Defining equations of modular curves
Yifan Yang
Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan
Received 8 October 2004; accepted 27 May 2005 Communicated by Michael Hopkins
Available online 3 August 2005
Abstract
We obtain defining equations of modular curves X0(N ), X1(N ), and X(N ) by explicitly constructing modular functions using generalized Dedekind eta functions. As applications, we describe a method of obtaining a basis for the space of cusp forms of weight 2 on a congruence subgroup. We also use our model of X0(37) to find explicit modular parameterization of rational elliptic curves of conductor 37.
© 2005 Elsevier Inc. All rights reserved.
MSC: primary 11F03; secondary 11G05; 11G18; 11G30
Keywords: Modular curves; Generalized Dedekind eta-functions; Cusp forms; Modular parameterization
of rational elliptic curves
1. Background
1.1. Defining equations of modular curves
Let be a congruence subgroup of SL2(R). The classical modular curves X() are
defined to be the quotients of the extended upper half-plane H∗ = { ∈ C : Im > 0} ∪ Q ∪ {∞} by the action of . In this note we will mainly concern ourselves with the congruence subgroups of the types
0(N ) = ∈ SL2(Z) : ≡ ∗ ∗ 0 ∗ mod N ,
E-mail address: [email protected].
0001-8708/$ - see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2005.05.019
1(N ) = ∈ SL2(Z) : ≡ ± 1 ∗ 0 1 mod N , (N) = ∈ SL2(Z) : ≡ ± 1 0 0 1 mod N ,
and the modular curves\H∗ associated with the above congruence subgroups will be denoted by X0(N ), X1(N ), and X(N ), respectively.
It turns out that a modular curve has the structure of a compact Riemann surface. Thus, a modular curve can be interpreted as a non-singular irreducible projective alge-braic curveC (see [10, Appendix B]). Equivalently, the field of rational functions on C is isomorphic to the field of meromorphic functions on the modular curve. Hence, the
homogeneous polynomials definingC are often referred to as defining equations of the
corresponding modular curve. In practice, however, we find that it is more convenient to drop the non-singular condition, and call any polynomials that yield an isomorphic function field defining equations of a modular curve.
When the genus g of a modular curve is less than 5 or the curve is hyperelliptic (that is, a 2-fold covering ofP1(C) branched at 2g + 2 points), there are standard forms for
defining equations. For example, if the genus is 0, the curve is isomorphic to P1(C),
and the defining equation is the zero polynomial. When the genus is 1, the curve is an elliptic curve, and an affine defining equation takes the form y2+ a1xy+ a3y = x3+a2x2+a4x+a6. When the genus is 2 or the curve is hyperelliptic, an affine defining
equation can be taken to be y2 = f (x) for some polynomial f. (Note that when the
degree of f is greater than 3, the curve y2 = f (x) has a singularity at infinity.) A non-hyperelliptic curve of genus 3 has a plane quartic as a defining equation, while a non-hyperelliptic curve of genus 4 is the complete intersection of a degree 2 surface
with a degree 3 surface in P3 (see [10]). When the genus exceeds 4, the geometry
becomes more complicated, and there are no single standard forms.
When a modular curve is of the type X0(N ), there is a canonical equation for it (the so-called modular equation of level N). Namely, let j () be the classical modular
j-function. Then the function field of X0(N ) is generated by j () and j (N), and a
defining equation of X0(N ) is FN(X, Y ) = 0, where FN is a symmetric polynomial
such that FN(j, Y ) is the minimal polynomial of j (N) over C(j). This model of
X0(N ) is of theoretical use, but has several practical drawbacks. Firstly, the degree of FN is very large, which means that the curve has many singular points. Secondly, the
coefficients are gigantic. For example, when N = 2, the largest coefficient in F2 is
already 157 464 000 000 000.
1.2. Obtaining equations using the canonical embedding
Let C be an algebraic curve, and let g be its genus. Let {1, . . . ,g} be a basis
of the space of holomorphic differentials. Suppose that g > 2. Then we can define a canonical map C −→ Pg−1 by P −→ [1(P ), . . . ,g(P )], where P denotes a point
onC. When the curve is non-hyperelliptic, this map is in fact injective, and we call it the canonical embedding (see[10, p. 341]).
In our modular curve setting, the above projective map is equivalent to the map
−→ [f1(), . . . , fg()], where {f1, . . . , fg} is a basis of the space S2() of cusp
forms of weight 2 on . Since any homogeneous polynomial of f1, . . . , fg of
de-gree k is a cusp form of weight 2k and dim S2k() grows roughly in the speed
of 2gk, there is linear dependence among homogeneous polynomials of f1, . . . , fg
of the same sufficiently large degree. In many cases, these relations give a projec-tive model of a modular curve. This approach has been adopted by Galbraith [8], Murabayashi [17], Shimura [21], and others to obtain defining equations for modu-lar curves of the type X0(N ). (Note that this method requires the knowledge of the Fourier coefficients of cusp forms of weight 2. One may obtain such information from Stein’s modular form database [22], whose method of computing the Fourier coeffi-cients in turn is originated from Merel [15,16].) This approach, however, has several drawbacks.
Firstly, ironically, the above method does not work for modular curves of genus 1 or 2, which presumably should be easier than those of higher genus, because there are not sufficient data. The method does not work for any hyperelliptic modular curve either because the map is two to one. (Note that equations of hyperelliptic modular
curves X0(N ) are also obtained by Galbraith [8], González [9], and Shimura [21].
Their methods are similar, except [9].) Secondly, in general, it is difficult to determine whether one has enough equations for a given curve of large genus.
1.3. Other methods of determining defining equations
Explicit equations of modular curves X1(N ) have been studied by several authors. Using the fact that X1(N ) can be interpreted as moduli spaces of isomorphic classes of elliptic curves with level N structures, Reichert [19] computed equations of X1(N ), for N= 11, 13, . . . , 18, and then used them to determine torsion structures of elliptic curves over quadratic number fields. However, the computation becomes tedious as
N gets large. Furthermore, the calculation is symbolic, and does not reveal what the
corresponding modular functions that generate function fields are.
Explicit equations of X1(13), X1(16), and X1(25) have also been computed by
Lecacheux [14], Washington [24], and Darmon [5], respectively, for the purpose of constructing cyclic extension of Q. Their methods used the Hauptmoduls of 0(N ).
(The curve X0(13), X0(16), and X0(25) are all of genus 0.) Thus, the methods cannot
be generalized immediately to other N.
Another method of computing equations of X1(N ) is due to Ishida and Ishii [12].
They showed that the function field of X1(N )can be generated by two certain products
of Weierstrass -functions. Thus, the relation between these two functions defines the curve X1(N ). A similar method is also used to obtain defining equations of X(N ) by Ishida [11]. In general, though, the degree of the equations obtained in this fashion is not optimal. For example, the modular curves X1(14) and X1(15) are both of genus
1. Thus, the defining equations can be taken to be y2 = x3+ ax + b. However, the
remedied by finding suitable birational maps. But it is still something to be taken care of.)
1.4. Goals of the present note
In this note we will describe a systematic way of constructing modular functions on congruence subgroups with desired behavior at cusps using the generalized Dedekind
-functions. (See the next section for the definition of these functions.) Our method
of constructing modular functions enables us to solve a variety of problems related to the theory of modular functions and modular curves, including the main theme of the present note, namely, determining defining equations of modular curves.
A distinct feature of our method is that the modular functions constructed all have poles only at infinity. (Thus, they can be regarded as analogs of Hauptmoduls for congruence subgroups of higher genus.) This feature makes the computation of defining equations relatively simple (see the discussion in Section 2). Furthermore, the equations obtained using our method are all plane curves, which may be more preferable in applications than those obtained from the canonical embedding.
Our method of finding defining equations works for curves of all types X0(N ),
X1(N ), and X(N ), regardless of the genus or whether the curve is hyperelliptic. (At
least in theory. To actually obtain equations for modular curves of large level in the range of hundreds, the solving of the related integer programming problem could take hours of computer time. Though, for the curves listed in the end of the article the computation takes only seconds.) Our method does not require knowledge of cusp forms of weight 2 either. On the contrary, our method in fact provides a way of finding a basis for the space of cusp forms of weight 2 on congruence subgroups. Furthermore, our model of X0(N ), in many cases, can be used to determine explicitly
the modular functions parameterizing a rational elliptic curve. In this note, we will work out the cases of elliptic curves of conductor 37.
The rest of the paper is organized as follows. In Section 2, we will give the definition and properties of the generalized Dedekind -functions, and describe our method of finding defining equations of modular curves using them. In Section 3, we will give details of the applications mentioned above. In Section 4, we list defining equations up to N = 50 for X0(N ), up to N = 22 for X1(N ), and up to N = 12 for X(N).
(We have also computed a few more curves of higher level. They are available upon request.)
2. A new approach
LetC be a modular curve of non-zero genus, and let K(C) denote the function field ofC. Our method of finding defining equations of C use the following basic idea, which is also used in [12]. Here, for f ∈ K(C), we let deg∞f denote the total number of poles of f.
Lemma 1. Suppose that X and Y are in K(C) such that gcd(deg∞X,deg∞Y ) = 1. Then one has K(C) = C(X, Y ), and thus a defining equation of C can be taken to be F (x, y) = 0, where F (x, y) is a polynomial such that F (X, y) is the minimal polynomial for Y over C[X]. Moreover, F (x, y) is a polynomial of degree n in x and of degree m in y.
Proof. Let m= deg∞X and n= deg∞Y, and assume that gcd(m, n) = 1. Then we have [K(C) : C(X)] = m and [K(C) : C(Y )] = n (see, for example, [7, p. 194]). It follows that[K(C) : C(X, Y )] divides both m and n. Since gcd(m, n) = 1, we conclude
that [K(C) : C(X, Y )] = 1. That is, K(C) = C(X, Y ), [C(X, Y ) : C(X)] = m, and
[C(X, Y ) : C(Y )] = n. Then the assertion about F (x, y) follows immediately. This
proves the lemma.
As mentioned in the introduction, the functions we construct will have poles only at infinity. In this case, the polynomial F (x, y) in Lemma 1 can be described as follows.
Lemma 2. Suppose that X and Y are functions onC with a unique pole of orders m
and n, respectively, at infinity such that gcd(m, n)= 1 and that the leading Fourier coefficients are both 1. Then the polynomial F (x, y) in Lemma 1 takes the form
xn− ym+
a,b0,am+bn<mn
ca,bxayb.
Proof. By Lemma 1, the polynomial F (x, y) takes the forman,b mca,bxayb. Let
a0 and b0 be non-negative integers such that
a0m+ b0n= max{am + bn : ca,b= 0}.
That is, Xa0Yb0 has the largest degree among all terms with c
a,b = 0. In order to
cancel the pole of order a0m+ b0n at infinity, there must be another pair (a1, b1) of
non-negative integers such that a0m+ b0n = a1m+ b1n. Since gcd(m, n) = 1, we
have n|(a0− a1)and m|(b1− b0). Now suppose that none of the integers a0 and a1 is
equal to zero. Then we will have a0> n or a1> n. This contradicts to the fact from
Lemma 1 that F (x, y) is a polynomial of degree n in x. Therefore, we have a0= 0,
b0= m or a1= n, b1= 0. This shows that the polynomial F (x, y) takes the claimed
form.
In practice, Lemma 2 means that, to find a relation between given X and Y with the prescribed properties, we can compute the Fourier expansion of Xn− Ym and use suitable products XaYb to cancel the poles at infinity recursively until we reach the constant term.
In light of Lemmas 1 and 2, to obtain defining equations of modular curves, it suffices to find functions with poles only at infinity. We now describe our method of constructing such functions.
2.1. Generalized Dedekind -functions
Let ∈ H, and set q = e2i. The ordinary Dedekind -function is defined to be
() = q1/24 ∞
n=1
(1− qn).
This classical function has been extensively used to construct modular functions and modular forms on congruence groups containing0(N ). For example, a table of
Haupt-moduls expressed in terms of the -functions enabled Conway and Norton [3] to dis-cover and describe explicitly the monstrous moonshine phenomena. However, in general,
-functions alone cannot yield all modular functions on a congruence group
contain-ing 0(N ). For example, there is no way to express a Hauptmodul for +0(23) :=
0(23)+ w23 in terms of() and (23), where w23 denotes the Atkin–Lehner
invo-lution. Furthermore, when a congruence group does not contain0(N ), the associated
function field has to be generated by something other than the Dedekind -functions,
and we find that generalized Dedekind-functions are suitable for this purpose. Following the notation by Yang [25], we fix a positive integer N, and define two classes of generalized Dedekind-functions by
Eg,h() = qB(g/N )/2 ∞ m=1 1− e2ih/Nqm−1+g/N 1− e−2ih/Nqm−g/N
for g and h not congruent to 0 modulo N simultaneously and
Eg() = qNB(g/N )/2 ∞ m=1 1− q(m−1)N+g 1− qmN−g
for g not congruent to 0 modulo N, where B(x) = x2− x + 1/6. In Yang [25] we illustrated how to find Hauptmoduls for torsion-free genus 0 congruence subgroups of
SL2(Z) using Eg. Moreover, generalizing the above result, we successfully determined
Hauptmoduls for all genus 0 congruence subgroups of SL2(Z) (up to conjugation) in
Chua et al. [2]. In this note we will make use of the above functions to construct modular functions that parameterize modular curves. Here, we recall the properties of
Eg relevant to our consideration.
Proposition 1 (Yang [25, Theorem 1]). The functions Eg,h satisfy
Moreover, let =
a b c d
∈ SL2(Z). Then we have, for c = 0,
Eg,h( + b) = eibB(g/N)Eg,bg+h(),
and, for c= 0,
Eg,h() = (a, b, c, d)ei Eg ,h (),
where (a, b, c, d) = ei bd(1−c2)+c(a+d−3) /6 if c is odd, −iei ac(1−d2)+d(b−c+3) /6 if d is odd, = g2ab+ 2ghbc + h2cd N2 − gb+ h(d − 1) N and (g h )= (g h) a b c d .
Proposition 2 (Yang [25, Corollary 2]). The functions Eg satisfy
Eg+N = E−g= −Eg. (2) Moreover, let = a b cN d ∈ 0(N ). We have, for c= 0, Eg( + b) = eibNB(g/N)Eg(), and, for c= 0,
Eg() = (a, bN, c, d)ei(g
2ab/N−gb) Eag(), (3) where (a, b, c, d) = ei bd(1−c2)+c(a+d−3) /6 if c is odd, −iei ac(1−d2)+d(b−c+3) /6 if d is odd.
Proposition 3 (Yang [25, Corollary 3]). Consider the function f () = g Eg()eg,
where g and eg are integers with g /≡ 0 mod N. Suppose that one has
g eg≡ 0 mod 12, g geg≡ 0 mod 2. (4)
Then f is invariant under the action of (N). Moreover, if, in addition to (4), one also
has
g
g2eg ≡ 0 mod 2N. (5)
Then f is a modular function on 1(N ).
Furthermore, for the cases where N is a positive odd integer, conditions (4) and (5)
can be reduced to g eg≡ 0 mod 12 and g g2eg ≡ 0 mod N, respectively.
Proposition 4 (Yang [25, Lemma 2]). The order of the function Eg at a cusp a/c with
(a, c)= 1 is (c, N)P2(ag/(c, N ))/2, where P2(x)= {x}2− {x} + 1/6 and {x} denotes
the fractional part of a real number x.
We now show that modular functions with poles only at infinity can be constructed using the above functions. This requires a result of Yu [26].
In [26] the cusps of X1(N ) that lies above 0 on X0(p) for all primes p|N are
referred to as the cusps of the first type. Let F10(N ) denote the group of functions on X1(N ) that have divisors supported on the cusps of the first type. Moreover, let
F1 (N )be the group generated by functions of the type
N−1
h=1 E0,h()eh satisfying the
conditions
N−1 h=1
h2eh≡ 0
mod N for odd N,
mod 2N for even N,
and
h≡±a mod N/p
eh= 0 for all p|N and for all congruence classes a.
Proposition 5 (Yu [26, Theorem 4]). We have F10(N )= F1 (N ), and they are of rank
(N)/2 − 1.
Now observe that the action of the Atkin–Lehner involution N sends the cusps
of the first type to the cusps that are equivalent to ∞ under 0(N ), and that, by
Proposition 1,
E0,g(−1/N) = e−ig/NEg,0(N) = e−ig/NEg().
Thus, we have the following result.
Proposition 6. Assume N3. Let F1∞(N ) denote the group of modular functions on
1(N ) that have divisors supported by the cusps lying above ∞ on X0(N ), and let
F
1(N ) denote the group generated by functions of the type
N−1 g=1 Eg()eg satisfying N−1 g=1 g2eg≡ 0
mod N for odd N,
mod 2N for even N, (6)
and
g≡±a mod N/p
eg= 0 for all p|N and for all congruence classes a. (7)
Then one has F1∞(N )= F1 (N ), and they are of rank(N)/2 − 1.
We remark that, by Proposition 3, conditions (6) and (7) imply that the product is a modular function on 1(N ), and, by Proposition 4, condition (7) implies that the function has neither poles nor zeroes at the cusps that are not equivalent to infinity under 0(N ).
We now prove a result stating that we can always find functions X and Y satisfying the assumptions in Lemma 2. The proof requires the following lemma.
Lemma 3. Let V ⊂ Zn be a Z-module of rank n − 1 such that a1+ · · · + an = 0
for all v = (a1, . . . , an) ∈ V . Let d be the greatest common divisor of all a1 in
v = (a1, . . . , an) ∈ V . Then there is an element(−md, a2, . . . , an) in V such that
a2, . . . , an0 for all sufficiently large integer m.
Proof. We first choose any vector v0 in V with v0 = (−d, b2, . . . , bn), and let b =
max2k n|bk|. Now consider the vector v1= (1 − n, 1, . . . , 1) ∈ Zn. It is contained
in the subspace W ⊂ Zn consisting of all vectors whose sums of entries are equal to zero. Since W is also of rank n− 1, there is a positive integer a such that av1 ∈ V .
V and they are of the form (−md, a2, . . . , an) with a2, . . . , an0. Then the assertion
follows immediately.
Proposition 7. The groupF1∞(N ) contains at least two functions that have poles only at infinity and whose orders of poles are relatively prime.
Proof. Assume that N3. By Proposition 6 and Lemma 3, it suffices to prove that
the group F1∞(N )contains a function having a simple pole at infinity.
When N is a prime greater than 3, we find (E22/E1E3)N is such a function. When N is
a prime power pa8, a 2, we consider functions of the type fk=Ek2+N/p/EkEk+2N/p,
k /≡ N/p mod N. It is easy to verify that the divisors are supported at cusps equivalent
to infinity under 0(N ). If k is an integer such that k+ 2N/p > N > k + N/p, then
the order of fk at infinity is
N (2B2(k/N+ 1/p) − B2(k/N )− B2(k/N+ 2/p − 1)) /2 = k − N/p2+ 2N/p + N,
where B2(x)= x2−x +1/6 is the second Bernoulli polynomial. Thus, if k is an integer
such that k+ 2N/p > N > k + N/p − 1, then the function fk/fk+1 has a simple pole
at infinity.
When N is a product paqb, p < q, of two prime powers, we consider the
func-tion fk = Ek+N/pEk+N/q/(EkEk+N/p+N/q), k /≡ −N/p, −N/q, −N/p − N/q mod N.
Again, these functions have poles and zeroes only at the cusps equivalent to infinity under 0(N ). When k is chosen such that k+ N/p + N/q > N > k + N/p, then the
order of fk at infinity is
k+ N/p + N/q − N/(pq) − N.
Thus, if k+ N/p + N/q > N > k + N/p − 1, then fk/fk+1 has a simple pole at
infinity.
When N is a product pa1
1 p
a2
2 . . . of at least three prime powers, the exact description
of construction becomes complicated, and we shall only sketch our idea. Let P0 denote
the set of primes dividing N. For a subset P of P0we let cP denote the sum
p∈P 1/p.
We consider the function fk of the form
fk = P⊂P0 Ek(−1)+Nc|P | P = Ek p Ek+N p −1 p1,p2 Ek+N p1+ N p2 p1,p2,p3 Ek+N p1+ N p2+ N p3 −1 . . . ,
where the products run over all subsets P of P0, and let k vary. Let m(x) denote
simplifying, is equal to C− k P⊂P0 m(k/N+ cP)+ P⊂P0 N cPm(k/N+ cP) +N 2 m(k/N+ cP)2+ m(k/N + cP) ,
where C is a constant depending only on N. Now choose k1 and k2 such that the
integers m(k1/N + cP)= m(k2/N+ cP)for all P ⊂ P0 with only one exception P1,
where m(k1/N+cP1)= 0 and m(k2/N+cP1)= 1. Then the function fk2/fk1 has order
(k1− k2)
P=P1
m(k/N+ cP)− k2+ C1
at infinity, where C1 is a constant depending only on N and P1. Finally, if k1+ 1 and
k2+ 1 also satisfy the property that m((k1+ 1)/N + cP)= m((k2+ 1)/N + cP) for
P = P1 and m((k1+ 1)/N + cP1)= 0, m((k2+ 1)/N + cP2)= 1, then the function
fk2+1fk1/(fk1+1fk2) has a simple pole at infinity. This concludes the proof of the
theorem.
Remarks. For the curves X1(N ) we have computed so far, we find that it is always possible to find a product of Eg that is modular on1(N ) and have a unique pole of
order m at infinity for each non-gap integer m. It is reasonable to conjecture that it is always the case, but we are unable to prove it at this point.
We also remark that since 1(N ) is normal in 0(N ), if f is a function on 1(N ),
then
∈0(N )/1(N )
f ()
is a modular function on 0(N ). Thus, Proposition 7 implies that we can always find
modular functions on 0(N )with a unique pole of order m at infinity for sufficiently
large m. Furthermore, since (N) is conjugate to a congruence subgroup containing 1(N2), suitable products of Eg will generate the function field on X(N ) as well.
In the following sections we will work out some simple examples to illustrate the procedures of constructing modular functions using our method.
2.2. Equations for X1(N )
Let us take the genus 1 curve X1(11) for example. From Property (2) in Proposition 2
we see that there are essentially only five distinct Eg. In order to fulfill the conditions
setting of Wk = E4k/E2k instead of E2k/Ek is to get rid of the factor involving eib in
formula (3) so that the transformation formula for Wk becomes simpler.) It is obvious
that any product of Wk will satisfy condition (4) in Proposition 3 automatically. Thus,
if ek are integers such that
k2ek ≡ 0 mod 11, then
Wek
k is modular on 1(11).
Furthermore, from Proposition 4 we see that the only poles or zeroes of Wk are at
cusps equivalent to cj = j/11, j = 1, . . . , 5. Let vk(cj)denote the order of Wk at cj.
The values of vk are given in the following table.
c1 c2 c3 c4 c5 11v1 −5 2 10 −3 −4 11v2 2 −3 −4 10 −5 11v3 10 −4 2 −5 −3 11v4 −3 10 −5 −4 2 11v5 −4 −5 −3 2 10
Thus, finding a function X with a pole of order 2 at infinity and analytic elsewhere is equivalent to solving the integer programming problem
−5x1+ 2x2+ 10x3− 3x4− 4x5 = −22,
2x1− 3x2− 4x3+ 10x4− 5x5 0,
10x1− 4x2+ 2x3− 5x4− 3x5 0,
−3x1+ 10x2− 5x3− 4x4+ 2x5 0,
−4x1− 5x2− 3x3+ 2x4+ 10x5 0,
and we find that one of the solutions is (x1, x2, x3, x4, x5)= (3, 2, 0, 1, 2). Hence, we
can choose
X= −W13W22W4W52= q−2+ 2q−1+ 4 + 5q + 6q2+ 5q3+ 3q4− q5+ · · · ,
where q= e2i. Similarly, we can choose a degree 3 function Y to be
Y = W14W2W4W53= q−3+ 3q−2+ 7q−1+ 13 + 19q + 24q2+ 25q3· · · .
Now consider Y2− X3, which has a Fourier expansion
Y2− X3= −q−4− 3q−3− 9q−2− 19q−1− 35 − 94q + · · · .
Using X2 to cancel the pole of order 4, we find
Y2− X3+ X2= q−3+ 3q−2+ 7q−1+ 13 + 19q + · · · = Y.
In general, to find an equation for X1(N ) we solve integer programming problems
analogous to that for X1(11) and find two modular functions X and Y with minimal
orders of pole at infinity so that gcd(deg∞X,deg∞Y ) = 1. Then we compute the
relation between X and Y as above.
2.3. Equations for X0(N )
For curves X0(N )the basic idea is the same, though the implementation is different
and in many cases we can just use the Dedekind eta function. (See[18] for properties of the Dedekind eta function.)
To construct a modular function with a pole of order k at infinity and analytic elsewhere, we first find a function F on 1(N ) that has a pole of order k at infinity,
poles of order < k at other cusps equivalent to infinity under 0(N ), and regular at
any other points. Then the function
X=
∈0(N )/1(N )
F
is modular on 0(N ) with the desired properties, where runs over a set of coset
representatives of0(N )/1(N ). Take X0(11) for example. We solve the integer
pro-gramming problem −5x1+ 2x2+ 10x3− 3x4− 4x5 = −22, 2x1− 3x2− 4x3+ 10x4− 5x5 −11, 10x1− 4x2+ 2x3− 5x4− 3x5 −11, −3x1+ 10x2− 5x3− 4x4+ 2x5 −11, −4x1− 5x2− 3x3+ 2x4+ 10x5 −11 and set X = − ∈0(11)/1(11) W12W53= ∈0(11)/1(11) E42E2 E310 = E24E2 E310 + E28E4 E203 + E212E6 E303 + E216E8 E403 + E220E10 E503 = E24E2 E13 − E23E4 E23 + E21E5 E33 − E52E3 E43 + E22E1 E53 = q−2+ 2q−1+ 4 + 5q + 8q2+ q3+ · · · .
Likewise, we let
Y =
∈0(11)/1(11)
W3−3W4= q−3+ 3q−2+ 7q−1+ 12 + 17q + 26q2+ · · · .
Then the functions satisfy Y2− X3+ X2+ 3Y + 10X + 22 = 0, which we take as a defining equation of X0(11). (In the result section we modify the choice of Y so that
the equation is in conformity with that of Birch and Swinnerton-Dyer [23] or that of Cremona [4].)
A modification of the above method is to utilize the fact that any intermediate
subgroup between 1(N ) and 0(N ) is also normal in 0(N ). Thus, to find a
modular function on0(N ) with a unique pole of order k at infinity, we can proceed
as above with the only difference being 1(N ) replaced by . For example, to find
a modular function on 0(31) with a unique pole of order 3 at infinity, we choose
to be the subgroup generated by 1(31) and
5 −1
31 −6
. It is easy to verify that
Wk = E6kE26kE30k/(E2kE10kE12k) is a modular function on for any integer k not
divisible k. There are five essentially distinct Wk, and they are W1, W2, W3, W4, and
W8. Moreover, the cusp ∞ splits into five cusps 1/31, 2/31, 3/31, 4/31, and 8/31 in
. The orders of Wk at those cusps are as follows:
1/31 2/31 3/31 4/31 8/31 W1 3 0 −4 2 −1 W2 0 2 3 −1 −4 W3 −4 3 −1 0 2 W4 2 −1 0 −4 3 W8 −1 −4 2 3 0 (8)
It follows that the function
∈0(31)/
W3W4W8
is invariant under 0(31) and has a unique pole of order 3 at infinity.
2.4. Equations for X(N )
The method is identical to that for X1(N ). We take (7) for example, and let
Wk = E4k/E2k. From Propositions 2 and 3 we see that Wk is a modular function on
(7). Moreover, the only possible poles of Wk occur at the cusps 1/7, 2/7, and 3/7,
and Wk is regular at any other points. Solving integer programming problems similar
to those mentioned earlier, we set
and Y = W1W32= q7−5+ 2q 2 7+ 2q 9 7+ q 16 7 − q 23 7 + · · · ,
where q7 = e2i/7 is a local parameter at infinity. (Note that the gap sequence is
{1, 2, 4}.) Thus, a defining equation of X(7) can be taken to be Y3− XY = X5.
(Setting Y = yx, X = −x, we obtain a non-singular model xy3+ x3+ y = 0, which is the famous Klein curve.)
2.5. Remarks
Since the complexity of integer programming problems mainly depend on the number and the range of variables, the amount of time needed to find required functions depends on the level, not the type of congruence subgroups. (That is, it will be easier to find modular functions that generate the function field on X(29), which is of genus 806 than that of X0(227), whose genus is only 19 because the integer programming problem for
the former curve involves only 14 variables, while the latter involves 113 variables.) It seems to us that to successfully apply our methods on curves of large level, one would need to take the symmetry of the integer programming problems involved into account.
3. Applications
3.1. Cusp forms of weight 2 on congruence subgroups
An immediate application of our result is the determination of cusp forms of weight 2 on congruence subgroups.
From[20, Proposition 2.16], we know that if = f d is a holomorphic differential 1-form on a modular curve X(), then f is necessarily a cusp form of weight 2 on
. Thus, to determine a basis for the space S2() of cusp forms of weight 2 on a
congruence subgroup , we can compute a defining equation using our method first,
and compute a basis {1, . . . ,g} for the space of holomorphic differential 1-forms.
Then {1/d, . . . , g/d} generates S2().
Let us take X1(17) for example. The genus is 5, and the gap sequence is 1, . . . , 4, 6. Choose
X = E62E7E8/(E12E2E3)= q−5+ 2q−4+ 4q−3+ 7q−2+ 11q−1+ · · · , Y = E62E7E82/(E31E22)= q−7+ 3q−6+ 8q−5+ 16q−4+ 30q−3+ · · · .
A defining equation is hence
Y5− (4X − 1)Y4+ (6X2− 3X)Y3− (X4+ 4X3− 5X2+ X)Y2
+ X3(
From the defining equation we deduce that the space of cusp forms of weight 2 are spanned by −X(2X2− 2X3− Y + X2Y )q dX/dq f (X, Y ) = q − q 2− 2q3+ 3q4− 2q5− q6+ · · · , (−5X3+ 3X4+ 3XY − Y3)q dX/dq f (X, Y ) = q 2− 4q3+ 7q4− 5q5− 4q6+ 10q7+ · · · , X(X2− X3− Y + XY )q dX/dq f (X, Y ) = q 3− 2q4+ q6− q7+ 3q8− q9+ · · · , −X(X − Y )2q dX/dq f (X, Y ) = q 4− 2q5− q6+ 3q7− q9+ q10+ · · · , (X3− X2Y − XY + Y2)q dX/dq f (X, Y ) = q 6− 3q7+ q8+ 3q9− q11− 4q12+ · · · , where f (X, Y )= 4X5− 2X4Y − 5X4+ X3− 8X3Y + 18Y2X2+ 10X2Y −2XY − 16Y3X− 9Y2X+ 5Y4+ 4Y3.
3.2. Modular parameterization of rational elliptic curves
The well-known Taniyama–Shimura conjecture states that every rational elliptic curve can be parameterized by modular functions. The truth of this conjecture has been established by A. Wiles and others. However, in general, it is difficult to explicitly write down modular functions that parameterize an elliptic curve. Here we will demonstrate how to obtain modular parameterization of rational elliptic curves of conductor 37 using our model of X0(37).
The modular curve X0(37) is of genus 2, and thus hyperelliptic. The hyperelliptic
involution is defined over Q, but it does not come from the normalizer of 0(37) in
SL2(R). Let w37 denote the Atkin–Lehner involution and whthe hyperelliptic involution.
Then the curves X0(37)/w37 and X0(37)/(w37wh) are of genus 1. We now construct
modular functions to parameterize these two elliptic curves.
Let be the intermediate subgroup between 1(37) and0(37) with [0(37): ] =
6, and set X= () 2 (37)2+ 37 and Y = ∈0(37)/ E6E8E14 E3E4E7 − 5X + 174.
Then one has
Y3+ (7X − 259)Y2− (7X2− 259X)Y = X2(X− 36)(X − 37), (9) which we take as the defining equation of X0(37).
From Kenku [13] we know that there are four rational points on X0(37). In the above model we can easily locate four rational points, namely, ∞, (0, 259), (36, 0), and (37, 0). (The singular point (0, 0) is not a rational point. Blowing up the point
(0, 0) we obtain a non-singular model y− tx = 0, t3x− x2+ 7t2x − 7tx + 73x −
259t2+ 259t − 1332 = 0. We can easily see that the point corresponding to X = 0,
Y = 0 is not a rational point.) The point ∞ corresponds to the cusp ∞. Using the
transformation formula for the Dedekind eta function we obtain
X
w37
= 37(37)()22 + 37, (10)
and thus X(0) = 37. Hence, the rational points (37, 0) corresponds to the cusp 0. The other two points (0, 259) and (36, 0) must be the image of the cusps under the hyperelliptic involution. Since the birational map
u= Y
X, v=
Y3+ 7XY2− 7X2Y + 73X3− 518Y2+ 518XY − 2664X2
X3 , X= 74(7u 2− 7u + 36) u3+ 7u2− 7u − v + 73, Y = 74u(7u2− 7u + 36) u3+ 7u2− 7u − v + 73
transforms (9) into the normal form
v2= u6+ 14u5+ 35u4+ 48u3+ 35u2+ 14u + 1,
the hyperelliptic involution wh sends the point (37, 0) to (36, 0) and the point ∞
to (0, 259). Thus, to find explicit modular parameterization of X0(37)/w37 we first
construct functions s and t with poles only at ∞ and (37, 0) such that s has a double pole at∞ and a pole of order at most 2 at (37, 0) and t has a triple pole at ∞ and a pole of order at most 3 at (37, 0). Then the functions x= s + sw
37 and y= t + tw37
yield an equation for the elliptic curve X0(37)/w37. Likewise, to obtain explicit modular
parameterization of X0(37)/(w37wh), we construct functions s and t with poles of order
2 and 3, respectively, at ∞ and (36, 0), and then proceed as usual. For the purpose of constructing such functions, we shall first study the behavior of X and Y under wh,
w37, and w37wh.
The involution wh sends u to u and v to −v. It follows that
X wh = 74(7u2− 7u + 36) u3+ 7u2− 7u + v + 73 = 37(7Y2− 7XY + 36X2) X3 (11)
and
Y
wh
= 74u(7u2− 7u + 36)
u3+ 7u2− 7u + v + 73 =
37Y (7Y2− 7XY + 36X2)
X4 . (12) From (10) we have X w37 =37(X− 36) X− 37 . (13) To express Yw
37 in terms of X and Y, we utilize Proposition 1. We have
Eg w37
= Eg,0(37)
w37
= Eg,0(−1/) = eig/37E0,g().
From this we deduce that
Y
w37
= 37(q + 3q2+ 2q3+ 7q4+ 11q5+ 25q6+ · · ·).
At the cusp 0, the function X− 37 has a triple zero, the function Y has a simple zero, and the function Yw
37 has a quadruple pole. Hence, Yw37·(X−37)Y is a function with
a unique pole of order 6 at ∞. Using the Fourier expansions of the above functions we find that
Y
w37
=37X(X− 36)
Y (X− 37) . (14)
Therefore, by (11), (12), (13), and (14), we have
X w37wh =(7X2− 7XY + 36Y2)(X− 37) Y2(X− 36) (15) and Y w37wh =X(7X2− 7XY + 36Y2)(X− 37) Y3(X− 36) . (16)
(Alternatively, we can use divisors of the functions X, X− 37, and Y to guess that
Y
w37 = cX(X−36)/((X−37)Y ) for some constant c. Then, since the choice of c = 37
makes the map (X, Y )−→ (37(X − 36)/(X − 37), 37X(X − 36)/((X − 37)Y )) an invo-lution on the curve (9), we conclude that Yw
37 has indeed the indicated expression.) We now construct functions to parameterize X0(37)/w37. For a given function f on a
div X= −3(∞) + 2(0, 0) + (0, 259) and div Y = −4(∞) + 2(0, 0) + (36, 0) + (37, 0). It follows that the function s= X(X − 36)/Y has poles of order 2 at ∞ and a simple pole at (37, 0), and regular everywhere. Thus, s+ sw
37 is a function on X0(37)/w37
with a unique pole of order 2 at∞. Using (9), (13), and (14), we express s+ sw 37 as
s+ s
w37
= X3− 73X2+ 1332X + Y2
(X− 37)Y .
Furthermore, the function X has a unique pole of order 3 at∞ on X0(37). Therefore,
X+ X w37 = X +37(X− 36) X− 37 = X2− 1332 X− 37
is a function with a unique pole of order 3 at ∞ on X0(37)/w37. Finally, setting
x = s + s w37 + 13 = X3− 73X2+ 1332X + Y2 (X− 37)Y + 13 = q−2+ 2q−1+ 5 + 9q + 18q2+ 29q3+ 51q4+ 82q5+ · · · and y = X + X w37 + 5x − 80 = X2− 1332 X− 37 + 5x − 80 = q−3+ 3q−2+ 9q−1+ 21 + 46q + 92q2+ 180q3+ 329q4+ · · · ,
we obtain the modular parameterization of the elliptic curve 37A1: y2+ y = x3− x. As a check on our computation we calculate the Fourier expansion of
−q dx/dq
2y+ 1 = q − 2q
2− 3q3+ 2q4− 2q5+ 6q6− q7+ 6q9+ 4q10+ · · · ,
which is indeed the Fourier expansion of the unique normalized eigenform of weight 2 on 0(37)+ w37.
We now construct functions to parameterize the elliptic curve X0(37)/(w37wh). Under
the quotient map X0(37)→ X0(37)/(w37wh), the points ∞ and (36, 0) are identified
together, and (37, 0) and (0, 259) together. Thus, to find a function on the quotient curve with a unique pole of order 2 at∞, we first look for a function on X0(37) that
has a double pole at ∞ and a pole of order at most 2 at (36, 0). From the divisors div X= −3(∞) + 2(0, 0) + (0, 259) and Y = −4(∞) + 2(0, 0) + (36, 0) + (37, 0) we easily see that X(X− 37)/Y has the desired properties. By (15) and (16), we have
X(X− 37) Y + X(X− 37) Y w37wh =X3− 66X2+ 1073X − 7XY + 259Y − Y2 Y (X− 36) .
This is a function on X0(37)/(w37wh)with a double pole at∞. Likewise, the function
X+ X
w37wh
= X +(7X2− 7XY + 36Y2)(X− 37)
Y2(X− 36)
is a function with a triple pole at∞. Finally, setting
x = X 3− 66X2+ 1073X − 7XY + 259Y − Y2 Y (X− 36) + 8 = q−2− 1 + q + 5q2− q3+ 10q4− 4q5+ 15q6+ · · · and y = X + (7X 2− 7XY + 36Y2)(X− 37) Y2(X− 36) + 2x − 72 = q−3− q−1+ 1 − 4q − 2q2− 12q3+ 4q4− 36q5+ · · · ,
we have y2+ y = x3+ x2− 23x − 50. This is the elliptic curve 37B1 in Cremona’s table. Again, we check that
−q dx/dq
2y+ 1 = q + q
3− 2q4− q7− 2q9+ 3q11− 2q12− 4q13+ · · ·
agrees with the Fourier expansion of the normalized eigenform f of weight 2 on0(37)
with fw
37 = −f .
We remark that the above method will certainly work for all rational elliptic curves that are in fact quotient curves of X0(N ) by Atkin–Lehner involutions.
4. Results
In this section, we list equations for modular curves of small level obtained using our method. The computer softwares we used include lp_solve, Ampl, and Maple. The first two are used to solve the integer programming problems for finding required modular functions. (We note that the use of Ampl is not essential in our computation because it serves mainly as a user-solver interface. In fact, the software lp_solve alone will suffice for our purpose.) Once required modular functions X and Y are found, we use the computer algebra software Maple to determine the equation satisfied by X and Y, which by the remark following Lemma 2 is nothing more than computing the
q-expansions of X and Y and finding suitable combination of X and Y to cancel the
negative powers of q in the expression Xn− Ym, where m and n are the orders of pole of X and Y at infinity, respectively. To give the reader a clearer idea of what kind of computation is involved, we shall work out the case0(31) in details.
Let be the congruence subgroup generated by 1(31) and the matrix 5 −1 31 −6 , as given in the last paragraph of Section 2.3. Then the index of in 0(31) is[0(31):
] = 5, and a set of coset representatives is given by {k : k = 0, . . . , 4}, where =
2 −1
31 −16
. For an integer k not divisible by 31, we let Wk= E6kE26kE30k/(E2kE10k
E12k). The functions Wk are modular on and have poles and zeroes only at 1/31,
2/31, 3/31, 4/31, and 8/31. There are only five essentially different Wk and their
orders at the above cusps are given in (8). Moreover, the action of on those Wk is
verified to be W1 = W2, W2 = W4, W4 = W8, W8 = W3, W3 = W1.
Now the genus of 0(31) is 2. Thus we need to find modular functions X and Y on
0(31) with a pole of order 3 and 4 at infinity (or equivalently 1/31), respectively.
The corresponding inequalities are
3x1+ 0x2− 4x3+ 2x4− 1x5 = −m,
0x1+ 2x2+ 3x3− 1x4− 4x5 −m + 1,
−4x1+ 3x2− 1x3− 0x4+ 2x5 −m + 1,
2x1− 1x2+ 0x3− 4x4+ 3x5 −m + 1,
−1x1− 4x2+ 2x3+ 3x4+ 0x5 −m + 1
with m= 3 and 4. We find that (using lp_solve) we can choose (x1, x2, x3, x4, x5)=
(0, 0, 1, 1, 1) and (0, 0, 1, 0, 0), respectively. Now we set X = 4 k=0 W3W4W8 k− 10 = W3W4W8+ W1W8W3+ W2W3W1+ W4W1W2+ W8W2W4− 10 = E4E7E11 E1E5E6 + E8E9E14 E2E10E12 − E3E13E15 E4E7E11 + E1E5E6 E8E9E14 − E2E10E12 E3E13E15 − 10 = q−3+ 2q−2− 8 − q + 3q2+ 2q3+ q4+ 2q5− 3q7+ 2q8+ 2q9− q10+ · · · and Y = 4 k=0 W3 k+ 3X + 50
= q−4+ 4q−3+ 7q−2+ q−1− 5 − 2q + 12q2+ 7q3+ 4q4+ 6q5+ 4q6
−10q7+ 10q8+ 8q9− 2q10+ · · · .
By Lemma 2, the functions X and Y satisfy
Y3− X4+
a,b0,3a+4b<12
ca,bXaYb= 0
for some rational numbers ca,b. To find the coefficients ca,b, we start from the Fourier
expansion
Y3− X4 = 4q−11+ 45q−10+ 235q−9+ 672q−8+ 948q−7− 108q−6− 2378q−5
−1709q−4+ 5501q−3+ 10958q−2+ 2382q−1
−11257 − 7145q + 6637q2+ · · · .
From this we see that the coefficient c1,2 must be −4. Computing the q-expansion of
Y3− X4− 4XY2 = 5q−10+ 51q−9+ 232q−8+ 556q−7+ 616q−6− 22q−5
−201q−4+ · · · , we get c2,1= −5. Continuing this way, we find
Y3− X4− 4XY2− 5X2Y − 11X3− 31Y2− 31XY − 31X2= 0.
This concludes the demonstration of our method.
4.1. Equations for X0(N )
In this section, we list defining equations for X0(N ). Here, in general, we choose
functions X and Y with leading Fourier coefficients 1. However, starting from X0(34),
there are a few cases where we make a slight adjustment to make the coefficients of the equations smaller. For example, in the case N= 34, we choose X = q−4/17+ · · · and
Y = q−5/17+ · · ·. In those cases, we will see a rational number in front of a product of Dedekind-functions or a sum of products of generalized Dedekind -functions.
For brevity, a product of Dedekind -functions (ai)bi will be abbreviated as
abi
i . The symbol Egis the generalized Dedekind-function introduced in Section 2.1.
The notation k Egeg represents
∈0(N )/
Egeg ,
where is the intermediate subgroup between 1(N )and0(N )with[0(N ): ] = k.
(In all the cases where this notation occurs, 0(N )/1(N ) is cyclic, and there is no
ambiguity about.)
Whenever the genus of X0(N ) is 1, we adjust the choice of X and Y so that the
equation is in agreement with Cremona’s table. When the genus is greater than 1, the equation is always singular. In those cases, we adjust the functions X and Y so that the (0, 0) is one of the singularities, provided that this adjustment will preserve the rationality of the coefficients.
Special attention should be given to the curve X0(43). The genus is 3, and the cusp
∞ is not a Weierstrass point. Thus, up to a constant displacement, there is only one modular function with a unique pole of order 4 at∞ with leading Fourier coefficient 1. We find that this function is
q−4+12q−3+21q−2+ c +12q+ q2+ · · ·
whose coefficients are not all integral. We have no explanation for this phenomenon.
N Functions Equation 11 X= 5 E2E24 E31 , Y= 5 E45 E31E3+ 1 Y2+ Y = X3− X2− 10X − 20 14 X= 2· 7 7 1· 147 + 1, Y = 28· 74 14· 148− 3X + 1 Y 2+ XY + Y = X3+ 4X − 6 15 X= 3· 5 5 1· 155 − 1, Y= 3 9· 53 13· 159− 32· 510 12· 1510− 3X − 2 Y2+ XY + Y = X3+ X2− 10X − 10 17 X= 8 E3E8 E1E2− 4, Y = 8 E26E8 E22E3 + 1 Y2+ XY + Y = X3− X2− X − 14 19 X= 9 E7E8 E1E6− 3, Y = 9 E62E8 E2E32 + X − 6 Y2+ Y = X3+ X2− 9X − 15 20 X= 4· 10 5 2· 205, Y= 4· 55 1· 205− X − 2 Y 2= (X + 1)(X2+ 4) 21 X= 3 3· 7 1· 213 − 2, Y= 3 6· 72 12· 216− 3· 77 1· 217− 2X − 4 Y2+ XY = X3− 4X − 1 22 X= 5 E8E9 E2E3+ 2, Y = 1 11 28· 114 14· 228 − 17· 113 23· 227 Y3+ (3X − 11)Y2+ X2Y = X4− 9X3+ 22X2 23 X= 11 E8E10 E1E5 − 15, Y= 11 E8E210E112 E4E52E62 + 7X + 85 Y3− (7X + 69)Y2− (12X2+ 230X)Y = X4+ 37X3+ 345X2
24 X= 6 3· 8 2· 243, Y= 4· 82· 125 2· 6 · 246 Y 2= (X − 1)(X − 2)(X + 2) 26 X= 2 4· 132 12· 264 − 13, Y = 3 E3E11 E1E5 + 3X + 35 Y3− (4X + 52)Y2− (4X2+ 52X) = X4+ 25X3+ 156X2 27 X= 9 4 3· 273, Y= 33 273 Y 2+ Y = X3− 7 28 X= 4 4· 142 22· 284, Y= 4· 147 2· 287− 1 Y 3+ 5X2Y= X4− 7X2 29 X= 7 E8E9 E2E5− 4, Y= 7 E4E6E10E14 E2E3E5E7 + 4X + 25 Y3− (5X + 29)Y2− X2Y = X4+ 10X3+ 29X2 30 X= 1· 6 6· 102· 153 22· 33· 5 · 306, Y= 6 3· 103· 156 2· 52· 159 −1· 2 · 5 · 6 · 10 · 153 3· 307 − 5X − 20 Y4+ (3X + 15)Y3+ (3X3+ 15X2)Y = X5+ 11X4+ 30X3 31 X= 5 E4E7E11 E1E5E6 − 10, Y= 5 E3E13E15 E1E5E6 + 3X + 50 Y3− (4X + 31)Y2− (5X2+ 31X)Y = X4+ 11X3+ 31X2 32 X= 16 6 82· 324, Y= 84· 162 42· 324 Y 2= X3+ 4X 33 X= 10 E7E10 E1E4 + 1, Y = 10 E13E16 E2E5 + 1 Y4+ (5X2− 11X)Y2− (4X3− 11X2)Y = X5− 11X4+ 22X3 34 X= 1 17 24· 172 12· 344, Y= 1 17 8 E12E15 E2E5 + 3 17 Y4+ 10XY3+ (21X2− 13X)Y2 +(6X3− 14X2+ 6X)Y = 17X5+ 2X4− 3X3+ 2X2− X 35 X= 1 35 12 E8E14 E1E7 − 19 35, Y= 1 35 12 E12E14E15 E5E6E7 + 5X + 76 35 Y4− (6X + 2)Y3+ (7X2+ 2X)Y2 −(12X3+ 5X2)Y = 35X5+ 31X4+ 7X3 36 X= 12· 18 3 6· 363, Y= 124· 182 62· 364 Y 2= X3+ 1 37 X= 1 2 372+ 37, Y = 6 E6E8E14 E3E4E7 − 5X + 174 Y3+ (7X − 259)Y2− (7X2− 259X)Y = X4− 73X3+ 1332X2 38 X= 1 38 9 E3E12E15E18 E1E4E7E16 − 9 38, Y= 1 38 3 E2E9E13E14E15E16 E3E4E5E6E10E17 − 17 38 2Y5+ (36X − 87)Y4+ (148X2+ 18X − 148)Y3 +(28X3+ 217X2+ 32X − 84)Y2 −(66X4− 12X3− 148X2− 48X + 16)Y = 76X6+ 148X5+ 128X4+ 24X3 −36X2− 16X
39 X= 1 13 33· 13 1· 393 − 1, Y= 1 13 12 E11E19 E2E7 + 5X + 51 13 Y4− (3X + 3)Y3− (3X2+ 3X)Y2 −(3X3+ 3X2)Y= 13X5+ 25X4+ 12X3 40 X=4 3· 20 8· 403, Y= 2· 8 · 204 10· 405 Y 4+ (4X2+ 20X)Y2= X5+ 9X4+ 20X3 41 X= 10 E16E20 E2E18 − 16, Y= 10 E11E17 E4E5 + 4X + 32 Y4− (6X + 41)Y3+ (6X2+ 41X)Y2 −(5X3+ 41X2)Y= X5+ 18X4+ 82X3 42 X=1 7 1· 66· 142· 213 22· 33· 7 · 426 − 1, Y=1 7 6 E16E19 E2E5 + 6 7 Y6− (X + 7)Y5+ (7X2+ 28X + 36)Y4 +(14X3+ 48X2+ 18X − 36)Y3 +(16X4+ 55X3+ 18X2− 36X)Y2 +(18X5+ 60X4+ 48X3) = 7X7+ 18X6+ 12X5 43 X= 1 43 7 E5E8E13 E1E6E7 − 15 43, Y= 1 43 7 E2E9E11E12E14E20 E1E4E6E7E15E19 − 9 43
32Y4− (88X − 1)Y3+ (166X2+ 34X + 5)Y2 −(147X3+ 49X2+ 7X)Y = 43X5− 16X4− 11X3− 2X2 44 X= 1 11 44· 222 22· 444, Y= 1 11 5 E16E18 E4E6 + 2 11 Y5+ 12X2Y3− 14X2Y2+ (13X4+ 6X2)Y = 11X6+ 6X4+ X2 45 X=9 3· 15 3· 453, Y=9· 15 5 3· 455− 1· 5 · 92· 15 3· 454 − X + 1 Y4+ 10XY2+ X3Y= X5− 25X2 46 X=1 2 11 E1E14E15E16E17E18 E5E6E7E8E9E22 − 19 2, Y= 11 E16E21 E2E7 − 2X − 19 Y6+ (5X + 23)Y5+ (12X2+ 46X)Y4 +(23X3+ 138X2)Y3+ (22X4+ 115X3)Y2 +(26X5+ 184X4)= X7+ 8X6 47 X= 1 47 23 E12E17E19E21 E6E10E13E15 − 17 47, Y= 1 47 23 E21E22E23 E6E11E13 + 3X + 102 47 Y5+ (2X − 2)Y4− (X2+ 9X)Y3 −(14X3+ 22X2)Y2− (40X4+ 35X3)Y = 47X6+ 81X5+ 35X4 48 X= 8 7· 12 43· 162· 24 · 482, Y= 84· 242 42· 484 Y 4= X5− 7X4+ 12X3 49 X= 1 49+ 2, Y=E21 E7 + E7 E14 − E14 E21 − 2X + 1 Y2+ XY = X3− X2− 2X − 1 50 X=2 2· 25 1· 502− 5, Y=1 2 104· 252 52· 504 − 12· 10 · 253 2· 5 · 504 + 2X +152 Y3− (2X + 10)Y2− (2X2+ 5X)Y = X4+ 9X3+ 20X2
4.2. Equations for X1(N )
Here the notation abi
i represents Ebi ai. N Functions Equation 11 X=3· 4 · 5 12· 2 , Y= 43· 5 13· 2− 1 Y 2+ Y = X3− X2 13 X=4 2· 5 · 6 12· 2 · 3, Y= 4· 63 13· 2 Y 3− (X − 1)Y2− XY = X4+ X3 14 X=3· 4 2· 7 1· 22· 5− 1, Y = 4· 52· 6 1· 22· 3− 1 Y 2+ XY + Y = X3− X 15 X=4· 7 1· 2− 1, Y = 4· 5 · 62 1· 2 · 32− 1 Y 2+ XY + Y = X3+ X2 16 X=5· 6 · 7 1· 2 · 3, Y= 4· 72· 8 1· 22· 3+ 1 Y 3+ (X − 1)Y2− X2Y= X4− X3 17 X=6 2· 7 · 8 12· 2 · 3, Y= 62· 7 · 82 13· 22 Y5− (4X − 1)Y4+ (6X2− 3X)Y3 −(X4+ 4X3− 5X2+ X)Y2 +X3(4X− 1)(X − 1)Y = X6(X− 1) 18 X=4· 5 · 9 1· 2 · 3, Y= 5· 6 · 7 · 8 1· 2 · 3 · 4− 1 Y3+ XY2+ (2X2− 2X)Y = X4− 3X3+ 2X2 19 X=6· 8 · 9 2 12· 2 · 3+ 1, Y=4· 6 2· 72· 82· 92 13· 23· 32· 5 Y6− (5X − 3)Y5− (3X3− 15X2+ 14X − 3)Y4 +(X − 1)(9X4− 18X3+ 7X2− 1)Y3 −X2(X− 1)(9X4− 20X3+ 13X2− X − 2)Y2 +X4(X− 1)2(4X3− 6X2+ 2X + 1)Y = X7(X− 1)4 20 X=6· 8 · 9 1· 2 · 4, Y= 5· 8 · 9 · 10 1· 2 · 3 · 4 + 1 Y4+ XY3+ X(2X − 3)Y2 −X(2X2− 1)Y = X4(X− 1) 21 X=6· 7 · 8 · 10 1· 2 · 3 · 5, Y= 4· 7 · 8 · 102 12· 22· 5 Y5− (6X − 4)Y4+ (2X − 1)(7X − 6)Y3 −3(X − 1)(X3+ 3X2− 4X + 1)Y2 +3X2(X− 1)2(2X− 1)Y = X4(X− 1)3 22 X=7· 8 · 9 · 10 1· 2 · 3 · 4, Y= 8· 92· 10 1· 22· 3 Y6+ (X + 5)Y5− (4X2+ 2X − 8)Y4 −(2X3+ 16X2+ 14X − 4)Y3 +(6X4+ 11X3− 6X2− 12X)Y2 +2X2(X+ 1)(X2+ 6X + 6)Y = X3(X+ 1)2(X+ 2)2 4.3. Equations for X(N )
Again, the notation abi
i represents
Ebi ai.
N Functions Equation 6 X= (2)(3) 3 ()(6)3, Y= (2) 4(3)2 ()2(6)4 Y2= X3+ 1 7 X= 3/1, Y = 2 · 3/12 Y3− XY = X5 8 X= 3/1, Y = 2 · 4/12 Y4= X(X − 1)(X + 1)(X2+ 1)2 9 X= 4/1, Y = 3 · 4/12 Y6− X(X3+ 1)Y3= X5(X3+ 1)2 10 X= 3· 4 1· 2, Y= 43· 5 12· 2 · 3 Y 10= X(X + 1)2(X− 1)8(X2+ X − 1)5 11 X= 4· 5 12 , Y= 4· 52 12· 3 Y10(Y+ 1)9= X22− Y (Y + 1)4
×(6Y4+ 13Y3+ 12Y2+ 5Y + 1)X11
12 X= 5/1, Y = 4 · 6/12 Y12= X(X − 1)2(X+ 1)6(X2+ 1)4(X2− X + 1)3
Acknowledgements
The integer programming problems occurring in this work were solved using the lp_solve software. The author would like to thank M. Berkelaar, the author of lp_solve, for making it freely available on the Internet. The author would also like to thanks Professor Chiang-Hsieh of the National Center for Theoretical Sciences (Taiwan) for reading earlier versions of the paper and providing valuable comments. The author’s use of the Fine functions Wk and their analogs is inspired by the work
of Chan et al. [1]. The author would like to thank Professor Chan of the National University of Singapore for drawing his attention to the Fine functions. Finally, the author would like to thank the anonymous referee, whose constructive criticisms and valuable comments result in a great improvement of the paper. The author is supported by the National Science Council (Taiwan) Grant 92-2119-M-009-001.
References
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