April 16, 2001 – 13 : 54 Revision : 1.67 ON THE IMAGE OF Λ-ADIC GALOIS REPRESENTATIONS

AMI FISCHMAN

Abstract. We explore the question of how big the image of a Galois representation attached to a Λ-adic modular form with no complex multiplication is and show that for a “generic” set of Λ-adic modular forms (normalized, ordinary eigenforms with no complex multiplication), all but a density 0 subset have large image.

1. Introduction

F. Momose has proved that the image of a restricted l-adic Galois representation attached to an appropriately generic (normalized, ordinary, no CM) modular form is full in the sense that it contains the special linear group (SL_{2}) for all but finitely many l [Mom81]. In this paper, we generalize that result to the Λ-adic setting developed by Hida in [Hid86a], [Hid86b], and [Hid86c]. We show that the image of the subgroup determined by the twists of the form is full (Thm. 4.8) and further determine the exact image of the Galois group of Q under the Λ-adic representation.

We then show that in some sense, of all generic Λ-adic Galois representations, all but a density 0 subset have full image as in Thm. 4.8 (Thm. 5.5).

In order to obtain analogs of the classical results in the Λ-adic situation, we determine the exact structure of the Λ-adic Hecke algebra and coefficient ring. We then lift the classical results of Momose to the Λ-adic setting using a proposition of N. Boston from the appendix to [MW86].

This work was done under the supervision of Professor Hida as part of my Ph.D. thesis. I am very grateful to Dr. Hida for all the time, patience, and energy he has devoted to my research and learning, and for the knowledge he so unsparingly shared with me.

1.1. Layout. In section 2, we discuss the case of weight 2, and the results already known for it. In particular, we quote the result of Momose which guarantees that the restricted l-adic representation attached to a generic weight 2 modular form is full for all but a finite set of primes l. Then in section 3, we lift the weight 2 modular form to an I-adic modular form and prove that in fact the Λ-adic Hecke algebra is of the form of a power series ring over the classical Hecke algebra. This allows us to compute the coefficient ring of the lifted form, I, explicitly, and subsequently to set up the proposition of N. Boston. In section 4, we prove that the restricted Λ-adic Galois representation is full, and in section 5 we show that for all but a density 0 subset of generic modular forms, the attached representations are full.

1.2. Notation. Throughout, A_{m} will denote the completion at m and A_{(m)} will denote the localization at m, for a ring A and a prime ideal m.

We recall the following standard definitions and notation for an odd prime l:

Let μ_{n} := \{ζ ∈ \overline{Q} | ζ^{n} = 1\} be the set of nth roots of unity. Define the Teichmüller character at l, ω_{l}, to be the first component of the canonical isomorphism Z_{l}^{×} →^{∼} μ_{l−1} × \overbrace{(1 + lZ_{l})}^{Γ}, and define ⟨⟩ : Z_{l}^{×} → 1 + lZ_{l} via ⟨x⟩ := ω_{l}(x)^{−1}x. Then x ↦ (ω_{l}(x), ⟨x⟩) is the canonical isomorphism above.

The cyclotomic character at l is defined by letting n → ∞ in the following canonical sequence:

Gal(\overline{Q}/Q) →^{res} Gal(Q(μ_{l^{n}})/Q) →^{∼} (Z/l^{n}Z)^{×}

to get the character

ν_{l} : Gal(\overline{Q}/Q) →^{res} Gal(Q(μ_{l^{∞}})/Q) →^{∼} Z_{l}^{×}

We define ι : Gal(\overline{Q}/Q) →^{res} Gal(Q(μ_{l^{∞}})/Q) →^{res} Gal(Q_{∞}/Q) →^{∼} 1 + lZ_{l} ↪^{κ} Λ to be the canonical character (so ι is onto 1 + lZ_{l}, and κ maps (1 + l) ↦ (1 + X)). We note that ι(σ) = κ(⟨ν_{l}(σ)⟩) by definition. In particular, for a prime q ≠ l, ι(Frob_{q}) = κ(⟨q⟩).

[figure: Gal(\overline{Q}/Q); ν_{p}; ι; Gal(Q(μ_{p^{∞}})/Q); Gal(Q_{∞}/Q); _{Z× p}; ∼; μ_{p−1} × (1 + pZ_{p}); ⟨⟩; 1 + pZ_{p}; κ; Z_{p}[[X]]]

2. Weight 2 Situation

2.1. Notation. Throughout l will denote a (variable) prime number greater than 3. Let f = ∑_{n=1}^{∞} a(n, f)q^{n} ∈ S_{2}(Γ_{0}(N), ε) be a weight 2 newform without complex multiplication. Fix an embedding \overline{Q} ⊂ C once and for all. Let E_{f} := Q(a(n, f)|n ∈ N) be the subfield of \overline{Q} ⊂ C generated by the coefficients of f, and note that E_{f} is finite dimensional over Q. Let O_{f} := O_{E_{f}} be the integer ring of E_{f}, and O_{f,l} := O_{f} ⊗_{Z} Z_{l} be the completion at l (for each l). Similarly for each l, complete E_{f} to E_{f,l} := E_{f} ⊗_{Q} Q_{l}.

Let ρ_{f,l} : G = G_{Q} = Gal(\overline{Q}/Q) → GL_{2}(O_{f,l}) ⊂ GL_{2}(E_{f,l}) be the continuous, unramified outside lN Galois representation attached to f such that T r(ρ_{f,l}(φ_{p})) = a(p, f) and det(ρ_{f,l}(φ_{p})) = ε(p)p for all p ∤ lN, where φ_{p} is a Frobenius element at p. Note that this determines ρ_{f,l} uniquely as an E_{f,l}-representation, but not necessarily as an O_{f,l}-representation.

For an automorphism γ ∈ Aut(E_{f}), a Dirichlet character χ may exist such that

γ(a(p, f)) = χ(p)a(p, f)

for almost all p. If it does, call this character χ_{γ}. Let Γ_{f} be the set of γ ∈ Aut(E_{f}) for which there exists a χ_{γ}.

2.2. Known Results. Work of Momose and Ribet allows us to make the following definitions with implicit claims:

1. Set H_{f} := ∩_{γ∈Γ_{f}} ker χ_{γ};

2. Γ_{f} ⊂ Aut(E_{f}) is an abelian subgroup;

3. Set F_{f} := E_{f}^{Γ_{f}} to be the the fixed subfield;

4. Set n := [E_{f} : F_{f}] < ∞;

5. Set R_{f} := O_{F_{f}} to be the integer ring;

6. Set R_{f,l} := R_{f} ⊗_{Z} Z_{l} to be the completion at l, for each l; and

7. Set A_{f,l} := {x ∈ GL_{2}(R_{f,l})| det(x) ∈ Z_{l}^{×}}.

These definitions come from [Rib85].

These definitions allow us to state [Rib85, Thm. 3.1]:

Theorem 2.1 (Momose). For all but finitely many primes l, ρ_{f,l}(H_{f}) = A_{f,l}, and in particular, ρ_{f,l}(H_{f}) ⊃ SL_{2}(R_{f,l}).

2.3. p-Ordinarity. We will be dealing with two similar, but not equivalent, notions of “ordinarity”. The first notion is for rational primes (say l) where we say that f is l-ordinary if l ∤ a(l, f). The second notion is for primes lying over l, say 𝔩. Then we say that f is 𝔩-ordinary if 𝔩 ∤ a(l, f).

When do the two notions clash? Clearly if l | a(l, f) then 𝔩|a(l, f) for any prime 𝔩 | l, so 𝔩ordinarity implies l-ordinarity. However, it is possible that 𝔩 | a(l, f) (i.e. f is not 𝔩-ordinary) and that l ∤ a(l, f) (i.e. f is l-ordinary). We will see shortly that this will not be a problem for us.

We recall the bound on the size of the coefficients of an eigenform of weight k: |a(l, f)| ≤ 2l^{(k−1)/2}, which in weight 2 gives: |a(l, f)| ≤ 2\sqrt{l}. This implies that f is l-ordinary if and only if a(l, f) ≠ 0 for l > 3 (since if it is non-zero, it is between 0 and 2\sqrt{l} < l in absolute value and not l-divisible).

Serre shows that for a real number x, the number of primes l less than x not dividing the level N such that a(l, f) = 0 is P_{f,0} = O(x/(log(x)^{3/2−δ})) for any δ > 0 [Ser81, Thm. 15, p.174]. Using the prime number theorem, one sees that the density of non-ordinary (rational) primes is d_{f} := lim_{x→∞} \frac{P_{E,0}}{x/log(x)} = 0, since δ can be taken to be less than a half. Thus the density of ordinary primes must be 1. We summarize this as:

Proposition 2.2 (Serre). Any form f as above of weight 2 is l-ordinary for a set of primes of density 1.

For any form f as above, let Σ_{f} be the set of primes l for which f is l-ordinary and which avoid the union of the following finite sets of primes:

1. the finitely many primes excluded by Thm. 2.1;

2. the primes dividing the discriminant of the reduced Hecke algebra h_{2}(Γ_{0}(N), ε, Z)^{red}

where

h_{2}(Γ_{0}(N), ε, Z) := Z[T (n) | n = 1 …] ⊂ End(S_{2}(Γ_{0}(N), ε))

and

3. the primes dividing 6N and

4. the primes l such that l|a(l, f)^{2} − ε(l), which by Ramanujan’s bound means that l ≤ 2\sqrt{l} + 1, so l ≤ 5 (i.e. we’re only excluding 2, 3, and 5).

Then Serre’s result (Prop. 2.2) can be thought to show that Σ_{f} is a density 1 set of primes. The reason we exclude the primes dividing the discriminant of the Hecke algebra is that

those might ramify in O_{f}. Now, for a prime l ∈ Σ_{f}, we are guaranteed that there is an 𝔩 lying over l such that f is 𝔩-ordinary. This follows since if all of the primes lying over l divided a(l, f), then so would l, because it is unramified. The notion of 𝔩-ordinarity will be the key to lifting f to a Λ-adic eigenform F later on.

Lemma 2.3. For a prime l, l ∤ disc(h_{2}^{red}(Γ_{0}(N), ε, Z)) implies that l doesn’t ramify in O_{f}.

Proof. We have the map λ : h_{2}(Γ_{0}(N), ε, Z) → O_{f} mapping a Hecke operator T (n) to f’s eigenvalue at n. We know that disc(Im(λ)) = disc(O_{f})(O_{f} : Im(λ))^{2} by [FT93, (2.3), p.121], so a prime l ramifying in O_{f} must divide the discriminant of Im(λ). But the map h_{2}(Γ_{0}(N), ε, Z) → Im(λ) is surjective, so the the pre-image of the different of Im(λ) must contain l and thus l divides the discriminant of h_{2}(Γ_{0}(N), ε, Z) which shows the lemma. ■

Note that by excluding the primes dividing 6N, we exclude the primes dividing the conductors of the χ_{γ} by [Mom81, Rmk. 1.6].

3. Machinery

3.1. Lifting the Eigenform. Throughout this section, fix f_{2} as in the previous section, a prime l ∈ Σ_{f_{2}}, and an 𝔩 | l for which f_{2} is 𝔩-ordinary. Then there is a unique 𝔩-ordinary eigenform f ∈ S_{2}(Γ_{0}(N) ∩ Γ_{1}(l), ε, Z) such that f and f_{2} have the same eigenvalues for all T (n), n ∤ Nl. Note that the choice of 𝔩 determines an embedding Q_{l} ↪ \overline{Q_{l}}. We note that O_{f,l} = O_{f}⊗_{Z}Z_{l} has O_{f,𝔩} as a direct summand and we will be mostly concerned with O_{f,𝔩} in the sequel. Similarly, we let R_{f,𝔩} be the corresponding direct summand of the fixed subring.

Let Λ := Z_{l}[[X]], L be its quotient field, and let \overline{L} be the algebraic closure of L. From [Hid89, Thms. 4.5 and 4.6] we know that there is a finite extension of L, call it K ⊂ \overline{L}, with integral closure I of Λ in K such that there is an I-adic normalized Hecke eigenform F ∈ S^{ord}(Nl^{∞}, χ) that specializes to f at weight 2 of character χ, where χ := εω_{l}^{2}. We use here the notation S^{ord}(Nl^{∞}, χ) to denote the space of (ordinary) Λ-adic modular cusp forms of outside level N with character χ (see [Hid93, Sec. 7.6] for definitions and examples). For the proof of the existence of a Λ-adic lift the reader should refer to the proof of [Hid86c, Cor. 3.7].

Let F = ∑_{n=1}^{∞} a(n, F)(X)q^{n} be the q-expansion of F.

We adapt here an argument of Gouvêa’s from [Gou92] to the weight 2 situation to show that:

Theorem 3.1. The coefficient ring I is a power series ring in one variable. Specifically, I ≅ O_{f,𝔩}ˆ⊗_{Z_{l}}Z_{l}[[X]] ≅ O_{f,𝔩}[[X]].

To prove this theorem, we set up some notation. Because f and F are eigenforms for their respective Hecke algebras, we can define maps

λ_{f} : h := h_{2}(Γ_{0}(N) ∩ Γ_{1}(l), ε, Z) → O_{f}

and

λ_{F} : h := h^{ord}(Nl^{∞}, χ) → I

mapping a Hecke operator to its eigenvalue on f and F, respectively. Let m_{f} := λ_{f}^{−1}(𝔩) be the inverse image of 𝔩. Then m_{f} is a maximal ideal of h and contains ker(λ_{f}). We will see shortly that there is a unique maximal ideal containing ker(λ_{F}), and we will call it m_{F}. Since the m’s contain the kernels, λ_{f} factors through h_{m_{f}} and λ_{F} factors through h_{m_{F}}. When no confusion can arise, we will write h_{m} for h_{m_{f}} and h_{m} for h_{m_{F}}.

Lemma 3.2. Extend the scalars of h to Z_{l} by h′ := h⊗_{Z}Z_{l}. Then m′ := m_{f}⊗_{Z}Z_{l} is the unique maximal ideal of h′ which contains ker(λ_{f}).

Proof. Since h′ is an algebra of finite rank over the local ring Z_{l}, it is semi-local, and in particular has only finitely many prime ideals. Write h′ = ∏_{n⊂h′} h′_{n} where the product ranges over the maximal ideals of h′. Then note that in general, Spec(A×B) = Spec(A)⊔Spec(B), so a prime ideal in the product must only actually show up in one of the components, proving the lemma. ■

Since h is already semi-local, the same proof yields:

Corollary 3.3. There is a unique maximal ideal m_{F} containing ker(λ_{F}).

Claim 3.4. m_{f} does not contain primes dividing the discriminant of h_{2}(Γ_{0}(N), ε, Z)^{red}.

Proof. Note that m_{f} ∩ Z is a maximal ideal of Z, so m_{f} ∩ Z = (q) for some rational prime q. This shows that m_{f} can contain only one rational prime. But by definition l ∈ m_{f}, so the claim follows. ■

Lemma 3.5. For l ∤ N, k ≥ 2, and m ⊂ h_{2}(Γ_{0}(N), ε, Z) the maximal ideal associated to the primitive form f, the completion h_{2}(Γ_{0}(N), ε, Z)_{m} of the Hecke algebra is an unramified discrete valuation ring if m doesn’t contain a prime dividing the discriminant of the reduced part of the Hecke algebra.

Proof. For simplicity of notation, let \underline{h} := h_{2}(Γ_{0}(N), ε, Z) and let K be its total fraction ring, and regard K as a finite dimensional algebra over Q. Then let U ⊂ K^{red} be the integral closure of Z in K^{red} and \underline{h}^{red} = Z[T (n) | n = 1, ⋯]^{red} ⊂ U. Note that Q(T (n)) = K, so \underline{h}^{red} is a lattice in U and (U : \underline{h}^{red}) is finite. Further, if q ∤ (U : \underline{h}^{red}) then U_{q} = \underline{h}_{q}^{red}. Now K^{red} is a commutative semi-simple finite-dimensional algebra over Q, so we can write K^{red} = K_{1} × ⋯ × K_{r} for some r, with each K_{i} a number field, and U = U_{1} × ⋯ × U_{r}, with each U_{i} an order in a number field. Then we have that \underline{h}_{q}^{red} ≅ U_{1,q} × ⋯ × U_{r,q}, and each component is a discrete valuation ring.

Since the discriminant of \underline{h}^{red} is disc(\underline{h}^{red}) = disc(U)(U : \underline{h}^{red})^{2} ([FT93, (2.3) on p.121]), the assumption on m implies in particular that it contains no primes dividing (U : \underline{h}^{red}). In particular, let q be a rational prime in m and obtain \underline{h}_{m}^{red} = (\underline{h}_{q}^{red})_{m} is a completion at one of the components of U_{1,q} × ⋯ × U_{r,q}, and so a discrete valuation ring. Since the q we chose is in fact outside of disc(\underline{h}^{red}), it does not divide disc(U) and so each U_{i,q} is unramified, so \underline{h}_{m}^{red} is unramified. Since f is primitive, \underline{h}_{m} is reduced ([Hid00, Prop. 3.23]) and the lemma is shown. ■

Lemma 3.6. If a(l, f) ≢ ±\sqrt{ε(l)} mod m (which is always the case if l ∤ N and f_{2} is of level N) then h_{2}(Γ_{0}(N) ∩ Γ_{1}(l), ε, Z)_{m} ≅ h_{2}(Γ_{0}(N), ε, Z)_{m}, and the former is an unramified discrete valuation ring.

Proof. For the proof of this lemma we refer the reader to [Hid00, p. 106]. ■

Proposition 3.7. h_{m} ≅ h_{m} ˆ⊗_{Z_{l}}Z_{l}[[X]]. In particular, we have

Aut_{Λ}(h_{m}) ≅ Aut_{Z_{l}}(h_{m})

Proof. From [Hid86c, Cor. 3.2], we know that h_{m}/P_{2}h_{m} ≅ h_{m}. Since h_{m} is an unramified discrete valuation ring over Z_{l}, h_{m} is a regular local ring of dimension 2 with a regular sequence (P_{2}, l). Now, h_{m}/m_{F} ≅ (h_{m}/P_{2})/(l) ≅ h_{m}/(l) ≅ F, and h_{m} is unramified over Z_{l}, so we get h_{m} ≅ W (F) ↪ h_{m} (by the universal properties of Witt vectors). Then together with the map given by the diamond action Z_{l}[[X]] → h_{m} we get a map h_{m} ˆ⊗_{Z_{l}}Z_{l}[[X]] → h_{m} which induces an isomorphism on residue fields and maps the regular sequence (P_{2}, l) for h_{m} ˆ⊗_{Z_{l}}Z_{l}[[X]] into a regular sequence for h_{m} and is thus an isomorphism. ■

In order to finish the proof of Thm. 3.1, it suffices to show that the local Hecke algebras are isomorphic to the coefficient rings:

Lemma 3.8. We can identify the local Hecke algebras with coefficient rings as: h_{m} ≅ O_{f,𝔩} and h_{m} ≅ I.

Proof. From the definition of m_{f}, we know that for any b ∈ h \ m, λ_{f}(b) ≠ l and more generally, λ_{f}(b) ∈ O_{f,𝔩}^{×}. This allows us to define a map h_{(m)} → O_{f,𝔩} by mapping

\frac{a}{b} ↦ \frac{λ_{f}(a)}{λ_{f}(b)}

which is now well-defined and a homomorphism. By the proof of lemma 3.5 we know that this map is surjective because λ_{f} maps onto Z[a(n, f)] and l (the prime at which we are localizing and then completing) is outside the index of (O_{f} : Z[a(n, f)]). Taking the completion of the left-hand-side gives us a surjection λ̃ : h_{m} → O_{f,𝔩} of discrete valuation rings. Furthermore h_{m} is of dimension 1 because h is a finite rank Z-module, h and Z have the same dimension, and localizing and completing doesn’t affect the dimension. O_{f,𝔩} is also of dimension 1, so we have a surjection of discrete valuation rings of equal dimension, hence an isomorphism.

Similarly, the map h_{m} → I is surjective, and both are finite rank Λ-modules, so they have the same dimension (2). But both are integral domains, so there must be no kernel and the rings are isomorphic. ■

Proof of Thm. 3.1. Given the results above, the theorem follows trivially by noting that I ≅ h_{m} ≅ h_{m} ˆ⊗_{Z_{l}}Z_{l}[[X]] ≅ O_{f,𝔩}ˆ⊗_{Z_{l}}Z_{l}[[X]] ≅ O_{f,𝔩}[[X]]. ■

Let 𝔪 := 𝔪(I) be the maximal ideal of I. Because O is unramified over Z_{l} we know that (l) ⊂ O is a prime ideal of O. Further, O is a discrete valuation ring (so local), so we know that the maximal ideal of O is (l). Using this and the structure of I above, we deduce that the maximal ideal of I is 𝔪 = (l, X). For each k ≥ 2 set P_{k} := ((1 + X) − (1 + l)^{k}) to be the prime ideal of “weight k” in I.

3.2. Setting up Notation. We now define Γ_{F} and H_{F} , similarly to the weight 2 analogues Γ_{f}, H_{f}, A_{f,l}. Let

Γ_{F} := {γ ∈ Aut(K) | γ(𝔩) = 𝔩 and ∃ a Dirichlet character χ_{γ} such that
γ(a(p, F)) = χ_{γ}(p)a(p, F) for almost all p}
Γ′_{f} := {γ ∈ Γ_{f} | γ(𝔩) = 𝔩}
R_{F} := I^{Γ_{F}}
H_{F} := ∩_{γ∈Γ_{F}} ker(χ_{γ})

Further, as we will see shortly, more important than Γ_{f} for us will be Γ′_{f}.

We will show that in fact as far as the fixed rings R_{F} and R_{f,𝔩} go, there is no new information in the Λ-adic setting over the weight 2 setting: R_{F} mod P_{2} ≅ R_{f,𝔩}. To prove this we record the following lemmas:

Lemma 3.9. Recall that we defined h′ = h⊗_{Z}Z_{l}. Then let h^{ord} be the ordinary part of h′, i.e. the product of local rings of h′ in which the image of T (l) is a unit. Then the set of local rings of h^{ord} is in bijection with the set of local rings of h.

Proof. For brevity in the proof we use h to denote h^{ord}. Write the decompositions as h = ∏_{i=1}^{s} h_{\underline{m}_{i}} and h = ∏_{j=1}^{r} h_{m_{j}}. Clearly the map h → h modulo the prime ideal of weight 2 is surjective, and each local component of h gets mapped onto a local component of h, so the only thing to show is that no two local components of h get mapped to the same component of h. Let \overline{e_{i}} : h → h_{m_{i}} be the canonical projections, and note that the \{\overline{e_{i}}\} form a complete set of orthogonal idempotents. Then by Hensel’s lemma (for instance [Eis95, Cor. 7.5 on p.187]), we can lift this set of idempotents to {e_{i}} : h → h, and h = ∏_{i=1}^{r} e_{i}h. Then the lemma will follow if each e_{i}h is local. Suppose n_{1} ≠ n_{2} ⊂ e_{i}h are both maximal ideals. Then we may decompose e_{i}h = R_{1} × R_{2}, and both R_{j}’s are Λ-algebras. Since e_{i}h is semi-local, P_{2}e_{i}h is contained in the Jacobson radical of e_{i}h, and so by Nakayama’s lemma neither of the factors in \overline{e_{i}}h = e_{i}h/P_{2}e_{i}h = R_{1}/P_{2}R_{1} × R_{2}/P_{2}R_{2} is trivial, contradicting the fact that \overline{e_{i}}h = h_{m_{i}} is local. ■

Recall that we know that Aut_{Λ}(h_{m}) ≅ Aut_{Z_{l}}(h_{m}) from Prop. 3.7. If γ ∈ Aut_{Z_{l}}(h_{m}) then we will denote by γ̃ ∈ Aut_{Λ}(h_{m}) the isomorphic image of γ.

Lemma 3.10. Let γ ∈ Γ_{f} ⊂ Aut_{Z_{l}}(h_{m}) and let γ̃ ∈ Aut_{Λ}(h_{m}) be the isomorphic automorphism. Then the following diagram commutes for P = P_{2}:

\xymatrix{O_{f,𝔩}[[X]] ≅ I ≅ h_{m} \ar[d]^{γ̃} \ar[r]^{modP} & O_{f,𝔩} ≅ h_{m} \ar[d]^{γ} \\ O_{f,𝔩}[[X]] ≅ I ≅ h_{m} \ar[r]^{modP} & O_{f,𝔩} ≅ h_{m}}

Proof. Note that for P = (X) the diagram obviously commutes. Now consider t := 1 + X − (1 + l)^{2} ∈ O_{f,𝔩}[[X]]. Clearly O_{f,𝔩}[[X]] ≅ O_{f,𝔩}[[t]], and substituting O_{f,𝔩}[[t]] for O_{f,𝔩}[[X]] = h_{m} in the diagram shows that it does indeed commute for P = (t) = P_{2}, since the difference between X and t is 1 − (1 + l)^{2} ∈ Z_{l} and thus fixed by both γ and γ̃. ■

Now we make precise the notion that there are no twists in the Λ-adic setting that were not present in weight 2:

Proposition 3.11. The Λ-adic twists are those twists that fix 𝔩. More precisely,

Γ_{F} ≅ Γ′_{f} ⊂ Γ_{f}

Proof. We know that

Γ_{F} ⊂ Aut_{Λ}(I) ≅ Aut_{Z_{l}}(O_{f,𝔩}) ⊃ Γ′_{f}

so the proposition follows if we can show that the isomorphism between the automorphism groups maps each Γ onto the other.

Let γ ∈ Γ′_{f}, and let γ̃ ∈ Aut_{Λ}(I) be its isomorphic image. Then to show that γ̃ ∈ Γ_{F} it suffices to show that there is a Dirichlet character that is compatible with it. By lemma (3.10), γ̃F mod P_{2} ≐ f⊗χ_{γ}, where the ≐ symbol signifies that all but finitely many of the prime-index Fourier coefficients on both sides agree. But F ⊗χ_{γ} mod P_{2} ≐ f⊗χ_{γ} as well. Since eigenforms that agree on all but finitely many prime-index coefficients are equal, F ⊗χ_{γ} mod P_{2} ≐ γ̃F mod P_{2}. But then the uniqueness of the Λ-adic lift yields that F ⊗χ_{γ} ≐ γ̃F, so χγ̃ = χ_{γ}, and γ̃ ∈ Γ_{F}. So Γ′_{f} ↪ Γ_{F}.

Conversely, let γ̃ ∈ Γ_{F}, and γ ∈ Aut_{Z_{l}}(O_{f,𝔩}) be its isomorphic image. Then

f⊗χγ̃ = F ⊗χγ̃ mod P_{2} ≐ γ̃F mod P_{2} = γ(F mod P_{2}) = γf

and γ ∈ Γ′_{f}, so Γ_{F} ↪ Γ′_{f}. ■

The last proposition gives us an injection φ : Γ_{F} ↪ Γ_{f}, so we expect the corresponding Galois groups to have the opposite relationship:

Proposition 3.12. Under the above assumptions, H_{f} ⊂ H_{F}.

Proof. Let g ∈ H_{f} = ⋂_{γ∈Γ_{f}} ker(χ_{γ}). Then to prove the proposition, it suffices to show that for any γ′ ∈ Γ_{F}, χ′_{γ} is trivial on g. This is equivalent to saying that γ′(a(p, F)) = a(p, F) for almost all p. g ∈ H_{f} means that γ′(a(p, F)) mod P_{2} = φ(γ′)(a(p, f)) = a(p, f) for almost all p. So \frac{γ′(a(p,F))}{a(p,F)} − 1 ∈ P_{2} for almost all p. But χ_{γ′}(p) is a finite order character, so \frac{γ′(a(p,F))}{a(p,F)} is an element of finite order in 1 + P_{2}, and thus must be 1 for almost all p. ■

Now we are ready to prove:

Theorem 3.13. R_{F} mod P_{2} ≅ R_{f,𝔩}

Proof. Given the previous results, this follows immediately:

(R_{F} mod P_{2}) = (I^{Γ_{F}} mod P_{2}) ≅ (O_{f,𝔩}[[X]]^{Γ_{F}} mod P_{2}) ≅ O_{f,𝔩}^{Γ′_{f}} = O_{f,𝔩}^{Γ_{f}} = R_{f,𝔩}

Note that we are abusing notation slightly here, where Γ_{f} doesn’t act on O_{f,𝔩} (an element of Γ_{f} might permute the components of O_{f,l}). Instead we take O_{f,𝔩}^{Γ_{f}} to mean that subset of O_{f,𝔩} which is fixed by Γ_{f} when embedded in O_{f,l}. ■

3.3. A Proposition of N. Boston. In the appendix to [MW86], Boston gives a criterion for lifting the property of being full from a residual representation to the original one. We suspend for this section our notation from above and use Boston’s.

If I is an ideal of a ring A, define Γ(I) := ker(SL_{n}(A) → SL_{n}(A/I)) (n will be clear from the context). Also, for any group D, let Z(D) denote the center of D.

Call an element T ∈ GL_{n}(W) a transvection if T d = d for all d ∈ D for some hyperplane D, and for all x ∈ V = W^{n}, T (x) = x + d_{x}, for some d_{x} ∈ D.

Let R be a complete, Noetherian, local ring, with maximal ideal m, and residue characteristic p ≥ 5. We further assume that R/m is finite, that R is regular and of Krull dimension 2, and that m = (p, t). Suppose p ∤ n. Let I_{1}, …, I_{d} be minimal ideals of R/m^{2} that generate m/m^{2}, where d := dim_{R/m}(m/m^{2}).

We use the following proposition:

Proposition 3.14. [MW86, Cor. in appendix] Let D be a closed subgroup of SL_{n}(R) projecting onto SL_{n}(R/m), such that for 1 ≤ i ≤ d, there exists x_{i} ∈ Γ(I_{i}) \ Z(SL_{n}(R/m^{2})) normalizing the image of D in SL_{n}(R/m^{2}). Then D = SL_{n}(R).

Then a slightly modified form of Boston’s result is:

Proposition 3.15 (Boston). Let ρ : G → GL_{2}(R) be a continuous representation, inducing \overline{ρ} : G → GL_{2}(R/m). Let L ⊂ G be a subgroup. Suppose \overline{ρ} is full. Then if:

1. ρ(L) ⊂ \{( \begin{matrix}1 & ∗ \\ 0 & ∗ \end{matrix})\};

2. There exists a matrix of the form ( \begin{matrix}1 & ∗ \\ 0 & (1+p)^{−1}(1+t) \end{matrix} ) in ρ(L); and

3. for each b ∈ F_{p}^{×} ⊂ (R/m)^{×}, there exists a matrix of the form ( \begin{matrix}1 & ∗ \\ 0 & b \end{matrix}) ∈ \overline{ρ}(L) then ρ is full.

Proof. The proof here basically follows the one given in the appendix to [MW86], with the exception that we make it more explicit here, and use the condition 2 above instead of Boston’s condition that there exists a matrix of the form ( \begin{matrix}1 & ∗ \\ 0 & (1+t) \end{matrix} ) in ρ(L). Also, we use the more general setting of the groups L ⊂ G instead of Boston’s specific use of the inertia group at p: I_{p} ⊂ Gal(\overline{Q}/Q). Let F := F_{l^{r}} ≅ R/m be the residue field of R. Let D := ρ(G) ∩ SL_{2}(R) and D_{2} := D mod m^{2} ⊂ SL_{2}(R/m^{2}) be its projection. Then we take as the ideals for Prop. 3.14 above J_{1} := (p, m^{2})/m^{2} ⊂ R/m^{2} and J_{2} := (t, m^{2})/m^{2} ⊂ R/m^{2}. These obviously generate m/m^{2} and are minimal in R/m^{2}.

If we can find elements x_{i} ∈ Γ(J_{i}) \ Z(SL_{2}(R/m^{2})) for i := 1, 2 that normalize D_{2}, we will be done by Prop. 3.14.

Claim 3.16. Γ(J_{1}) ⊂ D_{2}

Proof. Let 1 + u ∈ D_{2} be a lift of a non-scalar transvection to D_{2} ⊂ SL_{2}(R/m^{2}) from SL_{2}(R/m) (possible since p ≥ 5 and D →^{onto} SL_{2}(R/m)). Let x_{1} := (1 + u)^{p} = 1 + pu ∈ D_{2}. Note that x_{1} mod J_{1} = 1, so that x_{1} ∈ D_{2} ∩ Γ(J_{1}). But Γ(J_{1}) is minimal (since J_{1} is) so D_{2} ∩ Γ(J_{1}) = Γ(J_{1}), i.e. Γ(J_{1}) ⊂ D_{2}. So the claim is done ■

This gives us the x_{1} we were looking for (being in D_{2} guarantees normalizing it...). So only need to find an x_{2} now:

By the second hypothesis, we know that there exists at least one matrix of the form

( \begin{matrix}1 & r \\ 0 & (1+p)^{−1}(1+t) \end{matrix} ) in ρ(L). Two cases arise:

1. one of these r’s satisfies r ∈ m. Then choose a, b ∈ R so that r = pa + tb and consider the matrix

A′ := (1 + t)^{−1/2}( \begin{matrix}1 & pa+tb \\ 0 & (1+p)^{−1}(1+t) \end{matrix} )( \begin{matrix}1 & −pa(1+p) \\ 0 & 1+p \end{matrix} )
= (1 + t)^{−1/2}( \begin{matrix}1 & tb(1+p) \\ 0 & 1+t \end{matrix} ) = ( \begin{matrix}(1+t)^{−1/2} & tb(1+p) \\ 0 & (1+t)^{1/2} \end{matrix} )

Note that A′ has determinant 1 and mod t is the identity matrix (so A′ ∈ Γ(J_{2})), and non-scalar (so non-central in SL_{2}(R/m^{2})). But A′ is just a product of a matrix in ρ(L) mod m^{2} and a matrix in Γ(J_{1}) ⊂ D_{2} with a scalar, so it certainly normalizes D_{2} , and we have our desired x_{2} := A′, so the proof of the proposition is done in this case.

2. None of the r’s that arise in this manner are inside m. Thus we get a matrix ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) ∈ \overline{ρ}(L)

such that a ≠ 0 (r ≡ a mod m).

p ≥ 5, so we can choose n ∈ {2, …, p − 2} such that when viewed as an element of F_{p} ⊂ R/m, n is a unit and n^{−1} ≠ −1. Let c := n^{−1} + 1 ≠ 0 (c ∈ F_{p}). Then by the third assumption of the proposition, there exists s ∈ R/m such that ( \begin{matrix}1 & s \\ 0 & c \end{matrix}) ∈ \overline{ρ}(L). Then (letting L′ denote the commutator of L):

\overline{ρ}(L′) ∋ (1/c)( \begin{matrix}1 & na \\ 0 & 1 \end{matrix} )( \begin{matrix}1 & s \\ 0 & c \end{matrix})( \begin{matrix}1 & −na \\ 0 & 1 \end{matrix} )( \begin{matrix}c & −s \\ 0 & 1 \end{matrix} )
= (1/c)( \begin{matrix}1 & s+nac \\ 0 & c \end{matrix})( \begin{matrix}c & −s−na \\ 0 & 1 \end{matrix} )
= ( \begin{matrix}1 & s−s+(c−1)na \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) =: \overline{B}

Now we consider what \overline{B} might lift to: we know that det(ρ(L′)) = \{1\}, so by the first hypothesis, \overline{B} must be the reduction of some B = ( \begin{matrix}1 & r′ \\ 0 & 1 \end{matrix} ) ∈ ρ(L′) such that r′ ≡ a mod m. In particular, we get that ( \begin{matrix}1 & r−r′ \\ 0 & (1+p)^{−1}(1+t) \end{matrix} ) ∈ ρ(L), and r−r′ ∈ m. This is a contradiction to the assumption of this case, so the proof of the proposition is done.

■

Having set up this machinery, we are ready to lift the weight 2 results.

4. Λ-adic Situation

4.1. Lifting the Representation Fullness. Let ρ := ρ_{F} : G_{Q} → GL_{2}(I) be the Galois representation attached to F from section 3.1. Let \overline{ρ} be the reduction ρ mod 𝔪. Then we set ρ_{k} := ρ mod P_{k} for each weight k to be the weight k specialization of ρ. Note that this ρ_{2} coincides with the ρ_{2,l} from section 2.

Proposition 4.1. Up to conjugation ρ_{F} (H_{F}) ⊂ GL_{2}(R_{F})

Proof. By the Cebotarev density theorem, it is in enough to show that for any Frob_{q} ∈ H_{F}, ρ_{F} (Frob_{q}) ∈ GL_{2}(R_{F}). From the properties of ρ_{F}, we know that tr(ρ_{F} (Frob_{q})) = a(q, F) and Frob_{q} ∈ H_{F} implies that χ_{γ}(q) = 1 for all γ ∈ Γ_{F} so tr(ρ_{F} (Frob_{q})) ∈ R_{F}. Using Wiles’ theory of pseudo-representations (for instance, [Hid00, Prop. 2.16]), we know that there exists a representation π : H_{F} → GL_{2}(R_{F}) whose trace agrees with that of ρ_{F} |_{H_{F}}. But (ρ_{F} mod m_{I})|_{H_{F}} is irreducible, so using a result of Carayol and Serre ([Hid00, Prop. 2.13]) we see that ρ_{F} |_{H_{F}} and π are conjugate, and the proposition follows. ■

Hereafter we restrict our attention to the image of H_{F}. Thus we let ρ′_{f} := ρ_{f}|_{H_{f}} and ρ′_{F} = ρ_{F} |_{H_{F}}. In this section we let ρ := ρ′_{F} to simplify notation where no confusion can arise. Note also that because we have shown that H_{f} ⊂ H_{F}, the image of ρ mod P_{2} will contain the image of ρ′_{f}.

In order to apply Boston’s proposition to our situation we record the following observations:

Lemma 4.2. Specializing to the weight 2 representation and then reducing mod (l) is equivalent to reducing the I-adic representation mod 𝔪.

Note that the former is what corollary 2.1 gives us, and the latter is what Prop. 3.15 requires.

Proof. This is trivial by noting that the following diagram commutes:

\xymatrix{H_{F} \ar[r]^{ρ} \ar[dr]_{\overline{ρ}} & GL_{2}(R_{F}) \ar[r]^{(1+X)↦(1+l)^{2}}_{modP_{2}} \ar[d]_{mod𝔪} & GL_{2}(R_{f,𝔩}) \ar[dl]^{mod(l)} \\  & GL_{2}(F_{l^{r}})}

■

Combining lemma 4.2 with corollary 2.1 we get:

Lemma 4.3. \overline{ρ} is full: Im(\overline{ρ}) ⊃ SL_{2}(F_{l^{r}}).

Thus, to apply Prop. 3.15, we only need to verify the three technical conditions of Prop. 3.15. We make the following substitutions for the proposition (where I_{l} is the inertia at l in Gal(\overline{Q}/Q)): L = I_{l} ∩H_{F}, p = l, R = I, 𝔪 = (l, X), and R/𝔪 = F_{l^{r}}. We will let J := I_{l} ∩H_{F}.

Lemma 4.4. The first condition of Prop. 3.15 is satisfied: ρ(J) ⊂ \{( \begin{matrix}1 & ∗ \\ 0 & ∗ \end{matrix})\}

Proof. The definition of ρ_{F} being l-ordinary is that ρ_{F} |_{D_{l}} ≅ ( \begin{matrix}δ & ∗ \\ 0 & ψ \end{matrix} ) where δ is unramified at l (i.e., δ |_{I_{l}}= 1). So ρ_{F} |_{I_{l}} ≅ ( \begin{matrix}1 & ∗ \\ 0 & ψ \end{matrix} ) for some character ψ. An I-adic representation is always ordinary at the prime over which I is defined, so we are done. ■

In order to verify the other two criteria for Prop. 3.15, we make the following observation:

Lemma 4.5. J ⊂ Gal(\overline{Q}/Q) maps onto

Gal(\overline{Q}/Q)/ Gal(\overline{Q}/Q(μ_{l^{∞}})) ≅ Gal(Q(μ_{l^{∞}})/Q)

Call this latter Galois group L.

Proof. As noted in section 2.3, l ∤ cond(χ_{γ}), so in particular, l ∤ ord(χ_{γ}), and thus

Gal(\overline{Q}/Q)/ ker(χ_{γ})

is prime-to-l (contains no non-trivial homomorphic image of Z/lZ). Fix for the moment γ. Let K := ker(χ_{γ}). Then K′ := K/ Gal(\overline{Q}/Q(μ_{l^{∞}})) ⊂ L must be pro-l because L is, so L/K′ is pro-l. But L/K′ ≅ Gal(\overline{Q}/Q)/K which is prime-to-l as shown above, so L/K′ must be trivial, i.e. K′ = L. Varying γ now, we see that H_{F} / Gal(\overline{Q}/Q(μ_{l^{∞}})) must also be equal to L. Thus the lemma follows if we show that I_{l} surjects onto L.

Note that in G^{(n)} := Gal(Q(μ_{l^{n}})/Q), the inertia at l is the entire group G^{(n)} (because l is completely ramified, there is only one prime 𝔩 lying over it, which must be fixed by the entire Galois group). Consequently, one sees that the inertia at l in L = Gal(Q(μ_{l^{∞}})/Q) must be the entire Galois group L, and thus that I_{l} surjects onto L. ■

Lemma 4.6. There exists a matrix of the form ( \begin{matrix}1 & ∗ \\ 0 & (1+l)^{−1}(1+X) \end{matrix} ) in ρ(J)

Proof. The defining properties of ρ as an I-adic representation are laid out in [Hid86a, Thm. 2.1 and following remarks]. In particular, det(ρ) |_{I_{l}}= χν_{l}^{−1}ι. As shown above, J surjects onto L, so ι (which factors through L) must have a surjective image on J ⊂ Gal(\overline{Q}/Q), so that ι(J) = 1 + lZ_{l} ↪ Λ. For the same reason, ν_{l}(J) = Z_{l}^{×}.

Let σ ∈ J ⊂ Gal(\overline{Q}/Q). Let a_{σ} := ω_{l}(ν_{l}(σ)) ∈ (Z/lZ)^{×}, b_{σ} := ⟨ν_{l}(σ)⟩ ∈ 1 + lZ_{l}, and s ∈ Z_{l} such that b_{σ} = (1 + l)^{s}. Then

det(ρ_{F} (σ)) = χ(σ)ν_{l}(σ)^{−1}ι(σ) = χ(σ)ν_{l}(σ)^{−1}κ(⟨ν_{l}(σ)⟩) = χ(σ)b_{σ}^{−1}a_{σ}^{−1}κ(b_{σ})
= χ(σ)(1 + l)^{−s}a_{σ}^{−1}(1 + X)^{s} = ε(σ)ω_{l}^{2}(σ)(1 + l)^{−s}a_{σ}^{−1}(1 + X)^{s}
= ε(σ)(1 + l)^{−s}a_{σ}(1 + X)^{s}

By the surjectivity of ν_{l} and ι on J it follows that we can choose an element σ ∈ J to hit any given a_{σ} and any s ∈ Z_{l} (independently). Let M := cond(ε), and notice that by [Mom81, Rmk. 1.6], M|N. Note that since l ∤ N, Q(μ_{l^{∞}}) and Q(μ_{M}) are linearly disjoint inside Q(μ_{l^{∞}}, μ_{M}) ⊂ \overline{Q}, we can choose a σ ∈ Gal(\overline{Q}/Q) to yield any given s ∈ Z_{l}, a_{σ} ∈ (Z/lZ)^{×} and keeping ε(σ) = 1. So we can choose σ ∈ J so that s = 1 and ε(σ) = a_{σ} = 1, yielding the desired element of J. ■

Lemma 4.7. for each b ∈ F_{l}^{×} ⊂ (R/𝔪)^{×}, there exists a matrix of the form ( \begin{matrix}1 & ∗ \\ 0 & b \end{matrix}) in \overline{ρ}(J)

Proof. Using the proof of the previous lemma, one notes that the choice of a_{σ} is independent if the choice of s, so one may choose σ ∈ J so that s = 0 and χ(σ)a_{σ} hits any desired b ∈ \overline{ρ}(J). The lemma follows. ■

Putting lemmas 4.3, 4.4, 4.6, and 4.7 all together with Prop. 3.15, we obtain:

Theorem 4.8. The restricted Galois representation attached to F is full, i.e. ρ_{F} (H_{F}) ⊃ SL_{2}(R_{F}).

4.2. The Exact Image of ρ_{F} (H_{F}). We now know that SL_{2}(R_{F}) ⊂ ρ_{F} (H_{F}) ⊂ GL_{2}(R_{F}), and are interested in finding out just where between the two matrix groups the image of ρ_{F} lies.

Lemma 4.9. If

SL_{2}(R) ⊂ B ⊂ GL_{2}(R)

is an inclusion of groups and

det(B) = A^{×}

for some ring A, then

B = {X ∈ GL_{2}(R) | det(X) ∈ A^{×}}

Proof. The inclusion ‘⊂’ is obvious. To show ‘⊃’, let X ∈ GL_{2}(R) be such that a = det(X) ∈ A^{×}. Then by the second assumption there is a Y ∈ B such that det(Y) = a, so det(Y^{−1}X) = 1, and Y^{−1}X ∈ B. But Y, Y^{−1}X ∈ B, so X ∈ B, and the lemma is shown. ■

As shown in the proof of lemma 4.6,

det(ρ_{F} (σ)) = χ(σ)ν_{l}(σ)^{−1}ι(σ) = ε(σ)(1 + l)^{−s}a_{σ}(1 + X)^{s}

and the σ ∈ J = I_{l} ∩ H_{F} ⊂ Gal(\overline{Q}/Q) can be chosen to produce any (independently) chosen ε(σ) ∈ O_{f,𝔩}, a_{σ} ∈ (Z/lZ)^{×}, and s ∈ Z_{l}.

Let

D := {ca(1 + l)^{−s}(1 + X)^{s} ∈ O_{f,𝔩}[[X]] | c^{M} = a^{l} = 1, s ∈ Z_{l}}

Then we get

Proposition 4.10. Recalling that M is the conductor of the weight 2 nebentype and l ∈ Σ_{f} for the weight 2 specialization of F, we have:

det(ρ_{F} (H_{F})) = D ≅ μ_{M} × μ_{l} × Γ′

where

Γ′ := {(1 + l)^{−s}(1 + X)^{s} | s ∈ Z_{l}} ≅ Γ ⊂ Z_{l}[[X]]

Imitating the weight 2 situation, we let A_{F} := {x ∈ GL_{2}(R_{F}) | det(x) ∈ D} and putting the previous two lemmas together we get:

Corollary 4.11. The image of ρ_{F} (H_{F}) is as large as it can be given the determinant condition on it, i.e. ρ_{F} (H_{F}) = A_{F}

This is the analogous result to (Momose’s) Thm. 2.1 in weight 2.

Recalling that we made arbitrary choices for f_{2} and for l ∈ Σ_{f_{2}}, we reformulate Thm. 4.8:

Corollary 4.12. Let f ∈ S_{2}^{new}(Γ_{0}(N), ε) be a normalized eigenform without complex multiplication. Let Σ′_{f} be the set of ordinary primes for f (which is of density 1 by Prop. 2.2). Let

B_{f} := {F_{l} ∈ S^{ord}(Nl^{∞}, εω_{l}^{2}) | F_{l} specializes to f at weight 2 and l ∈ Σ′_{f}}

Then the restricted Galois representations (in the sense of Thm. 4.8) attached to all but finitely many of the elements of B_{f} are full.

4.3. The Entire Image ρ_{F} (G). The computation of the exact image of ρ_{F} (G) follows almost word for word the argument of E. Papier as presented in [Rib85, Sec. 4]. We include the proof here for the reader’s convenience.

The basic result of this section is that the difference between ρ_{F} (G) and ρ_{F} (H_{F}) is similar to the difference between G and H_{F}.

Lemma 4.13. Given γ ∈ Γ_{F}, γρ_{F} and ρ_{F} ⊗χ_{γ} are irreducible.

Proof. Suppose 0 ≠ V ⊊ K^{2} is a non-trivial stable subspace under γρ_{F} (G). Note that γρ_{F} (g)( \begin{matrix}x \\ y \end{matrix}) = γ (ρ_{F} (g)γ^{−1}( \begin{matrix}x \\ y \end{matrix})), so if (γρ_{F} (G))V = V then γ(ρ_{F} (G)γ^{−1}(V)) = V, and ρ_{F} (G)γ^{−1}(V) = γ^{−1}(V), contradicting the irreducibility of ρ_{F}.

That ρ_{F} ⊗χ_{γ} is irreducible follows immediately from the fact that ρ_{F} is irreducible and the fact that tensoring with a character has the effect of multiplying by a scalar, so clearly a character will preserve any linear subspace. ■

By the definition of χ_{γ}, it is clear that γρ_{F} and ρ_{F} ⊗χ_{γ} have the same trace on Frob_{p} for any p ∤ lN, so they have equal traces everywhere (by the Cebotarev density theorem). The two representations must then be equivalent (being semi-simple and of equal trace). So there is a matrix X ∈ GL_{2}(K) such that Xγρ_{F} X^{−1} = γ⊗χ_{γ}. Considering the restriction to H_{F}, we see that X commutes with ρ_{F} (H_{F}) ⊃ SL_{2}(R_{F}), so X is a scalar matrix and in fact γρ_{F} (g) = ρ_{F} (g)χ_{γ}(g) for any g ∈ G.

Fix for the moment γ ∈ Γ_{F} and g ∈ G. From the structure of I and R_{F} which we have already computed, we know that I/m_{I} and R_{F} /m_{R_{F}} are finite. Call these fields E and F respectively. Then we note that Γ_{F} ≅ Gal(E/F) and we can view γ as acting on elements of E. Notice that being a root of unity, χ_{γ}(g) has the same order when projected into E, so we view it as being both an element of I^{×} and E^{×}. Using Hilbert’s theorem 90 we can find an element \overline{α(g)} ∈ E such that γ(\overline{α(g)})/\overline{α(g)} = χ_{γ}(g). I is unramified, so we can lift \overline{α(g)} to W (E) ↪ I and call this lift α(g) ∈ I. Since E is finite, \overline{α(g)} has finite order, so α(g) has finite order, and α(g) ∈ I^{×}. Note that the choice of g modulo H_{F} is irrelevant, since χ_{γ} is trivial on H_{F}, so there are only finitely many α(g)’s.

Lemma 4.14. When χ is the nebentype of F, we have: γ(χ(g)) = χ_{γ}(g)χ(g)^{2}

Proof. It suffices to prove the lemma for g = Frob_{p} for all p ∤ lN. Let p be a prime outside lN. We consider the difference of Hecke operators: T (p)^{2} − T (p^{2}). Specifically, we apply the two operators to F and take the p’th Fourier coefficient in the q-expansion of the resulting eigenform:

a(p, F |T (p)) =a(p^{2}, F) + κ(⟨p⟩)χ(p)p^{−1}
a(p^{2}, F |T (p)) =a(p^{3}, F) + κ(⟨p⟩)χ(p)p^{−1}a(p, F)
a(p, F |T (p)^{2}) =a(p, (F |T (p))|T (p)) = a(p^{2}, f|T (p)) + κ(⟨p⟩)χ(p)p^{−1}
=a(p^{3}, F) + κ(⟨p⟩)χ(p)p^{−1}a(p, F) + κ(⟨p⟩)χ(p)p^{−1}
a(p, F |T (p^{2})) =a(p^{3}, F) + κ(⟨p⟩)χ(p)p^{−1}a(p, F)

So

a(p, F |T (p)^{2}) − a(p, F |T (p^{2})) = κ(⟨p⟩)χ(p)p^{−1}

but F is a normalized eigenform so a(p, F)^{2} = a(p, F |T (p)^{2}) and a(p^{2}, F) = a(p, F |T (p^{2}) yielding:

a(p, F)^{2} − a(p^{2}, F) = κ(⟨p⟩)χ(p)p^{−1}

Applying γ to both sides and using the fundamental relationship between γ and χ_{γ} we get:

χ_{γ}(p)^{2}a(p, F)^{2} − χ_{γ}(p)^{2}a(p^{2}, F) =γ(κ(⟨p⟩)χ(p)p^{−1})
χ_{γ}(p)^{2}(a(p, F)^{2} − a(p^{2}, F)) =γ(χ(p))κ(⟨p⟩)p^{−1}
χ_{γ}(p)^{2}κ(⟨p⟩)χ(p)p^{−1} =γ(χ(p))κ(⟨p⟩)p^{−1}
χ_{γ}(p)^{2}χ(p) =γ(χ(p))

and the lemma is proven. ■

The preceding lemma tells us that γ(\frac{α(g)^{2}}{χ(g)} ) = \frac{χ_{γ}(g)^{2}α(g)^{2}}{χ_{γ}(g)^{2}χ(g)} = \frac{α(g)^{2}}{χ(g)} , so \frac{α(g)^{2}}{χ(g)} ∈ R_{F}.

Re-write ρ_{F} (g) as

ρ_{F} (g) = (\begin{matrix}α(g) & 0 \\ 0 & χ(g)/α(g) \end{matrix}) \{(\begin{matrix}1 & 0 \\ 0 & α(g)^{2}/χ(g) \end{matrix})(α(g)^{−1}ρ_{F} (g))\}

Then the product in curly brackets is in GL_{2}(R_{F}) and has determinant in D, so is in A_{F}. This allows us to give a full characterization of the image of ρ_{F}:

Theorem 4.15. The image of ρ_{F} is the subgroup of GL_{2}(I) generated by A_{F} and the finite

set of matrices: (\begin{matrix}α(g) & 0 \\ 0 & χ(g)/α(g) \end{matrix}), where g ∈ G/H_{F}.

5. Density Analysis

Now that we have the general machinery for lifting the fullness of a weight-2 Galois representation, we wish to consider how often this machinery succeeds in telling us that an I-adic representation is full.

To avoid repetition, call an ordinary, normalized eigenform which has no complex multiplication a generic form.

A naive approach to the analysis problem might be to decompose the set of all generic I-adic forms into a disjoint union of lift families indexed over f ∈ S_{2}(N, χ), which is a finite set, and in each family only finitely many elements are non-full, so we would get that only finitely many I-adic forms are not full. This reasoning is not correct, since some generic I-adic forms don’t come from a weight-2 form of level N, but rather of level Nl, for some prime l.

Instead, we note that an arbitrary generic I-adic form with level N outside its structure prime (call it l) will be a lift of some generic form in S_{2}(Nl, χω_{l}^{a}). To put this observation into a usable form, we make the following definitions:

Definition 5.1. Fix an integer N ≥ 1, a prime l ∤ N, and a primitive character χ mod N. Then define:

C(N, l, χ) := ⨆_{a∈{0...l−1} g∈S_{2}(Nl,χω_{l}^{a}) g is generic} B_{g}

Further, varying l, define:

B(N, χ) := ⋃_{l∤N}C(N, l, χ)

In this setup, each of the B_{g}’s is an infinite set (indexed by a density 1 set of primes), and the Galois representations attached to all but finitely many of the elements of each B_{g} are full. Moreover, the disjoint union is taken over a finite set. Thus we obtain:

Corollary 5.2. The Galois representations attached to all but finitely many of the elements of C(N, l, χ) is full.

Now, to get a density theorem on the totality of B(N, χ), we need to define partial subsets of B and C as follows: for a positive integer M let

B_{g}^{(M)} := {F ∈ B_{g} | the structure prime of F < M}

and notice that this set is finite. Similarly, define

C(N, l, χ)^{(M)} := ⨆_{a∈\{0...l−1\} g∈S_{2}(Nl,χω_{l}^{a}) g is generic} B_{g}^{(M)} and B(N, χ)^{(M,M′)} := ⋃_{\begin{matrix}l∤N \\ l<M′ \end{matrix}} C(N, l, χ)^{(M)}

and note that both of the above sets are finite.

For any of the preceding sets of modular forms (B, C, etc.), let the superscript + on the letter denote the subset with full attached Galois representations. Then define:

d_{N,l,χ}^{(M)} := \frac{|C^{+}(N, l, χ)^{(M)}|}{|C(N, l, χ)^{(M)}|}

Then corollary 5.2 implies:

Corollary 5.3. d_{N,l,χ}^{(M)} → 1 as M → ∞.

Similarly, we may define

b_{N,χ}^{(M,M′)} := \frac{|B^{+}(N, χ)^{(M,M′)}|}{|B(N, χ)^{(M,M′)}|}

Then the previous corollary gives us:

Corollary 5.4. For any M′ > 1

b_{N,χ}^{(∞,M′)} := lim_{M→∞} b_{N,χ}^{(M,M′)} = 1

But the density of B^{+}(N, χ) in B(N, χ) can be thought of precisely as b_{N,χ}^{(∞,∞)}, and by the preceding corollary, we get:

Theorem 5.5. Keeping the outside level and character fixed, the set of generic I-adic forms that have full attached Galois representations has density 1 in the set of all generic I-adic modular forms. Since this is true for any level and character, the same density statement holds for all generic I-adic modular forms.

Note that calculating “along the diagonal”, i.e. b_{N,χ}^{(X,X)} → 1 as X → ∞ works just as well by the above reasoning. Note also that this reasoning does not work if one attempts to calculate b_{N,χ}^{(M,∞)} (since we do not have a good understanding of how d_{N,l,χ}^{(M)} behaves as l → ∞ and M is fixed).

It should also be noted that the density results here are optimal in the sense that it is not the case that only a finite number of generic I-adic modular forms have non-full representations. Each irregular prime gives rise to a Λ-adic form whose representation is reducible modulo the maximal ideal of the coefficient ring, so clearly it cannot be full. Then the infinitude of irregular primes guarantees the infinitude of non-full representations attached to generic I-adic modular forms.

Appendix A. Complex Multiplication

In this section we consider the relationship between a Λ-adic modular form F and its specializations f_{k} to weight k ≥ 2 with respect to complex multiplication. In particular, we show (Prop. A.1) that in our setting, if F has no CM then neither do the f_{k}.

We say of a modular form f = ∑_{n=0}^{∞} a_{n}q^{n} (classical or Λ-adic) that it has complex multiplication (CM) if there exists a non-trivial Dirichlet character φ such that φ(p)a_{p} = a_{p} for a density 1 set of primes p. If φ is defined mod D then we know that φ(p)a_{p} = a_{p} holds for all p ∤ DN when f is a classical modular form of level N. Further, φ has to be a quadratic character. If the kernel of φ in G = Gal(\overline{Q}/Q) is the quadratic field F we say f has CM by F.

In this section, F ∈ S_{I}^{ord}(Nl^{∞}, χ) will denote a normalized Hecke eigenform with no CM, and f_{k} ∈ S_{k}(Nl, ε) will denote F ’s specialization modulo the prime P_{k} (so χ = εω_{l}^{k}). Then f_{k} is a normalized Hecke eigenform (though not a newform for level Nl if k > 2).

Proposition A.1. If F has no complex multiplication, then neither does f_{k} for any k ≥ 2.

Proof. The key point is that the weight k is integral and greater than 1. Denote by the subscript ·^{0} the primitive part of a Hecke algebra. Then we have h^{0}/P_{k}h^{0} ≅ h_{k}^{0,ord} and so the completions obey:

h_{P_{k}}^{0}/P_{k}h_{P_{k}}^{0} ≅ h_{k}^{0,ord}⊗Q ≅ U_{1} × ⋯ × U_{v}

for some fields U_{i}. But this means that

h_{P_{k}}^{0} ≅ U_{1} × ⋯ × U_{v}

where the U’s and the U’s are in bijection (by the proof of Lemma 3.9). Each of the U’s is an irreducible component of the completed Hecke algebra.

Now suppose f_{k} has CM for integral k ≥ 2. As proven by Hida in [Hid93, Sec. 7.6], a classical eigenform with CM gives rise to a Λ-adic CM eigenform, so there is a Λ-adic eigenform F′ that specializes to f_{k} mod P_{k}. But the bijection between the Λ-adic and classical local decompositions in the previous paragraph, this means that F′ and F must belong to the same U_{i}. But since F′ has CM and F does not, they can’t be conjugate and thus can’t belong to the same irreducible component of the completed Hecke algebra. ■

Note that further, in the “worst” possible case, F can only specialize to finitely many CM classical forms:

Lemma A.2. f_{k} does not have CM for all but finitely many values of k.

Proof. Fix a character θ and let 𝔞_{θ} be the ideal generated by a(l, F) − θ(l)a(l, F) for all primes l ∤ DN (where D is the conductor of θ). Then for f_{k} to have CM by θ is equivalent to P_{k} ⊃ 𝔞_{θ}. This implies that P_{k} is in V := Spec(Λ/𝔞_{θ})(\overline{Q_{p}}), which is a proper closed subset of Spec(Λ)(\overline{Q_{p}}). But the latter is 1-dimensional, so V must be 0-dimensional, i.e. only contain finitely many points. Thus only finitely many P_{k} ∈ V, and only finitely many f_{k} have CM by θ.

On the other hand, the level of f⊗θ is the product of the level of f and the square of the conductor of θ. So if f is to have CM by θ, θ’s conductor must be bounded, so only finitely many such θ’s exist and the lemma is shown.

■

A.1. Galois Representations. In this section we consider the effect CM has on the Galois representations attached to a modular form. For f classical or Λ-adic , let ρ_{f} be the attached l-adic or Λ-adic Galois representation. Proposition A.3. f has CM by a character θ if and only if there is a non-trivial quadratic character θ such that ρ_{f} ≅ ρ_{f}⊗θ. Writing F = Q(\sqrt{−D}) for the kernel of θ, f having CM by θ and F is equivalent to ρ_{f} = Ind_{Gal(\overline{Q}/F)}^{Gal(\overline{Q}/Q)} ψ for some character ψ.

Proof. Note that in the first assertion, the (⇐) direction is trivial (since the traces of ρ_{f} on Frobenius elements are the a(p, f)).

The second assertion follows from the first by [DHI98, Lemma 3.2].

Notice that in [DHI98, Lemma 3.2] there as an apparent requirement that \overline{ρ_{f}} be absolutely irreducible. This is not necessary. The only place this is used is in determining that C^{r} is a scalar matrix. But this can be obtained separately in the cases we deal with. In the classical case, Ribet has shown that ρ_{f} is irreducible, which is enough to give that C^{r} is scalar (see for instance [Hid00, p.99]). In the Λ-adic case, the situation is only slightly more complicated, and comes from the classical case.

Letting Λ_{(P)} be the localization at a prime P, consider the diagram:

\xymatrix{Gal(\overline{Q}/Q) \ar[r]^{ρ} \ar[dr]_{ρ_{P}} & GL_{2}(Λ) \ar[r] \ar[d]^{modP} & GL_{2}(Λ_{(P)}) \ar[d]^{modP} \\  & GL_{2}(Z_{p}) \ar[r] & GL_{2}(Q_{p})}

Then ρ_{P} is irreducible from the classical case. Viewing Z(ρ) ⊂ M_{2}(Λ_{(P)}), we get a short exact sequence: \xymatrix{Λ_{(P)} \ar[r] & Z(ρ) \ar[r]^{modP} & Q_{p}^{×}} Tensoring with Q_{p} over Λ_{(P)} yields Λ_{(P)}⊗_{Λ_{(P)}}Q_{p} =

Q_{p} → Z(ρ)⊗_{Λ_{(P)}}Q_{p} = Q_{p} so by Nakayama’s lemma Z(ρ) = Λ_{(P)}, and we get that the centralizer of ρ is trivial again.

All that remains is to prove the direct direction of the first assertion. From the definition of CM, we know that tr(ρ_{f})(φ_{q}) = tr(ρ_{f}⊗θ)(φ_{q}) for q ∤ DN and φ_{q} the Frobenius element at q. Thus by Cebotarev’s density theorem, we know that the traces of the two representations must be equal. But then we can use the Brauer-Nesbitt theorem ([Hid00, Cor. 2.8]) to see that ρ_{f} ≅ ρ_{f}⊗θ (equivalence is over the field of fractions).

■

Proposition A.4. If f has CM by θ and F (F = Q(\sqrt{−D}) for D > 0 square-free), Im(ρ_{f}|_{Gal(\overline{Q}/F)}) ⊂ \{( \begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix})\}. Further, ρ(Gal(\overline{Q}/Q)) ⊂ \{( \begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix}), ( \begin{matrix}0 & ∗ \\ ∗ & 0 \end{matrix} )\}.

Proof. Consider the Galois tower: _{\overline{Q} _{H}_{G} Q(\sqrt{−D}) = F_{⟨θ⟩} Q}

By Prop. A.3 we know that ρ_{f} = Ind_{Gal(\overline{Q}/F)}^{Gal(\overline{Q}/Q)} ψ for some character ψ of H. Writing the

coset decomposition of G/H as G = ⊔_{i=1}^{[G:H]}Hσ_{i} we get a matrix representation of ρ_{f} as: ρ_{f}(g) = Ind_{Gal(\overline{Q}/F)}^{Gal(\overline{Q}/Q)} ψ(g) = (ψ(σ_{i}gσ_{j}^{−1}))_{[G:H]} (where ψ takes the value 0 on any argument not in H). Concretely, we have that [G : H] = 2 and we take σ_{1} = id, σ_{2} ∈ G \ H.

Consider now the image of Frobenius elements: let φ_{q} be a Frobenius element at q. Then

ρ_{f}(φ_{q}) = ( \begin{matrix}θ(g) & θ(gσ_{2}^{−1}) \\ θ(σ_{2}g) & θ(σ_{2}gσ_{2}^{−1}) \end{matrix} ). If φ_{q} ∈ H, then θ(g) and θ(σ_{2}gσ_{2}^{−1}) are non-zero, and θ(gσ_{2}^{−1}) =

θ(σ_{2}g) = 0. If φ_{q} ∈ G \ H then θ(gσ_{2}^{−1}) and θ(σ_{2}g) are non-zero, and θ(g) and θ(σ_{2}gσ_{2}^{−1}) are zero. Since Frobenius elements generate the Galois group, this shows the proposition. ■

References

[DHI98] Koji Doi, Haruzo Hida, and Hidenori Ishii. “Discriminant of Hecke fields and twisted adjoint L-values for GL(2).” Invent. Math., 134(3):547–577, 1998.

[Eis95] David Eisenbud. Commutative algebra with a view toward algebraic geometry. Springer-Verlag, New York, 1995.

[FT93] A. Fröhlich and M. J. Taylor. Algebraic number theory. Cambridge University Press, Cambridge, 1993.

[Gou92] Fernando Q. Gouvêa. “On the ordinary Hecke algebra.” J. Number Theory, 41(2):178–198, 1992.

[Hid86a] Haruzo Hida. “Galois representations into GL_{2}(Z_{p}[[X]]) attached to ordinary cusp forms.” Invent. Math., 85(3):545–613, 1986.

[Hid86b] Haruzo Hida. “Hecke algebras for GL_{1} and GL_{2}.” In Séminaire de théorie des nombres, Paris 1984–85, pp. 131–163. Birkhäuser Boston, Boston, Mass., 1986.

[Hid86c] Haruzo Hida. “Iwasawa modules attached to congruences of cusp forms.” Ann. Sci. École Norm. Sup. (4), 19(2):231–273, 1986.

[Hid87] Haruzo Hida. “Galois representations and the theory of p-adic Hecke algebras.” Sûgaku, 39(2):124– 139, 1987. in Japanese.

[Hid89] Haruzo Hida. “p-adic Hecke Algebras and Galois Representations.” Sugaku Expositions, 2(1):75– 102, 1989. English translation of [Hid87].

[Hid93] Haruzo Hida. Elementary theory of L-functions and Eisenstein series. Cambridge University Press, Cambridge, 1993.

[Hid00] Haruzo Hida. Modular Forms and Galois Cohomology. Cambridge University Press, Cambridge, 2000.

[Miy89] Toshitsune Miyake. Modular forms. Springer-Verlag, Berlin, 1989. Translated from the Japanese by Yoshitaka Maeda.

[Mom81] Fumiyuki Momose. “On the l-adic representations attached to modular forms.” J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28(1):89–109, 1981.

[MW86] B. Mazur and A. Wiles. “On p-adic analytic families of Galois representations.” Compositio Math., 59(2):231–264, 1986.

[Rib85] Kenneth A. Ribet. “On l-adic representations attached to modular forms II.” Glasgow Math. J., 27:185–194, 1985.

[Ser81] Jean-Pierre Serre. “Quelques applications du théorème de densité de Chebotarev.” Inst. Hautes Études Sci. Publ. Math., (54):323–401, 1981.

Department of Mathematics, UCLA, CA 90095-1555, USA

  E-mail address: fischman@math.ucla.edu