Modular equations and the genus zero property of moonshine functions.

C J Cummins^{1} and T Gannon^{2,3}

Abstract. In this paper we obtain the following result: Let f be an analytic function on the upper half plane with Fourier expansion f(z) = q^{−1} + ∑_{n=1}^{∞} a_{n}q^{n}, q = exp(2πiz). If a_{i} ∈ ZZ[ζ_{K}], i = 1, 2, …, for some integer K > 0, then the following are equivalent:

1 f satisfies a modular equation of order n for all n ≡ 1 (mod K). 2 f is either q^{−1} + ζq where ζ^{gcd(24,K)+1} = ζ, or is a Hauptmodul for a

subgroup G of SL(2, IR) satisfying:

a G contains Γ_{0}(N) with finite index for some N | K^{∞} (i.e. N divides some power of K).

b G contains ( \begin{matrix}1 & k \\ 0 & 1 \end{matrix} ) if and only if k ∈ ZZ.

c The Riemann surface X(G) has genus zero.

In Borcherds’ proof of the moonshine conjectures the moonshine functions satisfy condition 1 of this theorem, for a suitable choice of K, by virtue of the twisted denominator formulae of the Monster Lie algebra and hence are Hauptmoduln. A conceptual explanation is thus provided of the genus zero property of these functions.

§1. Introduction.

Let IM be the Monster simple group. Let o(g), g ∈ IM, be the order of g. Define

Γ_{0}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ZZ) | c ≡ 0 (mod N)\},
Γ(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ZZ) | ( \begin{matrix}a & b \\ c & d \end{matrix} ) ≡ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) (mod N)\}

and let Λ(N) be the normalizer of Γ_{0}(N) in SL(2, IR). We shall call a discrete subgroup Δ of SL(2, IR) a congruence group if it contains Γ(N) for some N. Necessarily the index of Γ(N)

1 email: cummins@abacus.concordia.ca 2 email: gannon@mpim-bonn.mpg.de 3 Address from September 1996:

Department of Mathematics, York University, North York, CANADA, M3J 1P3

Work supported by NSERC and FCAR grants AMS subject classification: 11F03, 11F22, 30F35 in Δ is finite and Δ acts on the extended upper half plane H^{∗} = H ∪ Q ∪ {i∞} by fractional linear transformations. The quotient Δ\H^{∗} has the structure of a compact Riemann surface and will be denoted by X(Δ). Conway and Norton [CN] conjectured relationships between certain congruence groups and IM known as the moonshine conjectures. These have been proved by Borcherds:

Theorem 1.1. [B] For each g ∈ IM there exists a formal q-expansion

j_{g}(q) = q^{−1} + ∑_{n=1}^{∞} a_{n}(g)q^{n},

a_{n}(g) ∈ ZZ, n ≥ 1, such that :

1 For all n ≥ 1 the map g ↦ a_{n}(g) is a character of IM.

2 For each g there exists h | gcd(24, o(g)) and a congruence group Δ(g) of SL(2, IR) such

that

Γ_{0}(ho(g)) ≤ Δ(g) ≤ Λ(ho(g))

3 The genus of X(Δ(g)) is zero.

4 Replacing q by exp(2πiz) in j_{g} yields the Fourier expansion of a function f_{g}(z) which is analytic on H. The field of automorphic functions* of Δ(g) is C (f_{g}).

Conway and Norton also conjectured that the series of Theorem 1.1 are related by replication formulae. These are also proved by Borcherds:

Theorem 1.2. [B] Let ζ_{d} = exp(2πi/d). For each g ∈ IM and for all n ≥ 1 there exists a polynomial Q_{n,g} such that

∑_{{ad=n \atop 0≤b<d}} j_{g^{a}}(ζ_{d}^{b}q^{\frac{a}{d}}) = Q_{n,g}(j_{g}(q)).  (1.1)

Note that Q_{n,g} is the unique polynomial such that

Q_{n,g}(j_{g}(q)) = q^{−n} + terms of positive degree.

Polynomials with this property are known as the Faber polynomials [D, chapter 4]. We shall use the notation ζ_{m} = exp(2πi/m) throughout this paper.

The outline of Borcherds’ proof is as follows: First using vertex operator techniques a generalized Kac-Moody algebra, called the Monster Lie algebra, is constructed on which the Monster acts [B]. This Lie algebra has a ZZ × ZZ grading which is respected by IM. The (m, n) ≠ (0, 0) graded piece of the Monster Lie algebra affords the representation V_{mn} of IM, where V_{n} is the representation of IM of conformal weight n+1 appearing in the Monster vertex operator algebra [FLM]. Let a_{n}(g) be the trace of g ∈ IM on V_{n}. By using a generalization of the Weyl denominator formula Borcherds finds a denominator identity for each conjugacy class in IM. For the identity element of IM the formula is:

* All automorphic functions in this paper are of weight zero. A modular function of level N is an automorphic function for Γ(N).

p^{−1} ∏_{m>0, n∈Z Z}(1 − p^{m}q^{n})^{c_{mn}} = j(p) − j(q),

where the c_{n} are the coefficients of the q-expansion of

j(q) = q^{−1} + 196884 q + 21493760 q^{2} + 864299970 q^{3} + …,

the normalized generator or Hauptmodul for SL(2, ZZ). For the other conjugacy classes the formula is:

p^{−1} exp(− ∑_{i>0} ∑_{m>0,n∈Z Z} a_{mn}(g^{i})p^{mi}q^{ni}/i) = ∑_{m∈Z Z} a_{m}(g)p^{m} − ∑_{m∈Z Z} a_{m}(g)q^{m}.  (1.2)

Consider next the formal q-series j_{g}(q) = ∑_{m∈Z Z} a_{m}(g)q^{m}, g ∈ IM. We want to show that these j_{g} satisfy the conditions of Theorem 1.1. By construction property 1 is satisfied and so it remains to show that each j_{g} is an automorphic function with the required properties. Borcherds’ proof of this is to note firstly that equation (1.2) implies that the j_{g}, g ∈ IM satisfy the replication identities of Theorem 1.2 as formal series. Koike [Koi] has shown that the modular functions attached to the elements of IM by Conway and Norton satisfy the replication formulae (this is shown in more generality in [CuN]). Any formal series which satisfies the replication formulae must obey recurrence relations which determine the series from a finite number of terms [N1,B; cf. Mah]. Thus, by checking a finite number of coefficients, it can be verified that each j_{g} is the expected automorphic function.

Unfortunately this last step in the proof does not offer any conceptual understanding of why the series involved are automorphic functions of genus zero. In this paper we address this problem which was first posed by Norton [N1] in terms of completely replicable functions (see section 8).

Let

h(q) = q^{−1} + ∑_{n=1}^{∞} a_{n}q^{n}

with a_{n} ∈ C, be a formal q-series and let ψ(n) = n ∏_{{p|n \atop p prime}} (1 + \frac{1}{p}). A modular polynomial

of order n > 1 for h is a polynomial F_{n}(x, y) ∈ C [x, y] such that*:

M.1 F_{n}(x, y) = F_{n}(y, x).

M.2 F_{n}(x, y) is a monic polynomial of degree ψ(n) in x (and y).

M.3 For all a, b, d ∈ ZZ such that ad = n, gcd(a, b, d) = 1 and 0 ≤ b < d, F_{n}(h(q), h(ζ_{d}^{b}q^{\frac{a}{d}})) = 0 as formal q-series.

We call F_{n}(h(q), h(ζ_{d}^{b}q^{\frac{a}{d}})) = 0 a modular equation of order n for h. The study of modular equations, particularly those of the j function, have a long history, for example see [Sh, section 4.6] and [L, chapter 5 §2]. We note that one consequence of Theorem 1.4 below is that many of

* Although we have included M.1 as a separate property, it is in fact a consequence of M.2 and M.3, cf. [K, Proposition 3.2]. the classical results on singular values of the j function can be extended to any Hauptmodul. Some partial results along these lines are contained in [CY].

Mahler [Mah, Theorem 8] has shown that if h satisfies a modular equation for some prime p, then it is the Laurent expansion of a meromorphic function defined in some neighbourhood of q = 0. Kozlov [K, Proposition 3.3] shows that if h satisfies a modular equation of order p for infinitely many primes p, then it is the Laurent expansion of a function analytic on the interior of the unit disc |q| < 1, except for a pole at q = 0 and so, in this case, the function f(z) = h(exp(2πiz)) extends to an analytic function on the upper half plane. Moreover f satisfies:

M.3′ For all a, b, d ∈ ZZ such that ad = n, gcd(a, b, d) = 1 and 0 ≤ b < d, F_{n}(f(z), f((az + b)/d)) = 0.

Thus in the rest of this paper we may, without loss of generality, consider f to be an analytic function on the upper half plane and we shall set q = exp(2πiz) so that f has Fourier expansion q^{−1} + ∑_{n=1}^{∞} a_{n}q^{n}. Let

G(f) = {m ∈ SL(2, IR) | f(m(z)) = f(z)}.

We shall call G(f) the symmetry group of f. Our main results are stated in Theorems 1.3 and 1.4. In this paper if a and b are integers such that a divides some power of b, then we write a|b^{∞}.

Theorem 1.3. Let f be an analytic function on H with Fourier expansion f(z) = q^{−1} + ∑_{n=1}^{∞} a_{n}q^{n}, where q = exp(2πiz) and a_{i} ∈ C, i = 1, 2, . . .. Let K > 0 be an integer and suppose f satisfies a modular equation of order n for all n ≡ 1 (mod K). Then

1 If G(f) ≠ \{± ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) | t ∈ ZZ\}, then f is is a Hauptmodul for G(f) which

satisfies:

a G(f) contains Γ_{0}(N) with finite index for some N |K^{∞}.

b G(f) contains ( \begin{matrix}1 & k \\ 0 & 1 \end{matrix} ) if and only if k ∈ ZZ.

c X(G(f)) has genus zero.

2 If G(f) = \{± ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) | t ∈ Z\} and the coefficients a_{n}, n = 1, 2, … are algebraic

integers, then f(z) = q^{−1} + ζq where ζ^{gcd(24,K)+1} = ζ.

As observed by Norton [N1] and shown by Kozlov [K], if a_{m}(g) ∈ Q for all m ∈ ZZ* and all g ∈ IM then equation (1.2) implies the existence of a modular polynomial of order n for j_{g}(q) for all n coprime to o(g). Thus setting Δ(g) = G, Theorem 1.3 implies that Δ(g) is a congruence group which satisfies properties 3 and 4 of Theorem 1.1 and contains Γ_{0}(N) for some N, where N | o(g)^{∞}. It has been shown by Thompson [T] that there are only finitely many groups G satisfying conditions a, b and c of Theorem 1.3.

* That the coefficients of the moonshine functions are rational can be seen without a case by case analysis, for example from the existence of a ZZ[1/2] form of the moonshine vertex operator algebra [BR].

In order to state Theorem 1.4 we must introduce generalized modular equations. If f has rational coefficients, then a generalized modular equation is a modular equation, but in general this is not the case. Let N ∈ ZZ^{>0}. For n coprime to N, ∗n will denote the Galois automorphism of Q (ζ_{N}) such that ζ_{N} ∗ n = ζ_{N}^{n}. With the same notation as above, if a_{i} ∈ Q (ζ_{N}), i = 1, 2, … then we define a generalized modular polynomial of order n > 1 for h to be a polynomial F_{n}(x, y) ∈ Q (ζ_{N})[x, y] such that**:

MI.1 F_{n}(x, y) = (F_{n} ∗ n)(y, x)

MI.2 F_{n}(x, y) is a monic polynomial of degree ψ(n) in x (and y).

MI.3 For all a, b, d ∈ ZZ such that ad = n, gcd(a, b, d) = 1 and 0 ≤ b < d, F_{n}((h ∗ n)(q), h(ζ_{d}^{b}q^{\frac{a}{d}})) = 0 as formal q-series.

In this definition F_{n} ∗ n is the polynomial obtained from F_{n} by applying ∗n to each of its coefficients and h ∗ n is obtained from h by applying ∗n to each of its coefficients. If f(z) = h(exp(2πiz)) and (f ∗ n)(z) = (h ∗ n)(exp(2πiz)) extend to analytic functions on H, then a generalized modular polynomial for f(z) satisfies MI.1 and MI.2 while MI.3 is replaced by:

MI.3′ For all a, b, d ∈ ZZ such that ad = n, gcd(a, b, d) = 1 and 0 ≤ b < d, F_{n}((f ∗ n)(z), f((az + b)/d)) = 0.

We then have the following:

Theorem 1.4.

1 If f be a Hauptmodul for a subgroup G of SL(2, IR) satisfying a G contains Γ_{0}(N) with finite index for some N.

b G contains ( \begin{matrix}1 & k \\ 0 & 1 \end{matrix} ) if and only if k ∈ ZZ.

c X(G) has genus zero.

d H is a subfield of Q (ζ_{N}), where H is the field generated over Q by the coefficients

of f. Then the exponent of the Galois group Gal(H/Q) is 1 or 2, i.e. H is a compositum of quadratic fields, and there exists a generalized modular equation F_{n}(x, y) for f of order n for all n coprime to N. Also F_{n}(f ∗ n, y) is irreducible over C (f ∗ n).

2 Let K > 0 be an integer. If f = q^{−1} + ζq where ζ^{gcd(24,K)+1} = ζ, then there exists a generalized modular equation for f of order n for all n coprime to K.

It is perhaps worthwhile to mention that one consequence of Theorem 1.3 is that if f satisfies the hypothesis of Theorem 1.4 1, then for all n coprime to N, f ∗ n is a Hauptmodul for the group G(f ∗ n) which satisfies conditions a, b and c of Theorem 1.4.

A full discussion of property 2 of Theorem 1.1 is beyond the scope of this paper. However, if j_{g} is a Hauptmodul for a group G such that G contains Γ_{0}(N) and p/| N for some prime p, then, since the Monstrous functions have rational coefficients, Theorem 1.4 implies that j_{g} satisfies a modular equation of order p. Since the sum of the roots of this modular equation

** MI.1 is implied by MI.2, MI.3 and the condition that Gal(H/Q) has exponent 1 or 2 (H as in Theorem 1.4), see Proposition 6.18. is a polynomial in j_{g}, by Theorem 1.2 j_{g} = j_{g^{p}}. In other words, if j_{g} ≠ j_{g^{p}} and G contains Γ_{0}(N) then p|N.

Results related to ours include the work of Martin [Mar], who uses the behaviour of completely replicable functions at “bad” primes to derive invariance under certain congruence groups, and Cohn and McKay [CM], who conjecture, based on computational evidence, that the hypotheses of Theorem 1.3 can be considerably weakened and show, using computer algebra, that if f has integer coefficients and satisfies modular equations of order 2 and 3 then f is either a Hauptmodul or trivial.

In the next four sections we shall take f to be an analytic function on H, with Fourier expansion f(z) = q^{−1} + ∑_{n=1}^{∞} a_{n}q^{n}, q = exp(2πiz) and we assume that f has a modular polynomial F_{n}(x, y) for all n ≡ 1 (mod K). Our aim is to show that f is one of the trivial functions or a Hauptmodul; a result that is obtained in section 7.

The proof is in several stages. In section 2 we show that if f(z_{1}) = f(z_{2}), then there is an analytic bijection α from a neighbourhood D_{1} of z_{1} to a neighbourhood D_{2} of z_{2} such that f(α(z)) = f(z) for all z in D_{1}. We call such an α a local symmetry of f. If α extends to an automorphism of H, then we say it is a global symmetry, or simply a symmetry, of f. In section 3 we show that any local symmetry has a unique maximal domain to which it can be extended. Section 4 shows that this maximal domain must be H and so every local symmetry of f is a global symmetry. In section 5 we address the problem of showing that the symmetry group of f contains Γ_{0}(N) for some N. In section 6 a proof of Theorem 1.4 1 is given. In section 7 the results are combined to obtain Theorem 1.3 and Theorem 1.4 2. In the last section we make some conjectures and comments. Acknowledgements We thank Yves Martin for many interesting and useful discussions. T.G. appreciates the hospitality shown to him by the Department of Mathematics and Statistics of Concordia University and the Max-Planck-Institut für Mathematik. We also thank Simon Norton, John McKay and Pieter Moree for comments on an earlier version of this paper and Hershy Kisilevsky for comments on section 6.

§2. Existence of local symmetries.

We shall make repeated use of Definition 2.1 and Lemma 2.2 in the rest of this paper:

Definition 2.1. Let N_{K} be the set of all positive integers n ≡ 1 (mod K). We will denote elements of P GL(2, Q)^{+}, where the + denotes positive determinants, by angular brackets. In particular, if m is a nonsingular 2 × 2 integer matrix with positive determinant, then ⟨m⟩

will denote the corresponding element of P GL(2, Q)^{+}. Any element ⟨ \begin{matrix}a & b \\ c & d \end{matrix} ⟩ of P GL(2, Q)^{+}

acts on H by the fractional linear transformation z ↦ (az + b)/(cz + d). For any positive

integer n define A(n) = \{⟨ \begin{matrix}a & b \\ 0 & d \end{matrix} ⟩ | ad = n, gcd(a, b, d) = 1, 0 ≤ b < d\}.

Lemma 2.2. There exists a t ∈ IR such that:

1 If Im (z_{1}) > t and Im (z_{2}) > t and f(z_{1}) = f(z_{2}), then z_{1} − z_{2} ∈ ZZ.

2 If s ∈ IR and s satisfies 1, then s ≥ t.

Proof: Let f̄ be the function meromorphic on the interior of the unit disk defined by the relation f̄(exp(2πiz)) = f(z) for all z ∈ H. Then the coefficients of the Laurent expansion of f̄ are the Fourier coefficients a_{n} of f. The Lemma follows immediately from the fact that f̄(q) has a simple pole at q = 0.

Define H_{t} = {z ∈ H | Im (z) > t}. Recall the definition of a local symmetry of f given in the introduction.

Proposition 2.3. If z_{1}, z_{2} ∈ H and f(z_{1}) = f(z_{2}) then there exists a local symmetry α : D_{1} → D_{2} of f for which α(z_{1}) = z_{2}.

The proof of Proposition 2.3 is a generalization of that of Corollary 3.7 in [K], obtained by restricting attention to those n ∈ N_{K}. The only place where this generalization is not immediate is in establishing the following:

Lemma 2.4. If z_{1}, z_{2} ∈ H are such that f(z_{1}) = f(z_{2}) and f′(z_{1}) = 0, then also f′(z_{2}) = 0.

As reference [K] is relatively inaccessible, for completeness we give the whole proof of Proposition 2.3. in the remainder of this section. We start with 3 lemmas which will be used to prove Lemma 2.4.

Lemma 2.5.

a Choose any n ∈ N_{K}. If Im (z_{0}) > nt, then all roots y of F_{n}(f(z_{0}), y) will be simple roots. b Choose any n ∈ N_{K} and β ∈ A(n). If f′(z_{0}) = 0 then either f′(β(z_{0})) = 0, or f(β(z_{0}))

is a multiple root of F_{n}(f(z_{0}), y). c Choose any n ∈ N_{K} and β ∈ A(n). If f(β(z_{0})) is a multiple root of F_{n}(f(z_{0}), y), then

either f′(z_{0}) = 0, or f(z_{0}) is a multiple root of F_{n}(f(β(z_{0})), y).

Proof: Part a is immediate from property MI.3′ of F_{n} and Lemma 2.2. For any n ∈ N_{K},

and any β = ⟨ \begin{matrix}n/d & r \\ 0 & d \end{matrix} ⟩ ∈ A(n), differentiating F_{n}(f(z), f(β(z))) = 0 gives:

\frac{∂F_{n}}{∂x} (f(z_{0}), f(β(z_{0}))) f′(z_{0}) + \frac{∂F_{n}}{∂y} (f(z_{0}), f(β(z_{0}))) \frac{n}{d^{2}} f′(β(z_{0})) = 0.  (2.1)

Part b follows from equation (2.1), while c follows from equation (2.1) and property MI.1 of F_{n}.

Lemma 2.6. If f′(z_{0}) ≠ 0, then for all sufficiently large primes p ∈ N_{K} and for all ℓ ∈ ZZ^{>0}, f(p^{ℓ}z_{0}) is a simple root of F_{p^{ℓ}}(f(z_{0}), y).

Proof: Choose any prime p ∈ N_{K} and ℓ ∈ ZZ^{>0} such that f(p^{ℓ}z_{0}) is a multiple root of F_{p^{ℓ}}(f(z_{0}), y): \frac{∂F_{p^{ℓ}}}{∂y} (f(z_{0}), f(p^{ℓ}z_{0})) = 0. By Lemma 2.5 c, we see that f(z_{0}) is a multiple root of F_{p^{ℓ}}(f(p^{ℓ}z_{0}), y). This means that f(z_{0}) = f(β(p^{ℓ}z_{0})) for some β ∈ A(p^{ℓ}), β(p^{ℓ}z_{0}) ≠ z_{0}.

If there were infinitely many such primes p, then we would have a sequence w_{i} = p_{i}^{2m_{i}}z_{0}+ s_{i}, 0 ≤ s_{i} < 1, i = 1, 2, ... of distinct numbers w_{i} at which f(w_{i}) = f(z_{0}). By Lemma 2.2 all but finitely many of these w_{i} have m_{i} = 0, and so the w_{i} lie in a bounded region bounded away from the real axis. The analyticity of f would then force f(z) = f(z_{0}) everywhere, a contradiction.

Lemma 2.7. Suppose z_{1}, z_{2} ∈ H, f(z_{1}) = f(z_{2}), f′(z_{1}) = 0 and f′(z_{2}) ≠ 0. Then there exists an N_{0} such that for all primes p > N_{0}, p ∈ N_{K}, and any ℓ ∈ ZZ^{>0}, there is some z_{p^{ℓ}} = β(z_{1}), where β ∈ A(p^{m}), for which both f(z_{p^{ℓ}}) = f(p^{ℓ}z_{2}) and f′(z_{p^{ℓ}}) = 0.

Proof: Take N_{0} sufficiently large so that for all primes p > N_{0}, f(p^{ℓ}z_{2}) is a simple root of F_{p^{ℓ}}(f(z_{2}), y) = F_{p^{ℓ}}(f(z_{1}), y). That such an N_{0} exists follows from Lemma 2.6. Then f(p^{ℓ}z_{2}) = f(z_{p^{ℓ}}) for some z_{p^{ℓ}} = β(z_{1}) for some β ∈ A(p^{ℓ}) and hence f(z_{p^{ℓ}}) must be a simple root of F_{p^{ℓ}}(f(z_{1}), y) and Lemma 2.5 b implies f′(z_{p^{ℓ}}) = 0.

Proof of Lemma 2.4: Assume that f′(z_{2}) ≠ 0, we shall show that this leads to a contradiction.

First note that if p ∈ N_{K} is a prime such that p > N_{0} and Im (pz_{2}) > t, then by Lemma 2.7 z_{p} and pz_{2} satisfy f(z_{p}) = f(pz_{2}), f′(z_{p}) = 0 and, since pz_{2} is in H_{t}, f′(pz_{2}) ≠ 0. Thus, by redefining z_{1} to be z_{p} and z_{2} to be pz_{2} if necessary, we may assume in our search for a contradiction that Im (z_{2}) > t.

With notation as in Lemma 2.7, choose any primes p, p′ ∈ N_{K}, p, p′ > N_{0}. We may find increasing sequences ℓ_{i}, ℓ′_{i} ∈ ZZ^{>0} such that 1 < p^{ℓ_{i}}/p′^{ℓ′_{i}} < 2 (see Lemma 4.2 below). Let n_{i} = p^{ℓ_{i}}, m_{j} = p′^{ℓ′_{j}}. By Lemma 2.5 a each root of F_{m_{i}}(f(z_{n_{i}}), y) = F_{m_{i}}(f(n_{i}z_{2}), y) will be simple. Then by Lemma 2.5 b, f′(m_{i}z_{n_{i}}) = 0. Thus we obtain an infinite number of distinct points z′_{i} = m_{i}z_{n_{i}}, such that f′(z′_{i}) = 0. Note that Im(z_{1}/2) < Im(z′_{i}) and the fact that f′(z′_{i}) = 0 means that Im(z′_{i}) ≤ t. Thus, translating by integers if necessary, all the z′_{i} lie in a compact region in which f is analytic. This forces f to be constant, which is impossible.

As noted above, the remaining results in this section are contained in [K].

Lemma 2.8. Let F (x, y) ∈ C [x, y] and let f and g be two functions analytic on the upper half plane. Then for any positive integer k:

\frac{d^{k}F}{dz^{k}} (f, g) = \frac{∂F}{∂x} (f, g)\frac{d^{k}f}{dz^{k}} + \frac{∂^{k}F}{∂y^{k}} (f, g)(\frac{dg}{dz})^{k} + ∑_{m=1}^{k−1} \frac{∂^{m}F}{∂y^{m}} (f, g)A_{m,k} + ∑_{m=1}^{k−1} \frac{d^{m}f}{dz^{m}} B_{m,k}

where A_{m,k} is a polynomial in \frac{dg}{dz}, \frac{d^{2}g}{dz^{2}} , …, \frac{d^{k}g}{dz^{k}} and B_{m,k} is a polynomial in derivatives of f, g and F.

Proof: This is easily established by induction.

Lemma 2.9. If \frac{d^{m}f}{dz^{m}} (z_{0}) = 0 for 1 ≤ m < k and \frac{d^{k}f}{dz^{k}} (z_{0}) ≠ 0, then for all sufficiently large primes p ∈ N_{K}, f(pz_{0}) is a root of F_{p}(f(z_{0}), y) of multiplicity at most k.

Proof: Suppose p is a prime such that f(pz_{0}) is a root of multiplicity greater than k of F_{p}(f(z_{0}), y) so that \frac{∂^{m}F}{∂y^{m}} (f(z_{0}), f(pz_{0})) = 0, 1 ≤ m ≤ k. Then Lemma 2.8 and the hypotheses on the derivatives of f at z_{0} give \frac{∂F}{∂x} (f(z_{0}), f(pz_{0})) = 0. So by M.1 f(z_{0}) is a multiple root of F_{p}(f(pz_{0}), y). Now the argument in the proof of Lemma 2.6 yields the result.

Proposition 2.10. Assume that for z_{1}, z_{2} ∈ H 1 f(z_{1}) = f(z_{2}), 2 \frac{df}{dz} (z_{1}) = \frac{d^{2}f}{dz^{2}} (z_{1}) = … = \frac{d^{k}f}{dz^{k}} (z_{1}) = 0,

3 \frac{df}{dz} (z_{2}) = \frac{d^{2}f}{dz^{2}} (z_{2}) = … = \frac{d^{k−1}f}{dz^{k−1}} (z_{2}) = 0.

Then \frac{d^{k}f}{dz^{k}} (z_{2}) = 0. Proof: By Lemma 2.9 there is some prime p such that f(pz_{2}) is a root of F_{p}(f(z_{2}), y) of multiplicity at most k and, by choosing p sufficiently large, such that f′(pz_{2}) ≠ 0. Since F_{p}(f(z_{1}), y) = F_{p}(f(z_{2}), y) we have f(z_{2}) = f(β(z_{1})) for some β ∈ A(p). Let z_{3} = β(z_{1}). If f′(z_{3}) = 0 then by Lemma 2.4 f′(pz_{2}) = 0: a contradiction. So f′(z_{3}) ≠ 0 and hence also (f ∘ β)′(z_{1}) = f′(z_{3})β′(z_{1}) ≠ 0. Applying Lemma 2.9 with k = 1 and g = f ∘ β gives

0 = \frac{∂F}{∂x} (f(z_{1}), f ∘ β(z_{1}))f′(z_{1}) + \frac{∂F}{∂y} (f(z_{1}), f ∘ β(z_{1}))(f ∘ β)′(z_{1})

which yields \frac{∂F}{∂y} (f(z_{1}), f(z_{3})) = 0. Continuing in this fashion yields \frac{∂^{m}F}{∂y^{m}} (f(z_{1}), f(z_{3})) = 0 for 1 ≤ m ≤ k. This is a contradiction since f(pz_{2}) = f(z_{3}) is a root of multiplicity at most k of F_{p}(f(z_{2}), y) = F_{p}(f(z_{1}), y).

Proposition 2.3 now follows by noting that if f(z_{1}) = f(z_{2}) = c, then by Proposition 2.10 there exists some k ∈ ZZ^{≥0} and analytic functions g_{1} and g_{2} such that f(z) = c + g_{1}(z)^{k+1} in a neighbourhood of z_{1} and f(z) = c + g_{2}(z)^{k+1} in a neighbourhood of z_{2}. The required local symmetry is then g_{2}^{−1} ∘ g_{1}.

§3. Existence of a unique maximal domain.

We know from Proposition 2.3 that if f(z_{1}) = f(z_{2}), then there exists a local symmetry α : D_{1} → D_{2} with z_{i} ∈ D_{i} and α(z_{1}) = z_{2}. Our goal in this section is to show this α has a unique maximal domain to which it can be analytically continued. Throughout this section let σ, σ′ : [0, 1] → H be any two continuous maps which are injective except possibly at finitely many points of [0, 1].

Definition 3.1. Write σ ≈ σ′ if for each s ∈ [0, 1] there exists a local symmetry α_{s}, defined in an open disc D_{s} about σ(s), such that α_{s} ∘ σ = σ′ in some neighbourhood V_{s} of s.

Note that ≈ is an equivalence relation. Also the equation α_{s} ∘ σ = σ′ implies that α_{s}(z) = α_{s′}(z) for all s′ ∈ V_{s}, and for all z in D_{s} ∩ D_{s′}.

Lemma 3.2.

a If σ ≈ σ′, n ∈ N_{K} and β ∈ A(n), then there exists a β′ ∈ A(n) such that β ∘ σ ≈ β′ ∘ σ′. b If α : D_{1} → D_{2} is a local symmetry, n ∈ N_{K} and β ∈ A(n), then there exists a β′ ∈ A(n)

such that β′ ∘ α ∘ β^{−1} : β(D_{1}) → β′(D_{2}) is also a local symmetry. c Let α : D_{1} → D_{2} be a local symmetry which is not a translation. Then for all sufficiently

large primes p ∈ N_{K}, there exists an β = ⟨ \begin{matrix}1 & r \\ 0 & p \end{matrix} ⟩ ∈ A(p) such that β ∘ α is a local

symmetry on pD_{1}.

Proof: First note that the points z ∈ H at which F_{n}(f(z), y) has multiple roots – say z ∈ Z_{n} – are isolated. To see this, note that y = f(β(z)) will be a multiple root if and only if

\frac{∂F_{n}}{∂y} (f(z), f(β(z))) = 0.  (3.1)

The left side of equation (3.1) is an analytic function of z. It cannot be identically zero by Lemma 2.2 (take Im (z) > nt); therefore its zeros must be isolated. Taking the union over all β ∈ A(n) of the solutions z of equation (3.1), we find that indeed the points of Z_{n} are isolated.

To prove a, choose any s ∈ [0, 1], and pick any z ∈ D_{s}, z ∉ Z_{n}. Then the equation

f(β(z)) = f(β′ ∘ α_{s}(z))  (3.2)

uniquely determines some β′ ∈ A(n). By continuity, equation (3.2) continues to hold for the same β′, for all other z ∈ D_{s}, z ∉ Z_{n}; thus again by continuity and the fact the points in Z_{n} are isolated, it holds for the same β′, for all z ∈ D_{s}. That equation (3.2) holds for all s now follows from uniqueness of β′. Part a is now immediate (the local symmetry is β′ ∘ α_{s} ∘ β^{−1}).

The proof of b is similar. Part c follows from b and Lemma 2.2.

Lemma 3.3. If σ ≈ σ′ and σ is closed (i.e. σ(0) = σ(1)), then so is σ′.

Proof: Choose any prime p ∈ N_{K} for which pσ′(0) and pσ′(1) both lie in H_{t}. By Lemma 3.2 a there exists a β ∈ A(p) such that β ∘ σ ≈ pσ′. Then f(pσ′(0)) = f(β ∘ σ(0)) = f(β ∘ σ(1)) = f(pσ′(1)); by Lemma 2.1 we must have σ′(0) − σ′(1) ∈ \frac{1}{p}ZZ. Choosing a different prime p then forces σ′(0) − σ′(1) to be an integer, n say.

Now choose any ℓ, m ∈ N_{K} for which \frac{ℓ}{m}σ′ ⊂ H_{t}. Again by Lemma 3.2 a, there exists some β ∈ A(ℓm) such that β∘σ ≈ \frac{ℓ}{m}σ′. This means, from the above paragraph, that n\frac{ℓ}{m} ∈ ZZ which, varying ℓ and m, forces n = 0.

Proposition 3.4. Every local symmetry has a unique maximal domain D.

Proof: Let α : D_{1} → D_{2} be a local symmetry of f. Choose a point z_{0} ∈ D_{1} and let D be the set of all points z ∈ H such that there exists a curve σ([0, 1]) from z_{0} to z along which α can be analytically continued. Clearly D is path connected and open and hence connected. By Lemma 3.3 α extends to a well-defined analytic function on D. If D′ is any other domain to which α can be analytically continued, then by construction all the points of D′ are contained in D and hence D is the unique maximal domain for α.

§4. Local symmetries are global symmetries.

In this section we show that the local symmetry of Proposition 2.3 extends to an automorphism of H.

Lemma 4.1. Let D_{1} and D_{2} be the maximal domains for local symmetries α_{1} and α_{2} respectively. If there exists a nonempty open set E ⊂ D_{1} ∩ D_{2} such that α_{1}(E) ⊂ H_{t} and α_{2}(E) ⊂ H_{t}, then D_{1} = D_{2}. (The set H_{t} was defined in section 2.)

Proof: We have a map α_{2} ∘ α_{1}^{−1} : α_{1}(E) ⟶ α_{2}(E). This is a local symmetry of f and so, since both α_{1}(E) and α_{2}(E) are in the H_{t}, it must be a translation, θ say. Set α_{3} = θ ∘ α_{1}. Then D_{1} is also a maximal domain for α_{3} and, moreover, the analytic functions α_{2} and α_{3} are equal on the open set E. Proposition 3.4 now forces D_{1} = D_{2}.

Lemma 4.2. [HW, Theorem 438] If s ∈ IR^{>0} is irrational, then the set {m−ns | m, n ∈ ZZ^{>0}} is dense in IR.

Proposition 4.3. Let α : B ⟶ A be a local symmetry which is not a translation, with B the maximal domain of α. Then B is invariant under two transformations z ↦ a_{i}z + b_{i}, i = 1, 2 where a_{1} and a_{2} are positive and not equal to 1 and a_{1} ≠ a_{2}^{r} for any rational r.

Proof: Choose B′ ⊆ B to be an open disc of radius η and centre z_{0} = x_{0} + iy_{0} and let A′ = α(B′). Choose any primes p, q ∈ N_{K}. By Lemma 4.2 there exist strictly increasing sequences ℓ_{i}, m_{i} ∈ ZZ^{>0}, i = 1, 2, …, for which 1 < p^{ℓ_{i}}/q^{m_{i}} < 2. Put p_{i} = p^{w}p^{ℓ_{i}} and q_{i} = q^{m_{i}} where w is an integer chosen sufficiently large so that p^{w}A′ ⊂ H_{t}. By Lemma 3.2 b there are local symmetries between p_{i}A′ and B′_{i} = β_{i}(B′) for some β_{i} ∈ A(p_{i}). Similarly there exist local symmetries between B_{i} = q_{i}B′_{i} and suitable images A_{i} of p_{i}A′. By the above inequalities all the A_{i} are contained in H_{t}. If B_{i} ∩ H_{t} ≠ ∅, then the local symmetry between A_{i} and B_{i} must be a translation, which implies that α is also a translation: a contradiction. So B_{i} ∩H_{t} = ∅. Let z_{i} = x_{i} +iy_{i} and η_{i} be the centre and radius of B_{i}. Since q_{i}/p_{i} > 1/(2p^{w}) we have y_{i} > y_{0}/(2p^{w}) and, since B_{i} ∩ H_{t} = ∅, y_{i} < t. Thus, translating by integers if necessary, all the z_{i} will lie in a compact subset of H. In addition η_{i} > η/(2p^{w}) and so by computing areas, we see that intersections must occur: say B_{i} ∩ (B_{j} + s) is nonempty for some integer s. By Lemma 4.1, the maximal domains corresponding to B_{i} and B_{j} +s must be equal. This gives us some transformation a_{1}z + b_{1} which must fix B. Since the sequence m_{i} is strictly increasing, a straightforward calculation shows that a_{1} = p^{σ}q^{τ} with τ ≠ 0. To obtain the second transformation a_{2}z + b_{2} with a_{1} ≠ a_{2}^{r} for any rational r, repeat the construction with two different primes.

Lemma 4.4. Let a_{1}, a_{2}, b_{1}, b_{2} be rational numbers such that:

1 a_{1} and a_{2} are positive and not equal to 1.

2 a_{1} ≠ a_{2}^{r} for any rational r.

3 b_{1}/(1 − a_{1}) − b_{2}/(1 − a_{2}) ≠ 0.

Let G be the subgroup of GL(2, IR) generated by g_{1} = ( \begin{matrix}a_{1} & b_{1} \\ 0 & 1 \end{matrix} ) and g_{2} = ( \begin{matrix}a_{2} & b_{2} \\ 0 & 1 \end{matrix} ). Then

the closure of G in GL(2, IR) is the subgroup \{( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) | a, b ∈ IR, a > 0\}.

Proof: Let c = b_{1}/(1 − a_{1}) − b_{2}/(1 − a_{2}), then an easy calculation shows that

(g_{1}^{−n}g_{2}^{−m}g_{1}^{n}g_{2}^{m})^{s} = ( \begin{matrix}1 & sc(1 − a_{1}^{−n})(1 − a_{2}^{−m}) \\ 0 & 1 \end{matrix} ) .

Taking the limit m → ±∞ as appropriate, we have that the closure of G contains

( \begin{matrix}1 & sc(1 − a_{1}^{n}) \\ 0 & 1 \end{matrix} ) and hence ( \begin{matrix}1 & sc(a_{1}^{n_{1}} − a_{2}^{n_{2}}) \\ 0 & 1 \end{matrix} ) for all n_{1}, n_{2}, s ∈ ZZ. By a suitable choice

of n_{1} and n_{2} we can make |c(a_{1}^{n_{1}} − a_{2}^{n_{2}})| arbitrarily small and nonzero. Since s is any integer the closure of the set of all these translations gives all real translations.

Next, note that by conditions 1 and 2 we may apply Lemma 4.2 to log(a_{1}) and log(a_{2}) to conclude that the set {a_{1}^{n}a_{2}^{m} | n, m ∈ ZZ} is dense in the positive reals. Now g_{1}^{n}g_{2}^{m} =

( \begin{matrix}a_{1}^{n}a_{2}^{m} & d \\ 0 & 1 \end{matrix} ) for some d. Thus the closure of G contains ( \begin{matrix}1 & −d \\ 0 & 1 \end{matrix} ) g_{1}^{n}g_{2}^{m} = ( \begin{matrix}a_{1}^{n}a_{2}^{m} & 0 \\ 0 & 1 \end{matrix} )

and so the closure of G contains all dilations by positive reals.

As \{( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) | a, b ∈ IR, a > 0\} is generated by real translations and dilations by positive

reals the result follows.

Corollary 4.5. The maximal domain B of a local symmetry α is either the upper half plane or the interior of a region bounded by two lines which meet at a common point on the real axis.

Proof: Let z ↦ a_{i}z + b_{i}, i = 1, 2 be the two transformations produced by Proposition 4.3. If b_{1}/(1 − a_{1}) ≠ b_{2}/(1 − a_{2}), then by Lemma 4.4 these two transformations generate a group that is dense in a group that acts transitively on the upper half plane. Hence B is the upper half plane and the result follows. If b_{1}/(1 − a_{1}) = b_{2}/(1 − a_{2}), then the two transformations generate a subgroup of the group Θ of transformations z ↦ wz + (1 − w)c, w ∈ IR^{>0}, c = b_{1}/(1 − a_{1}). In fact this subgroup is dense by the same argument as used in Lemma 4.4. Thus if z_{0} is a point in B, then B also contains the Θ-orbit of z_{0}. As the orbits of Θ are rays with tail at the point c on the real axis the result follows.

Proposition 4.6. Any local symmetry of f extends to a global symmetry.

Proof: Let α be a local symmetry of f. If α is a translation then we are done, so assume otherwise. Clearly α^{−1} is also a local symmetry of f. Let B_{1} be the maximal domain of α and B_{2} be the maximal domain for α^{−1}. Applying Lemma 3.3 to α^{−1} shows that there are no points of B_{1} at which α is not invertible. Thus α(B_{1}) ⊂ B_{2} and reversing the roles of α and α^{−1} gives α^{−1}(B_{2}) ⊂ B_{1} and hence α(B_{1}) = B_{2}. By Corollary 4.5 there are four cases to consider. If neither B_{1} nor B_{2} is H then note first that we can construct an explicit isomorphism γ_{1} : B_{1} → H given by γ_{1}(z) = (a_{1}(z − a_{2}))^{a_{3}} for suitable choices of the real parameters a_{2} and a_{3} and a complex parameter a_{1}. Similarly there is an isomorphism γ_{2} : B_{2} → H. Thus α = γ_{2}^{−1} ∘ M ∘ γ_{1} for some M ∈ SL(2, IR). We can thus find a continuous curve σ : [0, 1) → B_{1} such that lim_{t→1} σ(t) = z_{0} ∈ ∂B_{1} ∩ H and σ′ = α ∘ σ such that σ′ : [0, 1) → B_{2} ∩ H. Let w_{0} = lim_{t→1} σ′(t). By continuity f(w_{0}) = f(z_{0}) and so there is a local symmetry from a neighbourhood of z_{0} to a neighbourhood of w_{0}. This however is a contradiction since it gives an analytic continuation of α across the boundary of B_{1} which was assumed to be the maximal domain of α.

Consider next the case where B_{1} ≠ H and B_{2} = H. As above we know that α = M ∘ γ_{1}. Consider an open disc in H, bounded away from the real axis such that its centre lies on one of the boundary lines of B_{1}. Let D_{1} be the intersection of this disc with B_{1}. Let α(D_{1}) = D_{2}. Under the map γ_{1}, D_{1} maps to a simply connected region which has a bounded interval of the real axis as part of its boundary. Under M this interval must remain bounded, since if not D_{2} would contain points arbitrarily close to i∞ and so f would be unbounded on D_{2}. This is a contradiction since f is bounded on D_{1} and α is a local symmetry of f.

Thus we have α : D_{1} → D_{2} with D_{1} bounded away from the real axis and D_{2} such that part of the boundary of D_{2} is an interval of the real line. Now by Lemma 3.2 c for all

sufficiently large primes p ∈ N_{K} we have a local symmetry ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) ∘ α ∘ β^{−1} : β(D_{1}) → pD_{2}, where β = ( \begin{matrix}1 & k \\ 0 & p \end{matrix} ) for some 0 ≤ k < p. Thus taking p sufficiently large and applying a

translation if necessary, we can find a local symmetry α′ : D′_{1} → D′_{2} with D′_{1} ⊂ D′_{2}, D′_{1} ≠ D′_{2} and D′_{1} bounded and bounded away from the real axis. Let w_{0} be a point of D′_{2} that is not in D′_{1}. The points α^{′−k}(w_{0}), k ∈ ZZ^{>0} are distinct, are in D′_{1} and f has the same value at each point: a contradiction as f is analytic on H and not constant. The case B_{1} = H and B_{2} ≠ H is similar.

Thus the only possible case is B = B′ = H and so α is an automorphism of H.

Recall that the group of analytic automorphisms of H is P SL(2, IR). Our goal in the next section is to show that the subgroup of P SL(2, IR) generated by the symmetries of f is a congruence group.

§5. Modular invariance.

In this section 1_{2} will denote the identity 2×2 matrix and for any subgroup G of SL(2, IR) G_{∞} will denote the subgroup of G that stabilizes i∞. We shall obtain the following:

Proposition 5.1. Let G be a subgroup of SL(2, IR) and K be a positive integer. Suppose 1 G is a discrete subgroup.

2 G_{∞} = ⟨−1_{2}, ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )⟩.

3 For all n ∈ N_{K} and all m = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ G, there exist integers ℓ and k such that ℓ | n,

0 ≤ −k < n/ℓ and such that

( \begin{matrix}n & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}ℓ & −k \\ 0 & n/ℓ \end{matrix} )^{−1} = ( \begin{matrix}na/ℓ & ka + ℓb \\ c/ℓ & (kc + ℓd)/n \end{matrix} )  (5.1)

is in G.

Then either G = ⟨−1_{2}, ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )⟩ or G contains ±Γ_{0}(N) where N |K^{∞}.

We start with some lemmas. For m ∈ SL(2, IR) define c_{m} to be the lower left entry of m. Note that for any discrete subgroup Γ of SL(2, IR), |c_{m}| is an invariant of the double coset Γ_{∞}mΓ_{∞}.

Lemma 5.2. [Sh, p.11] Let Γ be a discrete subgroup of SL(2, IR). Given M > 0, there are only finitely many double cosets Γ_{∞}mΓ_{∞} with m ∈ Γ and |c_{m}| ≤ M.

Lemma 5.3. [Sh, p.11]

a Let Γ be a discrete subgroup of SL(2, IR). There exists an r > 0, depending only on Γ,

such that if m ∈ Γ − Γ_{∞} then |c_{m}| ≥ r. b For any r satisfying the condition of a let U = {z ∈ H | Im(z) > 1/r}. Then Γ_{∞} = {m ∈

Γ | m(U) ∩ U ≠ ∅}.

We shall not require Lemma 5.3 b until Section 8.

Lemma 5.4. If G is as in Proposition 5.1 and m ∈ G then m = λm′ where m′ ∈ GL(2, Q)^{+} (the superscript denotes positive determinants).

Proof: Let m = ( \begin{matrix}a & b \\ c & d \end{matrix} ). If c = 0 then the result follows from property 2 of G. Otherwise

define r_{m} ≡ a/c (mod 1). As G_{∞} = ⟨−1_{2}, ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )⟩, r_{m} is an invariant of the double coset

G_{∞}mG_{∞}. Also, by property 3 of G and Lemma 5.3 a, for all but finitely many primes p ∈ N_{K},

G contains m(p) = ( \begin{matrix}pa & ∗ \\ c & ∗ \end{matrix} ), with r_{m(p)} = pr_{m} and c_{m(p)} = c_{m}. If a/c is irrational, then for

distinct primes p_{1}, p_{2} ∈ N_{K}, we have p_{1}a/c ≢ p_{2}a/c (mod 1). So we obtain infinitely many double cosets G_{∞}m(p)G_{∞} with the same value of c_{m(p)}, which is a contradiction by property

1 of G and Lemma 5.2. Thus m = ( \begin{matrix}λa′ & b \\ λc′ & d \end{matrix} ) with a′ and c′ rational. Applying the same

argument to m^{−1} gives m = ( \begin{matrix}λa′ & b \\ λc′ & λd′ \end{matrix} ) with d′ rational. Finally applying the argument to

m^{2} ( or (( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) m)^{2} if a′ + d′ = 0) we get m = λm′ with m′ ∈ GL(2, Q)^{+} as required.

Definition 5.5. Call an integer matrix ( \begin{matrix}a & b \\ c & d \end{matrix} ) primitive if gcd(a, b, c, d) = 1. Let M^{∗}

be the set of primitive integer 2 × 2 matrices. For m ∈ G, by Lemma 5.4 we can write

m = λ ( \begin{matrix}a & b \\ c & d \end{matrix} ) with ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M^{∗}. Define |m| = ad − bc. This is well-defined, since the

matrix ( \begin{matrix}a & b \\ c & d \end{matrix} ) is unique up to multiplication by −1.

Lemma 5.6. |m| is an invariant of G_{∞}mG_{∞}.

Proof: Clear.

In the rest of this section we use the notation a || b if a and b are integers such that a divides b and a is coprime to b/a.

Lemma 5.7. Let m_{1} ∈ G with λ^{−1}m_{1} = ( \begin{matrix}a_{1} & b_{1} \\ c_{1} & d_{1} \end{matrix} ) an integer matrix with c_{1} ≠ 0. Choose

any n ∈ N_{K} and let m_{i} = λ ( \begin{matrix}a_{i} & b_{i} \\ c_{i} & d_{i} \end{matrix} ) for i ≥ 2 be a sequence in G obtained recursively by

applying property 3 of Proposition 5.1 to m_{i−1}. Let ℓ_{i}, k_{i} be the relevant parameters in (5.1). Let c_{∞} = c_{1}/ ∏_{i=1}^{∞} ℓ_{i} , then

a If c_{∞} ∈ ZZ then for each i ≥ 1 d_{i} is an integer; b If c_{∞} ∈ ZZ then c_{∞} ≠ 0 and if p divides gcd(n, c_{∞}) then p divides d_{i} for all i ≥ 1; c There exists a W, depending only on λ and c_{1}, such that if all prime divisors of n are

larger than W, then ℓ_{i} = 1 and d_{i} ∈ ZZ for all i ≥ 1.

Proof: Part a: by Lemma 5.3 a, ℓ_{i} = 1 for all but finitely i and so, since c_{∞} ∈ ZZ, ℓ_{i} will divide c_{i} for all i ≥ 1. Suppose d_{i_{0}} ∉ ZZ for some i_{0}. Then, since d_{i+1} = (k_{i}c_{i} + ℓ_{i}d_{i})/n and ℓ_{i} divides both n and c_{i}, it follows that d_{i} ∉ ZZ for all i > i_{0}. Thus we can choose i_{0} so that d_{i_{0}} ∉ ZZ and ℓ_{i} = 1 for all i ≥ i_{0}.

By construction d_{i} = x_{i}/y_{i} where x_{i} and y_{i} are coprime integers and y_{i} | n^{∞}. Since by hypothesis d_{i_{0}} is not an integer, there is a prime p such that p | y_{i_{0}}. Let p^{η} || n and p^{s} || y_{i_{0}}. For i ≥ i_{0} we have d_{i+1} = (k_{i}c_{i} + d_{i})/n and so p^{(i−i_{0})η+s} || y_{i}. Let p^{s′} || det(m_{1}). To find the exact power of p which divides |m_{i}| note that det(m_{i}) = det(m_{1}) and that a_{i}, b_{i} and c_{i} are integers for all i ≥ 1 and so p^{2((j−i)η+s)+s′} || |m_{i}| for all i ≥ i_{0}. It then follows from Lemma 5.6 that G_{∞}m_{i}G_{∞}, i ≥ i_{0} are infinitely many distinct double cosets, but c_{m_{i}} = c_{m_{i_{0}}} which contradicts Lemma 5.2. This establishes a.

To prove b, as noted above ℓ_{i} = 1 for all but finitely many i and as c_{1} ≠ 0 we must have c_{∞} ≠ 0 as required. If p | gcd(c_{∞}, n) then p | c_{i} for all i. If there is some i_{0} for which p/| d_{i_{0}} then since d_{i+1}(n/ℓ_{i}) = k_{i}c_{i+1} + d_{i} it follows that p/| d_{i} for all i > i_{0}. However for i sufficiently large d_{i+1}n = k_{i}c_{∞} + d_{i} so that p|d_{i}: a contradiction and so p|d_{i} for all i ≥ 1.

For part c take W > λc_{1}/r. If ℓ_{i} ≠ 1 for some i then there is some prime p | ℓ_{i} and the condition on W implies that λc_{1}/ℓ_{i} ≤ λc_{1}/p < r: a contradiction by Lemma 5.3 a. That d_{i} is an integer for all i ≥ 1 now follows from part a since c_{∞} = c_{1} is an integer.

Proof of Proposition 5.1: If all m ∈ G have c_{m} = 0 then G = ⟨−1_{2}, ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )⟩. Otherwise

G contains

m′ ( \begin{matrix}1 & −|m′| \\ 0 & 1 \end{matrix} ) m′^{−1} = ( \begin{matrix}1 + a′c′ & −a′^{2} \\ c′^{2} & 1 − a′c′ \end{matrix} ) ∈ G ∩ SL(2, ZZ)  (5.2)

and c′ ≠ 0.

Now consider any prime p which divides c′ but not K. Choose any η ≥ 1 so that p^{η} ≡ 1 (mod K), and take n = p^{η}. Let notation be as in Lemma 5.7, then ∏_{i} ℓ_{i} = p^{j} for some j and let p^{j′} ||c′^{2}. If j′ > j then c_{∞} is an integer divisible by p and as d ≡ 1 (mod p) Lemma 5.7 b yields a contradiction. Hence we have j ≥ j′. Moreover if we replace λ by λp^{j′−j} and make the appropriate rescaling of the entries of m′ = m_{1}, then the hypotheses of Lemma 5.7 a are satisfied and so p^{j−j′}d_{i} ∈ ZZ for all i. As noted in the proof of Lemma 5.7 there is some i_{0} such that c_{i_{0}} = c_{∞}. If we construct m_{i_{0}} and apply the transformation of equation (5.2) then from the above comments the result is an integer matrix for which p does not divide the lower left entry. Now iterate this process until all such primes are exhausted. Finally applying the transformation of equation (5.2) and take the Kth power of the resulting matrix. This is equivalent to applying the transformation of equation (5.2) with |m′| replaced by K|m′| and

yields a matrix m = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ G ∩ SL(2, ZZ) such that K |c, c ≠ 0 and d ≡ 1 (mod K).

Let W (m) the constant of Lemma 5.7 c. As c and d are coprime, by considering

m ( \begin{matrix}1 & k \\ 0 & 1 \end{matrix} ) for a suitable choice of k we find that G contains g = ( \begin{matrix}a & b′′ \\ c & d′′ \end{matrix} ) with d′′ ≡ 1

(mod K), d′′ positive and divisible only by primes larger than W (m). As m and g both have λ = 1 and the same bottom left entry we have W (g) = W (m) and so applying Lemma 5.7

c to g with n = d′′ and property 3 of Proposition 5.1 we find that G contains ( \begin{matrix}ad′′ & b′′ \\ c & 1 \end{matrix} ).

All operations used preserve determinants, hence, premultiplying by a suitable translation,

we see that G contains ( \begin{matrix}1 & 0 \\ c & 1 \end{matrix} ).

Consider any nonzero a, b ∈ ZZ with gcd(a, bc) = 1 and hence gcd(a, K) = 1 since K | c. Choose any prime q > bc/r, with r as in Lemma 5.3 a such that aq ≡ 1 (mod K), and any

prime p > qbc/r such that p ≡ a (mod bc). G contains ( \begin{matrix}1 & 0 \\ qbc & 1 \end{matrix} ) so, postmultiplying by a

suitable translation, we know G contains an element of the form ( \begin{matrix}1 & ∗ \\ qbc & p∗ \end{matrix} ) with determinant

equal to 1. By our choice of primes p, q, Lemma 5.7 a with n = pq implies in equation (5.1)

we must have ℓ = q and k = 0, so G contains an element of the form ( \begin{matrix}p & ∗ \\ bc & ∗ \end{matrix} ) and hence also

( \begin{matrix}a & ∗ \\ bc & ∗ \end{matrix} ), both with determinant equal to 1. As these matrices are a complete set of coset

representatives for the subgroup ⟨( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )⟩ in Γ_{0}(c) the result follows with N = c.

§6. Generalized modular equations and Hauptmoduln.

The aim of this section is to prove Theorem 1.4. For the j function, which is the Hauptmodul for SL(2, ZZ), the results are well known and may be found, for example, in [Knapp, p.335]. However, we have been unable to locate a discussion of the more general case and so we include a self-contained treatment here.

Let F_{N} be the field of automorphic functions of Γ(N) with Fourier coefficients in Q (ζ_{N}). For n coprime to N, ∗n will denote the Galois automorphism of Q (ζ_{N}) such that ζ_{N} ∗n = ζ_{N}^{n}. In Sections 6.1 to 6.5 of [Sh] it is shown that:

Theorem 6.0.

a F_{N} = Q (j, f_{a} | a ∈ (ZZ/NZZ)^{2}, a ≠ (0, 0)), for functions f_{a}(z), known as the Fricke

functions, related to the the division points of order N of the elliptic curve C /(ZZ +zZZ). b F_{N} is a Galois extension of Q (j) = F_{1} with Galois group GL(2, ZZ/NZZ)/{±1}. The

action of α ∈ GL(2, ZZ/NZZ) is given by f_{a} ↦ f_{aα}. If m ∈ SL(2, ZZ) then f_{a} ∘ m = f_{am}.

c Let n be an integer coprime to N. Then ( \begin{matrix}1 & 0 \\ 0 & n \end{matrix} ) ∈ GL(2, ZZ/NZZ) acts on F_{N} by

h ↦ h ∗ n where h ∗ n is obtained from h by applying ∗n to its Fourier coefficients. d Let M_{2}(ZZ) be the set of 2 × 2 integer matrices. Let U = ∏_{p} GL(2, ZZ_{p}) × GL(2, IR)^{+},

where the superscript denotes positive determinants. For every u ∈ U and every N, there exists an element α of M_{2}(ZZ) ∩ GL(2, Q)^{+} such that u_{p} ≡ α (mod N · ZZ_{p}), where u_{p} is the pth component of u. Set au = aα for all a ∈ (ZZ/NZZ)^{2}. Then f_{a} ↦ f_{au} defines an element of Gal(F_{N}/F_{1}), call it τ(u).

Proof:

a See [Sh, Proposition 6.9 (1)] and [Sh, p.134]. b See [Sh, Proposition 6.6 (1) and (2)] and [Sh, equation (6.1.3)]. c This follows from the explicit form of the q^{1/N}-expansion of the f_{a} in [Sh, equation

(6.2.1)] and parts a and b. d See the proof of [Sh, Proposition 6.21] and the discussion in [Sh, section 6.4].

In the rest of this section f will be a nonconstant element of F_{N} and where no confusion can arise we shall write G for G(f) and G ∗ n for G(f ∗ n). The initial results of this section hold without further restrictions on f. However later results require additional hypotheses. For example, in Proposition 6.8 we require that G contains Γ_{0}(N) and that G ∗ n^{2} = G for all n coprime to N. From Proposition 6.14 to the end of the section we shall require that f is a Hauptmodul of G and that G contains Γ_{0}(N).

Lemma 6.1. G is a discrete subgroup of SL(2, IR) which contains Γ(N) with finite index.

Proof: As f ∈ F_{N}, Γ(N) is a subgroup of G and the index must be finite, since if not f would give rise to a nonconstant function on X(Γ(N)) taking the same value at infinitely many distinct points: a contradiction. It then follows that G is discrete by [Sh, Proposition 1.11].

Lemma 6.2. There is a group homomorphism ϕ : G ⟶ P GL(2, Q)^{+} with ker(ϕ) = {±1_{2}}. \overline{G} = ϕ(G) is a discrete subgroup of P GL(2, Q)^{+}.

Proof: As G is commensurable with SL(2, ZZ), it follows ([Sh, Proposition 1.30]) that the set of cusps of G is the same as the set of cusps of SL(2, ZZ), i.e. Q ∪ {i∞}. Clearly if w is a cusp and m is an element of G then m(w) is also a cusp. It is not difficult, for example by considering m(i∞), m(0), m^{−1}(i∞) and m^{−1}(0), to use this fact to show that any element of G is a multiple of a matrix with rational entries. We may define ϕ to be the corresponding map to P GL(2, Q)^{+}. The only elements of SL(2, IR) which are multiples of the identity are ±1_{2}, however by definition −1_{2} ∈ G(f) and so ker(ϕ) = \{±1_{2}\}. As \overline{G} contains \overline{Γ_{0}(N)} = ϕ(Γ_{0}(N)) with finite index and \overline{Γ_{0}(N)} is a discrete subgroup of P GL(2, Q)^{+}, again by [Sh, Proposition 1.11] \overline{G} is a discrete subgroup of P GL(2, Q)^{+}.

Recall from section 2 that elements of P GL(2, Q)^{+} by angular brackets and that if m is a nonsingular 2 × 2 integer matrix with positive determinant then ⟨m⟩ will denote the corresponding element of P GL(2, Q)^{+}. Note also that by Lemma 6.2 any m ∈ G is a multiple of a primitive integer matrix. Recall the definition of |m| given in 5.5.

Lemma 6.3. [N2] If ⟨m⟩ ∈ \overline{G} then every prime p dividing |m| also divides N.

Lemma 6.4. If gcd(n, N) = 1 and j = (\overline{G} : \overline{Γ(N)}) then

\overline{G} = ⋃_{i=1}^{j} ⟨ \begin{matrix}a_{i} & nb_{i} \\ nc_{i} & d_{i} \end{matrix} ⟩\overline{Γ(N)} = ⋃_{i=1}^{j} \overline{Γ(N)}⟨ \begin{matrix}d_{i} & −nb_{i} \\ −nc_{i} & a_{i} \end{matrix} ⟩,

where ( \begin{matrix}a_{i} & nb_{i} \\ nc_{i} & d_{i} \end{matrix} ), 1 ≤ i ≤ j is a primitive integer matrix.

Proof: Take ⟨B⟩ ∈ \overline{G} with B a primitive integer matrix, so that by Lemma 6.3 its determinant is coprime to n. Since reduction of SL(2, ZZ) mod nN is surjective onto

SL(2, ZZ/nNZZ) we can find m ∈ SL(2, ZZ) such that m ≡ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) (mod N) and m ≡

B^{−1} ( \begin{matrix}det(B) & 0 \\ 0 & 1 \end{matrix} ) (mod n). Then ⟨Bm⟩ is in the same \overline{Γ(N)} coset as ⟨B⟩ and Bm is diago-

nal mod n.

The second equality follows from the first and the observation that for any group G and subgroup H if {g_{i}} is a complete set of left coset representatives of H in G, then {g_{i}^{−1}} is complete set of right coset representatives.

Theorem 6.5. [Sh, p.147] a For every α ∈ GL(2, Q)^{+} and every h ∈ F_{N}, h ∘ α ∈ F_{N′} for some N′. b If α ∈ GL(2, Q)^{+}, β ∈ GL(2, Q)^{+}, u ∈ U, v ∈ U and αu = vβ then (j ∘ α)^{τ(u)} = j ∘ β

and (f_{a} ∘ α)^{τ(u)} = f_{av} ∘ β.

Corollary 6.6.

a If h ∈ F_{N} and α ∈ GL(2, Q)^{+} ∩ M_{2}(ZZ) then h ∘ α ∈ F_{N det(α)}. b If G contains Γ_{0}(N), then for any n coprime to N, G ∗ n contains Γ_{0}(N).

Proof: For part a consider first f_{a} ∈ F_{N}. From Theorem 6.5 a, f_{a} ∘ α is a modular function of some level. Moreover, since αΓ(N det(α))α^{−1} ⊂ Γ(N), f_{a} ∘ α is a modular function of level N det(α). We may find γ ∈ SL(2, ZZ) such that γ^{−1}α = α′ is upper triangular . Then by Theorem 6.0 b, f_{a} ∘ α = f_{a} ∘ (γα′) = f_{aγ} ∘ α′. Since f_{aγ} is an element of F_{N}, it has a Fourier expansion with respect to exp(2πiz/N) with coefficients in Q (ζ_{N}), thus f_{aγ} ∘ α′ has a Fourier expansion with respect to exp(2πiz/N det(α)) with coefficients in Q (ζ_{N det(α)}) and so it, and hence f_{a} ∘ α, is in F_{N det(α)}. The corresponding result for j(z) follows similarly and so the corollary follows by Theorem 6.0 a.

To show part b note first that for any γ ∈ Γ_{0}(N) there exists a γ′ ∈ Γ_{0}(N) such that:

( \begin{matrix}1 & 0 \\ 0 & n \end{matrix} ) γ ≡ γ′ ( \begin{matrix}1 & 0 \\ 0 & n \end{matrix} ) (mod N).

So for any a ∈ (ZZ/NZZ)^{2}, a ≠ (0, 0), we have, by Theorem 6.0 b and c, (f_{a}∗n)∘γ = (f_{a}∘γ′)∗n. As F_{N} is generated over Q (j) by these f_{a}, this relation holds for any element of F_{N}, in particular (f ∗ n) ∘ γ = (f ∘ γ′) ∗ n = f ∗ n since Γ_{0}(N) ⊂ G.

A proof of the following corollary and an outline of the proof of Proposition 6.14 are given in [N2]. They are, however, essentially corollaries of Shimura’s more general result contained in Theorem 6.5 and we give proofs based on this fact. We believe that this approach is easier to follow than that adopted by Norton.

Corollary 6.7. Let n be coprime to N and let α = ( \begin{matrix}a & b \\ nc & d \end{matrix} ) and β = ( \begin{matrix}a & nb \\ c & d \end{matrix} ). If α is

a primitive integer matrix and ⟨α⟩ ∈ \overline{G}, then ⟨β⟩ ∈ \overline{G ∗ n}. Also if β is a primitive integer matrix and ⟨β⟩ ∈ \overline{G ∗ n}, then ⟨α⟩ ∈ \overline{G}.

Proof: Define u, v ∈ U by

u_{p} = \{ \begin{matrix}( \begin{matrix}1 & 0 \\ 0 & n \end{matrix} ) & for p|N \\ β & for p/|N and p = ∞ \end{matrix}

and

v_{p} = \{ \begin{matrix}( \begin{matrix}1 & 0 \\ 0 & n \end{matrix} ) & for p|N \\ α & for p/|N and p = ∞ \end{matrix}

Here u and v are well-defined since det(β) = det(α) and by Lemma 6.3 det(α)|N^{∞}. If

N′|N^{∞} then since u_{p} ≡ v_{p} ≡ ( \begin{matrix}1 & 0 \\ 0 & n \end{matrix} ) (mod N′ · ZZ_{p}) for all primes p we have, by Theorem 6.0 c and d, that if h ∈ F_{N′} then h^{τ(u)} = h^{τ(v)} = h∗n. In particular for any f_{a} ∈ F_{N} we have f_{av}∘β = (f_{a}^{τ(v)})∘β = (f_{a}∗n)∘β and by Lemma 6.3 and Corollary 6.6 a (f_{a}∘α)^{τ(u)} = (f_{a}∘α)∗n. Thus from Theorem 6.5 b we have (f_{a} ∘ α) ∗ n = (f_{a} ∗ n) ∘ β. Similarly (j ∘ α) ∗ n = j ∘ β. So by Theorem 6.0 a, for any h ∈ F_{N} we have (h ∘ α) ∗ n = (h ∗ n) ∘ β. Since f ∈ F_{N} and ⟨α⟩ ∈ \overline{G(f)}, we have f ∗ n = (f ∘ α) ∗ n = (f ∗ n) ∘ β and so ⟨β⟩ ∈ \overline{G ∗ n}.

The proof of the second part of the Corollary is very similar to the first part.

Let A(n) be as in section 2. Note that card(A(n)) = ψ(n) where ψ(n) = n ∏_{{p|n \atop p prime}} (1+ \frac{1}{p}).

Also fix α_{n} = ⟨ \begin{matrix}n & 0 \\ 0 & 1 \end{matrix} ⟩ ∈ A(n).

Proposition 6.8. Let T = \{( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) ∈ SL(2, IR) | t ∈ ZZ\} and \overline{T} = ϕ(T). Let n be any

integer coprime to N. If G contains Γ_{0}(N), then

⋃_{α∈A(n)} \overline{T} α\overline{G ∗ n^{−1}} = ⋃_{α∈A(n)} \overline{G}α = \overline{Γ_{0}(N)}α_{n}\overline{G ∗ n^{−1}}.

Proof: We begin by showing

⋃_{α∈A(n)} \overline{T} α\overline{Γ_{0}(N)} = ⋃_{α∈A(n)} \overline{Γ_{0}(N)}α = \overline{Γ_{0}(N)}α_{n}\overline{Γ_{0}(N)}.  (6.1)

Define

M^{∗}(n, N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M^{∗} | c ≡ 0 (mod N), ad − bc = n\}.

Let ⟨M^{∗}(n, N)⟩ be the corresponding subset of P GL(2, Q)^{+}. By Lemma 9.14 of [Knapp], we get

⟨M^{∗}(n, N)⟩ = ⋃_{α∈A(n)} \overline{Γ_{0}(N)}α.

¿From this equality it is clear that ⋃_{α∈A(n)} \overline{Γ_{0}(N)}α^{−1}\overline{T} ⊆ ⋃_{α∈A(n)} \overline{Γ_{0}(N)}α. Moreover, the reverse inclusion also holds, since for each α ∈ A(n) we have α = α^{′−1}t for some translation t and some α′ ∈ A(n) and hence

⋃_{α∈A(n)} \overline{Γ_{0}(N)}α = ⋃_{α∈A(n)} \overline{Γ_{0}(N)}α^{−1}\overline{T} .

The first equality of equation (6.1) now follows by taking inverses. The equality

⟨M^{∗}(n, N)⟩ = \overline{Γ_{0}(N)}α_{n}\overline{Γ_{0}(N)}

is established in [Sh, Proposition 3.32(1)], completing the proof of equation (6.1).

Now choose any α ∈ A(n) and g ∈ \overline{G}. By equation (6.1) we know α = hα_{n}h′ for some h, h′ ∈ \overline{Γ_{0}(N)}. By Lemma 6.4 we may write gh ∈ \overline{G} as h′′m_{i} where m_{i} equals some ⟨ \begin{matrix}a_{i} & nb_{i} \\ nc_{i} & d_{i} \end{matrix} ⟩ as in Lemma 6.4 and h′′ ∈ \overline{Γ_{0}(N)}. Now m_{i}α_{n} = α_{n}m′_{i} where

m′_{i} = ⟨ \begin{matrix}a_{i} & b_{i} \\ n^{2}c_{i} & d_{i} \end{matrix} ⟩ ∈ \overline{G ∗ n^{−1}} by Corollary 6.7. Thus we have shown that

⋃_{α∈A(n)} \overline{G}α ⊂ \overline{Γ_{0}(N)}α_{n}\overline{G ∗ n^{−1}},

where we absorbed the h′ because by Corollary 6.6 b \overline{Γ_{0}(N)} ⊆ \overline{G ∗ n^{−1}}. The reverse inclusion follows by identical arguments. Finally, multiplying equation (6.1) on the right by \overline{G ∗ n^{−1}}, we get

⋃_{α∈A(n)} \overline{T} α\overline{G ∗ n^{−1}} = \overline{Γ_{0}(N)}α_{n}\overline{G ∗ n^{−1}},

as required.

The following result is contained in the proof of [N2, Theorem 3].

Corollary 6.9. Let n be any integer coprime to N. Suppose G contains Γ_{0}(N) and c is a cusp of \overline{G}. Then \overline{G}(α(c)) = \overline{G}(i∞) for some α ∈ A(n) if and only if \overline{G ∗ n^{−1}}(c) = \overline{G ∗ n^{−1}}(i∞).

Proof: If \overline{G}(α(c)) = \overline{G}(i∞) then gα(c) = i∞ for some g ∈ \overline{G} and so by Proposition 6.8 tα′g′(c) = i∞ for some t ∈ \overline{T}, for some α′ ∈ A(n) and for some g′ ∈ \overline{G ∗ n^{−1}}. Since tα′ fixes i∞ this yields g′(c) = i∞.

For the reverse implication, if g′(c) = i∞ for some g′ ∈ \overline{G ∗ n^{−1}} then α_{n}g′(c) = i∞ and so by Proposition 6.8 there is some g ∈ \overline{G} and some α ∈ A(n) such that gα(c) = i∞ and so \overline{G}(α(c)) = \overline{G}(i∞) as required.

The following result is a small generalization of a result contained in the proof of [Sh, Proposition 6.9]. It will be used in the proof of Proposition 6.16.

Proposition 6.10. Let G be a discrete subgroup of SL(2, IR) which contains Γ(N) and let A_{0} be the field of automorphic functions of G. If A_{0} = C (f_{1}, …, f_{m}) for some functions f_{1}, …, f_{m} ∈ F_{N}, then A_{0} ∩ F_{N} = Q (ζ_{N}, f_{1}, …, f_{m}).

Proof: The inclusion Q (ζ_{N}, f_{1}, ..., f_{m}) ⊂ A_{0} ∩ F_{N} is obvious. For the reverse inclusion, consider any g ∈ A_{0} ∩ F_{N} and expand

g(q) = ∑_{i}^{∞} a_{i}q^{i/N} = ∑_{i=1}^{m} c_{i}f_{i}(q) = ∑_{i=1}^{m} ∑_{j}^{∞} c_{i}b_{i,j}q^{j/N},

where each a_{i}, b_{i,j} ∈ Q (ζ_{N}), c_{i} ∈ C. This equality yields a linear system for the c_{i} with coefficients in Q (ζ_{N}) which has at least one solution in C and hence a solution in Q (ζ_{N}).

Definition 6.11. For any n coprime to N we define w(n) ∈ GL(2, ZZ/NZZ) by w(n) ≡

( \begin{matrix}n & 0 \\ 0 & n \end{matrix} ) (mod N). By Theorem 6.0 b, w(n)\{±1\} is an element of the Galois group of F_{N}

over Q (j) and we shall write its action on any h ∈ F_{N} as h ↦ h!n.

The next three results show that in the case that f is a Hauptmodul for G and n is coprime to N then G ∗ n^{2} = G.

Lemma 6.12. If h ∈ F_{N}, m ∈ \overline{G(f)} and n is coprime to N then (h ∘ m)!n = (h!n) ∘ m.

Proof: By Lemma 6.2, m = ⟨m′⟩, where m′ is a primitive integer matrix. The proof is essentially identical to that of Corollary 6.7 with α = β = m′ and u = v where

v_{p} = \{ \begin{matrix}( \begin{matrix}n & 0 \\ 0 & n \end{matrix} ) & for p|N, \\ α & for p/|N and p = ∞. \end{matrix}

Lemma 6..13. If \overline{G} contains \overline{Γ_{0}(N)} then f!n = f ∗ n^{2} for all n coprime to N.

Proof: Any upper triangular matrix in SL(2, ZZ/NZZ) is the image under reduction modulo N of some element of Γ_{0}(N). So for any n coprime to N we can find m(n) ∈ Γ_{0}(N) such that

m(n) ≡ ( \begin{matrix}n^{−1} & 0 \\ 0 & n \end{matrix} ) (mod N). By Theorem 6.0 b and c we find (f_{a}∘m(n))!n = (f_{am(n)})!n =

f_{am(n)w(n)} = f_{a} ∗ n^{2}. Hence, by Theorem 6.0 a, for any h ∈ F_{N}, (h ∘ m(n))!n = h ∗ n^{2}. In particular, since m(n) ∈ G, f!n = f ∗ n^{2}.

In the remainder of this section will take f ∈ F_{N} to be a Hauptmodul of G which will contain Γ_{0}(N). We shall also normalize f to have zero constant term.

Proposition 6.14. f ∗ n^{2} = f for any n coprime to N.

Proof: Let n be an integer coprime to N. Since ∗n^{2} is an automorphism, f ∗ n^{2} ∈ F_{N}. Also, by Lemma 6.12 and Lemma 6..13, f ∗ n^{2} is fixed by G and so by [Sh, Proposition 2.6] it is an automorphic function for G. Since f is a Hauptmodul the field of automorphic functions of G is C (f) and hence by Proposition 6.10, f ∗ n^{2} ∈ Q (ζ_{N}, f). Similarly f ∗ n^{−2} can be expressed as a rational function of f and so applying ∗n^{2} to this expression we find that f ∈ C (f ∗ n^{2}) and hence C (f) = C (f ∗ n^{2}). Thus f ∗ n^{2} is also a normalized Hauptmodul for G and since there is only one such function we have f ∗ n^{2} = f.

Lemma 6.15. For any n coprime to N, with A(n) as in equation (6.1), the functions f ∘α, α ∈ A(n) are distinct.

Proof: Since f is a Hauptmodul its q-expansion is of the form f = 1/q + . . .. Now use the same proof as for the j function, see for example [Knapp, Lemma 11.31].

We now follow the standard proof of the existence of modular polynomials for the j function. See for example [Knapp, p.334-6] or [Sh, p.108-110]. Note that the possibility that the coefficients of f are irrational introduces a slight complication into the proof of the following result, as we do not know a priori that f ∗ n is a Hauptmodul. In fact, once we have proven Theorem 1.3 we can use it to show that f ∗ n is a Hauptmodul for G ∗ n; the condition that the coefficients of f are algebraic integers is not required since, for example by Corollary 6.6 b, G ∗ n is not trivial.

Proposition 6.16. For any n coprime to N, any symmetric polynomial of {f ∘α | α ∈ A(n)} is an element of H[f ∗ n] where H is the field generated over Q by the coefficients of f.

Proof: Let s be any symmetric polynomial with rational coefficients of the {f ∘α | α ∈ A(n)}. By Corollary 6.6 a each f ∘ α, α ∈ A(n) is a modular function of level nN, and hence so is s. By Proposition 6.8 and Lemma 6.15, s(z) is invariant under \overline{G ∗ n} and it follows from [Sh, Proposition 2.6] that s(z) is an automorphic function for \overline{G ∗ n}. In particular s(z) has a q-expansion (rather than a q^{1/n}-expansion) as it is invariant under z ↦ z + 1. Each coefficient in this q-expansion lies in Q (exp(2πi/nN)), the Galois group of which is (ZZ/nNZZ)^{∗} ≅ (ZZ/nZZ)^{∗} × (ZZ/NZZ)^{∗}. The action of r × 1 is to replace f ∘ α by f ∘ α′ where

if α = ⟨ \begin{matrix}a & b \\ 0 & d \end{matrix} ⟩ then α′ = ⟨ \begin{matrix}a & b′ \\ 0 & d \end{matrix} ⟩ where b′ ≡ rb (mod d). The effect of 1 × w is to replace

f ∘ α by (f ∗ w) ∘ α. It follows that the q coefficients of s(z) are fixed by (ZZ/nZZ) × Fix(H) and so lie in H. In particular we have s(z) ∈ F_{N}.

By Corollary 6.6 b, if G contains Γ_{0}(N) then so does G ∗ n and so G ∗ n satisfies the hypotheses of Proposition 6.8. The proof of Lemma 6.15 also works for f ∗ n. Thus we can repeat the argument at the start of this proof to deduce s ∗ n is invariant under \overline{G}. By Proposition 6.10, s ∗ n = (P ∗ n)(f)/(Q ∗ n)(f) where P (x), Q(x) ∈ Q (ζ_{N})[x] and so s = P (f ∗ n^{−1})/Q(f ∗ n^{−1}) = P (f ∗ n)/Q(f ∗ n).

It remains to show that Q is a constant. Suppose s = P (f ∗ n)/Q(f ∗ n) and that Q is not a constant. Then Q has some nontrivial factor f ∗ n − c. Since f is analytic in the upper half plane so is s and so there can be no point z_{0} ∈ H with (f ∗ n)(z_{0}) = c. Also since f ∗ n is a modular function of level N it must take the value c at some point in H^{∗} (see, for example, [Mu, Corollary 1.2]). Thus c must be a cusp value of f ∗ n. Let z_{0} be one of the corresponding cusps, so lim_{z→z_{0}}(f ∗ n)(z) = c (limits at cusps are taken in the topology described, for example, in [Sh, Chapter 1]). We have (f ∗ n)(i∞) = ∞ and so there is no ⟨m⟩ ∈ \overline{G ∗ n} such that ⟨m⟩(z_{0}) = i∞ and so by Corollary 6.9 for each α ∈ A(n), there is no ⟨m′⟩ ∈ \overline{G} such that ⟨m′⟩(α(z_{0})) = i∞. Since f is a Hauptmodul it is finite at any cusp not in G(i∞) and it follows that for each α ∈ A(n), lim_{z→z_{0}} f(α(z)) is finite and hence lim_{z→z_{0}} s(z) is finite. This is a contradiction and so Q must be a constant and s is a polynomial in f ∗ n as required.

Finally, if we write s = ∑ c_{i}(f ∗ n)^{i}, then equating the nonpositive q coefficients on each side yields a linear system for the c_{i} with coefficients in H. It follows that s(z) ∈ H[f ∗ n].

Let

F_{n}(y) = ∏_{α∈A(n)}(y − f ∘ α)

Proposition 6.17. For each n coprime to N, the polynomial F_{n}(y) has coefficients in H[f ∗n] and is irreducible over C (f ∗ n).

Proof: By Proposition 6.16 the coefficients of F_{n}(y) are in H[f ∗ n] ⊂ C (f ∗ n). Also F_{n}(y) splits in C (f ∘ α|α ∈ A(n)); \overline{G ∗ n} acts on this field, fixes C (f ∗ n) and permutes the roots of F_{n}(y). By Proposition 6.8, each α ∈ A(n) is in \overline{Γ_{0}(N)}α_{n}\overline{G ∗ n} so we have α = γα_{n}γ′, γ ∈ \overline{Γ_{0}(N)} and γ′ ∈ \overline{G ∗ n}. Hence γ^{−1}α = α_{n}γ′ and so \overline{G ∗ n} acts transitively on the roots. By Lemma 6.15 the roots are distinct and so F_{n}(y) is irreducible over C (f ∗ n).

Since the ring H[f ∗ n] is isomorphic to H[x] there is a unique polynomial F_{n}(x, y) ∈ H[x, y] such that F_{n}(f ∗ n, y) = F_{n}(y).

Proposition 6.18. For any n > 1 coprime to N, F_{n}(x, y) is a generalized modular polynomial for f.

Proof: By Proposition 6.17 F_{n}(x, y) ∈ H[x, y] and by the definition of F_{n}(x) properties MI.2 and MI.3′ are satisfied. Thus it remains to show that F_{n}(x, y) = (F_{n} ∗ n)(y, x). We follow the proof in [K, Proposition 3.2]. Let a, b, d ∈ ZZ with ad = n, gcd(a, b, d) = 1 and

0 ≤ b < d. Also let b′ = a − b, β_{1} = ⟨ \begin{matrix}d & b′ \\ 0 & a \end{matrix} ⟩, β_{2} = ⟨ \begin{matrix}a & b \\ 0 & d \end{matrix} ⟩ and z′ = β_{2}(z). We have that

F_{n}((f ∗ n)(z′), f(β_{1}(z′))) = 0. Since β_{1}∘β_{2}(z) = z+1 it follows that F_{n}((f ∗n)(β_{2}(z)), f(z)) = 0. Let n′ be such that n′ ≡ n (mod N) and n′ ≡ 1 (mod n). Then applying ∗n′ we find (F_{n} ∗ n)(f(β_{2}(z)), (f ∗ n)(z)) = 0. Thus the roots of χ_{n}(y) = (F_{n} ∗ n)(y, (f ∗ n)(z)) include those of F_{n}(y) = F_{n}((f ∗ n)(z), y). So c(y)F_{n}(y) = χ_{n}(y) for some polynomial c(y). We can compute c as follows: a straightforward calculation shows that the leading term in the q-expansion of F_{n}(0) is q^{t}, t = −ψ(n). Since F_{n}(0) is a polynomial in f ∗ n we can deduce that its leading term is (f ∗ n)^{t}. The other coefficients of F_{n}(y) are lower degree polynomials in f ∗ n. It follows that the leading term of χ_{n}(y) is y^{t} and as the leading term of F_{n}(y) is y^{t} the result follows. Proof of Theorem 1.4 1: This follows from Proposition 6.14 and Proposition 6.18.

§7. The main result.

In this section we obtain Theorem 1.3 and Theorem 1.4 2; before giving the proof we start with two Lemmas.

Lemma 7.1. If f satisfies the hypotheses of Theorem 1.3 and if G is the group of symmetries of f, then G satisfies the hypotheses of Proposition 5.1.

Proof: First, assume for contradiction that G is not discrete. Pick a sequence of distinct elements α_{i} ∈ G, i ∈ ZZ^{>0} which converges to α ∈ SL(2, IR), such that for all i and j, α_{i} ≠ ±α_{j}. For each i, j, α_{i}α_{j}^{−1} has at most 1 fixed point in H, and thus we can choose some z_{1} ∈ H which avoids those countably many points. It will obey α_{i}(z_{1}) ≠ α_{j}(z_{1}) ∀i, j. We have lim_{i→∞} α_{i}(z_{1}) = α(z_{1}) and for all i f(α_{i}(z_{1})) = f(z_{1}). But the points α_{i}(z_{1}) are distinct, so f takes on the same value infinitely many times in a compact neighbourhood of α(z_{1}), which is impossible since f is not constant. Thus G must be discrete.

Next, suppose that the transformation z ↦ (az + b)/d is in G. Then a ≠ 0 and for Im (z) sufficiently large we have that both (az + b)/d ∈ H_{t} and z ∈ H_{t}. As f((az + b)/d) = f(z) by Lemma 2.2 we must have (az + b)/d − z ∈ ZZ. It follows that a = d and d divides b so that z ↦ (az + b)/d is a translation by an integer as required.

Finally, note that the third condition of Proposition 5.1 is just Lemma 3.2 b in the case that α is a global symmetry.

Lemma 7.2. With f as in Lemma 7.1, suppose \overline{G}(f) is trivial (i.e. consists only of translations by integers) and that all the coefficients a_{n} of f are algebraic integers. Then f = q^{−1}+ζq where ζ^{gcd(24,K)+1} = ζ.

Proof: If f = q^{−1} then we are done, otherwise let a_{j} be the first nonzero coefficient of f. Then |a_{j}^{σ}| ≥ 1 for some σ ∈ Gal(Q̄ /Q), as can be seen by looking at the constant term of the minimal polynomial of a_{j}. Let h be the formal q-series corresponding to f as in the introduction. It is straightforward to show that as a formal series h^{σ} satisfies a modular equation of order n for all n ≡ 1 (mod K). In fact this modular equation is just F_{n}^{σ}. Thus, as discussed in the introduction, results of Mahler and Kozlov imply that (h^{σ})(exp(2πiz)) extends to an analytic function on the upper half plane which we shall call f^{σ}. It follows that f^{σ} satisfies the hypotheses of Theorem 1.3. By Lemma 7.1 and Proposition 5.1 either \overline{G(f^{σ})} is trivial or f^{σ} is a modular function of level N for some N | K^{∞}. The latter case, however, is impossible: By [Sh; Proposition 6.1] we have f^{σ} = P/Q, where P and Q are polynomials with complex coefficients in j and the Fricke functions (notation as in Theorem 6.0). Using a similar argument to that of Proposition 6.10 we can take the coefficients of P and Q to lie in Q̄. But then applying σ^{−1} to f^{σ} it is clear that f is also a modular function of level N so that \overline{Γ(N)} ⊂ \overline{G}: a contradiction since by hypothesis \overline{G} is trivial.

As in Lemma 2.2 let \overline{f^{σ}} be the function meromorphic on the interior of the unit disk defined by the relation \overline{f^{σ}}(exp(2πiz)) = f^{σ}(z) for all z ∈ H. We have shown that \overline{f^{σ}} is injective on |q| < 1. By Theorem 1.4 of [Mark],

∑_{i=j}^{∞} i|a_{i}^{σ}|^{2} ≤ 1.  (7.1)

Since |a_{j}^{σ}| ≥ 1, it follows that |a_{1}^{σ}| = 1 and a_{i} = 0 for i > 1. But then examining the sum of the roots of the modular equation of order n = K + 1 for f shows that 1/q^{K+1} + a_{1}q^{K+1} = Q_{K+1}(f(q)) and comparing the terms in q^{K+1} yields a_{1}^{K+1} − a_{1} = 0 and so a_{1} is a primitive sth root of 1 for some s|K. It remains to show that s|24.

Let n, s ∈ ZZ satisfy n ≡ 1 (mod K) and s|K. Also let R = Q (ζζ_{n})[q^{−1/n}, q^{1/n}] where ζ is a primitive sth root of 1. Let W be the group of automorphisms of R generated by g_{1} : q^{1/n} ↦ ζ_{n}q^{1/n}, and g_{2} : q^{1/n} ↦ ζ^{−1}q^{−1/n}. Let f = q^{−1} + ζq and R^{W} = {r ∈ R | g(r) = r, ∀g ∈ W }. Then it is not difficult to show that R^{W} = Q (ζζ_{n})[f].

Thus if f satisfies a modular equation of order n, then the group W permutes the roots of P (x) = F_{n}(f, x). In particular if r_{1} = ζ_{n}^{−ab}q^{−a^{2}/n} + ζζ_{n}^{ab}q^{a^{2}/n} is one such root, then so is g_{2}(r_{1}), this can only be the case, however, if a^{2} ≡ 1 (mod s). Repeating this argument for sufficiently many n we find that the exponent of the group (ZZ/sZZ)^{∗} is either 2 or 1 and so s|24.

Proof of Theorem 1.3: If f is such that for all z_{1}, z_{2} ∈ H f(z_{1}) = f(z_{2}) implies that z_{1} − z_{2} ∈ ZZ and the coefficients of f are algebraic integers, then by Lemma 7.2 f = q^{−1} + ζq where ζ^{gcd(24,K)+1} = ζ. If not, choose two points z_{1}, z_{2} ∈ H such that f(z_{1}) = f(z_{2}) and z_{1} − z_{2} ∉ ZZ. By Proposition 2.3 and Proposition 4.6 there exists an element m ∈ SL(2, IR) such that m(z_{1}) = z_{2} and f(m(z)) = f(z). Let m(z) = (az + b)/(cz + d). If c = 0 then by Lemma 7.1 m is a translation by an integer: a contradiction. Thus we must have c ≠ 0. From Proposition 5.1 and Lemma 7.1 the symmetry group G of f contains Γ_{0}(N) for some N, with N |K^{∞}.

Thus f(z) gives rise to a function f̂ on X(G) that is analytic except possibly at the cusps. At the cusp corresponding to infinity f̂ has a simple pole. At the other cusps, if any, f̂ is bounded and hence has removable singularities. It follows that f is an automorphic function. By Proposition 2.3 and Proposition 4.6 f̂ is injective on X(G), which is hence of genus zero and f is a Hauptmodul for G, as required. Proof of Theorem 1.4 2: If f(z) = q^{−1}+ζq with ζ a primitive sth root of 1 and s|gcd(24, K) then using the techniques in the proofs of Lemma 6.16 and Lemma 7.2 it can be shown that any symmetric polynomial in {f ∘α | α ∈ A(n)} lies in Q (ζ)[f]. Defining F_{n}(x, y) as in Proposition 6.18, the proof of Proposition 6.18 shows that F_{n} is a generalized modular polynomial for f.

In the case that f(z) = q^{−1} the proof is identical, except that in this case the group W of Lemma 7.2 is generated by g_{1} : q^{1/n} ↦ ζ_{n}q^{1/n}.

§8. Comments and conjectures.

Replicable functions In [N1] Norton introduced the notion of a replicable function. A formal q-series

h(q) = q^{−1} + ∑_{n=1}^{∞} h_{n}q^{n}

is said to be replicable if for each a ∈ ZZ^{>0} there exists a formal q-series

h^{(a)}(q) = q^{−1} + ∑_{n=1}^{∞} h_{n}^{(a)}q^{n}

such that the analogue of equation (1.1) holds, namely:

∑_{{ad=n \atop 0≤b<d}} h^{(a)}(ζ_{d}^{b}q^{\frac{a}{d}}) = Q_{n}(h(q)),

where Q_{n} depends on h and is the unique polynomial such that

Q_{n}(h(q)) = q^{−n} + terms of positive degree.

One aim of this definition is to investigate to what extent the moonshine conjectures can be extended to include other groups and automorphic functions. In the case that h has rational integer coefficients Norton conjectured that h is replicable if and only if h is the q-expansion of a function f analytic on the upper half plane and satisfying condition 2 of Theorem 1.3. In [CuN] it was shown that any Hauptmodul with rational integer coefficients is replicable.

Our results do not provide a proof of Norton’s conjecture. However Norton also introduced the notion of a completely replicable function of order K. These are replicable functions which satisfy the additional constraints * that for all a, b ∈ ZZ^{>0} h^{(a)} is replicable, h^{(a)}^{(b)} = h^{(ab)} and h^{(a)} = h^{(gcd(a,K))}. These properties are satisfied by the moonshine functions. Completely replicable functions are discussed in more detail by Kozlov [K]. He proves in particular that if f is a completely replicable function of order K then f satisfies a modular equation of order n for all n coprime to K ([K, Proposition 4.1] and [K, Proposition 3.2]), see

* Note although these conditions can be imposed on any h(q), Norton originally proposed them only for the case where the coefficients h_{n} are rational. A conjecture has been made as to how the definition of a replicable function should be modified when the h_{n} lie in a compositum of quadratic fields [N2]. That such a modification is necessary if Hauptmoduln with irrational coefficients are to be replicable can be seen, for example, by considering the sum of the roots of a generalized modular equation F_{n}((h ∗ n)(q), y) when n ≢ 1 (mod N). also [Mar]. Thus from Theorem 1.3 we deduce that every completely replicable function of order K with coefficients which are algebraic integers satisfies condition 2 of Theorem 1.3. The converse is not true even if we retrict to the case where h(q) has rational coefficients: there are Hauptmoduln satisfying condition 2 of Theorem 1.3 which are not completely replicable [N1].

A generalization of the Mahler recurrence relations Mahler [Mah] used the modular equation of order 2 to give recurrence relations for the coefficients of the j function. More generally these recurrences determine the coefficients of any odd level Hauptmodul, f = q^{−1} + ∑_{n=1}^{∞} a_{n}q^{n}, a_{n} ∈ Q, n = 1, 2, …, once a_{1}, a_{2}, a_{3} and a_{5} are given. Using Theorem 1.4 we can derive a similar set of recurrence relations that hold for any Hauptmodul of level N whose coefficients lie in Q (ζ_{N}) when N is odd. These recurrences are, for k ≥ 1:

a_{4k} = a_{2k+1} ∗ 2 + ∑_{j=1}^{k−1}(a_{j}a_{2k−j}) ∗ 2 + \frac{1}{2}((a_{k} ∗ 2)^{2} − a_{k}),

a_{4k+1} = a_{2k+3} ∗ 2 + ∑_{j=1}^{k}(a_{j}a_{2k+2−j}) ∗ 2 + \frac{1}{2}((a_{k+1} ∗ 2)^{2} − a_{k+1}) + \frac{1}{2}(a_{2k}^{2} + a_{2k} ∗ 2)

− (a_{2}a_{2k}) ∗ 2 + ∑_{j=1}^{k−1} a_{j}a_{4k−4j} + ∑_{j=1}^{2k−1}(−1)^{j}a_{j}a_{4k−j},

a_{4k+2} = a_{2k+2} ∗ 2 + ∑_{j=1}^{k}(a_{j}a_{2k+1−j}) ∗ 2,

and

a_{4k+3} = a_{2k+4} ∗ 2 + ∑_{j=1}^{k+1}(a_{j}a_{2k+3−j}) ∗ 2 − \frac{1}{2}(a_{2k+1}^{2} − a_{2k+1} ∗ 2)

− (a_{2}a_{2k+1}) ∗ 2 + ∑_{j=1}^{k} a_{j}a_{4k+2−4j} + ∑_{j=1}^{2k}(−1)^{j}a_{j}a_{4k+2−j}.

Weakening the hypotheses of Theorem 1.3 It seems probable that the condition in part 2 of Theorem 1.3 that the coefficents of f are algebraic integers can be weakened. We make the following:

Conjecture 8.1. Let f be an analytic function on H with Fourier expansion f(z) = q^{−1} + ∑_{n=1}^{∞} a_{n}q^{n}, q = exp(2πiz), a_{n} ∈ C, n ∈ ZZ^{>0} and let K be a positive integer. Then the following are equivalent:

1 f satisfies a modular equation of order n for all n ≡ 1 (mod K).

2 f is either q^{−1} + ζq where ζ^{gcd(24,K)+1} = ζ, or is a Hauptmodul for a subgroup

G of SL(2, IR) satisfying: a G contains Γ_{0}(N) with finite index for some N |K^{∞}.

b G contains ( \begin{matrix}1 & k \\ 0 & 1 \end{matrix} ) if and only if k ∈ ZZ. c X(G) has genus zero. d a_{i} ∈ ZZ[ζ_{N}], i ∈ ZZ^{>0}.

Some evidence is provided for this conjecture by the following:

Proposition 8.2. Let K ∈ ZZ^{>0} and P ⊂ N_{K} be any infinite set of primes. Then there are only finitely many formal q-series h of the form h = q^{−1} + ∑_{n=1}^{∞} a_{n}q^{n}, a_{n} ∈ C such that h satisfies a modular equation of order p for all p ∈ P. Moreover the coefficients of any such h lie in a finite extension of Q.

Proof: In sections 13 and 29 of [Mah] it is shown that if h satisfies a modular equation of order ℓ for some prime ℓ and m = ℓ^{2} + ℓ − 1, then the a_{i}, i > m are polynomials in a_{1}, …, a_{m}. The coefficients of these polynomials are in Q by a argument similar to that of Proposition 6.16. Consider p ∈ P and F_{p}(x) = F_{p}(h(q), x) = c_{0}(q) + c_{1}(q)x + … + c_{t}(q)x^{t} where F_{p}(x, y) is a modular polynomial for h and t = p + 1. Initially we only know that the coefficients of F_{p}(x, y) are complex numbers. Once again, however, an argument similar to the one in the proof of Proposition 6.16 shows that the c_{i}(q) are q-series with coefficients in Q [a_{1}, a_{2}, …] and as noted above Q [a_{1}, a_{2}, …] ⊆ Q [a_{1}, …, a_{m}]. Since F_{p}(x, y) is a modular polynomial each c_{i}(q) is a polynomial P_{i}, say, in h(q) which can be calculated by equating the coefficients and the nonpositive terms in the q-expansions of c_{i}(q) and P_{i}. The fact that the leading term of h(q)^{i} is q^{−i} means that P_{i} has coefficients in Q [a_{1}, …, a_{m}]. Hence each modular polynomial F_{p}(x, y) for h(q) has coefficients in Q [a_{1}, …, a_{m}]. It is not difficult to see that h(q) will satisfy M.3 if and only if F_{p}(h(q), h(q^{p})) = 0. Thus h will satisfy a modular equation of order p for all p ∈ P if and only if (a_{1}, …, a_{m}) is a zero of some ideal I in Q [x_{1}, …, x_{m}].

As noted in the introduction, the results of Mahler and Kozlov imply that f(z) = h(exp(2πiz)) extends to an analytic function on the upper half plane. Not all of our previous results apply. However the results of section 2 are still available by the following argument: For any analytic function g(z) on H such that g(z) = g(z + 1) define the Hecke operators

T_{n}(g) = ∑_{ad=n, 0≤b<d} g((az + b)/d), T_{n}^{∗}(g) = ∑_{{ad=n, 0≤b<d \atop gcd(a,b,d)=1}} g((az + b)/d).

The standard algebra of Hecke operators gives

T_{n}(T_{m}(g)) = ∑_{d| gcd(m,n)} d T_{mn/d^{2}}(g)  (8.1)

and we also have

T_{n}(g) = ∑_{d^{2}|n} T_{n/d^{2}}^{∗}(g).  (8.2)

We observe that f(z) obeys a modular equation of order n if and only if T_{n}^{∗}(f^{j}) is a polynomial in f(z) for all j ∈ ZZ^{>0}. If all the prime divisors of n are in P, then equations (8.1) and (8.2) by induction show that T_{n}^{∗}(f^{j}) is a polynomial in f(z) for all j ∈ ZZ^{>0} and so f(z) satisfies a modular equation of order n.

The results of sections 3 and 4 still apply and together with the first part of Lemma 7.1 show that if f(z_{1}) = f(z_{2}) then there exists m ∈ SL(2, IR) such that m(z_{1}) = z_{2} and that G(f) satisfies conditions 1 and 2 of Proposition 5.1. Let U be as in Lemma 5.3 b. If z_{1}, z_{2} ∈ U, f(z_{1}) = f(z_{2}) and m(z_{1}) = z_{2} then by Lemma 5.3 b we must have m ∈ G_{∞}.

Thus f(z) is injective modulo 1 on U. Moreover if m ∈ G then m′ = m ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) m^{−1} ∈ G

and c_{m′} = c_{m}^{2}. Iterating this procedure we see from Lemma 5.3 a that for all m ∈ G we must have |c_{m}| ≥ 1. Thus we may choose r = 1 in Lemma 5.3 so that f(z) is injective modulo 1 on {z ∈ H | Im(z) > 1}. Making an appropriate rescaling and applying equation (7.1) we find that the q coefficients of f(z) are bounded by

|a_{j}| ≤ e^{2π(j+1)}, j = 1, 2, ….  (8.3)

It can be shown, for example by constructing a Gröbner basis using a reverse lexicographic term ordering, that either there are only finitely many zeros of I all of which lie in some finite extension of Q or there are zeros (a_{1}, …, a_{m}) of I with max_{j} |a_{j}| arbitrarily large (see for example [BW, Propositions 7.42 and 7.52]). However the latter case is impossible by equation (8.3). The result now follows.

References

[B] R.E. Borcherds, Monstrous Moonshine and monstrous Lie superalgebras, Invent. Math. 109, 405-444 (1992).

[BR] R.E. Borcherds and A. J. E. Ryba, Modular Moonshine II, Duke Math. J. 83, 435-459 (1996).

[BW] T. Becker and V. Weispfenning, Gröbner Bases, Springer-Verlag, (1993).

[CM] H. Cohn and J. McKay, Spontaneous generation of modular invariants, Math. of Comp. 65, 1295-1309 (1996).

[CN] J.H. Conway and S.P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11, 308-339 (1979).

[CuN] C.J. Cummins and S.P. Norton, Rational hauptmoduls are replicable, Can. J. Math. 47, 1201-1218 (1995).

[CY] I. Chen and N. Yui, Singular values of Thompson series, in Groups, Difference sets and the Monster, eds. K.T. Arusu et al, de Gruyter (1995).

[D] P.L. Duren, Univalent Functions, Springer-Verlag (1983).

[FLM] I.B. Frenkel, J. Lepowsky and A. Meurman, Vertex operators and the monster, Academic Press: Boston (1988).

[HW] G.H. Hardy and E.M. Wright An introduction to the theory of numbers, Fifth Edition, Clarendon Press: Oxford, (1979).

[Knapp] A. W. Knapp, Elliptic Curves, Princeton University Press, (1992).

[Koi] M. Koike, On replication formula and Hecke operators, Nagoya University, (preprint).

[K] D. N. Kozlov, On completely replicable functions and extremal poset theory, M.Sc. thesis, Department of Mathematics, University of Lund, Sweden, (1994).

[L] S. Lang, Elliptic functions, 2nd edition, Addison-Wesley: Reading Mass (1987).

[Mah] K. Mahler, On a class of non-linear functional equations connected with modular functions, J. Austral. Math. Soc. 22A, 65-118 (1976).

[Mar] Y. Martin, On modular invariance of completely replicable functions, in Moonshine, the Monster, and related Topics, eds. C Dong and G. Mason, Contemporary Mathematics 193, 263–286, Amer. Math. Soc., Providence, RI, (1996).

[Mark] A. Markushevich, Theory of functions of a complex variable, Vol. III, Chelsea Publishing Company, (1977).

[M] G. Mason (with an appendix by S.P. Norton), Finite groups and modular functions, Proc. Symp. Pure Math. 47, Part 1, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), 181–210, Amer. Math. Soc., Providence, RI, (1987).

[Mu] V. K. Murty, Introduction to Abelian Varieties, CRM Monograph Series, Amer. Math. Soc., Providence, RI, (1993).

[N1] S. P. Norton, More on Moonshine, in Computational Group Theory ed. M. D. Atkinson, Academic Press, 185-193 (1984).

[N2] S. P. Norton, Non-monstrous Moonshine, in Groups, Difference Sets, and the Monster, eds. K. T. Arasu et al, de Gruyter, 433-441 (1996).

[Sh] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press, (1971).

[T] J.G. Thompson, A finiteness theorem for subgroups of P SL(2, IR) which are commensurable with P SL(2, ZZ), Proc. Sym. Pure. Math., 37, Santa Cruz Conference on finite groups, 533–555, Amer. Math. Soc., Providence RI, (1980).

Centre interuniversitaire en calcul mathématique algébrique (CICMA) Department of Mathematics and Statistics Concordia University 1455 de Maisonneuve Blvd West Montréal, H3G 1M8 Québec CANADA

Max-Planck-Institut für Mathematik Gottfried-Claren-Strasse 26 53225 Bonn GERMANY