ON FAMILIES OF 9-CONGRUENT ELLIPTIC CURVES

TOM FISHER

Abstract. We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over ℚ, i.e. pairs of non-isogenous elliptic curves over ℚ whose 9-torsion subgroups are isomorphic as Galois modules.

Introduction

Elliptic curves E_{1} and E_{2} are n-congruent if their n-torsion subgroups E_{1}[n] and E_{2}[n] are isomorphic as Galois modules. They are directly n-congruent if the isomorphism ϕ : E_{1}[n] → E_{2}[n] respects the Weil pairing e_{n} and reverse ncongruent if e_{n}(ϕP, ϕQ) = e_{n}(P, Q)^{−1} for all P, Q ∈ E_{1}[n]. The elliptic curves directly and reverse n-congruent to a given elliptic curve E are parametrised by the modular curves Y_{E}(n) = X_{E}(n) \ {cusps} and Y_{E}^{−}(n) = X_{E}^{−}(n) \ {cusps}. Finding equations for these modular curves can help with finding non-trivial pairs of n-congruent elliptic curves. Some of the potential applications are described in [1], [3], [8], [13], [19].

Equations for X_{E}(n), and the family of curves it parametrises, have been computed for n = 2, 3, 4, 5, 6, 7, 8, 11 by Rubin and Silverberg [20], [21], [22], [23], Papadopoulos [18], Halberstadt and Kraus [14], Chen [6], and Fisher [12]. The analogous problem for n-congruences that do not respect the Weil pairing has been solved for the same values of n. The additional references for this include papers by Fisher [10], [11], Bruin and Doerksen [4, Section 7], and Poonen, Schaefer and Stoll [19, Section 7.2].

In this paper we treat the case n = 9. That is, we give equations for X_{E}(9) and X_{E}^{−}(9), and for the families of curves they parametrise. We use these formulae to exhibit non-trivial triples of 9-congruent elliptic curves over ℚ, and non-trivial pairs of 9-congruent elliptic curves over ℚ(T). We also give equations for the modular diagonal quotient surfaces whose points parametrise pairs of 9-congruent elliptic curves.

We work over a field K of characteristic 0. We write \overline{K} for the algebraic closure, and ζ_{n} for a primitive nth root of unity.

Date: 2nd June 2015.

1. Statement of results

If n = 3 or 9 then every element of (ℤ/nℤ)^{×} is either plus or minus a square. Therefore any pair of n-congruent elliptic curves are either directly or reverse ncongruent. In the case n = 3, we have X_{E}^{±}(3) ≅ ℙ^{1} and the families of curves parametrised are as follows.

Theorem 1.1. Let E/K be the elliptic curve y^{2} = x^{3} − 27c_{4}x − 54c_{6}. Then the family of elliptic curves parametrised by Y_{E}^{±}(3) is

(1)  y^{2} = x^{3} − 27A^{±}(r, s)x − 54B^{±}(r, s)

where

A^{+}(r, s) = c_{4}r^{4} + 4c_{6}r^{3}s + 6c_{4}^{2}r^{2}s^{2} + 4c_{4}c_{6}rs^{3} − (3c_{4}^{3} − 4c_{6}^{2})s^{4},
B^{+}(r, s) = c_{6}r^{6} + 6c_{4}^{2}r^{5}s + 15c_{4}c_{6}r^{4}s^{2} + 20c_{6}^{2}r^{3}s^{3}
+ 15c_{4}^{2}c_{6}r^{2}s^{4} + 6(3c_{4}^{4} − 2c_{4}c_{6}^{2})rs^{5} + (9c_{4}^{3}c_{6} − 8c_{6}^{3})s^{6},

and

A^{−}(r, s) = −4(r^{4} − 6c_{4}r^{2}s^{2} − 8c_{6}rs^{3} − 3c_{4}^{2}s^{4})/(c_{4}^{3} − c_{6}^{2}),
B^{−}(r, s) = −8B^{+}(r, s)/(c_{4}^{3} − c_{6}^{2})^{2}.

Proof. In the direct case this family of curves was first computed by Rubin and Silverberg [20]. The above formulae are taken from [10, Sections 8, 9 and 13], with (r, s) = (λ, μ), respectively (r, s) = (c_{6}ξ + c_{4}^{2}η, −c_{4}ξ − c_{6}η). These formulae are available in Magma [2] via the function HessePolynomials. □

Our new results are in the case n = 9, where the curves X_{E}^{±}(9) have genus 10. We write these curves as the complete intersection of two cubics in ℙ^{3}.

Theorem 1.2. Let E/K be the elliptic curve y^{2} = x^{3} + ax + b. Then X_{E}^{±}(9) = {F_{1}^{±} = F_{2}^{±} = 0} ⊂ ℙ^{3} where

F_{1}^{+}(x, y, z, t) = x^{2}t + 6xyz + 6bxt^{2} + 6y^{3} − 9ay^{2}t + 6a^{2}yt^{2} − 3bz^{3}
+ 3a^{2}z^{2}t + 9abzt^{2} − (a^{3} − 12b^{2})t^{3},
F_{2}^{+}(x, y, z, t) = x^{2}z + 6xy^{2} − 6axyt + 2a^{2}xt^{2} − 9ay^{2}z − 18byz^{2} + 12a^{2}yzt
+ a^{2}z^{3} + 9abz^{2}t − 3a^{3}zt^{2} + a^{2}bt^{3},

and

F_{1}^{−}(x, y, z, t) = 9x^{2}y + 3x^{2}z − 6axyt + 6bxt^{2} − 6ay^{3} + 27by^{2}t + 3ayz^{2}
+ 18byzt + 3a^{2}yt^{2} + az^{3} + 3bz^{2}t + a^{2}zt^{2} − abt^{3},
F_{2}^{−}(x, y, z, t) = x^{3} + 6axyz + 18bxyt + 3axz^{2} + 6bxzt + a^{2}xt^{2} + 9by^{3} + 6a^{2}y^{2}t
− 9byz^{2} + 6a^{2}yzt − 3abyt^{2} − 4bz^{3} + 2a^{2}z^{2}t + 2b^{2}t^{3}.

The families of curves parametrised by X_{E}(9) and X_{E}^{−}(9) are given by Theorem 1.1 and the following. By abuse of notation we write P both for a point in ℙ^{3} and for a vector representing this point.

Theorem 1.3. Let X_{E}^{±}(3) and X_{E}^{±}(9) be as described in Theorems 1.1 and 1.2, with a = −27c_{4} and b = −54c_{6}. For P ∈ X_{E}^{±}(9) with tangent line P +λQ we write F_{i}^{±}(P + λQ) = γ_{i}λ^{2} + δ_{i}λ^{3} for i = 1, 2. Then the forgetful map X_{E}^{±}(9) → X_{E}^{±}(3) is given by (r : s) = (γ_{2} : 3γ_{1}).

Let Z(n) be the modular diagonal quotient surface whose points parametrise pairs of directly n-congruent elliptic curves, and likewise Z^{−}(n) in the reverse case. It is shown in [15, Theorem 4] that the surfaces Z^{±}(9) are each birational over ℂ to an elliptic surface. Using Theorem 1.2 we are able to determine these elliptic surfaces explicitly.

Theorem 1.4. The surface Z(9) is birational over ℚ to the elliptic surface

y^{2} + (6T^{2} + 3T + 2)xy + T^{2}(T + 1)(4T^{3} + 9T + 9)y
= x^{3} − (16T^{4} + 12T^{3} + 9T^{2} + 6T + 1)x^{2}.

The surface Z^{−}(9) is birational over ℚ to the elliptic surface

y^{2} + (12T^{3} + 3T^{2} − 6)xy + (T − 1)^{3}(T^{3} − 1)(4T^{3} − 3T − 7)y
= x^{3} − 3(T + 1)(T^{3} − 1)(9T^{2} + 2T + 1)x^{2}.

We use these results to prove [15, Conjecture 5] in the case n = 9.

Theorem 1.5. There are infinitely many non-trivial pairs of directly 9-congruent elliptic curves over ℚ, and these have infinitely many distinct pairs of j-invariants. By “non-trivial” we mean that the elliptic curves are not isogenous. The same result holds in the reverse case.

Proof. For each positive integer m there are finitely many curves on Z^{±}(9) corresponding to pairs of elliptic curves related by a cyclic isogeny of degree m. Each of these curves comes with a non-constant morphism to X_{0}(m) and so has positive genus for m sufficiently large. The elliptic surfaces in Theorem 1.4 each have a ℚ-rational section of infinite order given by (x, y) = (0, 0). Therefore the surfaces Z^{±}(9) each contain infinitely many curves birational to ℙ^{1} over ℚ. Since these curves have genus 0, we know by the above remarks that only finitely many correspond to pairs of isogenous elliptic curves. It follows that there are non-trivial pairs of 9-congruent elliptic curves over ℚ(T). In Section 6 we give a more explicit version of this first part of the proof by using Theorem 1.2 to exhibit some non-trivial pairs of 9-congruent elliptic curves over ℚ(T).

The proof is completed by specialising T to a rational number. In this way we obtain infinitely many pairs of elliptic curves that are not m-isogenous for any m ≤ d (for any fixed d). We are done by the theorem of Mazur [17], extended to composite m by Kenku [16], that any cyclic isogeny defined over ℚ has degree m ≤ 163. □

2. The modular curves X(3) and X(9)

Let M = μ_{n} × ℤ/nℤ equipped with the pairing

⟨(ζ, a), (ξ, b)⟩ = ζ^{b}ξ^{−a}.

The modular curve Y (n) = X(n)\{cusps} parametrises pairs (E, ϕ) where E is an elliptic curve and ϕ : E[n] → M is a symplectic isomorphism, i.e. an isomorphism that matches up the Weil pairing on E[n] with the pairing ⟨ , ⟩ on M. We identify SL_{2}(ℤ/nℤ) with the group of symplectic automorphisms of M. There is then a natural action of SL_{2}(ℤ/nℤ)/{±I} on X(n) with quotient the j-line.

In the case n = 3 it is well known that Y (3) = 𝔸^{1} \ {t^{3} = 1} parametrises the non-singular fibres in the Hesse pencil of plane cubics:

(2)  {x^{3} + y^{3} + z^{3} − 3txyz = 0} ⊂ ℙ^{2}.

We need an analogous result in the case n = 9.

Lemma 2.1. The modular curve X(9) has equations

(3)  X(9) = \{ \begin{matrix}a^{2}b + b^{2}c + c^{2}a & = 0 \\ ab^{2} + bc^{2} + ca^{2} & = d^{3} \end{matrix}\} ⊂ ℙ^{3}.

Moreover the forgetful map X(9) → X(3) is given by

(4)  t = −(a^{3} + b^{3} + c^{3} + 6abc)/(3d^{3}).

Proof. In [12, Section 2] we showed, following work of Klein, Vélu and others, that if n ≥ 5 is an odd integer then Y (n) ⊂ Z(n) where Z(n) ⊂ ℙ^{n−1} is defined as follows. We take co-ordinates (a_{0} : a_{1} : …: a_{n−1}) on ℙ^{n−1} and agree to read all subscripts mod n. Then Z(n) is defined by a_{0} = 0, a_{n−i} = −a_{i} and

rank(a_{i−j}a_{i+j})_{i,j=0}^{n−1} ≤ 2.

In the case n = 9 we put

(5)  (a_{0} : a_{1} : …: a_{8}) = (0 : a : −b : d : c : −c : −d : b : −a).

Then Z(9) ⊂ ℙ^{3} is defined by

rank (\begin{matrix}0 & −a^{2} & −b^{2} & −d^{2} & −c^{2} \\ a^{2} & 0 & −ad & bc & cd \\ b^{2} & ad & 0 & ac & −bd \\ d^{2} & −bc & −ac & 0 & −ab \\ c^{2} & −cd & bd & ab & 0 \end{matrix}) ≤ 2,

equivalently

(a^{2}b + b^{2}c + c^{2}a)d = 0  bc^{3} − ba^{3} − cd^{3} = 0
ab^{3} − ac^{3} − bd^{3} = 0  ca^{3} − cb^{3} − ad^{3} = 0.

Adding together the last three equations and factoring shows that Z(9) is the union of the curve defined in (3) and four isolated points

(0 : 0 : 0 : 1), (1 : 1 : 1 : 0), (1 : ζ_{3} : ζ_{3}^{2} : 0), (1 : ζ_{3}^{2} : ζ_{3} : 0).

Since X(9) is a curve, it must therefore be as defined in (3).

Now writing (x_{0} : x_{1} : …: x_{n−1}) for our co-ordinates on ℙ^{n−1}, and again agreeing to read all subscripts mod n, it is shown in [9, Section 3] that the family of elliptic curves parametrised by Y (n) is defined by

rank(a_{i−j}x_{i+j})_{i,j=0}^{n−1} ≤ 2.

In the case n = 9 the elliptic curve E corresponding to (a : b : c : d) ∈ Y (9) is the curve of degree 9 in ℙ^{8} defined by the equations

adx_{0}^{2} − b^{2}x_{1}x_{8} + a^{2}x_{2}x_{7} = 0
bcx_{0}^{2} + d^{2}x_{1}x_{8} − a^{2}x_{3}x_{6} = 0
cdx_{0}^{2} + c^{2}x_{1}x_{8} − a^{2}x_{4}x_{5} = 0

and their cyclic permutes. Thus E is defined by a total of 27 quadrics.

Let 0_{E} be the point on E given by (5). In principle we could complete the proof by putting the elliptic curve (E, 0_{E}) in Weierstrass form. However it is simpler to argue as follows. The action of E[9] on E by translation is generated by the maps x_{i} ↦ x_{i+1} and x_{i} ↦ ζ_{9}^{i}x_{i}. From this we see that the morphism

(x_{0} : x_{1} : …: x_{8}) ↦ (x : y : z) = (x_{0}x_{3}x_{6} : x_{1}x_{4}x_{7} : x_{2}x_{5}x_{8})

quotients out by the action of E[3]. We may therefore identify this morphism with the multiplication-by-3 map on E. On the other hand, we find by direct calculation that the image takes the form (2) with t as specified in (4). The forgetful map X(9) → X(3) is therefore as claimed. □

Remark 2.2. The action of SL_{2}(ℤ/9ℤ) on X(9) ⊂ ℙ^{3} is described by a projective representation \overline{ρ} : SL_{2}(ℤ/9ℤ) → PGL_{4}(\overline{K}). According to [12, Section 2] it is given on the generators S = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and T = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) for SL_{2}(ℤ/9ℤ) by

\overline{ρ}(S) = (\begin{matrix}ζ_{9} − ζ_{9}^{8} & ζ_{9}^{7} − ζ_{9}^{2} & ζ_{9}^{4} − ζ_{9}^{5} & ζ_{9}^{3} − ζ_{9}^{6} \\ ζ_{9}^{7} − ζ_{9}^{2} & ζ_{9}^{4} − ζ_{9}^{5} & ζ_{9} − ζ_{9}^{8} & ζ_{9}^{3} − ζ_{9}^{6} \\ ζ_{9}^{4} − ζ_{9}^{5} & ζ_{9} − ζ_{9}^{8} & ζ_{9}^{7} − ζ_{9}^{2} & ζ_{9}^{3} − ζ_{9}^{6} \\ ζ_{9}^{3} − ζ_{9}^{6} & ζ_{9}^{3} − ζ_{9}^{6} & ζ_{9}^{3} − ζ_{9}^{6} & 0 \end{matrix}) and \overline{ρ}(T) = (\begin{matrix}ζ_{9} & 0 & 0 & 0 \\ 0 & ζ_{9}^{4} & 0 & 0 \\ 0 & 0 & ζ_{9}^{7} & 0 \\ 0 & 0 & 0 & ζ_{9}^{6} \end{matrix}) .

In particular, the action of ker ( SL_{2}(ℤ/9ℤ) → SL_{2}(ℤ/3ℤ)) ≅ (ℤ/3ℤ)^{3} is generated by

(a : b : c : d) ↦ (a : b : c : ζ_{3}d)
(a : b : c : d) ↦ (b : c : a : d)
(a : b : c : d) ↦ (ζ_{3}a + b + c : a + ζ_{3}b + c : a + b + ζ_{3}c : (ζ_{3} − 1)d).

It may be checked that the map X(9) → X(3) in Lemma 2.1 quotients out by this action.

The pencil of cubics defining X(9) ⊂ ℙ^{3} is naturally a copy of X(3) ≅ ℙ^{1}. To explain this, let F_{1} = a^{2}b + b^{2}c + c^{2}a and F_{2} = ab^{2} + bc^{2} + ca^{2} − d^{3} be the cubics defining X(9). We write H(F) for the Hessian matrix of a form F, that is, the matrix of second partial derivatives. Then

(6)  det H(tF_{1} − F_{2}) = −48(t^{3} − 1)(a^{3} + b^{3} + c^{3} − 3abc)d.

Therefore the Hessian vanishes for just 4 cubics in the pencil, and these correspond to the cusps of X(3). The forgetful map X(9) → X(3) has the following geometric description.

Lemma 2.3. For P ∈ X(9) with tangent line P + λQ we write F_{i}(P + λQ) = γ_{i}λ^{2}+δ_{i}λ^{3} for i = 1, 2. Then the forgetful map X(9) → X(3) is given by t = γ_{2}/γ_{1}.

Proof. We temporarily write a_{1}, a_{2}, a_{3}, a_{4} for a, b, c, d and let Λ be the 4 × 4 alternating matrix with entries

Λ_{ij} = \frac{∂F_{1}}{∂a_{k}} \frac{∂F_{2}}{∂a_{l}} − \frac{∂F_{1}}{∂a_{l}} \frac{∂F_{2}}{∂a_{k}}

where (i, j, k, l) is an even permutation of (1, 2, 3, 4). Then specialising Λ at P = (a : b : c : d) ∈ X(9) gives a matrix whose rows (or columns) span the tangent line at P. Let D = (a^{3} + b^{3} + c^{3} − 3abc)d. We find by direct calculation that

Λ H(F_{i}) Λ ≡ γ_{i} D (\begin{matrix}a^{2} & ab & ac & ad \\ ab & b^{2} & bc & bd \\ ac & bc & c^{2} & cd \\ ad & bd & cd & d^{2} \end{matrix}) mod (F_{1}, F_{2})

for i = 1, 2, where γ_{1} = −18d^{3} and γ_{2} = 6(a^{3} + b^{3} + c^{3} + 6abc). By Lemma 2.1 we have t = γ_{2}/γ_{1}. □

3. Remarks on twisting

In this section we make some general remarks on computing twists of X(n). These will be used to find equations for X_{E}^{−}(9) once we have found equations for X_{E}(9) by another method. We suppose that X(n) has been embedded in (and spans) ℙ^{N−1} and that the action of SL_{2}(ℤ/nℤ) is given by

\overline{ρ} : SL_{2}(ℤ/nℤ) → PGL_{N}(\overline{K}).

We write ∝ for equality in PGL_{N}(\overline{K}), and a superscript −T for the inverse transpose of a matrix. Let ι = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). We further suppose that

\overline{ρ}(ιγι) ∝ \overline{ρ}(γ)^{−T}

for all γ ∈ SL_{2}(ℤ/nℤ). This condition is satisfied in the case of X(9) ⊂ ℙ^{3} since the matrices \overline{ρ}(S) and \overline{ρ}(T) in Remark 2.2 are symmetric.

Lemma 3.1. Let E/K be an elliptic curve and ϕ : E[n] → M a symplectic isomorphism defined over \overline{K}. Suppose h ∈ GL_{N}(\overline{K}) satisfies

σ(h)h^{−1} ∝ \overline{ρ}(σ(ϕ)ϕ^{−1})

for all σ ∈ Gal(\overline{K}/K). Then X_{E}(n) ⊂ ℙ^{N−1} and X_{E}^{−}(n) ⊂ ℙ^{N−1} are the twists of X(n) ⊂ ℙ^{N−1} given by X_{E}(n) ≅ X(n); x ↦ hx and X_{E}^{−}(n) ≅ X(n); x ↦ h^{−T} x. Moreover these isomorphisms are compatible with the maps to the j-line.

Proof. This is a special case of [12, Lemma 3.2]. □

The following lemma generalises [19, Proposition 7.5].

Lemma 3.2. Let E/K be an elliptic curve. If h ∈ GL_{N}(\overline{K}) describes X_{E}(n) ⊂ ℙ^{N−1} as a twist of X(n) ⊂ ℙ^{N−1} via X_{E}(n) ≅ X(n); x ↦ hx, and this isomorphism is compatible with the maps to the j-line, then X_{E}^{−}(n) ⊂ ℙ^{N−1} is the twist of X(n) ⊂ ℙ^{N−1} via X_{E}^{−}(n) ≅ X(n); x ↦ h^{−T} x.

Proof. If h is the same as in Lemma 3.1 then the result is clear. In general if h_{1} and h_{2} are two such maps then there is a commutative diagram

[figure: X_{E}(n); h_{1}; X(n); β; X_{E}(n); h_{2}; X(n)]

where β is an automorphism of X(n) defined over \overline{K}. Since h_{1} and h_{2} are compatible with the maps to the j-line, it follows that h_{2} ∝ \overline{ρ}(γ)h_{1}α^{−1} for some γ ∈ SL_{2}(ℤ/nℤ) and α ∈ GL_{N}(K). The change from h_{1} to h_{2} makes no difference to the desired conclusions. □

4. Formulae in the case of a rational 3-torsion point

In this section we find formulae for X_{E}^{±}(9) and X_{E}^{±}(9) → X_{E}^{±}(3) in the case E has a rational 3-torsion point.

Lemma 4.1. Let E/K be an elliptic curve with a rational 3-torsion point T, and discriminant Δ. Then E has a Weierstrass equation of the form

(7)  y^{2} + a_{1}xy + a_{3}y = x^{3}

with T = (0, 0). Moreover K(E[3]) = K(ζ_{3}, \sqrt[3]{Δ}).

Proof. Moving T to (0, 0) gives a Weierstrass equation y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x. Since 2T ≠ 0 we have a_{3} ≠ 0. By a substitution y ← y + rx we may assume a_{4} = 0. Then E meets the line y = 0 in divisor 2(0, 0) + (−a_{2}, 0). So for 3T = 0 we need a_{2} = 0. For the last statement we note that Δ = a_{3}^{3}(a_{1}^{3} − 27a_{3}) and E[3] has basis T = (0, 0) and T′ = (3a_{3}/(δ − a_{1}), a_{3}(ζ_{3}δ − a_{1})/(δ − a_{1})) where δ = \sqrt[3]{a_{1}^{3} − 27a_{3}}. □

Lemma 4.2. Let E/K be an elliptic curve with a rational 3-torsion point, and discriminant Δ.

(i) The family of elliptic curves parametrised by Y (3) = 𝔸^{1} \ {t^{3} = 1} is

y^{2} + 3txy + (t^{3} − 1)y = x^{3}.

(ii) The family of elliptic curves parametrised by Y_{E}^{±}(3) = 𝔸^{1} \ {t^{3} = Δ^{±1}} is

y^{2} + 3txy + (t^{3} − Δ^{±1})y = x^{3}.

Proof. The family of elliptic curves in (i) is the same as that in (2), after a quadratic twist by −3 or a substitution t ← (t + 2)/(t − 1). Let c_{4}, c_{6}, Δ be the usual quantities associated to the Weierstrass equation (7). Then part (ii) is the special case of Theorem 1.1 with (r, s) = (a_{1}^{2}t − a_{1}^{3}a_{3} + 36a_{3}^{2}, t − a_{1}a_{3}), respectively (r, s) = ((a_{1}^{3}a_{3} − 36a_{3}^{2})t + 2a_{1}^{2}, a_{1}a_{3}t + 2). Alternatively the lemma may be proved directly by an argument similar to the proof of Lemma 4.1. □

For a ∈ K we write \sqrt[3]{a} for the image of x in K[x]/(x^{3} − a). Each equation involving \sqrt[3]{a} in the next theorem is written as a short-hand for three equations, one for each K-algebra homomorphism K[x]/(x^{3} − a) → \overline{K}.

Theorem 4.3. Let E/K be the elliptic curve

(8)  y^{2} + 3t_{E}xy + (t_{E}^{3} − Δ)y = x^{3}.

Then X(9) and X_{E}^{±}(9) have equations in 𝔸^{4} (with co-ordinates t, u, v, w) given by

(9)  X(9)  t − \sqrt[3]{1} = 9 (u + v\sqrt[3]{1} + w(\sqrt[3]{1})^{2})^{3} ,
(10)  X_{E}(9)  t − \sqrt[3]{Δ} = (t_{E} − \sqrt[3]{Δ}) (u + v\sqrt[3]{Δ} + w(\sqrt[3]{Δ})^{2})^{3} ,
(11)  X_{E}^{−}(9)  t − (\sqrt[3]{Δ})^{−1} = \frac{3}{t_{E} − \sqrt[3]{Δ}} (u + v\sqrt[3]{Δ} + w(\sqrt[3]{Δ})^{2})^{3} .

Moreover the maps to X(3) and X_{E}^{±}(3) are given by (t, u, v, w) ↦ t, where t is the co-ordinate in Lemma 4.2.

Remark 4.4. The equations in Theorem 4.3 may be re-written as follows. First we expand and equate coefficients of 1, \sqrt[3]{1}, (\sqrt[3]{1})^{2}, respectively 1, \sqrt[3]{Δ}, (\sqrt[3]{Δ})^{2}. We then eliminate t, and homogenise (introducing a new variable s) to give

X(9) = \{ \begin{matrix}u^{2}v + v^{2}w + w^{2}u & = s^{3} \\ u^{2}w + v^{2}u + w^{2}v & = 0 \end{matrix}\} ⊂ ℙ^{3}

X_{E}(9) = \{ \begin{matrix}f_{0}(u, v, w) − t_{E}f_{1}(u, v, w) & = s^{3} \\ f_{1}(u, v, w) − t_{E}f_{2}(u, v, w) & = 0 \end{matrix}\} ⊂ ℙ^{3}

X_{E}^{−}(9) = \{ \begin{matrix}f_{0}(u, v, w) + t_{E}f_{1}(u, v, w) & = 9s^{3} \\ Δf_{2}(u, v, w) + 9t_{E}s^{3} & = 0 \end{matrix}\} ⊂ ℙ^{3}

where

f_{0}(u, v, w) = u^{3} + Δv^{3} + Δ^{2}w^{3} + 6Δuvw
f_{1}(u, v, w) = 3(u^{2}v + Δv^{2}w + Δw^{2}u)
f_{2}(u, v, w) = 3(u^{2}w + v^{2}u + Δw^{2}v).

Proof of Theorem 4.3. The formulae for X(9) and X(9) → X(3) were already established in Lemma 2.1.

Let 𝕂_{n} be the function field of X(n) over \overline{K}. Let B ⊂ 𝕂_{3}^{×}/(𝕂_{3}^{×})^{3} be the subgroup generated by all rational functions on X(3) with zeros and poles only at the cusps. Since X(3) ≅ ℙ^{1} has 4 cusps, B has dimension 3 as an 𝔽_{3}-vector space. By (9) we have 𝕂_{9} = 𝕂_{3}(\sqrt[3]{B}). So if t is the co-ordinate on X_{E}^{±}(3) in Lemma 4.2 then X_{E}^{±}(9) has equations

t − (\sqrt[3]{Δ})^{±1} = c_{±}(E) (u + v\sqrt[3]{Δ} + w(\sqrt[3]{Δ})^{2})^{3}

for some constant c_{±}(E). Since on X_{E}(9) there is a rational point (corresponding to E) above the point t = t_{E} on X_{E}(3), we can take c_{+}(E) = t_{E} − \sqrt[3]{Δ}.

The equations (9) and (10) for X(9) and X_{E}(9) differ by a change of co-ordinates defined over \overline{K}. In writing down this change of co-ordinates it is important to remember that each of (9) and (10) is really three equations. Let α, β, γ, δ ∈ \overline{K} satisfy

α^{3} = \frac{t_{E} − δ}{9δ} , β^{3} = \frac{t_{E} − ζ_{3}δ}{9ζ_{3}δ} , γ^{3} = \frac{t_{E} − ζ_{3}^{2}δ}{9ζ_{3}^{2}δ} , δ^{3} = Δ.

Then an isomorphism X_{E}(9) → X(9) is given by

(\begin{matrix}u \\ v \\ w \end{matrix}) ↦ (\begin{matrix}1 & 1 & 1 \\ 1 & ζ_{3} & ζ_{3}^{2} \\ 1 & ζ_{3}^{2} & ζ_{3} \end{matrix})^{−1} (\begin{matrix}α \\ & β \\ & & γ \end{matrix}) (\begin{matrix}1 & 1 & 1 \\ 1 & ζ_{3} & ζ_{3}^{2} \\ 1 & ζ_{3}^{2} & ζ_{3} \end{matrix}) (\begin{matrix}1 \\ & δ \\ & & δ^{2} \end{matrix}) (\begin{matrix}u \\ v \\ w \end{matrix}) .

By Lemma 3.2 an isomorphism X_{E}^{−}(9) → X(9) is given by

(\begin{matrix}u \\ v \\ w \end{matrix}) ↦ (\begin{matrix}1 & 1 & 1 \\ 1 & ζ_{3}^{2} & ζ_{3} \\ 1 & ζ_{3} & ζ_{3}^{2} \end{matrix})^{−1} (\begin{matrix}α^{−1} \\ & β^{−1} \\ & & γ^{−1} \end{matrix}) (\begin{matrix}1 & 1 & 1 \\ 1 & ζ_{3}^{2} & ζ_{3} \\ 1 & ζ_{3} & ζ_{3}^{2} \end{matrix}) (\begin{matrix}1 \\ & δ^{−1} \\ & & δ^{−2} \end{matrix}) (\begin{matrix}u \\ v \\ w \end{matrix}) .

Therefore X_{E}^{−}(9) has equations

(12)  t − (\sqrt[3]{Δ})^{−1} = \frac{9^{2}}{t_{E} − \sqrt[3]{Δ}} (u + v(\sqrt[3]{Δ})^{−1} + w(\sqrt[3]{Δ})^{−2})^{3} .

The equations (11) and (12), though different, are related by a change of coordinates defined over K. □

5. Completion of proofs

In this section we complete the proofs of Theorems 1.2, 1.3 and 1.4. We start by using the results of the last section to prove Theorems 1.2 and 1.3 in the case E(K)[3] ≠ 0. By Lemma 4.1 we may assume E takes the form (8). Then E has shorter Weierstrass equation y^{2} = x^{3} + ax + b where

a = −24Δt_{E} − 3t_{E}^{4},
b = 16Δ^{2} + 40Δt_{E}^{3} − 2t_{E}^{6}.

The equations for X_{E}(9) in Theorem 1.2 and Remark 4.4 are related by

(\begin{matrix}u \\ v \\ w \\ s \end{matrix}) = (\begin{matrix}1 & −3t_{E}^{2} & 12Δt_{E} − 3t_{E}^{4} & 36Δt_{E}^{3} − 9t_{E}^{6} \\ 0 & −12t_{E} & 24Δ + 12t_{E}^{3} & −216Δt_{E}^{2} \\ 0 & −12 & 36t_{E}^{2} & −144Δt_{E} − 72t_{E}^{4} \\ 1 & 9t_{E}^{2} & −36Δt_{E} + 9t_{E}^{4} & 96Δ^{2} + 132Δt_{E}^{3} + 15t_{E}^{6} \end{matrix}) (\begin{matrix}x \\ y \\ z \\ t \end{matrix}) .

The equations for X_{E}^{−}(9) in Theorem 1.2 and Remark 4.4 are related by

(\begin{matrix}u \\ v \\ w \\ s \end{matrix}) = (\begin{matrix}Δt_{E} & −12Δ^{2} + 12Δt_{E}^{3} & −4Δ^{2} + 7Δt_{E}^{3} & −12Δ^{2}t_{E}^{2} + 3Δt_{E}^{5} \\ −2Δ & 0 & −6Δt_{E}^{2} & 18Δt_{E}^{4} \\ t_{E}^{2} & 0 & 4Δt_{E} − t_{E}^{4} & −16Δ^{2} + 8Δt_{E}^{3} − t_{E}^{6} \\ −Δt_{E} & −4Δ^{2} + 4Δt_{E}^{3} & −4Δ^{2} + Δt_{E}^{3} & 12Δ^{2}t_{E}^{2} − 3Δt_{E}^{5} \end{matrix}) (\begin{matrix}x \\ y \\ z \\ t \end{matrix}) .

These 4 by 4 matrices have determinants −2^{10}3^{3}(t_{E}^{3} − Δ)^{3} and 2^{10}Δ^{3}(t_{E}^{3} − Δ)^{4}, and so are non-singular by the discriminant condition for E. Theorem 1.2 in the case E(K)[3] ≠ 0 now follows from Theorem 4.3.

Theorem 1.3 follows almost immediately from Lemma 2.3. The one detail we must check is how the pencil of cubics defining X_{E}^{±}(9) in Theorem 1.2 matches up with the co-ordinates (r : s) on X_{E}^{±}(3) ≅ ℙ^{1} in Theorem 1.1. Computing the discriminant of the Weierstrass equation (1), we find that the cusps of X_{E}^{±}(3) are the roots of f_{±}(r, s) = 0 where

f_{+}(r, s) = r^{4} − 6c_{4}r^{2}s^{2} − 8c_{6}rs^{3} − 3c_{4}^{2}s^{4}
f_{−}(r, s) = c_{4}r^{4} + 4c_{6}r^{3}s + 6c_{4}^{2}r^{2}s^{2} + 4c_{4}c_{6}rs^{3} − (3c_{4}^{3} − 4c_{6}^{2})s^{4}.

On the other hand, writing H(F) for the Hessian matrix of F, we find that

det H(3rF_{1}^{±} − sF_{2}^{±}) = f_{±}(r, s)D_{±}(x, y, z, t)

for some quartic form D_{±}(x, y, z, t). Comparing with (6) and Lemma 2.3 it follows that the forgetful map is as claimed in Theorem 1.3.

To extend the proofs of Theorems 1.2 and 1.3 to the case E(K)[3] = 0 we use the following two lemmas.

Lemma 5.1. Let X, Y, Y′ be curves defined over K. Suppose there is a commutative diagram

[figure: Y; π; ψ; Y′; π′; X; X]

where π and π′ are morphisms defined over K, and ψ is an isomorphism defined over a finite extension L/K. Suppose that π is the map that quotients out by a finite K-rational subgroup A ⊂ Aut(Y). If [L : K] and |A| are coprime then Y and Y′ are isomorphic over K.

Proof. The curve Y′ is the twist of Y by a class ξ ∈ H^{1}(K, A) whose restriction to H^{1}(L, A) is trivial. Since the composition

H^{1}(K, A) ⟶^{res} H^{1}(L, A) ⟶^{cores} H^{1}(K, A)

is multiplication by [L : K], and this degree is coprime to |A|, it follows that ξ is trivial. □

Lemma 5.2. Let E/K be an elliptic curve and p a prime. Then E(L)[p] ≠ 0 for some finite extension L/K with [L : K] coprime to p.

Proof. The orbits sizes for the action of Galois on E[p] \ {0} add up to p^{2} − 1. Therefore some orbit has size coprime to p. □

To prove Theorem 1.2 we use Lemma 5.2 to find L/K an extension of degree coprime to 3 with E(L)[3] ≠ 0. We already know that the theorem is true over L. We apply Lemma 5.1 with X = X_{E}^{±}(3), Y = X_{E}^{±}(9) and Y′ ⊂ ℙ^{3} the curve defined in the statement of Theorem 1.2. We further take π to be the forgetful map, and π′ the map defined in the statement of Theorem 1.3. It follows that Theorem 1.2 is true over K. The proof of Theorem 1.3 goes through exactly the same as before.

Remark 5.3. If elliptic curves E and E′ are quadratic twists, then it is easy to see that the modular curves X_{E}^{±}(n) and X_{E′}^{±}(n) are isomorphic. With F_{1}^{±} and F_{2}^{±} as defined in Theorem 1.2, this is borne out by the identities

F_{1}^{+}(λ^{2}a, λ^{3}b; λ^{3}x, λ^{2}y, λz, t) = λ^{6}F_{1}^{+}(a, b; x, y, z, t),
F_{2}^{+}(λ^{2}a, λ^{3}b; λ^{3}x, λ^{2}y, λz, t) = λ^{7}F_{2}^{+}(a, b; x, y, z, t),

and

F_{1}^{−}(λ^{2}a, λ^{3}b; λ^{2}x, λy, λz, t) = λ^{5}F_{1}^{−}(a, b; x, y, z, t),
F_{2}^{−}(λ^{2}a, λ^{3}b; λ^{2}x, λy, λz, t) = λ^{6}F_{2}^{−}(a, b; x, y, z, t).

Proof of Theorem 1.4. Treating a and b as additional variables, the equations for X_{E}^{±}(9) in Theorem 1.2 define a threefold. Our task is to quotient out by the action of 𝔾_{m} in Remark 5.3, to give a surface birational to Z^{±}(9). One way to do this is by setting a = b, but this gives a highly singular model for Z^{±}(9), and does not seem to help in finding an elliptic fibration. Instead we make the following substitutions.

We start with the direct case. Putting a = 12s − 3w^{2} and b = us + 2w^{3} in Theorem 1.2, and expanding in powers of s, we find

F_{1}^{+}(6vs − 3w^{3}, 2sT − w^{2}, w, 1) = 12s^{2}(g_{0} + 4sg_{1}),
F_{2}^{+}(6vs − 3w^{3}, 2sT − w^{2}, w, 1) = 36s^{2}(h_{0} + 4sh_{1}),

where

g_{0} = u^{2} + 3uv + 3v^{2} + 9uw + 6T vw + 3(T^{2} − 12T + 24)w^{2},
g_{1} = T^{3} − 9T^{2} + 36T − 36,
h_{0} = v^{2}w − (T − 1)uw^{2} + 2(T − 6)vw^{2} + (T^{2} − 24T + 48)w^{3},
h_{1} = u + (T^{2} − 6T + 12)v − 3(T − 2)(T − 6)w.

The plane cubic {g_{0}h_{1} − g_{1}h_{0} = 0} ⊂ ℙ^{2} is a smooth curve of genus one, defined over ℚ(T), with rational point (u : v : w) = (12 : T − 6 : −1). The elliptic surface in Theorem 1.4 is obtained by putting this curve in Weierstrass form, for example using the method in [5, Chapter 8]. To simplify the final answer we also made a substitution T ← 2T + 3.

In the reverse case we put a = 3us − 3w^{2} and b = 3vs^{2} − 3uws + 2w^{3}. We then compute

F_{1}^{−}(−3s^{2}T − w^{2}, s − w, 2w, 1) = 9s^{3}q_{1},
F_{2}^{−}(−3s^{2}T − w^{2}, s − w, 2w, 1) = 9s^{3}(wq_{1} − sq_{2}),

where

q_{1} = 3u^{2} − uv − 3uw + 2(3T − 1)us − 6vw − 3(2T − 3)vs + 2w^{2} − 3T^{2}ws + 9T^{2}s^{2},
q_{2} = 3(T − 2)u^{2} + 3uv + uw − 2v^{2} − 6(2T − 3)vw + 3(6T − 1)vs + 9T^{2}ws + 3T^{3}s^{2}.

The quadric intersection {q_{1} = q_{2} = 0} ⊂ ℙ^{3} is a smooth curve of genus one, defined over ℚ(T), with rational point (u : v : w : s) = (2 : 1 : 1 : 0). The elliptic surface in Theorem 1.4 is obtained by putting this curve in Weierstrass form, for example using the method in [5, Chapter 8]. To simplify the final answer we also made a substitution T ← (2T − 3)/(2T + 1). □

6. Examples

We use Theorems 1.2 and 1.3 to find some non-trivial pairs of 9-congruent elliptic curves over ℚ and ℚ(T). By “non-trivial” we mean that the elliptic curves are not isogenous. The examples may be checked independently of our work by comparing traces of Frobenius.

6.1. Examples over ℚ. We refer to elliptic curves over ℚ by their labels in Cremona’s tables [7]. For elliptic curves beyond the range of Cremona’s tables we simply write the conductor followed by a ∗.

Example 6.1. Let E be the elliptic curve 47775z1. The equations for X_{E}(9) in Theorem 1.2 with a = −41489280 and b = 102867483600 may be simplified by substituting

(\begin{matrix}x \\ y \\ z \\ t \end{matrix}) ← (\begin{matrix}2520473760 & 937149484320 & −1998984627360 & −152410870080 \\ 0 & 79644600 & −185343480 & −3827880 \\ 0 & −22932 & 47040 & 6468 \\ 0 & −6 & 13 & 1 \end{matrix}) (\begin{matrix}x \\ y \\ z \\ t \end{matrix}) .

This gives a model for X_{E}(9) with equations

−x^{2}z + x^{2}t + 4xyz + 2xyt − 3xz^{2} + 2xzt − 3xt^{2} + 6y^{3} + 14y^{2}z

+ y^{2}t + 6yz^{2} − 4yzt + 9yt^{2} − 6z^{3} + 27z^{2}t − 13zt^{2} − t^{3} = 0

−3x^{2}y + 4x^{2}z + 3x^{2}t + 3xy^{2} + 20xyz − 12xyt − 3xz^{2} − 32xzt + 25xt^{2} + 21y^{3}

+16y^{2}z − 24y^{2}t − 12yz^{2} + 100yzt + 34yt^{2} + 39z^{3} − 21z^{2}t − 56zt^{2} − 11t^{3} = 0.

On this curve we find rational points P_{1} = (1 : 0 : 0 : 0), P_{2} = (4 : −1 : −1 : 0) and P_{3} = (1 : 2 : −1 : 0). The corresponding elliptic curves directly 9-congruent to E are

P_{1} 47775z1  y^{2} + y = x^{3} − x^{2} − 32013x + 2215478
P_{2} 429975∗  y^{2} + y = x^{3} − 314688780x − 2148671872069
P_{3} 494901225∗ y^{2} + y = x^{3} − 23634650164230x − 21037908383222056594

Since X_{E}^{−}(9) is not locally soluble at p = 7 there are no elliptic curves reverse 9-congruent to E.

In addition to Example 6.1 we have found two further triples of directly 9congruent non-isogenous elliptic curves over ℚ. These are

\begin{matrix}4650j1 & y^{2} + xy = x^{3} + x^{2} − 2700x + 54000 \\ 553350∗ & y^{2} + xy = x^{3} + x^{2} − 10472207700x − 455228489646000 \\ 1966950∗ & y^{2} + xy = x^{3} − x^{2} − 20654522386242x − 36130051534030639084 \\ 27606c1 & y^{2} + xy = x^{3} − 10289707x + 12703497719 \\ 358878∗ & y^{2} + xy = x^{3} + 2940333x − 1416695391 \\ 1242270∗ & y^{2} + xy + y = x^{3} − x^{2} − 359912x − 322105301 \end{matrix}

The elliptic curves 1701a1, 1701g1 and 22113c1 are also 9-congruent but only the last two of these are directly 9-congruent.

Example 6.2. Let E be the elliptic curve 201c1. The equations for X_{E}^{−}(9) in Theorem 1.2 with a = −1029699 and b = 402173694 may be simplified by substituting

(\begin{matrix}x \\ y \\ z \\ t \end{matrix}) ← (\begin{matrix}−26471709 & −23136696 & 20106774 & −20376135 \\ −45147 & −39828 & 33990 & −34509 \\ 90294 & 79332 & −68304 & 69342 \\ 77 & 68 & −58 & 59 \end{matrix}) (\begin{matrix}x \\ y \\ z \\ t \end{matrix}) .

This gives a model for X_{E}^{−}(9) with equations

−x^{3} + 4x^{2}y + 3x^{2}z − x^{2}t + 6xy^{2} + 2xyz − 2xyt − 6xz^{2} + 4xzt
−11xt^{2} + y^{3} + 7y^{2}t − 2yz^{2} + 4yzt − 4yt^{2} + 6z^{3} − 7z^{2}t + 4zt^{2} + t^{3} = 0
2x^{3} − x^{2}y + 5x^{2}t − 10xy^{2} − 2xyz + 16xyt − 3xz^{2} + 4xzt + 8xt^{2}
−5y^{3} − y^{2}z − 3y^{2}t − yz^{2} − 2yzt + 12yt^{2} + 3z^{3} − 4z^{2}t + 2zt^{2} − 3t^{3} = 0.

On this curve we find the rational point (1 : −2 : −1 : 0). The corresponding elliptic curve reverse 9-congruent to E is

374865∗  y^{2} + xy = x^{3} + x^{2} − 60068738107x + 4858035498982726.

6.2. Examples over ℚ(T). Again we start our investigations with the equations for X_{E}^{±}(9) in Theorem 1.2. To find some interesting examples over ℚ(T), we set a = b = −27j/(4(j − 1728)) to obtain a model for Z^{±}(9), fibred over the jline. We then looked for some rational curves on this surface, by intersecting with coordinate hyperplanes.

The following example gives a proof of Theorem 1.5 in the direct case, without going via Theorem 1.4.

Example 6.3. Let E/ℚ(T) be the elliptic curve y^{2} = x^{3} + a(T)x + b(T) where

a(T) = \frac{1}{2}(39T^{4} − 60T^{3} − 162T^{2} + 60T + 39),
b(T) = 47T^{6} + 120T^{5} + 21T^{4} + 21T^{2} − 120T + 47.

Then on X_{E}(9), with equations as in Theorem 1.2, we find the rational point

(x : y : z : t) = (\frac{15}{2}(3T^{4} + 8T^{3} − 2T^{2} − 8T + 3) : T^{2} + 1 : 1 : 0).

The corresponding elliptic curve directly 9-congruent to E is the elliptic curve directly 3-congruent to E constructed in Theorem 1.1 with c_{4} = −a(T)/27, c_{6} = −b(T)/54 and (r : s) = (47T^{6} − 78T^{5} − 153T^{4} + 244T^{3} + 153T^{2} − 78T − 47 : 18(T^{2} + 1)(T^{2} + 6T − 1)). Specialising to T = 0 gives a pair of elliptic curves with conductors 80640 and 5886720. In particular these curves are not isogenous.

The following example gives a proof of Theorem 1.5 in the reverse case, without going via Theorem 1.4.

Example 6.4. Let E/ℚ(T) be the elliptic curve y^{2} = x^{3} + a(T)x + b(T) where

a(T) = 3(3T + 1)(6T^{3} − 3T − 1)(9T^{3} − 9T − 4)^{2},
b(T) = 2(3T^{3} + 27T^{2} + 21T + 4)(6T^{3} − 3T − 1)^{2}(9T^{3} − 9T − 4)^{2}.

Then on X_{E}^{−}(9), with equations as in Theorem 1.2, we find the rational point

(x : y : z : t) = (−(6T^{3} − 3T − 1)(9T^{3} − 9T − 4) : T : 1 : 0).

The corresponding elliptic curve reverse 9-congruent to E is the elliptic curve reverse 3-congruent to E constructed in Theorem 1.1 with c_{4} = −a(T)/27, c_{6} = −b(T)/54 and (r : s) = ((3T + 1)(9T^{3} − 9T − 4)(6T^{3} − 3T − 1)(180T^{4} + 321T^{3} + 216T^{2} + 66T + 8) : 3(369T^{6} + 1107T^{5} + 1431T^{4} + 1017T^{3} + 414T^{2} + 90T + 8)). Specialising to T = −1/4 gives the pair of elliptic curves 2304o1 and 343296g1. In particular these curves are not isogenous.

Acknowledgements

I would like to thank Zexiang Chen for some useful discussions that in particular helped to simplify the proof of Theorem 1.2. All computer calculations in support of this work were carried out using Magma [2].

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University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK

E-mail address: T.A.Fisher@dpmms.cam.ac.uk