In [16] Y. Yang gave a general method of finding defining equations of modular curves of the type X0(N ), X1(N ), and X(N ). Here we will first review his method for X0(p), and then we will refine the method to obtain modular parameterizations for X0+(p) that have genus 1. We will also describe an alternative method using the fact that the holomorphic 1-forms on a modular curve are actually cusp forms of weight 2 in disguise.
4.2.1 Equations for X
0(p)
In [16], it was shown that for any positive integer N it is always possible to find modu-lar functions X and Y that generate the function field on X0(N ) using the generalized Dedekind eta functions.
To be more explicit, recall the basic fact that the congruence subgroup Γ1(N ) is a normal subgroup of Γ0(N ). Let Γ be an intermediate subgroup between Γ0(N ) and Γ1(N ). We have the fact that Γ is a normal subgroup of Γ0(N ). Thus, if f (τ ) is a modular function on Γ, then the function
X
γ∈ Γ0(N )/Γ
f (γτ )
is modular on Γ0(N ). Furthermore, assume that f has only poles at cusps that are equivalent to ∞ under Γ0(N ). Assume also that the order of f at ∞ is m, while at the other poles the orders are less than m. Then the above function has exactly one pole of order m at ∞ and is holomorphic at any other point of X0(N ). Thus, the problem of finding generators of the function field on X0(N ) reduces to that of finding modular functions on Γ1(N ) that have the required analytic behaviors. In [16], Yang demonstrated a method how one can achieve this using generalized Dedekind eta functions. Here we will work out the case N = 17 to demonstrate the whole procedure.
First of all, let us set Wk = E4k/E2k. The purpose of this setting is to get rid of the factor involving eπib in Proposition 2.3.8. Then any product of Wk will satisfy
condition (2.6) in Proposition 2.3.9 automatically. Thus, if ek are integers such that P
kk2ek = 0 mod 17, then Q
kWkek is modular on Γ1(17). Furthermore, it is easy to see that the infinite products defining Ek converge absolutely for any τ ∈ H. Thus, the only possible poles or zeroes of Ek are all at cusps. In fact, using Proposition 2.3.10, we see that the poles and zeroes can happen at cusps k/17, 17 - k.
There are eight distinct Wk, k = 1 . . . 8. The cusp ∞ of X0(17) splits into eight inequivalent cusps k/17, k = 1 . . . 8, in X1(17). The orders of Wkat these cusps, multiplied by 17, are as follows.
3/17 8/17 7/17 4/17 5/17 2/17 6/17 1/17
W3 −7 −12 28 14 −10 −5 −11 3
We need a function F with a pole of order 2 at infinity and poles of order less than 2 at other cusps equivalent to infinity under Γ0(17), and holomorphic at any other points. To find F is equivalent to solving the integer programming problem
−7x1 −12x2 +28x3 +14x4 −10x5 −5x6 −11x7 +3x8 ≥ −17,
then X is modular on Γ0(17) with those properties which we want, where γ runs over a set of coset representatives of Γ0(17)/Γ1(17). Similarly, we can choose a degree 3 function
Y to be equation. This concludes the demonstration of the case X0(17). For other two X0(p) of genus 1, the same method also applies. We list the results in Section 4.3.
4.2.2 Equations for X
0+(p)
For curves X0+(p) the basic idea is the same. Since the curves X0+(p) are assumed to be of genus one, there are two modular functions x and y on X0+(p) such that they have poles only at the cusp ∞ with orders 2 and 3, respectively. Now consider x and y as modular functions on Γ0(p). Since the Atkin-Lehner involutions ωp on X0(p) sends the cusps ∞ and 0 to each other, the function x has double poles at cusps ∞ and 0, and the function y has triple poles only at cusps ∞ and 0. Thus, our goal here is to find modular functions that satisfy these requirement. One way to achieve this is as follows.
Suppose that s is a modular function on Γ0(p) such that s has a double pole at ∞, a pole of order less than or equal to 2 at 0, and holomorphic at any other points. Then the function
x = s + s|ωp,
considered as a function on Γ+0(p) will have a pole of order 2 at ∞. Similarly, if t is a modular function on Γ0(p) with a triple pole at ∞, a pole of order less than or equal to 3 at 0, then a possible choice of y is
y = t + t|ωp.
We take X0+(61) for example. Now we construct modular functions to parameterize this elliptic curve.
is a modular function on Γ for any positive integer k. There are six distinct Wk, and they are W2, W4, W8, W5, W10, and W1. Moreover, the cusp ∞ splits into six inequivalent
cusps 2/61, 4/61, 8/61, 5/61, 10/61, and 1/61 in Γ. The orders of Wk at these cusps are as follows.
2/61 4/61 8/61 5/61 10/61 1/61
W2 −4 4 −7 13 0 −6
is a modular function on Γ0(61) and has a unique pole of order 7 at infinity.
Now we set
where X and Y have pole at infinity of order 5 and 7, respectively. Then we have
Y5−23XY4+149X2Y3−9(X4+31X3+61X2)Y2+33(X5+X4+61X3)Y = X3(X2+X+61)2, which we takes as the defining equation of X0(61).
The points ∞ and (0, 0) correspond to the cusps ∞ and 0, respectively. This is because if we use the transformation formula for the Dedekind eta function (Proposition 2.3.4), then we get that
From Proposition 2.3.11, we deduce that Y (τ )|ω
61 = 61q3(1 + 3q + 10q2 + 24q3+ 57q4+ 120q5+ 246q6+ · · · ). (4.2) If we consider the Fourier expansions of these functions, then we obtain that the function
Y |ω
To find modular parameterization of X0+(61), we need to construct functions s and t with poles only at cusps ∞ and 0 such that s has double poles at cusps ∞ and 0, and t has triple poles at cusps ∞ and 0. According to equations (4.1), (4.2) and (4.3), since the Atkin-Lehner involution ω61 sends the cusps ∞ and 0 to each other, we have
div(X) = −5(∞) + 5(0, 0),
div(Y ) = −7(∞) + 3(0, 0) + 2(α, 0) + 2(β, 0), div(X|ω61) = 5(∞) − 5(0, 0),
div(Y |ω61) = 3(∞) − 7(0, 0) + 2(α, 0) + 2(β, 0),
where div(f ) means the divisor of the function f , and α, β are the roots of X2+X +61 = 0.
Thus, the function Xω61Y = 61Y /X has double poles only at ∞ and 0. Using the equations (4.1) and (4.3), we can find that
Y X
ω61
= 61Y X2 × X
61 = Y X, that is, Y /X is invariant under Atkin-Lehner involutions.
Also, one has
div(X2+ X + 61) = −10(∞) + 2(α, 0) + 2(β, 0) + other six simlpe zeros.
Hence the function (X2 + X + 61)/Y has triple poles only at ∞ and 0. Similarly, we can show that (X2+ X + 61)/Y is invariant under Atkin-Lehner involutions by directly computation.
By setting s = Y /X and t = (X2+ X + 61)/Y , we have t2+ 9st − 33t + 270 = s3− 23s2+ 149s, which we takes as the defining equation of X0+(61). Let
x =s − 1 = Y X − 1,
y =t + 4x − 12 = X2+ X + 61
Y + 4x − 12.
Hence we have modular parameterization of the elliptic curve y2+ xy = x3− 2x + 1.
4.2.3 An alternative method for X
0+(p)
In this section we describe an alternative method for finding modular parameterizations of elliptic curves of the type X0+(p), where p is one of the primes 37, 43, 53, 61, 79, 83, 89, 101, and 131.
First of all, for such a given modular curves X0+(p), there are two pieces of infor-mation available to us in the tables of [3] (The tables can be seen in the web site http://modular.fas.harvard.edu/Tables/. [14]). One piece of information is the equation
y2+ a1xy + a3y = x3+ a2x2+ a4x + a6
of the elliptic curve given in Table 1 of [3]. The other is the Fourier expansion q + b2q2+ b3q3+ · · ·
of the unique normalized Hecke eigenform given in Table 3. Now let x be a modular function on Γ+0(p) with a unique pole of order 2 at infinity with leading coefficient 1 and y be a function with a triple at infinity with leading coefficient 1. We may assume that they satisfy the equation y2 + a1xy + a3y = x3 + a2x2 + a4x + a6. Furthermore, recall that if Γ is a subgroup of P SL2(Z) of finite index, there is an isomorphism ω = f dτ between two vector spaces {f : meromorphic modular forms of weight 2 on Γ} and {ω:
meromorphic differential 1-forms on X(Γ)}. By this one-to-one correspondence, ω is holomorphic on X(Γ) if and only if f vanishes at every cusps on X(Γ). Thus, if ω is a holomorphic differential 1-form on X0+(p), then ω/dτ is a cusp form of weight 2 on X0+(p),
where τ denotes the standard local parameter of X0+(p). Since the holomorphic 1-form of the Fourier expansions of x and y.
Let us take the curve X0+(43) for example. According to Table 1 of [3], it has an equation
y2+ y = x3+ x2.
Furthermore, from Table 3 of [3], we find that b2 = −2, b3 = −2, b5 = −4, b7 = 0, b11= 3, and so on. Using Proposition 2.4.14, we then deduce that the unique normalized Hecke eigenform on Γ+0(43) has the Fourier expansion
q − 2q2 − 2q3+ 2q4− 4q5+ 4q6+ q9+ 8q10+ 3q11− 4q12+ · · · .
for the coefficients ci and di, we conclude that
x = q−2+ 2q−1+ 4 + 7q + 13q2+ 20q3+ 33q4+ 50q5+ 77q6+ 112q7+ 166q8+ · · · and
y = q−3+ 3q−2+ 8q−1+ 16 + 34q + 63q2+ 115q3+ 197q4+ 336q5+ 549q6+ 885q7+ · · · . The remaining task is to find a closed form representation for x and y.
Observe that x, considered as a modular function on Γ0(43), has two double poles at
∞ and 0 and holomorphic at any other points. Thus, if f is a modular function with a double zero at 0 and having poles only at ∞, then the function xf has only poles at ∞, and we can express it as a sum of functions having poles only at ∞. We now work out the details in the following computation.
Let Γ be the intermediate subgroup between Γ1(43) and Γ0(43) with [Γ0(43) : Γ] = 7,
Z = X
The Fourier expansions for them are
X =2q−4+ q−4+ q−2− 12 + q + 2q2+ 2q3+ q4+ 2q6 − 2q7+ 4q9+ · · · , (4.4)
Thus, the modular function X has pole of order 4 at cusp ∞ and zero of order 1 at cusp 0, the modular function Y has pole of order 5 at cusp ∞ and zero of order 2 at cusp 0, the modular function Z has pole of order 6 at cusp ∞ and zero of order 1 at cusp 0, and the modular function V has pole of order 7 at cusp ∞ and zero of order 1 at cusp 0 on Γ0(43). It follows that the function
xY = q−7+ 2q−6+ 2q−5+ 2q−4+ q−3+ 2q−2+ 2q−1+ 3 + 3q + 6q2+ · · · ,
has only a pole at ∞, and thus can be represented as a linear sum of X, Y , Z, and V . To be precise, we use the function V to cancel the pole of order 7 at cusp ∞. Then we have xY − V = q−6+ q−5+ 3q−4+ q−3+ 2q−2+ 15 + 3q + 5q2+ 4q3+ 2q4+ 6q6− 2q7+ 3q8+ · · · , which is a function with pole at cusp ∞ of order 6. We then use Z, Y , and X to cancel q−6, q−5, and q−4 sequentially. We arrive at
x = (X + V + Z + 43)/Y.
By a similar procedure, we find that
y = (2Zx − V x + Z + 2V + Y − 2X)/(3Y − Z + 2X).
Hence we have modular parameterization of the rational elliptic curve y2+ y = x3+ x2. For the other X0+(p) of genus 1, these two methods also apply. We list the results in Section 4.3.