The Recursive Structures of Manin Symbols over $\mathbb{Q}$, Cusps and Elliptic Points on $X_0(N)$

SanMin Wang

doi:10.20944/preprints202305.1118.v1

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The Recursive Structures of Manin Symbols over $\mathbb{Q}$, Cusps and Elliptic Points on $X_0(N)$

This version is not peer-reviewed.

SanMin Wang^*

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Submitted:

15 May 2023

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16 May 2023

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Abstract

Firstly, we present a more explicit formulation of the complete system $D(N)$ of representatives of Manin's symbols over $\mathbb{Q}$, which was initially given by Shimura. Then we establish a bijection between $D(M)\times D(N)$ and $D(MN)$ for $(M,N)=1$, which reveals a recursive structure between Manin's symbols of different levels. Based on Manin's complete system $\Pi (N)$ of representatives of cusps on $X_0(N)$ and Cremona's characterization of the equivalence between cusps, we establish a bijection between a subset $C(N)$ of $D(N)$ and $\Pi (N)$, and then establish a bijection between $C(M)\times C(N)$ and $C(MN)$ for $(M,N)=1$. We also provide a recursive structure for elliptical points on $X_0(N)$. Based on these recursive structures, we obtain recursive algorithms for constructing Manin symbols over $\mathbb{Q}$, cusps and elliptical points on $X_0(N)$. This gives rise to a more efficient algorithms for modular elliptic curve. As direct corollaries of these recursive structures, we present a recursive version of the genus formula and an elementary proof of formulas of the numbers of $D(N)$, cusps and elliptical points on $X_0(N)$.

Keywords:

modular curve

;

elliptic curve

;

recursive structure

;

Manin's symbols over $\mathbb{Q}$

;

cusps

;

elliptical points

;

algorithmic number theory

Subject:

Computer Science and Mathematics - Algebra and Number Theory

1. Introduction

In his seminal monograph [5], G. Shimura defined a complete set

D (N)

of representatives for the projective line

P^{1} (Z / N Z)

over

Z / N Z

to be all couples

{c, d}

of positive integers satisfying

(*) (c, d) = 1, d | N, 1 ⩽ c ⩽ N / d (o r c i n a n y s e t o f r e p r e s e n t a t i v e s f o r Z m o d u l o (N / d)),

where

(c, d)

denote the greatest common divisor of integers c and d.

Let

[x]

to be the greatest integer less than or equal to x. For two integers

a, b

, define

{[\frac{a}{b}]}^{'} = \{\begin{matrix} \frac{a}{b} - 1 if b | a, \\ [\frac{a}{b}] otherwise, \end{matrix}

Then

1 ⩽ a - b {[\frac{a}{b}]}^{'} ⩽ b

. In this paper, we define

\begin{matrix} D (N) = {(c, d) : c, d \in Z, c, d ⩾ 1, c | N, (c, d) = 1 and \\ (c, d - \frac{N}{c} ({[\frac{c d}{N}]}^{'} - n)) ⩾ 2 for 0 ⩽ n < {[\frac{c d}{N}]}^{'}} . \end{matrix}

(1)

We then establish a bijection between

D (M) \times D (N)

and

D (M N)

for

(M, N) = 1

in Section 2. This result gives a recursive algorithm to construct the projective line

P^{1} (Z / N Z)

over

Z / N Z

Let

Π (N) = {[δ; a mod (δ, N δ^{- 1})] : a, δ \in Z, δ ⩾ 1, δ | N, 1 ⩽ a ⩽ (δ, N δ^{- 1})}

. In [2], Ju. I. Manin proved that there exists a bijection between

Π (N)

and the set of cusps on

X_{0} (N)

. Based on Manin’s result and Cremona’s characterization(See Proposition 3), we identify

Π (N)

with

\begin{matrix} C (N) = {(c, d) : c, d \in Z, 1 ⩽ c ⩽ N, c | N, (c, d) = 1 and \\ (c, d - (c, N c^{- 1}) {[\frac{d}{(c, N c^{- 1})}]}^{'} + \frac{N n}{c}) ⩾ 2 for 0 ⩽ n < \frac{c (c, N c^{- 1})}{N} {[\frac{d}{(c, N c^{- 1})}]}^{'}}, \end{matrix}

(2)

which is a subset of

D (N)

. In Section 3, we establish a bijection between

C (N_{1} N_{2})

and

C (N_{1}) \times C (N_{2})

for

(N_{1}, N_{2}) = 1

. This result gives a recursive algorithm to construct the complete set of representatives of

Γ_{0} (N)

-inequivalent cusps.

Define

\begin{matrix} E_{2} (N) & = {(1, d) : (1, d) \in D (N), 1 + d^{2} \equiv 0 (\mod N)}, \\ E_{3} (N) & = {(1, d) : (1, d) \in D (N), 1 - d + d^{2} \equiv 0 (\mod N)} . \end{matrix}

(3)

Then there exist bijections between

E_{2} (N), E_{3} (N)

and complete sets of representatives of

Γ_{0} (N)

-inequivalent elliptic points of order 2, 3, respectively. In Section 4, we establish bijections between

E_{2} (N_{1} N_{2})

and

E_{2} (N_{1}) \times E_{2} (N_{2}), E_{3} (N_{1} N_{2})

and

E_{3} (N_{1}) \times E_{3} (N_{2})

, for

(N_{1}, N_{2}) = 1

. These results give a recursive algorithm for constructing the complete set

E_{3} (N)

and

E_{2} (N)

Γ_{0} (N)

-inequivalent elliptic points of order 2, 3.

The elements in

P^{1} (Z / N Z)

are called Manin symbols and there exists a bijection between the set of right cosets of

Γ_{0} (N)

S L (2, Z)

and

P^{1} (Z / N Z)

. The important steps in the modular elliptic algorithm are to construct the complete set

D (N)

of representatives for the projective line

P^{1} (Z / N Z)

and the complete set

C (N)

of representatives of

Γ_{0} (N)

-inequivalent cusps. The recursive structure of

D (N), C (N), E_{2} (N)

and

E_{3} (N)

may give rise to a more efficient modular elliptic algorithm.

2. The recursive structure of Manin symbols over $Q$

We firstly give some necessary notations and facts, for details, See [1].

Definition 1.

(a): $D_{2} (N) = {(c, d) : c, d \in Z, (c, d, N) = 1}$ ;
(b): $\forall (c_{1}, d_{1}), (c_{2}, d_{2}) \in D_{2} (N),$ define $(c_{1}, d_{1}) \sim (c_{2}, d_{2})$ if $c_{1} d_{2} \equiv d_{1} c_{2} (mod N)$ ,

then ∼ is an equivalence relation on $D_{2} (N)$ ;
(c): $\forall (c, d) \in D_{2} (N)$ , define $(c : d) = \{(c^{'}, d^{'}) : (c^{'}, d^{'}) \in D_{2} (N), (c^{'}, d^{'}) \sim (c, d)\}$ ;
(d): $D (N) = D_{2} (N) / \sim = {(c : d) : (c, d) \in D_{0} (N)}$ ;
(e): $D_{1} (N) = \{(c, d) : c, d \in Z, c, d ⩾ 1, c | N, (c, d, \frac{N}{c}) = 1, c d ⩽ N\}$ ;
(f): $D (N)$ is defined in (1);
(g): $μ (N)$ , $v_{\infty} (N)$ , $v_{2} (N)$ and $v_{3} (N)$ are the numbers of elements in $D (N), C (N), E_{2} (N)$ and $E_{3} (N)$ , respectively.

Lemma 1.

Let

c, d, h \in Z

(c, d, h) = 1

c, d ⩾ 1

and

d ⩽ h

then there exists an integer k such that

(c, d + h k) = 1

and

0 ⩽ k < c

Proof.

c = 1

, take

k = 0

then

(c, d + h k) = 1

. Thus let

c ⩾ 2

in the following. Let

c = p_{1}^{α_{1}} \dots p_{s}^{α_{s}}

be the standard factorization of c. The proof is by induction on the numbers of distinct prime divisors in c. Suppose that

c = p_{1}^{α_{1}}

. Assume that

(p_{1}^{α_{1}}, d) ⩾ 2

and

(p_{1}^{α_{1}}, d + h) ⩾ 2

then

p_{1} | d

and

p_{1} | d + h

. Thus

p_{1} | d

and

p_{1} | h

, this contradicts with

(c, d, h) = 1

and hence

(c, d + h k) = 1

for some

0 ⩽ k ⩽ 1 < c

Let

c_{1} = p_{1}^{α_{1}} \dots p_{s - 1}^{α_{s - 1}}

. By the induction hypothesis, there exists an integer

k_{1}

such that

(c_{1}, d + h k_{1}) = 1

and

0 ⩽ k_{1} < c_{1}

. Then

(c_{1}, d + h k_{1} + h c_{1}) = 1

. Assume that

(p_{s}^{α_{s}}, d + h k_{1}) ⩾ 2

and

(p_{s}^{α_{s}}, d + h k_{1} + h c_{1}) ⩾ 2

then

p_{s} | d + h k_{1}

and

p_{s} | d + h k_{1} + h c_{1}

. Thus

p_{s} | h c_{1}

and hence

p_{s} | h

(p_{s}, c_{1}) = 1

. Therefore

p_{s} | d

. This contradicts with

(c, d, h) = 1

and hence

(c, d + h k_{1}) = 1

(c, d + h k_{1} + h c_{1}) = 1

. Take

k = k_{1}

k = c_{1} + k_{1}

, then

(c, d + h k) = 1

for some

0 ⩽ k_{1} ⩽ k ⩽ c_{1} + k_{1} < 2 c_{1} ⩽ c

. This completes the proof by the induction principal. □

Corollary 1.

Let

a, b, c \in Z

(a, b, c) = 1

then the equation

a x + b y + c y z = 1

has solutions in

Z

Lemma 2.

There exists a bijection between

D (N)

and

D_{1} (N)

Proof.

Let

(c, d) \in D (N)

. Define

d_{n} = d - \frac{N}{c} {[\frac{c d}{N}]}^{'} + \frac{N n}{c}

for all

n \in Z

. Then

1 ⩽ d_{0} ⩽ \frac{N}{c}

and

(c, d_{0}, \frac{N}{c}) = 1

(c, d) = 1

. Thus

(c, d_{0}) \in D_{1} (N)

. Define

Φ : D (N) \to D_{1} (N)

by sending

(c, d)

(c, d_{0})

Let

(u, v) \in D (N)

such that

Φ (c, d) = Φ (u, v)

. Define

v_{n} = v - \frac{N}{u} {[\frac{u v}{N}]}^{'} + \frac{N n}{u}

for all

n \in Z

. Then

c = u

and

d_{0} = v_{0}

. Thus

d_{n} = v_{n}

for all

n \in Z

. Let

e = {[\frac{c d}{N}]}^{'}

and

w = {[\frac{c v}{N}]}^{'}

. Then

d = d_{e}

and

v = v_{w}

. Suppose that

e < w

then

(c, d_{e}) = 1

(c, d) = 1

but

(c, d_{e}) ⩾ 2

(c, v) \in D (N)

and

d_{e} = v_{e}

, a contradiction and thus

e ⩾ w

e ⩽ w

holds by a similar proof and thus

e = w

and

(c, d) = (u, v)

. Therefore

Φ

is an injection from

D (N)

D_{1} (N)

Let

(c, d_{0}) \in D_{1} (N)

. By Lemma 1, there exists an integer k such that

(c, d_{0} + \frac{N k}{c}) = 1

and

0 ⩽ k ⩽ c - 1

. Let

0 ⩽ k_{0} ⩽ k

such that

(c, d_{0} + \frac{N k_{0}}{c}) = 1

and

(c, d_{0} + \frac{N n}{c}) = 1

for all

0 ⩽ n < k_{0}

. Define

d = d_{0} + \frac{N k_{0}}{c}

. Then

(c, d) \in D (N)

and

Φ ((c, d)) = (c, d_{0})

. Therefore

Φ

is a surjection from

D (N)

D_{1} (N)

. □

Lemma 3.

There exists a bijection between

D (N)

and

D (N)

, i.e.,

D (N)

is a complete system of the representatives of elements of

D (N)

Proof.

Define

Φ : D (N) \to D (N)

by the natural map, i.e.,

Φ ((c, d)) = (c : d)

Let

(c : d) \in D (N)

. Then

(c, d, N) = 1

. Define

c_{1} = (c, N)

d_{0}

to be the unique solution of the congruence equation

\frac{c}{c_{1}} x \equiv d (mod \frac{N}{c_{1}})

such that

1 ⩽ d_{0} ⩽ \frac{N}{c_{1}}

. Then there exists an integer y such that

\frac{c}{c_{1}} d_{0} + \frac{N}{c_{1}} y = d

. Assume that there exists a prime p such that

p | (c_{1}, d_{0}, \frac{N}{c_{1}})

. Then

p | d

and

p | (c, N)

, this contradicts with

(c, d, N) = 1

and thus

(c_{1}, d_{0}, \frac{N}{c_{1}}) = 1

. Hence

(c_{1}, d_{0}) \in D_{1} (N)

. Then there exists the unique

(c_{1}, d_{1}) \in D (N)

which corresponds to

(c_{1}, d_{0})

. Hence

(c_{1}, d_{1}) \in (c : d)

, i.e.,

Φ ((c_{1}, d_{1})) = (c : d)

Assume that

(c_{1}, d_{1}), (c_{2}, d_{2}) \in D (N)

such that

Φ ((c_{1}, d_{1})) = Φ ((c_{2}, d_{2}))

. Then

(c_{1} : d_{1}) = (c_{2} : d_{2})

and thus there exists an integerk such that

c_{1} d_{2} - c_{2} d_{1} = N k

. Thus

c_{1} | c_{2} d_{1}

c_{1} | N

and

c_{2} | c_{1} d_{2}

c_{2} | N

. Hence

c_{1} | c_{2}

(c_{1}, d_{1}) = 1

and

c_{2} | c_{1}

(c_{2}, d_{2}) = 1

. Therefore

c_{1} = c_{2}

and

d_{1} = d_{2}

d_{1} \equiv d_{2} (mod \frac{N}{c_{1}})

and the definition of

D (N)

. Thus

Φ

is a bijection between

D (N)

and

D (N)

. This completes the proof. □

Theorem 1.

Let

M, N \in Z, M, N ⩾ 1

(M, N) = 1

. Then there exists a bijection between

D (M) \times D (N)

and

D (M N)

Proof.

Let

(a, b) \in D (M)

and

(c, d) \in D (N)

. Assume that there exists a prime p such that

p | (a c, b N + d M - \frac{M N}{a c} {[\frac{a c (b N + d M)}{M N}]}^{'}, \frac{M N}{a c})

. Then

p | a c, p | \frac{M}{a} \frac{N}{c}

and

p | b N + d M - \frac{M N}{a c} {[\frac{a c (b N + d M)}{M N}]}^{'} .

Then

p | a, p | \frac{M}{a}

p | c, p | \frac{N}{c}

(M, N) = 1

a | M, c | N

. If

p | a, p | \frac{M}{a}

then

p | b N

and thus

p | N

(a, b) = 1

, which contradicts with

(M, N) = 1

. The case of

p | c, p | \frac{N}{c}

is tackled by a similar way. Therefore

(a c, b N + d M - \frac{M N}{a c} {[\frac{a c (b N + d M)}{M N}]}^{'}, \frac{M N}{a c}) = 1

and

(a c, b N + d M - \frac{M N}{a c} {[\frac{a c (b N + d M)}{M N}]}^{'}) \in D_{1} (M N) .

Define

e = a c

f = b N + d M - \frac{M N}{a c} ({[\frac{a c (b N + d M)}{M N}]}^{'} - k)

for some k such that

(a c, b N + d M - \frac{M N}{a c} ({[\frac{a c (b N + d M)}{M N}]}^{'} - n)) ⩾ 2

for all

0 ⩽ n < k

. Then

(e, f) \in D (M N)

. Define

Φ : D (M) \times D (N) \to D (M N)

by sending

((a, b), (c, d))

(e, f)

Assume that

Φ ((a, b), (c, d)) = Φ ((a_{1}, b_{1}), (c_{1}, d_{1}))

for some

(a, b), (a_{1}, b_{1}) \in D (M)

and

(c, d), (c_{1}, d_{1}) \in D (N)

. Then

(a c, b N + d M - \frac{M N}{a c} ({[\frac{a c (b N + d M)}{M N}]}^{'} - k))

= (a_{1} c_{1}, b_{1} N + d_{1} M - \frac{M N}{a_{1} c_{1}} ({[\frac{a_{1} c_{1} (b_{1} N + d_{1} M)}{M N}]}^{'} - k_{1})) .

Thus

a c = a_{1} c_{1}

and

b N + d M - \frac{M N}{a c} ({[\frac{a c (b N + d M)}{M N}]}^{'} - k)

= b_{1} N + d_{1} M - \frac{M N}{a_{1} c_{1}} ({[\frac{a_{1} c_{1} (b_{1} N + d_{1} M)}{M N}]}^{'} - k_{1}) .

Hence

a = a_{1}

c = c_{1}

(M, N) = 1

a | M, a_{1} | M, c | N, c_{1} | N

. Therefore

b N + d M - \frac{M N}{a c} ({[\frac{a c (b N + d M)}{M N}]}^{'} - k)

= b_{1} N + d_{1} M - \frac{M N}{a c} ({[\frac{a c (b_{1} N + d_{1} M)}{M N}]}^{'} - k_{1}) .

Thus

d \equiv d_{1} (mod \frac{N}{c})

and

b \equiv b_{1} (mod \frac{M}{a})

(M, N) = 1

. Hence

b = b_{1}

d = d_{1}

. Then

((a, b), (c, d)) = ((a_{1}, b_{1}), (c_{1}, d_{1}))

Let

(e, f) \in D (M N)

. Then

e | M N

(e, f) = 1 a n d (e, f - \frac{M N}{e} ({[\frac{e f}{M N}]}^{'} - n)) ⩾ 2 f o r 0 ⩽ n < {[\frac{e f}{M N}]}^{'}

. Let

a = (e, M)

c = (e, N)

, then

e = a c

a | M

and

c | N

. Let

x_{0}, y_{0}, z_{0}

be a particular solution of the equation

N x + M y + \frac{M N}{a c} z = f

(4)

then

x = \frac{M}{a} X + x_{0}, y = \frac{N}{c} Y + y_{0}, z = - c X - a Y + z_{0}

are solutions of

(4)

for all integers

X, Y

. Take

b_{1} = x_{0} - \frac{M}{a} {[\frac{a x_{0}}{M}]}^{'}, d_{1} = y_{0} - \frac{N}{c} {[\frac{c y_{0}}{N}]}^{'}

, then

N b_{1} + M d_{1} + \frac{M N}{a c} (c {[\frac{a x_{0}}{M}]}^{'} + a {[\frac{c y_{0}}{N}]}^{'} + z_{0}) = f, 1 ⩽ b_{1} ⩽ \frac{M}{a}, 1 ⩽ d_{1} ⩽ \frac{N}{c} .

Then

(a, b_{1}, \frac{M}{a}) = 1

a | M, (e, f) = 1

and

(c, d_{1}, \frac{N}{c}) = 1

c | N, (e, f) = 1

. Hence

(a, b_{1}) \in D_{1} (M), (c, d_{1}) \in D_{1} (N)

. Let

(a, b) \in D (M)

and

(c, d) \in D (N)

which correspond to

(a, b_{1})

and

(c, d_{1})

, respectively. Then

b = b_{1} + \frac{M}{a} k_{1}

and

d = d_{1} + \frac{N}{c} k_{2}

for some

k_{1}, k_{2}

. Then

N b + M d + \frac{M N}{a c} (c {[\frac{a x_{0}}{M}]}^{'} + a {[\frac{c y_{0}}{N}]}^{'} - c k_{1} - a k_{2} + z_{0}) = f

. Then

(e, f) = Φ ((a, b), (c, d))

Thus

Φ

is a bijection between

D (M) \times D (N)

and

D (M N)

. □

Proposition 1.

Let p be a prime and l a positive integer. Then

(a) D (p^{l}) = {(1, d) : 1 ⩽ d ⩽ p^{l}} \cup {(p^{l}, 1)} \cup

{(p^{α}, k p + d) : 1 ⩽ α ⩽ l - 1, 1 ⩽ d ⩽ p - 1, 0 ⩽ k ⩽ p^{l - α - 1} - 1};

(b) μ (p^{l}) = p^{l} (1 + \frac{1}{p});

(c) μ (N) = N \prod_{p | N} (1 + (\frac{1}{p})) .

Proof. (c) is immediately from (b) and Theorem 1. □

Algorithm 1.

(1): Construct D( $p^{l})$ by Proposition 1(a);
(2): Given $D (M)$ and $D (N)$ for $(M, N) = 1$ , $D (M N)$ is constructed as follows. For all $(a, b) \in D (M)$ , $(c, d) \in D (N)$ , define $e = a c$ , $f = b N + d M - \frac{M N}{a c} ({[\frac{a c (b N + d M)}{M N}]}^{'} - k)$ for some $k \in Z$ such that $(e, f) = 1$ and $(a c, b N + d M - \frac{M N}{a c} ({[\frac{a c (b N + d M)}{M N}]}^{'} - n)) ⩾ 2$ for all $0 ⩽ n < k$ . Then $(e, f) \in D (M N)$ and all elements in $D (M N)$ are constructed if all pairs in $D (M) \times D (N)$ are processed.

3. The recursive structure of cusps

In order to describe the cusps on

X_{0} (N)

, Ju. I. Manin in [2] introduced the set

Π (N)

, which consists of pairs of the form

[δ; a mod (δ, N δ^{- 1})]

. Here

δ

runs through all positive divisors of N, and the second coordinate of the pair runs through any invertible class of residues modulo the greatest common divisor of

δ

and

N δ^{- 1}

. If

(δ, N δ^{- 1}) = 1

we sometimes put simply 1 in place of the second coordinate.

Proposition 2.

Let

δ | N, u, v \in Z

;

(u, v δ) = (v, N δ^{- 1}) = 1

. The map

Q \cup {i \infty} \to Π (N)

of the form

\frac{u}{v δ} \mapsto [δ; u v mod (δ, N δ^{- 1})]

gives an isomorphism of the set of cusps on

X_{0} (N)

with

Π (N)

Proof.

See Proposition 2.2 in [2]. □

In [1], J. E. Cremona gives the following characterization of cusps of

X_{0} (N)

Proposition 3.

For

j = 1, 2

let

α_{j} = p_{j} / q_{j}

be cusps written in lowest terms. The following are equivalent:

(a): $α_{2} = M (α_{1})$ for some $M \in Γ_{0} (N)$ ;
(b): $q_{2} \equiv u q_{1} (mod N)$ and $u p_{2} \equiv p_{1} (mod (q_{1}, N))$ , with $(u, N) = 1$ ;
(c): $s_{1} q_{2} \equiv s_{2} q_{1} (mod (q_{1} q_{2}, N))$ , where $s_{j}$ satisfies $p_{j} s_{j} \equiv 1 (mod q_{j})$ .

Proof.

See Proposition 2.2.3 in [1]. □

Definition 2.

(a) C_{1} (N) = {(c, d) : c, d \in Z, 1 ⩽ c ⩽ N, c | N, 1 ⩽ d ⩽ (c, N c^{- 1}), (c, d, N c^{- 1}) = 1},

(b) C (N) i s d e f i n e d i n (2) .

Lemma 4.

There exists a bijection between

C_{1} (N)

and

C (N)

Proof.

It holds by

C_{1} (N) \subseteq D_{1} (N)

C (N) \subseteq D (N)

and Lemma 2. □

Lemma 5.

There exists a bijection between

Γ_{0} (N) ∖ Q \cup {i \infty}

and

C_{1} (N)

Proof.

Let

γ_{i} = (\begin{matrix} a_{i} & b_{i} \\ c_{i} & d_{i} \end{matrix}), γ_{j} = (\begin{matrix} a_{j} & b_{j} \\ c_{j} & d_{j} \end{matrix}) \in {S L}_{2} (Z)

such that

(c_{i}, d_{i}), (c_{j}, d_{j}) \in D (N)

for

1 ⩽ i < j ⩽ μ (N)

then

{S L}_{2} (Z) = Γ_{0} (N) γ_{1} \cup \dots \cup Γ_{0} (N) γ_{μ (N)}

and

Γ_{0} (N) γ_{i} \neq Γ_{0} (N) γ_{j}

\forall c, a \in Z, (c, a) = 1, c ⩾ 1

, let

(\begin{matrix} a & b \\ c & d \end{matrix}) \in {S L}_{2} (Z)

for some

a, b

. Then there exists

γ \in Γ_{0} (N), 1 ⩽ i ⩽ μ (N)

such that

(\begin{matrix} a & b \\ c & d \end{matrix}) = γ γ_{i}

. Thus

(\begin{matrix} a & b \\ c & d \end{matrix}) (\infty) = γ γ_{i} (\infty)

γ (\frac{a_{i}}{c_{i}}) = \frac{a}{c}

and

Γ_{0} (N) \frac{a}{c} = Γ_{0} (N) \frac{a_{i}}{c_{i}}

. Then

Γ_{0} (N) ∖ Q \cup {i \infty} = {Γ_{0} (N) \frac{a_{i}}{c_{i}} : 1 ⩽ i ⩽ μ}

. Define

Φ : Γ_{0} (N) ∖ Q \cup {i \infty} \to C_{1} (N)

Γ_{0} (N) \frac{a}{c} \mapsto (c_{i}, d_{i} - (c_{i}, c_{i}^{- 1} N) {[d_{i} {(c_{i}, c_{i}^{- 1} N)}^{- 1}]}^{'}), Γ_{0} (N) \cdot i \infty \mapsto (N, 1) .

By Proposition 3,

Γ_{0} (N) \frac{a_{i}}{c_{i}} = Γ_{0} (N) \frac{a_{j}}{c_{j}}

iff

c_{i} d_{j} \equiv c_{j} d_{i} (mod (c_{i} c_{j}, N))

. Then

c_{i} d_{j} = c_{j} d_{i} + (c_{i} c_{j}, N) h

for some

h \in Z

. Thus

c_{i} = c_{j}

c_{i} | N

c_{j} | N

(c_{i}, d_{i}) = 1

and

(c_{j}, d_{j}) = 1

. Hence

c_{i} d_{j} \equiv c_{j} d_{i} (mod (c_{i} c_{j}, N))

iff

d_{i} \equiv d_{j} (mod (c_{i}, c_{i}^{- 1} N))

. Therefore

Φ

is a bijection between

Γ_{0} (N) ∖ Q \cup {i \infty}

and

C_{1} (N)

. □

Lemma 6.

There exists a bijection between

Γ_{0} (N) ∖ Q \cup {i \infty}

and

C (N)

Proof.

It is immediately from Lemma 4 and 5. □

Lemma 7.

Let

(N_{1}, N_{2}) = 1

. Then there exists a bijection between

C_{1} (N_{1} N_{2})

and

C_{1} (N_{1}) \times C_{1} (N_{2})

Proof.

Let

(c, d) \in C_{1} (N_{1} N_{2})

then

c | N_{1} N_{2}, d ⩽ (c, N_{1} N_{2} c^{- 1}), (d, c, N_{1} N_{2} c^{- 1}) = 1

. Let

c_{1} = (c, N_{1}), c_{2} = (c, N_{2})

then

c = c_{1} c_{2}, (c_{1}, c_{2}) = 1

and

(d, c_{1} c_{2}, N_{1} c_{1}^{- 1} N_{2} c_{2}^{- 1}) = 1

. Thus

(d, (c_{1}, N_{1} c_{1}^{- 1})) = 1

(d, (c_{2}, N_{2} c_{2}^{- 1})) = 1

(c, N_{1} N_{2} c^{- 1}) = (c_{1}, N_{1} c_{1}^{- 1}) (c_{2}, N_{2} c_{2}^{- 1})

. Let

d_{1} = d - (c_{1}, N_{1} c_{1}^{- 1}) {[d {(c_{1}, N_{1} c_{1}^{- 1})}^{- 1}]}^{'}

and

d_{2} = d - (c_{2}, N_{2} c_{2}^{- 1}) {[d {(c_{2}, N_{2} c_{2}^{- 1})}^{- 1}]}^{'}

then

(d_{1}, (c_{1}, N_{1} c_{1}^{- 1})) = 1

and

(d_{2}, c_{2}, N_{2} c_{2}^{- 1}) = 1

. Thus

(c_{1}, d_{1}) \in C_{1} (N_{1})

and

(c_{2}, d_{2}) \in C_{1} (N_{2})

. Define

Φ : C_{1} (N_{1} N_{2}) \to C_{1} (N_{1}) \times C_{2} (N_{2})

(c, d) \mapsto ((c_{1}, d_{1}), (c_{2}, d_{2}))

For any

((c_{1}, d_{1}), (c_{2}, d_{2})) \in C_{1} (N_{1}) \times C_{1} (N_{2})

, let

c = c_{1} c_{2}

there exists an integer d such that

d \equiv d_{1} (mod (c_{1}, N_{1} c_{1}^{- 1}))

d \equiv d_{2} (mod (c_{2}, N_{2} c_{2}^{- 1}))

and

1 ⩽ d ⩽ (c_{1}, N_{1} c_{1}^{- 1}) (c_{2}, N_{2} c_{2}^{- 1}) = (c, N_{1} N_{2} c^{- 1})

((c_{1}, N_{1} c_{1}^{- 1}), (c_{2}, N_{2} c_{2}^{- 1})) = 1

. Thus

(c, d) \in C_{1} (N_{1} N_{2})

and hence

Φ

is a surjective map.

Let

Φ ((c, d)) = Φ ((c^{'}, d^{'}))

. Then

((c_{1}, d_{1}), (c_{2}, d_{2})) = ((c_{1}^{'}, d_{1}^{'}), (c_{2}^{'}, d_{2}^{'}))

(c_{1}, d_{1}) = (c_{1}^{'}, d_{1}^{'})

and

(c_{2}, d_{2}) = (c_{2}^{'}, d_{2}^{'})

. Thus

c_{1} = c_{1}^{'}, c_{2} = c_{2}^{'}, d_{1} = d_{1}^{'}

and

d_{2} = d_{2}^{'}

. Hence

c = c_{1} c_{2} = c_{1}^{'} c_{2}^{'} = c^{'}

and

d = d^{'}

d \equiv d_{1} (mod (c_{1}, N_{1} c_{1}^{- 1}))

d \equiv d_{2} (mod (c_{2}, N_{2} c_{2}^{- 1}))

d^{'} \equiv d_{1}^{'} (mod (c_{1}, N_{1} c_{1}^{- 1}))

and

d^{'} \equiv d_{2}^{'} (mod (c_{2}, N_{2} c_{2}^{- 1}))

. Therefore

Φ

is an injective map. Then

Φ

is a bijection between

C_{1} (N_{1} N_{2})

and

C_{1} (N_{1}) \times C_{1} (N_{2})

. □

Theorem 2.

Let

(N_{1}, N_{2}) = 1

. Then there exists a bijection between

C (N_{1} N_{2})

and

C (N_{1}) \times C (N_{2})

Proof.

It is immediately from Lemma 4 and 7. □

Proposition 4.

Let p be a prime and l a positive integer. Then

(a) C (p^{l}) = {(1, 1), (p^{l}, 1)} \cup

{(p^{α}, k p + d) : 1 ⩽ α ⩽ l - 1, 1 ⩽ d ⩽ p - 1, 0 ⩽ k ⩽ p^{min {α, l - α} - 1} - 1};

(b) v_{\infty} (p^{l}) = \{\begin{matrix} (p + 1) p^{\frac{l}{2} - 1} i f 2 | l, \\ 2 p^{\frac{l - 1}{2}} o t h e r w i s e \end{matrix};

(c) v_{\infty} (N) = \prod_{p | N} v_{\infty} (p^{l}) .

Proof. (c) is immediately from (b) and Theorem 2. □

Algorithm 2.

(1): Construct C( $p^{l}$ ) by Proposition 4(a);
(2): Let $N = N_{1} N_{2}$ for $(N_{1}, N_{2}) = 1$ . Given $C (N_{1})$ and $C (N_{2})$ . $C (N)$ is constructed as follows. For all $(c_{1}, d_{1}) \in C (N_{1})$ , $(c_{2}, d_{2}) \in C (N_{2})$ , define $c = c_{1} c_{2}$ . Determinate $d_{0}$ such that

$d_{0} \equiv d_{1} (mod (c_{1}, N_{1} c_{1}^{- 1}))$ , $d_{0} \equiv d_{2} (mod (c_{2}, N_{2} c_{2}^{- 1}))$ and

$1 ⩽ d_{0} ⩽ (c_{1}, N_{1} c_{1}^{- 1}) (c_{2}, N_{2} c_{2}^{- 1}) .$

Determinate $d = d_{0} + \frac{N k}{c}$ such that $(c, d) = 1$ and $(c, d_{0} + \frac{N n}{c}) ⩾ 2 f o r 0 ⩽ n < k$ . Then $(c, d) \in C (N_{1} N_{2})$ and all elements in $C (N_{1} N_{2})$ are constructed if all pairs in $C (N_{1}) \times C (N_{2})$ are processed.

4. The recursive structure of elliptic points of $X_{0} (N)$

Let

ρ = \frac{- 1 + \sqrt{3} i}{2}

E_{2} (N)

and

E_{3} (N)

is defined in (3). Then

{γ (i) : (1, d) \in E_{2} (N), γ = (\begin{matrix} a & b \\ 1 & d \end{matrix}) \in S L (2, Z) f o r s o m e f i x e d a, b} a n d

{γ (ρ) : (1, d) \in E_{3} (N), γ = (\begin{matrix} a & b \\ 1 & d \end{matrix}) \in S L (2, Z) f o r s o m e f i x e d a, b}

are complete sets of representatives of

Γ_{0} (N)

-inequivalent elliptic points of order 2, 3, respectively.

Theorem 3.

Let

N_{1}, N_{2} \in Z, N_{1}, N_{2} ⩾ 1

and

(N_{1}, N_{2}) = 1

. Then

(a): there exists a bijection between $E_{3} (N_{1}) \times E_{3} (N_{2})$ and $E_{3} (N_{1} N_{2})$ ;
(b): there exists a bijection between $E_{2} (N_{1}) \times E_{2} (N_{2})$ and $E_{2} (N_{1} N_{2})$ .

Proof. (a) Let

(1, d_{1}) \in E_{3} (N_{1})

and

(1, d_{2}) \in E_{3} (N_{2})

. Let d be the unique integer such that

d \equiv d_{1} (m o d N_{1})

d \equiv d_{2} (m o d N_{2})

and

1 ⩽ d ⩽ N_{1} N_{2}

then

d^{2} - d + 1 \equiv 0 (m o d N_{1} N_{2})

Hence

(1, d) \in E_{3} (N_{1} N_{2})

. Define

Φ : E_{3} (N_{1}) \times E_{3} (N_{2}) \to E_{3} (N_{1} N_{2}), ((1, d_{1}), (1, d_{2})) \mapsto (1, d) .

Then

Φ

is a bijection between

E_{3} (N_{1}) \times E_{3} (N_{2})

and

E_{3} (N_{1} N_{2})

. The proof of (b) is similar to that of (a) and omitted. □

Proposition 5.

Let

p \in Z

be a prime and

l \in Z, l ⩾ 1

. Then

v_{2} (p^{l}) = \{\begin{matrix} 0 i f p \equiv 3 (m o d 4) o r 4 | p^{l}, \\ 1 i f p = 2, \\ 2 i f p \equiv 1 (m o d 4) . \end{matrix}

Proof.

Let

(1, d) \in E_{2} (p^{l})

then

d^{2} + 1 \equiv 0 (m o d p^{l})

. Since the system of two equations

x^{2} + 1 \equiv 0 (m o d p)

and

2 x \equiv 0 (m o d p)

has a common solution iff

p = 2

, the number of solutions of

x^{2} + 1 \equiv 0 (m o d p^{l})

is equal to that of

x^{2} + 1 \equiv 0 (m o d p)

p \neq 2

. The cases of

p = 2

4 | p^{l}

are trivial and we then let

p ⩾ 3

in the following. Then

x^{2} + 1 \equiv 0 (m o d p)

has a solution iff

(\frac{- 1}{p}) = 1

iff

p \equiv 1 (m o d 4)

(\frac{- 1}{p}) = {(- 1)}^{\frac{p - 1}{2}}

. In addition,

x^{2} + 1 \equiv 0 (m o d p)

has two and only two solutions when it is solvable. This completes the proof. □

Proposition 6.

Let

p \in Z

be a prime and

l \in Z, l ⩾ 1

. Then

v_{3} (p^{l}) = \{\begin{matrix} 0 i f p \equiv 2 (m o d 3) o r 9 | p^{l}, \\ 1 i f p = 3, \\ 2 i f p \equiv 1 (m o d 3) . \end{matrix}

Proof.

Let

(1, d) \in E_{3} (p^{l})

then

d^{2} - d + 1 \equiv 0 (m o d p^{l})

. Since the system of two equations

x^{2} - x + 1 \equiv 0 (m o d p)

and

2 x - 1 \equiv 0 (m o d p)

has a common solution iff

p = 3

, the number of solutions of

x^{2} - x + 1 \equiv 0 (m o d p^{l})

is equal to that of

x^{2} - x + 1 \equiv 0 (m o d p)

p \neq 3

. The cases of

p = 2, 3

9 | p^{l}

are trivial and we then let

p ⩾ 5

in the following.

x^{2} - x + 1 \equiv 0 (m o d p)

has a solution iff

y^{2} + 3 \equiv 0 (m o d p)

has a solution by taking

x = \frac{y + 1}{2}

and substituting

p - y

for y when

y \equiv 0 (mod 2)

. Then

x^{2} - x + 1 \equiv 0 (m o d p)

has a solution iff

(\frac{- 3}{p}) = 1

iff

p \equiv 1 (m o d 3)

(\frac{- 3}{p}) = (\frac{3}{p}) (\frac{- 1}{p}), (\frac{3}{p}) = {(- 1)}^{\frac{p - 1}{2}} (\frac{p}{3}), (\frac{- 1}{p}) = {(- 1)}^{\frac{p - 1}{2}}

and

(\frac{- 3}{p}) = (\frac{p}{3})

. In addition,

x^{2} - x + 1 \equiv 0 (m o d p)

has two and only two solutions if it is solvable. This completes the proof. □

As an application of Theorem 4, we give an elementary proof of the following well-known results by Proposition 5 and 6 (See Proposition 1.43 in [5]).

Corollary 2.

(1) v_{2} (N) = \{\begin{matrix} 0 i f 4 | N, \\ \prod_{p | N} (1 + (\frac{- 1}{p})) o t h e r w i s e . \end{matrix}

(2) v_{3} (N) = \{\begin{matrix} 0 i f 4 | N, \\ \prod_{p | N} (1 + (\frac{- 3}{p})) o t h e r w i s e . \end{matrix}

Corollary 3.

Let

g (N)

be the genus of modular curve

X_{0} (N)

. Then for any

(N_{1}, N_{2}) = 1

g (N_{1} N_{2}) = 1 + \frac{μ (N_{1}) μ (N_{2})}{12} - \frac{v_{2} (N_{1}) v_{2} (N_{2})}{4} - \frac{v_{3} (N_{1}) v_{3} (N_{2})}{3} - \frac{v_{\infty} (N_{1}) v_{\infty} (N_{2})}{2} .

Proof.

It is immediately from Theorem 1, 2, 3 and the formula for the genus of

X_{0} (N)

g (N) = 1 + \frac{μ (N)}{12} - \frac{v_{2} (N)}{4} - \frac{v_{3} (N)}{3} - \frac{v_{\infty} (N)}{2} .

□

Algorithm 3.

(1): Construct $E_{3}$ ( $p^{l}$ ) by the general method;
(2): Let $N = N_{1} N_{2}$ for $(N_{1}, N_{2}) = 1$ . Given $E_{3} (N_{1})$ and $E_{3} (N_{2})$ . $E_{3} (N)$ is constructed as follows. For all $(1, d_{1}) \in E_{3} (N_{1})$ , $(1, d_{2}) \in E_{3} (N_{2})$ , Determinate d such that

$d \equiv d_{1} (mod N_{1}), d \equiv d_{2} (mod N_{2}) a n d 1 ⩽ d ⩽ N .$

Then $(1, d) \in E_{3} (N)$ and all elements in $E_{3} (N)$ are constructed if all pairs in $E_{3} (N_{1}) \times E_{3} (N_{2})$ are processed.

5. Concluding Remarks

In [6], Stein mentioned that another approach to list

P^{1} (Z / N Z)

is to use that

P^{1} (Z / N Z) ≅ \prod_{p | N} P^{1} (Z / p^{v_{p}} Z),

where

v_{p} = or d_{p} (N)

, and that it is relatively easy to enumerate the elements of

P^{1} (Z / p^{n} Z)

for a prime power

p^{n}

. However, this approach had never been implemented by anyone as for as I know. Thus the results in this paper could been regarded as an explicit implementation of Stein’s ideas. The implementations of all the algorithms described in this paper have been completely written in Wolfram Language. We plan rewrite these programs in the free open source computer algebra system SAGE and integrate them into Stein’s program [3] and Walker’s program [8].

References

Cremona, J. E. Algorithms For Modular Elliptic Curves; Cambridge University Press: Cambridge, 1997; pp. 99–103. [Google Scholar]
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Sage: open source mathematical software(Version 9.8). 2023. Available online: https://doc.sagemath.org/html/en/reference/ modsym/sage/modular/modsym/p1list.html (accessed on 12 May 2023).
Schoeneberg, B. Elliptic modular functions: an introduction; Vol.203; Springer Science & Business Media: Germany, 2012; pp. 99–103. [Google Scholar]
Shimura, G. Introduction to the arithmetic theory of automorphic functions; No.11; Math. Soc.Japan: Japan, 1971; pp. 99–103. [Google Scholar]
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Walker, J. Sage: open source mathematical software(Version 9.8). 2023. Available online: https://doc.sagemath.org/html/en/reference/modsym/sage/modular/modsym/p1list.html (accessed on 12 May 2023).

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The Recursive Structures of Manin Symbols over $\mathbb{Q}$, Cusps and Elliptic Points on $X_0(N)$

Abstract

Keywords:

Subject:

1. Introduction

2. The recursive structure of Manin symbols over Q

3. The recursive structure of cusps

4. The recursive structure of elliptic points of X 0 ( N )

5. Concluding Remarks

References

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2. The recursive structure of Manin symbols over $Q$

4. The recursive structure of elliptic points of $X_{0} (N)$