Elliptic Curves with Large Intersection of Projective Torsion Points

TL;DR

This paper constructs pairs of elliptic curves over number fields that share a large number of projective torsion points, highlighting new intersections in their torsion structures.

Contribution

It introduces a method to explicitly construct elliptic curve pairs with significantly large intersections of their projective torsion points.

Findings

Pairs of elliptic curves with large torsion point intersections constructed

Demonstrates the existence of elliptic curves with unexpectedly large shared torsion structures

Provides explicit examples over number fields

Abstract

We construct pairs of elliptic curves over number fields with large intersection of projective torsion points.

Equations

$${\pi }_1(E_1[\infty ])\cap {\pi }_2(E_2[\infty ]),$$

$$\#{\pi }_1(E_1[\infty ])\cap {\pi }_2(E_2[\infty ])\ge 14.$$

$$\underset{\{(E_1,E_2):{\pi }_1(E_1[2])\ne {\pi }_2(E_2[2])\}}{\sup }\#{\pi }_1(E_1[\infty ])\cap {\pi }_2(E_2[\infty ])\ge 22.$$

$$E_{\delta }:y^2=x^4-\left({\delta }^2+\frac{1}{{\delta }^2}\right)x^2+1,$$

$$\hat{\pi }(E_{\delta }[2])=\{\pm {\delta }^{\pm 1}\},\hat{\pi }(E_{\delta }^{\ast }[4])=\{0,\infty ,\pm 1,\pm i\},$$

$$E_{\delta }^{ns}:Y^2=X(X-1)\left(X-\frac{1}{4}{\left(\delta +\frac{1}{\delta }\right)}^2\right),$$

$$X=\frac{({\delta }^2+1)(\delta x-1)}{2\delta (x-\delta )},Y=\frac{({\delta }^4-1)y}{4\delta (x-\delta {)}^2}.$$

$$F_3(x,\delta )$$

$$=$$

$$F_7(x,\delta )$$

$$=$$

$$u\in \hat{\pi }(E_{{\delta }_1}^{\ast }[3])\cap \hat{\pi }(E_{{\delta }_2}^{\ast }[3])\text{ and }v\in \hat{\pi }(E_{{\delta }_1}^{\ast }[7])\cap \hat{\pi }(E_{{\delta }_2}^{\ast }[7]).$$

$$F_3(u,{\delta }_1)=F_3(u,{\delta }_2)=0\text{ and }{F}_7(v,{\delta }_1)=F_7(v,{\delta }_2)=0.$$

$$2^{240}{u}^{900}{\left(u^4-1\right)}^{132}\left(128u^{48}+24352u^{44}+\cdots +24352u^4+128\right){\left(u^{72}+16u^{68}+\cdots +16u^4+1\right)}^3,$$

$$2^{320}{v}^{692}{\left(v^4-1\right)}^{132}\left(8v^{48}+4776v^{44}+\cdots +4776v^4+8\right)\left(v^{216}+690v^{212}+\cdots +690v^4+1\right),$$

$$\{\begin{array}{l}\hat{\pi }(E_{{\delta }_1}[2])\ne \hat{\pi }(E_{{\delta }_2}[2]),\hat{\pi }(E_{{\delta }_1}^{\ast }[4])=\hat{\pi }(E_{{\delta }_2}^{\ast }[4])=\{0,\infty ,\pm 1,\pm i\},\\ u\in \hat{\pi }(E_{{\delta }_1}^{\ast }[3])\cap \hat{\pi }(E_{{\delta }_2}^{\ast }[3]),-u,1/u,-1/u\in \hat{\pi }(E_{{\delta }_1}^{\ast }[6])\cap \hat{\pi }(E_{{\delta }_2}^{\ast }[6]),\\ v_k\in \hat{\pi }(E_{{\delta }_1}^{\ast }[7])\cap \hat{\pi }(E_{{\delta }_2}^{\ast }[7]),-v_k,1/v_k,-1/v_k\in \hat{\pi }(E_{{\delta }_1}^{\ast }[14])\cap \hat{\pi }(E_{{\delta }_2}^{\ast }[14]),k=1,2,3,\end{array}$$

$$\{\begin{array}{l}\text{the resultant of }{C}_{11,0}\text{ and }{C}_{11,1}\text{ w.r.t. }v/u=(\text{trivial factors})(\text{a factor of degree }12{)}^3\\ \cdot (\text{a factor of degree }30)(\text{a factor of degree }150{)}^3\in \mathbb{Z}[u^4],\\ \\ \text{the resultant of }{C}_{11,0}\text{ and }{C}_{11,1}\text{ w.r.t. }{u}^4=(\text{trivial factors})\\ \cdot (\text{a factor of degree }12)(\text{a factor of degree }24)\\ \cdot (\text{a factor of degree }30)(\text{a factor of degree }450)\in \mathbb{Z}[v/u];\end{array}$$

$$\{\begin{array}{l}\text{the resultant of }{C}_{13,0}\text{ and }{C}_{13,1}\text{ w.r.t. }v/u=(\text{trivial factors})(\text{a factor of degree }18{)}^3\\ \cdot (\text{a factor of degree }42)(\text{a factor of degree }324{)}^3\in \mathbb{Z}[u^4],\\ \\ \text{the resultant of }{C}_{13,0}\text{ and }{C}_{13,1}\text{ w.r.t. }{u}^4=(\text{trivial factors})\\ \cdot (\text{a factor of degree }18)(\text{a factor of degree }36)\\ \cdot (\text{a factor of degree }42)(\text{a factor of degree }972)\in \mathbb{Z}[v/u];\end{array}$$

$$\{\begin{array}{l}\text{the resultant of }{C}_{17,0}\text{ and }{C}_{17,1}\text{ w.r.t. }v/u=(\text{trivial factors})(\text{a factor of degree }54{)}^3\\ \cdot (\text{a factor of degree }72)(\text{a factor of degree }1008{)}^3\in \mathbb{Z}[u^4],\\ \\ \text{the resultant of }{C}_{17,0}\text{ and }{C}_{17,1}\text{ w.r.t. }{u}^4=(\text{trivial factors})\\ \cdot (\text{a factor of degree }54)(\text{a factor of degree }54)(\text{a factor of degree }54)\\ \cdot (\text{a factor of degree }72)(\text{a factor of degree }3024)\in \mathbb{Z}[v/u].\end{array}$$

$$J_k(n):=n^k\prod _{p|n}(1-p^{-k})$$

$$\hat{\pi }(E_{\delta }[2])=\{\pm {\delta }^{\pm 1}\}\text{ and }\hat{\pi }(E_{\delta }^{\ast }[4])=\{0,\infty ,\pm 1,\pm i\},$$

$$E:y^2=(x-a)(x+\sqrt{a})(x-\sqrt{a})=x^3-a{x}^2-ax+a^2.$$

$$F_3(x)$$

$$rc(x)$$

$$u\in \hat{\pi }(E_{{\delta }_1}^{\ast }[3])\cap \hat{\pi }(E_{{\delta }_2}^{\ast }[3]),$$

$$j(E_{{\delta }_2})=\frac{16{\left({\delta }_2^4+{\delta }_2^{-4}+14\right)}^3}{{\left({\delta }_2^4+{\delta }_2^{-4}-2\right)}^2}\in L_{{\delta }_1}\backslash \{\infty \}.$$

Full text

Elliptic Curves with Large Intersection

of Projective Torsion Points

Fedor Bogomolov $\cdot$ Hang Fu

Abstract. We construct pairs of elliptic curves over number fields with large intersection of projective torsion points.

Keywords. Elliptic curves $\cdot$ Torsion points $\cdot$ Division polynomials $\cdot$ Unlikely intersections

Mathematics Subject Classification. 14H52 $\cdot$ 14Q05

1 Introduction

Let $E$ be an elliptic curve over a number field,

• •

$E[n]\subseteq E(\bar{\mathbb Q})$ its $n$-torsion subgroup,

• •

$E^{*}[n]\subseteq E[n]$ the collection of torsion points of order exactly $n$,

• •

$E[\infty]=\cup_{n=1}^{\infty}E[n]$ the collection of all torsion points, and

• •

$\pi:E\to\mathbb{P}^{1}$ the standard double cover, identifying $\pm P\in E$.

In [1], the authors considered intersections

[TABLE]

for distinct elliptic curves $E_{1}$ and $E_{2}$. They observed the equivalence of conditions

• (A)

$\pi_{1}(E_{1}[2])=\pi_{2}(E_{2}[2])$,

• (B)

$\pi_{1}(E_{1}[\infty])=\pi_{2}(E_{2}[\infty])$, and

• (C)

$\#\pi_{1}(E_{1}[\infty])\cap\pi_{2}(E_{2}[\infty])=\infty$

and found pairs of elliptic curves $(E_{1},E_{2})$ failing these conditions but satisfying

[TABLE]

We view results of this type as instances of unlikely intersections, see, e.g., [7] for an extensive study of related problems. In this note, we provide some improvements of [1] and [3]:

Theorem 1**.**

[TABLE]

The proof will be given in Section 2.

In [3], the first author and Tschinkel studied intrinsic properties of subsets of $\mathbb P^{1}(\bar{\mathbb Q})$ generated by images of torsion points. Starting with distinct points $a,b,c,d\in\mathbb{P}^{1}(\bar{\mathbb{Q}})$, they defined $L[a,b,c,d]\supseteq\{a,b,c,d\}$ as the smallest subset of $\mathbb{P}^{1}(\bar{\mathbb{Q}})$ such that if $\pi(E[2])\subseteq L[a,b,c,d]$, then $\pi(E[\infty])\subseteq L[a,b,c,d]$. They proved:

Theorem 2**.**

If $\delta\not\in\{0,\infty,\pm 1,\pm i\}$, then $L[\pm\delta^{\pm 1}]\backslash\{\infty\}$ is a field. Moreover, it is closed under taking square roots.

A shortened version of their proof can be found in [2]. In Section 3, we establish new properties of this construction.

2 Large Intersection

In [1], we considered torsion points of order $3$ and $5$. Here, we will instead focus on torsion points of order $3$ and $p$, where $p$ is a different prime number. The calculations below were performed with Mathematica 11.0 [6].

Proof of Theorem 1.

Consider the family

[TABLE]

which is a curve of genus $1$ with a unique singularity at $(0:1:0)$, provided $\delta^{4}\neq 0,1$. Fix $O_{\delta}=(\delta,0)$ and $\hat{\pi}:\mathbb{P}^{2}\to\mathbb{P}^{1},(x,y)\mapsto x$. Recall that

[TABLE]

and that if $a\in\hat{\pi}(E_{\delta}[\infty])$, then $-a,1/a,-1/a\in\hat{\pi}(E_{\delta}[\infty])$. The nonsingular model of $E_{\delta}$ is

[TABLE]

where

[TABLE]

From the division polynomials [4, Chapter II; 5, Page 105, Exercise 3.7] of $E_{\delta}^{ns}$, we know that the third and seventh (modified) division polynomials of $E_{\delta}$ are

[TABLE]

Now we want to find $\delta_{1}$ and $\delta_{2}$ such that there exist

[TABLE]

In other words,

[TABLE]

Since $F_{3}(x,\delta)$ is a quadratic polynomial in $\delta$, any fixed $u$ such that $u^{4}\neq 0,1$ and $u^{8}+14u^{4}+1\neq 0$ gives exactly two roots satisfying $\delta_{1}^{4},\delta_{2}^{4}\neq 0,1$ and $\delta_{1}\neq\pm\delta_{2}^{\pm 1}$. Since $\delta_{1}$ and $\delta_{2}$ are also two roots of $F_{7}(v,\delta)=0$, we need $F_{3}(u,\delta)$ divides $F_{7}(v,\delta)$ as polynomials in $\delta$. By long division, this is equivalent to require both $C_{7,0}(u,v)=0$ and $C_{7,1}(u,v)=0$, where $C_{7,0}$ is a polynomial of degree $58$ with $177$ terms, and $C_{7,1}$ is a polynomial of degree $62$ with $202$ terms. The resultant of $C_{7,0}$ and $C_{7,1}$ w.r.t. $v$ is

[TABLE]

whose roots are the $u$-coordinates of their common points. The resultant of $C_{7,0}$ and $C_{7,1}$ w.r.t. $u$ is

[TABLE]

whose roots are the $v$-coordinates of their common points. Here the cube power in the first resultant plays the crucial role. There are $48+72=120$ nontrivial $u$-coordinates, and $48+216=264$ nontrivial $v$-coordinates, hence there exists some $u$ that is shared by at least three common points $(u,v_{k}),k=1,2,3$. Thus we have

[TABLE]

so the supremum is at least $22$. ∎

Remark 3*.*

The same phenomenon continues for larger primes $p=11$, $13$, and $17$. To accelerate our computer-aided calculation, we note that $C_{p,0}(u,v)$ and $C_{p,1}(u,v)$ are essentially polynomials in $u^{4}$ and $v/u$. When $p=11$,

[TABLE]

when $p=13$,

[TABLE]

when $p=17$,

[TABLE]

Here “a factor” always means an irreducible factor. From these facts, we note that for $p=11$, $13$, and $17$, the resultant of $C_{p,0}$ and $C_{p,1}$ with respect to $v/u$ only involves factors of multiplicity $1$ and $3$, the former has degree $(p^{2}-1)/4$, while the latter suggests that as $p\to\infty$, there might be infinitely many pairs $E_{\delta_{1}}$ and $E_{\delta_{2}}$ with at least $22$ intersection points, and this record might not be broken by the current approach. Due to the computational limitation, we have no information for $p\geq 19$.

Remark 4*.*

Let

[TABLE]

denote Jordan’s totient function. In Appendix of [1], the authors attempted to show that the values $J_{2}(n)$, $J_{4}(n)$, and $J_{6}(n)$ would suffice to determine $n$. However, as indicated by Prof. Kevin Ford in a personal email communication, this would contradict a plausible conjecture in analytic number theory.

3 The Fields Generated by the Projective Torsion

Points

For simplicity, we write $L_{\delta}=L[\pm\delta^{\pm 1}]$ and $L_{\mathbb{Q}}=L[0,\infty,1,-1]$. We have seen that

[TABLE]

so $L_{\mathbb{Q}}\subseteq L_{\delta}$.

Theorem 5**.**

The field $L_{\delta}\backslash\{\infty\}$ is closed under taking cube roots.

Proof.

We will show that if $a\in L_{\delta}\backslash\{\infty\}$, then $\sqrt[3]{a-1}\subseteq L_{\delta}\backslash\{\infty\}$. This is trivial for $a=1$. For $a=0$, since $L_{\delta}\backslash\{\infty\}$ contains all of the torsion points of $E_{\delta}$, it also contains all of the roots of unity [5, Page 96, Corollary 8.1.1]. For $a\neq 0,1$, since $\pm\sqrt{a}\in L_{\delta}\backslash\{\infty\}$ by Theorem 2, let us consider the elliptic curve

[TABLE]

The third (modified) division polynomial of $E$ is

[TABLE]

The resolvent cubic of $F_{3}(x)$ is

[TABLE]

The roots of $rc(x)=0$ are $x=-\dfrac{2}{3}a-\dfrac{4}{3}a\cdot\sqrt[3]{(a-1)^{2}}$, so $\sqrt[3]{a-1}\subseteq L_{\delta}\backslash\{\infty\}$. ∎

Corollary 6**.**

All quadratic, cubic, and quartic equations are solvable in $L_{\delta}\backslash\{\infty\}$.

Example 7**.**

For any $\delta\in L_{\mathbb{Q}}\backslash\{0,\infty,\pm 1,\pm i\}$, we have $L_{\delta}=L_{\mathbb{Q}}$, so $L_{\mathbb{Q}}$ is the smallest $L_{\delta}$.

Example 8**.**

In the constructions of Theorem 1 and Remark 3, we require that

[TABLE]

which implies $\delta_{1}\delta_{2}=-1/u^{2}$. Thus $L_{\delta_{1}}=L_{\delta_{2}}$.

Another example shows that $L_{\delta}$ is an isogeny invariant.

Corollary 9**.**

If $E_{\delta_{1}}$ and $E_{\delta_{2}}$ are isogenous over $\bar{\mathbb{Q}}$, then $L_{\delta_{1}}=L_{\delta_{2}}$.

Proof.

By [5, Page 74, Remark 4.13.2], the nonsingular model of $E_{\delta_{2}}$ can be defined over $\mathbb{Q}(E_{\delta_{1}}[\infty])$, which is a subfield of $L_{\delta_{1}}\backslash\{\infty\}$, so the $j$-invariant of $E_{\delta_{2}}$,

[TABLE]

Thus $\delta_{2}^{4}+\delta_{2}^{-4}$ can be obtained by solving a cubic equation with coefficients in $\mathbb{Q}(j(E_{\delta_{2}}))$. By Corollary 6, $\delta_{2}^{4}+\delta_{2}^{-4}\in L_{\delta_{1}}\backslash\{\infty\}$, and then $\delta_{2}\in L_{\delta_{1}}\backslash\{\infty\}$. ∎

Corollary 10**.**

The field $L_{\delta}\backslash\{\infty\}$ is Galois over $\mathbb{Q}(j(E_{\delta}))$. In particular, the field $L_{\mathbb{Q}}\backslash\{\infty\}$ is Galois over $\mathbb{Q}$.

Proof.

Take $\sigma\in G_{\mathbb{Q}(j(E_{\delta}))}$ and $a\in L_{\delta}\backslash\{\infty\}$, then $\sigma(a)\in L_{\sigma(\delta)}\backslash\{\infty\}$. Since $j(E_{\sigma(\delta)})=\sigma(j(E_{\delta}))=j(E_{\delta})$, Corollary 9 implies $L_{\sigma(\delta)}=L_{\delta}$. ∎

Corollary 11**.**

If $E$ is an elliptic curve in the Weierstrass form and defined over a number field $\mathbb{Q}(j(E))$, then $L[\hat{\pi}(E[2])]\backslash\{\infty\}$ is a Galois field extension of $\mathbb{Q}(j(E))$.

Proof.

Take $\delta$ such that $j(E)=j(E_{\delta})\in L_{\delta}$. By Corollary 6, $\hat{\pi}(E[2])\subseteq L_{\delta}$, so the linear fractional transformation between $L[\hat{\pi}(E[2])]$ and $L_{\delta}$ is defined over $L_{\delta}\backslash\{\infty\}$. Therefore, $L[\hat{\pi}(E[2])]=L_{\delta}$. ∎

Acknowledgments. The authors are grateful to Prof. Kevin Ford for indicating Remark 4. The first author was partially supported by the Russian Academic Excellence Project ‘5-100’, Simons Fellowship, and EPSRC programme grant EP/M024830. The second author was supported by the MacCracken Program offered by New York University.