Generators and number fields for torsion points of a special elliptic curve Open Access

Let $E$ be an elliptic curve with Weierstrass form $y2=x3−px$⁠, where $p$ is a prime number. Let $m$ be a positive number, we denote by $E[m]$ the $m$ -torsion subgroup of $E$⁠, by $ℚ(E[m])$ the number field generated by the coordinates of the $m$ -torsion points of $E$⁠, and by $ℚ(Ex[m])$ the number field generated by the abscissas of $m$ -torsion points of $E$⁠. Mazur proves the $m$ -torsion subgroup is isomorphic to one of 15 finite groups [5]. Let $p1=(x1,y1)$ and $p2=(x2,y2)$ be two points in $E$ forming a basis of $E[m]$⁠, then $ℚ(E[m])=ℚ(x1,x2,y1,y2)$⁠. By Artin’s primitive element theorem the extension $ℚ(x1,x2,y1,y2)/ℚ$ is monogeneous and we can find unique generator for $ℚ(x1,x2,y1,y2)/ℚ$ by combining the above coordinates. As usual, we denote by $μm$ the group of $m$th roots of unity and by $ξm$ one of its generators. By Weil pairing, we have $ξm∈ℚ(E[m])$⁠, so $ℚ(ξm)⊆ℚ(E[m])$ for all $m$ [5]. In [3] Paladino gives a family of elliptic curves such that $ℚ(E[3])=ℚ(ξ3)$ and in [4] finds the number fields generated by the 4th torsion points, degrees and Galois groups of an elliptic curve $y2=(x−α)(x−β)(x−γ)$ where $α,β,γ∈ℚ$⁠, and $α≠β≠γ$⁠. In [1] Bandini and Paladino determine the number fields generated by the 3-torsion points, degrees and Galois groups of an elliptic curve $y2=x3+c$ where $c∈ℚ∗$⁠. In [2] the result of Brau and Jones says that the rational points on the modular curve of level $6$ yield elliptic curve $E$ satisfying the given containment. In the first part of this paper we prove $ξm∈ℚ(Ex[m])$ and $ℚ(E[m])=ℚ(x1,x2,ξm,y1)$ for all $m$⁠. In the second part of this paper we find the number fields of torsion points $E[m]$ for cases $m=3,4$⁠, extensions and degrees. These theorems have applications in local–global divisibility problem [4] and modular curves [2].

Let $p1=(x1,y1)$ and $p2=(x2,y2)$ form a basis of $E[m]$⁠. We have $ℚ(E[m])=ℚ(x1,x2,y1,y2)$⁠. We will denote by $L$ the field $ℚ(x1,x2)$ and by $K$ the field $ℚ(E[m])$⁠. Suppose $(x3,y3)$ be the coordinates of the point $p3=p1+p2$ and $(x4,y4)$ be the coordinates of the point $p4=p1−p2$⁠. In next theorem we will prove $ξm∈ℚ(Ex[m])$ for all $m$⁠.

Lemma 2.1. Let ${P,Q}$ be a basis for $E[m]$. Then $em(P,Q)$ is a primitive $m$th root of unity.

Proof. We know that there are $S,T∈E[m]$ such that $em(S,T)=ξm$⁠, a primitive $m$th root of unity. Write $S=aP+bQ$ and $T=cP+dQ$⁠. Then the antisymmetry properties of the Weil pairing imply that

Since $em(P,Q)$ is an $m$th root of unity and a power of it is a primitive $m$th root of unity, it follows that $em(P,Q)$ is a primitive $m$th root of unity.□

Theorem 2.2. Let ${p1,p2}$ be a basis for $E[m]$, let $p3=p1+p2$ and $p4=p1−p2$, and write $pi=(xi,yi)$. Then

Proof. The second inclusion is by the definition of $ℚ(Ex[m])$⁠. For the first inclusion. Let $σ$ be an automorphism of $ℚ(E[m])$ that fixes $ℚ(x1,x2,x3,x4)$⁠. Then $σ(yi)=±yi$ since $σ(yi2)=yi2$⁠. The equation

shows that $σ(y1y2)=y1y2$⁠. This means that either $σ(yi)=yi$ for $i=1,2$⁠, or $σ(yi)=−yi$ for $i=1,2$⁠. These mean that either $σ(pi)=pi$ for $i=1,2$⁠, or $σ(pi)=−pi$ for $i=1,2$⁠. In the first case,

In the second case,

Since $em(p1,p2)$ is a primitive $m$th root of unity, we find that $ℚ(ξm)⊆ℚ(x1,x2,x3,x4)$⁠.□

We know that $ℚ(x1,x2,y1,y2)=ℚ(x1,x2,y1,y1y2)$⁠. In next theorem we will prove that $ℚ(E[m])$ is equal to the field $ℚ(x1,x2)$ by adding $ξm$ and $y1$⁠.

Theorem 2.3. 

Proof. We have $ℚ(x1,x2,ξm,y1,y2)=ℚ(E[m])$⁠. If we do not have the equality in the theorem, then $y2∉ℚ(x1,x2,ξm,y1)$⁠. Since $y22$ is in this field, there is an automorphism $σ$ such that $σ(y2)=–y2$ and $σ$ is the identity on $ℚ(x1,x2,ξm,y1)$⁠. Then

This implies that $em(p1,p2)2=1$⁠. Since $em(p1,p2)$ is a primitive $m$th root of unity, we must have $m=2$⁠. But then $y1=y2=0$⁠, in which case the theorem is true.□

It is well known that the abscissas of the 3-torsion points of an elliptic curve $y2=x3–px$ are the roots of the polynomial

then the roots $x1^,x2^,x3^,x4^$ of $ϕ3$ are:

In next theorems we will determine the field generated by 3 and 4 torsion points.

Theorem 3.1. Let $E$ be an elliptic curve with Weierstrass form $E:y2=x3−px$, where $p$ is a prime number. Then

Proof. We have $ℚ(x1^,x2^,x3^,x4^)=ℚ(x1^,x3^)$⁠. On the other hand we have

so $ℚ(x1^,x3^)=ℚ(x1^,x1^x3^)=ℚ(x1^,ξ3)=ℚ(p–2p3,ξ3)$⁠.

We have

Put $α=p–2p3$⁠, then

is irreducible over $ℚ(ξ3)$⁠, because the roots of $f(x)$ are $x1^,x2^,x3^,x4^$⁠. They are irrational, so either $f(x)$ is irreducible or it has a quadratic factor that has $x1^$ and some other $xi^$ as roots. Since $x1^x2^∉ℚ(ξ3)$⁠, the other root is not $x2^$⁠. Suppose the other root is $x3^$ or $x4^$⁠. Then (using $x3^$⁠)

is a square in $ℚ(ξ3)$⁠. But its norm to $ℚ$ is $16p23$⁠, which is not a square, so it cannot be a square. Therefore, there is no quadratic factor and $f(x)$ is irreducible. So $[ℚ(p−2p3,ξ3):ℚ(ξ3)]=4$⁠. It is easy to verify that $[ℚ(ξ3):ℚ]=2$⁠. Hence

By Theorem 2.2 we proved that $ℚ(E[3])=ℚ(x1^,x2^,ξ3,y1^)=ℚ(x1^,ξ3,y1^)$⁠, where $x1^=–x2^$⁠. As $y1^2=x1^3–px1^$⁠, then

and $[ℚ(x1^,ξ3,y1^):ℚ(x1^,ξ3)]=2$⁠. We found in previous case that $[ℚ(x1^,ξ3):ℚ]=8$⁠.

Hence

It is well known that the abscissas of the 4-torsion points of an elliptic curve $y2=x3–px$ are the roots of the polynomial

then the roots $x1^,x2^,x3^,x4^,x5^,x6^$ of $ϕ4$ are

Theorem 3.2. Let $E$ be an elliptic curve with Weierstrass form $y2=x3–px$, where $p$ is a prime number. Then

Proof. The points of exact order $4$ of $y2=x3–px$ are $±p1,±p2,±p3,±p4,±p5,±p6$⁠, where

We have:

with $[ℚ(Ex[4]):ℚ]=8$ if $p≠2$ and $[ℚ(Ex[4]):ℚ]=4$ if $p=2$⁠.□

Let ${p1,p2}$ be a basis for $E[4]$⁠, then

with $[ℚ(E[4]):ℚ]=16$ if $p≠2$ and $[ℚ(E[4]):ℚ]=[ℚ(i,84)]=8$ if $p=2$⁠.□

Declaration of Competing Interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The publisher wishes to inform readers that the article “Generators and number fields for torsion points of a special elliptic curve” was originally published by the previous publisher of the Arab Journal of Mathematical Sciences and the pagination of this article has been subsequently changed. There has been no change to the content of the article. This change was necessary for the journal to transition from the previous publisher to the new one. The publisher sincerely apologises for any inconvenience caused. To access and cite this article, please use Sankari, H., Bojakli, M. (2019), “Generators and number fields for torsion points of a special elliptic curve”, Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2, pp. 227-231. The original publication date for this paper was 29/10/2019