Global Quadratic Units and Hecke Algebras

Haruzo Hida^{1}

Received: May 12, 1998

Communicated by Don Blasius

Abstract. Let {ρ_{𝔭}}_{𝔭} be a compatible system of two dimansional 𝔭–adic Galois representations attached to a cusp form of Neben type ^{(\underline{D} )} (D > 0). We shall give an exact criterion, in terms of the fundamental unit ε of ℚ(\sqrt{D}), determining primes 𝔭 for which the image of ρ_{𝔭} mod 𝔭 is dihedral. Then we shall state a conjecture which gives an explicit description of the universal p– ordinary deformation ring of such mod 𝔭 dihedral representations.

0. Introduction. For a given 2–dimensional compatible system {ρ_{𝔭}}_{𝔭} of 𝔭–adic representations of Gal(\overline{ℚ}/ℚ) associated to an elliptic Hecke eigenform, if the image of one member ρ_{𝔭} at a prime 𝔭 is full containing the maximal compact subgroup of SL(2), then the image is full for almost all primes 𝔭 (cf. [R1]). Thus it is interesting to know for which primes the image shrinks to a proper subgroup of the maximal compact subgroup. This turns out to be quite an Arithmetic question; for example, if the system is associated to an elliptic Hecke eigenform of weight κ and of level 1, the image is reducible modulo 𝔭 only for irregular primes dividing the numerator of the Bernoulli number B_{κ} ([R]) if the prime p is large: p > κ + 1 (𝔭|p). This work of Ribet opened a possibility of a modular approach to the Iwasawa main conjecture, which was culminated by the proof of the conjecture by Mazur and Wiles 8 years later.

In this short note, we would like to determine when the image modulo 𝔭 is dihedral for non-dihedral systems. If it is the case for 𝔭|p, \overline{ρ} = (ρ_{𝔭} mod 𝔭) is isomorphic to an induced representation Ind_{F}^{ℚ} φ of a Galois character φ of a quadratic extension F over ℚ. We assume that F = ℚ(\sqrt{D}) is real (i.e. D > 0) to guarantee the non-dihedralness of the modular compatible systems. In the early 70’s, Shimura discovered, under certain conditions, that the primes for which \overline{ρ} is dihedral (for the system associated to an elliptic cusp form of weight 2 and of “Neben” type χ = ^{(\underline{D} )}) are given by prime factors of N_{F/ℚ}(ε − 1) for a positive fundamental unit ε of F ([S] and [S1]). Using this fact, he was able to show that the abelian extension of F associated to φ is generated by the coordinate of a certain torsion point of the Jacobian of a modular curve (solving Hilbert’s twelfth problem in this special case). The character φ as a Dirichlet character is just a ↦ (a mod 𝔭) for algebraic integers a ∈ F, and hence ε ≡ 1 mod 𝔭. Later some other Japanese mathematicians studied this phenomenon (cf. [O] and [K]), trying to eliminate some experimental nature of the argument of Shimura, and the general expectation was that the criterion holds for weight κ χ–Neben forms θ in terms of prime factors of N(ε^{κ−1} − 1) in place of N(ε−1) (see below Theorem 1). Although we have written θ for the Hecke eigenform with the required property for 𝔭, it is not a theta series. However the dihedralness modulo 𝔭 of the 𝔭–adic Galois representation of θ is equivalent to have a congruence modulo 𝔭 between θ and a theta series of weight 1 of a norm form of the quadratic field ℚ(\sqrt{D}).

1The author is partially supported by the NSF grant: DMS-9701017.

Recently I found with Maeda ([HM] Section 3) that a Hecke eigenform f of level N|D has a base-change to GL(2) over totally real fields E if p > 2κ − 1 and f has a congruence f ≡ θ mod 𝔩 for a prime 𝔩|D such that f is ordinary for 𝔩 and the mod 𝔩 Galois representation of f is irreducible. The field E is any totally real field in which all prime factors of pD are unramified. Thus it becomes increasingly important for us to know for what primes p the dihedral reduction \overline{ρ} shows up. This is the reason why we would like to record the exact criterion as stated below.

To make things precise, let us fix notation: Let F = ℚ(\sqrt{D}) ⊂ ℝ be a real quadratic field with discriminant D > 0 and Galois group Δ = Gal(F/ℚ). Let χ = ^{(\underline{D} )} be the Legendre symbol; thus, Δ̂ = \{id, χ\} for the Pontryagin dual Δ̂ of Δ. Let ψ ∈ Δ̂, 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. Let O be the 𝔪_{A}–adic completion for the maximal ideal 𝔪_{A} of A. We write ρ = ρ_{λ} : G = Gal(\overline{ℚ}/ℚ) → GL_{2}(O) for the Galois representation attached to λ. We put \overline{ρ} = (ρ_{λ} mod 𝔪_{O}) : G → GL_{2}(𝔽) for 𝔽 = O/𝔪_{O}. Let ε > 0 be a fundamental unit of F. Then it is easy to see that p|N(ε^{κ−1} − 1) (for even positive κ) implies χ(p) = 1 provided that p > 2 and N(ε) = −1. We would like to give a proof of the following fact:

Theorem 1. Let 𝔭 be a prime of ℚ(λ) associated to A. Suppose p ≥ 3. Then

(1) If λ(T (p)) ∈ A^{×} and the restriction \overline{ρ}_{F} of \overline{ρ} to H = Gal(\overline{ℚ}/F) is reducible but \overline{ρ} is absolutely irreducible, then ψ = χ, χ(p) = 1 and p|N(ε^{κ−1} − 1) for a fundamental unit ε of F which is positive at some real place of F;

(2) If ψ = χ, χ(p) = 1 and p|N(ε^{κ−1} − 1) for even κ and a prime p with κ > 2 or p ≥ 5, then there exist λ = λ_{κ} : h_{κ}(D, ψ; ℤ) → ℂ and 𝔭 such that (i) λ(T (p)) ∉ 𝔭, (ii) \overline{ρ} is absolutely irreducible, but (iii) \overline{ρ}|_{H} is reducible.

Moreover if χ(p) = 1, p|N(ε^{κ−1} − 1) and ψ = χ, then \overline{ρ} as in (2) is p–ordinary. Here we call a Galois representation ρ p–ordinary if its restriction to each decomposition group at p is isomorphic to ( \begin{matrix}δ & ∗ \\ 0 & ϵ \end{matrix} ) for an unramified character δ.

This should be known to specialists and is a consequence of a theory developed by the mathematicians quoted above ([S], [S1], [O] and [K]). However in these papers, some redundant assumptions are made, and it seems to me that the theorem is never stated in the literature in the above form. Although there is nothing essentially new in the proof, we shall give a proof based on my earlier works ([H86a,b]) and the theorems of Fontaine, Deligne and Mazur ([E] 2.5-6, 2.8) on classification of mod p modular Galois representations. Then we shall give a conjecture predicting the structure of the local component of the universal p–ordinary Hecke algebra through which λ_{κ} in the theorem factors (Conjecture 2.2). This conjecture is a Λ–adic version of the theorem and directly relates ε with the universal p–ordinary Hecke algebra (and hence with the universal p–ordinary deformation ring of \overline{ρ} by [W]; see also [HM] Section 4).

1. Divisibility of N(ε^{κ−1} − 1). Let χ be a quadratic character associated to a quadratic extension F/ℚ. Here first we study general properties of a 𝔭–adic Galois representation satisfying \overline{ρ} ⊗ χ ≅ \overline{ρ} (attached to an Hecke eigenform in S_{κ}(Γ_{0}(C(ψ)), ψ) for ψ ∈ {id, χ}), and after that, we shall prove the first statement of Theorem 1. We suppose that \overline{ρ}⊗χ ≅ \overline{ρ} throughout this section.

We assume p ≥ 3. For a while, we do not assume that F is real. Let ω_{p} be the Teichmüller character of G (at p). If ψ = χ, suppose first that \overline{ρ} is reducible:

\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 hence κ is odd if ψ = χ, F is an imaginary quadratic field, and \overline{δ} = ω_{p}^{(κ−1)/2}. If ψ = id and \overline{ρ} is reducible, then κ is even, χ = ω^{κ−1}δ^{−2} and hence F is again imaginary.

We now suppose that \overline{ρ} is absolutely irreducible. Let f = ∑_{n=1}^{∞} λ(T (n))q^{n} be the Hecke eigenform with eigenvalues λ. Then we look at the base change lift f̂ of f to GL(2)_{/F} (see [DN], [N] and [J]). Since f̂ is of level 1, \overline{ρ}_{F} is unramified outside p (cf [C] and [T]). Then we have a character φ : H → 𝔽^{×} such that \overline{ρ} ≅ Ind_{F}^{ℚ} φ (see Lemma 3.2 in [DHI]). Then by comparing the determinant, we get

φ · φ_{σ} = ω_{p}^{(κ−1)e},

where e is the ramification index of p in F/ℚ, φ_{σ}(g) = φ(σgσ^{−1}) for σ ∈ G which induces a non-trivial automorphism on F and ω_{p} is the Teichmüller character of G restricted to H. If F is real, this shows that −1 = det(\overline{ρ})(c) = ω_{p}^{(κ−1)e}(−1) for a complex conjugation c. Thus e = 1 if F is real. Let 𝔠 be the conductor of φ, which divides a high power of p. Since the conductor of ω_{p} is p, 𝔠 ∩ 𝔠^{σ} = p. The absolute irreducibility of \overline{ρ} implies that φ ≠ φ_{σ}.

Suppose that p is ramified in F. Thus F has to be imaginary. Then automatically, we have φ = φ_{σ} on the inertia group I_{𝔭} at 𝔭|p because φ is a character modulo 𝔭 for a unique prime 𝔭 of F over p. Thus \overline{ρ} becomes reducible if the class number of F is prime to |𝔽| − 1, contradicting to the irreducibility assumption. This also implies that φ^{2} = ω^{2(κ−1)} on I_{𝔭}. Thus φ = ω^{κ−1} on I_{𝔭}.

We hereafter assume that p ∤ D. Let A be a valuation ring of ℚ(λ) with residual characteristic p. Suppose that λ(T (p)) = a(p, f) ≢ 0 mod 𝔪_{A}. We fix an embedding i_{p} : \overline{ℚ} ↪ \overline{ℚ}_{p} associated to a prime 𝔓 of \overline{ℚ}, and assume that 𝔓|𝔪_{A}. Then by [H86b] Corollary 3.2 (see also [H88] when p = 3), we can find an algebra homomorphism λ′ : h_{κ′}(C(ψ), ψ; ℤ) → \overline{ℚ}_{p} of weight 2 ≤ κ′ ≤ p + 1 so that \overline{ρ}_{λ} ≅ \overline{ρ}_{λ′}, λ ≡ λ′ mod 𝔓 and κ ≡ κ′ mod p−1. Then by Deligne’s theorem ([E] 2.5), p = 𝔭𝔭^{σ} in F with 𝔭 ≠ 𝔭^{σ}, φ(x) = x^{κ−1} mod 𝔭, and C = 𝔭. Write 𝔯 for the integer ring of F. Regard φ as a Dirichlet character of (𝔯/𝔭)^{×} with values in 𝔽^{×}. Thus supposing that F is real (and hence that κ is even), ε_{+}^{κ−1} mod 𝔭 = φ(ε_{+}) = 1 for the totally positive fundamental unit ε_{+} of F. Thus 𝔭|ε_{+}^{κ−1} − 1. If ε ≠ ε_{+}, we may assume ε_{+} = ε^{2} and εε^{σ} = −1 for the generator σ of Δ. Since ε_{+}^{κ−1} − 1 = ε^{2(κ−1)} − 1 = (ε^{κ−1} − 1)(ε^{κ−1} + 1) = (−ε^{κ−1})N(ε^{κ−1} − 1), p|N(ε^{κ−1} − 1) ⟺ p|N(ε_{+}^{κ−1} − 1). The determinant of Ind_{F}^{ℚ} φ is given by φ_{ℤ}χ, where regarding φ as a Dirichlet character modulo 𝔭, φ_{ℤ} is the Galois character associated to the restriction of the Dirichlet character φ to ℤ. This shows that ψω^{κ−1} = det(\overline{ρ}) = det(Ind_{F}^{ℚ} φ) = χω^{κ−1}, and hence ψ = χ. Thus we get

Proposition 1.1. Suppose p ≥ 3 and that F = ℚ(\sqrt{D}) is a real quadratic field of discriminant D > 0. Let χ = ^{(\underline{D} )} be the Legendre symbol. If \overline{ρ} ≅ \overline{ρ} ⊗ χ for λ : h_{κ}(C(ψ), ψ; ℤ) → A with ψ ∈ Δ̂ and λ(T (p)) ∈ A^{×}, then ψ = χ, χ(p) = 1 and p|N(ε^{κ−1} − 1) for a fundamental unit ε of F which is positive at some real place of F. Moreover \overline{ρ} is p–ordinary and p ∤ D.

We remark that, by [DHI] Lemma 3.2, the following conditions are equivalent under the absolute irreducibility of \overline{ρ}:

(1) \overline{ρ} ⊗ χ ≅ \overline{ρ};

(2) \overline{ρ}_{F} is reducible;

(3) \overline{ρ} ≅ Ind_{F}^{ℚ} φ for φ with φ_{σ} ≠ φ. The first statement of Theorem 1 follows from this remark and the above proposition.

Since we have only dealt with the case where λ(T (p)) ≢ 0 mod 𝔪_{A}, we here add two remarks on what happens if λ(T (p)) ≡ 0 mod 𝔪_{A}. Suppose that λ(T (p)) ≡ 0 mod 𝔪_{A} and 2 ≤ κ ≤ p + 1. Then by Fontaine’s theorem ([E] 2.6), the restriction of \overline{ρ} to the decomposition group at p is irreducible, p has to be inert in F, and φ(x) = x^{κ−1} mod p for x ∈ 𝔯_{p}^{×}. If F is real, we take complex conjugation c ∈ Gal(\overline{ℚ}/F). Then we have det(\overline{ρ})(c) = (−1)^{2κ−2} = −1. This shows that F has to be imaginary to have λ(T (p)) = a(p, f) ≡ 0 mod 𝔪_{A} and 2 ≤ κ ≤ p + 1.

As is well known (cf. [E]), we can find an algebra homomorphism

λ′ : h_{κ′}(C(ψ), ψ; ℤ) → \overline{ℚ}_{p}

of weight 2 ≤ κ′ ≤ p + 1 such that \overline{ρ}_{λ} ⊗ ω^{a} ≅ \overline{ρ}_{λ′} for a suitable a. If the restriction of \overline{ρ} to the decomposition group at p is irreducible (that is, super-singular), twisting by ω_{p}^{a} does not change super-singularity. If the restriction to the decomposition group is reducible, (ρ_{λ′} mod 𝔭) has to be p–ordinary by Fontaine’s theorem and Deligne’s theorem combined.

2. Λ–adic version. Let p ≥ 3 be a prime and 𝔽 be a finite field of characteristic p. We start from a character

φ : H → 𝔽^{×} with φ(c)φ(σcσ^{−1}) = −1.

Thus \overline{ρ} = Ind_{F}^{ℚ} φ : G → GL_{2}(𝔽) is absolutely irreducible. Note that \overline{ρ} is p–ordinary if and only if p = 𝔭𝔭^{σ} for primes 𝔭 ≠ 𝔭^{σ} of 𝔯. In this case, we have that C(φ) = 𝔭. We take O to be the ring of Witt vectors of the finite field 𝔽 which is generated over 𝔽_{p} by the values of φ. Let K be the field of fractions of O. We use the same symbol φ for the Teichmüller lift of φ to O^{×}. On the inertia at 𝔭|p, φ = ω_{𝔭}^{κ−1} for some positive even integer κ, where ω_{𝔭} is the Teichmüller character modulo 𝔭. This implies that 𝔭|ε_{+}^{κ−1} − 1. Conversely, if 𝔭|ε^{κ−1} − 1 for an even positive integer κ and χ(p) = 1, ω_{𝔭}^{κ−1} gives rise to a class character modulo 𝔭∞ for an infinite place ∞ of F and hence to a character φ of H with the above property by class field theory. We fix an embedding i_{p} : \overline{ℚ} ↪ \overline{ℚ}_{p} and regard O as a subring of \overline{ℚ}_{p}. Thus we can think of φ having values in \overline{ℚ} ⊂ ℂ. Then we have a theta series θ(φ) = ∑_{𝔫⊂𝔯} φ(𝔫)q^{N(𝔫)} ∈ S_{1}(Γ_{0}(Dp), χω^{κ−1}) such that the associated ℓ–adic representation ρ_{θ(φ)} is isomorphic to Ind_{F}^{ℚ} φ for all ℓ [He].

We write Λ for the Iwasawa algebra O[[Γ]] for Γ = 1+pℤ_{p}. Let h^{ord}(Dp^{∞}, ϕ; O) be the universal p–ordinary Hecke algebra of tame character ϕ = χω_{p}^{κ} with coefficients in O as defined in [H86a]. Although it is assumed that p > 3 in [H86a], the result quoted above remains valid for p = 3 (see [H88] or [H93] Section 7.3). Let ℤ[ω_{p}^{a}] be the subalgebra of ℂ generated by the values of i_{p}^{−1}ω_{p}^{a}, and put

h_{κ}(Dp, χω_{p}^{a}; B) = h_{κ}(Dp, χω_{p}^{a}; ℤ[ω_{p}^{a}]) ⊗_{ℤ[ω_{p}^{a}]} B for B = O or K.

The algebra h^{ord}(Dp^{∞}, χω_{p}^{κ}; O) is a flat Λ–algebra. Let h_{κ}^{ord}(Dp, χω_{p}^{a}; O) be the maximal algebra direct factor on which the image of T (p) is invertible. We then put h_{κ}^{ord}(Dp, χω_{p}^{a}; K) = h_{κ}^{ord}(Dp, χω_{p}^{a}; O) ⊗_{O} K. The algebra homomorphism ϕ_{k} : Λ → O induced by the character: Γ ∋ γ ↦ γ^{k} gives rise to a surjective O–algebra homomorphism

π_{k} : h^{ord}(Dp^{∞}, χω_{p}^{κ}; O) ⊗_{Λ,ϕ_{k}} K → h_{k}^{ord}(Dp, χω_{p}^{κ−k}; K)

sending T (n) to T (n) for all n and all k ≥ 1, and π_{k} is an isomorphism for all k ≥ 2. In particular, for k = κ, we have

h^{ord}(Dp^{∞}, χω_{p}^{κ}; O) ⊗_{Λ,ϕ_{κ}} K ≅ h_{κ}^{ord}(Dp, χ; K) ≅ h_{κ}^{ord}(D, χ; K),

where the last isomorphism is only valid for κ > 2. If p > 3 the above isomorphisms are valid even for O in place of K. For k = 1, we have an algebra homomorphism λ_{1} : h_{1}(Dp, χω_{p}^{κ−1}; O) → O given by θ(φ)|T (n) = λ_{1}(T (n))θ(φ). Take a minimal prime ideal ℙ of h^{ord}(Dp^{∞}, χω_{p}^{κ}; O) such that ℙ ⊂ Ker(λ_{1}). Thus writing 𝕀 for h^{ord}(Dp^{∞}, χω_{p}^{κ}; O)/ℙ, we have a Λ–algebra homomorphism

λ_{𝕀} : h^{ord}(Dp^{∞}, χω_{p}^{κ}; O) → 𝕀

lifting λ_{1}. For each prime divisor P ∈ Spec(𝕀) with P ⊃ Ker(ϕ_{k}), we have λ_{P} : h_{k}(Dp, χω_{p}^{κ−k}; O) → \overline{ℚ}_{p} induced by λ_{𝕀} mod P. If k = κ > 2, λ_{P} is induced by a unique λ_{κ} : h_{κ}(D, χ; O) → \overline{ℚ}_{p}. Anyway we have a p–adic family of ordinary forms specializing to θ(φ) at weight 1.

Let 𝔓 be the prime associated to the embedding i_{p} : \overline{ℚ} ↪ \overline{ℚ}_{p}. Since \overline{a(q, f)} = χ(q)a(q, f) (q ∤ D) for the weight κ specialization f (associated to λ_{κ} as above), f has the property that

𝔓 ⊃ \{a(q, f) − \overline{a(q, f)}|q ∤ D\} ,

where z ↦ \overline{z} indicates complex conjugation. Thus writing ℚ(λ_{κ}) for the field generated by λ_{κ}(T (n)) for all n and ℚ(λ_{κ})^{+} for its subfield fixed by the complex conjugation, we see that [ℚ(λ_{κ}) : ℚ(λ_{κ})^{+}] = 2, and 𝔓 should divide the relative different of ℚ(λ_{κ})/ℚ(λ_{κ})^{+}.

Let h be the local ring of the Hecke algebra h^{ord}(Dp^{∞}, χω_{p}^{κ}; O) through which λ_{𝕀} factors. We have a bijection ([H86a] Section 1) for k ≥ 2:

Hom_{O−alg}(h ⊗_{Λ,ϕ_{k}} K, \overline{ℚ}_{p})

≅ {f ∈ S_{k}(Γ_{0}(Dp), χω_{p}^{κ−k})|f is a normalized eigenform with f ≡ θ(φ) mod 𝔓} .

In particular, if k = κ > 2, h ⊗_{Λ,ϕ_{κ}} K is isomorphic to an algebra direct factor of h_{κ}(D, χ; K) (cf. [H86a] Proposition 4.7), and hence λ_{κ} has to belong to Hom_{O−alg}(h ⊗_{Λ,ϕ_{k}} K, \overline{ℚ}_{p}).

We claim that if k = κ = 2 and p ≥ 5, then for some hight 1 prime P containing Ker(ϕ_{k}), λ_{P} is still induced by λ_{2} : h_{2}(D, χ; O) → \overline{ℚ}_{p}. To prove the claim, we introduce a notion of flatness of \overline{ρ}. Let L be a number field, and write O_{𝔩} for the 𝔩–adic completion of the integer ring of L. A mod p representation \overline{π} : Gal(\overline{ℚ}/L) → GL_{n}(𝔽) is called flat over L if its restriction to the decomposition group of each 𝔓|p is isomorphic to a representation realized on the special fiber of a finite flat group scheme (with a structure of 𝔽–vector space) defined over O_{𝔓}. Since ω_{𝔭} is flat over F, \overline{ρ} = Ind_{F}^{ℚ} φ is flat over ℚ. Then by a theorem of Mazur, see [E] 2.8, we can find λ_{2} as above. The theorem tells us that the q–expansion \overline{f} = ∑_{n} λ_{P} (T (n))q^{n} mod 𝔭 is the q–expansion of a mod p–modular form g on X_{1}(D)_{/𝔽_{p}}. Since the statement of [E] 2.8 only concerns mod p modular forms, the condition p ≥ 5 is not explicitely stated. Here we mean by a mod p modular form of weight k a global section of \underline{ω}^{⊗k} over X_{1}(D)_{/𝔽_{p}} (as in [E] 2.1). But for the given mod p modular form \overline{f} = g as above to be lifted to a classical modular form f ∈ H^{0}(X_{1}(D)_{/ℤ_{p}}, \underline{ω}^{⊗2}), one needs to have a characteristic 0 lift of the Hasse invariant A. Such a lift exists under the assumption p ≥ 5. This shows the assertion (2) of Theorem 1, and we finish the proof of Theorem 1.

Here we record what we have actually shown in the above proof of Theorem 1:

Proposition 2.1. Suppose p|N(ε^{κ−1} − 1) for an odd prime p with χ(p) = 1 and an even positive integer κ. Then

(1) There exists a finite order character φ : H → \overline{ℚ}^{×} of conductor 𝔭 such that (i) φ coincides with ω_{𝔭}^{κ−1} on the inertia group at 𝔭 for the Teichmüller character ω_{𝔭} and (ii) φ(c)φ(σcσ^{−1}) = −1 for complex conjugation c; (2) Let λ = λ_{κ} : h_{κ}(Dp, χ; ℤ) → \overline{ℚ} be a specialization at weight κ of the λ_{𝕀}. Then

\overline{ρ}_{λ} ≅ Ind_{F}^{ℚ} φ.

We now study the structure of the local ring h defined above. We write CNL_{O} for the category of complete noetherian local O–algebras with residue field 𝔽. Let F^{(p)}/F be the maximal extension inside \overline{ℚ} unramified outside \{p, ∞\} for the infinite place ∞ of ℚ. By a theorem of Wiles [W] Theorem 3.3, if p ≠ 2κ − 1, the ring h along with Galois representation ρ_{h} : G^{(p)} = Gal(F^{(p)}/F) → GL_{2}(h) of h represents the deformation functor F_{ℚ}^{ord} : CNL_{O} → SET S given by

F_{ℚ}^{ord}(B) = \{ρ : G^{(p)} → GL_{2}(B)|ρ ≡ \overline{ρ} mod 𝔪_{B} and ρ is p–ordinary\} / ≈,

where \overline{ρ} = (Ind_{F}^{ℚ} φ mod 𝔪_{O}) and “≈” is the strict equivalence (cf. [M]). The association: ρ ↦ ρ ⊗ χ gives a natural transformation of F_{ℚ}^{ord} onto itself, inducing a ring automorphism τ : h → h. To see this, we consider the involution W on S_{κ}(Γ_{0}(p), χ) induced by ( \begin{matrix}0 & −1 \\ D & 0 \end{matrix} ). Since W T (n)W = χ(n)T (n) for n prime to D, conjugation by W coincides with τ. Note that W T (n)W is the adjoint operator T^{∗}(n) of T (n) under the Petersson inner product, and T^{∗}(n) is an element in h_{κ}(D, χ; ℤ). Since this is true for all k with k ≡ κ mod p − 1, we have an involution τ on h such that W T W = τ(T) on h ⊗_{Λ,ϕ_{k}} O for all such k. We write h_{+} the subalgebra of h fixed by τ. The automorphism τ induces the complex conjugation on ℚ(λ_{κ}), which is the automorphism of ℚ(λ_{κ}) fixing ℚ(λ_{κ})^{+}.

We take p ≥ 3 as in Theorem 1 such that 𝔭|ε^{κ−1} −1 and χ(p) = 1. We now identify ℤ_{p} with 𝔯_{𝔭} via inclusion: ℤ ↪ 𝔯, and assume 𝔓 ∩ 𝔯 = 𝔭. In this way, we have Γ = 1 + pℤ_{p} ↪ 𝔯_{𝔭}^{×}. We fix a generator u of Γ and identify Λ ≅ O[[T]] via u ↦ 1 + T. Let “log” be the p–adic logarithm function. Then we write ⟨ε⟩ for (u^{−1}(1+T))^{log(ε)/ log(u)}, which is the unique element in Λ such that ϕ_{k}(⟨ε⟩) = ε^{k−1}ω_{𝔭}(ε)^{1−k}. In particular, ϕ_{κ}(ε) = ε^{κ−1}.

We write h^{ord}(p^{∞}, ϕ; O)_{/F} for the universal ordinary Hecke algebra for GL(2)_{/F} defined in [H88] for Hilbert modular forms (analogously to h^{ord}(Dp^{∞}, ϕ; O) for elliptic modular forms), which is again a Λ–algebra. This algebra is reduced, because it specializes to level 1 Hecke algebras (which is reduced) modulo Ker(ϕ_{k}) for all k > 2 with k ≡ κ mod p − 1. Let ĥ be the local ring of h^{ord}(p^{∞}, ω_{p}^{κ}; O)_{/F} through which λ̂_{κ} factors. We have a canonical Galois representation ρ_{h} : H → GL_{2}(F rac(ĥ)) such that T r(ρ_{h}(F rob_{𝔩})) is given by the projection of T (𝔩) to ĥ for all primes 𝔩 prime to p. Here F rac(ĥ) is the total quotient ring of ĥ. Then as in [DHI] Section 3.4, we can define the base change map β : ĥ → h so that β(T r(ρ_{h})) = T r(ρ_{h})|_{H}.

Conjecture 2.2. Suppose that p ≥ 3. Let h_{+} be the subalgebra of h fixed by τ. Then if O is sufficiently large, under the above assumption and the notation, we have

(1) h ≅ h_{+}[\sqrt{⟨ε⟩ − 1}],

(2) h(τ − 1)h = h\sqrt{⟨ε⟩ − 1},

(3) Im(β) = h_{+}.

Here h_{+}[\sqrt{Φ}] = h_{+}[X]/(X^{2} − Φ) for Φ ∈ h_{+}.

The reason why we need to assume O to be large is as follows: What we actually expect is that the ideal h(τ −1)h is generated by an element η such that η^{2} = x(⟨ε⟩−1) with x ∈ h^{×}. If h_{+} is a Gorenstein ring, the relative different h(τ − 1)h has to be principal (because, the Gorenstein-ness of h is known by Taylor-Wiles [W]). Since Hecke algebras tend to be Gorenstein (actually even a local complete intersection), expecting h_{+} would be Gorenstein may not be so outrageous. The unit x may not be a square in h. Since p is odd, replacing O by its quadratic extension if necessary, we may assume that x is a square in h and get the conclusion of the conjecture over O.

Related to the above reason, let us add one more remark. We have possibly 4 choices of ε: ε, ε^{−1}, −ε, −ε^{−1}. This yields two choices of ⟨ε⟩: ⟨ε⟩ and ⟨ε⟩^{−1}. Note that ⟨ε⟩^{−1} − 1 = ⟨ε⟩^{−1}(1 − ⟨ε⟩). Since ⟨ε⟩^{−1} is a square in Λ, if we add \sqrt{−1} to O if necessary, the statement of the conjecture does not depend on the choice of ε.

Out of this conjecture, we can prove Conjecture 3.8 of [DHI], and some other supporting evidences for this conjecture and the above are discussed in [DHI].

3. Examples. We compute the odd primes p appearing in N(ε^{κ−1} − 1) for even positive κ in some special cases. We take a real quadratic field F = ℚ(\sqrt{d}) for a square-free d. We assume that εε^{σ} = −1, which is equivalent to |N(ε^{κ−1} − 1)| = |T r_{F/ℚ}(ε^{κ−1})|. Then for each given odd prime 𝔭 of F, ε generates a subgroup ⟨ε⟩_{𝔭} of (𝔯/𝔭)^{×}. Let e = |⟨ε⟩_{𝔭}|. Then 𝔭|ε^{e} − 1. If 𝔭 does not split, then 𝔭|(ε^{σ})^{e} − 1. If e is odd, (ε^{σ})^{e}(ε)^{e} = −1. Thus 𝔭|(ε^{σ})^{e} − 1 = (ε^{σ})^{e}(1 + ε^{e}). This shows 𝔭|2 = 1 − ε^{e} + 1 + ε^{e}. This contradicts to the fact that 𝔭 is odd. Thus p must split in F. We consider the set S of all odd primes 𝔭 dividing ε^{e} −1 for some odd integer e. For each 𝔭 ∈ S, we write e(𝔭) for the minimum positive e such that 𝔭|ε^{e} − 1. We choose ε so that |ε| < 1 and |ε^{σ}| > 1. Thus

|N_{F/ℚ}(ε^{e} − 1)| = |T r_{F/ℚ}(ε^{e})| ⟶ ∞ as e → ∞.

Since e(𝔭) is the order of ε in (𝔯/𝔭)^{×}, the set of e such that ε^{e} ≡ 1 mod 𝔭 is an ideal of ℤ generated by e(𝔭). Let S_{e} = {𝔭|e(𝔭) = e}. Then

S = ⨆_{e:odd} S_{e} and S_{e} is a finite set.

Proposition 3.1. The set S is an infinite set of split primes. The set S_{1} is empty if and only if the integer d is the square-free part of 2^{2n} + 1 for a positive integer n (this implies that d ≡ 1 mod 8 or d = 5). Let q be an odd prime. If q is outside ⋃_{t|e} S_{t}, then S_{eq^{j}} ≠ ∅ for all j ≥ 1 unless 𝔽_{2}[ε] = 𝔽_{4} and q = 3 and 3 ∤ e. Any element in S_{e} is prime to e.

Proof. Let ξ_{e} = \frac{ε^{e}−1}{ε−1}. Then |N_{F/ℚ}(ξ_{e})| → ∞ as e → ∞. Suppose that S is a finite set. We write S = {𝔭_{1}, …, 𝔭_{r}} in which S_{1} = {𝔭_{t+1}, …, 𝔭_{r}}. We choose an even number e_{0} so that (i) e_{0} is prime to e_{1} = ∏_{i=1}^{r} e(𝔭_{i}), (ii) e = e_{0} + e_{1} is prime to all elements in S_{1} and (iii) |N_{F/ℚ}(ξ_{e})| > 1. Since ξ_{e} ≡ e mod 𝔭 for every 𝔭 ∈ S_{1}, any 𝔭 ∈ S_{1} does not divide ξ_{e}. Let 𝔔 be a factor of 2. If ε ≡ 1 mod 𝔔, ξ_{e} ≡ e mod 𝔔; thus, ξ_{e} is prime to 𝔔. If ε ≡ ζ mod 𝔔 for a cubic root ζ ∈ 𝔽_{4}, then we take e so that 3 ∤ e. Then ξ_{e} is prime to 𝔔. By (iii), there is a prime 𝔭 dividing ξ_{e}. As we have already seen, 𝔭 is a split prime. Since e(𝔭) is a factor of e, it is prime to e(𝔭_{i}) for 1 ≤ i ≤ r. We have already seen that any element in S_{1} does not divide ξ_{e}. Thus 𝔭 is not in S, which is a contradiction. We may assume |ε| < 1 and ε < 0. Thus |N_{F/ℚ}(ε^{e} − 1)| = |T r_{F/ℚ}(ε^{e})| = f(ε^{e}) for f(x) = x − \frac{1}{x}. We see for x ∈ [−1, 0) that f(x) = 1 ⟺ x = \frac{1−\sqrt{5}}{2} and that f(x) = 2^{n} ⟺ x = 2^{n−1} − \sqrt{2^{2n−2} + 1}. Thus d ≠ 5 is the unique square-free factor of 2^{2n−2} + 1 if and only if S_{1} = ∅. Since f(x) is increasing on [−1, 0), if ε^{e} < \frac{1−\sqrt{5}}{2}, then 0 < f(ε^{e}) < 1, which is impossible since f(ε^{e}) is a positive integer. If d ≠ 5, then \frac{1−\sqrt{5}}{2} < ε^{e} < 0 and f(ε^{e}) > 1 for any positive e. We now consider g_{n}(x) = \frac{f(x^{n})}{f(x)} = ∑_{m=0}^{n−1} x^{n−1−2m}. Since f(x) is increasing, g_{n}(x) = 1 ⟺ n = 1, and g_{n}(x) > 1 if n > 1. Let q be an odd prime. Then for each 𝔭 ∈ S_{qn}, N_{F/ℚ}𝔭 ≡ 1 mod q. Thus 𝔭 ∈ S_{qn} is prime to q. In particular, any element of S_{e} is prime to e. On the other hand, g_{q}(ε^{em}) ≡ q mod 𝔭 for all 𝔭 ∈ S_{t} with t|e. Suppose that q is outside ⋃_{t|e} S_{t}. If 𝔭|g_{q}(ε^{eq^{n}}) and 𝔭 ∈ S_{eq^{j}} with j > 0, then q ≡ g_{q}(ε^{eq^{n}}) ≡ 0 mod 𝔭, which contradicts to the fact that 𝔭 ∈ S_{eq^{j}} is prime to eq^{j}. This shows that any prime factor of g_{q}(ε^{eq^{n}}) outside ⋃_{t|e} S_{t} is outside ⋃_{j=0}^{n} S_{eq^{j}}. Therefore every odd factor of g_{q}(ε^{eq^{n}}) outside ⋃_{t|e} S_{t} gives an element of S_{eq^{n+1}}. Thus to prove S_{eq^{n+1}} ≠ ∅, we need to show that g_{q}(ε^{eq^{n}}) is odd. Let 𝔔 be a factor of 2. If ε ≡ 1 mod 𝔔, g_{n}(ε^{e}) ≡ n mod 𝔔. If ε ≡ ζ mod 𝔔 for a cubic root of unity in 𝔽_{4}, g_{n}(ε^{3e}) ≡ n mod 𝔔 and g_{n}(ε^{e}) ≢ 0 mod 𝔔 if 3 ∤ en. Thus if 𝔽_{2}[ε] = 𝔽_{2}, then S_{eq^{j}} ≠ ∅ for all j ≥ 1 and all odd prime q outside ⋃_{t|e} S_{t}. If 𝔽_{2}[ε] = 𝔽_{4}, then for q outside ⋃_{t|e} S_{t}, S_{eq^{j}} ≠ ∅ for all j ≥ 1 and all odd prime q provided that either 3|e or q ≠ 3.

The following result is supplied by Y. Maeda, to whom the author is grateful.

Proposition 3.2 (Y. Maeda). Let ε > 1 be a quadratic unit in ℝ satisfying ε^{2} − 2^{n}ε − 1 = 0 for a non-negative integer n. If n ≠ 2, then ε is a fundamental unit in K = ℚ[ε]. Thus, we have

(1) If n ∉ \{0, 2\}( ⟺ K ≠ ℚ(\sqrt{5})), for odd e > 2, ⋃_{t|e} S_{t} ≠ ∅ for K;

(2) If K = ℚ[\sqrt{5}], then ⋃_{t|e} S_{t} ≠ ∅ ⟺ e ≥ 5 for odd e.

Proof. Let K = ℚ[ε] be a real quadratic field for a unit ε as above. For an odd integer ℓ, we have T r(ε^{ℓ}) = −N(ε^{ℓ} − 1), and hence

(∗)  T r(ε^{ℓ})|T r(ε^{k})  if ℓ|k and ℓk is odd.

Let ε_{0} be a fundamental unit of K so that ε = ε_{0}^{ℓ} for ε as in the proposition. Then ε_{0} > 1. Since N(ε) = −1, ℓ is odd, and N(ε_{0}) = −1. By (∗), we find T r(ε_{0}) = 2^{k} with

0 ≤ k ≤ n, and hence ε_{0} = ^{\frac{2^{k}+\sqrt{2^{2k}+4}}{2}}. We divide our argument into the following three cases:

(i) k ≥ 2, (ii) k = 1 and (iii) k = 0.

(i) We first suppose k ≥ 2 and write ℓ = 2s + 1. We have ε_{0} = 2^{κ} + \sqrt{D} for κ = k − 1 and D = 2^{2κ} + 1. From the binomial theorem, we get

2^{n} = T r(ε_{0}^{ℓ}) = 2^{k} ∑_{r=0}^{s} \binom{ℓ}{2r}2^{κ(ℓ−2r−1)}D^{r}.

From this, we conclude

(∗∗) 2^{n−k} = ∑_{r=0}^{s} \binom{ℓ}{2r}2^{κ(ℓ−2r−1)}D^{r}.

Since D ≡ 1 mod 2 (k ≥ 2), we get from (∗∗)

2^{n−k} ≡ \binom{ℓ}{2s}D^{s} ≡ 1 mod 2.

This shows n = k, and the assertion follows.

(ii) Suppose k = 1 ( ⟺ κ = 0). Then D = 2, and the formula (∗∗) is still valid. Therefore,

2^{n−k} = ∑_{r=0}^{s} \binom{ℓ}{2r}2^{r} ≡ 1 mod 2,

and the conclusion again holds.

(iii) Suppose k = 0. Then ε_{0} = \frac{1+\sqrt{5}}{2}. Since we have T r(ε_{0}^{3}) = 2^{2} = T r(2 + \sqrt{5}), we need to show

T r(ε_{0}^{ℓ}) = 2^{n} ⟺ ℓ = 1 or 3.

We are going to show that

ℓ ≥ 5 ⇒ T r(ε_{0}^{ℓ}) is not a 2–power.

By (∗), we may assume that ℓ is either a prime or equal to 9. By computation, T r(ε_{0}^{9}) is not a 2–power. So we may assume that ℓ ≥ 5 is a prime. Then by Proposition 3.1, S_{ℓ} ≠ ∅ because S_{1} = ∅. This shows the result.

There are infinitely many d such that S_{1} = ∅. We list some of them:

d = 5, 17, 41, 257, 4097 = 17 · 241, 16385 = 5 · 29 · 113, 65537,

where 41 · 5^{2} = 2^{10} + 1. We give a way of computing S_{e}. Since f(ε^{e}) = |(ε^{e} − 1)(−ε^{−e} − 1)| = |T r_{F/ℚ}(ε^{e})|,

writing a_{e} = T r_{F/ℚ}(ε^{e}) and ε^{2} − aε − 1 = 0 for the equation of ε, a_{e} satisfies a_{0} = 2, a_{1} = a and a_{n} = aa_{n−1} + a_{n−2}. Thus {a_{n}} is a Fibonacci type sequence. Using the above recurrence relation, it is easy to compute. We list here some: Case d = 5: S_{1} = S_{3} = ∅, S_{5} = {11}, S_{7} = {29}, S_{9} = {19}, S_{11} = {199},

S_{13} = {521}, S_{15} = {31}, S_{17} = {3571}, S_{19} = {9349},

S_{21} = {211}, S_{23} = {139, 461};

Case d = 13: S_{1} = {3}, S_{3} = ∅, S_{5} = {131}, S_{7} = {1429}, S_{9} = {433},

S_{11} = {23, 7393};

Case d = 17: S_{1} = ∅, S_{3} = {67}, S_{5} = {4421}, S_{7} = {127, 2297}; Case d = 29: S_{1} = {5}, S_{3} = {7}, S_{5} = {151}, S_{7} = {20357}; Case d = 37: S_{1} = {3}, S_{3} = {7}, S_{5} = {11, 1951}; Case d = 41: S_{1} = ∅, S_{3} = {4099}; Case d = 61: S_{1} = {3, 13}, S_{3} = {127}; Case d = 257: S_{1} = ∅, S_{3} = {13, 79}.

All the above primes show up in the relative discriminant D_{+} = D(ℤ(λ_{κ})/ℤ(λ_{κ})^{+}) by Theorem 1, and we refer to the table in [DHI] Section 2.2 for examples of the numerical value of D_{+}. Here ℤ[λ_{κ}] is the order of ℚ(λ_{κ}) generated over ℤ by λ_{κ}(T (n)) for all n, and ℤ[λ_{κ}]^{+} = ℚ(λ_{κ})^{+} ∩ ℤ[λ_{κ}].

In the above computation, we may change ε by −ε. Thus we may assume that a = T r_{F/ℚ}(ε) > 0. Then if d ≠ 5, we see that a > 1 and a_{n} = aa_{n−1} + a_{n−2}. Since a_{0} = 2 and a_{1} = a, a_{n} > 0 for all n, and thus a_{n} > aa_{n−1}. Thus by induction on n, we see that a_{n} > a^{n} for n > 1. On the other hand, choosing ε > 1, we see that if n is odd,

a^{n} ≤ a_{n} = ε^{n} − ε^{−n} < ε^{n} < (\frac{a + \sqrt{a^{2} + 4}}{2})^{n} < a^{n}(\frac{1 + \sqrt{1 + (2/a)^{2}}}{2} )^{n}.

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Haruzo Hida,

Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA hida@math.ucla.edu