pacific journal of mathematics Vol. 181, No. 3, 1997

NON-ABELIAN BASE CHANGE FOR TOTALLY REAL FIELDS

Haruzo Hida and Yoshitaka Maeda^{1}

In memory of Olga Taussky Todd

1. Introduction.

Let f ∈ S_{k}(Γ_{0}(N), χ) with k ≥ 2 and N ≥ 1 be a primitive Hecke eigenform normalized so that the first Fourier coefficient a(1, f) is equal to 1. Let ρ : Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{ℚ}_{p}) be the p-adic Galois representation attached to f. Then writing f|T (n) = λ(T (n))f for the Hecke operators T (n), we have λ(T (n)) = a(n, f) and

L(s, ρ) = L(s, f) = L(s, λ) = ∑_{n=1}^{∞} λ(T (n))n^{−s}.

Let F be a number field. A cohomological Hecke eigen cusp form f̂ on GL_{2}(F_{𝔸}) for the adele ring F_{𝔸} of F is called a base change of f if L(s, f̂) = L(s, ρ_{F}) for ρ_{F} = ρ|_{Gal(\overline{ℚ}/F)}. When f̂ exists, we call f “liftable to F ” or “F -liftable”. When F is totally real, a Hilbert cusp form is cohomological if and only if it is holomorphic of weight k ≥ 2 at every archimedean place of F. Since the definition of the base change f̂ is given in purely algebraic terms, it is a non-trivial problem to find the base-change form f̂ in the space of cohomological forms characterized by analytic means. This problem was first studied by Doi and Naganuma [DN] and [N] for quadratic extensions F/ℚ, and the existence of f̂ was later proved by Langlands [L] for relative prime cyclic cases (so eventually covering all solvable cases).

In this paper, under a simple ramification condition on totally real F, we prove the existence of infinitely many F –liftable holomorphic elliptic cusp forms of a given odd prime power level (including 1). Our idea is as follows. Let p > 2 be a prime. We start from a (minimally) modular Galois representation \overline{ρ} : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝔽) unramified outside \{p, ∞\}. Here the word: “modular” means that there exists a Hecke eigenform f ∈ S_{k}(SL_{2}(ℤ)) such that its p-adic Galois representation ρ = ρ_{f} : Gal(\overline{ℚ}/ℚ) → GL_{2}(O) is a deformation of \overline{ρ}, that is, \overline{ρ} ≅ ρ mod 𝔪_{O}, where O is a valuation ring with maximal ideal 𝔪_{O} finite flat over the p-adic integer ring ℤ_{p}. Suppose that \overline{ρ} is minimally modular again over F in an obvious sense generalizing the above definition. Under certain assumptions on F and on \overline{ρ} (like, p-ordinarity and absolute irreducibility), we can deduce from the Hilbert modular generalization by Fujiwara [F] of a result of Taylor and Wiles [TW] (see also [W]), any p-ordinary deformation ϱ of \overline{ρ}_{F} = \overline{ρ}|_{Gal(\overline{ℚ}/F)} unramified outside \{p, ∞\} is associated to a Hilbert Hecke eigenform provided that det(ϱ) is an arithmetic character (Proposition 4.4). In this ideal situation, any p-ordinary deformation g of f is liftable to F, and if g is classical (not just a p-adic modular form), ĝ is actually a classical Hilbert cusp form. Thus the idea is simple, but it is a bit hard to verify if \overline{ρ}_{F} is modular or not. Anyway our approach to Langlands’ functoriality is from the Galois side not from the automorphic side, reversing the direction. This type of approach was first realized in a paper of Clozel [Cl], in which he applied the technique of Wiles to the descent problem of Galois invariant Hilbert modular forms.

1Research of the first author is partially supported by an NSF grant; this paper was written while the second named author was visiting UCLA, and he likes to thank the Mathematics department of UCLA for their hospitality.

We now ease the modularity condition a bit. Starting with f, try to find a sequence of elliptic Hecke eigenforms f_{j} ∈ S_{k_{j}}(Γ_{0}(N_{j}), χ_{j}) for a suitable level N_{j} such that f_{j} ≡ f_{j−1}^{σ_{j}} mod 𝔩_{j} for σ_{j} ∈ Gal(\overline{ℚ}/ℚ) and for a prime 𝔩_{j} of \overline{ℚ} dividing an odd prime ℓ_{j}. We write f ∼ f_{j} if such a sequence of congruence (and Galois conjugation) exists. If we can find F -liftable f_{j}, then applying the above argument to the prime ℓ_{j} in place of p, the base change lift f̂_{j−1} would exist. Then applying the above argument for ℓ_{j−1} in place of p to f_{j−2} and f̂_{j−1}, f_{j−2} would be liftable to F, and so on, getting eventually the existence of f̂ by induction. However at the present stage of knowledge, to make the above argument work, we need to assume the following conditions on χ = χ_{j}, ℓ = ℓ_{j}, 𝔩 = 𝔩_{j} and ϕ = f_{j} for all j (see Corollary 4.5 in the text):

(1) ϕ is 𝔩-ordinary, that is, a(ℓ, ϕ) is prime to 𝔩;

(2) The Galois representation \overline{ρ}_{𝔩,ϕ} of ϕ modulo 𝔩 is absolutely irreducible

as a representation of Gal (\overline{ℚ}/F (\sqrt{(−1)^{(ℓ−1)/2}ℓ}));

(3) ℓ is odd and is unramified in F/ℚ;

(4) χ has conductor equal to the level of ϕ and has order prime to ℓ. Anyway, by the p-adic density of weight 2 modular forms in the space of p-adic modular forms [H86], we may assume that k_{j} = 2 for all j. Then by an observation of Mazur showing the connectedness of the spectrum of the Hecke algebra of weight 2 over ℤ of a given level N, any two common eigenforms f, g of weight 2 are related, that is, f ∼ g. Thus we can choose g to be a Hecke eigenform congruent to a theta series θ(φ) for a Hecke character φ of a quadratic extension M/ℚ. If MF/F remains to be quadratic, θ̂(φ) = θ(φ ∘ N_{MF/F}), and our goal should be achieved if either we could remove the four assumptions mentioned above or could prove the connectedness of the Hecke algebra removing the primes not satisfying the four conditions. We have made an experiment with cusp forms of a prime power level to find a good θ(φ), since at this moment, it is hard to come by a proof of Fujiwara’s result removing all the four conditions. In our experiment, it is enough to use three step congruence for forms of level one with weight ≤ 468 except for the case of weight 26, in which we need four step congruence. To state a typical example of our result, we introduce some notation: For a Hecke eigenform f ∈ S_{k}(SL_{2}(ℤ)), let Σ_{f} be the (finite) set of odd primes ℓ for which the Galois representation \overline{ρ}_{𝔩,f} for 𝔩|ℓ is reducible as a Galois representation

of Gal (\overline{ℚ}/ℚ (\sqrt{(−1)^{(ℓ−1)/2}ℓ})). By Serre [Se1] Theorem 8 and [Se2] 3.2, if ℓ ∉ Σ_{f}, the Galois representation \overline{ρ}_{𝔩} satisfies the above condition (2) for ϕ = f assuming (3). We consider the following condition on F for a positive integer m:

(Hyp_{m}) m is prime to D_{F},

where D_{F} is the discriminant of F. Here is the example of our result.

Proposition 1.1. Let f ∈ S_{k}(SL_{2}(ℤ)) be a Hecke eigenform for 12 ≤ k ≤ 468. Let ℓ ∉ Σ_{f} be an odd prime. Then there exists an explicitly computable positive integer m with the following property: Suppose

(i) g ∈ S_{κ}(Γ_{1}(ℓ^{α})) is a Hecke eigenform for α ≥ 0 and κ ≥ 2; (ii) g is 𝔩-ordinary for 𝔩|ℓ; (iii) g ≡ f mod 𝔩. Then g is liftable to F provided that F is totally real and satisfies (Hyp_{mℓ}).

The set Σ_{f} is a finite set. When f = Δ ∈ S_{12}(SL_{2}(ℤ)), Σ_{Δ} = {3, 5, 7, 23, 691} (see [Se2] 3.3) and m is given by m = 17 · 67 or 13 · 131.

Probably, the hypothesis (Hyp_{mℓ}) could be removed, replacing it by the condition that ℓ_{j} ∉ Σ_{f_{j}} for each j (in the sequence of congruences), after analysing more the method in [F]. If we could prove the following conjecture of Maeda, we could have substantially generalized our result:

Conjecture 1.2 (Y. Maeda). The Hecke algebra over ℚ of S_{k}(SL_{2}(ℤ)) is simple (that is, a single number field) whose Galois closure over ℚ has Galois group isomorphic to a symmetric group 𝔖_{n} (with n = dim S_{k}(SL_{2}(ℤ))).

The conjecture has been verified numerically to be true for weight k ≤ 468. If f ∈ S_{k}(SL_{2}(ℤ)) is liftable to F, then each Galois conjugate of f is F -liftable. Thus if the conjecture holds for k, to prove the proposition, we only need to find one form of weight k liftable to F. For small k ∈ {12, 16, 18, 20, 22, 24, 28, 32, 34}, we choose one (small) prime p and 𝔭|p for which f is 𝔭-ordinary. By computation, we find a Hecke eigenform θ ∈ S_{κ} (Γ_{0}(p),_{( \overline{p} )}) such that θ ≡ θ(φ) mod 𝔮 for another prime 𝔮 of \overline{ℚ} and θ ≡ f mod 𝔭. Then under (Hyp_{pq}) for rational primes p, q with 𝔭|p and 𝔮|q, f is liftable to F. For k = 26, we need to use a four step congruence a bit different from the above (see Section 3 for details). For large 26 < k ≤ 107126, we can find a prime ℓ such that ℓ − 1|k − k_{0} for k_{0} ∈ K = {12, 16, 18, 20, 22, 26}. If f is 𝔩-ordinary (𝔩|ℓ), by the theory of 𝔩-ordinary forms ([H86]), we can find f′ of weight k_{0} such that f ≡ f′ mod 𝔩. Since ℓ ∈ Σ_{f} ⟺ ℓ ∈ Σ_{f′}, we can easily check numerically that ℓ ∉ Σ_{f} by a table of Serre [Se2], and thus m = pqℓ if k_{0} ≠ 26. We have tested numerically the 𝔩-ordinarity up to ℓ < 10000 and verified the validity of our argument for k ≤ 44972 (only using known ordinary primes supposing the conjecture for k). For k = 44974, ℓ − 1 = k − 12 for the prime ℓ = 44963. Thus if 44963 is ordinary for Δ, the argument still works. Since we can take any even weight k_{0} less than a given k as an inductive step, our argument should presumably work for every weight, although we do not have a rigorous proof of the existence of ℓ and k_{0} with ℓ − 1|k − k_{0} > 0 satisfying the four conditions for the weight k_{0} Hecke eigenform ϕ.

Although the above proposition has some technical assumptions, we like to emphasize the rather simple nature of our principle we described above, which virtually guarantees the existence of base change for any holomorphic form of weight ≥ 2 to any totally real field if we can improve further our technology in identifying Hecke algebras with universal deformation rings.

In the following section, we introduce tools to find θ(φ) and θ for general k_{0} ≥ 2, and in Section 3, we compute θ and explicit values of m for eigenforms f_{0} of small weight k_{0}. Each ordinary prime ℓ ∉ Σ_{f_{0}} for such a low weight form f_{0} yields F -liftability of forms of weight k ≡ k_{0} mod ℓ − 1, as long as the conjecture is true for weight k. Then for a given k ≤ 468, we made a list of primes ℓ with ℓ−1|k−k_{0} for one of k_{0} ∈ K, and we verified by computation that we have a nice ℓ in the list such that ℓ ∉ Σ_{f′} and ℓ is ordinary with respect to f_{0}. This computation verifying Proposition 1.1 (for k ≤ 468) is done by hand (one day work) and also by computer calculation (in this case, up to k ≤ 44972 assuming the conjecture). Actually if ℓ ≥ k − 1 with ℓ ≠ 2k−1 is ordinary for f and ℓ ∈ Σ_{f}, then ℓ is an Eisenstein prime dividing the numerator of the Bernoulli number \frac{B_{k}}{2k} (see the proof of Theorem 5.3 in [H89c]). Some of the Hecke eigenvalues used in Section 3 are supplied to us from K. Doi and H. Ishii as indicated there. We are grateful to them for allowing us to use their numerical examples. In Section 4, we give a proof of the above result attributed to Fujiwara [F], reducing it to what is actually proven in [F], although it is a straightforward application of the theory in [H88] and [H89a]. In Section 5, we give numerical examples supporting Maeda’s conjecture and explain how we verify the conjecture for small weights. Proposition 5.1 is a key to checking the conjecture. It is supplied to one of the authors by K. Kubo, to whom we are grateful.

We like to thank the referee of this paper for his carefull reading.

2. Cusp forms congruent to θ(φ).

Let M = ℚ(\sqrt{D}) be a quadratic field with discriminant D and Galois group Γ = Gal(M/ℚ) generated by σ. Let χ = ^{(\underline{D} )} be the Legendre symbol, and put Γ̂ = {id, χ}. Let ψ ∈ Γ̂, and consider the space of elliptic cusp forms S_{κ}(Γ_{0}(C), ψ) of weight κ and of level given by the conductor C = C(ψ) of ψ. Let A be a subring of ℂ. We write h_{κ}(C(ψ), ψ; A) for the A-subalgebra of the linear endomorphism algebra of S_{κ}(Γ_{0}(C), ψ) generated over A by Hecke operators T (n) for all n. Let λ = λ_{κ} : h_{κ}(C(ψ), ψ; ℤ) → ℂ be an algebra homomorphism and A be a valuation ring of ℚ(λ) with residual characteristic p. Here ℚ(λ) is the number field generated by λ(T (n)) for all n. As is well known, ℚ(λ) is totally real or a CM field. We write ℚ(λ)_{+} for the maximal real subfield of ℚ(λ). Let O be the 𝔪_{A}-adic completion for the maximal ideal 𝔪_{A} of A. We write ρ = ρ_{λ} : Gal(\overline{ℚ}/ℚ) → GL_{2}(O) for the Galois representation attached to λ. We write \overline{ρ} for the semi-simplification of ρ_{λ} mod 𝔪_{O} : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝔽) with 𝔽 = O/𝔪_{O}. For the moment, we assume that M is real. Then ℚ(λ) ≠ ℚ(λ)_{+} if ψ = χ. Since the Neben character ψ is even, the weight κ is even and hence κ ≥ 2. We quote the following fact from [H97], which is a straightforward generalization of a well known result of Shimura [Sh] and [Sh1]:

Theorem 2.1. Suppose that M is real, and let p be a prime number with p ≥ 3 if κ > 2 and p > 3 if κ ≥ 2. Let 𝔭|p be a prime of ℚ(λ) associated to A. Let ε be a fundamental unit of M chosen to be positive at some real place of M.

(1) If λ(T (p)) ∈ A^{×}, \overline{ρ} ≅ Ind_{M}^{ℚ} \overline{φ} for a character \overline{φ} of Gal(\overline{ℚ}/M), and \overline{ρ} is absolutely irreducible, then ψ = χ, χ(p) = 1 and p|N(ε^{κ−1} − 1);

(2) If ψ = χ, p = 𝔓𝔓′ with primes 𝔓 ≠ 𝔓′ in M(⇔ χ(p) = 1) and p|N(ε^{κ−1} − 1) for even κ, then there exist λ : h_{κ}(D, χ; ℤ) → ℂ and 𝔭 such that (i) λ(T (p)) ∈ A^{×}, (ii) \overline{ρ} is isomorphic to Ind_{M}^{ℚ} \overline{φ} for a

character \overline{φ} : Gal(\overline{ℚ}/M) → 𝔽^{×} with conductor 𝔓, (iii) \overline{ρ} is irreducible;

(3) If ψ = χ, λ(T (p)) ∈ A^{×}, and 𝔭 divides the relative different 𝔡 = 𝔡_{λ} of ℚ(λ)/ℚ(λ)_{+}, then p|N(ε^{κ−1} − 1), χ(p) = 1 and \overline{ρ} ≅ Ind_{M}^{ℚ} \overline{φ}, which is absolutely irreducible.

Moreover if χ(p) = 1, p|N(ε^{κ−1} − 1) and ψ = χ, then \overline{ρ} associated to λ in (2) satisfying the three conditions (i), (ii) and (iii) is p-ordinary and regular. Here we call a Galois representation ρ with values in GL_{2}(A) for a p–adic local ring A “p-ordinary” (and regular) if its restriction to each decomposition group at p is isomorphic to ( \begin{matrix}δ & ∗ \\ 0 & ϵ \end{matrix}) for an unramified character δ (and δ ≢ ϵ mod 𝔪_{A}).

Proof. All but the assertion (3) of the theorem have been proven in [H97]. Note here that the complex conjugate of λ(T (n)) is equal to χ(n)λ(T (n)) for n prime to D (see [Sh], Proposition 3.56). Thus T r(\overline{ρ}) = T r(\overline{ρ} ⊗ χ), and by semi-simplicity of \overline{ρ}, we get \overline{ρ} ≅ \overline{ρ} ⊗ χ. Then by [H97] Proposition 1.1, p|N(ε^{κ−1} −1) and χ(p) = 1. Since M is real, \overline{ρ} cannot be reducible as shown in the beginning of Section 1 of [H97]. The regularity of \overline{ρ} associated to λ in (2) is not explicitely mentioned in [H97], but it follows from the fact that κ is even and ϵ = ω_{p}^{κ−1} (on the inertia subgroup) for the Teichmüller character ω_{p}.

Remark 2.1. If N(ε) = −1 for the fundamental unit ε in the theorem, it is easy to see that any odd prime dividing N(ε^{κ−1} − 1) for even κ splits in M/ℚ. Thus in this case, the assumption: χ(p) = 1 in Theorem 2.1 (2) is redundant.

Remark 2.2. We could have formulated the theorem a bit differently. Let W be the involution on S_{κ}(Γ_{0}(D), χ) induced by ( \begin{matrix}0 & −1 \\ D & 0 \end{matrix} ). Then h ↦ W hW induces an involution τ (that is, a ring automorphism of order 2) on R = h_{κ}(D, χ; ℤ). We write R_{+} for the subring of R fixed by τ. Let c denote complex conjgation on ℂ. Since λ ∘ τ = c ∘ λ for λ : h_{κ}(D, χ; ℤ) → ℂ, by the above proof of (3), \overline{ρ} ≅ \overline{ρ} ⊗ χ for an odd prime ideal 𝔭 ⊃ Ker(λ) of R ramifying in R/R_{+} (that is, 𝔭^{τ} = 𝔭 and τ acts trivially on R/𝔭). In particular, if N(ε) = −1, \overline{ρ} for 𝔭 is ordinary, and hence by theorems of Deligne and Fontaine [E] 2.5 and 2.6, λ(T (p)) ∉ 𝔭 if p ≥ κ − 1. Thus under the hypothesis: N(ε) = −1, we know from Theorem 2.1 that the following two conditions are equivalent for primes p ≥ max(5, κ − 1):

(1) p|N(ε^{κ−1} − 1)

(2) There exist an algebra homomorphism λ : R → ℂ, a prime ideal 𝔭 ⊃ Ker(λ) with 𝔭|p and a character \overline{φ} : Gal(\overline{ℚ}/M) → 𝔽^{×} such that \overline{ρ} ≅ Ind_{ℚ}^{M} \overline{φ} and λ(T (p)) ∉ 𝔭.

Thus if R is an integral domain, for any given algebra homomorphism λ : R → ℂ and p|N(ε^{κ−1} − 1) with p ≥ max(5, κ − 1), there exists a prime ideal 𝔭 in ℚ(λ) such that \overline{ρ} for 𝔭|p and λ satisfies \overline{ρ} ≅ Ind_{ℚ}^{M} \overline{φ} and λ(T (p)) ∉ 𝔭.

Now we assume that M = ℚ(\sqrt{−D}) is imaginary with discriminant −D < 0. In this case, κ is odd if S_{κ}(Γ_{0}(D), χ) is non-trivial.

Proposition 2.2. Let the notation be as in Theorem 2.1. Suppose that M is imaginary, λ(T (p)) ∈ A^{×}, ℚ(λ) ≠ ℚ(λ)_{+}, χ(p) = 1, 𝔭|𝔡_{λ} and κ + 2 < p. Then \overline{ρ} ≅ Ind_{M}^{ℚ} \overline{φ}, which is absolutely irreducible, p-ordinary and regular.

Proof. By the assumption: ℚ(λ) ≠ ℚ(λ)_{+}, we have ψ = χ. By the same argument as above, we see \overline{ρ} ≅ \overline{ρ} ⊗ χ. Let ω_{p} be the Teichmüller character of Gal(\overline{ℚ}/ℚ) (at p). We suppose that \overline{ρ} ⊗ χ ≅ \overline{ρ}. Suppose first that \overline{ρ}

is reducible. Then writing \overline{ρ} ≅ (\begin{matrix}\overline{δ} & ∗ \\ 0 & \overline{ϵ} \end{matrix}), we have \overline{δ}\overline{ϵ} = χω_{p}^{κ−1} and \overline{δ}χ = \overline{ϵ},

because \overline{δ}χ = \overline{δ} never happens if p is odd. This shows that \overline{δ}^{2} = ω_{p}^{κ−1} and \overline{δ} = ω_{p}^{(κ−1)/2}. Since λ(T (p)) ∈ A^{×} and κ < p − 2, by a theorem of Deligne [E] 2.5, \overline{ρ} is p–ordinary, \overline{ϵ} = ω_{p}^{κ−1} on the inertia subgroup I_{p} at p for the Teichmüller character ω_{p}. This shows that \overline{δ} ≠ \overline{ϵ} and \overline{ρ} is absolutely irreducible. Then by [DHI] Lemma 3.2, \overline{ρ} = Ind_{M}^{ℚ} \overline{φ}.

Remark 2.3. Under the assumption of Theorem 2.1 or Proposition 2.2, the characteristic polynomial of λ(T (q)) for a prime q with χ(q) = −1 has the form f(X^{2}) for a monic f(X) ∈ ℤ[X]. Each square-free factor of f(0) divides 𝔡_{λ} provided that it is unramified in ℚ(λ)_{+}. When χ(q) = 1, the characteristic polynomial of λ(T (q)) has the form f(X)^{2} for a monic f(X) ∈ ℤ[X].

Remark 2.4. Let p ≥ max(5, κ − 1) be a prime with p|N(ε^{κ−1} − 1) for an even integer κ. Then by Remark 2.2, we have an algebra homomorphism λ : R → ℂ and a prime 𝔭 ⊃ Ker(λ) such that \overline{ρ} = Ind_{M}^{ℚ} \overline{φ} for a character φ : Gal(\overline{ℚ}/M) → 𝔽^{×}. The restriction \overline{ρ}_{L} of \overline{ρ} = Ind_{M}^{ℚ} \overline{φ} to Gal(\overline{ℚ}/L) for L = ℚ (\sqrt{(−1)^{(p−1)/2}p}) can be reducible even if \overline{ρ} is absolutely irreducible. Since \overline{ρ}_{L} = Ind_{LM}^{L} \overline{φ}_{ML}, this happens when \overline{φ}_{σ} ≠ \overline{φ} but \overline{φ}_{σ} = \overline{φ} on Gal(\overline{ℚ}/L) (⇒ φ|_{I_{p}} = ω_{p}^{κ−1}|_{I_{p}} is of order 2 ⇒ p = 2κ − 1 if p > κ), where \overline{φ}_{σ}(τ) = \overline{φ}(στσ^{−1}). By a simple computation, for any character ξ : Gal(\overline{ℚ}/M) → 𝔽^{×} ramifying at p, the representation Ind_{M}^{ℚ} ξ is not p–ordinary if it is irreducible and p is inert in M/ℚ.

3. Base change of elliptic cusp forms.

In this section, a gothic letter represents a prime of \overline{ℚ} dividing the rational prime denoted by its roman equivalent. Our strategy to find θ with θ ≡ θ(φ) mod 𝔮 and θ ≡ f mod 𝔭 is as follows: Pick a normalized Hecke eigenform f ∈ S_{k}(SL_{2}(ℤ)) whose p-adic Galois representation is p-ordinary with a p-adic unit a(p, f). Then lift f to a p-adic family of ordinary forms {f_{κ} ∈ S_{κ}(Γ_{0}(p), ω_{p}^{k−κ})}_{κ} for the Teichmüller character ω_{p} modulo p (see [H86]). We choose κ so that k − κ ≡ (p − 1)/2 mod p − 1 and put θ = f_{κ}. Let M = ℚ (\sqrt{(−1)^{(p−1)/2}p}), and write χ for the Legendre symbol ω_{p}^{(p−1)/2} associated to M. Then compute the characteristic polynomial H_{ℓ}(X) of T (ℓ) in h_{κ}(p, χ; ℚ) by the trace formula in [Mi] and H_{ℓ}(0) to find an odd prime q such that 𝔮|𝔡_{λ}.

Suppose that M is real and that h_{κ}(p, χ; ℚ) is simple (as often happens). We define λ : h_{κ}(p, χ; ℚ) → ℂ by θ|T (n) = λ(T (n))θ. Then just to find a prime 𝔮|q with θ ≡ θ(\overline{φ}) for a character \overline{φ} : Gal(\overline{ℚ}/M) → 𝔽^{×}, we only need to compute N(ε^{κ−1} − 1) when M is real. Then we pick a prime factor q ≥ max(κ − 1, 5) of N(ε^{κ−1} − 1). Then by Remark 2.2, a prime factor 𝔮 of q in ℚ(λ) does the job as long as q ≠ 2κ − 1 (see Remark 2.4). According to the conjecture made in [H97], the prime factors q of N(ε^{κ−1} − 1) with q ≥ max(κ − 1, 5) should also divide 𝔡_{λ} if h_{κ}(p, χ; ℚ) is simple.

If q, which is either a square-free prime factor of H_{ℓ}(0) or a prime factor of N(ε^{κ−1} − 1) with q ≥ 5, exists for λ, we know from Theorem 2.1 and Proposition 2.2 that θ ≡ θ(φ) mod 𝔮 for a prime 𝔮|q and the Teichmüller lift φ of \overline{φ} with respect to 𝔮. By the theorems of Deligne and Fontaine [E] 2.5-6, if κ ≥ 2, χ(q) = 1 and q ≥ κ − 1, the representation \overline{ρ} is ordinary and λ(T (q)) ∉ 𝔮. Under the condition that h_{κ}(p, χ; ℚ) is a single field, without assuming q ≥ κ − 1, replacing 𝔮 by its Galois conjugate, we may assume further that λ(T (q)) ∉ 𝔮 (see Remark 2.2). Then by the argument we described in the introduction (Corollary 4.5), f_{w} for all weights w ≥ 2, in particular f, are F -liftable under (Hyp_{pq}) provided that p ∉ Σ_{f}, q > κ + 2 and q ≠ 2κ − 1 (see Remark 2.4). Similarly, primitive forms of level p congruent to θ modulo 𝔮 is F –liftable under (Hyp_{pq}). If h_{κ}(p, χ; ℚ) is not simple, we need to find to which simple factor f_{κ} belongs. The F –liftable forms proliferate as we change primes p (and hence q). The above argument proves the following fact:

Proposition 3.1. Let p be an odd prime. Let χ = _{( \overline{p} )} be the Legendre symbol and λ : h_{κ}(p, χ; ℤ) → \overline{ℚ} be an algebra homomorphism for κ ≥ 2. Let ϕ ∈ S_{κ}(Γ_{0}(p), χ) be the Hecke eigenform with ϕ|T (n) = λ(T (n))ϕ. Suppose ℚ(λ) ≠ ℚ(λ)_{+}. If a prime factor 𝔮 of 𝔡_{λ} satisfies q ≠ 2κ − 1, q > κ + 2 and χ(q) = 1, then each Hecke eigenform g ∈ S_{κ′}(Γ_{1}(pq^{α})) (κ′ ≥ 2, α ≥ 0) with g ≡ ϕ mod 𝔮 is liftable to totally real F satisfying (Hyp_{pq}).

Suppose that p ≡ 1 mod 4 and p > 5. Then as shown in [H97] Lemma 3.1, unless (2^{2n} + 1)/p is a square for an integer n > 1, we can find a nontrivial q. It is easy to see that χ(q) = 1 provided that q > 2, q|N(ε^{κ−1} − 1) for even positive κ and N(ε) = −1. Thus this proposition combined with Theorem 2.1 shows the existence of infinitely many primes q with a 𝔮-adic ordinary family (of level p cusp forms) liftable to totally real F under (Hyp_{pq}).

Here is a list of primes q, that is, the set S_{e} of odd primes dividing N(ε^{e}−1) (but not N(ε^{e′} − 1) for 0 < e′ < e) when p ≡ 1 mod 4 (see [H97] Section 3).

S_{1} = {3}, S_{5} = {131}, S_{7} = {1429}, S_{9} = {433}; (p = 13)

S_{3} = {67}, S_{5} = {4421}, S_{7} = {127, 2297}, S_{9} = {19, 15121}; (p = 17)
S_{1} = {5}, S_{3} = {7}, S_{5} = {151}, S_{7} = {20357}, S_{9} = {19603}; (p = 29)

S_{1} = {3}, S_{3} = {7}, S_{5} = {11, 1951}, S_{7} = {71, 44129}; (p = 37) S_{3} = {4099}, S_{5} = {16797701}, S_{7} = {197, 13469, 25943}; (p = 41) S_{1} = {7}, S_{3} = {13}, S_{5} = {11, 241}, S_{7} = {43, 449}; (p = 53)

S_{1} = {3, 13}, S_{3} = {127}, S_{5} = {41, 56611}, S_{7} = {3534959149}. (p = 61)

We now give a list of Σ_{k_{0}} = Σ_{Δ_{k_{0}}} (made out of the data given in [Se2]) for the unique Hecke eigenform Δ_{k_{0}} in S_{k_{0}}(SL_{2}(ℤ)) with k_{0} ∈ {12, 16, 18, 20, 22, 26}:

Σ_{12} = {3, 5, 7, 23, 691},  Σ_{16} = {3, 5, 7, 11, 31, 3617},
Σ_{18} = {3, 5, 7, 11, 13, 43867}, Σ_{20} = {3, 5, 7, 11, 13, 283, 617},
Σ_{22} = {3, 5, 7, 13, 17, 131, 593}, Σ_{26} = {3, 5, 7, 11, 17, 19, 657931}.

To identify these primes, we have used the fact that for small weights, \overline{ρ}_{p,Δ_{k_{0}}} is dihedral if and only if 2k_{0}−1 = p is a prime and ℚ(\sqrt{−p}) has class number > 1.

Here is a table of known ordinary primes for Δ_{k_{0}}:

\begin{matrix}k_{0} & p \\ 12 & 11 ≤ p < 2411, 2417 ≤ p ≤ 19597 \\ 16 & 17 ≤ p < 59, 61 ≤ p < 15271, 15277 ≤ p ≤ 19597 \\ 18 & 17 ≤ p ≤ 14879 \\ 20 & 19 ≤ p < 3371, 3373 ≤ p ≤ 9973 \\ 22 & p = 11, 23 ≤ p ≤ 9973 \\ 26 & 29 ≤ p ≤ 9973 \end{matrix}

We start the description of our experiment with the Ramanujan’s function f = Δ ∈ S_{12}(SL_{2}(ℤ)). The function Δ is p-ordinary for 11 ≤ p < 2411. We assume p ∉ Σ_{12}. Thus the weight k Hecke eigenform f_{k} belonging to the p–adic family of Δ resides in S_{k}(Γ_{0}(p), ω_{p}^{12−k}). We make p = 13 and κ = 6. Then θ = f_{6} ∈ S_{6}(Γ_{0}(13), (_{\overline{13}})). This space has dimension 6 (see [Mi] Numerical Tables). The Hecke polynomial H_{2}(X) for T (2) on this space is:

H_{2}(X) = X^{6} + 161X^{4} + 5856X^{2} + 18864 with 18864 = 2^{4} · 3^{2} · 131,

which is irreducible over ℚ. Since 131 is unramified in ℚ(λ)_{+} = ℚ[X]/(P (X)) with P (X) = H_{2}(\sqrt{X}), the square-free factor 131 of H_{2}(0) is a factor of 𝔡_{λ}. We put q = 131. Thus for a prime 𝔮|131 of \overline{ℚ}, θ ≡ θ(φ) mod 𝔮 for a character φ = ω_{q}^{5} of Gal(\overline{ℚ}/ℚ(\sqrt{13})). By Theorem 2.1 and Remark 2.2, θ has to be 𝔮-ordinary. Thus the 𝔮-ordinary family of θ is liftable to F satisfying (Hyp_{131}) (Proposition 3.1). Since \overline{ρ}_{𝔭,Δ} ≅ \overline{ρ}_{𝔭,θ} for 𝔭|13, the 𝔭-adic family of Δ is liftable to F satisfying (Hyp_{13·131}). We can use p = 17 and κ = 4 in place of p = 13 and κ = 6. In this case, on S_{4}(Γ_{0}(17), (_{\overline{17}}))

H_{3}(X) = X^{4} + 74X^{2} + 1072 with 1072 = 2^{4} · 67,

which is irreducible over ℚ. Thus 67|𝔡, and the 17-adic family of Δ is liftable to F satisfying (Hyp_{17·67}). Let ℓ ∉ Σ_{12} be an odd prime for which Δ is ℓordinary. Then each member of the ℓ-adic ordinary family of Δ is F –liftable if F is totally real and satisfies either (Hyp_{ℓ·17·67}) or (Hyp_{ℓ·13·131}).

We now take Δ_{16} ∈ S_{16}(SL_{2}(ℤ)). The form Δ_{16} is 19-ordinary. On S_{7}(Γ_{0}(19), (_{\overline{19}})), which is 9 dimensional, as computed by K. Doi, the characteristic polynomial H_{2}(X) of T (2) is given by

H_{2}(X) = X(X^{8} + 483X^{6} + 75582X^{4} + 4242376X^{2} + 71047680),

where 71047680 = 2^{9} · 3 · 5 · 11 · 29^{2}. The degree 8 factor is irreducible. Since 11 splits in ℚ(\sqrt{−19}), the form belonging to the degree 8-factor is 𝔮-ordinary for 𝔮 with 𝔮|11 and 𝔮|𝔡. Since the member θ = f_{7} of weight 7 in the 19-adic family of Δ_{16} is not of CM type, it belongs to the degree 8-factor. Thus by Propositions 2.2 and 3.1, each member of the 𝔮-adic family of θ is liftable to F satisfying (Hyp_{11}). Thus for odd primes ℓ ∉ Σ_{16} for which Δ_{16} is ℓordinary, each member of the ℓ-adic family of Δ_{16} is liftable to a totally real F satisfying (Hyp_{ℓ·11·19}).

We look at Δ_{18}. Then for p = 19, Δ_{18} is p–ordinary. Take p = 19. We have f_{9} ∈ S_{9}(Γ_{0}(19), (_{\overline{19}})). The Hecke algebra of this space is computed by H. Ishii, and we have

H_{2}(X) = X(X^{12} + 2323X^{10} + 2010462X^{8} + 803113072X^{6} + 150633270400X^{4} + 12173735396352X^{2} + 333034797957120).

We have 333034797957120 = 2^{15} · 3^{9} · 5 · 7 · 14753. Since (\frac{14753}{19}) = 1, we can take q = 14753. The above argument works for forms of S_{k}(SL_{2}(ℤ)) for k = 12, 16, 18, 20, 22, 24, 28, 32, 34. We list (κ, p, q) for these values of k:

Case k = 12: (κ, p, q) = (6, 13, 131), (4, 17, 67);

Case k = 16: (κ, p, q) = (7, 19, 11), (8, 17, 127), (8, 17, 2297), (2, 29, 5); Case k = 18: (κ, p, q) = (9, 19, 14753) Case k = 20: (κ, p, q) = (6, 29, 5), (6, 29, 151); Case k = 22: (κ, p, q) = (8, 29, 5), (8, 29, 20357); Case k = 24: (κ, p, q) = (6, 37, 1951), (4, 41, 4099); Case k = 28: (κ, p, q) = (2, 53, 7); Case k = 32: (κ, p, q) = (2, 61, 13); Case k = 34: (κ, p, q) = (4, 61, 13), (4, 61, 127).

See [DHI] Section 2.2 for a table of 𝔡 used to compute the above values when p ≡ 1 mod 4.

We take Δ_{26}. The only prime which seems to be small enough for actual computation is 47. We look at S_{3}(Γ_{0}(47), (_{\overline{47}})). According to H. Ishii, the Hecke polynomial is given by

H_{2}(X) = (X^{5} − 20X^{3} + 80X − 17)(X + 1)^{2}, H_{5}(X) = X^{5}(X^{2} + 78).

Let g ∈ S_{3}(Γ_{0}(47), (_{\overline{47}})) be a Hecke eigenform associated to the factor X^{2}+78 of H_{5}. The odd prime factor 13 of 78 in the second factor of H_{5}(X) remains prime in ℚ(\sqrt{−47}), and hence it is not an ordinary prime for f_{3} = g in the 47-adic family of Δ_{26}. Although the other odd factor 3 splits in ℚ(\sqrt{−47}), we can show that 3 ∈ Σ_{g}. Thus they do not work for our purpose, and we need to go a different way. Δ_{26} is 29-ordinary (a(29, Δ_{26}) ≡ 15 mod 29). Thus its member Δ_{54} (or rather, associated primitive form) of 29-adic family falls in S_{54}(SL_{2}(ℤ)). By computation (see Section 5) the Hecke algebra of S_{54}(SL_{2}(ℤ)) is simple (that is, a single field), and hence Δ_{54} = f^{σ} for a member f of 43-adic family of Δ. Thus we get 4 step congruence:

Δ_{26} ≡ Δ_{54} mod 𝔩 (𝔩|29), Δ_{54} ≡ Δ mod 𝔮 (𝔮|43),
Δ ≡ θ mod 𝔭 (𝔭|13), θ ≡ θ(φ) mod 𝔯 (𝔯|131).

Thus Δ_{26} is liftable to a totally real F satisfying (Hyp_{13·29·43·131}).

There is one more way of getting level 1 F -liftable forms. For each prime ℓ > 13, there is a unique ν = ν(ℓ) ∈ {−1, 1, 3, 5, 7} such that ℓ − ν ≡ 12 mod 10. Thus in S_{ℓ−ν}(SL_{2}(ℤ)), there is a unique Hecke eigenform f such that f ≡ Δ mod 𝔭 for a given prime 𝔭|11. For any given prime 𝔩|ℓ, every Hecke eigenform in S_{ℓ−ν}(SL_{2}(ℤ)) is 𝔩-ordinary (see [J] Corollary 7.7 and [Se] Theorem 11). Thus any Hecke eigenform g ∈ S_{κ}(Γ_{1}(ℓ^{α})) with g ≡ f mod 𝔩 is F –liftable under (Hyp_{mℓ}) for m = 11·17·67 or 11·13·131 as long as ℓ ∉ Σ_{f}. By 𝔩-ordinarity, we can conclude ℓ ∉ Σ_{f} if ℓ is prime to the numerator of \frac{B_{ℓ−ν}}{ℓ−ν}. We record this as:

Proposition 3.2. Let m = 11·17·67 or 11·13·131. Let ℓ > 13 be a prime and 𝔭|11 be a prime of \overline{ℚ}. Suppose that ℓ is prime to the numerator of \frac{B_{ℓ−ν}}{ℓ−ν}. If Conjecture 1.2 is true for the weight ℓ−ν, then for every 𝔩|ℓ, congruence g ≡ f mod 𝔩 of Hecke eigenforms g ∈ S_{κ}(Γ_{1}(ℓ^{α})) (κ ≥ 2) and f ∈ S_{ℓ−ν}(SL_{2}(ℤ)) implies the liftability of g to a totally real F satisfying (Hyp_{mℓ}). Without assuming Conjecture 1.2, the assertion holds for the unique Hecke eigenform f ∈ S_{ℓ−ν}(SL_{2}(ℤ)) with f ≡ Δ mod 𝔭.

The above proposition implies the existence of ℓ-adic families of ordinary cusp forms of level 1 F –liftable under (Hyp_{mℓ}), as long as ℓ ≡ 1, 3 mod 10. In the above argument, one could have used a prime p different from 11, for example 13.

If one supposes that ordinary primes for a given Hecke eigenform f (of level 1) have Dirichlet density 1, we can find a specific m so that f is liftable to F under (Hyp_{m}) and the conjecture.

Proposition 3.3. If the Dirichlet density of ordinary primes for a Hecke eigenform f ∈ S_{k}(SL_{2}(ℤ)) is equal to 1, then under the conjecture, we can always find m such that f is F -liftable under (Hyp_{m}).

Proof. Because of the density assumption, we can find an ordinary prime l ∈ Σ_{f} for f as in the proposition such that gcd(l − 1, 10) = 2. Choose positive integers i and j such that k − 2 = 10i − j(l − 1). Choose a prime 𝔩|l so that f is 𝔩-ordinary. Then we pick a member g of the 𝔩-adic ordinary family of f of weight k + j(l − 1). Then g ≡ Δ mod 𝔭 for a prime 𝔭|11 supposing the conjecture. Thus f ≡ g mod 𝔩 is F –liftable under (Hyp_{ml}) for m as in Proposition 3.2.

It is plain from the argument proving the above proposition, actually it is enough to assume, for the validity of the assertion of Proposition 3.3, the density of ordinary primes for f is bigger than \frac{1}{5}.

4. p-Ordinary deformation.

4.1. Deformation problems. We fix a finite set of places Σ of F and consider the maximal extension F_{Σ} of F unramified outside Σ ∪ {∞}. We write G = Gal(F_{Σ}/F). Then we fix a 2-dimensional continuous Galois representation \overline{ρ} : G → GL_{2}(𝔽) for a finite field 𝔽 of characteristic p for an odd prime p.

We fix a valuation ring O finite flat over ℤ_{p} with residue field 𝔽 and consider the category CNL = CNL_{O} of complete noetherian local O-algebras with residue field 𝔽. A deformation of \overline{ρ} is a continuous representation ρ : G → GL_{2}(A) for an object A of CNL such that ρ mod 𝔪_{A} = \overline{ρ}. We call a deformation ρ nearly p-ordinary, if for a decomposition subgroup D_{𝔓} of G at each p-adic place 𝔓, ρ|_{D_{𝔓}} ≅ ( \begin{matrix}δ & ∗ \\ 0 & ε \end{matrix}). Thus we have characters δ = δ_{𝔓,ρ} and ε = ε_{𝔓,ρ} of D_{𝔓} realized as diagonal entries. When δ_{𝔓,ρ} is unramified for all 𝔓, we call ρ p-ordinary.

Following Wiles [W], we consider the following conditions on deformations of \overline{ρ}. For a prime 𝔮 of F, we write D_{𝔔} for the decomposition subgroup of G at a prime 𝔔|𝔮 (of F_{Σ}) and let I_{𝔔} be its inertia subgroup. We split

Σ = {𝔭|p} ⨆ M ⨆ M′ ⨆ M′′.

For 𝔮 ∈ M′′, there is no restriction on deformations. For 𝔮 ∈ M ∪ M′, we impose minimality condition (χ_{𝔮}) (𝔮 ∈ M) and (N_{𝔮}) (𝔮 ∈ M′) described in [H96a] 2.1. Let us recall briefly these conditions. We fix a non-trivial character χ_{𝔮} of I_{𝔔} with values in 𝔽^{×}. We regard χ_{𝔮} having values in A^{×} for any A ∈ CNL_{O} through Teichmüller lift. We write 𝔣_{𝔮} for the conductor of χ_{𝔮} and put 𝔣 = ∏_{𝔮∈M} 𝔣_{𝔮}. Thus 𝔮|𝔣 ⟺ 𝔮 ∈ M. Writing V (ρ) for the representation space of ρ, we impose for 𝔮 ∈ M ∪ M′,

(χ_{𝔮}) As I_{𝔔}-modules,V (ρ) ≅ V (id) ⊕ V (χ_{𝔮}) with V (χ_{𝔮}) A-free of rank 1 for 𝔮 ∈ M, (N_{𝔮}) For 𝔮 ∈ M′, we have an exact sequence of D_{𝔔}-modules, non-split

over I_{𝔔}:

0 ⟶ V (ρ_{1,𝔮}) ⟶ V (ρ) ⟶ V (ρ_{2,𝔮}) ⟶ 0

where V (ρ_{2,𝔮}) is A-free of rank 1, ρ_{i,𝔮} (i = 1, 2) is unramified and ρ_{1,𝔮}ρ_{2,𝔮}^{−1} = N_{𝔮} for the p-cyclotomic character N_{𝔮} of D_{𝔮}.

We call such ρ a deformation of type D = D|_{F} for

D = {(χ_{𝔮})|𝔮 ∈ M} ⋃{(N_{𝔮})|𝔮 ∈ M′} ⋃ M′′.

We then consider the following two deformation functors: CNL → SET S given by

F_{F}^{∅} (A) = F_{F,\overline{ρ}}^{∅}(A) = \{ρ : G → GL_{2}(A) is a deformation of \overline{ρ} of type D\} / ≈,
F_{F} (A) = F_{F,\overline{ρ}}(A) = \{ρ ∈ F_{F}^{∅} (A) is nearly p-ordinary\} ,

where “≈” is the strict equivalence, that is, conjugation by elements in ĜL_{2}(A) = 1 + 𝔪_{A}M_{2}(A). For a finite extension E/F, we consider

(AI_{E}) \overline{ρ}_{E} = \overline{ρ}|_{Gal(\overline{ℚ}/E)} is absolutely irreducible.

It is shown by Mazur that F_{F}^{∅} is representable under (AI_{F}). In addition to the above condition, to assure the representability of F_{F}, we need to assume

(Reg_{F}) δ_{𝔓,\overline{ρ}} ≠ ε_{𝔓,\overline{ρ}} on the inertia group I_{𝔓}.

For assuring representability, it is enough to assume δ_{𝔓,\overline{ρ}} ≠ ε_{𝔓,\overline{ρ}} over the decomposition group. But the stronger condition listed above is unaffected under p-unramified base-change. Thus for our purpose, it is convenient to assume the above condition. We list here two more restrictions we like to study: Let 𝔭|p be a prime ideal of F. We hereafter write 𝔓 for a prime of \overline{ℚ} above 𝔭. We call a nearly p-ordinary deformation ρ 𝔭-ordinary if δ_{𝔓,ρ} is unramified for every decomposition subgroup D_{𝔓} of G for 𝔓|𝔭. We call ρ p-ordinary if it is 𝔭-ordinary for every 𝔭|p. For a given character η : G → O^{×}, we say that a deformation ρ has fixed determinant η if det ρ = η in A^{×}. Then we define the following subfunctors of F_{F}:

F_{F}^{ord}(A) = {ρ ∈ F_{F} (A)|ρ is p-ordinary} ,
F_{F}^{η} (A) = {ρ ∈ F_{F} (A)| det ρ = η} ,
F_{F}^{ord,η}(A) = F_{F}^{η} (A) ∩ F_{F}^{ord}(A).

It is easy to check that the above subfunctors of F_{F} are representable under (AI_{F}) and (Reg_{F}) (see [H96a] Appendix and [Ti]). When representable, write (R_{F} , ϱ_{F}) for the universal couple representing F_{F}. In this paper, “near p-ordinarity” is always put in force. When we consider a deformation problem with additional restriction “?”, (for example ? = ord), we write (R_{F}^{?} , ϱ_{F}^{?} ) for the universal couple with the condition “?” and the near p-ordinarity.

There are some relations between the above functors. Here we list one of them. We suppose that η = χ_{𝔮} on I_{𝔔} for all 𝔮 ∈ M and η is unramified at each 𝔮 ∈ M′ to make η compatible with the condition (χ_{𝔮}). Hereafter we suppose that p > 2. Since (det ρ)^{−1}η for ρ ∈ F_{F}^{?} has values in the p– profinite subgroup Ĝ_{m}(A) = 1+𝔪_{A}, it has a unique square root \sqrt{(det ρ)^{−1}η}. Thus ρ ↦ (ρ^{η} = ρ ⊗ \sqrt{(det ρ)^{−1}η}, det ρη^{−1}) gives a natural transformation: F_{F} ≅ F_{F}^{η} × F_{F,id}, where

F_{F,id}(A)

= {ϕ : G → A^{×} : a character|ϕ mod 𝔪_{A} = id, ϕ|_{I_{𝔔}} = id for all 𝔮 ∈ M} .

Let G^{M′′} = Gal(F_{Σ′}/F) for Σ′ = Σ − M ∪ M′, and W be the maximal p-profinite abelian quotient of G^{M′′}. Then the functor F_{F,id} is represented by (O[[W]], ν) for the projection ν of G onto W ⊂ O[[W]]^{×}. Thus we get

(R_{F} , ϱ_{F}) ≅ (R_{F}^{η} ˆ⊗_{O}O[[W]], ϱ_{F}^{η} ⊗ \sqrt{ν}). (T)

4.2. Nearly ordinary Hecke algebras. Fix an integral ideal N of F prime to p. Let χ′ be a character modulo N with conductor 𝔣|N. Suppose that N = 𝔣N′ with square-free N′ prime to 𝔣. Suppose that there exist a Hecke character χ : F^{×}\F_{𝔸}^{×} → O^{×} such that

(1) χ is of finite order, and its order is prime to p;

(2) the conductor of χ is a factor of 𝔣p;

(3) the restriction of χ to (O_{F} /𝔣)^{×} coincides with χ′. Let

D = {(χ_{𝔮})|𝔮|𝔣} ⋃ {(N_{𝔮})|𝔮|N′} .

We take a maximal ideal 𝔪 of the nearly ordinary Hecke algebra h^{n. ord}(N, p^{∞}; O) with certain properties. These properties should include that the residual modular Galois representation ρ_{𝔪} : G → GL_{2}(𝔽) associated to 𝔪 satisfies D. Then we set \overline{ρ} = ρ_{𝔪}.

We now describe the nearly ordinary Hecke algebra h^{n. ord}(N, p^{∞}; O) of (prime-to-p) level N, which corresponds to F_{F} if \overline{ρ} is minimally modular in the sense (of Fujiwara) described below 4.3. We follow the treatment of [H96b] Section 2 where the case with N = 1 is treated. We write Z (resp. C_{∞+}) for the center of G = Res_{F/ℚ} GL(2) (resp. the connected component of the standard maximal compact subgroup of G(ℝ)). We consider for each open compact subgroup U of G(𝔸^{(∞)}) the complex modular variety

X(U) = G(ℚ)\G(𝔸)_{+}/UZ(ℝ)C_{∞+},

where G(𝔸)_{+} = G(𝔸^{(∞)})G(ℝ)_{+} for the identity connected component G(ℝ)_{+} of the Lie group G(ℝ). We take O to be sufficiently large so that its field K of fractions contains all conjugates of F over ℚ. Let O = O_{F} be the integer ring of F. We define an open compact subgroup of G(𝔸^{(∞)}) for an ideal N of F by

U_{0}(N) = {x ∈ G(ℤ̂)|x mod N ∈ B(O/N)} ,

where B is the standard upper triangular Borel subgroup of GL(2)_{/ℤ}. Then for each field embedding σ : F ↪ K, we have the projection σ : G(ℚ_{p}) → GL_{2}(K). Hereafter we suppose that U ⊂ U_{0}(𝔣). Then ( \begin{matrix}a & b \\ c & d \end{matrix}) ↦ χ′_{𝔣}(d_{𝔣}) is a character of U. Now we consider the space of a polynomial representation of U

L(n, v, χ′; K) ≅ χ′_{𝔣} ⊗ {⨂_{σ}(det(σ)^{v_{σ}} ⊗ Sym^{⊗n_{σ}}(σ))}

for the symmetric m-th tensor representation Sym^{⊗m}(σ) of σ : G(ℚ_{p}) → GL_{2}(K). Thus we regard n and v as a linear combination of embeddings of F into K with coefficients n_{σ} and v_{σ}. To make things more precise, we take L(n, v, χ′; A) to be the space of polynomials in {(X_{σ}, Y_{σ})}_{σ}, with coefficients in a general O-module A, homogeneous of degree n_{σ} for each pair (X_{σ}, Y_{σ}). We let γ ∈ U act on P ∈ L(n, v, χ′; A) by

γP ((X_{σ}, Y_{σ})) = χ′_{𝔣}(d_{𝔣}) det(γ)^{v}P ((X_{σ}, Y_{σ})^{t}σ(γ)^{ι}),

where γ = (\begin{matrix}a & b \\ c & d \end{matrix}) and a^{v} = ∏_{σ} σ(a)^{v_{σ}}. Then we define a covering space

X (U) of X(U) by

G(ℚ)\(G(𝔸) × L(n, v, χ′; K))/UZ(ℝ)C_{∞+},

where the action is given by γ(x, P)u = (γxu, u_{p}^{ι}P) for γ ∈ G(ℚ) and u ∈ UZ(ℝ)C_{∞+} with u^{ι} = det(u)u^{−1}.

Now we consider the sheaf of locally constant sections of X (U) over X(U), which we write again as L(n, v, χ′; A). For q = [F : ℚ], we consider

S(U; A) = H_{cusp}^{q}(X(U), L(n, v, χ′; A)).

Writing simply L(A) for L(n, v, χ′; A), we recall the definition of H_{cusp}^{q}(X(U), L(A)). When n ≠ 0 or [F : ℚ] is odd, H_{cusp}^{q}(X(U), L(O)) is defined to be the natural image of the compactly supported cohomology group H_{c}^{q}(X(U), L(O)) in H^{q}(X(U), L(K)). When n = 0 and [F : ℚ] is even, in H^{q}(X(U), L(K)), we have the space Inv(U) of invariant classes Inv(U) spanned by cohomology classes of the connected components of X(U) (see [H88] Theorem 6.2). We then define

H_{cusp}^{q}(X(U), L(O)) = Im (H_{c}^{q}(X(U), L(O)) → \frac{H^{q}(X(U), L(K))}{Inv(U)}) .

Once H_{cusp}^{q}(X(U), L(O)) is defined, we just put

H_{cusp}^{q}(X(U), L(A)) = H_{cusp}^{q}(X(U), L(O)) ⊗_{O} A.

We suppose that S(U; K) ≠ 0 for sufficiently small U, and hence n+2v = [n + 2v]t for [n + 2v] ∈ ℤ and t = ∑_{σ} σ. On this space, Hecke operators

T (y) = [U (\begin{matrix}y & 0 \\ 0 & 1 \end{matrix}) U] naturally acts as described in [H88] Section 7; further,

𝕋(y) = y_{p}^{−v}T (y) preserves the image of H_{cusp}^{q}(X(U), L(n, v, χ′; O)).

We consider the O-subalgebra h_{n,v,χ′}(U) of End_{K}(S(U; K)) generated by 𝕋(y) for all integral ideles y, which is an algebra free of finite rank over O. Then it is easy to see that the central action ⟨z⟩ = N_{p}^{−[n+2v]}χ^{−1}(z)[UzU] of z ∈ Z(𝔸^{(∞)}) is naturally contained in h_{n,v,χ′}(U) where N_{p} is the p–adic cyclotomic character, and χ is the Teichmüller lift of a global Hecke character (of finite order) extending χ′χ_{p} with values in 𝔽^{×}. Thus h_{n,v,χ′}(U) is an algebra over O[[G]] via the character z ( \begin{matrix}y & 0 \\ 0 & 1 \end{matrix} ) ↦ ⟨z⟩𝕋(y), where G = Cl_{F} (p^{∞}) × O_{p}^{×} (O_{p} = O ⊗_{ℤ} ℤ_{p}) for the ray class group Cl_{F} (p^{∞}) modulo p^{∞}. Now we define more open compact subgroups of U_{0}(N). For that, we write U for the unipotent radical of B. For each ideal 𝔪 of O

U_{11}(𝔪) = {u ∈ G(ℤ̂)|u mod 𝔪 ∈ U(O/𝔪)} .

We put U_{α} = U_{11}(p^{α}) ⋂ U_{0}(N). Then we have a commutative diagram for 0 < α < β

S(U_{α}; K) ⟶ S(U_{β}; K)
T ↓  T ↓
S(U_{α}; K) ⟶ S(U_{β}; K)

where T = 𝕋(y) and ⟨z⟩, respectively. Thus restriction of operators gives surjective O-algebra homomorphisms: h(U_{β}) → h(U_{α}). We then define

h(N, p^{∞}; O)_{/F} = \underleftarrow{lim}_{α}h(U_{α}),

which is a compact semi local ring. Since the projection maps take 𝕋(y) to 𝕋(y), we have well defined 𝕋(y) in h(N, p^{∞}; O). In particular, for a prime element ϖ_{𝔮} of F_{𝔮} (𝔮 ∤ p), 𝕋(ϖ_{𝔮}) is independent of the choice of ϖ_{𝔮}, which we therefore write as 𝕋(𝔮).

Writing h(N, p^{∞}; O)_{/F} = ∏_{h} h_{/F} as a product of local rings h_{/F} with maximal ideal 𝔪_{h}, we write 𝕋_{h}(y) for the projection of 𝕋(y) to h. Then we define the nearly p–ordinary part of h(N, p^{∞}; O)_{/F} by

h^{n. ord}(N, p^{∞}; O)_{/F} = ∏_{𝕋_{h}(p)∈h^{×}} h_{/F} .

We write W (resp. 𝕎) for the maximal torsion-free quotient (resp. the maximal p-profinite quotient) of G. The projector e = lim_{n} 𝕋(p)^{n!} onto the nearly p-ordinary part kills the p-old part, and the prime-to-p part of N′ = N/𝔣 for the conductor 𝔣 of χ′ is square-free. From this, if \overline{ρ} = ρ_{𝔪} satisfies the conditions of D, we conclude that the local component h of \overline{ρ} of the algebra h^{n. ord}(N, p^{∞}, O) is reduced. For each character κ = (n, v, ε, ξ) : G → O^{×} given by

⟨z⟩𝕋(y) ↦ N_{p}^{−[n+2v]}ε(z)y^{−v}ξ(y),

we consider the κ-eigen subspace S(U_{α}; K)[κ] of S(U_{α}; K) for sufficiently large α so that εξ factors through Cl_{F} (p^{α}) × (O_{F} /p^{α})^{×}. Then we define the Hecke algebra h_{κ}^{n. ord}(O) by the O-subalgebra of End_{O}(S(U_{α}; K)[κ]) generated by 𝕋(y) and ⟨z⟩. Then this algebra is well determined only by κ and is independent of α. As shown in [H89a] Theorem 2.4 (see also Lemma 3.10), we have a canonical morphism

h^{n. ord}(N, p^{∞}, O) ⊗_{O[[G]],κ} O ↠ h_{κ}^{n. ord}(O), (κ)

which is a priori surjective and proven to be of finite kernel. Further, as seen in [H89a] Section 3, we have

(FF) There exists a O[[𝕎]]-free module X of finite rank on which h^{n. ord}(N, p^{∞}; O) acts faithfully; in particular h^{n. ord}(N, p^{∞}; O) is a torsion-free O[[W]]-module of finite type;

(red) h^{n. ord}(N, p^{∞}; O) is reduced; (Inj) O[[𝕎]] injects into h^{n. ord}(N, p^{∞}; O).

For each local ring h = h_{/F} of h^{n. ord}(N, p^{∞}; O)_{/F} and prime ideal P of h, we know from [Ta] and [BR] (see also [H89b]) the existence of a semisimple Galois representation ρ_{P} : Gal(\overline{ℚ}/F) → GL_{2}(k(P)) for the residue field k(P) = h_{P} /P satisfying the following six conditions:

(Ct) ρ_{P} preserves an h-submodule L of finite type generating k(P)^{2} over k(P) and is 𝔪_{h}-adically continuous as a map from Gal(\overline{ℚ}/F) into

Aut_{h}(L);

(unr) ρ_{P} is unramified outside Σ; (n.ord) ρ_{P} is nearly p–ordinary and satisfies: δ_{𝔓,ρ}([y; F_{𝔭}]) = 𝕋_{h}(y) for

y ∈ F_{𝔭}^{×},

where [y, F_{𝔭}] is the Artin symbol of local class field theory. For primes 𝔮 dividing 𝔣, write χ_{𝔮} for the character of I_{𝔔} corresponding to χ restricted to O_{𝔮}^{×}. Then

(M) ρ_{P} satisfies (χ_{𝔮}) for 𝔮|𝔣.

For primes 𝔮 ∈ M′ ⇔ 𝔮|N′,

(M′) ρ_{P} satisfies (N_{𝔮}) or is unramified for 𝔮|N′.

Finally for primes 𝔮 outside pN,

(Ch) det(1_{2} − ρ_{P} (Frob_{𝔮})X) = 1 − 𝕋(𝔮)X + χN_{p}(Frob_{𝔮})⟨𝔮⟩X^{2} ∈ k(P)[X]

for the p-cyclotomic character N_{p}. When ρ_{𝔪} for the maximal ideal 𝔪 of h is absolutely irreducible, we have a unique isomorphism class of representations ρ_{h} : Gal(\overline{ℚ}/F) → GL_{2}(h) satisfying the above conditions [C]. Thus supposing (N_{𝔮}) for \overline{ρ} = ρ_{𝔪} for all 𝔮 ∈ M′, we know ρ_{h} ∈ F_{F,\overline{ρ}}(h).

4.3. Universality of the Hecke algebras. We continue to use the notation and keep the assumption introduced in 4.1, and assume hereafter \overline{ρ} is “minimally modular” in the following sense:

(1) M′′ = ∅;

(2) There exists a connected component Spec(h) of Spec(h^{n. ord}(N, p^{∞}; O)) such that \overline{ρ} ≅ ρ_{h} mod 𝔪_{h} = ρ_{𝔪_{h}} for the maximal ideal 𝔪_{h} of h;

(3) \overline{ρ} = ρ_{𝔪_{h}} satisfies the conditions of

D = {(χ_{𝔮})|𝔮|𝔣} ⋃ {(N_{𝔮})|𝔮|N′}

described in the previous section. Here the word “minimally” means that there exists a Hilbert Hecke eigenform of weight 2, exact prime-to-p level N and the central character χ whose Galois representation is a deformation of \overline{ρ} of type D. This conditions follow from the above three conditions. Because of nearly ordinarity, we can always find such f of weight 2 with optimal p-level and optimal p-primary part of the central character, and by (3) and the construction of the Hecke algebra done in the previous section, the prime-to-p part of the central character is already optimal (cf. [H89a] Lemma 3.10). Note here that we have assumed that χ is of order prime to p. Thus χω is the Teichmüller lift of det \overline{ρ}.

Let κ ∈ Spec(O[[𝕎]])(O). Then by (κ), h ⊗_{O[[𝕎]],κ} O is an O-module of finite type, and the maximal torsion-free quotient (h⊗_{O[[𝕎]],κ}O)_{f} is canonically isomorphic to the local ring h_{κ} of h_{κ}^{n. ord}(O) corresponding to the deformation of \overline{ρ}. Write ρ_{κ} : Gal(\overline{ℚ}/F) → GL_{2}(h_{κ}) for the deformation. For the trivial character κ_{0} : 𝕎 → O^{×}, h_{κ_{0}} is the local ring associated to \overline{ρ} of the Hecke algebra of weight 2, prime-to-p level N and with central character χ. The following result is shown by K. Fujiwara [F]:

Theorem 4.1 (Fujiwara). Let p be an odd prime, and suppose that p is unramified in F/ℚ: (Hyp_{p}). Let χ be a Hecke character of order prime to p with prime-to-p conductor 𝔣. Let N = N′𝔣 for a square free ideal N′ of F prime to p. Suppose that \overline{ρ} is minimally modular and is associated to a local ring h of h^{n. ord}(N, p^{∞}; O) in the sense described above; in particular, we assume that \overline{ρ} satisfies the conditions of D = \{(χ_{𝔮})|𝔮|𝔣\}∪\{(N_{𝔮})|𝔮|N′\}. Assume

further the following two conditions on \overline{ρ} = ρ_{𝔪_{h}}: (AI_{F(\sqrt{(−1)^{(p−1)/2}p})}) and

(Reg_{F}). Let κ_{0} : 𝕎 → O^{×} be the trivial character. Then the couple (h_{κ_{0}},ρ_{κ_{0}}) represents the functor F_{F}^{ord,χN_{p}}.

In [F], the above assertion is proven by assuming that p is also prime to the class number of F. Under this assumption, the functor F_{F}^{ord,χN_{p}} is identical to the subfunctor of F_{F}^{ord} made of deformations ρ with det(ρ) = χN_{p} locally on each inertia group at primes dividing the level. Imposing the determinant condition globally, it is easy to check the argument given in [F] still works without this extra hypothesis. According to the lecture of Fujiwara given in Paris at the Poincaré Institute on June 23 (1997), a further progress has been made: the assertion of the theorem is valid even if we replace the assumption (Hyp_{p}) by the following weaker condition:

F is linearly disjoint from ℚ(μ_{p}) over ℚ and μ_{p}(F_{𝔭}) = {1} for all 𝔭|p. (LD_{p})

Anyway until a written version of the above theorem under this weaker hypthesis appears in print, we would only rely on the theorem as stated above.

We now extend the assertion of Fujiwara’s theorem to more bigger Hecke algebras. For that, we prepare some notation. Let 𝕎_{𝔭} ↪ O_{p}^{×} (resp. W_{𝔭}) be the maximal p-profinite part (resp. the maximal p-torsion-free quotient) of the image of the inertia group in the maximal abelian quotient of D_{𝔓} (𝔓|𝔭). Let 𝕎^{−} = ∏_{𝔭|p} 𝕎_{𝔭} and W^{−} = ∏_{𝔭|p} W_{𝔭}. Then 𝕎 ≅ 𝕎^{−} × 𝕎^{+}(p) (resp. W ≅ W^{−} × W^{+}(p)) for the maximal p-profinite part 𝕎^{+}(p) (resp. the maximal torsion-free quotient W^{+}(p)) of Cl_{F} (p^{∞}). Hereafter we identify 𝕎 with 𝕎^{−} × 𝕎^{+}(p). Then O[[W^{−}]] is isomorphic to a power series ring of [F : ℚ] variables over O. The ring O[[O_{p}^{×}]] is isomorphic to a product of copies of O[[𝕎^{−}]]. Let h^{χ} be the subalgebra of h generated by T r(ρ_{h}^{χN_{p}}), where ρ_{h}^{χN_{p}} = ρ_{h} ⊗ (χN_{p} det(ρ_{h})^{−1})^{1/2}. In the same manner as in [H96b] Proposition 2.1, we can prove, on the automorphic side,

(FF′) There exists a O[[𝕎^{−}]]-free module of finite type on which h^{χ} acts faithfully, in particular, h^{χ} is a torsion-free O[[W^{−}]]-module of finite type;

(T′) h ≅ h^{χ} ˆ⊗_{O}O[[𝕎^{+}(p)]].

On the Galois side, we have seen in 4.1

(T) R_{F} ≅ R_{F}^{χN_{p}} ˆ⊗_{O}O[[𝕎^{+}(p)]].

We now prove the universality of h^{χ} and h assuming that (h_{κ_{0}}, ρ_{κ_{0}}) represents the functor F_{F}^{ord,χN_{p}}. For simplicity, we further assume that μ_{p}(F_{𝔭}) = {1} for all 𝔭|p (⇔ 𝕎^{−} = W^{−}). By definition, ρ is p-ordinary if and only if δ_{𝔓,ρ} is trivial on 𝕎_{𝔭} for all 𝔭|p because \overline{ρ} is p-ordinary. Thus writing 𝔞_{ord} for the augmentation ideal of O[[𝕎^{−}]],

R_{F}^{ord,χN_{p}} ≅ R_{F}^{χN_{p}}/𝔞_{ord}R_{F}^{χN_{p}}.

Then we have the following commutative diagram for the identity character κ_{0} : 𝕎 → O^{×}:

ι : R_{F}^{χN_{p}}  ⟶  h^{χ}
↓  ↓
ι′ : R_{F}^{χN_{p}}/𝔞_{ord}R_{F}^{χN_{p}} ⟶ h^{χ}/𝔞_{ord}h^{χ}
↓^{∥≀}  ↓^{β}
ι′′ : R_{F}^{ord,χN_{p}}  ⟶ h_{κ_{0}}.

Suppose that ι′′ is an isomorphism. Then ι′ has to be an isomorphism because of the surjectivity of ι′ and β. The surjectivity of ι and ι′ follows from the properties (χ_{𝔮}) and (N_{𝔮}) (cf. [MT]). Hence h^{χ}/𝔞_{ord}h^{χ} ≅ h_{κ_{0}} is O-free of rank r < ∞. Choose an O-base \{\overline{θ}_{1}, …, \overline{θ}_{r}\} of h_{κ_{0}} and lift it to a subset {θ_{1}, …, θ_{r}} of h^{χ}. By Nakayama’s lemma, {θ_{1}, …, θ_{r}} generates h^{χ}. Since h^{χ} is O[[𝕎^{−}]]–torsion-free (and acts faithfully on an O[[𝕎^{−}]]-free module of finite type (FF′)), h^{χ} has to be O[[𝕎^{−}]]-free of rank r. This implies, again by Nakayama’s lemma, ι has to be an isomorphism. Then by (T) and (T′), we get R_{F} ≅ h. Thus we get:

Theorem 4.2 Let p be an odd prime, and suppose μ_{p}(F_{𝔭}) = {1} for all 𝔭|p. Let χ be a Hecke character of order prime to p with prime-to-p conductor 𝔣. Let N = N′𝔣 for a square free ideal N′ of F prime to p. Suppose that \overline{ρ} is minimally modular and is associated to a local ring h of h^{n. ord}(N, p^{∞}; O); in particular, we assume that \overline{ρ} satisfies the conditions of D = \{(χ_{𝔮})|𝔮|𝔣\} ∪ \{(N_{𝔮})|𝔮|N′\}. Assume further the following conditions on \overline{ρ} = ρ_{𝔪} : (AI_{F}) and (Reg_{F}). If h_{κ_{0}} for the identity character κ_{0} : 𝕎 → O^{×} as above represents the functor F_{F}^{ord,χN_{p}}, then (h^{χ}, ρ_{h}^{χN_{p}}) represents F_{F}^{χN_{p}}, and (h, ρ_{h}) represents F_{F}.

Corollary 4.3 Let the assumption be as in Theorem 4.2. Suppose that h_{κ_{0}} represents the functor F_{F}^{ord,χN_{p}}. Then h^{χ} (resp. h) is O[[𝕎^{−}]]-free (resp. O[[𝕎]]-free) of finite rank. For every arithmetic character κ = (n, v, ε, ξ) of 𝕎, we have h ⊗_{O[[𝕎]],κ} O ≅ h_{κ}.

We now fix field embeddings i_{p} : \overline{ℚ} ↪ \overline{ℚ}_{p} and i_{∞} : \overline{ℚ} ↪ ℂ. We write 𝔓 for the prime of ℚ over p corresponding to i_{p}. By (κ) combined with the Eichler-Shimura isomorphism proven by Harder (cf. [H93] Section 4), we recover a result in [H89a]:

Proposition 4.4 Let κ be an arithmetic character (n, v, ε, ξ) of 𝕎 with (ε, ξ) factoring through Cl_{F} (p^{α}) × (O/p^{α}O)^{×}. For any O-algebra homomorphism λ : h_{κ} → \overline{ℚ}_{p}, there exists a unique holomorphic Hilbert modular form f of weight κ on U_{α} such that f|𝕋(𝔮) = i_{∞} ∘ i_{p}^{−1}(λ(𝕋(𝔮)))f for all ideal 𝔮 prime to p.

4.4. Base change and ordinary deformation. We first apply the theory described in 4.3 when F = ℚ. In our application, we anyway assume M′ = ∅. We therefore fix a primitive Dirichlet character χ′ of conductor N prime to p. We fix a Dirichlet character χ modulo Np with χ|_{(ℤ/Nℤ)^{×}} = χ′. We consider the deformation data:

D_{ℚ} = {(χ_{q})|q|N} .

Let \overline{ρ} : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝔽) be a minimally modular representation of

type D_{ℚ} satisfying (AI_{ℚ(\sqrt{(−1)^{(p−1)/2}p})}) and (Reg_{ℚ}). By our choice, we have

Σ = {q|Np} ∪ {∞}.

We now take a totally real finite extension F/ℚ. As long as Np is unramified in F/ℚ, for the restriction \overline{ρ}_{F} of \overline{ρ} to Gal(\overline{ℚ}/F), we see that

(AI_{F(\sqrt{(−1)^{(p−1)/2}p})}) ⟺ (AI_{ℚ(\sqrt{(−1)^{(p−1)/2}p})}) and (Reg_{F}) ⟺ (Reg_{ℚ}).

We regard the Dirichlet character χ we have chosen for ℚ as a Galois character of Gal(\overline{ℚ}/ℚ). We write χ_{F} for the restriction of χ to Gal(\overline{ℚ}/F). Let N_{F} be the prime-to-p conductor of χ_{F}. We then define Σ_{F} by the set of infinite places of F, places of F appearing in N_{F} and places of F above p. We then consider the deformation data:

D_{F} = {(χ_{F,𝔮})|𝔮|N_{F} } .

Then combining all what we have said, we get:

Corollary 4.5 Let the notation be as above. Suppose f ∈ S_{k}(Γ_{1}(Np^{α})) with k ≥ 2 be a primitive form whose 𝔓-adic Galois representation ρ is a p-ordinary deformation of \overline{ρ} of type D_{ℚ}. Suppose (Hyp_{Np}) for a totally

real field F, (Reg_{ℚ}) and (AI_{ℚ(\sqrt{(−1)^{(p−1)/2}p})}) for \overline{ρ} and that \overline{ρ}_{F} is minimally

modular over F, that is, there exists a Hilbert primitive form g of primeto-p level N_{F} on GL(2)_{/F} such that its 𝔓-adic Galois representation ρ_{g} is a deformation of type D_{F} with det(ρ_{g}) = χ_{F} N_{p}. If p is unramified in F/ℚ, then there exists a holomorphic primitive Hilbert cusp form f̂ whose Mellin transform is L(s, ρ_{F}).

As already remarked, When p is unramified in F/ℚ, we know (Reg_{F}) for \overline{ρ}_{F} ⟺ (Reg_{ℚ}) for \overline{ρ}. Thus the four conditions in the introduction are enough to apply the above corollary.

We close this section giving an example of minimally modular Galois representations. The example is given by a theta series of a quadratic extension M/F. Thus in particular, we have M′ = ∅. We take a finite order character \overline{φ} : Gal(\overline{ℚ}/M) → 𝔽^{×}. Let φ : Gal(\overline{ℚ}/M) → O^{×} be the Teichimüller lift of \overline{φ}. We regard φ as a Hecke character φ : M^{×}\M_{𝔸}^{×} → O^{×}, and write φ̌ for the restriction of φ to F_{𝔸}^{×}. Let 𝔠 be the conductor of φ. Let N = (𝔠 ∩ F)𝔡 for the relative discriminant 𝔡 for M/F. We assume

(1) Every prime factor 𝔮 of (𝔠 ∩ F)p in F splits in M;

(2) For each prime factor 𝔮 of (𝔠 ∩ F)p in F, φ is unramified at one prime factor 𝔔 of 𝔮 in M, and φ is ramified for the other factor 𝔔′ of 𝔮;

(3) For each real place σ of F which remains real in M, φ is trivial at one factor of σ in M and is non-trivial at the other factor.

Then, as is well known, we can find a holomorphic theta series θ(φ) with L(s, θ(φ)) = L(s, φ). The modular form θ(φ) is of weight 1, has exact primeto-p level N and the central character φ̌α, where α is the quadratic character associated to M/F. By definition, θ(φ) is p–ordinary (θ(φ)|T (𝔮) = φ(𝔔)θ(φ) under the notation of (2)). The Galois representation of θ(φ) is given by Ind_{M}^{F} φ. Since θ(φ) is p-ordinary, we can lift θ(φ) to a p–adic family of pordinary forms. This is possible even for weight 1 forms θ(φ) because of the existence of a higher weight p-power level form E with E ≡ 1 mod p. Then weight 2 member θ of the family satisfies the conditions for Ind_{M}^{F} \overline{φ} being minimally modular.

Choose φ as above for F = ℚ. Then M is just a quadratic field. Now for a general totally real field F, we consider composite MF and the restriction φ_{MF} : Gal(\overline{ℚ}/MF) → O^{×} of φ. As long as F/ℚ is unramified at N, Ind_{MF}^{F} φ_{MF} remains absolutely irreducible and p–ordinary. In particular, θ(φ_{MF}) is the base change lift of θ(φ) to F. By Theorem 4.1, Theorem 4.2 and Proposition 4.4 combined (in short, Corollary 4.5), a weight k ≥ 2 member of the p-adic family of θ(φ_{MF}) is the base change lift of each weight k member of the p-adic family of θ(φ).

5. Characteristic polynomials for the Hecke operator T (2).

Write simply S_{k} for S_{k}(SL_{2}(ℤ)). For a primitive form f_{k} ∈ S_{k}, we denote by ℚ(f_{k}) the field generated over ℚ by all Fourier coefficients of f_{k}, by ℚ(f_{k})^{Gal} the Galois closure of ℚ(f_{k}), and by G(f_{k}) the Galois group Gal(ℚ(f_{k})^{Gal}/ℚ). We call S_{k} non-splitting if S_{k} is spanned by all Galois conjugates of f_{k}. Let ϕ_{k}(x) be the characteristic polynomial of the Hecke operator T (2) on S_{k}. If ϕ_{k}(x) is irreducible over ℚ, then ℚ(f_{k}) = ℚ(α) with a root α of ϕ_{k}(x) = 0, ℚ(f_{k})^{Gal} is the minimal splitting field of ϕ_{k}(x), and S_{k} is nonsplitting. We have calculated ϕ_{k}(x) using Eichler-Selberg trace formula (see [Mi] pp. 266-267) for k ≤ 468. About 20 years ago, Maeda observed that ϕ_{k}(x) are irreducible over ℚ for all the cases where dim_{ℂ} S_{k} ≤ 12 and later checked the fact up to k ≤ 468; thus S_{k} is non-splitting if k ≤ 468. We denote by 𝔖_{n} the symmetric group of degree n. Recently, first Lee and Hung [LH] and then Buzzard [Bu] have observed that G(f_{k}) is isomorphic to 𝔖_{n} with n = dim_{ℂ} S_{k} for several values of k, in particular, the cases where dim_{ℂ} S_{k} ≤ 7 and k = 12ℓ with prime ℓ ≤ 19 are treated in [Bu]. Learning this phenomenon, we have checked that G(f_{k}) ≅ 𝔖_{n} with n = dim_{ℂ} S_{k} for k ≤ 468. This is the origin of Conjecture 1.2. The validity of the conjecture is checked numerically for weight ≤ 468 by using the following fact which is pointed out to Maeda by K. Kubo.

Proposition 5.1 Let ϕ(x) ∈ ℤ[x] be a monic polynomial. Suppose that there exist three primes p_{1}, p_{2}, p_{3} satisfying the following conditions:

(1) ϕ(x) mod p_{1} is irreducible over 𝔽_{p_{1}};

(2) ϕ(x) ≡ φ_{1}(x) ⋯ φ_{h}(x) mod p_{2} with distinct irreducible polynomials φ_{i}(x) ∈ 𝔽_{p_{2}}[x] such that deg φ_{1}(x) = 2 and deg φ_{i}(x) are odd for i ≥ 2;

(3) ϕ(x) ≡ ψ_{1}(x)ψ_{2}(x) mod p_{3} with distinct irreducible polynomials ψ_{i}(x) ∈ 𝔽_{p_{3}}[x] such that deg ψ_{1}(x) = 1 and deg ψ_{2}(x) = deg ϕ(x) − 1.

Then ϕ(x) is irreducible over ℚ and the Galois group of the minimal splitting field of ϕ(x) is isomorphic to 𝔖_{n} with n = deg ϕ(x). Here 𝔽_{ℓ} stands for a finite field with ℓ elements.

This follows from the following lemmas, whose proof we leave to the reader.

Lemma 5.2 Let G be a transitive subgroup of 𝔖_{n}. If G contains a transposition and a cycle of length n − 1, then G = 𝔖_{n}.

Lemma 5.3 (van der Waerden). Let ϕ(x) ∈ ℤ[x] be a monic irreducible polynomial over ℚ and G be the Galois group of the minimal splitting field of ϕ(x). We consider G as a permutation group over the set of all roots of ϕ(x) = 0. Suppose for a prime p that ϕ(x) ≡ φ_{1}(x) ⋯ φ_{h}(x) mod p with distinct irreducible polynomials φ_{i}(x) ∈ 𝔽_{p}[x]. Then G contains a permutation of type (1, 2, ⋯, ℓ_{1})(ℓ_{1} + 1, ⋯, ℓ_{1} + ℓ_{2}) ⋯ (ℓ_{1} + ⋯ + ℓ_{h−1}, ⋯, ℓ_{1} + ⋯ + ℓ_{h}). Here ℓ_{i} = deg φ_{i}(x).

Here we list ϕ_{k}(x) and the prime factorization^{1} of their discriminants D_{k} for some of small weights with 2 ≤ dim_{ℂ} S_{k} ≤ 3.

Table 1. Characteristic Polynomials of T (2) and their discriminants.

\begin{matrix}k & ϕ_{k}(x) & D_{k} \\ 24 & x^{2} − 1080x − 20468736 & 2^{6} · 3^{2} · 144169 \\ 28 & x^{2} + 8280x − 195250176 & 2^{6} · 3^{6} · 131 · 139 \\ 30 & x^{2} − 8640x − 454569984 & 2^{12} · 3^{2} · 51349 \\ 32 & x^{2} − 39960x − 2235350016 & 2^{6} · 3^{2} · 67 · 273067 \\ 34 & x^{2} + 121680x − 8513040384 & 2^{8} · 3^{4} · 479 · 4919 \\ 36 & \begin{matrix}x^{3} − 139656x^{2} − 59208339456x \\ −1467625047588864 \end{matrix} & \begin{matrix}2^{30} · 3^{10} · 5^{2} · 7^{2} · 23 · 1259· \\ 269461929553 \end{matrix} \\ 38 & x^{2} + 194400x − 137403408384 & 2^{10} · 3^{2} · 181 · 349 · 1009 \\ 40 & \begin{matrix}x^{3} − 548856x^{2} − 810051757056x \\ +213542160549543936 \end{matrix} & \begin{matrix}2^{26} · 3^{12} · 5^{2} · 7^{2} · 13^{2} · 73· \\ 59077 · 92419245301 \end{matrix} \\ 42 & \begin{matrix}x^{3} + 344688x^{2} − 6374982426624x \\ −520435526440845312 \end{matrix} & \begin{matrix}2^{34} · 3^{10} · 5^{2} · 7^{2} · 1465869841· \\ 578879197969 \end{matrix} \end{matrix}

1Here and in other computations, we have used “UBASIC86” which was created by Y.

Kida.

As explained in [H] Theorem 5.2.1, S_{k}(SL_{2}(ℤ)) has a ℤ–base of ℤ-integral cusp forms {h_{1}, …, h_{d}} with a(i, h_{j}) = δ_{ij} as long as j ≥ i. From this, it is easy to conclude that the Hecke algebra h_{k}(1, id; ℤ) has a ℤ–base {T (1), …, T (d)}. Thus we have the formula of the exact discriminant of the Hecke algebra:

(disc) d(h_{k}(1, id; ℤ)/ℤ) = det(T r(T (i)T (j)))_{1≤i,j≤d},

where d = dim_{ℂ} S_{k}(SL_{2}(ℤ)). Using this, we can compute the discriminant of the Hecke algebra as follows:

Table 2. Discriminant of Hecke algebras.

weight dim Discriminant

24  2  2^{6} · 3^{2} · 144169
28  2  2^{6} · 3^{6} · 131 · 139
30  2  2^{12} · 3^{2} · 51349
32  2  2^{6} · 3^{2} · 67 · 273067
34  2  2^{8} · 3^{4} · 479 · 4919
36  3  2^{24} · 3^{6} · 5^{2} · 7^{2} · 23 · 1259 · 269461929553
38  2  2^{10} · 3^{2} · 181 · 349 · 1009
40  3  2^{20} · 3^{10} · 5^{2} · 13^{2} · 73 · 59077 · 92419245301
42  3  2^{22} · 3^{6} · 5^{2} · 7^{2} · 1465869841 · 578879197969
44  3  2^{22} · 3^{8} · 5^{2} · 7^{2} · 37 · 92013596772457847677
46  3  2^{31} · 3^{12} · 5^{2} · 227 · 454287770269681529
48  4  \begin{matrix}2^{40} · 3^{14} · 5^{6} · 7^{4} · 31 · 6093733 \\ ·1675615524399270726046829566281283 \end{matrix}
50  3  2^{22} · 3^{10} · 5^{4} · 7^{4} · 12284628694131742619401

We now give a table of prime factorizations of ϕ_{k}(x) mod p in 𝔽_{p} in the cases where 8 ≤ dim_{ℂ} S_{k} ≤ 9, which confirms the conjecture for these weights.

Table 3. ϕ_{k}(x) mod p.
k  p  ϕ_{k}(x) mod p
149 x^{8} + 52x^{7} + 132x^{6} + 40x^{5} + 3x^{4} + 38x^{3} + 83x^{2} + 92x + 11
96  23  (x + 3)(x + 5)(x + 8)(x + 10)(x + 13)(x + 22)(x^{2} + 8x + 5)
101 (x + 97)(x^{7} + 35x^{6} + 4x^{5} + 96x^{4} + 26x^{3} + 35x^{2} + 9x + 83)
179 x^{8} + 54x^{7} + 145x^{6} + 5x^{5} + 85x^{4} + 169x^{3} + 9x^{2} + 102x + 83
100  41  (x + 1)(x + 6)(x + 8)(x + 31)(x + 36)(x + 38)(x^{2} + 17x + 27)
107 (x + 51)(x^{7} + 21x^{6} + 41x^{5} + 50x^{4} + 72x^{3} + 103x^{2} + 62x + 25)
109 x^{8} + 58x^{7} + 98x^{6} + 83x^{5} + 64x^{4} + 39x^{3} + 21x^{2} + 42x + 10
102  37  (x + 5)(x + 7)(x + 13)(x + 23)(x + 27)(x + 33)(x^{2} + 21x + 32)
107 (x + 39)(x^{7} + 59x^{6} + 97x^{5} + 36x^{4} + 83x^{3} + 36x^{2} + 78x + 17)
127 x^{8} + 41x^{7} + 86x^{6} + 70x^{5} + 9x^{4} + 66x^{3} + 2x^{2} + 64x + 107
104  37  (x + 9)(x + 10)(x + 14)(x + 23)(x + 24)(x + 26)(x^{2} + 5x + 17)
107 (x + 50)(x^{7} + 45x^{6} + 83x^{5} + 90x^{4} + 50x^{3} + 74x^{2} + 63x + 14)
151 x^{8} + 142x^{7} + 138x^{6} + 98x^{5} + 122x^{4} + 135x^{3} + 87x^{2} + 2x + 141
106  41  (x + 1)(x + 2)(x + 4)(x + 7)(x + 23)(x + 29)(x^{2} + 13x + 6)
113 (x + 26)(x^{7} + 73x^{6} + 34x^{5} + 7x^{4} + 59x^{3} + 65x^{2} + 66x + 70)
223 x^{9} + 106x^{8} + 194x^{7} + 210x^{6} + 80x^{5} + 114x^{4} + 123x^{3} + 190x^{2} + 8x + 96
108  29  (x + 1)(x + 3)(x + 8)(x + 19)(x + 22)(x + 25)(x + 27)(x^{2} + 22x + 15)
127 x(x^{8} + 51x^{7} + 24x^{6} + 69x^{5} + 125x^{4} + 102x^{3} + 112x^{2} + 39x + 19)
127 x^{8} + 110x^{7} + 91x^{6} + 66x^{5} + 125x^{4} + 34x^{3} + 7x^{2} + 67x + 40
110  29  (x + 6)(x + 7)(x + 15)(x + 16)(x + 19)(x + 21)(x^{2} + 15x + 2)
137 (x + 96)(x^{7} + 59x^{6} + 136x^{5} + 91x^{4} + 89x^{3} + 40x^{2} + 33x + 84)
127 x^{9} + 45x^{8} + 27x^{7} + 121x^{6} + 47x^{5} + 104x^{4} + 108x^{3} + 60x^{2} + 89x + 37
112  23  (x + 1)(x + 4)(x + 7)(x + 9)(x + 15)(x + 16)(x + 17)(x^{2} + x + 22)
137 (x + 87)(x^{8} + 3x^{7} + 110x^{6} + 61x^{5} + 17x^{4} + 128x^{3} + 75x^{2} + 4x + 116)
137 x^{9} + 33x^{8} + 38x^{7} + 86x^{6} + 7x^{5} + 82x^{4} + 94x^{3} + 60x^{2} + 29x + 21
114  47  (x + 6)(x + 28)(x + 29)(x + 39)(x + 40)(x + 42)(x + 44)(x^{2} + x + 33)
199 (x + 134)(x^{8} + 172x^{7} + 144x^{6} + 11x^{5} + 110x^{4} + 37x^{3} + 58x^{2} + 20x + 59)
263 x^{9} + 246x^{8} + 70x^{7} + 260x^{6} + 14x^{5} + 40x^{4} + 90x^{3} + 118x^{2} + 68x + 194
116  53  (x + 3)(x + 14)(x + 24)(x + 34)(x^{2} + 48x + 51)(x^{3} + 7x^{2} + 29x + 49)
139 (x + 50)(x^{8} + 60x^{7} + 31x^{6} + 28x^{5} + 48x^{4} + 129x^{3} + 69x^{2} + 94x + 12)
163 x^{9} + 72x^{8} + x^{7} + 68x^{6} + 143x^{5} + 117x^{4} + 115x^{3} + 60x^{2} + 17x + 63
118  31  (x + 3)(x + 7)(x + 8)(x + 10)(x + 12)(x + 26)(x + 27)(x^{2} + 3x + 7)
109 (x + 69)(x^{8} + 24x^{7} + 30x^{6} + 29x^{5} + 4x^{4} + 79x^{3} + 15x^{2} + 50x + 6)
179 x^{9} + 61x^{8} + 80x^{7} + 146x^{6} + 163x^{5} + 107x^{4} + 154x^{3} + x^{2} + 51x + 27
122  37  (x + 4)(x + 9)(x + 11)(x + 14)(x + 20)(x + 27)(x + 29)(x^{2} + 7x + 20)
173 x(x^{8} + 87x^{7} + 56x^{6} + 121x^{5} + 18x^{4} + 17x^{3} + 7x^{2} + 83x + 11)

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