DIHEDRAL CONGRUENCE PRIMES AND CLASS FIELDS OF REAL QUADRATIC FIELDS

ALEXANDER F. BROWN AND EKNATH P. GHATE

Abstract. We show that for a real quadratic field F the dihedral congruence primes with respect to F for cusp forms of weight k and quadratic nebentypus are essentially the primes dividing expressions of the form ϵ_{+}^{k−1} ± 1 where ϵ_{+} is a totally positive fundamental unit of F. This extends work of Hida. Our result allows us to identify a family of (ray) class fields of F which are generated by torsion points on modular abelian varieties.

1. Introduction

Let F = ℚ(\sqrt{D}) denote a real quadratic field of discriminant D > 0. Let χ_{D} denote the corresponding quadratic character. Let S_{k}(D, χ_{D}) denote the cusp forms of weight k ≥ 2, level D and nebentypus χ_{D}. A cusp form f ∈ S_{k}(D, χ_{D}) is said to be primitive if it is a normalized newform that is a common eigenform of all the Hecke operators.

In [Hid98] Hida characterized the primes of congruence between f and f⊗χ_{D} as f varies through all primitive elements in S_{k}(D, χ_{D}) in terms of the primes dividing ϵ_{+}^{k−1} − 1 where ϵ_{+} is a totally positive fundamental unit of F.

The starting point of this paper was the following question: can the primes of congruence between f and f⊗χ be similarly characterized for arbitrary even quadratic characters χ that are ‘factors’ of the nebentypus χ_{D}? In the first part of this paper we answer this question affirmatively. This time the answer involves expressions of the form ϵ_{+}^{k−1} ± 1 where ϵ_{+} is a totally positive fundamental unit of the real quadratic field F_{χ} corresponding to χ. As in [Hid98] we work under certain technical assumptions of ordinarity and absolute irreducibility and do not elaborate upon this here.

Hilbert’s twelfth problem asks whether one can generate the (ray) class fields of a given number field explicitly, for instance by the torsion points on an abelian variety, or by the values of a modular function. The case where the base field is ℚ or an imaginary quadratic field is classical. For more general CM fields some partial results due to Shimura and Taniyama are available [Shi99]. However for many years the next simplest case, that of real quadratic fields, defied treatment. Then, in the early seventies Shimura [Shi71], [Shi72] invented a beautiful method to explicitly generate the (ray) class of real quadratic fields via torsion points on certain modular abelian varieties. His method has since been extended by other authors [DY73], [Koi76], [Oht77].

Date: August 9, 2001.

2000 Mathematics Subject Classification. Primary: 11F33, Secondary: 11G18, 11R37.

Key words and phrases. Dihedral congruence primes, fundamental units, real quadratic fields, class field theory.

In the second part of this paper we further extend Shimura’s method using our characterization of ‘dihedral congruence primes’ in the first part. We show that by considering congruences in a systematic way with twists that are not necessarily the full nebentypus we can, at least in principle, generate explicitly infinitely many (ray) class fields of the same real quadratic field by Shimura’s method. We say ‘in principle’ since, as we shall see in the text, our results are subject to the phenomenon of extra twisting. We also prove some results which combine Hida’s theory of families of ordinary modular forms with Shimura’s method, further increasing its scope.

2. Dihedral congruence primes and fundamental units

We start with some notation which is more general than that used in the Introduction. Let F = ℚ(\sqrt{D}) denote an arbitrary quadratic field of discriminant D. Let

χ_{D} : (ℤ/|D|ℤ)^{×} ⟶ {±1}

denote the quadratic character corresponding to F/ℚ. Let S_{k}(|D|, χ_{D}) denote the cusp forms of weight k ≥ 2, level |D| and nebentypus χ_{D}. Let f ∈ S_{k}(|D|, χ_{D}) be a primitive cusp form; i.e. a normalized newform that is a common eigenform of all the Hecke operators. Note that there are no oldforms in the space S_{k}(|D|, χ_{D}) since the conductor of χ_{D} is equal to the level |D|.

Let ℂ denote the field of complex numbers. Recall that f has a Fourier expansion

f = ∑_{n=1}^{∞} a(n, f)q^{n}

where a(n, f) ∈ ℂ for all n ≥ 1. Let \overline{ℚ} ⊂ ℂ denote the subfield of algebraic numbers. It is well known that the a(n, f) are algebraic integers. Let ℘ be a prime of \overline{ℚ} of residue characteristic p. f is said to be ordinary at ℘ if a(p, f) is a ℘-adic unit. (Warning: it is possible for f to be ordinary at some ℘ above p but not at another prime ℘′ above p).

Consider the factorization D = D_{1}D_{2} of D into two fundamental discriminants D_{1} and D_{2}. We allow the case D_{1} = D and D_{2} = 1. Let χ_{D_{1}} denote the quadratic character of conductor |D_{1}| and let F_{1} = ℚ(\sqrt{D_{1}}) denote the corresponding quadratic field. Throughout this paper we shall assume that

F_{1} is a real quadratic field.

The cusp form f⊗χ_{D_{1}} is again a primitive element of S_{k}(|D|, χ_{D}). We will call a prime ℘ of \overline{ℚ} a dihedral congruence prime for f with respect to F_{1} if there is a congruence f ≡ f⊗χ_{D_{1}} mod ℘. By a slight abuse of notation we sometimes call the prime p of ℚ lying under ℘ a dihedral congruence prime as well.

In this section we will describe the set of ordinary dihedral congruence primes with respect to F_{1} of the primitive elements of S_{k}(|D|, χ_{D}) in terms of the fundamental units of F_{1}. This generalizes a result of Hida [Hid98] where the case F_{1} = F was treated. We start with the following theorem:

Theorem 2.1. Let p be a prime such that (p, 2D) = 1. Let f be a primitive element of S_{k}(|D|, χ_{D}) and assume that k − 1 ≢ 0 mod p − 1. Suppose that f satisfies a congruence of the form

f ≡ f⊗χ_{D_{1}} mod ℘  (2.2)

for a prime ℘ of \overline{ℚ} over p. Suppose also that f is ordinary at ℘ and that the mod ℘ Galois representation attached to f is absolutely irreducible. Then

• p splits in F_{1},

• q splits in F_{1} for each prime q | D_{2}, and

• p | N_{F_{1}/ℚ}(ϵ_{+}^{k−1} − (−1)^{a}) where ϵ_{+} is a totally positive fundamental unit of F_{1} and a is an integer depending on F_{1} and D_{2} described in (2.10) below.

Proof. We remark that the condition p | N_{F_{1}/ℚ}(ϵ_{+}^{k−1} − (−1)^{a}) is independent of the choice of the totally positive fundamental unit ϵ_{+}.

We make the convention that all number fields K are subfields of \overline{ℚ}. Here and below we will write ℘ for the prime of the number field K which lies under the prime ℘ of \overline{ℚ}. Let K_{f} denote the Hecke field of f. It is the number field generated by the Fourier coefficients of f. Let K_{f,℘} denote the completion of K_{f} at ℘.

For a subfield N of \overline{ℚ} write G_{N} for Gal(\overline{ℚ}/N). Let

ρ_{f} : G_{ℚ} ⟶ GL_{2}(K_{f,℘})

denote the Galois representation attached to f constructed by Eichler and Shimura for k = 2 and by Deligne for k ≥ 2. Note that ρ_{f} is irreducible, unramified outside Dp and is characterized by

Tr(ρ_{f}(Frob_{ℓ})) = a(ℓ, f)

for all primes ℓ | D̸p. Moreover

det(ρ_{f}) = ω_{p}^{k−1} · χ_{D}

where ω_{p} : G_{ℚ} → ℤ_{p}^{×} is the cyclotomic character.

By choosing a G_{ℚ}-stable lattice we may assume that the representation ρ_{f} takes values in GL_{2}(O_{f,℘}) where O_{f,℘} is the ring of integers of K_{f,℘}. By reducing mod ℘ we obtain the mod ℘ representation attached to f, namely

ρ̄_{f} : G_{ℚ} ⟶ GL_{2}(𝔽)

where 𝔽 = O_{f,℘}/℘ is the residue field. By hypothesis, ρ̄_{f} is absolutely irreducible. In particular it is independent of the choice of lattice made above.

The congruence (2.2) along with the absolute irreducibility of ρ̄_{f} implies that

ρ̄_{f} ∼ ρ̄_{f} ⊗ χ_{D_{1}}

where we are now thinking of χ_{D_{1}} as a Galois character with values in ±1 ⊂ 𝔽^{×}. Write H_{1} = G_{F_{1}} for simplicity. Under the absolute irreducibility of ρ̄_{f} it is well known that there is a character ϕ : H_{1} → 𝔽̄^{×} such that

ρ̄_{f} = Ind_{H_{1}}^{G_{ℚ}}(ϕ).

Let σ denote the non-trivial element of Gal(F_{1}/ℚ). Extend σ to an element of G_{ℚ} and continue to denote this element by σ. The condition of absolute irreducibility implies that

ϕ ≠ ϕ^{σ}

where ϕ^{σ} is the conjugate character defined by ϕ^{σ}(h) = ϕ(σhσ^{−1}) for all h ∈ H_{1}. Moreover we have

ρ̄_{f} |_{H_{1}} ∼ (\begin{matrix}ϕ & 0 \\ 0 & ϕ^{σ} \end{matrix}) .  (2.3)

Let G_{p} ⊂ G_{ℚ} denote the decomposition group at ℘ and I_{p} the inertia subgroup. Let 𝔭 be the prime of F_{1} lying under ℘. Let H_{𝔭}, I_{𝔭}, denote the decomposition and inertia subgroups of H_{1} at ℘.

Since a(p, f) is a ℘-adic unit a theorem of Deligne says that the restriction of ρ_{f} to G_{p} is reducible. More precisely we have

ρ_{f} |_{G_{p}} ∼ (\begin{matrix}δ & ∗ \\ 0 & ϵ \end{matrix})  (2.4)

for characters δ, ϵ : G_{p} → O_{f,℘}^{×}. Moreover, it is known that ϵ is unramified. There is an exact sequence

\xymatrix{1 \ar[r] & \overline{F_{1}^{×}F_{∞+}^{×}} \ar[r] & 𝔸_{F_{1}}^{×} \ar[r]^{[ ,F_{1}]} & H_{1}^{ab} \ar[r] & 1}

where [ , F_{1}] is the Artin map. Composing ϕ with the Artin map allows us to think of ϕ as a continuous finite order Hecke character

ϕ : F_{1}^{×}\𝔸_{F_{1}}^{×} ⟶ 𝔽̄^{×}.

Let 𝔠 be the finite part of the conductor of ϕ. Denote by ϕ once more the associated Dirichlet character of conductor 𝔠:

ϕ : (O_{F_{1}}/𝔠)^{×} ⟶ 𝔽̄^{×}.

Now (2.3) shows that

ϕ · ϕ^{σ} = ω_{p}^{k−1} · χ_{D}  (2.5)

on H_{1}. Since we have assumed that k − 1 ≢ 0 mod p − 1 we see that ω_{p}^{k−1} ≠ 1 and therefore at least one of ϕ or ϕ^{σ} is ramified at 𝔭. On the other hand by comparing (2.3) and (2.4) on H_{𝔭} = H_{1} ∩ G_{p} we see that exactly one of ϕ or ϕ^{σ} is ramified at 𝔭. By relabelling if necessary we may assume that ϕ is ramified at 𝔭 and that ϕ^{σ} is unramified at 𝔭.

This forces p to split in F_{1}. Indeed if 𝔭 were inert then, since the conductor of ϕ^{σ} is 𝔠^{σ} and 𝔭 | 𝔠, one would have 𝔭 = 𝔭^{σ}|𝔠^{σ}, contradicting the fact that ϕ^{σ} is unramified at 𝔭.

For a prime λ|𝔠, let us write ϕ_{λ} for the component of the Dirichlet character ϕ at λ. We now describe the possibilities for the components ϕ_{λ} using (2.5) in various cases.

First assume that D_{1} = D so that D_{2} = 1. This case has already been treated by Hida but we repeat the argument here for completeness. In this case (2.5) gives

ϕ · ϕ^{σ} = ω_{p}^{k−1}  (2.6)

on H_{1} since χ_{D} is trivial on H_{1}. We conclude that ϕ_{𝔭} · ϕ_{𝔭}^{σ} = ω_{𝔭}^{k−1}. Since ϕ_{𝔭}^{σ} = 1 we have ϕ_{𝔭} = ω_{𝔭}^{k−1}. Also, since the following diagram commutes

\xymatrix{(O_{F_{1}}/𝔭)^{×} \ar[rr]^{σ}_{∼} \ar[dr]_{ϕ_{𝔭}^{σ}} &  & (O_{F_{1}}/𝔭^{σ})^{×} \ar[dl]^{ϕ_{𝔭^{σ}}} \\  & 𝔽̄^{×}}

we see that ϕ_{𝔭^{σ}} = 1. Thus ϕ = ϕ_{𝔭} · ϕ_{𝔭^{σ}} = ω_{𝔭}^{k−1} has conductor 𝔭 and is given by the map

(O_{F_{1}}/𝔭)^{×} ⟶ 𝔽̄^{×}, x ⟼ x^{k−1}.

Since ϕ(ϵ_{+}) = 1 we see that p|N_{F_{1}/ℚ}(ϵ_{+}^{k−1} − 1), as desired.

Let us now consider the case in which D_{1} ≠ D so that D_{2} ≠ 1. Since χ_{D_{1}} becomes trivial when restricted to H_{1}, (2.5) shows that

ϕ · ϕ^{σ} = ω_{p}^{k−1} · χ_{D_{2}}  (2.7)

on H_{1}. Note that since the conductor of χ_{D_{2}} is D_{2}O_{F_{1}} we obtain ϕ_{𝔭} = ω_{𝔭}^{k−1} and ϕ_{𝔭}^{σ} = 1 as before.

We now prove that q splits in F_{1} for each q | D_{2}. Let N be the prime to p part of the Artin conductor of ρ̄_{f}. It is known (see for instance Carayol [Car89] or Diamond-Taylor [DT94]) that N divides the level of f. Thus if e_{q} denotes the exponent of q dividing N then e_{q} = 1. On the other hand

e_{q} = ∑_{i=0}^{∞} \frac{1}{[I_{0} : I_{i}]} dim(V/V^{I_{i}})

where V is a model for ρ̄_{f} over 𝔽̄_{p} and I_{i} denotes the i^{th} ramification group at q. This shows that V^{I_{0}} ≠ 0 since otherwise e_{q} ≥ 2. Let (a, b) ∈ V denote a non-zero vector fixed by I_{0} = I_{q}. Let 𝔮 be a prime of F_{1} lying over q. Let I_{𝔮} denote the inertia subgroup at 𝔮. Then for h ∈ I_{𝔮} ⊂ H_{1} we have

ρ(h) (\begin{matrix}a \\ b \end{matrix}) = (\begin{matrix}ϕ_{𝔮}(h) & 0 \\ 0 & ϕ_{𝔮}^{σ}(h) \end{matrix}) · (\begin{matrix}a \\ b \end{matrix}) = (\begin{matrix}a \\ b \end{matrix}) .

Since at least one of a or b is non-zero, either ϕ_{𝔮} = 1 or ϕ_{𝔮}^{σ} = 1. Now (2.7) yields the relation

ϕ_{𝔮} · ϕ_{𝔮}^{σ} = χ_{𝔮}  (2.8)

where we have written χ_{𝔮} for (χ_{D_{2}})_{𝔮}. This shows that exactly one of ϕ_{𝔮} or ϕ_{𝔮}^{σ} is trivial. Since the following digram commutes

\xymatrix{(O_{F_{1}}/𝔮)^{×} \ar[rr]^{σ}_{∼} \ar[dr]_{ϕ_{𝔮}^{σ}} &  & (O_{F_{1}}/𝔮^{σ})^{×} \ar[dl]^{ϕ_{𝔮^{σ}}} \\  & 𝔽̄^{×}}

we deduce that q = 𝔮𝔮^{σ} must be split in F_{1}.

The above argument shows that for any prime 𝔮 of F dividing D_{2} either

ϕ_{𝔮} = χ_{𝔮} and ϕ_{𝔮^{σ}} = 1  (2.9)

or

ϕ_{𝔮} = 1 and ϕ_{𝔮^{σ}} = χ_{𝔮^{σ}}.

Now

1 = ϕ(ϵ_{+}) = ϕ_{𝔭}(ϵ_{+}) · ϕ_{𝔭^{σ}}(ϵ_{+}) · ∏_{q|D_{2}} ϕ_{𝔮}(ϵ_{+})ϕ_{𝔮^{σ}}(ϵ_{+}) = ω_{𝔭}^{k−1}(ϵ_{+}) · ∏_{q|D_{2}} χ_{𝔮}(ϵ_{+})

where 𝔮 in these products denotes the prime of F lying over q at which (2.9) holds. This shows that ϵ_{+}^{k−1} ≡ (−1)^{a} mod 𝔭 where

a is the number of primes q dividing D_{2} for which χ_{𝔮}(ϵ_{+}) = −1.  (2.10)

Hence we have p | N_{F_{1}/ℚ}(ϵ_{+}^{k−1} − (−1)^{a}).

As already remarked, Theorem 2.1 was proved by Hida [Hid98] in the situation where χ_{D_{1}} = χ_{D} is the full nebentypus. In this case k is even and so the condition that k − 1 is not a multiple of p − 1 is superfluous since p is an odd prime. Also in this case a = 0 and the conclusion of the theorem p | N_{F_{1}/ℚ}(ϵ_{+}^{k−1} − 1) may be checked to be equivalent to p | N_{F_{1}/ℚ}(ϵ^{k−1} − 1) where ϵ is a fundamental unit for F_{1}.

In [Hid98] Hida also establishes a converse: he shows that if (p, 6D) = 1, p splits in F = ℚ(\sqrt{D}) and p | N_{F/ℚ}(ϵ^{k−1} − 1) where ϵ is a fundamental unit of F, then for every prime ℘ of \overline{ℚ} lying over p, there is a primitive element f ∈ S_{k}(D, χ_{D}) such that f is ordinary at ℘, the mod ℘ Galois representation attached to f is absolutely irreducible and f ≡ f⊗χ_{D} mod ℘. We now establish a converse to Theorem 2.1. Our result includes Hida’s converse. Like Hida we avoid the prime p = 3 when k = 2; in fact a study of the space S_{2}(37, χ_{37}) shows that the converse is false for p = 3 and k = 2.

Theorem 2.11. Let p be a prime such that (p, 2D) = 1. Let k ≥ 2 be such that k − 1 ≢ 0 mod p − 1 and assume that p ≠ 3 if k = 2. Suppose that

• p splits in F_{1},

• q splits in F_{1} for each prime q | D_{2}, and

• p | N_{F_{1}/ℚ}(ϵ_{+}^{k−1} − (−1)^{a}) where ϵ_{+} is a totally positive fundamental unit of F_{1} and a is defined as in (2.10). For each prime ℘ of \overline{ℚ} lying over p there is a primitive element f ∈ S_{k}(|D|, χ_{D}) such that f is ordinary at ℘, the mod ℘ Galois representation attached to f is absolutely irreducible and f satisfies the congruence

f ≡ f⊗χ_{D_{1}} mod ℘.

Proof. The proof is similar to the one given in [Hid98] and uses Hida’s theory of families for ordinary forms.

Fix a prime 𝔭 be any prime of \overline{ℚ} lying over p. Let

𝔠 = 𝔭 ∏_{q|D_{2}} 𝔮,

where 𝔮 in this product denotes a prime of F_{1} over q and 𝔭 is the prime of F_{1} lying under ℘. Let 𝔽_{p} denote the field with p elements. Consider the Dirichlet character

ϕ_{f} : (O/𝔠)^{×} → 𝔽̄_{p}^{×}

whose components are ϕ_{𝔭} = ω_{𝔭}^{k−1} ≠ 1 and ϕ_{𝔮} = χ_{𝔮} for each 𝔮 dividing 𝔠. Under the hypothesis on ϵ_{+} we have ϕ_{f}(ϵ_{+}) = 1.

We claim that by choosing ϕ_{∞} to be the sign character ϕ_{∞,ν} at exactly one infinite place ν of F_{1} we can ensure that

ϕ_{f}(u)ϕ_{∞}(u) = 1  (2.12)

for all units u of F_{1}. Let ϵ be a fundamental unit of F_{1}. If ϵ is totally positive then ϵ ∈ ⟨ϵ_{+}⟩, hence ϕ_{f}(ϵ) = 1. Then for any infinite place ν of F_{1} we have ϕ_{f}(ϵ)ϕ_{∞,ν}(ϵ) = 1. If ϵ is not totally positive then ϵ^{2} ∈ ⟨ϵ_{+}⟩, hence ϕ_{f}(ϵ) ∈ {±1}. Choose ν such that ϕ_{∞,ν}(ϵ) = ϕ_{f}(ϵ). Furthermore we have ϕ_{f}(−1) = (−1)^{k−1}χ_{D_{2}}(−1), hence ϕ_{f}(−1)ϕ_{∞,ν}(−1) = 1. Since ϵ, −1 generate the units of F_{1} this establishes the claim.

Let Ô_{F_{1}} denote the pro-finite completion of O_{F_{1}} and let let U(𝔠) = 1 + 𝔠Ô_{F_{1}} ⊂ Ô_{F_{1}}^{×}. Let F_{1,∞}^{×} = (ℝ^{×})^{2} and let F_{1,∞+}^{×} = (ℝ_{+}^{×})^{2}. Then we have an inclusion of finite groups

O_{F_{1}}^{×}\(Ô_{F_{1}}^{×} × F_{1,∞}^{×})/U(𝔠) × F_{1,∞+}^{×} ↪ F_{1}^{×}\𝔸_{F_{1}}^{×}/U(𝔠) × F_{1,∞+}^{×}  (2.13)

where the index is the class number of F_{1}. The condition (2.12) shows that the characters (ϕ_{f}, ϕ_{∞}) give rise to a character on the smaller subgroup in (2.13). Extending this character to a character of the larger group we obtain a finite order Hecke character ϕ : F_{1}^{×}\𝔸_{F_{1}}^{×} ⟶ 𝔽̄_{p}^{×} of conductor 𝔠ν. Now let

f_{1} = ∑_{𝔞⊂O} ϕ(𝔞) q^{N_{F_{1}/ℚ}(𝔞)}

be the theta series attached to ϕ where ϕ here is thought of as a multiplicative function on the integral ideals of F_{1} which vanishes on those ideals that meet 𝔠. Then (see [Miy89, Theorem 4.8.3])

f_{1} ∈ S_{1}(|D|p, χ_{D}ω_{p}^{k−1}).

Moreover ρ̄_{f_{1}} ∼ ρ̄_{f_{1}} ⊗ χ_{D_{1}} so that f_{1} ≡ f_{1}⊗χ_{D_{1}} mod ℘.

Now a(p, f_{1}) = ϕ(𝔭) + ϕ(𝔭^{σ}) where σ is the non-trivial element in Gal(F_{1}/ℚ). Since ϕ(𝔭) = 0 and ϕ(𝔭^{σ}) is a root of unity we see that f_{1} is ordinary at ℘. Therefore f_{1} sits in a Hida family (see, for example, [Hid93, Chapter 7] where the prime power level is treated) whose weight l member is a ℘-ordinary element

f_{l} ∈ S_{l}(|D|p, χ_{D}ω_{p}^{k−l}).

The form f = f_{k} ∈ S_{k}(|D|p, χ_{D}) then almost satisfies the desired properties of the form in the conclusion of the theorem. It is ℘-ordinary by construction, ρ̄_{f} is absolutely irreducible since ϕ ≠ ϕ^{σ}, and f ≡ f⊗χ_{D_{1}} mod ℘ since f and f_{1} have the same mod ℘ Galois representation.

We say ‘almost’ in the sentence above since f has p in its level. But, as we now show, we may assume that f ∈ S_{k}(|D|, χ_{D}) by replacing f with a ℘-ordinary primitive form g ∈ S_{k}(|D|, χ_{D}) with the same mod ℘ Galois representation as f. Indeed, when k ≥ 3, there is an isomorphism

S_{k}^{ord}(|D|p, χ_{D}) = S_{k}^{ord}(|D|, χ_{D}),

where the superscript denotes the subspace of ℘-ordinary forms. This isomorphism is well known: it follows from the fact that a (p-new) primitive form g ∈ S_{k}(|D|p, χ_{D}) has p^{th} Fourier coefficient satisfying a(p, g) = ±p^{(k−2)/2} (see [Miy89, Theorem 4.6.17]) and so is not ordinary at ℘ if k ≥ 3.

When k = 2 the above argument fails. We use instead the following argument which will require the additional hypothesis that p ≠ 3. Recall that ϕ = ω_{𝔭} and ϕ^{σ} = 1 on I_{𝔭}. Thus (2.3) shows that

ρ̄_{f} |_{I_{p}} ∼ (\begin{matrix}ω_{p} & 0 \\ 0 & 1 \end{matrix})  (2.14)

since I_{p} = I_{𝔭} ⊂ H_{1}. The behaviour of ρ̄_{f} on I_{p} described by (2.14) above along with [Ser87, Proposition 3] shows that the Serre weight of ρ̄_{f} is k(ρ̄_{f}) = 2. By [Rib94, Theorem 3.3], ρ̄_{f} arises from S_{2}(Γ_{1}(D)). Actually Ribet’s theorem requires p ≥ 5, but the part of the theorem that we need (namely the equality of the sets N_{1} and N_{3} in his notation) is valid even for p = 3. Now [Car89, Proposition 3] (or [Rib94, Theorem 1.3]) shows that ρ̄_{f} arises from a form g ∈ S_{2}(D, χ_{D}). The result of Carayol is valid only for p ≥ 5 and it is here that we use the hypothesis that p ≠ 3. Note that the form g continues to be ℘-ordinary since if it were not a theorem of Fontaine [Edi92, Theorem 2.6] would say that ρ̄_{g} is irreducible on G_{p} whereas we already know that ρ̄_{g} ∼ ρ̄_{f} is reducible on G_{p} = H_{𝔭} ⊂ H_{1}.

Here is some numerical data to illustrate Theorems 2.1 and 2.11. This data was compiled with the aid of the modular forms calculator HECKE written by W. Stein,

available at http://modular.fas.harvard.edu.

We restrict to the case where D = D_{1}D_{2} is the product of two primes congruent to 1 modulo 4 with χ_{D_{1}}(D_{2}) = 1. Thus D is positive and therefore k ≥ 2 is even.

Column 2 lists the odd primes which split (or ramify) in ℚ(\sqrt{D}) and which divide the norm of ϵ_{+,D}^{k−1} − 1 for a totally positive fundamental unit ϵ_{+,D} of ℚ(\sqrt{D}). Let ϵ_{+,D_{i}} be an analogous unit for ℚ(\sqrt{D_{i}}) for i = 1, 2. Since D_{i} ≡ 1 mod 4, the norm of a fundamental unit of ℚ(\sqrt{D_{i}}) is −1. Consequently ϵ_{+,D_{i}} is a square and a = 0 in (2.10) above. Thus, in Columns 3 and 4 we list the odd primes which split (or ramify) in ℚ(\sqrt{D_{i}}) and which divide the norm of ϵ_{+,D_{i}}^{k−1} −1 for i = 1, 2. Column 5 lists the primes dividing the discriminant of the Hecke algebra.

As predicted by our theorems every prime (which is prime to 6D) that occurs in Columns 2-4 also occurs in Column 5 as a dihedral congruence prime (actually the Hecke discriminant only shows that it occurs as a congruence prime).

Column 6 gives the dimension of S_{k}(D, χ_{D}) partitioned according to G_{ℚ}-orbits of primitive forms. When there is only one orbit then the dihedral congruence primes arise because of the phenomenon of extra twisting as we shall explain in Section 3.

Level  ϵ_{+,D} ϵ_{+,D_{1}} ϵ_{+,D_{2}}  Hecke Discriminant  Orbits
145 = 5 · 29  3  5  2 3 5 17  12 = 4 + 4 + 4
205 = 5 · 41  2 13 41 59 853  20 = 20
221 = 13 · 17  3  2 3 5 7 13 733  20 = 4 + 4 + 6 + 6
305 = 5 · 61  3 13  2 3 7 13 23 61 1459 35201  28 = 28
377 = 13 · 29  3  5  2 3 5 29 109 2089 24551  32 = 32
445 = 5 · 89 3 7  5  2 3 5 7 4057 10187892099374809 44 = 44
689 = 13 · 53  3  7  2 3 7 13 107 1123 p_{689}  60 = 60
793 = 13 · 61 3  3  3 13 2 3 13 61 691 9525232953521 p_{789} 68 = 8 + 60
901 = 17 · 53 3 5  7  2 3 5 7 11 3923 p_{901} q_{901}  80 = 8 + 72

Dihedral congruence primes for S_{2}(D, χ_{D}).

Level  ϵ_{+,D} ϵ_{+,D_{1}}  ϵ_{+,D_{2}}  Hecke Discriminant  Orbits
145 = 5 · 29 3 193  5 7  2 3 5 7 193 p_{145}  44 = 44
205 = 5 · 41  4099  2 3 5 11 41 1259 2273 2447 4099 p_{205} q_{205} 60 = 60
221 = 13 · 17  3  67  2 3 7 13 19 67 p_{221}  60 = 60
305 = 5 · 61  3 13 127  2 3 5 11 13 61 89 127 n_{305}  92 = 92
377 = 13 · 29  3  5 7  2 3 5 7 29 467 2027 n_{377}  104 = 104
445 = 5 · 89 3 7 37  5 1000003  2 3 5 7 17 37 73 1000003 n_{445}  132 = 132
689 = 13 · 53 211  3  7 13  2 3 5 7 13 31 41 67 211 81569 137491 n_{689} 188 = 188

Dihedral congruence primes for S_{4}(D, χ_{D}).

Some large factors (denoted by p or q if they are known to be prime and by n if they are possibly composite) appear in the tables above. These factors are not important for our purposes but for completeness we list them:

p_{145} = 3423792162108426353056857548769671099
p_{689} = 1205496111864022253479367
p_{793} = 2324195671211370589
p_{901} = 129196963641464623
q_{901} = 3328968195415999986013
p_{205} = 22723708823
q_{205} = 10385895568903051794474791759280258119944561
p_{221} = 1111523239622632157524576477637325711959616179045094404902744467611
n_{305} = 443676377376494576412547377680235319562033952734766599190542418559674471632457873503/

2837424267253963633270020217800106601076493292511

n_{377} = 187620235161328728324464813781092054323517740380798079362035839768081101025161812558/

333395576383481646527023327932899249379830433647245293

n_{445} = 322443621973381479228334689663960724759803165509455916126140363537120347931114164815/

587217927678015411774335019525264133118010579825048350477643350341530765668934687352/ 182447455102808460902080151411156518591825207901933674167379977678963595990473416261/ 826061024951872852529048807677789891280944699586502529618779886298028491600257131855/ 3991880761030752020723081

n_{689} = 607032771483018670674948503661874951176272369487928437133644278567895112391058938993/

578356081283803175523883714504638851738428762525802070403608958918988949017429708613/ 516955449668899409076858297085496580573387780409951108015104989423622202209678906319/ 59968430130145338353915072457061381.

Two final remarks about the numerical data are in order. First, within the limits of the tables, Theorem 2.11 is valid when p = 3 and k = 2. However, as already mentioned, there are counterexamples, for instance when the level is 37. Second, although in this paper we have not investigated what happens at the odd primes dividing D, the data above shows that analogs of our results are likely to be true for such primes as well.

In Theorems 2.1 and 2.11 we assumed that k − 1 ≢ 0 mod p − 1. This condition is vacuously true if D > 0 since in this case k is even and p is odd. When k is odd this condition may fail for some odd primes p smaller than k. However the following proposition shows that the conclusions of Theorems 2.1 and 2.11 remain essentially the same even in this case.

Proposition 2.15. Let p be an odd prime with (p, 2D) = 1. Let ℘ be a prime of \overline{ℚ} lying over p. Let k ≥ 2 be an integer satisfying

k − 1 ≡ 0 mod p − 1.

Then, there is a primitive ℘-ordinary form f ∈ S_{k}(|D|, χ_{D}) with ρ̄_{f} absolutely irreducible and f ≡ f⊗χ_{D_{1}} mod ℘ if and only if

• p splits in F_{1},

• q splits in F_{1} for each prime q|D_{2}

• the integer a defined in (2.10) is even, and,

• the condition (2.16) below holds.

Proof. The proof is similar to the proofs of Theorems 2.1 and 2.11 and so we only outline the differences.

Assume that f is a primitive ℘-ordinary form satisfying the hypotheses of the proposition. Then as in the proof of Theorem 2.1 we obtain a Hecke character ϕ of F_{1} but this time both its components ϕ_{𝔭} = ω_{p}^{k−1} = 1 and ϕ_{𝔭^{σ}} are trivial. In particular we cannot use our old argument to show that p splits in F_{1}. But we can work around this. Let f_{1} be the theta series attached to ϕ. Since ϕ has prime to p conductor 𝔠ν with 𝔠 = ∏ 𝔮, the form f_{1} ∈ S_{k}(|D|, χ_{D}) has level prime to p. Assume towards a contradiction that p is inert in F_{1}. Let ρ_{f_{1}} be the Galois representation attached to f_{1} by Deligne and Serre [DS74]. Since both ρ̄_{f_{1}} and ρ̄_{f} = Ind(ϕ) have the same traces on the Frobenius element at ℓ | D̸p, namely

a(ℓ, f_{1}) = \begin{cases}ϕ(𝔩) + ϕ(𝔩^{σ}) & if ℓ = 𝔩𝔩^{σ} splits in F_{1} \\ 0 & if ℓ = 𝔩 is inert in F_{1}, \end{cases}

we have that ρ̄_{f_{1}} ∼ ρ̄_{f}. Since ϕ has prime to p conductor, ρ̄_{f} is unramified at p. Since f is ℘-ordinary we have Tr(ρ̄_{f}(Frob_{p})) ≠ 0. On the other hand this trace is a(p, f_{1}) = 0 since we assumed that p was inert in F_{1}. This is a contradiction, so p must split in F_{1}.

The argument that each q|D_{2} splits in F_{1} proceeds as before, and the computation just above (2.10) shows that (−1)^{a} = 1 so that a must be even.

Since p = 𝔭𝔭^{σ} in F_{1} we have a(p, f_{1}) = ϕ(𝔭) + ϕ(𝔭^{σ}) ≠ 0. Thus, since ϕ is a character of the larger group in (2.13), with 𝔠 = ∏_{q|D_{2}} 𝔮 now, the following condition holds automatically:

The character ∏_{q|D_{2}} χ_{𝔮} · sgn_{ν} extends to a character ϕ of the larger group in (2.13) satisfying ϕ(𝔭) + ϕ(𝔭^{σ}) ≠ 0.

(2.16)

Conversely, suppose that p and each q|D_{2} splits in F_{1}, and that a is even. Then as in the proof of Theorem 2.11 we may construct a character ϕ_{f} of conductor 𝔠 = ∏_{q|D_{2}} 𝔮 whose components are ϕ_{𝔮} = χ_{𝔮} for each 𝔮 dividing 𝔠. This time we set ϕ_{𝔭} = 1. As before we may choose a character ϕ_{∞} of F_{1,∞}^{×} such that the pair (ϕ_{f}, ϕ_{∞}) extends to a Hecke character ϕ of F_{1}. Again the associated theta series f_{1} has level prime to p, that is f_{1} ∈ S_{1}(|D|, χ_{D}).

Since p splits in F_{1} by assumption we have a(p, f_{1}) = ϕ(𝔭)+ϕ(𝔭^{σ}). The condition (2.16) implies that we can choose ϕ so that the corresponding theta series f_{1} is ℘- ordinary.

The rest of the proof, namely considering the weight k member of the Hida family to which f_{1} belongs and then lowering p from its level, proceeds exactly as before to yield a form f ∈ S_{k}(|D|, χ_{D}) with the desired properties.

3. Extra Twists

In this section we describe the relevance of extra twists to the construction of dihedral congruence primes. Let us fix a primitive form f ∈ S_{k}(|D|, χ_{D}) of weight k ≥ 2 and let K_{f} denote the Hecke field of f.

Lemma 3.1. Let γ ∈ Aut(K_{f}) be an element of order 2. Let K_{f}^{γ} denote the fixed field of γ.

1. If ℘ is a ramified prime in K_{f}/K_{f}^{γ} then

f^{γ} ≡ f mod ℘.  (3.2)

2. Conversely if ℘ satisfies (3.2) and the ring generated by the Fourier coefficients of f is the maximal order in K_{f} then ℘ is a ramified prime in K_{f}/K_{f}^{γ}.

Proof. Let O_{f} denote the maximal order of K_{f} and let O(f) ⊂ O_{f} denote the order generated by the a(n, f). Let O_{f}^{γ} denote the maximal order of K_{f}^{γ}. If ℘ is ramified in K_{f}/K_{f}^{γ} then γ ∈ I(℘, K_{f}/K_{f}^{γ}), the inertia subgroup at ℘. Thus

γ(x) ≡ x mod ℘  (3.3)

for all x ∈ O_{f}. In particular this holds for all x = a(n, f) ∈ O(f) and (3.2) follows. This proves the first statement.

Conversely, if ℘ satisfies (3.2) then (3.3) holds for all x ∈ O(f) and, under our hypothesis on O(f), for all x ∈ O_{f}. This implies that γ fixes ℘ so that ℘ is not a split prime. Moreover, ℘ is not an inert prime, for if it were one would have the absurdity that γ is the generator of the extension of residue fields yet is trivial mod ℘. Thus ℘ must be ramified.

Let γ ∈ Aut(K_{f}) be of arbitrary order and let χ be a Dirichlet character that takes values in K_{f}. We say that f has an extra twist or simply a twist by (γ, χ) if

f^{γ} = f⊗χ.

The prototype of an extra twist for f is (c, χ_{D}) where c denotes complex conjugation on the CM field K_{f} and χ_{D} is the nebentypus.

If f has a twist by (γ, χ) and

f has no complex multiplication  (3.4)

then χ is uniquely determined by γ, in which case we denote it by χ_{γ}. We assume that f satisfies the condition (3.4) for the rest of this paper.

Now suppose f has a twist by (γ, χ_{γ}). Then

χ_{D}^{γ} = χ_{D} · χ_{γ}^{2}.

Since χ_{D} takes values in {±1} we see that χ_{D}^{γ} = χ_{D}. Thus χ_{γ} is necessarily a quadratic character and γ has order 2. It may be checked that χ_{γ} = χ_{D_{1}} for a unique fundamental discriminant D_{1} such that D = D_{1}D_{2} is a product of fundamental discriminants.

The first part of Lemma 3.1 shows that each prime ℘ that divides the different of K_{f}/K_{f}^{γ} is a dihedral congruence prime for f with respect to F_{1}. The moral is that extra twists for f are directly responsible for the occurrence of dihedral congruence primes for f. In particular when there is only one Galois orbit in S_{k}(|D|, χ_{D}) then all dihedral congruence primes arise from extra twists (see the tables in the previous section for examples of spaces with one orbit).

4. Class fields of real quadratic fields

In this section we use Shimura’s method (see [Shi71, Chapter 7] and [Shi72]) to generate (ray) class fields of real quadratic fields explicitly in terms of torsion points on certain abelian varieties defined over these fields.

Shimura’s method has already been studied in various different contexts by other authors [DY73], [Koi76], [Oht77] and is in principle well understood. However, a new feature of our study is that, by considering in a systematic way congruences with twists that are not necessarily the full nebentypus we can, at least in principle, generate explicitly infinitely many class fields of the same real quadratic field by this method.

We start with some class field theoretic preliminaries. Let K denote a real quadratic field. Let 𝔪 be a modulus for K. Recall that the ray class field modulo 𝔪 is a finite abelian extension RCF(𝔪) of K of conductor dividing 𝔪, which is maximal with respect to this property: if L is another finite abelian extension of K whose conductor divides 𝔪 then L ⊂ RCF(𝔪).

Write 𝔪 = 𝔪_{f}𝔪_{∞} where 𝔪_{f} is supported at finite places of K and 𝔪_{∞} is supported at infinite places of K. Let

K_{𝔪} = {a/b | a, b ∈ O_{K}, (a), (b) relatively prime to 𝔪_{f}} ,
K_{𝔪,1} = {α ∈ K_{𝔪} | α ≡ 1 (mod 𝔪)} .

Here the condition α ≡ 1 (mod 𝔪) means the following: for each finite 𝔭 dividing 𝔪_{f}, we require that v_{𝔭}(α − 1) ≥ v_{𝔭}(𝔪_{f}), and for each real prime 𝔭 dividing 𝔪_{∞}, we require α to be positive at this place.

Let I_{K} denote the group of fractional ideals of K. Write I_{K}^{𝔪} for the subgroup of I_{K} generated by all the primes not in 𝔪_{f}. Let P_{K} denote the subgroup of principal ideals of K. Let i : K^{×} → P_{K} denote the map x ↦ (x). The class group of K is Cl(K) = I_{K}/i(K^{×}); in fact one may check that Cl(K) = I_{K}^{𝔪}/i(K_{𝔪}) for any modulus 𝔪. The ray class group of conductor 𝔪 is

RCG(𝔪) = \frac{I_{K}^{𝔪}}{i(K_{𝔪,1})}.

There is an exact sequence

1 ⟶ \frac{i(K_{𝔪})}{i(K_{𝔪,1})} ⟶ RCG(𝔪) ⟶ Cl(K) ⟶ 1.

This shows that the class number h_{K} of K is a factor of the order of the ray class group of conductor 𝔪. To compute the full order consider the following commutative diagram with exact rows

\xymatrix{1 \ar[r] & U_{𝔪,1} \ar[r] \ar@{^{(}->}[d] & K_{𝔪,1} \ar[r] \ar@{^{(}->}[d] & i(K_{𝔪,1}) \ar[r] \ar@{^{(}->}[d] & 1 \\ 1 \ar[r] & O_{K}^{×} \ar[r] & K_{𝔪} \ar[r] & i(K_{𝔪}) \ar[r] & 1}

where U_{𝔪,1} = O_{K}^{×} ∩ K_{𝔪,1}. By the snake lemma we get the exact sequence:

1 ⟶ \frac{O_{K}^{×}}{U_{𝔪,1}} ⟶ \frac{K_{𝔪}}{K_{𝔪,1}} ⟶ \frac{i(K_{𝔪})}{i(K_{𝔪,1})} ⟶ 1.

Now there is an isomorphism:

\frac{K_{𝔪}}{K_{𝔪,1}} =^{∼} (O_{K}/𝔪_{f})^{×} × ∏_{𝔭_{∞}|𝔪_{∞}} ℝ^{×}/ℝ_{+}^{×} =: A(𝔪).

Putting things together we see that the order of the ray class group modulo 𝔪, which is also the degree over K of the ray class field of K modulo 𝔪, is

|RCG(𝔪)| = \frac{2^{r}Mh_{K}}{u}  (4.1)

where r is the number of infinite places in 𝔪_{∞}, M is the order of (O_{K}/𝔪_{f})^{×}, and u is the order of the subgroup of A(𝔪) generated by the units.

Let now D be the discriminant of a quadratic field. Write D = D_{1}D_{2} and assume as usual that F_{1} := ℚ(\sqrt{D_{1}}) is a real quadratic field. Let f ∈ S_{2}(|D|, χ_{D}) be a primitive form and suppose that f has a twist by (γ, χ_{D_{1}}) where γ denotes an element of Aut(K_{f}) and χ_{D_{1}} is the quadratic character of conductor D_{1}. We assume that f satisfies the condition (3.4).

Let ℘ be a prime of \overline{ℚ} which divides the different of K_{f}/K_{f}^{γ}. Then as explained at the end of Section 3, ℘ is a dihedral congruence prime for f as in (2.2). Let p be the residue characteristic of ℘. To put us in the context of Theorem 2.1 assume that:

• (p, 2D) = 1

• f is ordinary at ℘

• ρ̄_{f} is absolutely irreducible. The proof of Theorem 2.1 shows that we may then write

ρ̄_{f} = Ind_{H_{1}}^{G_{ℚ}}ϕ

for a character ϕ : H_{1} → 𝔽̄_{p}^{×} with associated Dirichlet character ϕ of the form

ϕ_{f} = ω_{𝔭} · ∏_{q|D_{2}} χ_{𝔮}

where 𝔭 is a split prime of F_{1} lying under ℘ and 𝔮 is a split prime of F_{1} lying over q|D_{2}. Moreover since ϕ_{f}(−1) = −1 we see that ϕ_{∞}(−1) = −1 so that ϕ_{∞} is trivial at one infinite place ν of F_{1} and non-trivial at the other. Thus ϕ cuts out a cyclic extension N of F_{1} of degree

[N : F_{1}] = p − 1

and conductor 𝔭(∏ 𝔮)ν.

Let ϵ_{ν} be the fundamental unit of F_{1} with respect to ν. For a prime ideal λ of F_{1} let d_{λ,ν} denote the order of ϵ_{ν} modulo λ. Let 𝔪_{f} = 𝔠 = 𝔭 ∏ 𝔮 and 𝔪_{∞} = ν and let 𝔪 = 𝔪_{f}𝔪_{∞}. Using the formula (4.1) above one computes that the degree of RCF(𝔪), the ray class field of F_{1} modulo 𝔪, over F_{1}, is:

[RCF(𝔠ν) : F_{1}] = \frac{(p − 1) · ∏_{q|D_{2}}(q − 1) · h_{F_{1}}}{lcm_{λ|𝔠}(d_{λ,ν})} .  (4.2)

Let A_{f} denote the abelian variety attached to f. A_{f} is defined over ℚ, is simple over ℚ, and comes equipped with a map

ι : K_{f} ↪ End(A_{f}) ⊗ ℚ.

Here ι(a(n, f)) acts as the n^{th} Hecke operator on A_{f} and so is defined over ℚ. Let w = w_{D_{1}} denote the Atkin-Lehner involution of level D_{1}. Note that

f|_{w} = c · f⊗χ_{D_{1}}

for a constant c of absolute value 1 which lies in F_{1}. Since f has a twist by (γ, χ_{D_{1}}) we see that w induces an involution of A_{f} which is defined over F_{1}. The Atkin-Lehner action and the Hecke action do not commute but satisfy instead the well-known relation

w ∘ ι(a(n, f)) = ι(a(n, f)^{γ}) ∘ w.  (4.3)

Now set B_{f} = (1+w)A_{f}. Then B_{f} is an abelian sub-variety of A_{f} defined over F_{1} of half the dimension of A_{f}. Further since w^{σ} = −w we see that B_{f}^{σ} = (1 − w)A_{f}. Thus A_{f} = B_{f} + B_{f}^{σ} and the intersection of B_{f} and B_{f}^{σ} is contained in A_{f}[2]. Moreover (4.3) shows that there is a map

ι : K_{f}^{γ} ↪ End(B_{f})

and similarly for B_{f}^{σ}.

Write ℘ again for the the prime of K_{f} lying under the prime ℘ of \overline{ℚ}. Since ℘ is prime to 2, one can write

A_{f}[℘] = R ⊕ S

where

R = B_{f} ∩ A_{f}[℘],  S = B_{f}^{σ} ∩ A_{f}[℘].

Let O_{f}^{γ} be the ring of integers of K_{f}^{γ}. Assume that O(f) = O_{f} is the full ring of integers of K_{f}. One can always replace A_{f} by an isogenous abelian variety which satisfies this assumption (see the discussion in [Shi71, pages 198-199]). Then one may check that R =^{∼} S =^{∼} O_{f}^{γ}/℘ =^{∼} O_{f}/℘. Moreover we see that G_{F_{1}} acts on R by ϕ and on S by ϕ^{σ}. Fix P_{f} respectively Q_{f}, be a generator of R, respectively S, as an O^{γ}/℘-module. Then, for each h ∈ G_{F_{1}}, we have

P_{f}^{h} = ϕ(h) · P_{f}  Q_{f}^{h} = ϕ^{σ}(h) · Q_{f}.

We summarize the discussion above by stating the following theorem.

Theorem 4.4. Let the notation and assumptions be as above. The class field N of conductor 𝔭(∏ 𝔮)ν and degree p − 1 can be explicitly generated over F_{1} by adjoining the coordinates of the torsion point P_{f} to F_{1}.

In general N is too small to be the ray class field of conductor 𝔭(∏ 𝔮)ν. However we have the following theorem:

Theorem 4.5. Let the notation and assumptions be as in Theorem 4.4. Suppose also that F_{1} has class number 1, that D_{2} is divisible by only one prime q, and that the fundamental unit ϵ_{ν} ∈ F_{1} generates all of (O_{F_{1}}/𝔮)^{×}. Then N = F_{1}(P_{f}) is the ray class field modulo 𝔭𝔮ν.

Proof. Note that 1 = ϕ_{f}(ϵ_{ν})ϕ_{∞}(ϵ_{ν}) ≡ ±ϵ_{ν} mod 𝔭. This says that d_{𝔭,ν} ∈ {1, 2}. Since h_{F_{1}} = 1 and d_{𝔮,ν} = q − 1 the formula (4.2) shows that

[RCF(𝔭𝔮ν) : F_{1}] = p − 1 = [N : F_{1}].

Since N ⊂ RCF(𝔭𝔮ν) they are equal.

Theorems 4.4 and 4.5 show that, at least in principle, it is possible to generate infinitely many (ray) class fields of the same real quadratic field F_{1} by adjoining torsion points on modular abelian varieties.

For the purposes of illustration we summarize the method in a concrete situation. Let F_{1} be a fixed real quadratic field of prime discriminant D_{1} congruent to 1

modulo 4. This is compatible with the class number 1 hypothesis made in Theorem 4.5, the only known obstruction to the class number being 1 coming from genus theory. Let D_{2} be a prime congruent to 1 modulo 4 such that there is an f ∈ S_{2}(D, χ_{D}) of level D = D_{1}D_{2} such that

f has a twist by (γ, χ_{D_{1}}) for some γ ∈ Aut(K_{f}).  (4.6)

Then for each prime ℘ of \overline{ℚ} dividing the different of K_{f}/K_{f}^{γ} such that • p is prime to 2D,

• a(p, f) is a ℘-adic unit, and,

• ρ̄_{f} is absolutely irreducible, one can generate a class field of F_{1} modulo 𝔭𝔮ν by adjoining a ℘-torsion point on the modular abelian variety B_{f}. Here 𝔭 and p are the primes of F_{1} and ℚ lying under ℘, 𝔮 is a split prime above D_{2} and ν is an infinite place of F_{1}.

If F_{1} has class number 1 and D_{2} is chosen so that

⟨ϵ_{ν}⟩ ↠ (O_{F_{1}}/𝔮)^{×}  (4.7)

is surjective then the above class field is the ray class field modulo 𝔭𝔮ν.

Corollary 4.8. Let the notation be as above. If there are infinitely many pairs (D_{2}, f) satisfying (4.6) above, then infinitely many class fields of F_{1} can be generated by adjoining torsion points on modular abelian varieties. If F_{1} has trivial class number and there are infinitely many pairs (D_{2}, f) satisfying (4.6) and (4.7) then infinitely many of these class fields are ray class fields.

We now give an example to illustrate Corollary 4.8. Let F_{1} = ℚ(\sqrt{13}). Then h_{F_{1}} = 1. Further there is only one dihedral congruence prime for F_{1} namely p = 3. Thus [N : F_{1}] = 2. Let 𝔭_{3} be a split prime of F_{1} lying above p = 3. For the first twelve primes D_{2} satisfying the conditions of Corollary 4.8 we list in Column 2 the order of the class field of F_{1} of conductor 𝔠ν, where 𝔠 = 𝔭_{3}𝔮_{D_{2}} with 𝔮_{D_{2}} a split prime of F_{1} lying above D_{2}. In case this number depends on the infinite place ν we write one value for each infinite place. In Column 3 we indicate whether the class field N is the ray class field. In Column 4 we give the dimension of the space S_{2}(D, χ_{D}) partitioned according to Galois orbits. When there is just one orbit then the (ray) class field N can be generated over F_{1} by Shimura’s method described above.

Conductor 𝔠 |RCF(𝔠ν)| N = RCF(𝔠ν)? Orbits
𝔭_{3}𝔮_{17}  2  Yes  20 = 4+4+6+6
𝔭_{3}𝔮_{29}  2  Yes  32 = 32
𝔭_{3}𝔮_{53}  4/8  No  60 = 60
𝔭_{3}𝔮_{61}  4/8  No  68 = 8+60
𝔭_{3}𝔮_{101}  4/8  No  116 = 116
𝔭_{3}𝔮_{113}  2  Yes  132 = 132
𝔭_{3}𝔮_{157}  2  Yes  180 = 180
𝔭_{3}𝔮_{173}  4/8  No  200 = 200
𝔭_{3}𝔮_{181}  2  Yes  208 = 208
𝔭_{3}𝔮_{233}  4/8  No  272 = 272
𝔭_{3}𝔮_{257}  4/8  No  300 = 300
𝔭_{3}𝔮_{269}  2  Yes  312 = 312
Modular (Ray) Class Fields of F_{1} = ℚ(\sqrt{13}).

5. Higher weights

One of the limitations of Shimura’s method is that the underlying modular form f should have weight 2 since only in this case can one associate an abelian variety to f. As a consequence the set of split primes 𝔭 that occur in the conductors of the class fields is finite since the residue characteristic of these primes must divide N_{F_{1}/ℚ}(ϵ_{+} ± 1) where ϵ_{+} is a totally positive fundamental unit of F_{1}.

In [Shi71, page 211] Shimura observes that there is some numerical evidence connecting cusp forms of higher weight to the class field theory of real quadratic fields. To the best of our knowledge this connection has not been pursued in the literature, except for the recent work of Hida [Hid98], where, incidentally, the author was not directly concerned with generating class fields by torsion points on modular abelian varieties.

In this section we observe that Shimura’s weight 2 method can be combined with a key idea introduced into the subject in [Hid98], namely the use of Hida families, allowing us to consider f ∈ S_{k}(|D|, χ_{D}) for arbitrary weight k ≥ 2. This allows us, in principle, to construct (ray) class fields whose conductors are divisible by 𝔭 for an infinite set of split primes 𝔭. These primes are essentially the split primes 𝔭 whose residue characteristics divide N_{F_{1}/ℚ}(ϵ_{+}^{k−1}±1) as k varies through all integers. We insert the modifier ‘in principle’ in the sentence above since the success of the method depends as usual on the occurrence of extra twists.

We start by illustrating the method in the situation D_{1} = D so that F_{1} = F = ℚ(\sqrt{D}). That is we first consider dihedral congruence primes with respect to the full nebentypus.

Let p be a prime such that (p, 2D) = 1. Choose 𝔭|p and ν|∞ places of F such that 𝔭 is a split prime. Assume that d_{𝔭,ν} is odd. Write d_{𝔭,ν} = k − 1 for an even integer k ≥ 2. Then ϵ_{ν}^{k−1} ≡ 1 mod 𝔭.

Let ℘ be a prime of \overline{ℚ} lying over 𝔭. Then by Theorem 2.11 (in this case already proved by Hida) there exists a primitive modular form f_{k} ∈ S_{k}(D, χ_{D}) with

f_{k} ≡ f_{k}⊗χ_{D} mod ℘.

Let f_{2} ∈ S_{2}(Dp, χ_{D}ω_{p}^{k−2}) be the weight 2 member of the Hida family to which f_{k} belongs. Since f_{2} ≡ f_{k} mod ℘ we see that f_{2} satisfies a congruence similar to the one above. As usual this implies that there is a character ϕ of G_{F} of order (p − 1)/(k − 1) and conductor 𝔭ν. In particular N, the cyclic extension of F cut out by ϕ, has degree (p − 1)/(k − 1) over ℚ and conductor 𝔭ν. On the other hand if h_{F} = 1 then (4.2) shows that

[RCF(𝔭ν) : F] = \frac{(p − 1)h_{F}}{d_{𝔭,ν}} = \frac{p − 1}{k − 1}.

We conclude that N = RCF(𝔭ν).

Let us now assume that

f_{2} ⊗ χ_{D} = f_{2}^{γ}  (5.1)

for some γ ∈ Aut(K_{f_{2}}) of order 2. Then applying Shimura’s method to f_{2} (which no longer has nebentypus χ_{D} but this is of no matter) we can generate N = RCF(𝔭ν) explicitly by adjoining a torsion point on a modular abelian variety. We summarize what we have proved:

Theorem 5.2. Let F = ℚ(\sqrt{D}) be a real quadratic field of class number 1. Let p be a prime such that (p, 2D) = 1. Let 𝔭|p be a split prime of F and ν be an infinite place of F such that d_{𝔭,ν} is odd. Then under (5.1) the ray class field of F modulo 𝔭ν can be generated explicitly by adjoining to F a torsion point on a modular abelian variety.

Let us now describe the method without the restriction F_{1} = F made above. Thus we consider dihedral congruence primes with respect to the real quadratic field F_{1} = ℚ(\sqrt{D_{1}}) for cusp forms in S_{k}(|D|, χ_{D}) where D is an arbitrary quadratic discriminant with D_{1}|D.

Again let p be a prime satisfying (p, 2D) = 1. Let 𝔭|p and ν|∞ be places of F_{1} with 𝔭 a split prime. Again define k ≥ 2 by k = d_{𝔭,ν} + 1. This time k is not necessarily even. We therefore assume that k − 1 ≢ 0 mod p − 1 so that we may apply Theorem 2.11. Since d_{𝔭,ν} = (k − 1) | (p − 1) this assumption is equivalent to the assumption that

d_{𝔭,ν} = k − 1 ≠ p − 1.  (5.3)

In any case we have ϵ_{ν}^{k−1} ≡ 1 mod 𝔭. In particular

ϵ_{+}^{k−1} ≡ 1 mod 𝔭  (5.4)

where ϵ_{+} is a totally positive fundamental unit of F_{1}.

Let D_{2} denote an arbitrary fundamental discriminant such that each q|D_{2} splits in F_{1}. Fix a prime 𝔮|q for each q|D_{2}. Assume that D_{2} has been chosen so that the integer a in (2.10) satisfies

a ≡ 0 mod 2.  (5.5)

The conditions (5.3), (5.4) and (5.5) show that the hypotheses of Theorem 2.11 are satisfied.

Let ℘ be a prime of \overline{ℚ} lying over 𝔭. By Theorem 2.11 there exists a ℘-ordinary primitive form f_{k} ∈ S_{k}(|D|, χ_{D}) with

f_{k} ≡ f_{k}⊗χ_{D_{1}} mod ℘.

The methods used in Section 2 imply that there is a character ϕ of G_{F_{1}} of order

n := δ(p − 1)/(k − 1),

with δ = 1 or 2 depending on whether (p − 1)/(k − 1) is even or odd, and conductor 𝔭(∏_{q|D_{2}} 𝔮)ν. In particular N, the cyclic extension of F_{1} cut out by ϕ, has degree n over F_{1} and conductor 𝔭(∏_{q|D_{2}} 𝔮)ν.

Let f_{2} ∈ S_{2}(|D|p, χ_{D}ω_{p}^{k−2}) be the weight 2 member of the Hida family to which f_{k} belongs. Then f_{2} also satisfies a congruence of the form

f_{2} ≡ f_{2}⊗χ_{D_{1}} mod ℘.

We make the usual hypothesis of extra-twisting. Assume that

f_{2}⊗χ_{D_{1}} = f_{2}^{γ}  (5.6)

for some γ ∈ Aut(K_{f_{2}}). Then we have the following theorem:

Theorem 5.7. Let F_{1} = ℚ(\sqrt{D_{1}}) be a real quadratic field and let D_{2} be a quadratic discriminant such that each q|D_{2} splits in F_{1}. For each q|D_{2} fix a prime 𝔮|q. Assume that (5.5) holds. Let p be a prime such that (p, 2D_{1}D_{2}) = 1. Let 𝔭|p be a split prime of F_{1} and ν an infinite place of F_{1}. Assume that

d_{𝔭,ν} ≠ p − 1.

Under (5.6) the class field N of F_{1} modulo 𝔭(∏_{q|D_{2}} 𝔮)ν can be generated explicitly by adjoining to F_{1} a torsion point on a modular abelian variety.

Again the cyclic extension N of F_{1} may be too small to be the ray class field of conductor 𝔭(∏ 𝔮)ν. Let us therefore assume that D_{2} is divisible by only one prime q and that the split prime 𝔮|q of F_{1} satisfies

lcm(d_{𝔭,ν}, d_{𝔮,ν}) = (k − 1)(q − 1)/δ

where δ = 1 or 2 depending on whether (p − 1)/(k − 1) is even or odd. Also assume that h_{F_{1}} = 1. Then by (4.2) we have

[RCF(𝔭𝔮ν) : F_{1}] = \frac{(p − 1)(q − 1)}{lcm(d_{𝔭,ν}, d_{𝔮,ν})} = n

so that N = RCF(𝔭𝔮ν). We have:

Theorem 5.8. Let F_{1} = ℚ(\sqrt{D_{1}}) be a real quadratic field of class number 1. Let 𝔮 be a split prime of F_{1} lying over q such that χ_{𝔮}(ϵ_{+}) = 1. Let p be a prime such that (p, 2qD_{1}) = 1. Let 𝔭|p be a split prime of F_{1} and ν an infinite place of F_{1}. Assume that

d_{𝔭,ν} ≠ p − 1 and lcm(d_{𝔭,ν}, d_{𝔮,ν}) = d_{𝔭,ν}(q − 1)/δ

where δ = 1 or 2 depending on whether (p − 1)/d_{𝔭,ν} is even or odd. Then under (5.6) the ray class field N of F_{1} modulo 𝔭𝔮ν can be generated explicitly by adjoining to F_{1} a torsion point on a modular abelian variety.

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School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India.

E-mail address: abrown@math.tifr.res.in

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India.

E-mail address: eghate@math.tifr.res.in