國
立
交
通
大
學
應
用
數
學
系
碩
士
論
文
有理橢圓曲線的模參數化
Modular Parameterization of
Rational Elliptic Curves
研 究 生：凃芳婷
指導教授：楊一帆 教授
有理橢圓曲線的模參數化
Modular Parameterization of Rational Elliptic Curves
研 究 生：凃芳婷 Student：Fang-Ting Tu
指導教授：楊一帆 教授 Advisor：Yifan Tang
國 立 交 通 大 學
應用數學系
碩 士 論 文
A ThesisSubmitted to Department of Applied Mathematics College of Science
National Chiao Tung University in partial Fulfillment of the Requirements
for the Degree of Master
in
Applied Mathematics July 2006
Hsinchu, Taiwan, Republic of China
中華民國九十五年七月 i
自 序
待在交大應數已將近六個年頭，前些年，是看過不少書，也修了不少課，卻 也只能用不學無術來形容。有幸於大學的最後一年碰上我的指導教授楊一帆，使 我碩士這兩年的學習生涯增色不少。 教授個性溫和謙虛，縱使有我如此不材的學生，仍保有耐性傾囊相授。不論 是學習遇到的困難亦或是生活中的挫折，一路走來，始終盡其所能給我最大的幫 助。八股地如同小時候的作文，卻也是事實：教授是良師、是益友，他的諄諄教 誨，循循善誘，「搏我以文、約我以禮」，逐漸地扎實了我的認知。獻萬分的謝意 給我的指導教授，楊一帆。 除了感謝家人多年來的支持與鼓勵，在此感謝所有授予學識的老師、教授， 使我的學習更有組織性；特別感謝蔡孟傑教授，許世雄教授，陳燕美教授，以及 其餘兩位碩士論文的口試委員：潘戍衍教授及 Professor Michael Fuchs，使我 的學習更趨完整。也謝謝我的好友們、學長姐、學弟妹，陪我偷懶、陪我學習、 陪我喜怒哀樂，豐富了我的生活。 這篇論文的概念，延續了指導教授的論文，目前只完成了一部分的結果；未 來的日子裡，將繼續完成其餘部分，且讓這些結果更完美。 謹以此，獻給驟逝的父親，共同分享這份未能趕上的喜悅。 ii有理橢圓曲線的模參數化
研究生：凃芳婷 指導教授：楊一帆 教授
國 立 交 通 大 學
應用數學系
摘 要
二十世紀末，Andrew Wile 證明了 谷山-志村猜想，也間接地證明著名的費 瑪最後定理。而這個猜想的敘述是說：只要給定任何一個有理橢圓曲線，我們都 能夠找到一個自然數N以及模函數將其參數化。然而，目前沒有人將有理橢圓 曲線是如何模參數化的型式完整寫下。在楊一帆教授的論文"Defining equations of modular curves＂裡提到我
們如何利用廣義的 Dedekind eta 函數建構在模曲線X0(N)上模函數域的生成 元，提供了將有理橢圓曲線模參數化的方法。在這裡，我們利用相同的概念以及 類似的方法將本身就是模曲線且型式為X0(p)或X0+(p)的有理橢圓曲線模參數 化，其中 p 為質數。方法如下： 一.型式為X0(p)，我們找兩個模函數 X ，Y 在無限大有極點且次數分別為 二跟三。 二.型式為X0+(p)，我們有兩種方法： 1. 利用模曲線X0(p)上模函數域的生成元建構上述的兩個模函數。
2. 利用＂一個全形的( holomorphic )的 differential 1-form 本身 就是一個 cusp form of weight 2＂的性質以及給定的有理橢圓曲 線唯一決定我們要的這兩個模函數。
中華民國九十五年七月 iii
Modular Parameterization of
Rational Elliptic Curves
Fang-Ting Tu
Department of Applied Mathematics
National Chiao Tung University
Hsinchu, Taiwan
July 17, 2006
Modular Parameterization of
Rational Elliptic Curves
Student: Fang-Ting Tu Advisor: Yifan Yang
Department of Applied Mathematics National Chiao Tung University
July 2006 Abstract
The well-known Taniyama-Shimura conjecture states that for every rational elliptic curve
E : y2+ a1xy + a3y = x3+ a2x2+ a4x + a6,
there is a natural integer N and a rational map such that φ : X0(N ) → E.
To complete the proof of Fermat’s Last Theorem, this conjecture plays an important role. Although it was proved by Andrew Wiles, it is difficult to find a rational map φ for a given rational elliptic curve actually. In this thesis, we will find modular functions that parameterize elliptic curves that are actually modular curves of type X0(p) or X0+(p),
where p are prime numbers.
Our basis ideas are referred to Y. Yang’s Defining equations of modular curves. We use the Dedekind eta function and generalized Dedekind eta functions to construct generators of modular function fields on X0(p). Then we use distinct ways to parameterize the given
elliptic curves, the methods are as follows.
1. For X0(p), find X with pole at ∞ of order 2, and find Y with pole at ∞ of order 3.
2. For X0+(p),
(a) using the generators of the function field on X0(p) to construct the functions
with pole at ∞ of order 2 and 3.
(b) using the fact that the holomorphic 1-forms on a modular curves are actually cusp forms of weight 2.
Contents
1 Introduction 2
2 Modular curves and Modular forms 4
2.1 Congruence subgroups of P SL2(R) . . . 4
2.2 Modular curves . . . 5
2.3 Modular forms and Modular functions . . . 8
2.4 Petersson Inner Product and Hecke Operators . . . 11
3 Elliptic Curves 17 3.1 Definitions . . . 17
3.2 Taniyama-Shimura Conjecture . . . 20
4 Modular parameterizations 21 4.1 Genera of X0(p) and X0+(p) . . . 21
4.2 Methods for finding modular parameterizations . . . 22
Notations
1. We use the notations R, C, Q, Z, and N to stand for the real number field, the complex number field, the rational number field, the ring of integers, and the set of positive integers, respectively. And P1(C) = C ∪ {∞}, the Riemann sphere;
P1(Q) = Q∪{∞}; H = {τ ∈ C : Im τ > 0}, the upper half plane; H∗ = H∪Q∪{∞}. 2. For any commutative ring R with unity 1, the vector space Mn(R) is defined
to be the set of square matrices with degree n over R; the general linear group GLn(R) = {γ ∈ Mn(R) : det γ ∈ R×} , where R× = {a ∈ R : a is an unit} is the
group of invertible elements in R. The special linear group SLn(R) is defined
to be the subgroup of GLn(R) consisting of matrices of determine 1. In the
se-quel, we mostly consider the rings R = R, Z, or ZN, for N ∈ N. And we denote
Chapter 1
Introduction
In a famous story about Fermat it is said that Fermat once wrote in his copy of Diophantus’ Arithmetica that he had a truly marvelous proof of the fact that the equation
xn+ yn = zn
has no solutions in positive integers when n ≥ 3, but the proof was too long to be contained in the margin of the book. However, during Fermat’s lifetime, he never published the claimed proof. Afterwards, many great mathematicians tried to produce a legitimate proof, but to no avail, and this problem became widely known as Fermat’s Last Theorem. It was not until 1995 that an English mathematician Andrew Wiles, assisted by his student Richard Taylor, finally put an end to this 350 year old puzzle.
It turns out that what Wiles did was not a direct attack on Fermat’s Last Theorem, but an effort to prove the Taniyama-Shimura conjecture. In 1957, Yutaka Taniyama, a young mathematician in Japan, based on some numerical examples, made a daring conjecture that there is a one-to-one correspondence between rational elliptic curves and cusp forms. This conjecture was later made more rigorous by Goro Shimura. To be more precise, the conjecture states that the L-function of a rational elliptic function is equal to the L-function of a cusp form of weight 2 on a certain modular curve X0(N ). Putting it
an alternative way, this means that every rational elliptic curve can be parameterized by modular functions on X0(N ) for some N . In 1986 Ken Ribet proved that any non-trivial
solution to xn+ yn = zn, n ≥ 3, will give rise to an elliptic curve that is too weird to
be modular. Wiles then proceeded to prove the Taniyama-Shimura conjecture for the semistable cases, and hence establsihed Fermat’s Last Theorem. However, we should remark that in general it is difficult to find modular functions that parameterize a given rational elliptic curve. In this thesis we will address this problem. In particular, we will find modular functions that parameterize elliptic curves that are modular curves of type X0(p) or X0+(p) themselves, where p are prime numbers.
Our method is basically an extension of Yang’s method [16]. In [15] Y. Yang obtained transformation formulas for generalized Dedekind eta functions (see Section 2.3.3), from which he deduced criteria for a product of generalized Dedekind eta functions to be modular on X(N ) or X1(N ). Using these simple criteria, he then devised a systematic
method of finding generators of modular function fields on modular curves of type X(N ), X1(N ), and X0(N ). Since finding modular parameterizations of elliptic modular curves
X0+(p) is equivalent to finding generators of function fields on Γ+0(p), this thesis can naturally be considered as a continuation of Yang’s work [16]. (Here we should remark that our method works also in the cases where the levels are not primes. The main reason
why we consider only the prime cases here is that this thesis has to be submitted by a certain deadline.)
The rest of thesis is organized as follows. In Chapters 2 and 3 we will briefly review the definition and basic properties of modular curves and elliptic curves. Finally, in the last chapter of the thesis we will describe our method in more details and give the results of our computation.
Chapter 2
Modular curves and Modular forms
Almost all of this chapter are from the Lecture Notes on Modular Forms and Modular Functions by Y. Yang [17]. Some sources are given by T. Miyake’s Modular Forms [10] and T.M. Apostol’s Modular Functions and Dirichlet Series in Number Theory [1].
2.1
Congruence subgroups of P SL
2(R)
In general, we call P SL2(Z) the modular group. There are many subgroups of finite index
of P SL2(Z). Among them, we are interested in congruence subgroup.
Definition 2.1.1 Let Γ be a discrete subgroup of P SL2(R) commensurable with P SL2(Z).
If Γ contains the subgroup Γ(N ) = γ ∈ P SL2(Z) : γ ≡ ± 1 0 0 1 mod N
for some positive integer N , then Γ is a congruence subgroup. The smallest such positive integer N is the level of Γ. The group Γ(N ) is called the principal congruence subgroup of level N .
The following two types of congruence subgroups Γ0(N ) = a b c d ∈ P SL2(Z) : c ≡ 0 mod N , Γ1(N ) = a b c d ∈ P SL2(Z) : c ≡ 0, a ≡ d ≡ ±1 mod N
are most often encountered in number theory. The congruence subgroups Γ0(N ) are also
called the Hecke congruence subgroups and are conjugate to Γ0(N ) =a b c d ∈ P SL2(Z) : b ≡ 0 mod N in P SL2(R). Proposition 2.1.2 We have
(1) Γ1(N ) is a normal subgroup of Γ0(N ) and Γ0(N )/Γ1(N ) ' Z×N/ ± 1, where Z×N is
the multiplicative group of residue classes modulo N that are relatively prime to N . (2) Γ(N ) is a normal subgroup of P SL2(Z) and P SL2(Z)/Γ(N ) ' SL2(ZN)/ ± 1.
2.1.1
The index of a congruence subgroup in P SL
2(Z)
From the last section, we can determine the indices of the congruence groups. Proposition 2.1.3 We deduce that
(1) [Γ1(N ) : Γ(N )] = N , (2) Γ0(2) = Γ1(2) and [Γ0(N ) : Γ1(N )] = N 2 Y p|N 1 − 1 p , for N ≥ 3. (3) [P SL2(Z) : Γ0(N )] = N Y p|N (1 + 1/p),
where the products run over all prime divisors p of N .
2.1.2
Atkin-Lehner involutions
Let N ≥ 2 be any positive integer.
Definition 2.1.4 Let n be a divisor of N with gcd(n, N/n) = 1. The elements in wn= 1 √ n an b cN dn , adn2 − bcN = n
are the Atkin-Lehner involutions on Γ0(N ) \ H. The set of Γ0(N ) union all possible
Atkin-Lehner involutions is denoted by Γ+0(N ).
Proposition 2.1.5 The Atkin-Lehner involutions normalize Γ0(N ). Furthermore, we
have Γ+0(N )/Γ0(N ) ' Zk2, where k is the number of distinct prime divisors of N .
2.2
Modular curves
2.2.1
Group action of P SL
2(Z) on H
Recall that the linear fractional transformation of H is given by γ : τ 7→ aτ + b
cτ + d, a, b, c, d ∈ R, ad − bc > 0,
and a linear fractional transformation γ determines the matrix ac db ∈ GL+2(R) up to a scalar multiplication. Hence, dividing by a suitable scalar, we may represent γ by a matrix of determinant 1. The group SL2(R) contains I =
1 0
0 1
and −I = −10 −10 which act trivially, and the group P SL2(R) is identified with the group of linear
frac-tional transformations. It is not difficult to check that the definition of linear fracfrac-tional transformations gives a group action of P SL2(R) on H.
Let Γ be a congruence subgroup of finite index of P SL2(Z). Then H and H∗ =
the quotient space Γ \ H. Note that a group action gives rise to an equivalence relation given by x ∼ y if and only if there is an element γ lying in Γ such that γx = y. The equivalence class containing x is exactly Γx. The classical modular curve X(Γ) is defined by the quotient space Γ \ H∗. For the ease of notation, we abbreviate X(Γ0(N )) to X0(N ),
X(Γ1(N )) to X1(N ), and X(Γ(N )) to X(N ).
Now we consider the fix points of linear fractional transformations. Let γ =ac db∈ P SL2(R) be a representation of a linear fractional transformation. The points fixed by γ
are the roots of cτ2+ (d − a)τ − b = 0. When γ = ±I, every point is fixed by γ, and this
identity motion forms a class by itself. When γ 6= ±I, there are three possibilities, (1) γ has one fixed point on P1(R), and this γ is called parabolic;
(2) γ has two distinct fixed points on P1(R), and this γ is called hyperbolic;
(3) γ has a pair of conjugate complex numbers as fixed points, and this γ is called elliptic.
These classifications can be also described in terms of the trace of γ, Lemma 2.2.1 Let γ ∈ P SL2(R). Then
(1) γ is parabolic if and only if tr(γ) = 2; (2) γ is hyperbolic if and only if tr(γ) > 2; (3) γ is elliptic if and only if tr(γ) < 2.
Definition 2.2.2 A point on P1(R) fixed by a parabolic element is called a cusp, and a
point in H fixed by an elliptic element is called an elliptic point.
Proposition 2.2.3 The set of cusps of P SL2(Z) is P1(Q), and the cusps are all
equiva-lent to each other under P SL2(Z).
Proposition 2.2.4 Now we consider elliptic element and elliptic point of P SL2(Z)
(1) Every elliptic element of P SL2(Z) has order 2 or 3.
(2) An element of P SL2(Z) has order 2 if and only if its trace is 0. An element has
order 3 if and only if its trace has absolute value 1.
(3) Every elliptic element of order 2 is conjugate to01 −10 in P SL2(Z). Every elliptic
element of order 3 is conjugate to either 01 −1−1 or −1−1 10.
(4) P SL2(Z)\H has only two inequivalent elliptic points. One is represented by i, which
is of order 2; the other is represented by eπi/3, which is of order 3. That is ,if τ ∈ H
is of order 2, then τ is equivalent to i; if it is of order 3, it is equivalent to eπi/3.
2.2.2
Cusps and Elliptic points of congruence subgroups
Here we discuss the cusps and elliptic points of congruence subgroups.
Lemma 2.2.5 Let Γ be a congruence subgroup of P SL2(Z), then the set of cusps of Γ is
Definition 2.2.6 Let Γ be a congruence subgroup of P SL2(Z). Let a/c ∈ P1(Q) be a
cusp. The smallest positive integer m such that the matrix a b c d 1 m 0 1 d −b −c a =1 − acm a 2m −c2m 1 + acm (2.1) falls in Γ is called the width of the cusp a/c.
Remark. In fact, a cusp a/c is fixed by the matrix describing in (2.1). Proposition 2.2.7 A set of inequivalent cusps for Γ0(N ) is given by
na
c : c|N, a = 0, . . . , gcd(c, N/c) − 1, gcd(a, c) = 1 o
. Hence the number of inequivalent cusps is
X
c|N
φ(gcd(c, N/c)).
The number of inequivalent cusps for Γ1(N ), N ≥ 3, is
1 2
X
c|N
φ(c)φ(N/c),
where φ is the Euler totient function.
Proposition 2.2.8 The number v2 of inequivalent elliptic points of order 2 for Γ0(N ) is
equal to the number of solutions of x2+ 1 = 0 in Z
N. That is, when 4|N , v2 = 0 and
when 4 - N, v2 = Y p|N, p odd prime 1 + −1 p .
The number v3 of inequivalent elliptic points of order 3 for Γ0(N ) is equal to the number
of solutions of x2+ x + 1 = 0 in ZN. That is, when 9|N , v3 = 0, and
when 9 - N, v3 = Y p|N, p odd prime 1 + −3 p , where a b
is the Legendre symbols.
Proposition 2.2.9 When N ≥ 4, the congruence subgroups Γ1(N ) are torsion-free.
When N ≥ 2, the principal congruence subgroups Γ(N ) are torsion-free.
2.2.3
Genus
Let Γ be a subgroup of P SL2(Z) of index m. We now determine the genus of X(Γ). Let
v2, v3, v∞ be the numbers of Γ-inequivalent elliptic points of order 2, elliptic points of
order 3, and cusps, respectively. Then the genus g(Γ) of X(Γ) is given by the formula g(Γ) = 1 + m 12− v2 4 − v3 3 − v∞ 2 .
2.3
Modular forms and Modular functions
2.3.1
Definitions
Let γ = ac db∈ GL+2(R). For an integer k and a meromorphic function f : H 7→ C we let the notation f (τ ) [γ]k denote the slash operator
f (τ ) [γ]k = (det γ)k/2(cτ + d)−kf
aτ + b cτ + d
.
The factor cτ +d is called the automorphy factor. If the weight k is clear from the context, we often write simply f
γ.
Definition 2.3.1 Let Γ be a subgroup of P SL2(Z) of finite index, and k be an even
integer. A holomorphic function f : H 7→ C is called a modular form of weight k with respect to Γ if
(1) f (τ ) [γ]k= f (τ ) for all τ ∈ H and γ ∈ Γ, that is,
f (τ ) = (cτ + d)−kf aτ + b cτ + d
.
(2) f (τ ) is holomorphic at every cusp.
In addition to (1) and (2), if the function also satisfies (3) f vanishes at every cusp,
then the function f is a cusp form of weight k with respect to Γ. Let a/c ∈ P1(Q) be a cusp and choose σ = a b
c d
∈ P SL2(Z). Then a function
f satisfies condition (1) if and only if the function g(τ ) = f [σ]k is invariant under the
action of σ−1Γσ since f [σ] [σ−1γσ] = f [γ] [σ] = f [σ], for all γ ∈ Γ.
In particular, g(τ ) is invariant under the substitution τ 7→ τ + h, where h is the width of the cusp a/c. Let P
n∈Zane2πinτ /h be the Fourier expansion of g(τ ). Then we say f is
holomorphic at a/c provided that an= 0 for all n < 0, or equivalently, that g is bounded
in a neighborhood of a/c. Moreover, condition (3) means that an = 0 for all n ≤ 0 for
each cusp a/c.
Definition 2.3.2 A meromorphic modular form of weight 0 is called a modular function. That is a modular function on Γ is a meromorphic function f : H → P1(C) such that
f (γτ ) = f (τ ) for all γ ∈ Γ.
2.3.2
Dedekind eta function η(τ )
The Dedekind eta function plays a central role in number theory. It was introduced by Dedekind in 1877 and provides another convenient way of constructing modular forms and modular functions.
Definition 2.3.3 Let τ ∈ H, and write q = e2πiτ. The Dedekind eta function η(τ ) is
defined by η(τ ) = q1/24 ∞ Y n=1 (1 − qn) = eπiτ /12 ∞ Y n=1 (1 − e2πinτ). Proposition 2.3.4 For γ =a b c d ∈ SL2(Z),
the transformation formula for η(τ ) is given by, for c = 0, η(τ + b) = eπib/12η(τ ), and, for c > 0, η(γτ ) = (a, b, c, d) r cτ + d i η(τ ) with ε(a, b, c, d) = d c
i(1−c)/2eπi(bd(1−c2)+c(a+d))/12, if c is odd, c
d
eπi(ac(1−d2)+d(b−c+3))/12, if d is odd,
(2.2)
where d c
is the Legendre-Jacobi symbol.
Proposition 2.3.5 Let N be a positive integer. If f (τ ) =Y h|N η(hτ )eh satisfies (1) e =X h|N eh ≡ 0 mod 4, (2) Y h|N
heh is a square of a rational number,
(3) X h|N ehh ≡ 0 mod 24, (4) X h|N ehN/h ≡ 0 mod 24,
2.3.3
Generalized eta functions
In this section, what we discuss are refered to the paper [15] Transformation formulas for generalized Dedekind eta functions by Y. Yang.
Definition 2.3.6 Let N be a positive integer. For τ ∈ C with Im τ > 0, we set q = e2πiτ.
Let g and h be arbitrary real numbers not congruent to 0 modulo N simultaneously. We define the generalized Dedekind eta functions Eg,h(τ ) by
Eg,h(τ ) = qB(g/N )/2 ∞ Y m=1 1 − ζhqm−1+g/N 1 − ζ−hqm−g/N .
Let g be an arbitrary real number not congruent to 0 modulo N . We define the generalized Dedekind eta function Eg(τ ) by
Eg(τ ) = qN B(g/N )/2 ∞
Y
m=1
1 − q(m−1)N +g 1 − qmN −g .
Where ζ = e2πi/N and B(x) = x2 − x + 1/6. Proposition 2.3.7 The functions Eg,h satisfy
Eg+N,h= E−g,−h = −ζ−hEg,h, Eg,h+N = Eg,h. (2.3)
Moreover, let γ =ac db∈ P SL2(Z). Then we have, for c = 0,
Eg,h(τ + b) = eπibB(g/N )Eg,bg+h(τ ),
and, for c > 0,
Eg,h(γτ ) = ε(a, b, c, d)eπiδEg0,h0(τ ),
where
ε(a, b, c, d) = (
eπi(bd(1−c2)+c(a+d−3))/6, if c is odd, −ieπi(ac(1−d2)+d(b−c+3))/6, if d is odd, δ = g 2ab + 2ghbc + h2cd N2 − gb + h(d − 1) N , and (g0 h0) = (g h)a b c d . Using the fact that
N γτ = a(N τ ) + bN c(N τ ) + d = a bN c d (N τ )
and the special class of generalized Dedekind eta functions Eg(τ ) = Eg,0(N τ ), we have
Proposition 2.3.8 The functions Eg satisfy
Moreover, let γ =cNa db∈ Γ0(N ). Then we have, for c = 0,
Eg(τ + b) = eπibN B(g/N )Eg(τ ),
and, for c 6= 0,
Eg(γτ ) = ε(a, bN, c, d)eπi(g
2ab/N −gb)
Eag(τ ),
where
ε(a, b, c, d) = (
eπi(bd(1−c2)+c(a+d−3))/6, if c is odd, −ieπi(ac(1−d2)+d(b−c+3))/6, if d is odd. Proposition 2.3.9 Consider the function f (τ ) = Q
gEg(τ )
eg, where g and e
g are
inte-gers. Suppose that one has X g eg ≡ 0 mod 12, X g geg ≡ 0 mod 2. (2.4)
Then f is invariant under the action of Γ(N ). Moreover, if, in addition to (2.4), one also has
X
g
g2eg ≡ 0 mod 2N. (2.5)
Then f is a modular function on Γ1(N ).
Furthermore, for the cases where N is a positive odd integer, the conditions (2.4) and (2.5) can be reduced to X g eg ≡ 0 mod 12 (2.6) and X g g2eg ≡ 0 mod N, (2.7) respectively.
Proposition 2.3.10 The order of the function Eg at a cusp a/c with gcd(a, c) = 1 is
1
2gcd(c, N )P2(ag/gcd(c, N )),
where P2(x) = {x}2− {x} + 16 and {x} denotes the fractional part of a real number x.
Proposition 2.3.11 Observe that the action of the Atkin-Lehner involution ωN sends
the cusps to the cusps that are equivalent to ∞ under Γ0(N ). Then we have
E0,g
−1 N τ
= −ieπig/NEg,0(N τ ) = −ieπig/NEg(τ ).
2.4
Petersson Inner Product and Hecke Operators
2.4.1
Petersson inner product
Let Γ be a subgroup of P SL2(Z) of finite index. The vector space
Sk(Γ) = {f : f is a cusp form of wight k on Γ}
Definition 2.4.1 Let D be a fundamental domain for Γ. Then the Petersson inner product of f, g ∈ Sk(Γ) is defined as hf, giΓ = 1 [P SL2(Z) : Γ] Z Z D ykf (τ )g(τ )dxdy y2 ,
where for τ ∈ H we write τ = x + iy.
The Petersson inner product is defined only on Sk(Γ) because if f and g are not cusp
forms then the integral may not be finite. However, the integral will converge whenever at least one of f and g is a cusp form. We remark that
(1) The hyperbolic measure dxdy/y2 is invariant under the substitution τ 7→ γτ for any
γ ∈ GL+2(R).
(2) The factor 1/[P SL2(Z) : Γ] is inserted so that the inner product hf, gi will remain
the same if we consider f and g as modular forms on a smaller subgroup Γ0.
(3) The Petersson inner product is independent of the choice of the fundamental domain D.
2.4.2
Hecke Operators on modular forms on Γ
0(N )
To define the Hecke operators Tn on Γ0(N ) it is necessary to distinguish the cases
gcd(n, N ) = 1 from the cases gcd(n, N ) 6= 1. When n is a positive integer relatively prime to N , we consider the sets
M(N ) n = a b c d ∈ GL2(Z) : ad − bc = n / ± 1. When n = p is a prime divisor of N , we consider the set
M(N ) p = a b c d ∈ GL2(Z) : ad − bc = p, N |c, p|d, gcd(a, N) = 1 / ± 1
instead. Then the modular group Γ0(N ) acts on M (N )
n by multiplication from left. We
denote the set of orbits (equivalence classes) by Sn(N ).
Proposition 2.4.2 Let σ ∈ Γ0(N ). Then φσ : S (N )
n → Sn(N ) given by φσ([γ]) = [γσ] is a
well-defined permutation on the elements of Sn(N ).
Proposition 2.4.3 Let n be a positive integer relatively prime to N . A complete set of representatives of orbits in Sn(N ) is a b 0 d : ad = n, a > 0, b = 0, . . . , d − 1 .
When n = p is a prime divisor of N , a set of representatives of Sp(N ) is
1 b 0 p : b = 0, . . . , p − 1 .
Definition 2.4.4 Let k be an even integer and n a positive integer. Write n in the form mpe1
1 . . . perr such that gcd(m, N ) = 1 and pi|N are prime divisors of N . Then the nth
Hecke operator Tn on the space of modular forms of weight k is defined by
Tn= TmTpe11. . . T er pr, where Thf = hk/2−1 X [γ]∈ Sh(N ) f [γ]k.
Proposition 2.4.5 The Hecke operators Tn are linear transformations on the space of
modular forms on Γ0(N ). Moreover, if f is a cusp form, then so is Tnf .
Proposition 2.4.6 For all positive integers m and n with gcd(m, n) = 1 we have Tmn= TmTn.
Furthermore, suppose that m and n are positive integers such that both of them are rela-tively prime to N . Then we have
TmTn =
X
d| gcd(m,n)
dk−1Tmn/d2 = TnTm.
In the following contents we will determine the adjoints Tn∗ of the Hecke operators Tn
with respect to the Petersson inner product. For convenience, throughout this section the notation h·, ·i without any subscript denotes the inner product on the space of modular forms on P SL2(Z), while h·, ·iΓ carries the usual meaning.
Proposition 2.4.7 Let n be a positive integer relatively prime to N . The Hecke operators Tn are self-adjoint (also called hermitian) with respect to the Petersson inner product.
That is, we have
hTnf, gi = hf, Tngi.
for all modular forms f and g of weight k on Γ0(N ).
Since Tn are self-adjoint, an elementary result in linear algebra asserts that the
eigen-values of Tn are all real.
Proposition 2.4.8 For all positive integers n with gcd(n, N ) = 1, the eigenvalues of Tn
are all real.
Now we have a family of self-adjoint linear operators Tnthat are commuting with each
other on an inner product space. By a well-known theorem in linear algebra, These linear operators Tn are simultaneously diagonalizable. In other words, the vector space has a
basis consisting entirely of simultaneous eigenvectors.
Proposition 2.4.9 There is a decomposition of the vector space Sk(Γ0(N )) into a direct
sum
Sk(Γ0(N )) = ⊕ Vi
of orthogonal subspaces Vi such that each Vi is a simultaneous eigenspace for all Tn with
Moreover, if f is an eigenform in Vi, then so is Tpf for p|N . Therefore, each Tp, p|N ,
stabilizes Vi. However, in general, Vi may not have a basis consisting of simultaneous
eigenforms for all Tn. There does not exist a basis whose elements are all simultaneous
eigenforms for all Tn. Nevertheless, if f =
P∞
n=1cnqn is a simultaneous eigenforms for all
Tn, then f still enjoys the property that Tnf = cnf .
In general, let f be a non-vanishing modular form of weight k on P SL2(Z).
Definition 2.4.10 If f is a simultaneous eigenform for all Hecke operators Tnon P SL2(Z),
then we say f is a simultaneous eigenform or a Hecke eigenform. If the Fourier expansion of f has leading coefficient 1, then f is normalized.
Proposition 2.4.11 The space Sk(P SL2(Z)) of cusp forms of weight k on P SL2(Z) is
spanned by simultaneous eigenforms.
2.4.3
Properties of Hecke eigenforms
Throughout this section we let k be a positive even integer and d be the dimension of Sk(P SL2(Z)). We assume that {f1, . . . , fd} is a basis of Sk(P SL2(Z)) consisting of Hecke
eigenforms. In this section we will study properties of fi.
Proposition 2.4.12 Let f be a Hecke eigenform, and assume that the Fourier expansion of f is c1q + c2q2+ · · · . Then c1 6= 0.
Recall that we say a Hecke eigenform is normalized if the leading Fourier coefficient is 1, the Fourier coefficients of a normalized Hecke eigenform are multiplicative.
Proposition 2.4.13 Let f be a normalized Hecke eigenform with a Fourier expansion q + c2q2+ c3q3+ · · · . Then Tnf = cnf for all positive integers n.
Proposition 2.4.14 If f is a normalized Hecke eigenform with a Fourier expansion q + c2q2+ c3q3· · · , then we have
cmcn=
X
d| gcd(m,n)
dk−1cmn/d2
for all positive integers m and n. In particular, if gcd(m, n) = 1, then cmn = cmcn.
Proposition 2.4.15 Let f 6= g be two normalized Hecke eigenforms. Then f and g are orthogonal with respect to the Petersson inner product.
2.4.4
Newforms and oldforms
Some of the cusp forms in Sk(Γ0(N )) actually have level smaller than N . Namely, if M
is an integer dividing N , then any cusp forms on Γ0(M ) is also a cusp form on Γ0(N ).
Proposition 2.4.16 Let M be a positive integer dividing N , and f (τ ) be a cusp form on Γ0(M ). Then for any h|(N/M ), the function f (hτ ) is a cusp form on Γ0(N ). Moreover,
Definition 2.4.17 If f (τ ) ∈ Sk(Γ0(N )) satisfies f (τ ) = g(hτ ) for some simultaneous
eigen-form g(τ ) ∈ Sk(Γ0(M )) with M |N , M < N , and h|(N/M ), then f (τ ) is called an oldform.
The subspace spanned by all oldforms is called the space of oldforms and is denoted by Soldk (Γ0(N )). The orthogonal complement is called the space of newforms and is denoted by
Snewk (Γ0(N )).
Proposition 2.4.18 Each of Soldk (Γ0(N )) and Snewk (Γ0(N )) is stable under Tn for all n
with gcd(n, N ) = 1.
Proposition 2.4.19 The subspace Snewk (Γ0(N )) has a basis consisting of simultaneous
eigenforms for all Tn with gcd(n, N ) = 1.
Definition 2.4.20 Let f ∈ Snewk (Γ0(N )). If f is a non-vanishing simultaneous eigenform
for all Tn with gcd(n, N ) = 1, then f is a newform.
In fact, more about Snewk (Γ0(N )) is true. To facilitate further discussion. Let us recall
a result of Atkin and Lehner.
Proposition 2.4.21 (Atkin-Lehner) Let f (τ ) =P∞
n=1cnqn be a cusp form on Γ0(N ).
If there exists a positive integer M such that whenever gcd(n, M ) = 1 the coefficient an
vanishes, then f lies in the space of oldforms. Proposition 2.4.22 Let f (τ ) =P∞
n=1cnqn be a newform on Γ0(N ). Then c1 6= 0.
Proposition 2.4.23 Let Snewk (Γ0(N )) = ⊕ Vi be the decomposition of Snewk (Γ0(N )) into
a direct sum of simultaneous eigenspaces for all Tn with gcd(n, N ) = 1. Then each Vi has
dimension 1.
Before we state our last results in this chapter, let us recall that the Atkin-Lehner involutions wn= 1 √ n an b cN dn , n|N, gcd(n, N/n) = 1, adn2− bcN = n , normalize Γ0(N ). Thus if f is a modular form of weight k on Γ0(N ), then so is f
[wn]k.
For the ease of notation, given a modular form on Γ0(N ), we write
Wnf = f
[wn]. We remark that since w2
n ∈ Γ0(N ), the coset Γ0(N )wn is all the same for a fixed n,
regardless of which wnwe choose. Therefore, the definition of Wn does not depend on the
representative wn.
Proposition 2.4.24 If m is a positive integer such that gcd(m, N ) = 1, then TmWn =
WnTm.
Proposition 2.4.25 The space of newforms is spanned by cusp forms that are simulta-neous eigenforms for all Atkin-Lehner involutions and all Hecke operators.
Remark. Since w2n ∈ Γ0(N ), we have Wn2f = f . Thus, the eigenvalues of Wn must be
1 or −1.
Finally we study the eigenvalue of Tp, p|N , associated with a newform.
Proposition 2.4.26 Let f =P∞
n=1cnqn be a normalized newform on Γ0(N ). Let p be a
prime divisor of N . If p2|N , then c
p = 0. If p2 does not divide N , then cp = −pk/2−1,
Chapter 3
Elliptic Curves
An elliptic curve is a pair (E, O), where E is a smooth projective curve of genus 1 and O is a basepoint of E.
Let K be any perfect field. The elliptic curve (E, O) is said to be defined over K if the curve is defined over K, which is also denoted by E/K, and O is a rational point on E defined over K. Every such curve can be written as the locus in P2 of a cubic equation
with only one point on the line at infinity. That is, after scaling X and Y , as an equation of the form
Y2Z + a1XY Z + a3Y Z2 = X3+ a2X2Z + a4XZ2+ a6Z3.
Here O = [0, 1, 0] and ai ∈ K. And there exists a morphism
+ : E × E → E defined by (P1, P2) 7→ P1+ P2
giving the group law such that the set of rational points on E with identity O (point at infinity) forms an abelian group.
3.1
Definitions
In this chapter, we consider the special case K = Q. A cubic curve in normal form looks like
y2 = f (x) = x3+ ax2+ bx + c.
Assuming that the roots of f (x) are distinct, that is, this curve is nonsingular. Then such a curve is an elliptic curve, and every point on an elliptic curve has a well-defined tangent line. The reason is as follows. If we write the equation as F (x, y) = y2− f (x) = 0 and
take the partial derivatives, ∂F
∂x = −f
0
(x), and ∂F ∂y = 2y.
We can see that there is no point on the curve at which the partial derivatives vanish simultaneously, since the curve is nonsingular. The details can be found in Rational Points on Elliptic Curves by J.H. Silverman [13].
If this curve is singular, we classify the singular points depending on its tangent di-rections.
Definition 3.1.1 Let P be a singular point on the curve F (x, y) = 0. We say that P is a node if there are two distinct tangent directions at P . P is a cusp if there is one tangent direction at P .
3.1.1
Minimal Weierstrass Equation
We may assume that each elliptic curve have a Weierstrass equation of the form E : y2+ a1xy + a3y = x3+ a2x2+ a4x + a6.
And this equation has an associated discriminant
∆ = − (a21+ 4a2)2(a21a6+ 4a2a6− a1a3a4+ a2a23 − a 2
4) − 8(2a4+ a1a3)3
− 27(a2
3+ 4a6)2+ 9(a21+ 4a2)(2a4 + a1a3)(a23+ 4a6).
If we replace the variables (x, y) by (x/u2, y/u3), then each a
i in the Weierstrass
equation E : y2 + a1xy + a3y = x3+ a2x2+ a4x + a6 becomes uiai and the discriminant
is u12∆. In this way, we can choose u such that uia
i are all integers.
Let E0 be a new equation
E0 : y2+ b1xy + b3y = x3+ b2x2+ b4x + b6
of this elliptic curve E form changing variables such that bi are all integers. Let ∆0 be the
discriminant of E0. For each prime p, define vp(∆0) as the power of p such that pvp(∆
0)
| ∆0
but pk
- ∆0 if k > vp(∆0).
Definition 3.1.2 A Weierstrass equation is called a minimal Weierstrass equation E0 for E at p if vp(∆0) is minimized subject to the condition bi ∈ Z.
If we define
vp(∆) = min E0 vp(∆
0
), then the minimal discriminant of E is defined by
D =Y
p
pvp(∆).
Definition 3.1.3 A Weierstrass equation is called a global minimal Weierstrass equation for E if E is simultaneously minimal at all primes of Q. The discriminant ∆ of this global minimal Weierstrass equation is equal to the minimal discriminant D of E/Q.
Proposition 3.1.4 ([12], Corollary 8.3) Every elliptic curve E/Q has a global mini-mal Weierstrass equation.
We remark that we can find a global minimal Weierstrass equation for E/Q by finding local minimal equations(e.g. by using Tate’s algorithm).
3.1.2
Reduction
The reduction of E modulo p, denoted eE, is then the curve over Zp defined by the equation
e
E : y2+ae1xy +ae3y = x
3+
e
a2x2+ae4x +ae6,
whereaei denotes reduction modulo p. The curve eE may be singular; its non-singular part
is denoted eEns.
(1) E has good (stable) reduction if eE is non-singular.
(2) E has multiplicative (semi-stable) reduction if eE has a node. And the reduction is called split if the tangent directions are defined over Zp, otherwise it is non-split.
(3) E has additive (unstable) reduction if eE has a cusp. In cases (2) and (3), E is naturally said to have bad reduction.
3.1.3
Conductor
The minimal discriminant is a measure of the bad reduction of E. Another such measure is the conductor of E/Q.
Definition 3.1.6 The conductor of E/Q is defined by N (E/Q) =Y
p
pfp(E/Q),
where the exponents fp(E/Q) are given by
fp(E/Q) =
0, if E has good reduction at p,
1, if E has multiplicicative reduction at p, 2 + δp, if E has additive reduction at p,
where δp = 0 if p - 6.
Further, fp may be computed by using Ogg’s formula[11].
3.1.4
L-Series
The L-series of an elliptic curve is a generating function which records information about the reduction of the curve modulo every prime.
Let E be an elliptic curve. Set q = pk, for some prime p.
Definition 3.1.7 Let E(Fqr) be the set of points on E with coordinates in Fqr. The zeta
function of E over Fqr is given by the formal power series
Z(E/Fq; T ) = exp ∞ X r=1 (#E(Fqr)) Tr r ! , where exp(u) = ∞ X k=0 uk k!. Proposition 3.1.8 [12] There is an integer a so that
Z(E/Fq; T ) = 1 − aT + qT2 (1 − T )(1 − qT ) = (1 − αT )(1 − βT ) (1 − T )(1 − qT ) with |α| = |β| = √ q. Further more Z(E/Fq; T ) = Z(E/Fq; 1/(qT )).
For each prime p, if E has good reduction at p, let ap = p + 1 − # eE(Fp). The local
factor of the L-series of E at p is Lp(T ) = 1 − apT + pT2. We extend the definition of
Lp(T ) to the case that E has bad reduction by setting
Lp(T ) =
1 − T, if E has split multiplicicative reduction at p, 1 + T, if E has non-split multiplicicative reduction at p, 1, if E has additive reduction at p.
In all cases, the relation Lp(1/p) = # eE(Fp)/p holds.
Definition 3.1.9 We make substitution T = p−s in Z(E/Fp; T ), and define Hasse-Weil
L-series L(E, s) by
L(E, s) = ζ(s)ζ(s − 1) ΠpZ(E/Fp; p−s)
, where ζ(s) is the Riemann zeta function defined by
ζ(s) =X
n∈N
1
ns, for Re(s) > 1
and we can express ζ(s) as
ζ(s) = Y primes p 1 1 − p−s. Thus, we have L(E/Q, s) =Y p 1 1 − app−s+ p1−2s =Y p Lp(p−s)−1.
3.2
Taniyama-Shimura Conjecture
The conjecture says that every rational elliptic curve y2 = f (x) = x3 + ax2+ bx + c is a modular form in disguise.
Proposition 3.2.1 Taniyama-Shimura Conjecture
Let E/Q be an elliptic curve of conductor N , let L(E, s) = P cnn−s be its Hasse-Weil
L-series,and let f (τ ) =P cne2πinτ be the inverse Mellin transform of L(E, s).
1. f (τ ) is a cusp form of weight 2 on Γ0(N ), for some positive integer N .
2. For each prime p - N , let T (p) be the corresponding Hecke operator; and let W be the operator (W f )(τ ) = f (−1/N τ ). Then
T (p)f = cpf and W f = wf,
where w = ±1 is the sign of the functional equation
ξE(s) = wξE(2 − s), where ξE(s) = (2π)−sNs/2Γ(s)L(E, s), for s ∈ C,
with the well-know Gamma-function Γ(s).
3. Let ω be an invariant differential on E/Q. There exists a morphism φ : X0(N ) → E,
defined over Q, such that φ∗(ω) is a multiple of the differential form on X0(N )
Chapter 4
Modular parameterizations
In this chapter we will obtain modular parameterization of elliptic curves that are modular curves of the form X0(p) and X0+(p) = X(Γ
+
0(p)) = X0(p)/ωp, where the levels p are
primes. In Section 4.1 we will determine all such curves, and then describe methods to obtain parameterizations in Section 4.2. Afterward, we work out several examples to illustrate the procedures. Finally, we will tabulate the results in Section 4.3.
4.1
Genera of X
0(p) and X
0+(p)
First we observe that the genus of an elliptic curve is 1, so we need to find the modular curves X0(p) and X0+(p) of genus 1.
Since X0(p) is a Riemann surface, from the Riemann Hurwitz’s formula, we have
g(Γ0(p)) = 1 + m 12− v2 4 − v3 3 − v∞ 2 ,
which is given in section 2.2.3. Using Propositions 2.1.3, 2.2.7, and 2.2.8,
g(Γ0(p)) = bp/12c − 1, if p ≡ 1 mod 12, bp/12c, if p ≡ 5, 7 mod 12, bp/12c + 1, if p ≡ 11 mod 12,
the formula becomes simplifier. Hence we can deduce that only when p = 11, 17 and 19, the modular curves X0(p) are of genus 1.
P. Zograf [18] proved that
g(Γ+0(p)) + 1 > 3χ(Γ+0(p))/64, where
χ(Γ) = Vol(Γ\H) 6Vol(P SL2(Z)\H)
.
For Γ+0(p), the value of χ(Γ+0(p)) is (p + 1)/12. Thus, we only need to determine the genera of X0+(p) for p ≤ 511. To find the genus of X0+(p), we use the Fricke’s formula that the number of fixed points of the involution ωp on Γ0(p) is
ν(p) = (
h(−4p), if p ≡ 1 mod 4, h(−4p) + h(−p), if p ≡ 3 mod 4,
where h(d) is the number of
ax2+ bxy + cy2 : b2− 4ac = d /SL 2(Z).
(The general formula for Γ+0(N ) can be seen in [9].) Now the class numbers h(−p) and h(−4p) can be easily computed using Kronecker’s class number relations
X n2<4N H(4N − n2) =X d|N max(d, N/d) + ( 1/6, if N is a square, 0, else,
with the initial values H(1) = H(2) = 0, H(3) = 1/3. Here H(d) denotes the Hurwitz class number, and is essentially h(−d). In fact, when d 6= n2, 3n2, we have H(d) = h(−d).
Finally the Riemann-Hurwitz formula yields
g(Γ0(p)) = 2(g(Γ+0(p)) − 1) + 1 + ν(p)/2.
From this we deduce that only when p = 37, 43, 53, 61, 79, 83, 89, 101, and 131, the genus of g(Γ+0(p)) is 1.
4.2
Methods for finding modular parameterizations
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 W
condition (2.6) in Proposition 2.3.9 automatically. Thus, if ek are integers such that
P
kk 2e
k = 0 mod 17, then QkWkek 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 W8 −12 28 14 −10 −5 −11 3 −7 W7 28 14 −10 −5 −11 3 −7 −12 W4 14 −10 −5 −11 3 −7 −12 28 W5 −10 −5 −11 3 −7 −12 28 14 W2 −5 −11 3 −7 −12 28 14 −10 W6 −11 3 −7 −12 28 14 −10 −5 W1 3 −7 −12 28 14 −10 −5 −11
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, −12x1 +28x2 +14x3 −10x4 −5x5 −11x6 +3x7 −7x8 ≥ −17, 28x1 +14x2 −10x3 −5x4 −11x5 +3x6 −7x7 −12x8 ≥ −17, 14x1 10x2 −5x3 −11x4 +3x5 −7x6 −12x7 +28x8 ≥ −17, −10x1 −5x2 −11x3 +3x4 −7x5 −12x6 +28x7 +14x8 ≥ −17, −5x1 −11x2 +3x3 −7x4 −12x5 +28x6 +14x7 −10x8 ≥ −17, −11x1 +3x2 −7x3 −12x4 +28x5 +14x6 −10x7 −5x8 ≥ −17, 3x1 −7x2 −12x3 +28x4 +14x5 −10x6 −5x7 −11x8 = −34.
We find one of the solutions is (x1, x2, x3, x4, x5, x6, x7, x8) = (0, 1, 0, 0, 0, 0, 1, 2). That is,
F = W8W6W12 = q −2
+ q−1+ 2 + 2q + q2+ 2q3+ q4+ q5+ q6− q7 + · · · ,
where q = e2πiτ. Thus, if we choose
X = X γ∈ Γ0(17)/Γ1(17) F |γ = X γ∈ Γ0(17)/Γ1(17) E10E42 E2E16E12 γ = − E13E12 2 E6E14E2 + E12E15 2 E16E9E11 + E2E11 2 E16E14E10 − E6E16 2 E8E13E14 − E16E3 2 E9E10E12 − E3E8 2 E4E15E7 + E9E7 2 E12E11E4 + E10E4 2 E2E16E12 =q−2+ q−1+ 3 + q + 2q2+ 2q3+ 3q4− 4q5+ 4q6− 2q7− q8+ · · · ,
then X is modular on Γ0(17) with those properties which we want, where γ runs over a
Y to be Y = X γ∈ Γ0(17)/Γ1(17) E10E63E8 E4E143 E12 γ =q−3− 2q−2− 2q−1+ 8 − 2q2− 7q3 − 2q4+ 15q5− 6q6+ · · · .
Then the functions satisfy
Y2+ 7XY − 31Y = X3− 19X2+ 123X − 264. Now let x =X − 2 = X γ∈ Γ0(17)/Γ1(17) E10E42 E2E16E12 γ − 2, y =Y + 3x − 9 = X γ∈ Γ0(17)/Γ1(17) E10E63E8 E4E143 E12 γ + 3x − 9.
We have an elliptic curve y2+ xy + y = x3− x2− x − 14, which is a minimal Weierstrass
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.
Let Γ be the intermediate subgroup between Γ1(61) and Γ0(61) with [Γ0(61) : Γ] = 6.
Then Γ is generated by Γ1(61) and
3 2 61 41 , and Wk= E6kE18kE54kE40kE2k/E3kE9kE27kE20kEk
is a modular function on Γ for any positive integer k. There are six distinct Wk, and they
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 W4 4 −7 13 0 −6 −4 W8 −7 13 0 −6 −4 4 W5 13 0 −6 −4 4 −7 W10 0 −6 −4 4 −7 13 W1 −6 −4 4 −7 13 0 It follows that X γ∈ Γ0(61)/Γ E30E32E26E44E10 E46E16E48E22E56 γ
is a modular function on Γ0(61) and has a unique pole of order 7 at infinity.
Now we set X(τ ) = η(τ ) 2 η(61τ )2 and Y (τ ) = X γ∈ Γ0(61)/Γ E30E32E26E44E10 E46E16E48E22E56 (τ ) γ + 9,
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
X(τ )|ω 61 = 61 η(61τ )2 η(τ )2 = 61 X(τ ) (4.1) and thus X(0) = 0.
From Proposition 2.3.11, we deduce that Y (τ )|ω
61 = 61q
3(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
61X
2 has a pole at cusp ∞ of order 7, so we use the function Y to cancel it. Thus,
we have
Y |ω
61 =
61Y
X2 . (4.3)
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),
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 = X
2+ 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 + a
1xy + 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 y2+ a 1xy + a3y = x3+ a2x2+ a4x + a6 is given by dx 2y + a1x + a3 , we have − qdx/dq 2y + a1x + a3 = q + ∞ X n=2 bnqn.
This relation, together with y2+ a
1xy + a3y = x3+ a2x2+ a4x + a6, uniquely determine
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+ · · · .
Thus, assuming that
x = q−2+ c−1q−1+ c0+ · · · , y = q−3+ d−2q−2+ d−1q−1+ · · · , and solving −qdx/dq 2y + 1 = q − 2q 2− 2q3 + 2q4− 4q5+ 4q6+ q9+ 8q10+ 3q11− 4q12+ · · · , y2+ y = x3+ x2,
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,
and set X = X γ∈ Γ0(43)/Γ E34E32E20E12E14E2 E24E28E4E6E36E42 −E26E16E10E12E14E2 E36E42E6E32E20E34 γ − 16, Y = X γ∈ Γ0(43)/Γ E26E16E10E12E14E2 E36E42E6E32E20E34 γ + 7,
Z = X γ∈ Γ0(43)/Γ E24E28E4 E6E36E42 γ − 18, and V = X γ∈ Γ0(43)/Γ E22E40E18E26E16E10E12E14E2 E32E20E34E30E8E38E6E36E42 γ − 15. The Fourier expansions for them are
X =2q−4+ q−4+ q−2− 12 + q + 2q2+ 2q3+ q4+ 2q6 − 2q7+ 4q9+ · · · , (4.4)
Y =q−5− 2q−3− q−2− 2q−1+ 4 + 2q + 2q3− 3q4− 2q5+ 2q7− q9+ · · · , (4.5) Z =q−6+ q−5+ q−4+ q−2− 16 + 2q + 3q2 + 2q3+ q4+ 4q6 + 3q8+ · · · , (4.6)
and
V = q−7+ q−6+ q−5− q−4+ 2q−1− 12 + q2− 2q3+ 2q5+ · · · . (4.7)
Now we consider the behavior of X, Y , Z and V under ω43. We can deduce that
X|ω 43 =43(q + 3q 2+ 7q3+ 16q4+ 32q5+ 63q6+ 117q7+ · · · ), Y |ω 43 =43(q 2+ 4q3+ 12q4+ 31q5+ 71q6 + 154q7+ 314q8 + · · · ), Z|ω 43 =43(2q + 8q 2+ 24q3+ 65q4 + 159q5+ 366q6 + 794q7+ 1654q8+ · · · ), and V |ω 43 = 43(3q + 13q 2+ 47q3+ 141q4+ 385q5+ 963q6+ 2270q7+ 5074q8+ · · · ).
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.
4.3
Results
N Functions Elliptic Curve
11 x =X 5 E2E42 E103 , y =X 5 E54 E13E3 + 1 y2+ y = x3− x2− 10x − 20 17 x =X 8 E10E42 E2E16E12 − 2, y =X 8 E10E63E8 E4E143 E12 + 3x − 9 y2+ xy + y = x3− x2− x − 14 19 x =X 9 E12E8 E18E6 − 3, y = X 9 E62E8 E2E162 + x − 6 y2+ y = x3+ x2− 9x − 15 37 X = η(τ ) 2 η(37τ )2, Y = X 6 E6E8E14 E3E4E7 − 11 x = −47X 2+ X3+ 6Y2− 7XY + 185Y + 222X X(Y + 185) y = 37/X + X + 5x − 7 y2+ y = x3− x 43 X =X 7 E34E32E20E12E14E2 E24E28E4E6E36E42 − 9 − Y Y =X 7 E26E16E10E12E14E2 E36E42E6E32E20E34 + 7 Z =X 7 E24E28E4 E6E36E42 − 18 V =X 7 E22E40E18E26E16E10E12E14E2 E32E20E34E30E8E38E6E36E42 − 15 x = (X + V + Z + 43)/Y, y = 2Zx − V x + Z + 2V + Y − 2X 3Y − Z + 2X y2+ y = x3+ x2 53 X =X 13 E24E22E36E20 E12E42E18E10 − 4, Y =X 13 E14E4E24E22 E46E2E12E42 , Z =X 13 E24E22 E16E50 − 16 x = 3Y (Y − 53) + 5Z(Z − 53) + 2173 (53 + 3X)X +−9Y + 29X − 25Z + 547 53 + 3X y = (x − 7)Z + (x − 3)Y + 37 X + 10 − 3x y2+ xy + y = x3− x2 61 X = η(τ ) 2 η(61τ )2, Y = X 6 E30E32E26E44E10 E46E16E48E22E56 + 9 x = Y X − 1, y = X2+ X + 61 Y + 4x − 12 y2+ xy = x3 − 2x + 1
N Functions Elliptic Curve 79 X9 = X 13 E54E22E32 E24E56E78 − 16 X10= X 13 E42E60E18E56E78E24 E72E10E76E34E26E8 − 24 X12= X 13 E54E22E32 E62E66E4 − 20 X13= X 3 E54E36E24E16E42E28E34E30E20E66E44E76E2 E52E18E12E8E58E14E62E64E10E46E22E38E78 − 7 X15= X 3 E54E36E24E16E42E28E34E30E20E66E44E76E2 E50E72E48E32E74E56E68E60E40E26E70E6E4 − 10 x = X15+ X13+ X12− 2X10+ X9 X13− 2X12+ X10 y = X13− X10+ X9 X10− X9 − x y2+ xy + y = x3+ x2− 2x
N Functions Elliptic Curve 83 X19= X 41 E38E10E70E62E56E40E24E66E18 E76E78E48E20E4E50E74E6E26 − 215 Y19= X 41 E38E46E20E30E82E28E66E18E12 E76E78E48E60E80E68E4E50E74 − 37 X20= X 41 E46E20E42E16E28E66E18E12E52 E60E10E80E62E4E14E50E74E26 − 111 Y20= X 41 E38E46E20E30E82E16E28E66E18E12E52 E64E76E78E48E60E80E62E68E4E50E74 − 400 Z20 = X 41 E48E44E22E40E74E4E16E52 E46E10E80E62E14E70E72E2 + 12 U20 = X 41 E48E44E642E62E40 E16E80E14E52E10E70 + 13 V20= X 41 E38E48E58E68E44E22E40E74E4E16E52 E64E76E78E54E34E24E46E80E62E72E2 − 208 T20= X 41 E38E64E44E62E40E30E26E82 E76E78E70E68E16E52E80E42 + 79 X21= X 41 E38E44E46E20E72E2E16E28E12E52 E64E76E78E24E56E22E80E62E4E74 + 27 Y21= X 41 E38E14E70E54E34E36E44E46E20E6E42 E64E76E78E48E56E66E18E22E60E80E62 − 508 Z21 = X 41 E38E54E34E44E20E28E12 E76E78E48E56E80E68E58 + 86 V21= X 41 E48E44E64E40 E80E14E10E70 − 12 U21 = X 41 E38E44E40E30E26E82 E76E78E70E68E80E42 + 42 T21= X 41 E38E64E44E22E62E40E74E4 E76E78E16E52E46E80E72E2 − 162 X22= X 41 E54E34E44E20E42E16E28E12E52 E56E10E80E62E58E30E82E14E26 − 240 Y22= X 41 E54E34E44E20E38E16E28E12E52 E64E76E78E48E56E80E62E68E58 − 405 Z22 = X 41 E48E44E40E16E52 E10E80E62E14E70 − 40 U22 = X 41 E38E44E22E40E74E4 E76E78E46E80E72E2 − 125 V22= X 41 E642E622E38E44E40 E522E162E76E78E80 − 585 y2+ xy + y = x3+ x2+ x
N Functions Elliptic Curve 83 X18= T20− X20 X17= X20− U20 X16= Y20− Z20+ Y19+ X18− X17 Y16= V22− X22− X18− X17 X15= X20− Z20+ X19+ 2X18+ 3X16 Y15= X20− V20+ Y19+ 2X18+ 2X17+ 3X16 X14= Z21− U21+ X20+ 2Y19+ 2X18+ 2X17+ 2X16 Y14= Z21− Y21− 2U21+ 2T21+ 2X20+ X19+ 3X18+ 2X17 + Y16 Z14 = (X15− Y15)/2 Y13= (V21− Z21− Y19− X17+ X16− X15+ 3X14)/21 R13= Y22− V22+ 2U22− 2Z22+ X21− 3X20− X19− 2X18 − X16+ 2X15− Y15− X14− Y14− 4Z14 T13= (V21− Z21− Y19− X17+ X16− Y15+ X14)/2 U13 = 4Y15− 4X15+ 3X14+ 5Y14 Y12= 3(Z21− X21+ X20+ X19− X18+ 2X17− 8X16 + 7X15− 34X14+ 301Y13) − R13+ T13 Z12 = 4X22− 2U22− 2V22− 2X21+ 3Y20+ 2X20− 3U20 − 4X19+ X18− 12X17− X16− 5X15+ 58Z14− 22R13 V12= 21(X21− Z21− X20− X19+ X18− 2X17+ 8X16 − 7X15+ 34X14− 301Y13)/2 + R13− V13 X11= 10X15− 10Y15− 3X14− 17Y14− 65T13+ Z12+ 3Y12 Z11 = (X15− Y15− 2Y14− 11T13− 17Y12)/4 X10= (2U22− Y22− Z22− X20− X18− X17+ 2X16− 4X15 + 24X14− 210Y13)/25 Y9 = (5X21− 11U21+ 6T21− X19+ 7X18− 4X17+ 20X16 − 59X15+ 52Y15+ 17X14+ 98Y14+ 5R13+ 393T13 − U13+ 290Y12− 77/2X11+ 48035/2X10)/48 R9 = (2T13− 2R13− 2V12− 18Y12+ X11− Z11 − 2157/4X10)/4 X8 = 5(R9− Y9)/313 x = −Y19− 2X18+ 2X17− 6X16+ 11X15− 62Z14− 39T13 X16− X17 −464Y12− 391/2X1115131/2X10+ 109580R9 X16− X17 −15055231/10X8 + 498 X16− X17 y = 4X19+ 2X18+ 6X17+ 3X16+ 84X15− 472Z14 3X16− 2X15 −496R13+ 181(V12− Z12) − 9240Y12− 326X11 3X16− 2X15 −937109/2X10+ 739424R9− 20418507/2X8 3X16− 2X15 −x(X17+ 4X15+ 1826) + 10292 3X16− 2X15 y2+ xy + y = x3+ x2+ x
N Functions Elliptic Curve 89 X11 = X 12 E26E6E46E38 E76E86E66E70 + 4, Y11= X 12 E26E6E32E20 E76E86E78E18 − 32, X12 = X 12 E24E74E46E38E42E4 E8E84E52E12E66E70 − 1 X13 = X 11 E28E32E62E20 E74E68E24E2 − 4 X14 = X 11 E44E26E72E6E28E32E62E20 E22E76E36E86E14E16E58E10 − 32 x = (X13+ X12)/X11 y = (X14+ X13+ X12+ 2Y11+ 89)/X11− 1 y2+ xy + y = x3+ x2− x
N Functions Elliptic Curve 101 X14= X 10 E40E46E38E74E26 E20E78E82E64E88 − 17 X15= X 10 E62E60E32E44E10E28E12E34E72E2 E56E24E68E58E4E14E6E84E36E100 − 41 Y15 = X 10 E90E48E66E86E8E40E46E38E74E26 E28E12E34E72E2E78E82E64E88E20 − 36 X16= X 10 E22E96E70E30E16E62E60E32E44E10 E28E12E34E72E2E42E18E50E94E98 E40E46E38E74E26E84E36E100E14E6 E782E822E642E882E202 + 41 Z17 = X 10 E56E24E68E58E4E62E60E32E44E10 E28E12E34E72E2E70E30E16E22E96 E50E94E98E42E18 E20E78E82E64E88 − 1 V17 = X 10 E62E60E32E44E10E78E82E64E88E20 E28E12E34E72E2E90E48E66E86E8 + 14 Y18 = X 10 E40E46E38E74E26 E28E12E34E72E2 − 133 Z18 = X 10 E56E24E68E58E4E62E60E32E44E10 E70E30E16E22E96E20E78E82E64E88 − 14 V18 = X 10 E56E24E68E58E4E22E96E70E30E16 E28E12E34E72E2E90E48E66E86E8 E40E46E38E74E26 E76E54E52E80E92 + 17 X13= X 10 E40E46E38E74E26 E70E30E16E22E96 −E40E46E38E74E26 E90E48E18E66E8 + X14+ 24 X12= X15− Y15+ 2X14 X11= Z17− V17− X16− 2X15+ 4X14− 5X13 X10= (V18− Y18+ Z17− X16+ X15+ 3X14− X13 + 2X12+ X11)/6 X9 = (V18− Z18+ V17− X16+ X15− X14+ 3X13 − X12− 5X11− 33X10)/18 x = X15− X12− X14− X11− 3X10 X13 + 2 y = 3X12+ X16+ X15+ 101 + 6X11+ 39X10+ 27X9 X13 + 1 y2+ y = x3+ x2 − x − 1
N Functions Elliptic Curve 131 X18= X 13 E18E50E94E30E4E68E44E64E26E14E104E56E10E86 E122E106E84E116E2E52E28E126E88E128E102E66E96E92 E6 E110 − 33 X20= X 5 E52E92E62E112E84E68E80E60E86E130E32E24E18 E96E72E54E106E14E76E74E10E58E22E82E4E128 − 79 Y20 = X 13 E58E130E70E78E42 E22E34E32E118E124 − 44 Z20 = X 13 E36E100E74E60E96E8E92E110E102E66 E16E72E62E114E120E54E112E20E90E12 − 50 X24= X 13 E36E100E74E60E68E44E64E26E8E14E104E56E10E86 E32E118E124E34E22E126E88E52E28E108E38E40E82E24 E6 E128 − 87 Y24 = X 13 E36E100E74E8E60 E122E106E84E116E2 − 50 Z24 = X 13 E18E50E94E30E4E102E66E96E92E110 E122E106E84E116E2E54E112E20E90E12 + 21 V24 = X 13 E108E38E40E82E24E58E130E70E78E42 E122E106E84E116E2E68E44E64E26E14 − 28 R24= X 13 E36E100E74E60E8E102E66E96E92E110E58E130E70E78 E106E84E2E112E20E90E12E76E122E116E54E80E98E48 E42 E46 − 116 Q24 = X 13 E54E112E20E90E12E76E80E98E48E46 E18E50E94E4E72E30E62E114E120E16 − 23 X25= X 13 E68E44E64E26E14E104E56E10E86E6 E32E118E124E34E22E126E88E128E52E28 − 50 Y25 = X 13 E122E106E84E116E68E44E64E26E14E108E38E40E24 E32E118E124E34E22E126E88E128E52E28E18E50E94 E82E2 E30E4 − 28 R25= X 13 E108E38E40E82E24 E122E106E84E116E2 − 98 T25= X 13 E18E50E94E30E4E68E44E64E26E14 E122E106E84E116E2E54E112E20E90E12 − 125 X21= X24− V24 X22= V24− T24 X23= Z24− X24 X19= Z20− X20 X17= X20− Y20− X18 X16= X24− Q24+ X23− X22+ X19+ 4X18+ X17 X15= Q24− Y24+ X20+ X19+ X18− 2X16 X14= (Y16− X16− 2X15)/6 X13= 3(Y25− R25+ 2X24+ 4X23− 3X22+ X21+ 3X20 + 9X19+ 16X18+ 8X17− 3X16+ 7X15+ 31X14)/5 y2+ y = x3− x2+ x
N Functions Elliptic Curve 131 x = 3X25− 3T25+ 3X23+ 6X20+ 18X19+ 18X18+ 21X17 3(Z20 − X18) +6X16+ 27X15+ 102X14− 5X13 3(Z20 − X18) y = 8R25− 3Y25− 5T25+ 2X24− 2Y24− 12Q24+ R24− 4X23 X18+ Y20− X20 −8X21+ 6X20− 2Y20− 3Z20− 5X18 X18+ Y20− X20 y2+ y = x3− x2+ x
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