PRINCIPAL MODULI AND CLASS FIELDS

DAVID COX, JOHN McKAY, AND PETER STEVENHAGEN

Abstract. We study the values taken by Γ_{0}(n)-modular functions at elliptic points of order 2 for the Fricke group Γ_{0}(n)^{†}. In the case of a principal modulus (‘Hauptmodul’) for Γ_{0}(n) or Γ_{0}(n)^{†}, we determine the class fields generated by these values.

1. Introduction

The Fricke group Γ_{0}(n)^{†} of level n > 1 is the subgroup of PSL_{2}(ℝ) generated by the congruence subgroup Γ_{0}(n) ⊂ PSL_{2}(ℤ) and the Fricke involution w_{n} = \sqrt{n} ( \begin{matrix}0 & − \frac{1}{n} \\ 1 & 0 \end{matrix} ) , which acts on the upper half plane 𝔥 by w_{n}(τ) = \frac{−1}{nτ}. As explained in [2, Section 4], Γ_{0}(n)^{†} is a subgroup of the full normalizer Γ_{0}(n)^{+} of Γ_{0}(n) in PSL_{2}(ℝ), which may be generated over Γ_{0}(n) by a finite number of Atkin-Lehner involutions. The subgroup Γ_{0}(n) has index 2 in Γ_{0}(n)^{†}, and the elements of the coset Γ_{0}(n)^{†} \ Γ_{0}(n) are the matrices \sqrt{n} ( \begin{matrix}A & \frac{B}{n} \\ C & D \end{matrix} ) (mod ± 1) with A, B, C, D ∈ ℤ satisfying nAD − BC = 1. Such a matrix is elliptic of order 2 if and only if it has trace 0, i.e., if it is represented by

(1.1)  α = \sqrt{n} (\begin{matrix}A & \frac{B}{n} \\ C & −A \end{matrix}) ∈ SL_{2}(ℝ)

with nA^{2} + BC = −1. We always pick α in (1.1) in such a way that we have B < 0 and C > 0. Then the fixed point of α in 𝔥 is the element

(1.2)  τ_{α} = \frac{nA + \sqrt{−n}}{nC} ∈ 𝔥,

which is a zero of the polynomial

(1.3)  f_{α} = nCX^{2} − 2nAX − B ∈ ℤ[X]

1991 Mathematics Subject Classification. Primary 10D12, 11R37; Secondary 11G15, 20D08, 20H10.

2 DAVID COX, JOHN McKAY, AND PETER STEVENHAGEN

of discriminant 4n(nA^{2} + BC) = −4n. As B is coprime to nA, the polynomial f_{α} is irreducible in ℤ[X] unless B and C are both even. In the latter case we have n = (−1 − BC)\frac{1}{A^{2}} ≡ 3 mod 4, and \frac{1}{2}f_{α} is irreducible in ℤ[X] of discriminant −n. It follows (cf. [3, Thm. 7.7]) that the complex lattice

(1.4)  I_{α} = [τ_{α}, 1] = \frac{1}{nC} [nA + \sqrt{−n}, nC] ⊂ ℂ

corresponding to the fixed point of α is an invertible ideal for the quadratic order O_{α} = {λ ∈ ℂ : λI_{α} ⊂ I_{α}}, which has discriminant −n if B and C are both even, and discriminant −4n otherwise. More explicitly, we have

(1.5)  O_{α} = \begin{cases}ℤ[\frac{−n+\sqrt{−n}}{2}] & for n ≡ 3 mod 4, B and C even \\ ℤ[\sqrt{−n}] & otherwise. \end{cases}

The goal of this paper is to study the Galois theoretic properties of the values f(τ_{α}) over ℚ(τ_{α}) when f is a Γ_{0}(n)-modular function and α is an elliptic element of order 2 in Γ_{0}(n)^{†} \ Γ_{0}(n). If f is modular for the full modular group PSL_{2}(ℤ) and has rational q-expansion, i.e., if f is an element of the field ℚ(j) of modular functions of level 1, then the classical theory of complex multiplication tells us that if f(τ_{α}) is finite, then it is contained in the ring class field of the order O_{α}. This class field is an abelian extension of ℚ(τ_{α}) = ℚ(\sqrt{−n}), and its Galois group over ℚ(τ_{α}) is canonically isomorphic under the Artin isomorphism to the ideal class group C(O_{α}) of the order O_{α}. We will write σ_{α} for the automorphism corresponding to the ideal class [I_{α}] ∈ C(O_{α}).

For f ∈ ℚ(j) of level 1 as above, it is a classical result that we have σ_{β}(f(τ_{α})) = f(τ_{γ}) when α, β and γ are elliptic elements as in (1.1) satisfying O_{α} = O_{β} = O_{γ} and [I_{α}I_{β}^{−1}] = [I_{γ}] ∈ C(O_{α}).

Our main result below shows that Γ_{0}(n)-modular functions behave like level 1 functions when evaluated at the elliptic points of order 2 of the Fricke group Γ_{0}(n)^{†}. The proof given in Section 3 uses the Shimura reciprocity law [11, Thm. 6.31] as explained in [5, 12].

PRINCIPAL MODULI AND CLASS FIELDS 3

Theorem 1.1. Fix n > 1 and let f be a Γ_{0}(n)-modular function with rational q-expansion at ∞. Also let α be an elliptic element of order 2 in Γ_{0}(n)^{†} \Γ_{0}(n) as in (1.1), and assume that f is defined at τ_{α}. Then:

(1) f(τ_{α}) lies in the ring class field of O_{α}.

(2) Let β, γ be elliptic elements as in (1.1) with O_{α} = O_{β} = O_{γ}, and suppose [I_{α}I_{β}^{−1}] = [I_{γ}] ∈ C(O_{α}). Then we have

σ_{β}(f(τ_{α})) = f(τ_{γ}).

Our motivation for proving Theorem 1.1 goes back to the special case where f is a principal modulus (‘Hauptmodul’) of the kind that arises in the moonshine conjectures for the monster group [2] and its generalizations to replicable functions [4]. These principal moduli are known to correspond to genus zero subgroups of PSL_{2}(ℝ) that lie between Γ_{0}(n) and its normalizer Γ_{0}(n)^{+} for some n. In Section 4 we apply Theorem 1.1 to this situation, and explain how our results relate to those of Chen and Yui [1]. The main corollary, which is proved in Section 4, is the following.

Corollary 1.2. Fix n > 0 and let f be a principal modulus with rational q-expansion at ∞ for either Γ_{0}(n) or Γ_{0}(n)^{†}. Suppose that f is defined at the fixed point τ_{α} from (1.2). Then ℚ(τ_{α}, f(τ_{α})) is the ring class field of the order O_{α}.

2. Ideals, Quadratic Forms, and Conjugacy Classes

This section collects what we need about the ideals I_{α} arising in (1.4) and their classes in C(O_{α}). While part of this material is well known (see [7, 8]), we include the details for the convenience of the reader.

In (1.3), we denoted by f_{α} the quadratic polynomial in ℤ[X] having the fixed point τ_{α} of the elliptic element α as one of its zeroes. Its homogeneous form

(2.1)  F_{α}(x, y) = nCx^{2} − 2nAxy − By^{2} ∈ ℤ[x, y]

4 DAVID COX, JOHN McKAY, AND PETER STEVENHAGEN

is a binary quadratic form of discriminant −4n. Since we take B < 0 and C > 0, the form F_{α} is positive definite. Depending on the case we are in in (1.5), either F_{α} or \frac{1}{2}F_{α} is primitive of discriminant disc(O_{α}) ∈ {−n, −4n}. Under the standard bijection [3, Thm. 7.7] between the set of SL_{2}(ℤ)-orbits of primitive positive definite binary quadratic forms of discriminant disc(O_{α}) and the class group C(O_{α}), this primitive form corresponds to the ideal class [I_{α}] ∈ C(O_{α}).

Lemma 2.1. Let α and α′ be as in (1.1). Then the quadratic forms F_{α} and F_{α′} are SL_{2}(ℤ)-equivalent if and only if α and α′ are conjugate in Γ_{0}(n)^{†}. If α and α′ are conjugate in Γ_{0}(n)^{†}, there exists δ ∈ Γ_{0}(n) with δ^{−1}αδ = α′.

Proof. Let M_{α} denote the matrix representing the form α from (2.1):

M_{α} = (\begin{matrix}nC & −nA \\ −nA & −B \end{matrix}) = \sqrt{n} (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) α.

Since every δ ∈ SL_{2}(ℤ) satisfies

(\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) δ^{t} (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) = δ^{−1},

we have

δ^{t}M_{α}δ = M_{α′} ⟺ δ^{−1}αδ = α′.

When F_{α} and F_{α′} are equivalent via δ ∈ SL_{2}(ℤ), working modulo n gives

δ^{t} (\begin{matrix}0 & 0 \\ 0 & −B \end{matrix}) δ ≡ (\begin{matrix}0 & 0 \\ 0 & −B′ \end{matrix}) mod n.

If δ = ( \begin{matrix}a & b \\ c & d \end{matrix}), this easily implies

−Bc^{2} ≡ −Bcd ≡ 0 mod n.

Since gcd(B, n) = gcd(c, d) = 1 (because det(α) = det(δ) = 1), we see that c ≡ 0 mod n, so that δ ∈ Γ_{0}(n). Then the above equivalence implies that α and α′ represent conjugate elements in Γ_{0}(n)^{†}.

Conversely, suppose we have α′ = ±δ^{−1}αδ (remember that conjugacy classes are computed in Γ_{0}(n)^{†} ⊂ PSL_{2}(ℝ)). If δ represents an

PRINCIPAL MODULI AND CLASS FIELDS 5

element of Γ_{0}(n), then the above equivalence shows that F_{α} and ±F_{α′} are properly equivalent, and the sign must be + since F_{α} and F_{α′} are positive definite. The final statement of the lemma says that this is the only case we need to consider. Indeed, if δ represents an element of Γ_{0}(n)^{†} \ Γ_{0}(n), then α^{−1}δ represents an element of Γ_{0}(n) since Γ_{0}(n) has index 2 in Γ_{0}(n)^{†}, and we may replace δ by α^{−1}δ:

α′ = ±δ^{−1}αδ = ±(α^{−1}δ)^{−1}α(α^{−1}δ).

This completes the proof of the lemma. □

It follows from Lemma 2.1 that Γ_{0}(n)-modular functions have a welldefined value on the Γ_{0}(n)^{†}-orbits of the fixed points in (1.2), and that elliptic elements α, β in (1.1) with O_{α} = O_{β} are conjugate in Γ_{0}(n)^{†} if and only if σ_{α} and σ_{β} coincide on the ring class field of O_{α}.

Our main result (Theorem 1.1) gives a Galois action in terms of Artin automorphisms σ_{α} corresponding to the ideal classes [I_{α}]. It is therefore useful to know that every ideal class in C(O_{α}) is of this form.

Lemma 2.2. Let n be a positive integer and fix O to be either ℤ[\sqrt{−n}] or ℤ[\frac{n+\sqrt{−n}}{2}], where in the latter case we assume n ≡ 3 mod 4. Then the ideals I_{α} = \frac{1}{nC} [nA + \sqrt{−n}, nC] from(1.4) with O_{α} = O represent all ideal classes in the ideal class group C(O).

Proof. It suffices to show that a primitive positive definite quadratic form G of discriminant D = −4n, or D = −n with n ≡ 3 mod 4, is properly equivalent to F_{α} or \frac{1}{2}F_{α} respectively. By [3, Lemmas 2.3 and 2.25], we may assume that G(x, y) = ax^{2} +bxy +cy^{2} has gcd(c, D) = 1.

First suppose that D = −4n. Since b is even in this case, we can find an integer k such that

ck ≡ −b/2 mod n.

Using this k to replace G(x, y) with the equivalent form

(2.2)  G(x, y + kx) = (a + bk + ck^{2})x^{2} + (b + 2kc)xy + cy^{2},

6 DAVID COX, JOHN McKAY, AND PETER STEVENHAGEN

we may assume that b is divisible by 2n. Then −4n = b^{2} − 4ac and gcd(c, −4n) = 1 imply that a is divisible by n. It follows that G is of the form F_{α}, as desired.

Next assume D = −n and n ≡ 3 mod 4. Then we can find k such that

2ck ≡ −b mod n.

Using this k and (2.2), we may assume that b is divisible by n. Since n is odd, −n = b^{2} − 4ac and gcd(c, −n) = 1 imply that a is divisible by n. Then G = \frac{1}{2}F_{α} follows easily. □

For O = ℤ[\sqrt{−n}], the form F_{α} = nx^{2}+y^{2} corresponding to the principal ideal generated by τ = \frac{−1}{\sqrt{−n}} is in the unit class in C(O). For O = ℤ[\frac{−n+\sqrt{−n}}{2}], we can start from G(x, y) = \frac{n+1}{4}x^{2} + xy + y^{2} in the unit class and apply the proof of (2.2) (with k = \frac{n−1}{2}) to obtain the element F_{α}(x, y) = \frac{n(n+1)}{4}x^{2} + nxy + y^{2} in the unit class. It corresponds to the principal ideal generated by τ = \frac{−2n+2\sqrt{−n}}{n(n+1)}. Note that in both cases, τO is the lattice [τ, 1].

3. Proof of the Main Theorem

We now prove Theorem 1.1 using the method explained in [12].

Let f be a modular function for Γ_{0}(n) with rational q-expansion and assume that f is defined at the fixed point τ_{α} from (1.2). Our first task is to show that f(τ_{α}) lies in the ring class field of O_{α}. We first do this when the corresponding ideal I_{α} is in the unit class in C(O_{α}). This leads to the following special choice for τ = τ_{α} and in each of the two cases for O = O_{α} provided by (1.5).

n  α  τ  O
Case 1 arbitrary  \sqrt{n} (\begin{matrix}0 & −\frac{1}{n} \\ 1 & 0 \end{matrix})  \frac{−1}{\sqrt{−n}}  ℤ[nτ]
Case 2 n ≡ 3 mod 4 \sqrt{n} (\begin{matrix}1 & −\frac{2}{n} \\ \frac{n+1}{2} & −1 \end{matrix}) \frac{−2n+2\sqrt{−n}}{n(n+1)} ℤ[\frac{n(n+1)}{4}τ]

PRINCIPAL MODULI AND CLASS FIELDS 7

Standard results in complex multiplication imply that f(τ) lies in the maximal Abelian extension K^{ab} of K = ℚ(τ) = ℚ(\sqrt{−n}). Class field theory describes the absolute abelian Galois group Gal(K^{ab}/K) as the surjective image under the Artin map of the group K̂^{∗} = (K ⊗_{ℤ} ℤ̂)^{∗} of finite K-ideles. In order to show that f(τ) lies in the ring class field H_{O} of O, it suffices to show that every automorphism of K^{ab} over H_{O} leaves f(τ) invariant. As Gal(K^{ab}/H_{O}) is the image under the Artin map of the subgroup Ô^{∗} = (O ⊗_{ℤ} ℤ̂)^{∗} ⊂ K̂^{∗}, we need to show that the Artin symbol any idele x ∈ Ô^{∗} leaves f(τ) invariant.

Shimura’s reciprocity law tells us how (the Artin symbol of) the idele x ∈ Ô^{∗} acts on f(τ): we have

(3.1)  f(τ)^{x} = f^{g_{τ} (x^{−1})}(τ),

where g_{τ} : Ô^{∗} → GL_{2}(ℤ̂) is the map that sends x ∈ Ô^{∗} to the transpose of the matrix of multiplication by x on the free ℤ̂-module ℤ̂τ + ℤ̂ with respect to the basis {τ, 1}. From the irreducible polynomial of τ in ℤ[X] one easily computes g_{τ} as in [12, (3.3)] to be

(3.2)  g_{τ}(x) = g_{τ}(a + bnτ) = (\begin{matrix}a & −b \\ nb & a \end{matrix})

for the order O = ℤ[nτ] in Case 1, and

(3.3)  g_{τ}(a + b\frac{n(n+1)}{4}τ) = (\begin{matrix}a − nb & −b \\ \frac{n(n+1)}{4}b & a \end{matrix})

for the order O = ℤ[\frac{n(n+1)}{4}τ] in Case 2. In both cases, the matrix g_{τ}(x) is upper triangular modulo n for every choice of x ∈ Ô^{∗}. This means that modulo n, it is a product

M_{1}M_{2} = (\begin{matrix}1 & 0 \\ 0 & d \end{matrix}) M_{2} ∈ GL_{2}(ℤ/nℤ),

where M_{2} is the reduction modulo n of a matrix from Γ_{0}(n). But M_{2} acts trivially on f since f is a modular function for Γ_{0}(n), and M_{1}, which acts on f via its Fourier coefficients, also acts trivially since the q-expansion is rational. It follows that f is invariant under Ô^{∗}, so that

8 DAVID COX, JOHN McKAY, AND PETER STEVENHAGEN

f(τ) lies in the desired ring class field H_{O}. In particular, the class group C(O), which is naturally isomorphic to the Galois group Gal(H_{O}/K) under the Artin map, now acts on f(τ).

Our next task is to show that for elliptic elements β and γ satisfying O_{β} = O_{γ} = O and [I_{β}^{−1}] = [I_{γ}] ∈ C(O), the Galois action of σ_{β} on f(τ) is in each of the two cases given by

(3.4)  σ_{β}(f(τ)) = f(τ_{γ}).

This shows that we have σ_{γ}(f(τ_{γ})) = f(τ) for all f(τ_{γ}) with γ as in (1.1) having O_{γ} = O. In particular, all these values f(τ_{γ}) are conjugate over K as soon as one of them is known to be finite, and one easily derives the second statement in (1.1) from (3.4). We may therefore finish the proof of (1.1) by proving (3.4).

To prove (3.4), we apply (3.1) once more, with x ∈ K̂^{∗} an Ôgenerator of Î_{γ} = I_{γ} ⊗ Ẑ and g_{τ} : K̂^{∗} → GL_{2}(ℚ̂) the natural ℚ-linear extension of the map g_{τ} : Ô^{∗} → GL_{2}(ℤ̂) we had before. As in [12, Section 6], we let M ∈ GL_{2}^{+}(ℚ) be the transpose of a matrix mapping the lattice [τ, 1] = τO to the lattice τI_{γ}. Write I_{γ} = \frac{1}{nC} [nA+\sqrt{−n}, nC] as in (1.4). In Case 1 we have τ\sqrt{−n} = −1, so

τI_{γ} = τ[\frac{nA+\sqrt{−n}}{nC}, 1] = [\frac{nAτ−1}{nC}, τ],

and we can take M ∈ GL_{2}^{+}(ℚ) to be

M = (\begin{matrix}A/C & −1/nC \\ 1 & 0 \end{matrix}) = (\begin{matrix}1/C & A/C \\ 0 & 1 \end{matrix}) (\begin{matrix}0 & −1/n \\ 1 & 0 \end{matrix}) .

In Case 2 our element τ satisfies τ(n + \sqrt{−n}) = −2, so we have

τI_{γ} = τ[\frac{nA+\sqrt{−n}}{nC}, 1] = [\frac{n(A−1)τ−2}{nC}, τ]

and we can take the matrix M ∈ GL_{2}^{+}(ℚ) to be

M = (\begin{matrix}(A − 1)/C & −2/nC \\ 1 & 0 \end{matrix}) = (\begin{matrix}2/C & (A − 1)/C \\ 0 & 1 \end{matrix}) (\begin{matrix}0 & −1/n \\ 1 & 0 \end{matrix}) .

By construction, we have M(τ) = τ_{γ} in both cases. Note that at this point we use the fact that the lattice [τ, 1] is a principal O-ideal.

PRINCIPAL MODULI AND CLASS FIELDS 9

Further M and g_{τ}(x) are transposes of matrices mapping the ℤ̂-module ℤ̂τ + ℤ̂ ⊂ K̂ onto Î_{γ}, so the matrix g_{τ}(x)M^{−1} is in GL_{2}(ℤ̂) as its transpose fixes ℤ̂τ + ℤ̂. We can now compute

(3.5)  σ_{β}(f(τ)) = f^{g_{τ} (x)}(τ) = f^{g_{τ} (x)M^{−1}}(M(τ)) = f^{g_{τ} (x)M^{−1}}(τ_{γ})

by evaluating g_{τ}(x)M^{−1} ∈ GL_{2}(ℤ̂) modulo the level n of f. All we need to know for this are the p-components x_{p} of a generator x ∈ K̂^{∗} of Î_{γ} for the primes p dividing n. Finding a generator x_{p} of the (O ⊗_{ℤ} ℤ_{p})- ideal I_{γ} ⊗ ℤ_{p} is easy at primes p|n, since for these p the identity BC = −1 − nA^{2} implies that C is a unit in ℤ_{p}. This yields

(3.6)  \begin{aligned}I_{γ} ⊗_{ℤ} ℤ_{p} & = \frac{nA+\sqrt{−n}}{nC}ℤ_{p} + ℤ_{p} = (A − \frac{1}{\sqrt{−n}})ℤ_{p} + ℤ_{p} \\ & = \frac{−1}{\sqrt{−n}}ℤ_{p} + ℤ_{p} = \frac{−1}{\sqrt{−n}}ℤ_{p}[\sqrt{−n}]. \end{aligned}

In Case 1 we have O⊗_{ℤ}ℤ_{p} = ℤ_{p}[\sqrt{−n}] and we can choose x_{p} = \frac{−1}{\sqrt{−n}} = τ for all p|n. Applying (3.2) then yields

g_{τ}(x_{p}) = (\begin{matrix}0 & −1/n \\ 1 & 0 \end{matrix}) ∈ GL_{2}(ℚ_{p})

at all p|n, so g_{τ}(x)M^{−1} has upper triangular reduction

(\begin{matrix}1/C & A/C \\ 0 & 1 \end{matrix})^{−1} = (\begin{matrix}C & −A \\ 0 & 1 \end{matrix})

modulo n. It therefore leaves f invariant, and (3.5) reduces to (3.4).

In Case 2 only odd primes can divide n, so again we have O ⊗_{ℤ}ℤ_{p} = ℤ_{p}[\sqrt{−n}] for p dividing n, and we may take x_{p} = \frac{n+1}{4}τ = \frac{1+\sqrt{−n}}{2} \frac{−1}{\sqrt{−n}} since \frac{1+\sqrt{−n}}{2} is a p-adic unit if p divides n. Applying (3.3) now yields

g_{τ}(x_{p}) = (\begin{matrix}−1 & −1/n \\ \frac{n+1}{4} & 0 \end{matrix}) = (\begin{matrix}1 & −1 \\ 0 & \frac{n+1}{4} \end{matrix}) (\begin{matrix}0 & −1/n \\ 1 & 0 \end{matrix}) ∈ GL_{2}(ℚ_{p})

at all p|n, so g_{τ}(x)M^{−1} has upper triangular reduction

(\begin{matrix}1 & −1 \\ 0 & \frac{n+1}{4} \end{matrix}) (\begin{matrix}2/C & (A − 1)/C \\ 0 & 1 \end{matrix})^{−1}

10 DAVID COX, JOHN McKAY, AND PETER STEVENHAGEN

modulo n. As in Case 1, we find that g_{τ}(x)M^{−1} leaves f invariant. This finishes the proof of Theorem (1.1).

By Lemma 2.2, Theorem 1.1 furnishes a complete description of the Galois theoretic properties of f(τ_{α}) over ℚ(τ_{α}). This will be particularly useful in the cases in the next section, where we know exactly which elements in PSL_{2}(ℝ) fix f.

4. Principal moduli

A modular function f for a discrete group G ⊂ PSL_{2}(ℝ) commensurable with PSL_{2}(ℤ) is a principal modulus (in German: Hauptmodul) if it generates the field of all modular functions for G. This implies G\𝔥^{∗} ≃ ℙ^{1} via f. In particular, G has genus 0.

Corollary 1.2, which we will prove now, states that if the modular function f in Theorem 1.1 is a principal modulus for Γ_{0}(n) or Γ_{0}(n)^{†}, then we get a primitive element of the ring class field.

Proof of Corollary 1.2. The case n = 1 is obvious, so we may assume n > 1. Lemma 2.2 shows that the ideals I_{α} represent all ideal classes in C(O_{α}). Then Lemma 2.1 and the correspondence between ideals and quadratic forms imply that we get |C(O_{α})| different Γ_{0}(n)^{†}-inequivalent points of the form τ_{α}. If f is a principal modulus for Γ_{0}(n)^{†}, it follows that the complex numbers f(τ_{α}) have |C(O_{α})| distinct values as we vary α, and the same is true if f is a principal modulus for Γ_{0}(n) since Γ_{0}(n)^{†}-inequivalent points are automatically Γ_{0}(n)-inequivalent. Thus the minimal polynomial of f(τ_{α}) over ℚ(τ_{α}) has degree |C(O_{α})| by Theorem 1.1. The theorem follows since this is the degree of the ring class field over ℚ(τ_{α}). □

It is well known that Γ_{0}(n) has genus 0 for the 15 numbers

(4.1)  n = 1–10, 12, 13, 16, 18, 25.

Turning to Γ_{0}(n)^{†}, Ogg [10] showed that Γ_{0}(n)^{†} has genus 0 for the following 37 values of n > 1 (we exclude n = 1 since Γ_{0}(1)^{†} = Γ_{0}(1)):

(4.2) n = 2–21, 23–27, 29, 31, 32, 35, 36, 39, 41, 47, 49, 50, 59, 71.

However, in order to use Corollary 1.2 for either Γ_{0}(n) or Γ_{0}(n)^{†}, we also need to know that the group has a principal modulus with a rational q-expansion. As we are dealing with a finite list of groups, one can simply check the tables in [2], which give explicit formulas for principal moduli in all cases when Γ_{0}(n) or Γ_{0}(n)^{†} has genus 0. These all have rational q-expansions. Alternatively, as S. Norton pointed out to us, Theorem 1 of [9] easily implies the existence of a principal modulus with rational q-expansion whenever Γ_{0}(n) or Γ_{0}(n)^{†} has genus 0. It follows that Corollary 1.2 applies to all values of n in (4.1) for Γ_{0}(n) and (4.2) for Γ_{0}(n)^{†}.

The primitive elements for the ring class fields obtained from principal moduli are often much smaller than those provided by the jfunction. For example, take f to be the function listed as 71A in [2, p. 337], which is a principal modulus for the Fricke group Γ_{0}(71)^{†}. There are 14 elliptic fixed points of order 2 for Γ_{0}(71)^{†}, of which seven are of discriminant −71 and seven are of discriminant −4 · 71 = −284. This is in accordance with the fact that the two quadratic orders having these discriminants have class number 7. In both cases the corresponding ring class field is the Hilbert class field H of ℚ(\sqrt{−71}).

When n = 71, the two special elliptic points of order 2 used in the proof of Theorem 1.1 are

τ_{−284} = \frac{−1}{\sqrt{−71}} and τ_{−71} = \frac{−71 + \sqrt{−71}}{71 · 36}.

corresponding to the orders of discriminant −284 and −71 respectively. One can compute that the minimal polynomial of f(τ_{−284}) is

h_{−284} = x^{7} − 7x^{5} − 11x^{4} + 5x^{3} + 18x^{2} + 4x − 11,

while the minimal polynomial of f(τ_{−71}) is

h_{−71} = x^{7} + 4x^{6} + 5x^{5} + x^{4} − 3x^{3} − 2x^{2} + 1.

Each of these polynomials is a small generating polynomial for H over ℚ(\sqrt{−71}) and compares favorably with the polynomial

w_{−71} = x^{7} − 2x^{6} − x^{5} + x^{4} + x^{3} + x^{2} − x − 1

found by Weber [13, Vol. III, p. 723] (this is the minimal polynomial of 𝔣(\sqrt{−71})/\sqrt{2}, where 𝔣 is the Weber function defined on page 114 of [13, Vol. III]). If β is a root of w_{−71}, then an easy calculation shows that

β^{2} − 1 − β^{−1} is a root of h_{−284}
−β^{6} + 3β^{5} − 2β^{4} + 1 is a root of h_{−71}.

Note that the values f(τ_{−71}) and f(τ_{−284}) are in this case integral, a general phenomenon for which we currently do not have a proof. For the importance of h_{−284} and h_{−71} for the Schwarzian differential equation satisfied by f, see [6].

As we mentioned in the introduction, the moonshine conjectures for the monster group provide us with principal moduli for genus zero subgroups of PSL_{2}(ℝ) in the form of so-called McKay-Thompson series. Their singular values have been studied by Chen and Yui in [1]. They restrict to fundamental McKay-Thompson series, i.e., the series T_{g} associated to a conjugacy class g of the monster group that is the principal modulus for a subgroup of PSL_{2}(ℝ) whose level n equals the order of g. If T_{g} is fundamental, τ ∈ 𝔥 is a CM-point and O = ℤ[aτ] is the multiplier ring of [τ, 1], they prove the following in [1, Theorem 3.7.5]:

(1) If T_{g} is a principal modulus for Γ_{0}(n), then ℚ(τ, T_{g}(τ)) is the ring class field of the order of index \frac{n}{gcd(a,n)} in O.

(2) If T_{g} is a principal modulus for Γ_{0}(n)^{†} with n prime and coprime to a, then ℚ(τ, T_{g}(τ)) is the ring class field of the order of index n in O.

Applying the first part of this theorem to T_{g}(τ_{α}), one can prove Corollary 1.2 when T_{g} is a principal modulus for Γ_{0}(n). So for T_{g}(τ_{α}), the Chen-Yui result covers all numbers in (4.1). On the other hand, if T_{g} is a principal modulus for Γ_{0}(n)^{†}, then the hypothesis gcd(a, n) = 1 means that the second part of Theorem 3.7.5 doesn’t apply to T_{g}(τ_{α}). So for T_{g}(τ_{α}), the Chen-Yui result covers none of the numbers in (4.2). The integrality results in [1] for the numbers T_{g}(τ), although interesting in their own right, do not apply to the fixed points τ = τ_{α} in this paper. Numerical computations do however suggest that T_{g}(τ_{α}) is an algebraic integer, so this question deserves further study.

Acknowledgements

The research of the second author is partially supported by NSERC. The first author is grateful to Heng Huat Chan for bringing [5, 12] to his attention, and the second author would like to thank Simon Norton for suggesting the relevance of [9].

References

[1] I. Chen and N. Yui, ‘Singular values of Thompson series’, Groups, Difference Sets, and the Monster (Columbus, OH, 1993 ) (eds K. T. Arasu, J. F. Dillon, K. Harada, S. Sehgal and R. Solomon), Ohio State Univ. Math. Res. Inst. Publ. 4 (de Gruyter, Berlin, 1996), pp. 255–326.

[2] J. H. Conway and S. P. Norton, ‘Monstrous moonshine’, Bull. Lond. Math. Soc. 11 (1979) 308–339.

[3] D. Cox, Primes of the Form x^{2} + ny^{2} (John Wiley & Sons, Inc., New York, 1989).

[4] C. J. Cummins and S. P. Norton, ‘Rational Hauptmoduls are replicable’, Can. J. Math. 47 (1995) 1201–1218.

[5] A. Gee and P. Stevenhagen, ‘Generating class fields using Shimura reciprocity’, Algorithmic number theory (Portland, OR, 1998 ) (ed J. Buhler), Lecture Notes in Comput. Sci. 1423 (Springer-Verlag, Berlin, 1998), pp. 441–453.

[6] J. Harnad and J. McKay, ‘Modular solutions to equations of generalized Halphen type’, Proc. R. Soc. Lond. A 456 (2000) 261–294.

[7] H. Helling, ‘On the commensurability class of the rational modular group’, J. London Math. Soc. 2 (1970) 67–72.

[8] C. Maclachlan, ‘Groups of units of zero ternary quadratic forms’, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981) 141–157.

[9] S. P. Norton, ‘Non-monstrous moonshine’, Groups, Difference Sets, and the Monster (Columbus, OH, 1993 ) (eds K. T. Arasu, J. F. Dillon, K. Harada, S. Sehgal and R. Solomon), Ohio State Univ. Math. Res. Inst. Publ. 4 (de Gruyter, Berlin, 1996), pp. 433–441.

[10] A. Ogg, ‘Hyperelliptic modular curves’, Bull. Soc. Math. France 102 (1974) 449–462.

[11] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions (Princeton U. Press, Princeton, NJ, 1971).

[12] P. Stevenhagen, ‘Hilbert’s 12th problem, complex multiplication, and Shimura reciprocity’, Class field theory—its centenary and prospect (Tokyo, 1998) (ed K. Miyake), Adv. Stud. in Pure Math. 30 (Math. Soc. Japan, Tokyo, 2001), pp. 161–176.

[13] H. Weber, Lehrbuch der Algebra, 2nd Edition (Vieweg, Braunschweig, 1908; Chelsea, New York, 1961.)

Department of Mathematics and Computer Science, Amherst College, Amherst, MA 01002-5000, USA

E-mail address: dac@cs.amherst.edu

Department of Mathematics and Statistics, 1455 de Maisonneuve West, Concordia University, Montréal, Québec, H3G 1M8, Canada

E-mail address: mckay@cs.concordia.ca

Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands

E-mail address: psh@math.leidenuniv.nl