is induced by a unique global section ϕ_{p,∗}ω_{can} of Ω_{E/A}. Tracking down the proof of the isomorphism: 0^{∗}Ω_{\overline{𝔼}/S_{m}} ≅ P_{m}⊗_{ℤ_{p}} O_{S_{m}}, we see that the inclusion β : A_{Γ}(A) ↪ V_{Γ/A} is given by

β(f)(E, ϕ_{p}, ϕ_{N}) = f(E, ϕ_{p,∗}ω_{can}, ϕ_{N}).

This shows that

If f is of weight k, then β(f) ∈ V_{Γ/A}[−k],  (20)

where we regard k ∈ ℤ as a character of G_{m} sending a to a^{k}. An element in A_{Γ} or its degree k–component H^{0}(\overline{M}_{Γ}, \underline{ω}^{k}) will be called (hereafter) a true modular form integral over A, if we would like to distinguish them from p–adic ones. For simplicity, we write G_{k}(Γ, A) for H^{0}(\overline{M}_{Γ}, \underline{ω}^{k}). Each element of f ∈ G_{k}(Γ, A) can be interpreted as a function of triples (E, ϕ_{N}, ω)_{/R} (for A–algebras R) with the following three properties:

(M_{Γ}0) (E, ϕ_{N}, ω)_{/R} ≅ (E′, ϕ_{N}, ω′)_{/R} ⇒ f(E, ϕ_{N}, ω) = f(E′, ϕ′_{N}, ω′);

(M_{Γ}1) f(E, ϕ_{N}, λω) = λ^{−k}f(E, ϕ_{N}, ω) for λ ∈ G_{m}(R);

(M_{Γ}2) If ρ : R → R′ be a continuous A–algebra homomorphism, then

f((E, ϕ_{N}, ω) ×_{R,ρ} R′) = ρ(f(E, ϕ_{N}, ω));

(M_{Γ}3) For all level N–structure ϕ_{N} of type Γ on the Tate curve T ate(q^{1/N}),

f(T ate(q^{1/N}), ϕ_{N}, ω_{∞}) ∈ A[[q^{1/N}]].

This is consistent with earlier definitions of the same space. The value of a true modular form is defined for any triple (E, ϕ_{N}, ω) defined over any A–algebra R, but for p–adic modular forms, it is restricted to p–ordinary triples (E, ϕ_{p}, ϕ_{N}) defined over a p–adic W–algebra R. Hereafter we often regard true modular forms as p–adic modular forms by β, without attaching β in front.

7.4 Hecke operators

To apply the theory of p–ordinary false modular forms to our setting, we need to construct operators U(p) and T (p) introduced in 7.2. These operators are supplied by Hecke operators acting on true modular forms and p–adic modular forms.

We would like to define Hecke operators T (n) indexed by positive integers n. The operator T (1) is the identity operator. Let E_{/A} be an elliptic curve and C_{/A} be a locally free étale subgroup in E of rank n. We can make the quotient elliptic curve E/C (see [H00b] I.8) defined over A. We have thus an exact sequence:

0 ⟶ C ⟶ E ⟶ E/C ⟶ 0

in the category of locally free group schemes over A.

We now define the Hecke operator T (n) on G_{k}(Γ, χ, A). Let (E, ϕ_{N}, ω) be a test object with a level N structure ϕ_{N} of Γ–type. Take a faithfully flat extension B/A such that all étale locally free subgroups of E[n] of rank n is rational over B. Supposing that n is prime to N, let S_{n} be the set of all locally free étale subgroups of rank n in E. As a group functor, the value of C is a subgroup of a group of order n, the set S_{n} is made of subgroups of E[n]. Since E[n] is locally free of finite rank, S_{n} is a finite set.

If n is prime to N, we have E[N] ≅ E/C[N] canonically over B, and hence ϕ_{N} induces a unique level N–structure on E/C, which we again write as ϕ_{N}. Since E is étale over E/C, Ω_{E/A} ≅ π^{∗}(Ω_{(E/C)/A}) canonically. This shows ω induces a unique nowhere vanishing differential ω_{C} on E/C, and we can think of the value f(E/C, ϕ_{N}, ω_{C}). We then define

f|T (n)(E, ϕ_{N}, ω) = \frac{1}{n} ∑_{C∈S_{n}} f(E/C, ϕ_{N}, ω_{C}).  (21)

It is easy to check that f|T (n) satisfies the axioms of geometric modular forms (M_{Γ}0-3) over B, and we get a linear map for n prime to N:

T (n) : G_{k}(Γ, A), → G_{k}(Γ, B[\frac{1}{n}]).

We will show later using q–expansion principle that the operator is actually an endomorphism of G_{k}(Γ, A) (see below Proposition 7).

When none of the prime factors of n is invertible in A, the set S_{n} can be empty, and T (n) is not well defined in appearance. However, by inverting n, we can define T (n), and we will again show that T (n) preserves G_{k}(Γ, A) if A[\frac{1}{n}] is flat over A. If A is a p–adic algebra, and we consider p–ordinary level structure, we have non-empty S_{p} made of p étale subgroups. If n is invertible in A, the number of elements in S_{n} is equal to the number of subgroups in (ℤ/nℤ)^{2} of order n.

We now suppose that n has a non-trivial common divisor with N. In this case, we restrict ourselves to Γ = Γ_{0}(N) and Γ_{1}(N); hereafter, Γ is one of the above two groups. We write C_{0} for the level Γ_{0}(N)–structure on E that is, a cyclic subgroup of order N, and C_{0} is given by Im(ϕ_{N}). We consider only étale subgroups C of order n outside C_{0} (that is, we assume that C_{0} ∩ C = {0}). Write S_{n} for the set of such subgroups. Then we again define T (n) by (21).

We can extend the definition of Hecke operator to p–adic modular forms. Since the connected component of E[p^{α}] is not affected by the process of making quotient by an étale subgroup, p–adic level structure ϕ_{p} : μ_{p^{∞}} ↪ E induces a unique p–adic level structure ϕ_{p} : μ_{p^{∞}} ↪ E/C. Thus T (p) can be defined for f ∈ V_{Γ/A}, as long as n is non-zero-divisor in the p–adic algebra A, by

f|T (n)(E, ϕ_{p}, ϕ_{N}) = \frac{1}{n} ∑_{C∈S_{n}} f(E/C, ϕ_{p}, ϕ_{N}).  (22)

When p is not invertible in A or p|N, we often write U(p^{α}) for T (p^{α}) because in this case |S_{p^{α}}| = p^{α}. Inverting p, if p ∤ N, we have another operator T (p^{α}) defined for the full set of subgroups of rank p^{r}, defined over A[\frac{1}{p}].

Proposition 7. Let the notation be as above, and let A be an algebra of characteristic 0. Let Γ be one of Γ_{0}(N) and Γ_{1}(N). Write F_{k} for G_{k} and S_{k}.

(1) The operators T (n) and U(p^{r}) for p|N or p ∉ A^{×} give endomorphisms of F_{k}(Γ, χ, A). Moreover for any A–algebra B (not necessarily of characteristic 0), if either n is invertible in B or k ≥ 2, T (n) and U(p^{r}) induce operators on F_{k}(Γ, χ, B);

(2) Suppose that A is a p–adic algebra of characteristic 0. Then U(ℓ^{r}) for a prime ℓ|Np and T (n) for n prime to p give uniformly continuous endomorphisms of V_{Γ/A} and V_{Γ,m,α/A_{m}} for A_{m} = A/p^{m}A.

Proof. We prove (1). Since the definition of the Hecke operators T (n) and U(p^{r}) is functorial, they commutes with base-extension, as long as they are well defined. If the operator is well defined over ℤ[χ], it is well defined over any algebra of characteristic 0 (by flat scalar extension).

When A = ℂ, we may identify E = ℂ/L for L = ℤz + ℤ with z ∈ ℌ. Here ℌ = \{z ∈ ℂ|\sqrt{−1}(\overline{z} − z) > 0\}. We identify L with row vectors in ℤ^{2} by (m, n) ↦ (m, n)^{t}(z, 1) for z ∈ ℌ. The matrix ring M_{2}(ℤ) acts on the set Lat of lattices in ℂ from the right in the following way: Lα = ℤ^{2}α^{t}(z, 1). Then E/C can be identified with ℂ/L_{C} for a lattice L_{C} ⊃ L with [L_{C} : L] = n, and nL_{C} = ℤ(az + b) + ℤ(cz + d) = Lα_{C} for an integral matrix α = α_{C} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) with det(α) = n. Any α in the left coset Γα_{C} has the same effect.

For the moment, we assume that Γ = Γ_{0}(N). The level structure C_{0} on E = ℂ/(ℤz + ℤ) is given by the subgroup \{\frac{i}{N} mod L|i ∈ ℤ\} ⊂ E[N]. Since the quotient process making E/C is supposed not to affect C_{0} and nα_{C}^{−1} = ( \begin{matrix}d & −b \\ −c & a \end{matrix} ), α_{C} has to satisfy N|c and a is prime to N. Let Δ(n) is the subset made of matrices α in M_{2}(ℤ) with det α = n satisfying this condition. Therefore S_{n} ≅ Γ\Δ(n). Regarding f as a function on ℌ, we get

f|T (n)(z) = \frac{1}{n} ∑_{α∈Γ\Δ(n)} n^{k/2}f(α(z))j(α, z)^{−k}.

This operator coincides with the Hecke operator defined by Hecke. See [Sh] Chapter 3 or [H00a] 3.1.6-7 for more details of the classical definition if Hecke operators. A complete representative set for Γ\Δ(n) is given by

R_{n} = \{( \begin{matrix}a & u \\ 0 & d \end{matrix}) |ad = n, a|d, u = 0, a, 2a, …, d − a\} .

Using this representative set and computing its effect to the q–expansion, we get

a(m, f|T (n)) = ∑_{d|m,d|n} χ(d)d^{k−1}a(\frac{mn}{d^{2}}, f),  (23)

where d runs over all positive common divisors of m and n prime to N. This shows that the operator T (n) (and U(p^{r}) if p|N) preserves F_{k}(Γ_{0}(N), χ, ℤ[χ]), and induces a unique operator on F_{k}(Γ_{0}(N), χ, ℤ[χ]) ⊗_{ℤ[χ]} B for any A–algebra B. This space is canonically isomorphic to F_{k}(Γ_{0}(N), χ, B) if k ≥ 2 or B is flat over ℤ[χ].

We define the diamond operator ⟨a⟩_{k} : F_{k}(Γ_{1}(N), B) → F_{k}(Γ_{1}(N), B) by f|⟨a⟩_{k}(E, ϕ_{N}, ω) = f(E, aϕ_{N}, ω). Over ℂ, we have

a(m, f|T (n)) = ∑_{d|m,d|n} d^{k−1}a(\frac{mn}{d^{2}}, f|⟨d⟩_{k}),  (24)

where d runs over all positive common divisors of m and n prime to N. This formula can be computed also using the Tate curve. For example if n is a prime ℓ, T ate(q)/μ_{ℓ} is induced by x ↦ x^{ℓ} on G_{m} so the Tate period of T ate(q)/μ_{ℓ} is q^{ℓ}, and E_{∞}/⟨ζq_{E}^{1/ℓ}⟩ for an ℓ–th roots of unity ζ has the Tate period ζq^{\frac{1}{ℓ}}. We note that E_{∞}/μ_{ℓ} has the differential ℓ^{−1}ω_{∞}, because ω_{∞} = \frac{dt^{ℓ}}{t^{ℓ}} = ℓ\frac{dt}{t}. Writing f(q) = f(E_{∞}, ϕ_{∞,N}, ω_{∞}), we have

f|T (ℓ)(E_{∞}, ϕ_{N}, ω_{∞}) = ℓ^{k−1}f(q^{ℓ}) + \frac{1}{ℓ} ∑_{ζ∈μ_{ℓ}} f(ζq^{\frac{1}{ℓ}}),  (25)

f|U(ℓ)(E_{∞}, ϕ_{N}, ω_{∞}) = \frac{1}{ℓ} ∑_{ζ∈μ_{ℓ}} f(ζq^{\frac{1}{ℓ}})  (26)

which gives rise to (24). This shows (1).

We now show (2). We can let ℤ_{p}^{×} act on V_{Γ/A} and V_{Γ,m,α/A_{m}} by

f|⟨a⟩(E, ϕ_{p}, ϕ_{N}) = f(E, a^{−1}ϕ_{p}, aϕ_{N}).

By a similar computation of the q–expansion, we get

a(m, f|T (n)) = ∑_{d|m,d|n} d^{−1}a(\frac{mn}{d^{2}}, f|⟨a⟩),  (27)

where d runs over all positive common divisors of n and m prime to Np. By our definition, the embedding of \overline{A}_{Γ}(A) into V_{Γ/A} is equivariant under the Hecke operators. Write \overline{A}′_{Γ}(A) for the sum of G_{k}(Γ, A) for k ≥ 2. The intersection \overline{A}′_{Γ}(A[\frac{1}{p}]) ∩ V_{Γ/A} is dense in V_{Γ/A} under the q–expansion norm (Theorem 20). Thus the assertion (1) implies (2) (or else, it also follows from the q–expansion principle: Corollary 4 for p–adic modular forms).

It is easy to verify the operators U(p^{r}) and T (n) commute each other either from their definition or from (23), (24) and (27). Similarly we can verify

U(p^{r}) = U(p)^{r} and T (m)T (n) = T (mn) if m and n are mutually prime. (28)

Note that, by (25) and (26), if k ≥ 2, f|T (p) ≡ f|U(p) mod p on G_{k}(Γ, A) if the level N of Γ is prime to p. Since A_{Γ}(K) ∩ V_{Γ/W} is dense (Theorem 20), the sequence {U(p)^{n!}}_{n} converges to a projector e acting on V_{Γ/W} (see Lemma 9). We can define V_{Γ}^{ord} by \underrightarrow{lim}_{α} \underrightarrow{lim}_{m}eV_{Γ,m,α} as in Theorem 18.

We write Λ = W[[ℤ_{p}^{×}]]. Let G_{Γ,Λ}^{∗} be the Pontryagin dual module of V_{Γ}^{ord}. We also define V_{Γ,cusp}^{ord} to be the subspace of V_{Γ}^{ord} made of elements whose q– expansion vanishes at the Tate curves. Define S_{Γ,Λ}^{∗} by the Pontyagin dual module of V_{Γ,cusp}^{ord}. Thus G_{Γ,Λ}^{∗} and S_{Γ,Λ}^{∗} are Λ–modules via the action of G_{m}(ℤ_{p}) = ℤ_{p}^{×} on p–adic modular forms by the diamond operators. We claim

Theorem 21. Let W be a discrete valuation ring finite flat over ℤ_{p} for a prime p ∤ N. Suppose that W contains μ_{N} if Γ = Γ(N). Let F = G and S. Then F_{Γ,Λ}^{∗} is a well controlled Λ–projective module of finite type. This means that we have the following canonical isomorphisms:

F_{Γ,Λ}^{∗} ⊗_{Λ,k} W ≅ Hom_{W}(F_{k}^{ord}(Γ, W), W) if k ≥ 3;
F_{Γ,Λ}^{∗} ⊗_{Λ,2} W ≅ Hom_{W}(F_{2}^{ord}(Γ ∩ Γ_{0}(p), W), W),

where p = p or 4 according as p > 2 or p = 2.

Proof. As remarked at the end of 7.2, the assertion for k ≥ 2 follows from the argument proving Theorem 18 in 7.2 and

rank_{W} F_{k}^{ord}(Γ ∩ Γ_{0}(p), χω_{p}^{−k}, W) = rank_{W} F_{2}^{ord}(Γ ∩ Γ_{0}(p), χω_{p}^{−2}, W), (29)

for all k ≥ 2, where F_{k}^{ord} is the image under the projector e, χ : (ℤ/Nℤ)^{×} → W^{×} is a fixed character, ω_{p} is the p–adic Teichmüller character (ω_{2} = ^{(\underline{−1})}) and p is the conductor of ω_{p}. The assertion (29) follows from a group cohomology argument in [H93] page 217. When k ≥ 3 and χω^{−k} is trivial on ℤ_{p}^{×}, we can replace Γ ∩ Γ_{0}(p) by Γ. This finishes the proof.

7.5 Family of ordinary p–adic modular forms

We now state a result: Theorem 22 more general than Theorems 1 and 2 for primes p including p = 2, 3 and give a detailed proof of them. After proving this general result, we deduce Theorems 1 and 2 from Theorem 22.

Let W be a valuation ring finite flat over ℤ_{p}. Write p = 4 or p according as p = 2 or not as in Theorem 21. The group 1 + pℤ_{p} is the maximal torsion free subgroup of ℤ_{p}^{×} = G_{m}(ℤ_{p}). We fix a generator u = 1 + p of this group. Let Λ = W[[1+pℤ_{p}]]. Write the group element z ∈ 1+pℤ_{p} in Λ as [z]. The profinite ring Λ is isomorphic to the formal power series ring W[[T]] via [u] ↦ 1 + T.

Let Γ be a congruence subgroup of between Γ_{0}(N) and Γ_{1}(N). We have Hecke operators of level Np acting on V_{Γ/W}. In this section, we write T (n) for U(n) even if n is not prime to Np. Let X be the W–submodule of V_{Γ/W} stable under Hecke operators T (n) for all n. We define H(X, W) = H(X) to be the W–subalgebra of End_{W}(X) generated by T (n) for all n.

A p–adic modular form f ∈ V_{Γ/W} is called of weight s ∈ ℤ_{p} on Γ if f|⟨z⟩ = z^{s}f for all z ∈ 1 + pℤ_{p}. We consider formal q–expansions

Φ(T; q) = ∑_{n≥0} a(n, Φ)(T)q^{n} ∈ Λ[[q]]

such that Φ(u^{s} − 1; q) is a p–adic modular form on Γ of weight s, where u^{s} ∈ 1 + pℤ_{p} is the p–adic power given by

u^{s} = ∑_{n≥0} \binom{s}{n}p^{n}.

The formal expansion Φ is called a Λ–adic modular form and the set of p– adic forms Φ = {Φ(u^{s} − 1; q)}_{s} is called a family of p–adic modular forms. A family of p–adic modular forms (or a Λ–adic form) is called arithmetic if for any sufficiently large positive integers k, Φ(u^{k} − 1; q) is a true modular form. We call Φ a family of cusp forms if Φ(u^{s} − 1; q) is a p–adic cusp form for infinitely many s (which is equivalent to saying that Φ(u^{s} − 1) is a cusp form for all s). Plainly the totality of Λ–adic forms is a Λ–module.

Since each element ϕ(T) in Λ can be considered as a p–adic bounded measure on 1 + pℤ_{p} (see [H93] Chapter III) in the following way:

∫_{1+pℤ_{p}} x^{s}dϕ = ϕ(u^{s} − 1)

for all p–adic integer s. Here boundedness implies

|∫ f(z)dϕ(z)|_{p} ≤ |f|_{p} = Sup_{z}|f(z)|_{p}.

Since V_{Γ/W} and W[[q]] have well defined norm: |f|_{p} = Sup_{r}|a(r, f)|_{p}, we can think of bounded measures on 1 + pℤ_{p} with values in V_{Γ/W}. For a given Λ–adic form Φ, we can define a p–adic bounded measure with values in V_{Γ/W} by

∫ z^{s}dΦ = ∑_{n≥0} a(n, Φ)(u^{s} − 1)q^{n}.

Originally this is just a formal power series in q, but this value actually falls in V_{Γ/W} by the very definition of Λ–adic forms. For any bounded Λ–linear operator T acting on V_{Γ/W}, T ∘ dΦ is another bounded measure; so, corresponding to another Λ–adic form Φ|T. In this way, G_{m}(ℤ_{p}) acts on the space of Λ–adic forms by the diamond operators, and this action coincides with that of Λ on the subgroup 1 + pℤ_{p}. Similarly the Hecke operators T (n) act on the space of Λ–adic forms as Λ–linear operators. For a given Λ–adic modular from Φ, the effect of the Hecke operator T (n) on the q–expansion coefficients is given again by the following now familiar formula:

a(m, Φ|T (n)) = ∑_{d|n,d|m} d^{−1}a(\frac{mn}{d^{2}}, Φ|⟨d⟩).  (30)

where d runs over all common divisors of m and n prime to Np.

Let X be a Λ–submodule in the space of Λ–adic forms stable under Hecke operators. Similarly to the case for W, we define H(X) = H(X, Λ) by the Λ–subalgebra of End_{Λ}(X) generated by all Hecke operators T (n).

Lemma 10. We have the following assertions:

1. Let X be a W–submodule of finite rank in V_{Γ/W}. Suppose that the constant term of the q–expansion at ∞ of every element of X vanishes and that (X ⊗_{W} K) ∩ V_{Γ/W} = X. Then Hom_{W}(H(X, W), W) ≅ X and Hom_{W}(X, W) ≅ H(X, W) canonically.

2. Let X be a Λ–submodule of finite type in the space of Λ–adic forms. Suppose that the space of Λ–adic forms modulo X is Λ–torsion-free and that the constant term of the q–expansion of every element of X vanishes. Then Hom_{Λ}(H(X, Λ), Λ) ≅ X, canonically.

Proof. Since W is a discrete valuation ring, we only need to prove one isomorphism, say, Hom(H(X), W) ≅ X. We consider the following pairing:

( , ) : H(X) × X → W

given by (h, f) = a(1, f|h). By (27), we have (T (n), f) = a(n, f). By the q–expansion principle, if (h, f) = 0 for all h ∈ H(X), f = 0. This implies that X ↪ Hom_{W}(H(X), W) by this pairing. Since H(X, K) = H(X) ⊗_{W} K and X_{K} = X ⊗_{W} K are finite dimensional, this argument tells us that X_{K} ↪ Hom_{W}(H(X, K), K). Suppose that (h, f) = 0 for all f ∈ X_{K}. We see

a(n, f|h) = a(1, f|hT (n)) = (h, f|T (n)) = 0,

showing that h = 0. Thus X_{K} ≅ Hom_{W}(H(X, K), K). From this, for each W–linear map ϕ : H(X, W) → W, f = ∑_{n>0} ϕ(T (n))q^{n} ∈ X_{K} ∩ V = X is the p–adic modular form inducing ϕ under the pairing. This shows the desired surjectivity of the isomorphism.

The same argument works well for the Λ–module X, just replacing (K, W) by the field of fractions of Λ and Λ. This finishes the proof.

A p–adic modular form f is called p–ordinary if f ∈ eV_{Γ/A} for a p–adic ring A. A family of p–adic modular form is called p–ordinary if every member of the family is p–ordinary. Let G_{Γ,Λ} and S_{Γ,Λ} be the space of all p–ordinary Λ–adic forms on Γ.

Theorem 22. Let Γ be a congruence subgroup between Γ_{0}(N) and Γ_{1}(N) of level N for N prime to p. Let F = G and S. Let h = h(Γ, Λ) = H(S_{Γ,Λ}) and h_{k}^{ord}(Γ ∩ Γ_{1}(p), W) = H(S_{k}^{ord}(Γ ∩ Γ_{1}(p), W)). Then we have

(1) h ≅ S_{Γ,Λ}^{∗} and h ⊗_{Λ,k} W ≅ h_{k}^{ord}(Γ ∩ Γ_{1}(p), W);

(2) F_{Γ,Λ} ≅ Hom_{Λ}(F_{Γ,Λ}^{∗}, Λ); (3) F_{Γ,Λ} is free of finite rank over Λ;

(4) If k ≥ 2, then the specialization k : Φ ↦ Φ(u^{k} − 1; q) induces

F_{Γ,Λ} ⊗_{Λ,k} W ≅ F_{k}^{ord}(Γ ∩ Γ_{1}(p), W);

(5) All p–ordinary Λ–adic forms are arithmetic.

Proof. By q–expansion, we have a(n) : V_{ord,Γ} → K/W associating to f its coefficients of q^{n}. This is an element of G_{Γ,Λ}^{∗}. We consider a formal q–expansion

Hom_{Λ}(G_{Γ,Λ}^{∗}, Λ) → Λ[[q]]

given by ϕ ↦ Φ(T; q) = ∑_{n} ϕ(a(n))q^{n}. Since G_{Γ,Λ}^{∗} is Λ–free, we get

Hom_{Λ}(G_{Γ,Λ}^{∗}, Λ) ⊗_{Λ,k} W ≅ Hom_{W}(G_{Γ,Λ}^{∗} ⊗_{Λ,k} W, W)

≅ Hom_{W}(Hom(G_{k}(Γ ∩ Γ_{1}(p), W), W) ≅ G_{k}(Γ ∩ Γ_{1}(p), W).

Tracking down our construction, this isomorphism brings the formal q–expansion Φ to Φ(u^{k} − 1; q). Thus ϕ ↦ Φ induces

Hom_{Λ}(G_{Γ,Λ}^{∗}, Λ) ≅ G_{Γ,Λ},

and therefore G_{Γ,Λ} is Λ–free of finite rank. This shows (2), (3), (4) and (5) for F = G. The assertion for cusp forms can be proven similarly. Then (1) follows from Lemma 10 (1) and (2). In particular, it sends T (n) to a(n).

Proof of Theorems 1 and 2. When p is odd, we have

F_{k}^{ord}(Γ ∩ Γ_{1}(p), W) ≅ ⨁_{a=0}^{p−2} F_{k}^{ord}(Γ ∩ Γ_{0}(p), ω^{a}, W).

Taking χ–quotient of the above formula, by (4) of the above theorem, we conclude the proof of Theorem 2. By (1) of the above theorem, taking again the χ–quotient, we see the assertion of Theorem 1.

8 Applications

In this section, we describe some arithmetic applications of the vertical control theorem, the horizontal control theorem and the theorem of Wiles-Taylor-Diamond (Theorem 9).

We start with some generalities on F –motives following [R4]. Then, we describe a sufficient condition for the associated Galois representation to be residually dihedral non-dihedral (this condition is almost necessary as well; see [H00b] V.1.1). When the associated Galois representation is residually dihedral, the Galois representation is modular by Theorem 9 under the assumptions in this theorem. Moreover, since being residually dihedral is preserved under basechange, we get a base-change result at the same time.

8.1 F –motives

All motives we consider are absolute Hodge motives defined by Deligne in [DM].

Let M_{/\overline{ℚ}} be a pure motive of rank n with coefficients in a number field E. Let F be another number field. We say that M is an F –motive of rank n with coefficients in E, if

• End_{E}(M_{/\overline{ℚ}}) = E, and,

• there is an isomorphism μ_{σ} : σ(M) → M defined over \overline{ℚ} for all σ ∈ Gal(\overline{ℚ}/F) commuting with the action of E.

Here End_{E}(M_{/\overline{ℚ}}) is the E–linear endomorphism ring. Moreover, we assume that μ_{1} = id.

We say that a number field X is an anti-CM field if it does not contain a CM subfield. Hereafter we often assume that the F –motives M we consisder satisfy the following condition.

Either F or E is an anti-CM field.  (ACM)

Let G_{F} := Gal(\overline{ℚ}/F). Then (σ, τ) ↦ c(σ, τ) = μ_{σ} ∘^{σ}μ_{τ} ∘μ_{στ}^{−1} ∈ End(M)^{×} = E^{×} gives a locally constant 2–cocycle on G_{F} with values in the trivial G_{F} – module E^{×}. The following propositon was proved by Ribet [R4] 3.1 for F – HBAVs. The motivic version below, as well as the proof is a word for word reinterpretation of Ribet’s result.

Proposition 8. If Res_{K/F} [c] ∈ H^{2}(G_{K}, E^{×}) vanishes for an algebraic extension K/F, then M is defined over K.

Proof. Let L/F be a finite Galois extension such that there is a 1–cochain α : Gal(L/F) → E^{×} such that c(σ, τ) = α(σ)α(τ)/α(στ). Replacing μ_{σ} by α(σ)^{−1}μ_{σ}, we get c(σ, τ) = μ_{σ} ∘^{σ}μ_{τ} ∘ μ_{στ}^{−1} = 1.

Consider M̃ = Res_{L/F} M. The functor Res_{L/F} satisfies the following Frobenius reciprocity: for all motives N defined over F,

Hom(N_{/F} , M̃_{/F}) ≅ Hom(N_{/L}, M_{/L}),

where Hom(X_{/K}, Y_{/K}) for motives X and Y over a field K stands for the group of morphisms of motives defined over K. Supposing that N_{/F} also has coefficients in E. Then taking the commutant of E, we get

Hom_{E}(N_{/F} , M̃_{/F}) ≅ Hom_{E}(N_{/L}, M_{/L})

for E–linear morphisms Hom_{E}. Since the restriction of scalar has expression: M̃_{/L} ≅ ⨁_{σ∈Gal(L/F)} σ(M), writing μ_{σ} for the direct sum: ⊕_{τ∈Gal(L/F)}^{τ} μ_{σ} as

an endomorphism of M̃_{/L}, this morphism μ_{σ} corresponds to the projection of M̃_{/L} to σ(M) composed with μ_{σ} : σ(M) → M. We see clearly that

End_{E}(M̃_{/F}) = ⨁_{σ∈Gal(L/F)} Eμ_{σ},

and μ_{σ}μ_{τ} = c(σ, τ)μ_{στ}. Since c = 1, σ ↦ μ_{σ} gives rise to an isomorphism of the group algebra E[Gal(L/F)] onto End_{E}(M̃_{/F}). The idempotent

e = [L : F]^{−1} ∑_{σ∈Gal(L/F)} σ

gives a factor eM̃ defined over F, which becomes isomorphic to M_{/L} after base extension.

The following result deals with the case of F –motives of rank 1:

Proposition 9. Let N be an F -motive of rank 1 satisfying (ACM). Then N can be defined over F.

Proof. We take a finite Galois extension L/F over which N is defined. The isomorphism class of N is determined by the associated Hecke character of φ : L_{𝔸}^{×}/L^{×} → ℂ^{×} (see [De] and [Bl] Section 3.2). Let the formal sum m = ∑_{σ} m_{σ}σ denote the infinity type of φ. If E is an anti-CM field, we have φ(x^{(∞)}) ∈ E, and hence m_{σ} = m_{σc} for complex conjugation c. This shows that [m] := m_{σ} ∈ ℤ independent of σ.

Now suppose that F is an anti-CM field. The motive σ(N) is associated to the character φ^{σ}(x) = φ(σ(x)) for σ ∈ Gal(L/F). Then mσ = m for the infinity type of φ, and m is the inflated image of an infinity type of a Hecke character of F. Since F is anti-CM, again we conclude that m_{σ} is independent of σ.

The motive N ⊗ ℚ([m]) is a rank 1 Artin motive associated to a finite order character. Enlarging L, we may assume that N ≅ E ⊗_{ℚ} ℚ(−[m]) = E(−[m]). We fix such an isomorphism λ : N ≅ E(−[m]) and write ν_{σ} : σ(N) ≅ N for λ^{−1} ∘^{σ}λ : σ(N) ≅ N. We define γ(σ, τ) = ν_{σ} ∘^{σ}ν_{τ} ∘ ν_{στ}^{−1}. We have γ = 1 since ν_{σ} = λ^{−1} ∘^{σ}λ. Thus the obstruction class [γ] vanishes in H^{2}(G_{F} , E^{×}).

Corollary 5. Let M be a rank n F –motive satisfying (ACM). Then its obstruction class is killed by n.

Proof. Write c for the obstruction cocycle. Consider N = ⋀^{n} M, which is rank 1 F –motive satisfying (ACM). Note that c(σ, τ)^{n} gives rise to the obstruction for N. By the above proposition, we see n[c] = [c^{n}] = 0.

Now suppose that M is an F –motive of rank 2 with coefficients in E satisfying (ACM). Suppose that E does not contain \sqrt{−1}. By [R4] 3.3, [c] gets trivialized over G_{K} for a (2, 2, …, 2)–extension K/F. Although in [R4], the coefficient field E (written as F there) is supposed to be totally real, one can easily check that the proof works equally well for any field without \sqrt{−1}.

Proposition 10. Suppose that M is a rank 2 F –motive with coefficients in a field E. If M satisfies (ACM) and \sqrt{−1} ∉ E, then M can be defined over a (2, 2, …, 2)–extension of F.

Suppose that the rank 2 F –motive M with coeffcients in E is defined over a (2, 2, …, 2)–extension K/F. We assume that E^{×}/{±1} is torsion-free, that is, E does not have complex roots of unity. We suppose further that M is regular of Hodge type (p_{σ}, q_{σ}), (q_{σ}, p_{σ}) p_{σ} > q_{σ} for each place σ of K. Then M_{DR} has a Hodge filtration F ⊂ M_{DR} which is free of rank 1 over K ⊗_{ℚ} E (see [H96] 5.1). Choose a generator ω of F. For any δ ∈ G_{F}, there is e_{δ} ∈ (\overline{ℚ} ⊗ E)^{×} such that μ_{δ}^{∗}ω = e_{δ}δ(ω). For σ ∈ G_{K} we have

(^{σ}μ_{δ})^{∗}ω = ^{σ⊗1}e_{δ}δ(ω).

Thus

σ⊗1e_{δ}δ(ω) = (^{σ}μ_{δ})^{∗}ω = e_{δ}(μ_{δ}^{−1} ∘^{σ}μ_{δ})^{∗}δ(ω),

and

e_{δ}^{−1}(^{σ⊗1}e_{δ}) = μ_{δ}^{−1} ∘^{σ}μ_{δ} =: ζ_{σ} ∈ End_{E}(M_{/K}) = E.

Since σ ↦ e_{δ}^{−1}(σ ⊗ 1)(e_{δ}) is a character of G_{K} with values in E^{×}, we see that ζ_{σ} is a root of unity in E and that e_{δ} ∈ C_{δ} ⊗ 1(K ⊗ E) with C_{δ} ∈ \overline{ℚ}^{×}. Since the only roots of unity in E are ±1, ζ_{σ} = ±1. The map: σ ↦ ^{σ−1}C_{δ} ∈ {±1} gives an isomorphism α of Gal(K_{δ}/K) into {±1} for at most a quadratic extension K_{δ} = K[C_{δ}], which is the minimal field of definition of μ_{δ}. We have ^{σ}μ_{δ} = α(σ)μ_{δ} and ^{σ}e_{δ} = α(σ)e_{δ}.

Suppose that [K_{δ} : K] = 2. Since

(μ_{δ} ∘^{δ}μ_{δ})^{∗}ω = e_{δ}^{δ}e_{δ}ω,

we see that a = e_{δ}^{δ}e_{δ} ∈ E for any extension of δ to G_{F}. From this, we have

δ^{2}e_{δ} = δ(ae_{δ}^{−1}) = e_{δ},

and δ^{2} = 1 for all δ ∈ Gal(K_{δ}/K) inducing δ on K. Thus again Gal(K_{δ}/K) is a (2, …, 2)–group.

Writing Δ = Gal(K/F), we find that the minimal field of definition of all {μ_{δ}|δ ∈ Δ} is a (2, …, 2)–extension L of F. We have obtained the following theorem:

Theorem 23. Let K/F be a field of definition of rank 2 regular F –motive M with coefficients in a number field E. Suppose that Gal(K/F) is a (2, 2, …, 2)– group and that roots of unity in E are only ±1. Then the data {M, μ_{δ}|δ ∈ Δ} is defined over a (2, 2, …, 2)–extension L/F containing K.

Note that there are examples of rank 2 F –motives with coefficients in E whose endomorphism algebra End(M_{/\overline{ℚ}}) is bigger than E. We say that a motive M_{/\overline{ℚ}} is an F –motive of division type if End(M_{/\overline{ℚ}}) = D for a central division algebra D over an anti-CM field E′, and we have an isomorphism μ_{σ} : σ(M) ≅ M compatible with the action of D for all σ ∈ G_{F}. We defined the division rank of M to be the dimension over D of the Betti realization M_{B}. The obstruction cocycle c commutes with D and hence has values in E′^{×}. We suppose that D is isomorphic to its opposite algebra D^{∘}, and D has an involution ι with ι(xy) = ι(y)ι(x) fixing E′; so, D is a quaternion algebra over E′. An anti-CM quadratic extension E/E′ inside D is the commutant of E itself in D. Thus we may regard M_{/\overline{ℚ}} as an F –motive of rank 2 over E. We can construct an F –motive of this type from a simple abelian variety with multiplication by D if E′ is totally real.

8.2 Endomorphism algebras of F –motives

Let M_{/\overline{ℚ}} be an F –motive with coefficients in a number field E. We suppose that the rank of M over E is 2. We assume that there is a model (M, μ_{δ})_{/L} over a (2, …, 2)–extension L/F.

From now we werite Δ for Gal(L/F). We continue to write M̃ = Res_{L/F} M. As we have already seen,

B = End_{E}(M̃_{/F}) = Hom_{E}(M̃_{/F} , M̃_{/F}) = Hom_{E}(M̃_{/L}, M_{/L})

= ⨁_{δ∈Δ} Hom_{E}(δ(M_{/L}), M_{/L}) = ⨁_{δ} Eμ_{δ}. (31)

Here as an endomorphism of M̃_{/L} ≅ ⊕_{δ}δ(M), μ_{δ} brings (x_{δ})_{δ} with x_{δ} ∈ δ(M) to

(^{δ′}μ_{δ}(x_{δδ′}) ∈ δ′(M))_{δ′} .

This shows that μ_{σ} ∘ μ_{τ} ∘ μ_{στ}^{−1} = c(σ, τ) ∈ E in general and [B : E] = |Δ| = 2^{r}. In particular, μ_{δ}^{2} = c(δ, δ) = c_{δ} ∈ E.

Since μ_{σ}μ_{τ} = c(σ, τ)μ_{στ}, the algebra B = End_{E}(M̃_{/F}) contains the E– algebra given by the cocycle c. Since 2–coboundaries of 1–cochains on Δ satisfies ∂b(σ, τ) = ∂b(τ, σ) (by the commutativity of Δ), the commutativity c(σ, τ) = c(τ, σ) for all σ, τ ∈ Δ depends only on the obstruction class Ob_{F} (M) = [c] in H^{2}(Δ, E^{×}). We call Ob_{F} (M) commutative if c(σ, τ) = c(τ, σ) for all σ, τ ∈ Δ.

Remark 5. Suppose an elliptic curve E is defined over a quadratic field ℚ[\sqrt{D}] with an isogeny θ : ^{δ}E → E for the generator δ of Gal(ℚ[\sqrt{D}]/ℚ). If θ is not rational over ℚ[\sqrt{D}], then it is rational over a (2, 2)–extension L/ℚ containing ℚ[\sqrt{D}]. In this case, it is easy to check that Ob(M) is not commutative.

Lemma 11. Let (M_{/\overline{ℚ}}, μ_{σ}) be a rank 2 F –motive with coefficients in a field E defined over a (2, 2, …, 2)–extension L/F with Δ = Gal(L/F). Suppose that \sqrt{−1} ∉ E. Then there exists an F –motive (M′, μ′_{σ}) defined over a subfield L′ ⊃ F of L with coefficients in an extension E′/E such that

(1) M′ is isomorphic to M ⊗_{E} E′ over an abelian extension of L;

(2) Ob(M′) is commutative.

Moreover in the process of making M′, we may assume that for a given σ ∈ Δ, σ acts non-trivially on L′ and μ′_{σ} = μ_{σ} ⊗ id_{E′}.

Proof. The proof is by induction on [L : F].

By [Br] Exercise V.6 (5) (and Exercise IV.3 (8)), the map of 2–cocycle given by c(σ, τ) ↦ c(σ, τ) − c(τ, σ) induces a homomorphism

θ : H^{2}(Δ, A) → Hom(⋀^{2} Δ, A)

for all trivial Δ–modules A, and Ker(θ) is made of commutative classes. In particular, if A is 2–torsion-free, H^{2}(Δ, A) is spanned by commutative classes.

Let c : Δ → E^{×} be an obstruction cocycle. We have a split exact sequence

1 ⟶ {±1} ⟶ E^{×} ⟶ P ⟶ 1

for a 2–torsion-free module P, on which Δ acts trivially (cf. [R4] (3.5)). Thus Ob(M) = [c] ∈ H^{2}(Δ, {±1}) × H^{2}(Δ, P). We can split c = c_{±} × c_{P} for cocycles with values in {±1} and P, respectively. Since P is 2–torsion-free, the cocycle c_{P} is commutative.

We may assume that θ(c) = θ(c_{±}) ≠ 0, because otherwise there is nothing to prove. We regard θ(c) as a bilinear form Θ : Δ × Δ → 𝔽_{2} on the 𝔽_{2}–vector space Δ. Take σ and τ in Δ such that Θ(σ, τ) ≠ 0. Regarding Δ as a vector space over 𝔽_{2}, let X be the 2–dimensional subspace spanned by τ and σ. We have Δ = X ⊕ X^{⊥}, where X^{⊥} is the orthogonal complement of X under Θ.

We consider c_{X} = c|_{X}. By definition

μ_{σ}^{σ}μ_{τ} μ_{στ}^{−1} = c(σ, τ) = (−1)^{Θ(σ,τ)}c(τ, σ) = −μ_{τ}^{τ} μ_{σ}μ_{στ}^{−1}.

We have

μ_{σ}^{σ}μ_{τ} = −μ_{τ}^{τ} μ_{σ}  (32)

and μ_{σ} and μ_{τ} commute with μ_{δ} for all δ ∈ X^{⊥}. We replace the motive M by M ⊗_{E} E[\sqrt{c(τ, τ)}] and then E by E[\sqrt{c(τ, τ)}]. By our choice, c(τ, τ) is square and hence c|_{⟨τ⟩} = 0 in H^{2}(⟨τ⟩, E^{×}) = E^{×}/(E^{×})^{2}. We can descend M to the subfield K of L fixed by τ, and we may change μ_{τ} by the identity map of M_{/K}. We thus have μ_{σ}^{τ} = −μ_{σ} by (32).

We need to twist the action of G_{K} = Gal(\overline{ℚ}/K) by a character η so that μ_{σ} becomes K–rational under the twisted action. Writing the twisted action as g · x for g ∈ G_{K}, we have

g · (μ_{s}(x)) = η(g)g(μ_{σ}(x)) = η(g)^{g}μ_{σ}((σgσ^{−1})x)

= η(g)η(σgσ^{−1})^{−1}μ_{σ}((σgσ^{−1}) · x).

If we find a finite order character η of Gal(\overline{ℚ}/K) such that η^{1−σ}(τ) = −1, we get the desired motive M′ twisting M by η (and replacing E by the field generated by the values of η over E). Here η^{σ}(g) = η(σgσ^{−1}).

Let ε : K_{𝔸}^{×}/K^{×} → {±1} be the quadratic character corresponding to L/K. We want to have a finite order Hecke character η : K_{𝔸}^{×} → μ_{N} such that η^{1−σ} = ε, where η^{σ}(x) = η(σ(x)) for x ∈ K_{𝔸}^{×}. Let F′ be the subfield of K fixed by σ. We have [L : F′] = 4.

We now show that η with ε = η^{σ−1} always exists. Let k be a number field. By class field theory, any continuous character of Gal(\overline{ℚ}/k) can be regarded as a continuous idele character: C_{k} = k_{𝔸}^{×}/k^{×} → 𝕋, where

𝕋 = {z ∈ ℂ||z| = 1} .

A given continuous character of C_{k} is of finite order if and only if it is trivial on the identity component of the infinite part F_{∞}^{×} of F_{𝔸}^{×} (cf. [H00a] Proposition 2.2). By Artin reciprocity, any continuous character of C_{k} trivial on the identity component of k_{∞}^{×} ⊂ k_{𝔸}^{×} can be viewed as a (finite order) character of Gal(\overline{ℚ}/k) canonically.

Looking at the exact sequence:

1 → K^{×} → K_{𝔸}^{×} → C_{K} → 1,

by Hilbert’s theorem 90 applied to K^{×} and Gal(K/F′) = ⟨σ⟩, we find

H^{0}(Gal(K/F′), C_{K}) = C_{F′},

and the kernel of σ−1 : x ↦ x^{σ−1} is given by C_{F′}. A character ψ : C_{K} → 𝕋 is of the form ψ = η^{σ−1} if and only if ψ is trivial on C_{F′}. Since Gal(L/F′) ≅ (ℤ/2ℤ)^{2}, we find a quadratic character χ of C_{F′} such that ε = χ ∘ N_{K/F′}. This shows that ε(x) = χ(xx^{σ}) = χ(x^{2}) = 1 for x ∈ C_{F′}. Thus we can write ε = η^{σ−1} for a character η : C_{K} → 𝕋.

To have η factor through the Galois group of the maximal abelian extension of K, we need to show that η can be chosen so that its restriction to the identity component of K_{∞}^{×} is trivial. Since ε = η^{σ−1} is trivial on the identity component K_{∞+}^{×}, η is trivial on (K_{∞+}^{×})^{σ−1} = Ker(N_{K/F′} : K_{∞+}^{×} → F′_{∞}^{×}). Thus η|_{K_{∞+}^{×}}

factors through N_{K/F′} : K_{∞+}^{×} → F′_{∞+}^{×}. Replacing η by η(ξ ∘ N_{K/F′}) for a Hecke character ξ of F′, we may assume that η is trivial on K_{∞+}^{×}.

The motive M ⊗ η is of rank 2 with coefficients in E(η) and is defined over K. We identify Gal(K/F) with X^{⊥} ⊕ ⟨σ⟩. Then

(M ⊗ η_{/K}, μ′_{δ} = μ_{δ} ⊗ id_{E(η)})_{δ∈X^{⊥}⊕⟨σ⟩}

is an F –motive with coefficients in E(η) defined over K. Note that μ′_{σ} commutes with μ′_{δ} for all δ ∈ X^{⊥}.

Repeating the above process (replacing Δ and (M, μ_{δ}) by Gal(K/F) and (M ⊗ η_{/K}, μ′_{δ})), we can find eventually an extension E′/E(η) and an F –motive (M′, μ′_{δ})_{/L′} with coefficients in E′ satisfying the required property of the lemma.

The induction terminates after exhausting all (mutually orthogonal) pairs (σ, τ) with Θ(σ, τ) ≠ 0. Thus [L : F] ≥ [L′ : F] ≥ 2.

The last assertion is clear from our construction, because we kept μ_{σ} and μ_{σ} commutes with μ_{δ} for all δ ∈ X^{⊥}.

Lemma 12. Let ε be as in the proof of the above lemma. We can find a finite order character η so that ε = η^{1−σ} and η is unramified at a given finite set S of primes outside the ramification of ε.

Proof. Indeed, if η ramifies at 𝔩 ∈ S, for an inertia group I ⊂ G_{K} of a prime 𝔩, we have η = η^{σ} on I because ε is unramified at 𝔩, and η extends to the inertia group of G_{F′} for the subfield F′ ⊂ K fixed by σ (because Gal(K/F′) is cyclic:

[H00a] Corollary 4.37). Choose a finite order character ξ of G_{F′} unramified at S − {𝔩} so that ηξ is unramified at 𝔩 (by using a result of Chevalley [C]). Repeating this process, we find the desired character η.

By Lemma 11, we may assume that the F –motive M has commutative obstruction class.

Proposition 11. Let (M\overline{ℚ}, μ_{σ}) be a rank 2 F –motive with coefficients in a number field E and (M_{/L}, μ_{δ}) for δ ∈ Δ = Gal(L/F) be a model of (M\overline{ℚ}, μ_{σ}) over a (2, 2, …, 2)–extension L over F. Take a minimal set {δ_{1}, …, δ_{r}} of generators of Δ, and put c_{j} = c(δ_{j}, δ_{j}). If Ob(M) is commutative, we have

End_{E}(M̃_{/F}) ≅ E[X_{1}, …, X_{r}]/(X_{1}^{2} − c_{1}, …, X_{r}^{2} − c_{r}).  (33)

Proof. Write D for the algebra on the right-hand-side of equation (33). Since Ob(M) is commutative, the algebra generated by {μ_{δ}}_{δ∈Δ} in End(M̃_{/F}) is dimension |Δ| over E with relations: μ_{δ_{i}}^{2} = c_{i}. Therefore this algebra is a surjective image of D by sending X_{i} to μ_{δ_{i}}. Comparing the dimension, we get the result.

8.3 Modularity of Galois representations of ℚ–motives

In this section, we assume F = ℚ (though sometimes we do continue to assume that F is a general number field). Thus, we assume the following conditions:

(M1) (M_{/\overline{ℚ}}, μ_{σ}) is a ℚ–motive with coefficients in a number field E;

(M2) (M, μ_{σ}) has a model (M, μ_{δ})_{δ∈Δ} defined over a (2, 2, …, 2)–extension L/ℚ with Δ = Gal(L/ℚ);

(M3) Ob(M) is commutative;

(M4) rank_{E} M = 2.

By Proposition 10 and Theorem 23, (M2) follows from (M1) and (M4) if the only roots of unity in E are ±1. By Lemma 11, We may also twist M to assure that (M3) holds, under (M2).

We briefly recall the construction done in [H00b] V.1.1: For each δ ∈ Δ = Gal(L/F), we fix an extension of δ to \overline{ℚ}, still written as δ ∈ G_{F}. Let A be a discrete valuation ring finite flat over ℤ_{p}. Let V = A^{2} be a continuous A[H]–module for H = Gal(\overline{ℚ}/L) whose ramification is restricted to a finite set. Writing ρ_{H} for the Galois representation on V, we suppose (I0) V = V (ρ_{H}) is generated over A[H] by a single element;

(I1) ρ_{H} is absolutely irreducible over the quotient field of A;

(I2) We have an A–linear endomorphism T_{δ} : V → V such that T_{δ}(hx) = δhδ^{−1}T_{δ}(x) and det T_{δ} ≠ 0.

We then define T_{hδ} = hT_{δ} for h ∈ H. It is easy to check the property (I2) for T_{hδ} and hδ in place of T_{δ} and δ. Dividing T_{δ} by an element of A, we may assume that \overline{T}_{δ} = (T_{δ} mod 𝔪_{A}) ≠ 0 for the maximal ideal 𝔪_{A} of A.

We consider W = A[G]⊗_{A[H]} V for G = Gal(\overline{ℚ}/ℚ), which is again generated over A[G] by a single element. We see from (I2),

(ρ_{H}(δ^{−2})T_{δ}^{2}) (hx) = h (ρ_{H}(δ^{−2})T_{δ}^{2}) (x).

The operator T̃_{σ} : W → W given by T̃_{σ}(δ ⊗ v) = δσ^{−1} ⊗ T_{σ}(v) is firstly welldefined and secondly commutes with the action of H. To see well-definedness, we have, for h ∈ H

(hσ)^{−1} ⊗ T_{hσ}(v) = σ^{−1} ⊗ h^{−1}T_{hσ} = σ^{−1} ⊗ T_{σ},

because hT_{σ} = T_{hσ}. In particular, T̃_{σ} only depends on the restriction of σ to L. The commutativity of T_{σ} with the multiplication by g ∈ G from the left is obvious, because we have multiplied δ by σ from the right. The commutativity of T_{σ} and T_{τ} is more subtle. Since T_{σ}T_{τ} has the same effect as T_{στ}, by Schur’s lemma, we get c(σ, τ) ∈ A such that T_{σ}T_{τ} = c(σ, τ)T_{στ}. If this obstruction cocycle is commutative, the commutativity between T̃_{δ} with δ ∈ G follows from the following computation:

T̃_{τ} T̃_{σ}(δ ⊗ v) = δσ^{−1}τ^{−1} ⊗ T_{τ} T_{σ}(v) = δ(τσ)^{−1} ⊗ T_{τ} T_{σ}(v) = c(τ, σ)T̃_{τσ}(v).

We would like to apply the above argument to the Galois representation arising from ℚ–motive (M, μ_{δ})_{δ∈Δ} with coefficients in the field E.

First we choose a prime ideal 𝔭 of E with odd residual characteristic p, put A = O_{E,𝔭}. Take an A–lattice V stable under G_{L} in the 𝔭–adic étale realization M_{𝔭}.

Suppose (I0) and (I1) hold for this choice of H. The condition (I1) is automatic if M is associated to H_{1} of an abelian variety, by the solution of the Tate conjecture for abelian varieties by Faltings. If a similar conjecture holds for motives, (I1) follows from it. We can achieve (I0), replacing V by A[H]v for 0 ≠ v ∈ V under (I1).

For each δ ∈ Δ, we then define T_{δ}(v) = μ_{δ}(δ(v)). The pair (V, T_{δ}) satisfies (I0-2). Multiplying T_{δ} by scalar in F rac(A) = E_{𝔭}, we may assume that \overline{T}_{δ} = T_{δ} mod 𝔪_{A} is non-zero. The obstruction cocycle c(σ, τ) given by T_{σ}T_{τ} = c(σ, τ)T_{στ} under this normalization is called the obstruction cocycle normalized at 𝔭. We write 𝔽 for A/𝔪_{A}. Let

W = A[G] ⊗_{A[H]} V = ⨁_{δ∈Δ} δ ⊗ V.

By (I0), W = A[G]v for v ∈ V. Write ρ_{G}(σ) for the action of σ ∈ G on W as an A–linear operator. Choosing δ ∈ G inducing the original δ on L, we define T̃_{σ}(δ ⊗ v) = δσ^{−1} ⊗ T_{σ}.

Theorem 24. Let the assumption and the notation be as above. Assume (M1-4). Let δ_{1}, …, δ_{r} be a minimal set of generators of Δ = Gal(L/ℚ), and write c_{δ} ∈ A for T̃_{δ}^{2} (that is, c_{δ} is the value of c(δ, δ) for the obstruction cocycle normalized at 𝔭). In particular, we write c_{j} for c_{δ_{j}}. Then we have

(1) B = End_{A[G]}(W) ≅ A[X_{1}, …, X_{r}]/(X_{1}^{2} − c_{1}, …, X_{r}^{2} − c_{r}) via T̃_{δ_{j}} ↔ X_{j}.

(2) W is free of rank 2 over B; so, we write ρ_{G} : G → GL_{2}(B) for the Galois representation on W.

(3) Suppose c_{δ} ∈ 𝔪_{A} for one δ ∈ Δ. Then there exists a maximal ideal 𝔪 of B such that W/𝔪W restricted to H has a unique one dimensional subspace stable under H on which H acts by a character ξ. Let H′ ⊃ H be a maximal subgroup to which ξ can be extended to a character ξ_{H′} of H′. If H′ ≠ G, then H′ is of index two in G and ξ_{H′}^{δ} ≠ ξ_{H′} for δ ∉ H′, and W/𝔪W is isomorphic to the absolutely irreducible representation Ind_{H′}^{G} ξ_{H′} for a maximal ideal 𝔪 of B.

(4) Suppose that c_{δ} ∈ 𝔪_{A} for an element δ ∈ Δ. If ξ ≠ ξ^{δ} (for ξ^{δ}(h) = ξ(δhδ^{−1})) with the notation of (3), then ξ extends to a character ξ_{H′} of a subgroup H′ ⊃ H of index two in G, and W/𝔪W is isomorphic to the absolutely irreducible representation Ind_{H′}^{G} ξ_{H′} for a unique maximal ideal 𝔪 of B.

Proof. Choose a complete representative set Δ of G/H inside G. The family of linear operators \{\overline{t}_{δ} = T̃_{δ} mod 𝔪_{A}|δ ∈ Δ\} is linearly independent. Indeed, if \overline{t} = ∑_{δ} λ_{δ}\overline{t}_{δ} = 0, for each v ∈ \overline{V} with \overline{T}_{δ}v ≠ 0, the projection of \overline{t}v to δ ⊗ \overline{V} in \overline{W} = W/𝔪_{A}W is equal to

λ_{δ}δ^{−1} ⊗ \overline{T}_{δ}v = λ_{δ}δ ⊗ ρ_{H}(δ^{−2})\overline{T}_{δ}v;

so, λ_{δ} = 0. This shows that the 𝔽–subalgebra B of End(\overline{W}) generated by \overline{t}_{δ} (δ ∈ Δ) has rank 2^{r} = |Δ|. By our construction, we have

T̃_{σ}T̃_{τ} (δ ⊗ v) = δ(στ)^{−1} ⊗ T_{σ}T_{τ} (v) = c(σ, τ)T̃_{στ} (δ ⊗ v).

So, the algebra generated by T̃_{σ} over A is commutative (because c(σ, τ) is commutative). As we have already seen before stating the proposition, B commutes with the action of G; so, B ⊂ End_{A[G]}(W).

We now prove that B = End_{A[G]}(W). We let G acts on End_{A}(\overline{W}) by conjugation: T ↦ ρ_{G}(g)T ρ_{G}(g)^{−1}. We embed A into \overline{ℚ}_{p} and take an extension ϱ : G → GL_{2}(\overline{ℚ}_{p}) of ρ_{H}. Such an extension exists, because W ⊗_{ℤ_{p}} ℚ_{p} is free of rank 2 over B ⊗_{ℤ_{p}} ℚ_{p}. Note that we have W ⊗_{A} \overline{ℚ}_{p} ≅ ⨁_{χ∈Δ̂} ϱ ⊗ χ as representations of G over \overline{ℚ}_{p}, where χ runs over all characters of Δ. Since ϱ⊗χ ≅ ϱ if and only if χ = 1 (by (I1)), we know from the absolute irreducibility of ϱ that rank_{A} End_{A[G]}(W) = 2^{r}. Since we already know that B/𝔪_{A}B has dimension 2^{r} over 𝔽, we conclude by Nakayama’s lemma B = End_{A[G]}(W).

We now show that W is free of rank 2 over B. Over B_{ℚ} = B ⊗_{ℤ} ℚ, W_{ℚ} = W ⊗_{ℤ} ℚ is free of rank 2, as easily seen. Let Δ̂ be the character group Hom(Δ, {±1}). The algebra B has an automorphism associated to χ ∈ Δ̂ taking T̃_{δ} to χ(δ)T̃_{δ}. In this way, Δ̂ acts on B through A–algebra automorphisms, and H^{0}(Δ̂, B) = A by definition. Writing ϱ for the 2–dimensional representation of G into GL_{2}(B_{ℚ}) on W_{ℚ}, we see that the action of χ takes ϱ to ϱ ⊗ χ. The character ν = det ϱ is invariant under Δ̂; hence, ν has values in A ⊗_{ℤ} ℚ. Since ν is continuous, ν has values in A^{×}. We consider the alternating pairing: ⟨ , ⟩ : V ∧_{A} V → A, which satisfies ⟨hx, hy⟩ = ν(h)⟨x, y⟩ for all h ∈ H. We extend this pairing to W so that σ ⊗ V and τ ⊗ V are mutually orthogonal if σ ≠ τ in Δ, and otherwise

⟨σh ⊗ v, σh′ ⊗ v′⟩ = ν(σ)⟨hv, h′v′⟩

for h, h′ ∈ H and v, v′ ∈ V. This gives a well-defined pairing ⟨ , ⟩ : W ×W → A such that ⟨gx, gy⟩ = ν(g)⟨x, y⟩. By definition, ⟨T_{σ}x, T_{σ}y⟩ = det T_{σ}⟨x, y⟩, and hence

⟨T̃_{σ}(δ ⊗ x), T̃_{σ}(δ ⊗ y)⟩ = ν(δσ^{−1}) det T_{σ}⟨x, y⟩.

On the other hand, we have

⟨δ ⊗ x, T̃_{σ}^{2}(δ ⊗ y)⟩ = c_{σ}ν(δ)⟨x, y⟩.

Since ρ_{H}(σ^{−2})T_{σ} = c_{σ}, we know that ν(σ^{−1}) det T_{σ} = χ(σ)c_{σ} for a character χ : Δ → {±1}. In any case, the adjoint with respect to ⟨ , ⟩ on W induces an automorphism c of B over A.

Choose v ∈ V such that A[H]v = V. If we have t_{σ}(\overline{v}) = 0 for \overline{v} = (v mod 𝔪_{A}), then t_{σ}(\overline{W}) = t_{σ}(𝔽[G]\overline{v}) = 0, which is a contradiction to our choice of T_{σ}. The module Bv is A–free of rank 2^{r} and is a direct summand of W. By definition, Bv is a maximal totally isotropic subspace of W, W = Bv ⊕ (Bv)^{∗}, and (Bv)^{∗} ≅ Hom_{A}(Bv, A) as B–modules. Since B is a complete intersection, it is a Gorestein ring. Thus (Bv)^{∗} is B–free. In particular, W is B–free of rank 2.

Since T̃_{δ}^{2} = ρ_{H}(δ^{−2})T_{δ}^{2} = c_{δ} ∈ A, either \overline{T}_{δ} is nilpotent or invertible. Suppose that c_{δ} ∈ 𝔪_{A}. The kernel Ker(\overline{T}_{δ}) is one dimensional, because \overline{T}_{δ} ≠ 0. Since T_{δ}(hx) = δhδ^{−1}T_{δ}(x), Ker(\overline{T}_{δ}) is stable under H, on which H acts by a character ξ. We have an exact sequence:

0 → Ker(\overline{T}_{δ}) → \overline{V} → Im(\overline{T}_{δ}) → 0.  (34)

Since the semi-simplification of \overline{V} is stable under the conjugation by δ, we know that H acts on Im(\overline{T}_{δ}) by ξ^{δ}, where ξ^{δ}(h) = ξ(δhδ^{−1}). Since the semisimplification of \overline{V} is stable under conjugation by elements of Δ, Δ acts on {ξ, ξ^{δ}}. There are two possibilities:

(a) ξ = ξ^{δ}, (b) ξ ≠ ξ^{δ}.

Suppose that we are in Case (a). Then Δ acts trivially on ξ. Since \overline{V} = 𝔽[H]v, the exact sequence (34) is non-split as H–modules. The one dimensional subspace of \overline{V} stable under the action of H is unique. If c_{σ} ∈ 𝔪_{A}, then Ker(\overline{T}_{σ}) = Ker(\overline{T}_{δ}). If c_{σ} is a unit, \overline{T}_{σ}(Ker(\overline{T}_{δ})) is still stable under H; so, \overline{T}_{σ}(Ker(\overline{T}_{δ})) = Ker(\overline{T}_{δ}). This shows that the matrix of \overline{T}_{σ} with respect to the base compatible with (34) is upper triangular if c_{σ} is a unit, otherwise it is a nilpotent matrix with only non-trivial entry at the upper right corner. Writing the lower right corner element as t_{σ} when c_{σ} ∈ A^{×}, we find t_{σ}t_{τ} = \overline{c}(σ, τ)t_{στ} for \overline{c}(σ, τ) = (c(σ, τ) mod 𝔪_{A}). Note that Δ′′ = \{σ ∈ Δ|c_{δ} ∈ A^{×}\} is a subgroup; so, we write H′′ for the subgroup of G with G/H′′ = Δ/Δ′′. Then we conclude that ξ can be extended to H′′ (see [H00a] Theorem 4.35), because the cohomology class of \overline{c} in H^{2}(Δ′, 𝔽^{×}) is the obstruction class to have an extension of ξ to H′′. In any case, after tensoring 𝔽[G] over 𝔽[H], we get an exact sequence:

0 → 𝔽[G] ⊗_{𝔽[H]} Ker(\overline{T}_{δ}) → \overline{W} → 𝔽[G] ⊗_{𝔽[H]} Im(\overline{T}_{δ}) → 0.  (35)

This sequence is non-split as G–modules; so, tensoring B/𝔪 over B for any maximal ideal 𝔪 of B, we get the first half of the assertion (3). If a character of a normal subgroup X of a group Y is invariant under the conjugate action of Y/X, it can be extended to Y as long as Y/X is cyclic (see [H00a] Corollary 4.37). If we take maximal subgroup H′ ⊃ H′′ such that ξ can be extended to H′, writing ξ_{H′} for an extension, we must have ξ_{H′}^{δ} ≠ ξ_{H′} for all δ ∉ H′. Since ξ_{H′} appears in Ind_{H}^{H′} ξ, ξ_{H′} appears in the semi-simplification of W/𝔪W for a maximal ideal 𝔪 of B. Since the semi-simplification only contains two characters {ξ_{H′}, ξ_{H′}^{δ}} for δ ∉ H′, [G : H′] ≤ 2. If H′ is of index two, the representation on W/𝔪W has to be isomorphic to Ind_{H′}^{G} ξ_{H′}, which is absolutely irreducible.

We now suppose that we are in Case (b). The stabilizer Δ′ of ξ in Δ is of index 2 in Δ. If \overline{T}_{σ} for σ ∈ Δ′ is nilpotent, it takes ξ^{δ}–eigenspace to ξ– eigenspace (because it is upper triangular and nilpotent), which is impossible by ξ^{σ} = ξ ≠ ξ^{δ}. This implies that \overline{T}_{σ} is invertible. Hence ξ extends to the subgroup H′ of G generated by Δ′ and H (again by using [H00a] Theorem 4.35). In particular, for any maximal ideal 𝔪 of B, W/𝔪W is isomorphic to Ind_{H′}^{G} ξ_{H′} where ξ_{H′} is an extension of ξ to H′, which is absolutely irreducible.

Let Z^{2}(Δ, X) for an Δ–module X be the module of 2–cocycles with values in X.

Corollary 6. Let the notation be as in the theorem. We assume (M1-4). Suppose that there exists an odd prime 𝔭 of E with residual characteristic p such that the obstruction cocycle normlaized at 𝔭 has non-trivial image in Z^{2}(Δ, E_{𝔭}^{×}/A) for A = O_{E,𝔭}. Let 𝔪 be the maximal ideal of B as in (3) and (4) of the theorem and ϱ : G → GL_{2}(B_{𝔪}) be the representation of G on the localization of W at 𝔪. Suppose the following conditions for ϱ in addition to (I0-2):

(I3) det ϱ(c) = −1 for the complex conjugation c and det ϱ = ν^{k−1} for k ≥ 2 up to finite order characters, where ν is the p–adic cyclotomic character;

(I4) the character ξ of H appearing as a subquotient of W/𝔪W cannot be extended to G; (I5) \overline{ρ} = (ϱ mod 𝔪) remains absolutely irreducible over Gal(\overline{ℚ}/ℚ(μ_{p})).

Then if the Galois representation ϱ is p–ordinary, it is modular, that is, for each minimal prime P ⊂ 𝔪 of B, there exists a Hecke eigenform f ∈ S_{k}(Γ_{1}(N)) for a suitable positive integer N such that the p–adic Galois representation of f is isomorphic to ϱ mod P.

If det(ϱ) equals the p–adic cyclotomic character and ϱ is flat, the same assertion as above holds, without assuming p–ordinarity. We discuss sufficient conditions for the irreducibility (I5) after giving the proof of the above result.

Recently Wiles and Skinner announced a modularity result similar to Theorem 9 for many residually reducible (irreducible) representations. If their result is valid for any odd p, we can replace (I5) by irreducibility of \overline{ρ} over Gal(\overline{ℚ}/ℚ).

Let v_{𝔭} be the normalized valuation associated to each prime 𝔭 of E. For each cocycle c : Δ → E^{×}, we can define Div^{(2)}([c]) by the cohomology class of ⊕_{𝔭∤2}v_{𝔭} ∘ c in ⨁_{𝔭∤2} H^{2}(Δ, ℤ). Note that

H^{2}(Δ, ℤ) ≅ Ext^{1}(Δ, ℤ) ≅ Hom(Δ, ℤ/2ℤ)

canonically, because there is no non-abelian extension of Δ by a free module. If Div^{(2)}(Ob(M)) ≠ 0, there exists 𝔭 with non-zero image in Z^{2}(Δ, E_{𝔭}^{×}/O_{E,𝔭}^{×}) ( ⟺ c_{δ} ∈ 𝔭O_{E,𝔭}). However the non-vanishing of Div^{(2)}(Ob(M)) is stronger than the non-triviality of c in Z^{2}(Δ, E_{𝔭}^{×}/A^{×}).

Proof. By (I4), \overline{ρ} = ϱ mod 𝔪 is residually dihedral absolutely irreducible representation. Such a mod p Galois representation is modular by the existence of theta series. By (I5), \overline{ρ} remains irreducible over Gal(\overline{ℚ}/ℚ(μ_{p})). Then Theorem 9 tells us the modularity of ϱ.

Remark 6. If L is real, then (I4) follows from (I3), because det(\overline{ρ}(c)) = −1 tells us that det \overline{ρ} cannot be a square of a character of H. If M is associated to the homology group H_{1} of an elliptic curve E without complex multiplication, one can show that if the prime 𝔭 is unramified in L/ℚ and E has semi-stable reduction at the prime p ≥ 5, we can show that \overline{ρ} is p–ordinary and satisfies (I0-5); so, such an elliptic ℚ–curve is modular (see the following subsection and [H00b] V.2.3).

Let H′ be a subgroup of index 2 in G, and write F for the fixed field of H′. We now describe a sufficient condition for \overline{ρ} = Ind_{H′}^{G} ξ in (I4) to remain irreducible over Gal(\overline{ℚ}/ℚ(μ_{p})), where ξ : H′ → 𝔽^{×} is a character. By Frobenius reciprocity law (cf. [H00a] 2.1.6), reducibility of \overline{ρ} over Gal(\overline{ℚ}/ℚ(μ_{p})) is equivalent to the reducibility of \overline{ρ}|_{H′} for H′ = Gal(\overline{ℚ}/κ), where κ = ℚ(\sqrt{(−1)^{(p−1)/2}p}) is the unique quadratic extension in ℚ(μ_{p}). This happens only when \overline{ρ}(H′) modulo center is abelian.

Since the image of \overline{ρ} modulo center is a dihedral group of order 2n for the order n of ξ^{−1}ξ^{δ}, we study when a dihedral group D_{2n} of order 2n has an abelian normal subgroup of index 2. A dihedral group D_{2n} is abelian only when n ≤ 2.

A dihedral group of order 2n with n ≥ 3 has a unique normal cyclic subgroup C_{n} of order n. We see easily that the maximal abelian quotient D_{2n}^{ab} of D_{2n} is either isomorphic to ℤ/2ℤ or (ℤ/2ℤ)^{2} according as n is odd or even. Therefore when n ≥ 3 is odd, the cyclic subgroup C_{n} is the unique subgroup of index 2.

Suppose that n is even. Since D_{2n}^{ab} ≅ (ℤ/2ℤ)^{2}, there are three normal subgroups X of index 2. If X ≠ C_{n}, choose d ∈ X − C_{n}. Then d^{2} = 1 and dcd^{−1} = c^{−1} for c ∈ C_{n}, and X ≅ D_{n}, which is non-abelian if n > 4. Thus we have an abelian normal subgroup of index two in D_{2n} if and only if n = 2 and 4.

Write n = 2m. The conjugation action of D_{2n}/C_{n} on C_{n} fixes element by element the unique subgroup C_{n} of order 2. For any character ξ of C_{n} of order n, ξ^{−1}ξ^{δ} for δ ∈ D_{2n} −C_{n} has order m. If m ≥ 4, we still see that the restriction of Ind_{C_{n}}^{D_{2n}} ξ to any subgroup of index 2 is irreducible. This settles the case where the order of ξ^{−1}ξ^{δ} is equal to 4. We thus conclude

Proposition 12. Let the notation be as in (I4) and p be an odd prime. Suppose that F ⊄ ℚ(μ_{p}) and that \overline{ρ} = Ind_{H′}^{G} ξ is absolutely irreducible. Then if either the order of ξ^{−1}ξ^{δ} is larger than 2 or there exists σ ∈ Ker(\overline{ρ}) acting non-trivially on κ, \overline{ρ} remains absolutely irreducible over ℚ(μ_{p}).

8.4 Abelian varieties with real multiplication

We consider an abelian variety X defined over \overline{ℚ} with End_{E}(X_{/\overline{ℚ}}) = E for a totally real field E such that H_{1}(X) gives rise to a rank 2 ℚ–motive with coefficients in E. Such an abelian variety will be called a ℚ–abelian variety with real multiplication (ℚ–AVRM). Some authors refer to such abelian varieties as ℚ–HBAV (cf. [R4] but we do not assume that X_{/\overline{ℚ}} is simple). We study modularity of such abelian varieties.

The isomorphism of motives μ_{σ} : σ(H_{1}(X)) ≅ H_{1}(X) induces an isogeny μ_{σ} : σ(X) → X. We take a model (X, μ_{σ})_{/L} for a composite of quadratic fields L/ℚ. We suppose that L is minimal among such fields having a model of X. The obstruction cocycle c(σ, τ) = μ_{σ} ∘^{σ}μ_{τ} ∘ μ_{στ}^{−1} ∈ E^{×} gives rise to the obstruction class Ob(X) ∈ H^{2}(Δ, E^{×}). Regarding Ob(X) as a cohomology class in H^{2}(Δ, E_{𝔸^{(2)}}^{×}), we consider the following condition:

(NT) Div^{(2)}(Ob(X)) ≠ 0 in ⨁_{𝔭∤2} H^{2}(Δ, ℤ) = ⨁_{𝔭∤2} Hom(Δ, ℤ/2ℤ),

where Ô_{E}^{(2)} = ∏_{𝔭∤2} O_{E,𝔭} and E_{𝔸^{(2∞)}} is the adele ring away from 2 and ∞. This condition is equivalent to Ob(M) ≠ 0 in H^{2}(Δ, E^{×}/O_{E}[\frac{1}{2}]^{×}) if the class number of E is odd.

We may choose an abelian variety with O_{E} = End(X_{/L}) in the isogeny class of X. By the above argument, if (NT) holds, we find a prime ideal 𝔭 of O_{E} satisfying the following condition:

• The residual characteristic p of 𝔭 is odd;

• Writing A = O_{E,𝔭}, Ob(X) does not lie in the image of H^{2}(Δ, A^{×}) in H^{2}(Δ, E_{𝔭}^{×}).

Proposition 13. Let the notation and the assumption be as above. Suppose (NT), that p ≥ 5 and one of the following conditions:

(1) p is unramified in L/ℚ and X has good reduction at p;

(2) X has either ordinary good reduction or multiplicative reduction at p.

If Ob(X) = Ob(H_{1}(X)) is commutative, Res_{L/ℚ}X is isogenous to a ℚ–rational factor of the modular jacobian J_{1}(N) for a positive integer N. Even if Ob(X) is non-commutative, X is isogenous to a factor of J_{1}(N) over an abelian extension of L for a positive integer N.

The assumption p ≥ 5 can be removed if the assertion of Theorem 9 is valid under (ai_{p}(ℚ)) instead of (ai_{p}(κ)).

Proof. We use the notation introduced in the earlier subsections: The motive M is obtained from H_{1}(X) as in Lemma 11 to make Ob(M) commutative. The motive M is a rank 2 ℚ–motive with coefficients in E′ defined over L′, where E′ is an extension of E and L′ is a subfield of L. In this modification process, we can keep the condition (NT) for the prime 𝔭 intact; in other words, for a prime 𝔭′ of E′ over E, Ob(M) does not lie H^{2}(Δ′, O_{E′,𝔭′}^{×}) (Δ′ = Gal(L′/ℚ)). We use the symbols H = Gal(\overline{ℚ}/L) and G = Gal(\overline{ℚ}/ℚ).

We first assume that Ob(X) is commutative. Let V be the 𝔭–adic Tate module of X. As already remarked, V satisfies (I1-2). Since any two dimensional p–adic Galois representation associated to an abelian variety defined over ℚ is odd, the condition (I3) is satisfied ([R3] 3.2).

Replacing X in its isogeny class, we may assume (I0). For W = A[G]⊗_{A[H]}V, we have a maximal ideal 𝔪 of B = End_{A[H]}(W) and a subgroup H′ of index two in G such that W/𝔪W ≅ ξ ⊕ ξ^{δ} as H′–modules for a character ξ : H′ → 𝔽^{×}, where 𝔽 = B/𝔪 and G is generated by δ and H′.

Since End_{E}(M_{/\overline{ℚ}}) = E, the involution induced on E by any polarization of X is the identity map (because a totally real field has only one positive involution: the identity map). By the existence of a polarization λ : X → X^{t} of X, we see X[𝔭] is self dual under Cartier duality (induced by the polarization); so, we have X[𝔭] ∧ X[𝔭] ≅ ω_{p} as H–modules for the Teichmüller character ω_{p} acting on μ_{p} ([R1] 4.5.1). We have X[𝔭] ≅ ξ_{H} ⊕ ξ_{H}^{δ} as H–modules and ξξ^{δ} = ω_{p}.

Suppose that p is unramified in L/ℚ. Let 𝔓|p be a prime of L. Suppose that [L_{𝔓} : ℚ_{p}] = 2, and regard ξ as a character of L_{𝔓}^{×} by local class field theory. We may write, on O_{𝔓}^{×}, ξ(u) ≡ u^{a} mod 𝔓 and ξ^{δ}(u) ≡ u^{b} mod 𝔓 with 0 ≤ a, b ≤ p^{2} − 1. We have a + b = 1 + p. If δ ≡ Frob_{𝔓} mod H′, then b = pa. If Frob_{𝔓} ∈ H′, we may assume that ξ is a character of D_{p} = Gal(\overline{ℚ}_{p}/ℚ_{p}) and hence ξ ≠ ξ^{δ}, because ω_{p} cannot be a square character on D_{p}.

If X has good reduction at p, X[𝔭] is a locally free group scheme; so, by Raynaud’s classification theorem of such group schemes (e.g. [O] Theorem 1), we conclude {a, b} = {1, p} or {0, 1 + p}. If L_{𝔓} = ℚ_{p}, then similarly as above, we have {a, b} = {0, 1}; thus, X[𝔭] is p–ordinary. In any case, W/𝔪W satisfies (I4) and (ai_{p}(κ)) by Proposition 12.

Now we assume that p ramifies in L/ℚ and either has ordinary good reduction or multiplicative reduction at p. We see that {a, b} = {0, 2} and hence W/𝔪W is absolutely irreducible (⇔ ξ ≠ ξ^{δ}). Since ξ ≠ ξ^{δ} on O_{𝔓}^{×}, H′ contains the inertia group of 𝔓 in Δ. The fixed field of H′ is unramified at p, and from the same argument as above, we conclude (I5).

The requirement of Theorem 9 is satisfied by W localized at 𝔪. For a suitable N, we have a non-trivial Galois equivariant homomorphism of the p– adic Tate module of J_{1}(N) into W. By the solution of the Tate conjecture for abelian varieties by Faltings, this implies that we have a non-trivial ℚ–rational homomorphism π from J_{1}(N) into Res_{L/ℚ}(X). Since π covers X and hence all δ(X) for δ ∈ Δ (the minimality of L), π has to be a surjection.

We now assume that Ob(X) is non-commutative. Let M be the motive obtained in Lemma 11 out of H_{1}(X). The motive M is given by H_{1}(Y) for an abelian variety Y defined over a subfield L′ of L, because we can twist an abelian variety by a finite order character within the category of abelian varieties. We write A for O_{E′,𝔭′} and take V to be the 𝔭′–adic Tate module of Y. We play the same game as above, starting from V. We may assume that (I0-3) (by the same argument as above).

As we have shown, on X[𝔭], H acts by two characters ξ and ξ^{δ} such that the order of ξ^{1−δ} is larger than 2. In the construction of M out of H_{1}(X) in Lemma 11, we twisted X[𝔭] (and X) successively by a sequence of characters of H, and we kept μ_{δ} intact. Therefore we may assume that H′/H = ⟨δ⟩^{⊥} and ⟨δ⟩^{⊥} ⊃ Gal(L/L′) with the notation in the proof of Lemma 11. We can also choose twisting characters unramified at p if L/ℚ is unramified at p (so, ε in Lemma 11 is unramified at p: Lemma 12). The group Gal(\overline{ℚ}/L′) acts on Y [𝔭′] by two characters ξ′ and (ξ′)^{δ} such that the order of (ξ′)^{1−δ} is larger than 2. When we are in Case (1), this is enough to find a non-trivial morphism: J_{1}(N) → Y defined over ℚ (which may not be surjective because Y may not be ℚ–simple). If we are in Case (2) and p ramifies in L/ℚ, we need to tensor finite order (Dirichlet) character to bring the 𝔭′–adic Galois representation of Y to a p–ordinary representation. Since a character twist of a modular form is again a modular form, we find again a non-trivial morphism J_{1}(N) → Y defined over ℚ. By the absolute simplicity of H_{1}(X) as motive with coefficients in E, the variety X has to appear as a factor of J_{1}(N) over an abelian extension of L, since Y is isogenous to a product of copies of X over an abelian extension of L.

8.5 Cusp forms with quadratic “Neben” type.

Let N be a positive integer and χ : (ℤ/Nℤ)^{×} → {±1} be a character of order 2. Regarding χ as a character of Gal(ℚ(μ_{N})/ℚ) ≅ (ℤ/Nℤ)^{×}, we define a quadratic field L by the subfield of ℚ(μ_{N}) fixed by Ker(χ). We do not assume that χ is primitive for the moment. We look at the space S_{k}(N, χ, ℤ) for k ≥ 2 and its Hecke algebra h = h_{k}(N, χ, ℤ).

It is well known that we have an involution τ of h which brings the Hecke operator T (n) to its adjoint T^{∗}(n) and T^{∗}(n) = χ(n)T (n) if n is prime to N. For each Galois representation ρ_{λ} associated to an algebra homomorphism λ : h → O for a p–adic DVR O, we have τ ∘ ρ_{λ} ≅ ρ_{λ} ⊗ χ. Thus τ fixes λ if and only if ρ_{λ} = Ind_{H}^{G} φ (e.g. [H00a] Lemma 2.15). In this case, we call that λ has complex multiplication by L.

We pick λ without complex multiplication by L and write ℤ[λ] for the image of λ. The ring ℤ[λ] is an order of its field of fraction ℚ[λ]. Since λ does not have complex multiplication by L, τ induces a non-trivial involution on ℚ[λ]. The subfield ℚ_{+}[λ] of ℚ[λ] fixed by τ is smaller than ℚ[λ]. We write O_{+} for the integer ring of ℚ_{+}[λ] and consider the order O_{+}[λ] generated by ℤ[λ] and O_{+} in ℚ[λ].

Proposition 14. Let 𝔮 be an odd prime factor of the relative different 𝔡 of O_{+}[λ] over O_{+}. Write ρ be the 𝔮–adic Galois representation associated to λ. Write H = Gal(\overline{ℚ}/L) and G = Gal(\overline{ℚ}/ℚ). Then \overline{ρ}|_{H} is reducible; so, if \overline{ρ} = (ρ mod 𝔮) is absolutely irreducible, then \overline{ρ} ≅ Ind_{H}^{G} ξ for a character ξ of H.

Proof. We may assume that \overline{ρ} is absolutely irreducible. The isomorphism \overline{ρ} ≅ \overline{ρ} ⊗ χ is equivalent to \overline{ρ} ≅ Ind_{H}^{G} ξ (cf. [DoHI] Lemma 3.2) as long as \overline{ρ} is absolutely irreducible. Since τ ∘ ρ ≅ ρ ⊗ χ as we observed, we get immediately the result from the definition of the relative different.

Here are some sufficient conditions for the absolute irreducibility of \overline{ρ}:

Proposition 15. The Galois representation \overline{ρ} in Proposition 14 is irreducible if one of the following conditions are satisfied:

(1) L is real (⇔ χ(−1) = 1 ⇔ k is even).

(2) p is prime to N and p > k.

Proof. Since ρ is associated to modular form, it is odd: det ρ(c) = −1. If L is real, for a complex conjugation c, we have ξξ^{δ}(c) = −1. This shows that ξξ^{δ} cannot be a square; so, ξ ≠ ξ^{δ} assuring the absolute irreducibility of Ind_{H}^{G} ξ.

If p is prime to N, the Galois representation is crystalline. If p > k − 1, by a theorem of Fontaine [E], we know either \overline{ρ} is p–ordinary or its restriction to the decomposition group at p is absolutely irreducible. In the ordinary case, obviously we get ξ ≠ ξ^{δ} because the product ramifies at p if k ≢ 1 mod p−1.

The involution τ is induced by the Hecke involution W given by the action

of ( \begin{matrix}0 & −1 \\ N & 0 \end{matrix} ) on S_{k}(N, χ, ℂ); in other words, τ(h) = W hW^{−1} for h ∈ h_{k}(N, χ, ℤ).

We can split the motive M(λ)_{/ℚ} associated to λ by the action of W as M(λ) = M[+] ⊕ M[−] for the ±1–eigenspaces M[±] of M(λ). Since τ ∘ ρ ≅ ρ ⊗ χ, the morphism W is defined over L; so, M[±]_{/L} and T (n) with χ(n) = −1 induces an isomorphism μ_{δ} : δ(M[+]) = M[−] → M[+] (as long as it is non-zero) for δ inducing the non-trivial automorphism on L. If M[+] is simple over \overline{ℚ}, M = Res_{L/ℚ}M[+] and M[+] is a rank 2 ℚ–motive. This simplicity has been proved by Ribet if N is a prime and k = 2 [R] (more generally, it has been shown that all \overline{ℚ}–simple factors of J_{1}(N) for an integer N is a ℚ–AVRM: [R2] and [R3]). When k = 2 and N is a prime, Ob(M) cannot be in the image of H^{2}(Δ, O_{+}^{×}) if 𝔡 contains an odd prime factor. We should emphasize that 𝔡 is always non-trivial because O_{+}[λ] is the direct sum of ±1–eigenspace of τ. However there are some examples in which 𝔡 is a 2–power; so, a natural question we ask is:

Question: When does 𝔡 contain an odd prime factor?

In any case, we expect that N(𝔡) has a non-trivial odd factor p > k if the weight k is sufficiently large.

8.6 Base-change to totally real fields

We start with a Hecke eigenform f ∈ S_{k}(p, ψ) for an odd prime p. Define λ(T (n)) ∈ \overline{ℚ} by f|T (n) = λ(T (n))f, and write ℚ[λ] for the field generated by λ(T (n)) for all n. Pick a prime 𝔮 of ℚ[λ] and consider the 𝔮–adic Galois representation ρ_{λ,𝔮} of f acting on two dimensional 𝔮–adic vector space V. As we have already seen, this Galois representation is unramified outside pq for the residual characteristic q of 𝔮 and has the characteristic polynomial det(1 − ρ_{λ,𝔮}(Frob_{ℓ})|_{V^{I(ℓ)}}X) independent of 𝔮 as long as ℓ ∉ 𝔮, where I(ℓ) is the inertia group of the prime ℓ.

The L–function L(s, λ) = ∑_{n=1}^{n} λ(T (n))n^{−s} has Euler product expansion, which coincides with the Euler product of the system ρ_{λ} = {ρ_{λ,𝔮}}_{𝔮} of Galois representations:

L(s, ρ_{λ}) = ∏_{ℓ} det(1 − ρ_{λ,𝔮}(F rob_{ℓ})|_{V^{I(ℓ)}}ℓ^{−s})^{−1},

where we always choose the prime 𝔮 outside ℓ to have well-defined Euler factor det(1 − ρ_{λ,𝔮}(F rob_{ℓ})|_{V^{I(ℓ)}}ℓ^{−s})^{−1}. For a given number field F ⊂ \overline{ℚ}, we can think of the Euler product

L(s, ρ_{λ}|_{Gal(\overline{ℚ}/F)}) = ∏_{𝔩} det (1 − ρ_{λ,𝔮}(Frob_{𝔩})|_{V^{I(𝔩)}}N(𝔩)^{−s})^{−1} .

Here 𝔩 runs over all prime ideals of F. Multiplying them out, we can expand the Euler product L(s, ρ_{λ}|_{Gal(\overline{ℚ}/F)}) into a Dirichlet series

L(s, ρ_{λ}|_{Gal(\overline{ℚ}/F)}) = L(s, λ̂) = ∑_{𝔫} λ̂(T (𝔫))N(𝔫)^{−s}.

Numbers λ̂(𝔫) are well-defined for integral ideals 𝔫 of F. If there exists a cusp form f̂: GL_{2}(F)\GL_{2}(F_{𝔸}) → ℂ such that f̂|T (𝔫) = λ̂(T (𝔫))f̂, the cusp form f̂ is called the base-change of f to F.

Since the Hecke eigenvalue λ̂(T (𝔫)) is derived from λ(T (n)) in a purely arithmetic way, finding f̂ is a highly non-trivial question. This type of problem was first considered by Doi and Naganuma in the late 60’s and the existence of f̂ is now known by Langlands [L] for prime cyclic extensions F/ℚ as long as the restriction of ρ_{λ} to Gal(\overline{ℚ}/F) remains irreducible. In any case, the problem remains open for non-soluble (and non-Galois) extensions F/ℚ.

We study here the problem of base-change, assuming that f ∈ S_{k}(SL_{2}(ℤ)) is 𝔭–ordinary for an odd prime 𝔭 of ℚ[λ] with residual characteristic p. This automatically implies that p ≥ 11. Let O be the p–adic integer ring of the field generated by the Hecke eigenvalues λ(T (n)) of T (n) for f over ℚ_{p}. The Hecke eigenvalues λ(T (n)) of T (n) for f extend to an algebra homomorphism λ : h(1, 1) → O. We have by Theorem 2 the isomorphism

h(1, 1) ⊗_{O[[T(ℤ_{p})]],ν^{k−1}} O ≅ h_{k}^{ord}(1, 1, O)

for all even k ≥ 4. Since h_{k}^{ord}(1, 1, O) is reduced, the algebra h(1, 1) is reduced. In any case, we have a unique local ring 𝕋 of h(1, 1) through which λ–factors. For each 2 ≤ k ∈ ℤ, the ring 𝕋 ⊗_{Λ,ν^{k−1}} O is a local ring of h_{k}^{ord}(p, χω^{−k}, O) for a Dirichlet character χ modulo p. In other words, any algebra homomorphism λ_{k} : h_{k}^{ord}(p, χω^{−k}, O) → \overline{ℚ}_{p}, λ(T (n)) ≡ λ_{k}(T (n)) mod ℘ for all n > 0, where ℘ is the maximal ideal of the valuation ring of \overline{ℚ}_{p}.

In particular, if we choose k ≥ 2 so that χω^{−k} equals the Legendre symbol (\overline{p}), the Hecke eigenform f_{k} ∈ S_{k} (p,_{(\overline{p})}) with eigenvalue λ_{k} has quadratic

‘Neben’ type _{(\overline{p})}. Since ρ_{λ} ⊗_{(\overline{p})} ≅ ρ̸_{λ}, the cusp form f does not have complex

multiplication by κ = ℚ[\sqrt{(−1)^{(p−1)/2}p}]. We see ρ_{𝕋} ⊗_{(\overline{p})} ≅ ρ̸_{𝕋}, which implies the same property for λ_{k}. We conclude O_{+}[λ_{k}] ≠ O_{+}. Assume that the relative different 𝔡 = 𝔡_{k} of O_{+}[λ_{k}] over O_{+} has odd prime factor 𝔮 = 𝔮_{k}. We study the 𝔮–adic Galois representation ρ_{k,𝔮} associated to λ_{k}. If the reduction \overline{ρ}_{k,𝔮} of ρ_{k,𝔮} modulo 𝔮 is irreducible, \overline{ρ}_{k,𝔮} is isomorphic to Ind_{H}^{G} ξ for a character ξ of Gal(\overline{ℚ}/κ).

Since ρ_{k,𝔮} is known to be unramified outside q over κ (e.g. [H00b] Theorem IV.2.4), ξ is unramified outside q. Since ρ_{k,𝔮} ⊗ _{(\overline{p})} ≅ ρ̸_{k,𝔮}, ρ_{k,𝔮}|_{H} for H =

Gal(\overline{ℚ}/κ) is irreducible.

When p ≡ 1 mod 4, κ is real, and hence \overline{ρ}_{k,𝔮} is irreducible (by Proposition 15); so, we assume for simplicity that p ≡ 1 mod 4 (because then we do not need to worry about irreducibility under this assumption).

Proposition 16. Suppose that κ is real (⇔ p ≡ 1 mod 4). Let 𝔮 be an odd prime factor of 𝔡_{k} of residual characteristic q. Then q splits in κ, and \overline{ρ}_{k,𝔮} restricted to the decomposition group D_{q} at q is reducible. If q ≥ k − 1, then ρ_{k,𝔮} is 𝔮–ordinary.

Proof. Let H = Gal(\overline{ℚ}/κ). We realize ρ_{𝔮} = ρ_{k,𝔮} : H → GL_{2}(A) for A = O_{+,𝔮} on a A–lattice V stable under W in the 𝔮–adic étale realization of the motive attached to λ_{k}. By Proposition 14, \overline{ρ}_{k,𝔮}|_{H} is reducible over the residue field 𝔽_{𝔮} = A/𝔪_{A}. The Hecke involution W acts on the representation space V (≅ A^{2}) of ρ_{𝔮}. Since W commutes with the action of H, the one dimensional eigenspace \overline{V} [±] of W with eigenvalue ±1 for \overline{V} = V/𝔪_{A}V is stable under H, which again shows the reducibility of \overline{ρ}_{𝔮}|_{H}. Since δ(W) = −W for δ ∈ G = Gal(\overline{ℚ}/ℚ) inducing nontrivial automorphism on κ, δ interchange \overline{V} [±].

If q ramifies in κ, the character ξ of H acting on \overline{V} [+] is δ–invariant: ξ^{δ} = ξ, because ξ has to be tamely ramified, and \overline{ρ}_{𝔮} is reducible over I_{q}. Since \overline{ρ}_{𝔮} is reducible over H, it has to be reducible over G = HI_{q}. Thus this case cannot happen.

Suppose now that q remains prime in κ. Since p ≡ 1 mod 4, the number of strict ideal classes of κ is odd, and hence, κ has a fundamental unit ε with norm −1. The character ξ|_{I_{q}} (regarded as a character ξ : O_{κ,q}^{×} → 𝔽_{𝔮}^{×} by local class field theory) factors through 𝔽_{q^{2}}^{×} and satisfies ξ(ε) ≠ ξ^{δ}(ε). This shows irreducibility of \overline{ρ} on I_{q}. Since ξ^{δ} ≠ ξ on I_{q}, ξ(x) = x^{a} for 0 < a < q^{2} − 1 not divisible by 1 + q. Thus ε^{a} ≡ −1 mod q and δ(ε^{a}) ≡ 1 mod q, which is impossible if q remains prime.

If q ≥ k − 1, by the classification of mod q crystalline Galois representations by Fontaine-Laffaille [FL] 0.6 that ξ is unramified at one factor 𝔔 of q in κ. Since ρ_{𝔮} is crystalline at q, V has the crystalline Frobenius operator Φ acting on the corresponding Dieudonné module D(V). Then ϕ = lim_{n→∞} Φ^{n!} gives a projector to the subspace of D(V) which corresponds to a rank 1 subspace V_{I} of V such that V/V_{I} is unramified. Thus V is 𝔮–ordinary.

There is a generalization of Theorem 8 to totally real fields by Fujiwara [Fu]. Applying his result in this special case, as long as \overline{ρ}_{𝔮} is irreducible over Gal(\overline{ℚ}/F (μ_{q})) for the residual characteristic q of 𝔮 for a given totally real field F unramified at q, we conclude the existence of the base change f̂_{k} on GL_{2}(F_{𝔸}). In exactly the same manner as in the proof of Theorem 7, we can prove that the universal p–ordinary deformation ring (unramified outside p) of ρ_{λ,𝔭} restricted to Gal(\overline{ℚ}/F) is isomorphic to a local ring 𝕋_{F} of the universal p–ordinary Hecke algebra h(1, 1)_{F} for p–ordinary modular forms on GL_{2}(F_{𝔸}) of level 1. In particular, this implies the existence of the base-change f̂ also. It has been verified in [HM] numerically in this way that we can base-change lift infinitely many p–adic families of cusp forms.

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