Generators and Equations for Modular Function

Fields of Congruence Subgroups Γ(N) and Γ_{1}(N)

Nobuhiko Ishida and Noburo Ishii

1. Introduction.

In this note, we shall study a defining equation of the modular curves X(N) and X_{1}(N) for any positive integer N ≥ 7, as an affine (singular) plane curve.

In [5], Ishii defined a family of modular functions X_{r}(τ) (r ∈ ℤ, r ≢ 0 mod N) by

X_{r}(τ) = X_{r}(τ, N) = exp(−\frac{πi(r − 1)(N − 1)}{2N}) ∏_{s=0}^{N−1} \frac{K_{r,s}(τ)}{K_{1,s}(τ)},

where K_{u,v}(τ) are Klein forms of level N. (The functions K_{u,v}(τ) are modular forms with respect to Γ(2N^{2}) of weight −1.) It is noted that the Fourier expansion of the function X_{r}(τ) at the cusp i∞ has integral coefficients and its leading coefficient is ±1. Let A(N) be the field of all modular functions with respect to Γ(N) and 𝔉_{N} the subfield of A(N) consisting of the functions with k_{N}-rational Fourier coefficients at the cusp i∞, where k_{N} = ℚ(e^{2πi/N}). Then we know that X_{r}(τ) ∈ 𝔉_{N} (resp. X_{r}(τ)^{ϵ_{N}} ∈ 𝔉_{N}) if r is odd (resp. if r is even), where ϵ_{N} is 1 or 2 according to whether N is odd or even. Further, we can prove the following:

Theorem 1. Let N be an integer ≥ 7. Then

(1.1)  𝔉_{N} = k_{N}( X_{2}(τ)^{ϵ_{N}} , X_{3}(τ) ).

It is remarked that (1.1) imply that A(N) = ℂ(X_{2}(τ)^{ϵ_{N}} , X_{3}(τ)), since 𝔉_{N} and ℂ are linearly disjoint over k_{N} and A(N) = ℂ 𝔉_{N} (cf. §6.2 of Shimura [11]). Now we denote by F_{N}(X, Y) a polynomial of two variables X and Y such that F_{N}(X_{2}(τ)^{ϵ_{N}} , Y) = 0 is an irreducible equation of X_{3}(τ) over k_{N}(X_{2}(τ)^{ϵ_{N}}). Then the affine plane curve F_{N}(X, Y) = 0 gives a defining equation of the modular curve X(N).

In case of modular curves X_{0}(N) for the modular groups Γ_{0}(N), it is well-known that elliptic modular functions j(τ) and j(Nτ) generate over ℂ the field of all modular functions with respect to Γ_{0}(N), and modular equations Φ_{N}(X, Y) = 0 which are irreducible equations satisfying that Φ_{N}(j(τ), j(Nτ)) = 0, have many good properties, including the integrality of j(Nτ) over Z[j(τ)].

Typeset by AMS-TEX

Some of examples of F_{N}(X, Y) are as follows:

F_{7}(X, Y) = Y^{3} − X^{3}Y + X  (g_{7} = 3),
F_{8}(X, Y) = Y^{7} + 2Y^{5} + Y^{3} − X^{4}Y^{2} + X^{4} (g_{8} = 5),
F_{9}(X, Y) = Y^{6} − (X^{5} − X^{2})Y^{3} + X^{7} − 2X^{4} + X (g_{9} = 10),
F_{10}(X, Y) = Y^{14} + 4X^{2}Y^{10} + 2Y^{9} − X^{6}Y^{7} − 2X^{4}Y^{6} + 3X^{2}Y^{5} + Y^{4} + X^{8}Y^{3}
− 3X^{6}Y^{2} + 3X^{4}Y − X^{2} (g_{10} = 13),
F_{11}(X, Y) = Y^{12} − X^{7}Y^{8} + 2X^{6}Y^{7} − 4X^{5}Y^{6} + 5X^{4}Y^{5} − 2X^{3}Y^{4}
+ (X^{13} + X^{2})Y^{3} − (3X^{12} + X)Y^{2} + 3X^{11}Y − X^{10} (g_{11} = 26),
F_{12}(X, Y) = Y^{21} − 2Y^{18} + (6X^{4} + 1)Y^{15} − (X^{8} − 14X^{4})Y^{12} − (7X^{8} + X^{4})Y^{9}
+ (X^{12} + 6X^{8} + 9X^{4})Y^{6} − (2X^{12} − 4X^{8} + 2X^{4})Y^{3} + X^{12} − 2X^{8}
+ X^{4} (g_{12} = 25),
F_{13}(X, Y) = Y^{20} + XY^{18} − X^{2}Y^{16} − X^{9}Y^{15} + 2X^{3}Y^{14} + 2X^{10}Y^{13}
− 5X^{4}Y^{12} − 7X^{11}Y^{11} − X^{5}Y^{10} + 14X^{12}Y^{9} + (X^{19} + 6X^{6})Y^{8}

− 10X^{13}Y^{7} − (3X^{20} + 7X^{7})Y^{6} + (4X^{14} − X)Y^{5} + (3X^{21} + 5X^{8})Y^{4}

− 4X^{15}Y^{3} − X^{22}Y^{2} + 2X^{16}Y − X^{10} (g_{13} = 50).

Here we denote by g_{N} the genus of the modular curve X(N). Note that all these examples have very small integral coefficients. (Compare with the modular equations Φ_{N}(X, Y) = 0.)

In general, we know the following:

Theorem 2. For any integer N ≥ 7, X_{3}(τ) is integral over ℚ[X_{2}(τ)^{ϵ_{N}}]. Further, If N is a prime, then X_{3}(τ) is integral over ℤ[X_{2}(τ)^{ϵ_{N}}].

It seems that X_{3}(τ) is integral over ℤ[X_{2}(τ)^{ϵ_{N}}] for all integers N ≥ 7. However, we are currently unable to prove this conjecture.

For the modular curves X_{1}(N), we have the defining equation G_{N}(X, Y) = 0 in this case also. In fact, two generators are constructed from the functions X_{r}(τ) and the equations G_{N}(X, Y) = 0 are obtained from the equation F_{N}(X, Y) = 0 by straightforward transformation.

2. Klein Forms and Modular Functions X_{r}(τ, N)

For N ∈ ℤ, > 1 and (r, s) ∈ ℤ^{2}, ≢ (0, 0) (mod N), we define Klein forms by

K_{r,s}(τ) = e^{−(rη_{1}(τ)+sη_{2}(τ))(rτ+s)/2N^{2}}σ(\frac{rτ + s}{N}, L_{τ}),

where η_{1}(τ) and η_{2}(τ) are quasi-periods defined by the Weierstrass ζ-function, and σ(z, L_{τ}) the Weierstrass σ-function. Klein forms are modular forms with respect to Γ(2N^{2}) of weight −1. The next properties follow from Kubert-Lang [7] or Lang Proposition 1.

(1) For A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ), let (r′, s′) = (r, s)A. Then

K_{r,s}|[A]_{−1}(τ) = (cτ + d)K_{r,s}(A(τ)) = K_{r′,s′}(τ).

(2) For t, u ∈ ℤ,

K_{r+t,s+u}(τ) = ϵ · K_{r,s}(τ),

where

ϵ = (−1)^{(tu+t+u)}e^{−2πi(ts−ur)/2}.

(3) For A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ(N),

K_{r,s}|[A]_{−1}(τ) = ϵ(A) · K_{r,s}(τ),

where

ϵ(A) = −(−1)^{( \frac{a−1}{N} r+ \frac{c}{N} s+1)( \frac{b}{N} r+ \frac{d−1}{N} s+1)}e^{2πi(br^{2}+(d−a)rs−cs^{2})/2N^{2}}.

(4) In a neighborhood of the cusp i∞ of Γ(2N^{2}), the form K_{r,s}(τ) has an infinite product expansion:

K_{r,s}(τ) = −(2πi)^{−1}e^{2πi(r−N)s/2N^{2}}q^{r(r−N)/2N^{2}}

× (1 − ζ^{s}q^{\frac{r}{N}}) ∏_{n=1}^{∞} \frac{(1 − ζ^{s}q^{n+ \frac{r}{N}})(1 − ζ^{−s}q^{n− \frac{r}{N}})}{(1 − q^{n})^{2}},

where q = e^{2πiτ} and ζ = e^{2πi/N}.

Now we shall construct the modular functions X_{r}(τ, N) from Klein forms. They will constitute generators of modular function fields for modular group Γ(N) and Γ_{1}(N). For r ∈ ℤ, ≢ 0 (mod N), we put

X_{r}(τ) = X_{r}(τ, N) = e^{−2πi(r−1)(N−1)/4N} ∏_{s=0}^{N−1} \frac{K_{r,s}(τ)}{K_{1,s}(τ)}.

These functions were introduced by Ishii [5]. From Proposition 1, we deduce the following propreties of X_{r}(τ, N).

Proposition 2.

(1) If r is odd (resp. even), X_{r}(τ) ∈ A(N) (resp. X_{r}(τ)^{ϵ_{N}} ∈ A(N)).

(2) X_{r+kN}(τ) = (−1)^{k}X_{r}(τ), for k ∈ ℤ.

(3) X_{−r}(τ) = −X_{r}(τ).

(4) For A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ), c ≡ 0 (mod N),

X_{r}(A(τ)) = e(\frac{(r^{2} − 1)ab}{2N})(−1)^{(r−1)b} \frac{X_{ra}(τ)}{X_{a}(τ)}.

(5) In a neighborhood of the cusp i∞ of Γ(2N^{2}), the function X_{r}(τ) has an infinite product expansion:

X_{r}(τ) = q^{(r−1)(r+1−N)/2N}(\frac{1 − q^{r}}{1 − q}) ∏_{n=1}^{∞} \frac{(1 − q^{pn−r})(1 − q^{pn+r})}{(1 − q^{pn−1})(1 − q^{pn+1})},

where q = e^{2πiτ}.

(6) X_{r}(τ) has poles and zeros only at the cusps of Γ(N).

Proof. The statements (1) ∼ (5) are obtained from K 1 ∼ K 4 in §1 of Kubert and Lang [7] by elementary computation. The statement (6) is deduced from the fact that Klein forms K_{r,s}(τ) have neither poles nor zeros on the upper half complex plane H. □

Especially, the function X_{r}(τ) (or X_{r}(τ)^{2}) ∈ A(N) has a x-expansion at the cusp i∞ with integral coefficients and the leading coefficient is ±1, where x = e^{2πiτ/N} is a local parameter at the cusp i∞. So we know that X_{r}(τ) ∈ 𝔉_{N} (resp. X_{r}(τ)^{ϵ_{N}} ∈ 𝔉_{N}) if r is odd (resp. if r is even).

3. The order of X(τ) at cusps

Throughout this section we fix N and r, and by X(τ) let us denote the function X_{r}(τ, N) or X_{r}(τ, N)^{ϵ_{N}} according to whether r is odd or even. In this section, we shall compute the order ν_{P} (X(τ)) of X(τ) at the point of Riemann surface Γ(N)\H^{∗}, where H^{∗} = H ∪ ℚ ∪ {∞}. By Proposition 2, X(τ) has poles and zeros only at cusps of Γ(N). So, to our purpose, it is sufficient to calculate the order of X(τ) at the cusps.

Lemma 1. Let N be an integer ≥ 3. If N is odd, then we put

𝔖 = \{ (u, v) | 1 ≤ u ≤ \frac{N − 1}{2}, 1 ≤ v ≤ N, (u, v, N) = 1 \}

∪ \{ (N, v) | 1 ≤ v ≤ \frac{N − 1}{2}, (v, N) = 1 \},

and if N is even, then we put

𝔖 = \{ (u, v) | 1 ≤ u ≤ \frac{N}{2} − 1, 1 ≤ v ≤ N, (u, v, N) = 1 \}

∪ \{ (\frac{N}{2} , v) | 1 ≤ v ≤ \frac{N}{2} , (v, \frac{N}{2} ) = 1 \}

∪ \{ (N, v) | 1 ≤ v ≤ \frac{N}{2} , (v, N) = 1 \}.

For each (u, v) ∈ 𝔖, take a matrix B(u, v) ∈ SL_{2}(ℤ) so that

B(u, v) ≡ ( \begin{matrix}u & ∗ \\ v & ∗ \end{matrix} ) mod N.

Then the set of rational numbers {B(u, v)(i∞)| (u, v) ∈ 𝔖} form a system of representatives of inequivalent cusps of Γ(N).

Proof. See Lemma 1.42 of Shimura [11]. □

For each (u, v) ∈ 𝔖, let P (u, v) denote the cusp of Γ(N) represented by a rational number B(u, v)(i∞). Then the order ν_{u,v}(f(τ)) of the function f(τ) ∈ A(N) at the cusp P (u, v) is defined to be the order of the x-expansion of f(B(u, v)(τ)), where x = e(\frac{τ}{N}). To describe the order of the function X(τ) at cusp P (u, v), let us define a function α_{m}(w) (w, m ∈ ℤ) by α_{m}(w) = ⟨w⟩_{m}(⟨w⟩_{m} − m), where ⟨w⟩_{m} denotes the smallest non-negative integer congruent to w modulo m. Note that α_{m}(w) is determined by w mod m, and α_{m}(w) = α_{m}(−w).

Proposition 3. For any (u, v) ∈ 𝔖, let m = m_{v} = GCD(v, N). Then we have

ν_{u,v}(X(τ)) = \{ \begin{matrix}\frac{1}{2}(α_{m}(ru) − α_{m}(u)), if r is odd, \\ \frac{ϵ_{N}}{2} (α_{m}(ru) − α_{m}(u)), if r are even. \end{matrix}

Proof. Let either N or r be odd for instence. By Proposition 1 and 2, we have

X(B(u, v)(τ)) = c^{∗} ∏_{s=0}^{N−1} K_{ru+sv,ru′+sv′}(τ)/K_{u+sv,u′+sv′}(τ),

where c^{∗} is a non-zero constant. Therefore we know

ν_{u,v}(X(τ)) = \frac{1}{2N} ∑_{s=0}^{N−1}(α_{N}(ru + sv) − α_{N}(u + sv)).

Then by straightforward calculation in both cases GCD(v, N) = 1 and ≠ 1, we have the desired formula:

ν_{u,v}(X(τ)) = \frac{1}{2}(α_{m}(ru) − α_{m}(u)).

□ Corollary 1. Let (u, v) ∈ 𝔖. The functions X(τ) have poles only at the cusps P (u, v) of Γ(N) such that

m = GCD(v, N) > 3, GCD(u, m) = 1 and 0 < u < \frac{N}{2} .

Further, for r = 2, 3, the order of X(τ) at the cusps P (u, v) are given by

ν_{u,v}(X_{2}(τ)^{ϵ_{N}}) = ν_{ū,v}(X_{2}(τ)^{ϵ_{N}}) = \frac{ϵ_{N}}{2} (3ū^{2} − mū),
ν_{u,v}(X_{3}(τ)) = ν_{ū,v}(X_{3}(τ)) = \{ \begin{matrix}4ū^{2} − mū, & if ū < \frac{m}{3}, \\ (2ū − m)^{2}, & if ū ≥ \frac{m}{3}. \end{matrix}

where w̄ is defined by

w̄ = \{ \begin{matrix}⟨w⟩_{m}, if ⟨w⟩_{m} < m/2, \\ m − ⟨w⟩_{m}, otherwise. \end{matrix}

Proof. By the property of α_{m} and the straightforward calculation, we have our assertions immediately. □

Corollary 2.

(1) X_{3}(τ) has poles only at the points where X_{2}(τ)^{ϵ_{N}} has poles.

(2) If N is odd, then X_{3}(τ) has a zero at each point where X_{2}(τ)^{ϵ_{N}} has a zero.

(3) If 3 ∤ N, then X_{2}(τ)^{ϵ_{N}} has a pole or a zero at each point where X_{3}(τ) has a zero.

4. Generators of A(N)

In this section, we shall prove Theorem 1. In the following, for a modular function f(τ), we write simply f instead of f(τ) if there is no danger of confusion. Let N be an integer ≥ 7. ( Note that N ≥ 7 means the genus of X(N) is not 0. ) Since A(N) is an algebraic function field of one dimension over ℂ, if f ∈ A(N) is a non-constant function, then A(N) is finite over the subfield ℂ(f) of A(N). In this case, we denote by d(f) the degree of A(N) over ℂ(f). Our proof is based on the next lemma.

Lemma 2. Let L be a subfield of A(N) over ℂ such that [A(N) : L] < ∞. Let f_{1}, ⋯, f_{n} be non-constant functions of the subfield L. If GCD(d(f_{1}), ⋯, d(f_{n})) = 1, then L = A(N).

Proof. The degree [A(N) : L] is a divisor of d(f_{i}) = [A(N) : ℂ(f_{i})] for each i. Hence, GCD(d(f_{1}), ⋯, d(f_{n})) = 1 implies [A(N) : L] = 1. □

First, we assume N is odd. Note that ϵ_{N} = 1 in this case. Let L be the subfield of A(N) generated over ℂ by X_{2}(τ) and X_{3}(τ). By Lemma 2, to prove L = A(N),

Generators and Equations for Modular Function Fields 7

it suffices to show that there exist two pairs of positive integers (i_{1}, j_{1}), (i_{2}, j_{2}) such that

GCD( d(X_{2}), d(X_{2}^{i_{1}} + X_{3}^{j_{1}}), d(X_{2}^{i_{2}} + X_{3}^{j_{2}}) ) = 1.

It is well known that if f ∈ A(N) is a non-constant function,

d(f) = the total degree of poles of f on X(N).

(See for example Proposition 2.11 of Shimura [11].) Therefore, from the property of α_{m}(w) and Corollary 1, we have

d(X_{2}) = − ∑_{\begin{matrix}m|N \\ m>3 \end{matrix}} φ(\frac{N}{m})\frac{N}{m} ∑_{\begin{matrix}0<u<m/3 \\ (u,m)=1 \end{matrix}} \frac{3u^{2} − mu}{2},

where φ(n) is Euler’s function.

Further, we can compute the degree d(X_{2}^{i}+X_{3}^{j}) on the assumption that i < 2j, \frac{(2j−i)N}{8j−3i} ∉ ℤ and [\frac{(2j−i)N}{8j−3i} ] = 1. In fact, by Corollary 1, we know the following:

d(X_{2}^{i} + X_{3}^{j}) = i d(X_{2}) + \frac{1}{2}\{2(N − 4)j − (N − 3)i\}.

Now, for each N ≥ 7, we take two pairs of positive integers (i, j) satisfying above assumptions:

(i_{1}, j_{1}) = (N − 3, \frac{N − 1}{2}) and (i_{2}, j_{2}) = (N − 5, \frac{N − 3}{2}).

Then we have

GCD( d(X_{2}), d(X_{2}^{i_{1}} + X_{3}^{j_{1}}), d(X_{2}^{i_{2}} + X_{3}^{j_{2}}) ) = 1.

This shows Theorem 1 for odd integer N.

When N is even, taking (i, j) so that

(i_{1}, j_{1}) = (\frac{N − 2}{2}, \frac{N}{4} ) and (i_{2}, j_{2}) = (N − 4, \frac{N − 2}{2}), if N ≡ 0 (mod 4),
(i_{1}, j_{1}) = (\frac{N − 4}{2}, \frac{N − 2}{4}) and (i_{2}, j_{2}) = (N − 2, \frac{N}{2} ), if N ≡ 2 (mod 4),

we have by similar argument

GCD( d(X_{2}^{2}), d(X_{2}^{2i_{1}} + X_{3}^{2j_{1}}), d(X_{2}^{2i_{2}} + X_{3}^{2j_{2}}) ) = 1.

This shows A(N) = ℂ(X_{2}(τ)^{2}, X_{3}(τ)^{2}). We have thus proved the Theorem 1.

5. Equations for A(N)

In this section, we shall give a method to calculate the equations F_{N}(X, Y) = 0. Let X_{2} and X_{3} be the functions X_{2}(τ, N)^{ϵ_{N}} and X_{3}(τ, N) respectively. Put d_{2} = d(X_{2}), d_{3} = d(X_{3}). Since the degree 𝔉_{N} over k_{N}(X_{2}) is d_{2}, the function X_{3} has an irreducible equation F_{N}(X_{2}, Y) = 0 of degree d_{2} over k_{N}(X_{2}). Let

F_{N}(X, Y)
= Φ_{d_{2}}(X)Y^{d_{2}} + Φ_{d_{2}−1}(X)Y^{d_{2}−1} + ⋯ + Φ_{1}(X)Y + Φ_{0}(X) ∈ k_{N}[X, Y],

Φ_{j}(X) ∈ k_{N}[X] for all j, Φ_{d_{2}}(X) is monic, and Φ_{d_{2}}(X), …, Φ_{1}(X) and Φ_{0}(X) have no common factors. We also write

F_{N}(X, Y) = ∑_{i,j} C_{i,j}X^{i}Y^{j}, C_{i,j} ∈ k_{N}.

From studying behaviors of the functions X_{2} and X_{3} at some cusps, we deduce the following properties. ( See Ishida [2] and Ishida and Ishii [3] for more details.)

Lemma 3.

(1) Φ_{d_{2}}(X) = 1.

(2) max_{0≤k≤d_{2}} deg Φ_{k}(X) = d_{3}.

(3) If N is odd, then Φ_{k}(X) is divisible by X for all k < d_{2}.

(4) If 3 ∤ N, then Φ_{0}(X) is a monomial.

(5) If N is odd, then 3i + 8j ≢ 8d_{2} (mod N) implies that C_{i,j} = 0.

(6) If N is even, then 3i + 4j ≢ 4d_{2} (mod N) implies that C_{i,j} = 0.

(7) If (N − 3)ϵ_{N}i/2 + (N − 4)j > (N − 4)d_{2}, then C_{i,j} = 0.

Further, we have

Theorem 2. For any integer N ≥ 7, X_{3}(τ) is integral over ℚ[X_{2}(τ)^{ϵ_{N}}]. Further, If N is a prime, then X_{3}(τ) is integral over ℤ[X_{2}(τ)^{ϵ_{N}}].

Finally let us explain how to compute the coefficients C_{i,j} effectively. Since F_{N}(X_{2}, X_{3}) = 0 and a non-constant function of A(N) necessarily has poles, we get a finite system of linear equations in C_{i,j} by replacing X_{2} and X_{3} in F_{N}(X_{2}, X_{3}) with their x-expansions at the cusps where X_{2} or X_{3} has poles and by letting the coefficients of non-positive powers of x equal to 0. By solving these linear equations we will be able to determine all C_{i,j}. But, in general, the x-expansions of X_{r} are in ℤ[ζ_{N}]((x)). Therefore, to calculate the coefficients, (especially when we use a computer,) this method is not so good. Let us consider the x-expansions at the cusps P (u, N). At these cusps, by Proposition 2 (4), the x-expansions of X_{r} are essentially in ℤ((x)). Furthermore, by Lemma 3 (5), the elements of the coefficient matrix of the system of linear equations can be taken in ℤ. (See the proof of Lemma 7 in [3].) For some N, making sufficiently many linear equations obtained by equating the coefficients of powers of x included positive powers, we are able to determine all the coefficients of F_{N}(X, Y). The calculation were performed by means of ”Mathematica” on Unix machine.

6. Generators for A_{1}(N)

Let A_{1}(N) be the field of all modular functions with respect to Γ_{1}(N). In this section, we shall give some of generators of A_{1}(N). Let N be a positive integer ≥ 7, and consider the functions X_{2}(τ, N), X_{3}(τ, N) for a fixed N. We shall find out generators of A_{1}(N) among functions of the form X_{2}^{m}X_{3}^{n}, for m, n ∈ ℤ.

Lemma 4. Let m, n be integers. Then

X_{2}^{m}X_{3}^{n} ∈ A_{1}(N) ⟺ 3m + 8n ≡ 0 (mod ϵ_{N}N).

Proof. By Proposition 2, we know

X_{2}^{m}X_{3}^{n}(τ + 1) = exp (\frac{πi(3m + 8n)}{N}) (−1)^{m}X_{2}^{m}X_{3}^{n}(τ).

Thus

X_{2}^{m}X_{3}^{n} ∈ A_{1}(N) ⟺ 3m + 8n + Nm ≡ 0 mod 2N.

We note that X_{2}^{m}X_{3}^{n} ∈ A(N), since the congruence of right hand side implies that m is even in the case N is even. It is easy to see that the above congruence is equivalent to a congruence 3m + 8n ≡ 0 mod ϵ_{N}N. □

By Lemma 4, we have

Proposition 4. Let m_{0}, n_{0} be integers such that 3m_{0} + 8n_{0} = ϵ_{N}N. Then

X_{2}^{m_{0}}X_{3}^{n_{0}}, X_{2}^{8}X_{3}^{−3}, X_{2}^{ϵ_{N} N}, X_{3}^{N} ∈ A_{1}(N).

Next we shall calculate the order of X_{2}^{m}X_{3}^{n} ∈ A_{1}(N) at cusps of Γ_{1}(N). In

Ogg [9], all inequivalent cusps of Γ_{1}(N) are given by pairs of integers ( \begin{matrix}u \\ v \end{matrix} ), where

1 ≤ v ≤ N and if we write GCD(v, N) = d, then u runs all over integers such that 0 ≤ u ≤ \frac{d}{2}, GCD(u, d) = 1. Further we know that the ramification index of

the canonical covering map: X(N) → X_{1}(N) at the cusps lying over ( \begin{matrix}u \\ v \end{matrix} ) is d.

Therefore from Proposition 3, we easily deduce

Lemma 5. Let X_{2}^{m}X_{3}^{n} ∈ A_{1}(N). Then the function X_{2}^{m}X_{3}^{n} has zeros and poles

only at cusps of Γ_{1}(N) and its order ν_{u,v}(X_{2}^{m}X_{3}^{n}) at a cusp ( \begin{matrix}u \\ v \end{matrix} ) is given by

ν_{u,v}(X_{2}^{m}X_{3}^{n}) = \{ \begin{matrix}\frac{(3m+8n)u^{2}−(m+2n)du}{2d} & if u < \frac{d}{3}, \\ \frac{(3m+8n)u^{2}−(m+8n)du+2d^{2}n}{2d} & if u ≥ \frac{d}{3}, \end{matrix}

where d = GCD(v, N).

Now we have the next theorem by the slightly modified argument used to prove Theorem 1.

Theorem 3. Let N be an integer ≥ 7. Then the functions X_{2}^{ϵ_{N} N} and X_{3}^{N} generate A_{1}(N) over ℂ.

Further, we have another generators of A_{1}(N) over ℂ. These are used in the next section.

Theorem 4. Let m_{0}, n_{0} be integers such that 3m_{0} + 8n_{0} = ϵ_{N}N. Then the functions X_{2}^{m_{0}}X_{3}^{n_{0}}, X_{2}^{8}X_{3}^{−3} generate A_{1}(N) over ℂ.

Proof. By Proposition 4, we know X_{2}^{8}X_{3}^{−3}, X_{2}^{m_{0}}X_{3}^{n_{0}} ∈ A_{1}(N). Since

X_{2}^{ϵ_{N} N} = (X_{2}^{m_{0}}X_{3}^{n_{0}})^{3}(X_{2}^{8}X_{3}^{−3})^{n_{0}}, X_{3}^{N} = (X_{2}^{m_{0}}X_{3}^{n_{0}})^{8/ϵ_{N}} (X_{2}^{8}X_{3}^{−3})^{−m_{0}/ϵ_{N}} ,

Theorem 3 shows the assertion. □

6. Equations for A_{1}(N)

In this section, we shall give an equation for A_{1}(N) satisfied by X_{2}^{8}X_{3}^{−3} and X_{2}^{m_{0}}X_{3}^{n_{0}}.

Here, we replace the coefficients C_{i,j} in F_{N}(X, Y) by

Φ_{j}(X^{ϵ_{N}}) = ∑_{i=0}^{ϵ_{N} d_{3}} C_{i,j}X^{i}.

Then

F_{N}(X^{ϵ_{N}} , Y) = Y^{d_{2}} + ∑_{i,j} C_{i,j}X^{i}Y^{j},

where i, j of the sum run over integers such that 0 ≤ i ≤ ϵ_{N}d_{3}, 0 ≤ j < d_{2}. Let 𝔗 be a set of pairs of integers (i, j) such that 0 ≤ i ≤ ϵ_{N}d_{3}, 0 ≤ j < d_{2}, C_{i,j} ≠ 0. Then by Lemma 3, we have

Lemma 6. Let (i, j) ∈ 𝔗. Then (1) 3i + 8j ≡ 8d_{2} mod ϵ_{N}N.

(2) (N − 3)i/2 + (N − 4)j ≤ (N − 4)d_{2}.

Let m_{0}, n_{0} be integers such that 3m_{0} + 8n_{0} = ϵ_{N}N. We can always take m_{0}, n_{0} such that m_{0} ≥ 0, n_{0} ≤ 0. We shall deduce an equation of X_{2}^{m_{0}}X_{3}^{n_{0}}, X_{2}^{8}X_{3}^{−3} from a polynomial F_{N}(X, Y). To simplify the notation, put Θ = X_{2}^{8}X_{3}^{−3}, Λ = X_{2}^{m_{0}}X_{3}^{n_{0}}. Choose (i_{0}, j_{0}) ∈ 𝔗 so that 3i_{0} + 8j_{0} is the smallest value among 3i + 8j, (i, j) ∈ 𝔗. Then by (2) of Lemma 6, for (i, j) ∈ 𝔗, we have for a non-negative integer k, 3(i − i_{0}) + 8(j − j_{0}) = ϵ_{N}Nk. By the definition of m_{0}, n_{0}, we know there exists an integer ℓ such that

i − i_{0} = km_{0} − 8ℓ, j − j_{0} = kn_{0} + 3ℓ.

Thus we have X_{2}^{i}X_{3}^{j} = Θ^{−ℓ}Λ^{k}X_{2}^{i_{0}}X_{3}^{j_{0}}. By (2) of Lemma 6,

\begin{aligned}3i + 8j & ≤ \frac{6(N − 4)}{(N − 3)}(d_{2} − j) + 8j \\ & = \frac{6(N − 4)}{(N − 3)}d_{2} + (8 − \frac{6(N − 4)}{(N − 3)}) j. \end{aligned}

Since (8 − \frac{6(N−4)}{(N−3)}) ≥ 0, we have 3i + 8j ≤ 8d_{2} and the identity holds only for (i, j) = (0, d_{2}). Further since n_{0} ≤ 0 and ℓ = (j − j_{0} − n_{0}k)/3, ℓ takes the greatest value for (i, j) = (0, d_{2}). Put

(6.1)  k_{0} = \frac{−3i_{0} + 8(d_{2} − j_{0})}{ϵ_{N}N}, ℓ_{0} = \frac{d_{2} − j_{0} − n_{0}k_{0}}{3} .

For (i, j) ∈ 𝔗, let

(6.2)  k(i, j) = \frac{3(i − i_{0}) + 8(j − j_{0})}{ϵ_{N}N}, ℓ(i, j) = \frac{j − j_{0} − n_{0}k(i, j)}{3}.

Then we have

\begin{aligned}F_{N}(X_{2}^{ϵ_{N}} & , X_{3}) \\ & = X_{2}^{i_{0}}X_{3}^{j_{0}}(Θ^{−ℓ_{0}}Λ^{k_{0}} + ∑_{(i,j)∈𝔗} C_{i,j}Θ^{−ℓ(i,j)}Λ^{k(i,j)}) = 0. \end{aligned}

Therefore we have

Λ^{k_{0}} + ∑_{(i,j)∈𝔗} C_{i,j}Θ^{ℓ_{0}−ℓ(i,j)}Λ^{k(i,j)} = 0.

Put

(6.3)  G_{N}(X, Y) = Y^{k_{0}} + ∑_{(i,j)∈𝔗} C_{i,j}X^{ℓ_{0}−ℓ(i,j)}Y^{k(i,j)}.

By above argument, we obtain

Theorem 5. Let m_{0}, n_{0} be integers such that 3m_{0} + 8n_{0} = ϵ_{N}N, m_{0} ≥ 0, n_{0} ≤ 0. Then X_{2}^{m_{0}}X_{3}^{n_{0}} is integral over ℚ[X_{2}^{8}X_{3}^{−3}]. Further if N is prime, then X_{2}^{m_{0}}X_{3}^{n_{0}} is integral over ℤ[X_{2}^{8}X_{3}^{−3}]. Let G_{N}(X, Y) be the polynomial defined by (6.1) ∼ (6.3). Then G_{N}(X_{2}^{8}X_{3}^{−3}, Y) = 0 is a monic irreducible equation of X_{2}^{m_{0}}X_{3}^{n_{0}} over ℚ[X_{2}^{8}X_{3}^{−3}]. Further we have

F_{N}(X^{ϵ_{N}} , Y) = X^{i_{0}−8ℓ_{0}}Y^{j_{0}+3ℓ_{0}}G_{N}(X^{8}Y^{−3}, X^{m_{0}}Y^{n_{0}}).

12 Nobuhiko Ishida and Noburo Ishii

7. Examples Let us give examples of equations F_{N}(X, Y) and G_{N}(X, Y) for some N ≥ 7.

Take m_{0} and n_{0} as given in §6. Put Θ = X_{2}^{8}X_{3}^{−3}, Λ = X_{2}^{m_{0}}X_{3}^{n_{0}}. Remark. Let g_{1}(N) be the genus of X_{1}(N). Since

g_{1}(N) = 1 + \frac{N^{2}}{24} ∏_{\begin{matrix}p|N \\ p:prime \end{matrix}} (1 − \frac{1}{p^{2}} ) − \frac{1}{4} ∑_{d|N} φ(d)φ(N/d),

we know that g_{1}(N) > 0 ⟺ N = 11, N ≥ 13. □

(1) N = 7 (m_{0} = 5, n_{0} = −1).

F_{7}(X^{ϵ_{N}} , Y) = Y^{3} − X^{3}Y + X,
G_{7}(X, Y) = Y^{3} − XY^{2} + X^{2}.

Since G_{7}(Θ, Λ) = 0, we have Θ = \frac{1}{Ω^{2}−Ω^{3}} , where Ω = Λ/Θ. Therefore we know A_{1}(7) = ℂ(Ω).

(2) N = 8 (m_{0} = 8, n_{0} = −1).

F_{8}(X^{ϵ_{N}} , Y) = Y^{7} + 2Y^{5} + Y^{3} − X^{8}Y^{2} + X^{8},
G_{8}(X, Y) = Y^{2} − (X^{2} − 2X)Y + X^{3} + X^{2}.

Since G_{8}(Θ, Λ) = 0, we have Θ = \frac{(Ω+1)^{2}}{Ω−1} , where Ω = Λ/Θ. Therefore we know A_{1}(8) = ℂ(Ω).

(3) N = 9 (m_{0} = 3, n_{0} = 0).

F_{9}(X^{ϵ_{N}} , Y) = Y^{6} − (X^{5} − X^{2})Y^{3} + X^{7} − 2X^{4} + X,
G_{9}(X, Y) = Y^{5} − XY^{4} + XY^{3} + X^{2}Y^{2} − 2X^{2}Y + X^{2}.

Since G_{9}(Θ, Λ) = 0, we know Λ = \frac{1}{Ω−Ω^{2}} , where Ω = \frac{(Λ−1)Θ}{Λ^{3}}. Thus A_{1}(9) = ℂ(Ω).

(4) N = 10 (m_{0} = 12, n_{0} = −2).

F_{10}(X^{ϵ_{N}} , Y) = Y^{14} + 4X^{4}Y^{10} + 2Y^{9} − X^{12}Y^{7} − 2X^{8}Y^{6} + 3X^{4}Y^{5} + Y^{4}
+ X^{16}Y^{3} − 3X^{12}Y^{2} + 3X^{4}Y − X^{2},
G_{10}(X, Y) = Y^{5} − (X^{3} − 4X^{2})Y^{4} + (X^{5} − 2X^{4} + 2X^{3})Y^{3}
− 3(X^{6} − X^{5})Y^{2} + (3X^{7} + X^{6})Y − X^{8}.

Generators and Equations for Modular Function Fields 13

(5) N = 11 (m_{0} = 9, n_{0} = −2).

F_{11}(X^{ϵ_{N}} , Y) = Y^{12} − X^{7}Y^{8} + 2X^{6}Y^{7} − 4X^{5}Y^{6} + 5X^{4}Y^{5} − 2X^{3}Y^{4}

+ (X^{13} + X^{2})Y^{3} − (3X^{12} + X)Y^{2} + 3X^{11}Y − X^{10},

G_{11}(X, Y) = Y^{7} − X^{2}Y^{6} + 2X^{3}Y^{5} + (X^{5} − 4X^{4})Y^{4}

− (3X^{6} − 5X^{5})Y^{3} + (3X^{7} − 2X^{6})Y^{2} − (X^{8} − X^{7})Y − X^{8}.

Since G(Θ, Λ) = 0, by setting U = \frac{Λ}{Θ^{2}} , V = \frac{Θ}{Λ}, we deduce

U^{2} − (V − 1)^{2}(V^{4} + V^{3} + 3V^{2} + 1)U − V (V − 1)^{3} = 0.

From this, by setting T = \frac{1}{2}(\frac{2U−(V −1)^{2}(V^{4}+V^{3}+3V^{2}+1)}{(V −1)^{3}(V^{3}+V −1)} + 1), S = \frac{1}{V −1} + 1, we have a Weierstrass equation:

T^{2} − T = S^{3} − S^{2}.

(6) N = 12 (m_{0} = 8, n_{0} = 0).

F_{12}(X^{ϵ_{N}} , Y) = Y^{21} − 2Y^{18} + (6X^{4} + 1)Y^{15} − (X^{8} − 14X^{4})Y^{12}

− (7X^{8} + X^{4})Y^{9} + (X^{12} + 6X^{8} + 9X^{4})Y^{6} − (2X^{12} − 4X^{8} + 2X^{4})Y^{3} + X^{12} − 2X^{8} + X^{4},

G_{12}(X, Y) = Y^{6} − (X^{3} − 6X^{2} + 2X)Y^{5} + (X^{5} − 7X^{4} + 14X^{3} + X^{2})Y^{4}

− (2X^{6} − 6X^{5} + X^{4})Y^{3} + (X^{7} + 4X^{6} + 9X^{5})Y^{2} − (2X^{7} + 2X^{6})Y + X^{7}.

(7) N = 13 (m_{0} = 7, n_{0} = −1).

F_{13}(X^{ϵ_{N}} , Y) = Y^{20} + XY^{18} − X^{2}Y^{16} − X^{9}Y^{15} + 2X^{3}Y^{14} + 2X^{10}Y^{13}

− 5X^{4}Y^{12} − 7X^{11}Y^{11} − X^{5}Y^{10} + 14X^{12}Y^{9} + (X^{19} + 6X^{6})Y^{8} − 10X^{13}Y^{7} − (3X^{20} + 7X^{7})Y^{6} + (4X^{14} − X)Y^{5} + (3X^{21} + 5X^{8})Y^{4} − 4X^{15}Y^{3} − X^{22}Y^{2} + 2X^{16}Y − X^{10},

G_{13}(X, Y) = Y^{10} − (X^{2} − X)Y^{9} + (2X^{3} − X^{2})Y^{8}

+ (X^{5} − 7X^{4} + 2X^{3})Y^{7} − (3X^{6} − 14X^{5} + 5X^{4})Y^{6} + (3X^{7} − 10X^{6} − X^{5})Y^{5} − (X^{8} − 4X^{7} − 6X^{6})Y^{4} − (4X^{8} + 7X^{7})Y^{3} + (2X^{9} + 5X^{8})Y^{2} − X^{8}Y − X^{10}.

In [10], Reichert obtained the defining equation of X_{1}(N) explicitely for N = 11, 13, 14, 15, 16, 17 and 18. In fact, they are given by the following:

X_{1}(11) : V^{2} + V = U^{3} − U^{2},
X_{1}(13) : V^{2} + (U^{3} − U^{2} − 1)V − U^{2} + U = 0,
X_{1}(14) : V^{2} + UV + V = U^{3} − U,
X_{1}(15) : V^{2} + UV + V = U^{3} + U^{2},
X_{1}(16) : (U^{2} + 3U + 2)V^{2} + (U^{3} + 4U^{2} + 4U)V − U = 0,
X_{1}(17) : V^{4} + (U + 2)V^{3} + (U^{3} + 1)V^{2}
+ (−U^{5} − 2U^{4} − U^{3} − U^{2} − U)V − U^{5} − 2U^{4} − U^{3} = 0,
X_{1}(18) : (U^{2} − 2U + 1)V^{2} + (−U^{3} + U − 1)V + U^{3} − U^{2} = 0.

It seems to be very hard to transform G_{N}(X, Y) = 0 to such a simple defining equation.

References

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4. N. Ishida and N. Ishii, An equation for the modular curve X_{1}(N) derived from an equation for the modular curve X(N), preprint.

5. N. Ishii, Construction of generators of modular function fields, Math. Japon. 28 (1983), 655–681.

6. K. Iwasawa, Algebraic functions, MMONO 118, American Mathematical Society, Providence R. I., 1993.

7. D. Kubert and S. Lang, Units in the modular function fields, Math. Ann. 218 (1975), 175–189.

8. S. Lang, Elliptic functions, Springer, New York, 1987.

9. A. Ogg, Rational points on certain elliptic modular curves, Proc.Symp.Pure Math. 24 (1973), 221–231.

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11. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Iwanami Shoten Publisher and Princeton Univ. Press, Tokyo, 1971.

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Graduate School of Sciences, College of Integrated Arts and Science, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai-city, Osaka 599-8231, Japan

E-mail address: ishida@an.email.ne.jp

Department of Mathematics, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai-city,Osaka 599-8231, Japan

E-mail address: ishii@mi.cias.osakafu-u.ac.jp