ON FAMILIES OF 7 AND 11-CONGRUENT ELLIPTIC CURVES

TOM FISHER

Abstract. We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n = 7, 11. Searching for rational points on these twists enables us to find non-trivial pairs of n-congruent elliptic curves over ℚ, i.e. pairs of non-isogenous elliptic curves over ℚ whose n-torsion subgroups are isomorphic as Galois modules. We also find a non-trivial pair of 11-congruent elliptic curves over ℚ(T), and hence give an explicit infinite family of non-trivial pairs of 11-congruent elliptic curves over ℚ.

1. Introduction

Elliptic curves E_{1} and E_{2} over a field K 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 n-congruent if

e_{n}(ϕP, ϕQ) = e_{n}(P, Q)^{−1}

for all P, Q ∈ E_{1}[n]. The elliptic curves directly n-congruent to a given elliptic curve E are parametrised by the modular curve Y_{E}(n) = X_{E}(n) \ {cusps}.

For n ≤ 5 we have X_{E}(n) ≅ ℙ^{1} and the corresponding families of elliptic curves were computed by Rubin and Silverberg [27], [29], [30]. It was shown independently by Papadopoulos [25] and Rubin and Silverberg [28] that X_{E}(6) is the elliptic curve y^{2} = x^{3} + Δ_{E}, where Δ_{E} is the discriminant of E. However for n ≥ 7 the genus of X_{E}(n) is greater than 1. This prompted Mazur [23] to ask whether there are any pairs of non-isogenous elliptic curves over ℚ that are directly n-congruent for any n ≥ 7. This was answered by Kraus and Oesterlé [22] who gave the example of the directly 7-congruent elliptic curves 152a1 and 7448e1. The labels here are those in Cremona’s tables [4]. Nowadays it is easy to find further examples by searching in Cremona’s tables, for example

n = 11  190b1 and 2470a1,
n = 13  52a2 and 988b1,
n = 17 3675b1 and 47775b1.

Date: 15th April 2014.

In each case the n-congruence is proved by computing sufficiently many traces of Frobenius. See for example [22, Proposition 4].

Motivated by Mazur’s question, Kani and Schanz [18] studied the geometry of the surfaces that parametrise pairs of n-congruent elliptic curves. This prompted them to conjecture that for any n ≤ 12 there are infinitely many pairs of ncongruent non-isogenous elliptic curves over ℚ. It is understood that we are looking for examples with distinct pairs of j-invariants, since otherwise from any single example we could construct infinitely many examples by taking quadratic twists. The conjecture was proved in the case n = 7 by Halberstadt and Kraus [15], who subsequently [16] gave an explicit formula for X_{E}(7) and used it to show that there are infinitely many 6-tuples of directly 7-congruent non-isogenous elliptic curves over ℚ. In this paper we find a formula for X_{E}(11) and use it to construct a non-trivial pair of 11-congruent elliptic curves over ℚ(T). This proves the conjecture in the case n = 11. In contrast the proof by Kani and Rizzo [17] does not construct any explicit examples.

We briefly mention three further motivations for studying n-congruence of elliptic curves.

• The modular approach to solving Diophantine equations sometimes requires us to find all elliptic curves n-congruent to a given elliptic curve. For example the paper of Poonen, Schaefer and Stoll [26] makes essential use of the formula for X_{E}(7) due to Halberstadt and Kraus.

• There is a correspondence between pairs of reverse n-congruent elliptic curves and curves of genus 2 that admit a degree n morphism to an elliptic curve. See for example [14].

• It was observed by Cremona and Mazur [6] that if elliptic curves E and F are n-congruent then the Mordell-Weil group of F can sometimes be used to explain elements of the Tate-Shafarevich group of E.

As each of these motivations makes clear, we should also be interested in congruences that do not respect the Weil pairing. The elliptic curves reverse n-congruent to E are parametrised by the modular curve Y_{E}^{−}(n) = X_{E}^{−}(n) \ {cusps}. The families of elliptic curves parametrised by Y_{E}^{−}(3) and Y_{E}^{−}(4) were computed in [11], and the analogous problem for n = 5 was solved in [12]. An equation for X_{E}^{−}(7) was given in [26, Section 7.2]. In this paper we find equations for X_{E}^{−}(11).

In Section 1.1 we recall the definitions of X(n) and its twists. We then record the formulae for X_{E}(n) and X_{E}^{−}(n) for n = 7, 11 in Section 1.2. In the case n = 7 these are the formulae given in [16], [26], but our method for finding them is new. In the case n = 11 the formulae themselves are new.

In Section 2 we derive Klein’s equations for X(n) for n ≥ 5 an odd integer. The original approach of Klein was via theta functions, but our treatment is purely algebraic. We also give explicit formulae for the action of SL_{2}(ℤ/nℤ) on X(n).

Then in Section 3 we use invariant theory for SL_{2}(ℤ/nℤ) to compute the twists X_{E}(n) and X_{E}^{−}(n) for n = 7, 11.

In Section 4 we work out formulae for the families of elliptic curves parametrised by Y_{E}(n) and Y_{E}^{−}(n) for n = 7, 11. Computing the j-invariant maps j : X_{E}(n) → ℙ^{1} and j : X_{E}^{−}(n) → ℙ^{1} is reasonably straightforward. Finding the right quadratic twists takes considerably more work, although in specific numerical examples one can always fall back on the method in [15], [22]. In the case of Y_{E}(7) a formula is given in [16], but this formula does not quite cover all cases. We give a new proof leading to formulae that work in all cases. We then generalise to the families of elliptic curves parametrised by Y_{E}^{−}(7), Y_{E}(11) and Y_{E}^{−}(11).

Our formulae reduce the problem of finding elliptic curves n-congruent to E to that of finding rational points on X_{E}(n) and X_{E}^{−}(n). However before searching for rational points it helps to simplify the equations by making a change of coordinates. We have written programs in Magma [3] to do this in the case K = ℚ, using ideas of minimisation and reduction similar to those in [5]. We will report on this in future work. In fact we have written a program in Magma that given an elliptic curve E/ℚ and n ∈ {7, 11} searches for rational points (up to a specified height bound) on minimised and reduced models for X_{E}(n) and X_{E}^{−}(n), and returns the corresponding list of elliptic curves n-congruent to E. In Section 5 we give some examples over ℚ to illustrate how this works, and also some examples over ℚ(T), which by specialisation of T give infinite families of examples over ℚ. The examples over ℚ may be checked, independent of the methods we use to find them, by checking that the traces of Frobenius are congruent mod n for sufficiently many primes.

All computer calculations in support of this work were performed using Magma [3]. A Magma file checking all our formulae, together with a table of 11-congruent elliptic curves over ℚ, is available from the website [13]. We have used the same methods to study families of 9-congruent elliptic curves, and will report on this in future work. Our restriction to odd n is explained by our use of Klein’s equations (see Section 2.1).

1.1. Some modular curves. We work over a field K of characteristic 0 and write \overline{K} for the algebraic closure. Let n ≥ 3 be an integer and M a Galois module, isomorphic to (ℤ/nℤ)^{2} as an abelian group, and equipped with a nondegenerate alternating Galois equivariant pairing M × M → μ_{n}. We temporarily write Y_{M} for the algebraic curve defined over K whose L-rational points (L a field extension of K) parametrise the isomorphism classes of pairs (E, ϕ), where E is an elliptic curve defined over L and ϕ : E[n] ≅ M is a symplectic isomorphism (i.e. one that matches up the given pairing on M with the Weil pairing on E[n]) commuting with the action of Gal(\overline{L}/L). Two such pairs (E_{1}, ϕ_{1}) and (E_{2}, ϕ_{2}) are isomorphic if there is an L-isomorphism α : E_{1} → E_{2} such that ϕ_{1} = ϕ_{2} ∘ (α|_{E_{1}[n]}).

Let X_{M} be the smooth projective model of Y_{M}. We write X(n) and Y (n) for X_{M} and Y_{M} in the case M = μ_{n} × ℤ/nℤ with pairing

⟨(ζ, a), (ξ, b)⟩ = ζ^{b}ξ^{−a}.

Given an elliptic curve E/K, let X_{E}(n) be X_{M} in the case M is E[n] equipped with the Weil pairing. More generally let X_{E}^{(r)}(n) be X_{M} in the case M is E[n] equipped with the rth power of the Weil pairing for some r ∈ (ℤ/nℤ)^{×}. Since multiplication by m ∈ (ℤ/nℤ)^{×} is an automorphism of E[n] that raises the Weil pairing to the power m^{2}, the curve X_{E}^{(r)}(n) only depends on the class of r mod squares. Since we are interested in the cases n = 7, 11 it will suffice to take r = ±1. We write X_{E}^{−}(n) for X_{E}^{(−1)}(n).

Let ζ_{n} ∈ \overline{K} be a primitive nth root of unity. Over K(ζ_{n}) we may identify the Galois modules μ_{n} × ℤ/nℤ and (ℤ/nℤ)^{2}, and hence the group of symplectic automorphisms of μ_{n} × ℤ/nℤ with SL_{2}(ℤ/nℤ). There is then a natural action of PSL_{2}(ℤ/nℤ) := SL_{2}(ℤ/nℤ)/{±I_{2}} on X(n) with quotient map j : X(n) → ℙ^{1}. From the analytic theory we know that the j-map is ramified above 0, 1728 and ∞ with ramification indexes 3, 2 and n. Hence by the Riemann-Hurwitz formula the genus of X(n) is

g(n) = \frac{n − 6}{12n} # PSL_{2}(ℤ/nℤ) + 1

where for n ≥ 3 we have # PSL_{2}(ℤ/nℤ) = (n^{3}/2) ∏_{p|n}(1 − 1/p^{2}). For some small values of n the genus is as follows.

\begin{matrix}n & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\ g(n) & 0 & 0 & 0 & 0 & 1 & 3 & 5 & 10 & 13 & 26 & 25 & 50 & 49 & 73 & 81 & 133 \end{matrix}

1.2. Statement of results. A formula for X_{E}(7) was obtained by Halberstadt and Kraus [16]. Their method relies on studying the points on the Klein quartic X(7) = {x^{3}y + y^{3}z + z^{3}x = 0} ⊂ ℙ^{2} corresponding to an elliptic curve E and the elliptic curves E_{a}, E_{b}, E_{c} that are 2-isogenous to E. By combining this result with some classical invariant theory, Poonen, Schaefer and Stoll [26, Section 7.2] then gave a formula for X_{E}^{−}(7).

Theorem 1.1 ((Halberstadt, Kraus, Poonen, Schaefer, Stoll)). Let E be an elliptic curve with Weierstrass equation y^{2} = x^{3} + ax + b. Then X_{E}(7) ⊂ ℙ^{2} has equation F = 0 where

F = ax^{4}+7bx^{3}z+3x^{2}y^{2}−3a^{2}x^{2}z^{2}−6bxyz^{2}−5abxz^{3}+2y^{3}z+3ay^{2}z^{2}+2a^{2}yz^{3}−4b^{2}z^{4},

and X_{E}^{−}(7) ⊂ ℙ^{2} has equation G = 0 where

G = −a^{2}x^{4} + 2abx^{3}y − 12bx^{3}z − (6a^{3} + 36b^{2})x^{2}y^{2} + 6ax^{2}z^{2} + 2a^{2}bxy^{3} − 12abxy^{2}z

+ 18bxyz^{2} + (3a^{4} + 19ab^{2})y^{4} − (8a^{3} + 42b^{2})y^{3}z + 6a^{2}y^{2}z^{2} − 8ayz^{3} + 3z^{4}.

We give a new proof of Theorem 1.1 and then extend to the case n = 11. Although we believe our formulae in the case n = 11 are correct for all elliptic curves E, our proof does not naturally extend to the cases j(E) = 0, 1728. We therefore assume for simplicity that j(E) ≠ 0, 1728. It was observed by Klein [20] (see also [2, Example 22.3]) that X(11) may be embedded in ℙ^{4} as the singular locus of the Hessian of the cubic threefold

{v^{2}w + w^{2}x + x^{2}y + y^{2}z + z^{2}v = 0} ⊂ ℙ^{4}.

Theorem 1.2. Let E be an elliptic curve with Weierstrass equation y^{2} = x^{3} + ax + b. If j(E) ≠ 0, 1728 then X_{E}(11) ⊂ ℙ^{4} is the singular locus of the Hessian of

F = v^{3} + av^{2}z − 2avx^{2} + 4avxy − 6bvxz + avy^{2} + 6bvyz + a^{2}vz^{2} − w^{3}

+ aw^{2}z − 4awx^{2} − 12bwxz + a^{2}wz^{2} − 2bx^{3} + 3bx^{2}y + 2a^{2}x^{2}z + 6bxy^{2}

+ 4abxz^{2} + by^{3} − a^{2}y^{2}z + abyz^{2} + 2b^{2}z^{3},

and X_{E}^{−}(11) ⊂ ℙ^{4} is the singular locus of the Hessian of

G = v^{2}z + 2vwy + 4vxy + 2w^{2}x − aw^{2}z + 2wx^{2} − 2awy^{2} − 6bwyz
+ 6x^{3} − 6ax^{2}z + 2a^{2}xz^{2} + by^{3} − 2a^{2}y^{2}z − 5abyz^{2} − b^{2}z^{3}.

2. Equations for X(n)

We derive equations of Klein [19], [20], [21] for the modular curves X(n). Our treatment follows the survey in [8, Chapter 4], but see also [2], [33].

2.1. Klein’s equations. Suppose to begin with that ζ_{n} ∈ K. Then the modular curve Y (n) parametrises the triples (E, P, Q) where E is an elliptic curve and P, Q is a basis for E[n] with e_{n}(P, Q) = ζ_{n}. If we embed E ⊂ ℙ^{n−1} by a complete linear system |D| of degree n then the translation maps τ_{P} and τ_{Q} extend to automorphisms of ℙ^{n−1}. In fact we have the following lemma, as proved in [9, Section 2.1].

Lemma 2.1. (i) We may change co-ordinates on ℙ^{n−1} (over \overline{K}) so that τ_{P} and τ_{Q} are given by

M_{1} = ( \begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 0 & ζ_{n} & 0 & ⋯ & 0 \\ 0 & 0 & ζ_{n}^{2} & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & 0 & ⋯ & ζ_{n}^{n−1} \end{matrix} ) and M_{2} = ( \begin{matrix}0 & 0 & ⋯ & 0 & 1 \\ 1 & 0 & ⋯ & 0 & 0 \\ 0 & 1 & ⋯ & 0 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \end{matrix} ).

(ii) If n is odd and [−1]^{∗}D ∼ D then there is a unique choice of co-ordinates (over \overline{K}) such that τ_{P}, τ_{Q} and multiplication by −1 are given by M_{1}, M_{2} and

[−1] = ( \begin{matrix}1 & 0 & ⋯ & 0 & 0 \\ 0 & 0 & ⋯ & 0 & 1 \\ 0 & 0 & ⋯ & 1 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 1 & ⋯ & 0 & 0 \end{matrix} ).

It is well known that if n ≥ 4 then the image of E ⊂ ℙ^{n−1} is defined by quadrics. In fact the homogeneous ideal is generated by a vector space of quadrics of dimension n(n − 3)/2. See [10, Section 5.1] for a short proof, or [24] for a more general result.

We restrict to n ≥ 5 an odd integer. If we embed E ⊂ ℙ^{n−1} via the complete linear system |n.0_{E}|, and choose co-ordinates as in Lemma 2.1, then sending (E, P, Q) to the image of 0_{E} defines an embedding Y (n) ⊂ ℙ^{n−1}. We check injectivity as follows. If we know the co-ordinates of 0_{E} ∈ ℙ^{n−1} then M_{1} and M_{2} allow us to write down n^{2} points on E. By Bezout’s theorem any quadric not containing E meets E in at most 2n points. Therefore E is defined by the quadrics containing these n^{2} points, and P, Q ∈ E[n] are the translates of 0_{E} under M_{1} and M_{2}.

We now drop our assumption that ζ_{n} ∈ K. The subgroup of PGL_{n}(\overline{K}) generated by M_{1} and M_{2} is isomorphic to μ_{n} × ℤ/nℤ as a Galois module. In view of the definition of X(n) in Section 1.1, it follows that the embedding Y (n) ⊂ ℙ^{n−1} described in the last paragraph is defined over K, and not just over K(ζ_{n}).

We write (x_{0} : x_{1} : …: x_{n−1}) for our co-ordinates on ℙ^{n−1} and agree to read all subscripts mod n. Since n is odd we have

n.0_{E} ∼ 0_{E} + P + 2P + … + (n − 1)P.

The divisor on the right is a hyperplane section and is invariant under translation by P. It is also the only such divisor with 0_{E} in its support. Therefore 0_{E} belongs to exactly one of the hyperplanes fixed by M_{1}. But 0_{E} is fixed by [−1], so we have either

\begin{aligned}& 0_{E} & = & (0 : a_{1} : a_{2} : …: a_{2} : a_{1}) & (+) \\ or & 0_{E} & = & (0 : a_{1} : a_{2} : …: −a_{2} : −a_{1}) & (−) \end{aligned}

where a_{1}, a_{2}, … are non-zero.

Let W be the vector space of quadrics on ℙ^{n−1} and V the subspace of quadrics vanishing on E. Then dim W = n(n + 1)/2 and dim V = n(n − 3)/2. The action of M_{1} allows us to write these as direct sums V = ⊕V_{i} and W = ⊕W_{i} with

V_{i} ⊂ W_{i} = ⟨x_{i}^{2}, x_{i−1}x_{i+1}, …⟩.

The V_{i} and W_{i} are the subspaces on which M_{1} acts with eigenvalue ζ_{n}^{2i}. The action of M_{2} shows that V_{i} ≅ V_{i+1} and W_{i} ≅ W_{i+1} for all i. Therefore dim V_{i} = (n − 3)/2 and dim W_{i} = (n + 1)/2. The requirement that the quadrics in V_{0} vanish at 0_{E} = (a_{0} : a_{1} : …: a_{n−1}), and its translates under M_{2}, imposes some linear conditions on the coefficients of these quadrics. Since V_{0} ⊂ W_{0} has codimension 2 it follows that rank(a_{i−j}a_{i+j})_{i,j=0}^{n−1} ≤ 2.

If 0_{E} is of the form (+) then this matrix is symmetric, and the vanishing of the top left 3 × 3 minor contradicts that a_{1}, a_{2}, a_{3} are non-zero. Therefore 0_{E} must be of the form (−). This motivates the following definition.

Definition 2.2. For n ≥ 5 an odd integer let Z(n) ⊂ ℙ^{n−1} be the subvariety defined by a_{0} = 0, a_{n−i} = −a_{i} and

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

We note that (1) is equivalent to the vanishing of the 4 × 4 Pfaffians of this skew-symmetric matrix. Using minors instead of Pfaffians also works, but gives equations of larger degree. The above construction shows that Y (n) ⊂ Z(n). It is natural to ask whether X(n) = Z(n). Vélu [33] proved this in the case n = p is a prime. However if n is composite then Z(n) has extra components.

When n = 7 we put 0_{E} = (0 : a : b : −c : c : −b : −a) so that Z(7) ⊂ ℙ^{2} with co-ordinates (a : b : c). Then Z(7) is defined by

rank (\begin{matrix}0 & −a^{2} & −b^{2} & −c^{2} \\ a^{2} & 0 & ac & −bc \\ b^{2} & −ac & 0 & ab \\ c^{2} & bc & −ab & 0 \end{matrix}) ≤ 2.

Computing the Pfaffian of this matrix (i.e. the square root of its determinant) shows that X(7) = Z(7) is the Klein quartic {a^{3}b + b^{3}c + c^{3}a = 0} ⊂ ℙ^{2}.

When n = 11 we put 0_{E} = (0 : a : −c : b : e : d : −d : −e : −b : c : −a) so that Z(11) ⊂ ℙ^{4} with co-ordinates (a : b : c : d : e). Computing 4 × 4 Pfaffians shows that X(11) = Z(11) is the singular locus of the Hessian of the cubic threefold

{a^{2}b + b^{2}c + c^{2}d + d^{2}e + e^{2}a = 0} ⊂ ℙ^{4}.

In other words, X(11) is defined by the vanishing of the partial derivatives of the determinant of the matrix

(2)  (\begin{matrix}b & a & 0 & 0 & e \\ a & c & b & 0 & 0 \\ 0 & b & d & c & 0 \\ 0 & 0 & c & e & d \\ e & 0 & 0 & d & a \end{matrix}) .

We refer to [2] for further details. In fact, as we checked using Magma, the homogeneous ideal of X(11) is generated by the 4 × 4 minors of (2).

2.2. The action of SL_{2}(ℤ/nℤ). We suppose ζ_{n} ∈ K so that Y (n) parametrises the triples (E, P, Q) where E is an elliptic curve and P, Q is a basis for E[n] with e_{n}(P, Q) = ζ_{n}. The natural action of SL_{2}(ℤ/nℤ) on Y (n) is given by

(3)  ( \begin{matrix}a & b \\ c & d \end{matrix}) : (E, P, Q) ↦ (E, dP − cQ, −bP + aQ).

This extends to an action on X(n), and so defines a group homomorphism

(4)  \overline{ρ} : SL_{2}(ℤ/nℤ) → Aut(X(n)).

We now take n ≥ 5 an odd integer. In Section 2.1 we defined an embedding X(n) ⊂ ℙ^{m−1} where m = (n−1)/2. In this setting (4) becomes a projective representation \overline{ρ} : SL_{2}(ℤ/nℤ) → PGL_{m}(\overline{K}). We show that it lifts to a representation. See [2, Appendix I] for a discussion of how this relates to work of Weil. We write ∝ for equality in PGL_{n}(\overline{K}).

Proposition 2.3. The projective representation \overline{ρ} : SL_{2}(ℤ/nℤ) → PGL_{m}(\overline{K}) lifts to a representation ρ : SL_{2}(ℤ/nℤ) → GL_{m}(\overline{K}).

Proof. If we embed X(n) ⊂ ℙ^{n−1} as described in Section 2.1 then the action (3) extends to a projective representation \overline{π} : SL_{2}(ℤ/nℤ) → PGL_{n}(\overline{K}) where the image of γ = ( \begin{matrix}a & b \\ c & d \end{matrix}) is uniquely determined by the properties that

(5)  \overline{π}(γ)^{−1}M_{1}^{u}M_{2}^{v}\overline{π}(γ) ∝ M_{1}^{du−bv}M_{2}^{−cu+av}

for all u, v ∈ ℤ/nℤ, and \overline{π}(γ) commutes with [−1]. We regard \overline{π} as describing an action on ℙ^{n−1} = ℙ(W) where W is an n-dimensional vector space. The action of [−1] gives an eigenspace decomposition W = W_{+} ⊕ W_{−} with dim W_{±} = (n ± 1)/2. We may then identify \overline{ρ} with the restriction of \overline{π} to ℙ(W_{−}) = ℙ^{m−1}. To prove the proposition we prove the stronger result that \overline{π} lifts to a representation π : SL_{2}(ℤ/nℤ) → GL_{n}(\overline{K}).

Let S = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and T = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) be the usual generators of SL_{2}(ℤ/nℤ). In view of the relations (ST)^{3} = S^{4} = T^{n} = I_{2}, the only non-trivial 1-dimensional characters of SL_{2}(ℤ/nℤ) are the ones, in the case n is a multiple of 3, that factor via PSL_{2}(ℤ/3ℤ) ≅ A_{4}. Using (5) we compute

(6)  \overline{π}(S) ∝ (ζ_{n}^{ij})_{i,j=0}^{n−1}  \overline{π}(T) ∝ Diag(ζ_{n}^{i^{2}/2})_{i=0}^{n−1}

where the exponents are read as elements of ℤ/nℤ.

If M ∈ GL_{n}(\overline{K}) acts on each of the subspaces W_{±} then we write M_{±} for the endomorphisms obtained by restricting to W_{±}. Since

3(1^{2} + 2^{2} + … + m^{2}) ≡ 0 (mod n)

it is clear that if M = Diag(ζ_{n}^{i^{2}/2})_{i=0}^{n−1} then the determinants of M_{+} and M_{−} are cube roots of unity. So by (6) there is a lift π(T) of \overline{π}(T), and 1-dimensional characters χ_{±} of SL_{2}(ℤ/nℤ), such that det(π(T)_{±}) = χ_{±}(T). Next we lift \overline{π}(S) to a matrix π(S) such that

(7)  π(S)π(T)^{−1}π(S) = π(T)π(S)π(T).

Restricting to W_{±} and taking determinants it follows that det(π(S)_{±}) = 1 = χ_{±}(S). For each γ ∈ SL_{2}(ℤ/nℤ) we now let π(γ) be the unique lift of \overline{π}(γ) such that det(π(γ)_{±}) = χ_{±}(γ). These lifts exist since S and T generate SL_{2}(ℤ/nℤ) and are unique since dim W_{−} and dim W_{+} are coprime. It is evident that the map π so defined is a group homomorphism. □

Remark 2.4. (i) A calculation using (7) shows that π(S) = g_{n}^{−1}(ζ_{n}^{ij})_{i,j=0}^{n−1} where the Gauss sum g_{n} = ∑_{i=0}^{n−1} ζ_{n}^{−i^{2}/2} satisfies g_{n}^{2} = (−1)^{(n−1)/2}n. (ii) If we take 0_{E} = (0 : a_{1} : a_{2} : …: −a_{2} : −a_{1}) then with respect to co-ordinates (a_{1} : …: a_{m}) we may take

ρ(S) = g_{n}^{−1}(ζ_{n}^{ij} − ζ_{n}^{−ij})_{i,j=1}^{m} ρ(T) = Diag(ζ_{n}^{i^{2}/2})_{i=1}^{m}.

In particular ρ(−I_{2}) = (−1)^{(n+1)/2}I_{m}. (iii) If n is not divisible by 3 then SL_{2}(ℤ/nℤ) has no 1-dimensional characters, and so the lift we have constructed is unique. If n is divisible by 3 then m is not divisible by 3 and we can make ρ unique by demanding that det ρ(T) = 1, equivalently that ρ takes values in SL_{m}(\overline{K}).

3. Equations for X_{E}(n) and X_{E}^{−}(n)

We derive our equations for X_{E}(n) and X_{E}^{−}(n) by using invariant theory for the group SL_{2}(ℤ/nℤ) to twist the equations for X(n) in Section 2.1. We first make some general remarks about twisting and then split into the cases n = 7, 11.

3.1. Preliminaries on twisting. Let n ≥ 3 be an integer. We recall that Y (n) parametrises the pairs (E, ϕ) where E is an elliptic curve and ϕ : E[n] ≅ μ_{n}×ℤ/nℤ is a symplectic isomorphism. We temporarily write Γ for the group of symplectic automorphisms of μ_{n} × ℤ/nℤ. Then Γ acts on Y (n) by γ : (E, ϕ) ↦ (E, γ ϕ). This action extends to X(n), and so defines a group homomorphism

(8)  \overline{ρ} : Γ → Aut(X(n)).

If X_{1} and X_{2} are varieties defined over K, and α : X_{1} → X_{2} is a morphism defined over \overline{K}, then for each σ ∈ Gal(\overline{K}/K) we write σ(α) for the morphism X_{1} → X_{2} given on \overline{K}-points by P ↦ σ(α(σ^{−1}P)).

Lemma 3.1. Let E/K be an elliptic curve and ϕ : E[n] ≅ μ_{n} × ℤ/nℤ a symplectic, respectively anti-symplectic, isomorphism defined over \overline{K}. Then there is an isomorphism α : X_{E}(n) → X(n), respectively α : X_{E}^{−}(n) → X(n), defined over \overline{K}, such that

σ(α)α^{−1} = \overline{ρ}(σ(ϕ)ϕ^{−1})

for all σ ∈ Gal(\overline{K}/K).

Proof. The points of Y_{E}(n), respectively Y_{E}^{−}(n), correspond to pairs (F, ψ) where F is an elliptic curve and ψ : F [n] ≅ E[n] is a symplectic, respectively antisymplectic, isomorphism. For ϕ as in the statement of the lemma, the composite ϕ ψ : F [n] ≅ μ_{n} × ℤ/nℤ is a symplectic isomorphism. Let α : Y_{E}^{±}(n) → Y (n) be the isomorphism defined by (F, ψ) ↦ (F, ϕ ψ). Then σ(α) maps (F, ψ) ↦ (F, σ(ϕ) ψ). Therefore σ(α)α^{−1} maps (F, ψ′) ↦ (F, σ(ϕ)ϕ^{−1}ψ′). In our notation this automorphism of Y (n) is denoted \overline{ρ}(σ(ϕ)ϕ^{−1}). □

Fixing a primitive nth root of unity ζ_{n} ∈ \overline{K}, we identify μ_{n} × ℤ/nℤ with (ℤ/nℤ)^{2} via (ζ_{n}^{a}, b) ↦ (a, b). Then Γ = SL_{2}(ℤ/nℤ), and the maps \overline{ρ} defined in (4) and (8) are the same. We now suppose, as happened in Section 2 for n ≥ 5 an odd integer, that X(n) is embedded in ℙ^{m−1} for some m, and \overline{ρ} is realised as a projective representation (also denoted \overline{ρ} by abuse of notation)

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

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

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

for all γ ∈ SL_{2}(ℤ/nℤ). Equivalently, \overline{ρ}(S) and \overline{ρ}(T) are symmetric matrices, where S and T are the generators for SL_{2}(ℤ/nℤ) defined in Section 2.2. Our strategy for computing X_{E}(n) and X_{E}^{−}(n) as twists of X(n) is explained by the following lemma.

Lemma 3.2. Let E/K be an elliptic curve and ϕ : E[n] ≅ μ_{n} ×ℤ/nℤ a symplectic isomorphism defined over \overline{K}. Suppose h_{1}, h_{2} ∈ GL_{m}(\overline{K}) satisfy

σ(h_{1})h_{1}^{−1} ∝ \overline{ρ}(σ(ϕ)ϕ^{−1}) σ(h_{2})h_{2}^{−1} ∝ \overline{ρ}(σ(ϕ)ϕ^{−1})^{−T}

for all σ ∈ Gal(\overline{K}/K). Then X_{E}(n) ⊂ ℙ^{m−1} and X_{E}^{−}(n) ⊂ ℙ^{m−1} are the twists of X(n) ⊂ ℙ^{m−1} given by X_{E}(n) ≅ X(n); x ↦ h_{1}x and X_{E}^{−}(n) ≅ X(n); x ↦ h_{2}x, where x is a point in projective space written as a column vector.

Proof. Let X′ = {x ∈ ℙ^{n−1} | h_{1}x ∈ X(n)}. Since σ(h_{1})h_{1}^{−1} is an automorphism of X(n), we see that X′ is defined over K. Then by Lemma 3.1 the curves X_{E}(n) and X′ are twists of X(n) by the same cocycle. They are therefore isomorphic over K. The proof is the same for X_{E}^{−}(n), except that we apply Lemma 3.1 to the pair (E, ιϕ), and observe by (9) that

\overline{ρ}(σ(ιϕ) (ιϕ)^{−1}) = \overline{ρ}(ισ(ϕ)ϕ^{−1}ι) ∝ \overline{ρ}(σ(ϕ)ϕ^{−1})^{−T} .

For the first equality we use that ι corresponds to an automorphism of μ_{n} × ℤ/nℤ which is defined over K. □

Remark 3.3. If, as happened in Section 2 for n ≥ 5 an odd integer, the projective representation \overline{ρ} lifts to a representation

ρ : SL_{2}(ℤ/nℤ) → GL_{m}(\overline{K})

then the existence of a matrix h_{1} satisfying the conditions in Lemma 3.2 follows from the generalised form of Hilbert’s Theorem 90 which states that

H^{1}(Gal(\overline{K}/K), GL_{m}(\overline{K})) = 0.

We could then take h_{2} = h_{1}^{−T}. Nonetheless we find it more convenient to compute h_{1} and h_{2} using invariant theory for SL_{2}(ℤ/nℤ).

3.2. Formulae in the case n = 7. We saw in Section 2.1 that X(7) is the Klein quartic {F = 0} ⊂ ℙ^{2} where

F = a^{3}b + b^{3}c + c^{3}a.

Let G ≅ PSL_{2}(ℤ/7ℤ) be the image of ρ : SL_{2}(ℤ/7ℤ) → GL_{3}(\overline{K}). It is generated by

\frac{1}{g_{7}} (\begin{matrix}ζ_{7} − ζ_{7}^{6} & ζ_{7}^{2} − ζ_{7}^{5} & ζ_{7}^{4} − ζ_{7}^{3} \\ ζ_{7}^{2} − ζ_{7}^{5} & ζ_{7}^{4} − ζ_{7}^{3} & ζ_{7} − ζ_{7}^{6} \\ ζ_{7}^{4} − ζ_{7}^{3} & ζ_{7} − ζ_{7}^{6} & ζ_{7}^{2} − ζ_{7}^{5} \end{matrix}) and (\begin{matrix}ζ_{7} & 0 & 0 \\ 0 & ζ_{7}^{4} & 0 \\ 0 & 0 & ζ_{7}^{2} \end{matrix})

where g_{7} = 1 + 2(ζ_{7}^{3} + ζ_{7}^{5} + ζ_{7}^{6}) = \sqrt{−7}.

Definition 3.4. An invariant of degree m is a homogeneous polynomial I = I(a, b, c) of degree m such that I ∘ g = I for all g ∈ G.

Following Klein (see for example [7], [16], [19]) we put

H = (−1/54) × | \begin{matrix}\frac{∂^{2}F}{∂a^{2}} & \frac{∂^{2}F}{∂a∂b} & \frac{∂^{2}F}{∂a∂c} \\ \frac{∂^{2}F}{∂a∂b} & \frac{∂^{2}F}{∂b^{2}} & \frac{∂^{2}F}{∂b∂c} \\ \frac{∂^{2}F}{∂a∂c} & \frac{∂^{2}F}{∂b∂c} & \frac{∂^{2}F}{∂c^{2}} \end{matrix} | ,

c_{4} = (1/9) × | \begin{matrix}\frac{∂^{2}F}{∂a^{2}} & \frac{∂^{2}F}{∂a∂b} & \frac{∂^{2}F}{∂a∂c} & \frac{∂H}{∂a} \\ \frac{∂^{2}F}{∂a∂b} & \frac{∂^{2}F}{∂b^{2}} & \frac{∂^{2}F}{∂b∂c} & \frac{∂H}{∂b} \\ \frac{∂^{2}F}{∂a∂c} & \frac{∂^{2}F}{∂b∂c} & \frac{∂^{2}F}{∂c^{2}} & \frac{∂H}{∂c} \\ \frac{∂H}{∂a} & \frac{∂H}{∂b} & \frac{∂H}{∂c} & 0 \end{matrix} | , c_{6} = (1/14) × | \begin{matrix}\frac{∂F}{∂a} & \frac{∂F}{∂b} & \frac{∂F}{∂c} \\ \frac{∂H}{∂a} & \frac{∂H}{∂b} & \frac{∂H}{∂c} \\ \frac{∂c_{4}}{∂a} & \frac{∂c_{4}}{∂b} & \frac{∂c_{4}}{∂c} \end{matrix} | .

The ring of invariants K[a, b, c]^{G} is generated by F, H, c_{4} and c_{6} subject to a single relation which reduces when we set F = 0 to

c_{4}^{3} − c_{6}^{2} ≡ 1728H^{7} (mod F).

Since F, H, c_{4} and c_{6} have degrees 4, 6, 14 and 21 it is clear that every invariant of odd degree is divisible by c_{6}.

Lemma 3.5. The j-invariant X(7) → ℙ^{1} is given by j = c_{4}^{3}/H^{7}.

Proof. Both j and j_{0} = c_{4}^{3}/H^{7} define maps X(7) → ℙ^{1} that quotient out by the action of G ≅ PSL_{2}(ℤ/7ℤ). So they can differ by at most a Möbius map. We recall that j is ramified above 0, 1728 and ∞ with ramification indices 3, 2 and 7. Since

#\{F = c_{4} = 0\} ≤ 4 deg(c_{4}) = \frac{1}{3}|G|
#\{F = c_{6} = 0\} ≤ 4 deg(c_{6}) = \frac{1}{2}|G|
#\{F = H = 0\} ≤ 4 deg(H) = \frac{1}{7}|G|

and j_{0} − 1728 = c_{6}^{2}/H^{7}, we see that j_{0} is ramified above 0, 1728 and ∞ with ramification indices at least 3, 2 and 7. It follows that j = j_{0} as required. □

Definition 3.6. A covariant column, respectively contravariant column, of degree m is a column vector v = (v_{1}, v_{2}, v_{3})^{T} of homogeneous polynomials of degree m in variables a, b, c such that v ∘ g = gv, respectively v ∘ g = g^{−T} v, for all g ∈ G.

We note that x = (a, b, c)^{T} is a covariant column of degree 1, whereas if I is an invariant of degree m then ∇I = (\frac{∂I}{∂a}, \frac{∂I}{∂b}, \frac{∂I}{∂c})^{T} is a contravariant column of degree m − 1.

Lemma 3.7. Let E/K be an elliptic curve and ϕ : E[7] ≅ μ_{7} × ℤ/7ℤ a symplectic isomorphism defined over \overline{K}. Let (a : b : c) be the corresponding \overline{K}-point on X(7) ⊂ ℙ^{2} with co-ordinates (a, b, c) scaled so that

(10)  c_{4}(a, b, c) = c_{4}(E) and c_{6}(a, b, c) = c_{6}(E)

where E has Weierstrass equation y^{2} = x^{3}−27c_{4}(E)x−54c_{6}(E). If j(E) ≠ 0, 1728 and h ∈ GL_{3}(\overline{K}) is a matrix whose columns are covariant columns, respectively contravariant columns, of the same degree mod 7, evaluated at (a, b, c) then

σ(h)h^{−1} ∝ ρ(σ(ϕ)ϕ^{−1}),

respectively

σ(h)h^{−1} ∝ ρ(σ(ϕ)ϕ^{−1})^{−T} ,

for all σ ∈ Gal(\overline{K}/K).

Proof. Let ξ_{σ} = σ(ϕ)ϕ^{−1} ∈ SL_{2}(ℤ/7ℤ). Since ρ describes the action of SL_{2}(ℤ/7ℤ) on X(7) ⊂ ℙ^{2} we have

(11)  σ((a, b, c)^{T}) = λ_{σ}ρ(ξ_{σ})(a, b, c)^{T}

for some λ_{σ} ∈ \overline{K}^{×}. Now ρ(ξ_{σ}) ∈ G, whereas c_{4} and c_{6} are homogeneous polynomials of degrees 14 and 21 invariant under the action of G. Therefore

σ(c_{4}(a, b, c)) = λ_{σ}^{14}c_{4}(a, b, c) and σ(c_{6}(a, b, c)) = λ_{σ}^{21}c_{6}(a, b, c)

for all σ ∈ Gal(\overline{K}/K). We are given that c_{4}(E), c_{6}(E) ∈ K. So by (10), and our assumption j(E) ≠ 0, 1728, we have λ_{σ}^{14} = λ_{σ}^{21} = 1. Hence λ_{σ} is a 7th root of unity. Now suppose the columns of h are obtain by specialising polynomials whose degrees are all congruent to r mod 7. Then by (11) and Definition 3.6 we have

σ(h) = h ∘ (λ_{σ}ρ(ξ_{σ})) = λ_{σ}^{r}ρ(ξ_{σ})h,

respectively

σ(h) = h ∘ (λ_{σ}ρ(ξ_{σ})) = λ_{σ}^{r}ρ(ξ_{σ})^{−T} h.

Therefore σ(h)h^{−1} ∝ ρ(ξ_{σ}), respectively σ(h)h^{−1} ∝ ρ(ξ_{σ})^{−T}, as required. □

We use Lemmas 3.2 and 3.7 to compute equations for X_{E}(7) and X_{E}^{−}(7). First we classify the covariant and contravariant columns. It is evident that

• The dot product of a covariant column and a contravariant column is an invariant.

• The cross product of two covariant columns is a contravariant column.

• The cross product of two contravariant columns is a covariant column. We also write [v_{1}, v_{2}, v_{3}] = (v_{1} × v_{2}) · v_{3} for the scalar triple product. It is straightforward to solve for the covariant and contravariant columns of any given degree by linear algebra. As above we put x = (a, b, c)^{T}. Let e and f be the covariant columns of degrees 9 and 11 given by

e = \frac{I_{22}(∇F × ∇H) − c_{4}(∇F × ∇c_{4}) + 12H^{2}(∇H × ∇c_{4})}{14c_{6}}
f = \frac{I_{24}(∇F × ∇H) − (16F^{4} − 104F H^{2})(∇F × ∇c_{4}) + c_{4}(∇H × ∇c_{4})}{14c_{6}}

where

I_{22} = 448F^{4}H − 48F^{2}c_{4} − 2048F H^{3}
I_{24} = 128F^{6} − 160F^{3}H^{2} − 236F Hc_{4} − 336H^{4}.

We describe the covariant and contravariant columns as modules over the ring K[F, H, c_{4}] of invariants of even degree.

Lemma 3.8. (i) The covariant columns of odd, respectively even, degree form a free K[F, H, c_{4}]-module of rank 3 generated by x, e, f, respectively ∇F × ∇H, ∇F × ∇c_{4}, ∇H × ∇c_{4}.

(ii) The contravariant columns of odd, respectively even, degree form a free K[F, H, c_{4}]-module of rank 3 generated by ∇F, ∇H, ∇c_{4}, respectively x × e, x × f, e × f.

Proof. By direct calculation we have [x, e, f] = −c_{6}, whereas the definition of c_{6} may be rewritten as [∇F, ∇H, ∇c_{4}] = 14c_{6}. Since c_{6} is not identically zero it follows that x, e, f are linearly independent over K(a, b, c), and likewise for ∇F, ∇H, ∇c_{4}.

Let v be a covariant column of odd degree. We write v = I_{1}x + I_{2}e + I_{3}f where I_{1}, I_{2}, I_{3} are rational functions in a, b, c. Taking the dot product with e × f shows that [v, e, f] = I_{1}[x, e, f]. But [v, e, f] is an invariant of odd degree and therefore divisible by c_{6}. It follows that I_{1} is an invariant, and likewise for I_{2} and I_{3}.

The other cases are similar. □

Theorem 3.9. Let E/K be an elliptic curve with Weierstrass equation y^{2} = x^{3} − 27c_{4}x − 54c_{6} and let Δ = (c_{4}^{3} − c_{6}^{2})/1728. If j(E) ≠ 0, 1728 then X_{E}(7) ⊂ ℙ^{2} has equation F = 0 where

F = 12x^{3}z + 108x^{2}y^{2} + 3c_{4}x^{2}z^{2} + 72c_{4}xy^{2}z − 108c_{4}y^{4} − 12c_{6}xyz^{2}
+ 84c_{6}y^{3}z + c_{4}^{2}xz^{3} − 15c_{4}^{2}y^{2}z^{2} + c_{4}c_{6}yz^{3} + 768Δz^{4},

and X_{E}^{−}(7) ⊂ ℙ^{2} has equation G = 0 where

G = 3x^{4} + c_{4}x^{3}z − 18c_{4}x^{2}y^{2} − 3c_{6}x^{2}yz + 24c_{6}xy^{3} + 3c_{4}^{2}xy^{2}z
− 9c_{4}^{2}y^{4} − c_{4}c_{6}y^{3}z + 168Δxz^{3} + 1728Δy^{2}z^{2} + 5c_{4}Δz^{4}.

Proof. The covariant columns x, ∇F × ∇H, He have degrees 1, 8, 15, and the contravariant columns ∇F, x×e, H^{2}∇H have degrees 3, 10, 17. The determinants of the matrices formed from these columns are

(12)

det(x, (∇F × ∇H), He) = 72H^{4} − 4c_{4}F H
det(∇F, (x × e), H^{2}∇H) = 72H^{5} − 4c_{4}F H^{2}.

Therefore the matrices h_{1} and h_{2} obtained by evaluating at a point of Y (7) are non-singular.

The coefficients of the quartic F̃(x, y, z) = F (xx + y(∇F × ∇H) + zHe) are invariants. Using linear algebra to rewrite these invariants as polynomials in F, H, c_{4} and c_{6} we find

F̃(x, y, z) = F x^{4} + 12H^{3}x^{3}z + (108H^{3} − 6c_{4}F)x^{2}y^{2} − 8c_{6}F xy^{3}
+ 3c_{4}H^{3}x^{2}z^{2} + (72c_{4}H^{3} + 4128F^{2}H^{4} + 48c_{4}F^{3}H − 768F^{5}H^{2})xy^{2}z

+ (−108c_{4}H^{3} − 3c_{4}^{2}F − 11376F^{2}H^{4} + 32c_{4}F^{3}H + 3392F^{5}H^{2} − 256F^{8})y^{4}

− 12c_{6}H^{3}xyz^{2} + (84c_{6}H^{3} − 16c_{6}F^{3}H)y^{3}z + (c_{4}^{2}H^{3} + 688F H^{7}
+ 8c_{4}F^{2}H^{4} − 128F^{4}H^{5})xz^{3} + (−15c_{4}^{2}H^{3} − 10512F H^{7} − 384c_{4}F^{2}H^{4}
+ 6144F^{4}H^{5} + 96c_{4}F^{5}H^{2} − 768F^{7}H^{3})y^{2}z^{2} + (c_{4}c_{6}H^{3} − 8c_{6}F^{2}H^{4})yz^{3}

+ (768H^{10} − 36c_{4}F H^{7} − c_{4}^{2}F^{2}H^{4} + 176F^{3}H^{8} + 16c_{4}F^{4}H^{5} − 64F^{6}H^{6})z^{4}.

Likewise G̃(x, y, z) = F (x∇F + y(x × e) + zH^{2}∇H) becomes

G̃(x, y, z) = (3H^{2} + 28F^{3})x^{4} + (c_{4}H^{2} + 168F^{2}H^{3})x^{3}z + (−18c_{4}H^{2}
− 816F^{2}H^{3} − 24c_{4}F^{3} + 192F^{5}H)x^{2}y^{2} − 3c_{6}H^{2}x^{2}yz + 24c_{6}H^{2}xy^{3}
+ (222F H^{6} + 24F^{4}H^{4})x^{2}z^{2} + (3c_{4}^{2}H^{2} + 3744F H^{6} − 576F^{4}H^{4})xy^{2}z
+ (−9c_{4}^{2}H^{2} − 5184F H^{6} − 240c_{4}F^{2}H^{3} − 4c_{4}^{2}F^{3} + 2240F^{4}H^{4} + 64c_{4}F^{5}H
− 256F^{7}H^{2})y^{4} + (−c_{4}c_{6}H^{2} + 8c_{6}F^{2}H^{3})y^{3}z + (168H^{9} + 3c_{4}F H^{6}
+ 24F^{3}H^{7})xz^{3} + (1728H^{9} − 78c_{4}F H^{6} + 816F^{3}H^{7} + 24c_{4}F^{4}H^{4}
− 192F^{6}H^{5})y^{2}z^{2} + c_{6}F H^{6}yz^{3} + (5c_{4}H^{9} + 35F^{2}H^{10} − 4F^{5}H^{8})z^{4}.

Let (a : b : c) be the \overline{K}-point on X(7) corresponding to (E, ϕ) for some choice of symplectic isomorphism ϕ : E[7] ≅ μ_{7} × ℤ/7ℤ. By Lemma 3.5 we may scale (a, b, c) to satisfy (10). With this choice of scaling we also have H(a, b, c)^{7} = Δ. By Lemma 3.7 the matrix h_{1}, respectively h_{2}, formed by evaluating the covariant, respectively contravariant, columns at (a, b, c), satisfies the conditions of Lemma 3.2. Therefore a formula for X_{E}(7), respectively X_{E}^{−}(7), is given by specialising the coefficients of F̃, respectively G̃, to this choice of (a, b, c). Explicitly we set F = 0, divide through by H^{3}, respectively H^{2}, and replace H^{7} by Δ. □

Remark 3.10. It is not immediately clear how the equations for X_{E}(7) and X_{E}^{−}(7) found by our method (see Theorem 3.9) are related to those already in the literature (see Theorem 1.1). In fact writing a = −27c_{4} and b = −54c_{6} we have

F(x, y, z) = \frac{1}{4}F(6c_{4}z − \frac{1}{3}y, x, −18z), G(x, y, z) = G(9c_{4}y + z, 3x, 108y).

3.3. Formulae in the case n = 11. We saw in Section 2.1 that X(11) is the singular locus of the Hessian of the cubic threefold {F = 0} ⊂ ℙ^{4} where

F = a^{2}b + b^{2}c + c^{2}d + d^{2}e + e^{2}a.

Let G ≅ PSL_{2}(ℤ/11ℤ) be the image of ρ : SL_{2}(ℤ/11ℤ) → GL_{5}(\overline{K}). It is generated by

\frac{1}{g_{11}} (\begin{matrix}ζ_{11} − ζ_{11}^{−1} & ζ_{11}^{3} − ζ_{11}^{−3} & ζ_{11}^{9} − ζ_{11}^{−9} & ζ_{11}^{5} − ζ_{11}^{−5} & ζ_{11}^{4} − ζ_{11}^{−4} \\ ζ_{11}^{3} − ζ_{11}^{−3} & ζ_{11}^{9} − ζ_{11}^{−9} & ζ_{11}^{5} − ζ_{11}^{−5} & ζ_{11}^{4} − ζ_{11}^{−4} & ζ_{11} − ζ_{11}^{−1} \\ ζ_{11}^{9} − ζ_{11}^{−9} & ζ_{11}^{5} − ζ_{11}^{−5} & ζ_{11}^{4} − ζ_{11}^{−4} & ζ_{11} − ζ_{11}^{−1} & ζ_{11}^{3} − ζ_{11}^{−3} \\ ζ_{11}^{5} − ζ_{11}^{−5} & ζ_{11}^{4} − ζ_{11}^{−4} & ζ_{11} − ζ_{11}^{−1} & ζ_{11}^{3} − ζ_{11}^{−3} & ζ_{11}^{9} − ζ_{11}^{−9} \\ ζ_{11}^{4} − ζ_{11}^{−4} & ζ_{11} − ζ_{11}^{−1} & ζ_{11}^{3} − ζ_{11}^{−3} & ζ_{11}^{9} − ζ_{11}^{−9} & ζ_{11}^{5} − ζ_{11}^{−5} \end{matrix})

and Diag(ζ_{11}, ζ_{11}^{9}, ζ_{11}^{4}, ζ_{11}^{3}, ζ_{11}^{5}), where g_{11} = 1+2(ζ_{11} +ζ_{11}^{3} +ζ_{11}^{9} +ζ_{11}^{5} +ζ_{11}^{4}) = \sqrt{−11}.

We define the invariants, covariant columns and contravariant columns exactly as in Section 3.2. Let ∑ denote a sum over all cyclic permutations, so that for example F = ∑ a^{2}b. Other examples of invariants of small degree include

H = 3abcde + ∑(a^{3}c^{2} − a^{3}de),
I_{7} = ∑(a^{6}e + 3a^{5}d^{2} − 15a^{4}bce + 5a^{3}b^{3}d + 15a^{3}bcd^{2}),
I_{8} = ∑(a^{7}c − 7a^{4}bd^{3} − 7a^{4}de^{3} + 7a^{3}b^{2}c^{3} + 21a^{3}c^{2}d^{2}e).

Writing A and B for the matrices of second partial derivatives of F and H we find

det(A + tB) = 32H − 32I_{7}t − 24I_{9}t^{2} − 8c_{4}t^{3} + …

where I_{9} and c_{4} are invariants of degrees 9 and 11. Although we will not need a complete set of generators for the ring of invariants, we remark that such a set is given in [1], and may also be computed using Magma. Let I be the homogeneous ideal of X(11), i.e. the ideal generated by the 4 × 4 minors of the Hessian matrix of F. The degree 19 polynomial

c̃_{6} = a^{9}b^{10} − 509b^{18}d − 14107b^{14}d^{4}e + 510b^{9}c^{10} + 42326b^{7}d^{12} + 20669b^{3}d^{15}e
− 14107b^{2}d^{2}e^{15} − 277419bc^{2}d^{10}e^{6} − 248909bcd^{16}e − 209926bcd^{5}e^{12}
+ 762409bd^{11}e^{7} + be^{18} − 1018c^{18}e − 14107c^{16}de^{2} − 586835c^{12}d^{3}e^{4}

+ 197780c^{10}d^{4}e^{5} + 1019c^{9}d^{10} − 787130c^{8}d^{5}e^{6} + 15634c^{7}d^{11}e + 42326c^{7}e^{12}

+ 2007576c^{6}d^{6}e^{7} + 247382c^{5}d^{12}e^{2} − 528424c^{5}de^{13} − 616653c^{4}d^{7}e^{8}
+ 376744c^{3}d^{13}e^{3} + 1067732c^{3}d^{2}e^{14} − 225004c^{2}d^{8}e^{9} + 463659cd^{14}e^{4}
− 582142cd^{3}e^{15} + 70511d^{9}e^{10},

is not an invariant but satisfies

c̃_{6}^{2} ≡ abcde(c_{4}^{3} − 1728F^{11}) (mod I).

Lemma 3.11. The j-invariant X(11) → ℙ^{1} is given by j = c_{4}^{3}/F^{11}.

Proof. Both j and j_{0} = c_{4}^{3}/F^{11} define maps X(11) → ℙ^{1} that quotient out by the action of G ≅ PSL_{2}(ℤ/11ℤ). So they can differ by at most a Möbius map. We recall that j is ramified above 0, 1728 and ∞ with ramification indices 3, 2 and 11. It is shown in [2, Corollary 23.28] that X(11) ⊂ ℙ^{4} has degree 20. Since

#X(11) ∩ \{c_{4} = 0\} ≤ 20 deg(c_{4}) = \frac{1}{3}|G|
#X(11) ∩ {c̃_{6} = 0} ≤ 20 deg(c̃_{6}) < |G|
#X(11) ∩ \{F = 0\} ≤ 20 deg(F) = \frac{1}{11}|G|

and j_{0} − 1728 = c̃_{6}^{2}/((abcde)F^{11}) it follows that j = j_{0} as required. □

Lemma 3.12. Let E/K be an elliptic curve and ϕ : E[11] ≅ μ_{11} × ℤ/11ℤ a symplectic isomorphism defined over \overline{K}. Let (a : b : c : d : e) be the corresponding \overline{K}-point on X(11) ⊂ ℙ^{4} with co-ordinates (a, b, c, d, e) scaled so that

(13)  c_{4}(a, b, c, d, e) = c_{4}(E)

where E has Weierstrass equation y^{2} = x^{3} − 27c_{4}(E)x − 54c_{6}(E). If j(E) ≠ 0 and h ∈ GL_{5}(\overline{K}) is a matrix whose columns are covariant columns, respectively contravariant columns, of the same degree mod 11, evaluated at (a, b, c, d, e) then

σ(h)h^{−1} ∝ ρ(σ(ϕ)ϕ^{−1}),

respectively

σ(h)h^{−1} ∝ ρ(σ(ϕ)ϕ^{−1})^{−T} ,

for all σ ∈ Gal(\overline{K}/K).

Proof. The proof is similar to that of Lemma 3.7. Recall that c_{4} is a homogeneous polynomial of degree 11 and so (13) determines the scaling of (a, b, c, d, e) up to an 11th root of unity. □

We use Lemmas 3.2 and 3.12 to compute equations for X_{E}(11) and X_{E}^{−}(11). First we compute some covariant columns. Let x_{1} = (a, b, c, d, e)^{T}. If γ ∈ SL_{2}(ℤ/11ℤ) is diagonal then ρ(γ) cyclically permutes the co-ordinates a, b, c, d, e.

18 TOM FISHER

A covariant column is therefore uniquely determined by its first entry. By averaging over the group we found covariant columns x_{4}, x_{5}, x_{9} with first entries

f_{4} = 2a^{2}e^{2} + 4ab^{2}c − 4ac^{2}d + 4bce^{2} + d^{4},
f_{5} = −5a^{3}ce + 5a^{2}b^{2}d + 5a^{2}cd^{2} + 5abc^{2}e − 10abde^{2} + b^{5} − 5b^{3}cd + 5bd^{3}e + 5c^{2}e^{3},
f_{9} = −14a^{6}bde − 8a^{5}bd^{3} + 9a^{5}c^{2}e^{2} + 2a^{5}de^{3} + 8a^{4}b^{4}e + 5a^{4}b^{2}c^{3} + 63a^{4}b^{2}cde

+ 6a^{4}c^{4}d − 18a^{4}c^{2}d^{2}e + 8a^{4}d^{3}e^{2} + 31a^{3}b^{4}d^{2} − 21a^{3}b^{3}e^{3} + 47a^{3}b^{2}cd^{3} + 35a^{3}bc^{3}e^{2} + 14a^{3}bcde^{3} − 12a^{3}c^{2}d^{4} + 10a^{3}d^{5}e + 3a^{2}b^{5}ce − 26a^{2}b^{3}c^{4} − 42a^{2}b^{3}c^{2}de − 75a^{2}b^{3}d^{2}e^{2} + 3a^{2}b^{2}e^{5} + 18a^{2}bc^{5}d − 30a^{2}bc^{3}d^{2}e − 36a^{2}bcd^{3}e^{2} + 2a^{2}c^{3}e^{4} − 9a^{2}cde^{5} + a^{2}d^{7} − 2ab^{7}d − 6ab^{5}cd^{2} + 50ab^{4}ce^{3} − 7ab^{3}c^{2}d^{3} − 6ab^{3}d^{4}e − 54ab^{2}c^{4}e^{2} − 3ab^{2}c^{2}de^{3} − 9ab^{2}d^{2}e^{4} − 29abc^{3}d^{4} + 21abcd^{5}e + abe^{7} + 9ac^{5}de^{2} + 25ac^{3}d^{2}e^{3} − 7acd^{3}e^{4} − 10b^{6}c^{2}e − 2b^{6}de^{2} + 4b^{4}c^{5} + 40b^{4}c^{3}de − 6b^{4}cd^{2}e^{2} + 13b^{3}ce^{5} − 3b^{3}d^{6} − 15b^{2}c^{4}d^{2}e − 54b^{2}c^{2}d^{3}e^{2} + 31b^{2}d^{4}e^{3} − 11bc^{4}e^{4} + 3bc^{2}de^{5} − 2bcd^{7} − 7bd^{2}e^{6} − c^{7}d^{2} + 5c^{5}d^{3}e − 9c^{3}d^{4}e^{2} + 8cd^{5}e^{3} − e^{9}.

We temporarily write a_{1}, …, a_{5} for a, b, c, d, e and let Ξ be the 5 × 5 alternating matrix with entries

Ξ_{ij} = \frac{∂F}{∂a_{r}} \frac{∂I_{7}}{∂a_{s}} − \frac{∂F}{∂a_{s}} \frac{∂I_{7}}{∂a_{r}}

where r ≡ (i − j − 2)^{3} + i + 3 (mod 5) and s ≡ (j − i − 2)^{3} + j + 3 (mod 5). Then x_{14} = Ξ∇I_{7} is a covariant column of degree 14. Theorem 3.13. Let E/K be an elliptic curve with Weierstrass equation y^{2} = x^{3} − 27c_{4}x − 54c_{6} and let Δ = (c_{4}^{3} − c_{6}^{2})/1728. If j(E) ≠ 0, 1728 then X_{E}(11) ⊂ ℙ^{4} is the singular locus of the Hessian of

F = v^{3} + 3v^{2}w + c_{4}v^{2}y + 3vw^{2} + 2c_{4}vwy − c_{4}Δvx^{2} + 48Δvxy + 9w^{3}

+ 5c_{4}w^{2}y − c_{4}^{2}w^{2}z + c_{4}^{2}wy^{2} − 576Δwyz + 72c_{4}Δwz^{2} − 4Δ^{2}x^{3} − 72Δ^{2}x^{2}z + 4c_{4}Δxy^{2} + 2c_{4}^{2}Δxyz − (c_{4}^{3}Δ − 1728Δ^{2})xz^{2} + 64Δy^{3} − 72c_{4}Δy^{2}z + 12c_{4}^{2}Δyz^{2} + (c_{4}^{3}Δ − 3456Δ^{2})z^{3},

and X_{E}^{−}(11) ⊂ ℙ^{4} is the singular locus of the Hessian of

G = 5v^{3} − c_{4}v^{2}x − 60v^{2}y + 28c_{4}v^{2}z − 2c_{4}Δvw^{2} − 48Δvwx − 240Δvwz
− 16c_{4}vxy + 1680vy^{2} − 872c_{4}vyz + 121c_{4}^{2}vz^{2} + 8Δ^{2}w^{3} + 44c_{4}Δw^{2}y

− 11c_{4}^{2}Δw^{2}z + c_{4}Δwx^{2} + 336Δwxy − 122c_{4}Δwxz + 25c_{4}^{2}wy^{2} − 14160Δwyz

+ 817c_{4}Δwz^{2} − 20Δx^{3} + 5c_{4}^{2}x^{2}y − 1884Δx^{2}z − 364c_{4}xy^{2} + 160c_{4}^{2}xyz
− 34764Δxz^{2} + 19840y^{3} − 10268c_{4}y^{2}z + 1643c_{4}^{2}yz^{2} − 129220Δz^{3}.

Proof. The covariant columns x_{1}, x_{4}, x_{5}, x_{9}, x_{14} have degrees 1, 4, 5, 9, 14 and the contravariant columns ∇F, ∇I_{7}, ∇I_{8}, ∇I_{9}, ∇c_{4} have degrees 2, 6, 7, 8, 10. The determinants of the matrices formed from these columns satisfy

(14)

det(x_{1}, x_{4}, x_{5}, x_{9}, x_{14}) = c_{4}^{3} − 1728F^{11} (mod I)
det(∇F, ∇I_{7}, ∇I_{8}, ∇I_{9}, ∇c_{4}) = 55(c_{4}^{3} − 1728F^{11}) (mod I).

The coefficients of the cubic F̃(v, w, x, y, z) = F (vx_{1}+wx_{4}+xx_{5}+yx_{9}+zx_{14}) are invariants. Using the Gröbner basis machinery in Magma to rewrite the coefficients mod I as polynomials in c_{4} and F we find

F̃ = F v^{3} + 3F^{2}v^{2}w + c_{4}v^{2}y + 3F^{3}vw^{2} + 2F c_{4}vwy − c_{4}vx^{2} + 48F^{5}vxy + 9F^{4}w^{3}

+ 5F^{2}c_{4}w^{2}y − c_{4}^{2}w^{2}z + c_{4}^{2}wy^{2} − 576F^{9}wyz + 72F^{7}c_{4}wz^{2} − 4F^{5}x^{3} − 72F^{8}x^{2}z + 4F^{4}c_{4}xy^{2} + 2F^{2}c_{4}^{2}xyz − (c_{4}^{3} − 1728F^{11})xz^{2} + 64F^{9}y^{3} − 72F^{7}c_{4}y^{2}z + 12F^{5}c_{4}^{2}yz^{2} + (F^{3}c_{4}^{3} − 3456F^{14})z^{3}.

Likewise G̃(v, w, x, y, z) = F (v∇F + w∇I_{7} + x∇I_{8} + y∇I_{9} + z∇c_{4}) becomes

G̃ = 5F^{2}v^{3} − c_{4}v^{2}x − 60F^{4}v^{2}y + 28F c_{4}v^{2}z − 2F c_{4}vw^{2} − 48F^{5}vwx − 240F^{6}vwz

− 16F^{2}c_{4}vxy + 1680F^{6}vy^{2} − 872F^{3}c_{4}vyz + 121c_{4}^{2}vz^{2} + 8F^{6}w^{3} + 44F^{3}c_{4}w^{2}y − 11c_{4}^{2}w^{2}z + F^{3}c_{4}wx^{2} + 336F^{7}wxy − 122F^{4}c_{4}wxz + 25c_{4}^{2}wy^{2} − 14160F^{8}wyz + 817F^{5}c_{4}wz^{2} − 20F^{7}x^{3} + 5c_{4}^{2}x^{2}y − 1884F^{8}x^{2}z − 364F^{4}c_{4}xy^{2} + 160F c_{4}^{2}xyz − 34764F^{9}xz^{2} + 19840F^{8}y^{3} − 10268F^{5}c_{4}y^{2}z + 1643F^{2}c_{4}^{2}yz^{2} − 129220F^{10}z^{3}.

Let (a : b : c : d : e) be the \overline{K}-point on X(11) corresponding to (E, ϕ) for some choice of symplectic isomorphism ϕ : E[11] ≅ μ_{11} × ℤ/11ℤ. By Lemma 3.11 we may scale (a, b, c, d, e) to satisfy (13) and F (a, b, c, d, e)^{11} = Δ. Moreover the determinants (14) are non-zero by our assumption j(E) ≠ 1728. By Lemmas 3.2 and 3.12 we obtain cubic forms describing X_{E}(11) and X_{E}^{−}(11) by putting

(15)  \begin{aligned}F(v, w, x, y, z) & = \frac{1}{F^{4}} F̃(F v, w, F^{7}x, F^{2}y, F^{4}z) \\ G(v, w, x, y, z) & = \frac{1}{F^{8}} G̃(F^{2}v, F^{8}w, F^{4}x, y, F^{3}z) \end{aligned}

and replacing F^{11} by Δ. □

Remark 3.14. We simplify the cubic forms F and G in Theorem 3.13 by putting

F(v, w, x, y, z) = \frac{1}{2^{3}c_{6}^{3}} F(−v′, w′, −864x, −36c_{4}x − 108c_{6}z, 72y)
G(v, w, x, y, z) = \frac{1}{2^{5}3^{6}(55c_{6})^{3}} G(v′′, −427680y, x′′, −y′′, −z′′)

where a = −27c_{4}, b = −54c_{6} and

v′ = c_{6}v + 2c_{6}w − 6c_{4}^{2}x + 3c_{4}^{2}y − 9c_{4}c_{6}z  w′ = c_{6}v + 6c_{4}^{2}x + 3c_{4}^{2}y + 9c_{4}c_{6}z
v′′ = 44(2c_{4}v − 6c_{6}w + 33c_{6}x + 135c_{4}^{2}y + 810c_{4}c_{6}z) x′′ = 60(5v + 729c_{4}y + 2187c_{6}z)
y′′ = 11(c_{4}v − 3c_{6}w − 6c_{6}x)  z′′ = 60(v + 27c_{4}y + 81c_{6}z).

This gives the cubic forms F and G in Theorem 1.2.

4. Modular interpretation

In Section 3 we computed equations for X_{E}(n) and X_{E}^{−}(n) for n = 7, 11. In this section we compute equations for the families of curves they parametrise.

4.1. Computing the j-invariant. We give formulae for the j-maps X_{E}(n) → ℙ^{1} and X_{E}^{−}(n) → ℙ^{1} by adapting the formulae in Lemmas 3.5 and 3.11.

First we define the invariant Ψ(F) of a polynomial F of the form considered in Theorems 1.1 and 1.2. For F a polynomial in variables x_{1}, …, x_{m} and M = (M_{ij}) an m × m matrix we write F ∘ M for F (x′_{1}, …, x′_{m}) where x′_{i} = ∑_{j} M_{ij}x_{j}.

Definition 4.1. We split into the cases n = 7, 11.

(i) The invariant Ψ of a twisted form μ(F ∘ M) of F = x^{3}y + y^{3}z + z^{3}x is

Ψ(μ(F ∘ M)) := μ^{3}(det M)^{4}.

(ii) The invariant Ψ of a twisted form μ(F ∘ M) of F = v^{2}w + w^{2}x + x^{2}y +

y^{2}z + z^{2}v is

Ψ(μ(F ∘ M)) := μ^{5}(det M)^{3}.

Lemma 4.2. Let F be one of the twisted forms in Definition 4.1. Then (i) Ψ(F) is well-defined, i.e. it is independent of the choice of M ∈ GL_{m}(\overline{K}). (ii) If F has coefficients in K then Ψ(F) ∈ K.

Proof. (i) This is easy to check for M a scalar matrix. In general we appeal to the fact, proved in [2, Lemma 20.4], that Aut(X(n)) ≅ PSL_{2}(ℤ/nℤ). Therefore it suffices to consider M = ρ(γ) for γ ∈ SL_{2}(ℤ/nℤ). We then use that SL_{2}(ℤ/nℤ) has no non-trivial 1-dimensional characters. (ii) This follows from (i) by Galois theory. □

Remark 4.3. (i) In the case n = 7 it is shown in [26, Section 7.1] that Ψ(F) is an integer coefficient polynomial in the coefficients of F. We expect this is also true in the case n = 11, but we have not worked out the details.

(ii) By following the proofs in Section 3, the twisted forms in Theorems 1.1 and 1.2 have invariants

X_{E}(n)  X_{E}^{−}(n)
n = 7 −4(4a^{3} + 27b^{2}) 16(4a^{3} + 27b^{2})^{2}
n = 11 −4(4a^{3} + 27b^{2})^{2} 8(4a^{3} + 27b^{2}).

We now split into the cases n = 7, 11 and give formulae for the j-map.

Case n = 7. Let X = {F = 0} ⊂ ℙ^{2} be a twist of X(7). Starting with F in place of the Klein quartic F, the formulae in Section 3.2 define polynomials H(F), c_{4}(F) and c_{6}(F). If F = μ(F ∘ M) then Ψ(F) = μ^{3}(det M)^{4} and

(16)

H(F) = μ^{3}(det M)^{2}(H ∘ M)
c_{4}(F) = μ^{8}(det M)^{6}(c_{4} ∘ M)
c_{6}(F) = μ^{12}(det M)^{9}(c_{6} ∘ M)

As observed in [26, Lemma 7.2], the syzygy c_{4}^{3} − c_{6}^{2} ≡ 1728H^{7} (mod F) becomes

c_{4}(F)^{3} − c_{6}(F)^{2} ≡ 1728 Ψ(F) H(F)^{7} (mod F).

In particular the j-map X → ℙ^{1} is given by

j = \frac{c_{4}(F)^{3}}{Ψ(F)H(F)^{7}} .

Case n = 11. Let X ⊂ ℙ^{4} be a twist of X(11) given as the singular locus of the Hessian of a cubic form F = F(v, w, x, y, z). Starting with F in place of the cubic form F = v^{2}w + w^{2}x + x^{2}y + y^{2}z + z^{2}v, the formulae in Section 3.3 define polynomials H(F) and c_{4}(F). If F = μ(F ∘ M) then Ψ(F) = μ^{5}(det M)^{3} and

(17)  \begin{aligned}H(F) & = μ^{5}(det M)^{2}(H ∘ M) \\ c_{4}(F) & = μ^{17}(det M)^{8}(c_{4} ∘ M) \end{aligned}

By Lemma 3.11 the j-map X → ℙ^{1} is given by

j = \frac{c_{4}(F)^{3}}{Ψ(F)^{8}F^{11}} .

4.2. Modular interpretation of X(n). In Section 2.1 we gave equations for X(n). In [8, Chapter 4] it is shown that (analogous to Definition 2.2) the elliptic curve E ⊂ ℙ^{n−1} above (0 : a_{1} : a_{2} : …: −a_{2} : −a_{1}) ∈ Y (n) is defined by

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

The following theorem gives an equation for this curve in Weierstrass form. Notice that the coefficients are homogeneous polynomials of degrees 4t and 6t for some integer t. An alternative proof in the case n = 7 is sketched in [16, Section 3].

Theorem 4.4. We split into the cases n = 7, 11.

(i) The family of curves parametrised by X(7) = {a^{3}b + b^{3}c + c^{3}a = 0} ⊂ ℙ^{2} is

(18)  y^{2} = x^{3} − 27(abc)^{2}c_{4}(a, b, c)x − 54(abc)^{3}c_{6}(a, b, c)

where c_{4}, c_{6} ∈ K[a, b, c] are as defined in Section 3.2. (ii) The family of curves parametrised by X(11) ⊂ ℙ^{4} is

(19)  y^{2} = x^{3} − 27(abcde)c_{4}(a, b, c, d, e)x − 54(abcde)c̃_{6}(a, b, c, d, e).

where c_{4}, c̃_{6} ∈ K[a, b, c, d, e] are as defined in Section 3.3.

Proof. The modular curve Y_{1}(n) parametrises pairs (E, P) where E is an elliptic curve and P ∈ E is a point of order n. If n = 7 then we choose a coordinate λ on X_{1}(7) ≅ ℙ^{1}. If n = 11 then X_{1}(11) is the elliptic curve ν^{2} + ν = λ^{3} − λ^{2}. We write λ to indicate λ in the cases n = 7, and the pair λ, ν in the case n = 11. By [31, Exercise 8.13] the elliptic curves D_{λ} parametrised by Y_{1}(7) and Y_{1}(11) have Weierstrass equations

y^{2} − (λ^{2} − λ − 1)xy − (λ^{3} − λ^{2})y = x^{3} − (λ^{3} − λ^{2})x^{2},
y^{2} + (λν + 2λ − (ν + 1)^{2})xy − λ^{2}ν(ν + 1)(λ − ν − 1)y = x^{3} − λν(ν + 1)(λ − ν − 1)x^{2}.

On each of these curves P = (0, 0) is a point of order n. If we write the Weierstrass equation for D_{λ} as y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} then by Vélu’s formulae [32] the n-isogenous elliptic curve C_{λ} = D_{λ}/⟨P ⟩ has Weierstrass equation

(20)  y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} − 5tx − (a_{1}^{2} + 4a_{2})t − 7w

where t = 6s_{2} + (a_{1}^{2} + 4a_{2})s_{1} + a_{1}a_{3}s_{0}, w = 10s_{3} + 2(a_{1}^{2} + 4a_{2})s_{2} + 3a_{1}a_{3}s_{1} + a_{3}^{2}s_{0} and s_{k} = ∑_{j=1}^{(n−1)/2} x(jP)^{k}. The Weierstrass equations (20) have discriminant

(21) \begin{aligned}n & = 7 & Δ(C_{λ}) & = λ(λ − 1)(λ^{3} − 8λ^{2} + 5λ + 1)^{7} \\ n & = 11 & Δ(C_{λ,ν}) & = λ(λ − 1)(λν + 2λ^{2} − 2λ + 1)(ν + 1)^{6}f(λ, ν)^{11} \end{aligned}

where f(λ, ν) = (−3λν + 2ν − λ^{3} + 5λ^{2} − 5λ + 1)/(λ − 1).

Let ϕ : C_{λ} → D_{λ} and ϕ̂ : D_{λ} → C_{λ} be the dual isogenies of degree n with ker ϕ̂ = ⟨P ⟩. Then by properties of the Weil pairing ker ϕ is isomorphic to μ_{n} as a Galois module. Let Q ∈ C_{λ}(\overline{K}) with ϕ(Q) = P. Then σ ↦ σ(Q) − Q is a cocycle taking values in μ_{n}. By Hilbert’s Theorem 90 there exists q ∈ K^{×}/(K^{×})^{n} such that σ(Q) − Q = σ(\sqrt[n]{q})/\sqrt[n]{q} for all σ ∈ Gal(\overline{K}/K). Computing q = q(λ) as described in [9, Section 1.2] we find

(22)  q(λ) = \{ \begin{matrix}λ^{4}(λ − 1) & if n = 7 \\ λν^{2}(λ − 1)(λ − ν − 1)^{3} & if n = 11. \end{matrix}

Now X(n) is birational to {q(λ) = τ^{n}} ⊂ X_{1}(n) × 𝔾_{m}. In the case n = 7 an explicit birational map is given in [9, Section 2.2]. Applying the same method for n = 11 we obtain

n = 7  (a : b : c) ↦ (λ, τ) = (−ac^{2}/b^{3}, ac/b^{2}),
n = 11  (a : b : c : d : e) ↦ (λ, ν, τ) = (−abd/c^{2}e, ab^{3}/c^{3}e, −ab/c^{2}).

We checked directly that these are birational maps, and that the cusps of X(n), i.e. (1 : 0 : …: 0) and its translates under the action of SL_{2}(ℤ/nℤ), map to the cusps of X_{1}(n), i.e. the roots of (21).

Let c_{4}(λ) and c_{6}(λ) be the invariants of the Weierstrass equation for C_{λ}. Using Magma we compute

n = 7 \begin{aligned}c_{4}(−ac^{2}/b^{2}) & ≡ ξ_{7}^{4}(abc)^{2}c_{4}(a, b, c) mod (a^{3}b + b^{3}c + c^{3}a) \\ c_{6}(−ac^{2}/b^{2}) & ≡ ξ_{7}^{6}(abc)^{3}c_{6}(a, b, c) mod (a^{3}b + b^{3}c + c^{3}a) \end{aligned}

n = 11 \begin{aligned}c_{4}(−abd/c^{2}e, ab^{3}/c^{3}e) & ≡ ξ_{11}^{4}(abcde)c_{4}(a, b, c, d, e) mod I \\ c_{6}(−abd/c^{2}e, ab^{3}/c^{3}e) & ≡ ξ_{11}^{6}(abcde)c̃_{6}(a, b, c, d, e) mod I \end{aligned}

where ξ_{7} = a/b^{5}c and ξ_{11} = a^{3}b/c^{6}e^{2}. Since we are free to cancel 4th powers and 6th powers from the coefficients of a shorter Weierstrass equation, the result follows. □

4.3. An alternative projective embedding. We take p ≥ 5 a prime and let G = PSL_{2}(ℤ/pℤ) act on X(p) in the usual way.

Theorem 4.5 ((Adler, Ramanan)). The group of G-invariant divisor classes on X(p) is free of rank 1 generated by a divisor class [Λ] of degree (p^{2} − 1)/24.

Proof. See [2, Theorem 24.1]. □

Let m = (p − 1)/2. Klein showed there are embeddings X(p) ⊂ ℙ^{m−1} and X(p) ⊂ ℙ^{m} with linear G-action. The images are called the z-curve and the Acurve respectively. The corresponding hyperplane sections are (m − 1)Λ and mΛ, and indeed the divisor Λ in Theorem 4.5 is constructed by taking the difference of these. It is conjectured that each of these embeddings is via a complete linear system (the WYSIWYG Hypothesis in [2]) and this is known for p = 7, 11. The equations for X(p) introduced in Section 2.1 are for the z-curve. However in Sections 4.4 and 4.5 we also need the A-curve.

Case p = 7. The z-curve is the Klein quartic X(7) = {x^{3}y+y^{3}z+z^{3}x = 0} ⊂ ℙ^{2}. The cusps of X(7) are the 24 points of inflection. We recall from [26] that the cusps are naturally partitioned into eight sets of three {P_{1}, P_{2}, P_{3}} with

P_{1} + 3P_{2} ∼ P_{2} + 3P_{3} ∼ P_{3} + 3P_{1} ∼ H

where H ∼ 2Λ is the hyperplane section. We write T_{0}, …, T_{7} for the effective divisors of degree 3 of the form P_{1} +P_{2} +P_{3}. One of these divisors, T_{0} say, satisfies X(7) ∩ {xyz = 0} = 4T_{0}. As observed in [26, Section 11] we have 2T_{i} ∼ 2T_{j} for all 0 ≤ i, j ≤ 7. It follows by Theorem 4.5 that 2T_{0} ∼ 3Λ. Since 3Λ ∼ 3H − 2T_{0} and L(3H − 2T_{0}) has basis x^{2}y, y^{2}x, z^{2}x, xyz, the A-curve is the image of

X(7) → ℙ^{3} ; (x : y : z) ↦ (t_{1} : t_{2} : t_{3} : t_{4}) = (x^{2}y : y^{2}z : z^{2}x : xyz)

with equations

rank (\begin{matrix}t_{1} & 0 & t_{4} & −t_{2} \\ t_{2} & −t_{3} & 0 & t_{4} \\ t_{3} & t_{4} & −t_{1} & 0 \end{matrix}) ≤ 2.

Case p = 11. The z-curve is the singular locus of the Hessian of

{F = v^{2}w + w^{2}x + x^{2}y + y^{2}z + z^{2}v = 0} ⊂ ℙ^{4}.

We write H ∼ 4Λ for the hyperplane section. The cusps are the 60 points of intersection of X(11) with {F = 0}. They are naturally partitioned into twelve sets of five {P_{1}, …, P_{5}} with

P_{1} + 6P_{3} + 3P_{4} + 10P_{5} ∼ H

and likewise under all cyclic permutations of the P_{i}. We write T_{0}, …, T_{11} for the effective divisors of degree 5 of the form P_{1} + … + P_{5}. One of these divisors, T_{0} say, satisfies X(11) ∩ {vwxyz = 0} = 20T_{0}. It may be shown that 5T_{i} ∼ 5T_{j} for all 0 ≤ i, j ≤ 11 and hence 5T_{0} ∼ 5Λ by Theorem 4.5. Since 5Λ ∼ 5H − 15T_{0} we find by computing a basis for L(5H − 15T_{0}) that the A-curve is the image of the morphism X(11) → ℙ^{5} given by

(v : w : x : y : z) ↦ (t_{1} : …: t_{6}) = (v^{2}wxz : vw^{2}xy : wx^{2}yz : vxy^{2}z : vwyz^{2} : vwxyz).

It is shown in [2, Theorem 51.1], and we checked using Magma, that this is the singular locus of the quartic hypersurface

t_{6}^{4} − (t_{1}^{2}t_{2} + t_{2}^{2}t_{3} + t_{3}^{2}t_{4} + t_{4}^{2}t_{5} + t_{5}^{2}t_{1})t_{6} + t_{1}^{2}t_{3}t_{5} + t_{2}^{2}t_{4}t_{1} + t_{3}^{2}t_{5}t_{2} + t_{4}^{2}t_{1}t_{3} + t_{5}^{2}t_{2}t_{4} = 0.

4.4. Formulae in the case n = 7.

Theorem 4.6. Let X = {F = 0} ⊂ ℙ^{2} be a twist of the Klein quartic, with hyperplane section H. Let T = P_{1}+P_{2}+P_{3} where P_{1}, P_{2}, P_{3} are points of inflection on X with

P_{1} + 3P_{2} ∼ P_{2} + 3P_{3} ∼ P_{3} + 3P_{1} ∼ H.

Let d ∈ K[x, y, z] be a cubic form with {d = 0} meeting X in a divisor 2D with D ∼ 2T. Then there is a Gal(\overline{K}/K)-module M such that for every field extension L/K and rational point P = (x : y : z) ∈ X (L)\{d = 0}, not a point of inflection, the elliptic curve

(23)  Y^{2} = X^{3} − 27\frac{c_{4}(F)(x, y, z)}{d(x, y, z)^{2}}X − 54\frac{c_{6}(F)(x, y, z)}{d(x, y, z)^{3}}

has 7-torsion isomorphic to M as a Gal(\overline{L}/L)-module.

Proof. If d_{1}, d_{2} ∈ K[x, y, z] are cubic forms meeting X in divisors 2D_{1} and 2D_{2} with D_{1} ∼ D_{2} then d_{1}/d_{2} is the square of a rational function, and hence the elliptic surfaces (23) with d = d_{1} and d = d_{2} are isomorphic over \overline{K}. Since X is a twist of the Klein quartic it follows (by taking D = 2T_{0} as defined in the last section) that the elliptic surfaces (18) and (23) are isomorphic over \overline{K}. Notice it does not matter whether we write the terms d(x, y, z) in the numerator or in the denominator. We are done by [27, Proposition 2.1]. □

In Theorem 4.8 below we determine rational functions d satisfying the hypothesis of Theorem 4.6 in the cases X = X_{E}(7) and X = X_{E}^{−}(7). We also show how to scale these functions to give the quadratic twist with M ≅ E[7].

Remark 4.7. Recall that X_{E}(7) has a trivial K-rational point corresponding to E itself. Following [27] one method for finding the right quadratic twist would be to specialise at this point. However this approach fails when d vanishes at the trivial point, and also does not generalise to X_{E}^{−}(7).

Theorem 4.8. Let E/K be an elliptic curve with Weierstrass equation y^{2} = x^{3} − 27c_{4}x − 54c_{6} and let Δ = (c_{4}^{3} − c_{6}^{2})/1728. If j(E) ≠ 0, 1728 then the families of elliptic curves parametrised by Y_{E}(7) and Y_{E}^{−}(7) are given by (23) with (F, d) = (F, d_{1}) and (G, d_{2}) where F and G are the quartics in Theorem 3.9 and

d_{1}(x, y, z) = −6(3x^{2} + c_{4}xz − 3c_{4}y^{2} + c_{6}yz)z
d_{2}(x, y, z) = 2Δ(4x^{3} + c_{4}x^{2}z − 12c_{4}xy^{2} − 2c_{6}xyz + 8c_{6}y^{3} + c_{4}^{2}y^{2}z + 200Δz^{3}).

Proof. We fix a symplectic isomorphism ϕ : E[7] ≅ μ_{7} × ℤ/7ℤ and let (a : b : c) be the \overline{K}-point on X(7) corresponding to (E, ϕ). As in the proof of Theorem 3.9 we scale (a, b, c) so that c_{4}(a, b, c) = c_{4} and c_{6}(a, b, c) = c_{6}. The action of SL_{2}(ℤ/7ℤ) on both the z-curve and the A-curve suggests we start with the forms

s_{1}(x, y, z) = (a^{2}c^{3} − 2ab^{3}c)x^{2}y + (a^{3}b^{2} − 2abc^{3})y^{2}z
+ (b^{3}c^{2} − 2a^{3}bc)z^{2}x + (a^{3}c^{2} + a^{2}b^{3} + b^{2}c^{3})xyz,
s_{2}(x, y, z) = a^{2}bx^{2}y + b^{2}cy^{2}z + c^{2}az^{2}x + 2abcxyz.

We then let r_{1} and r_{2} be the unique cubic forms satisfying

(24)  r_{i}(x, y, z)xyz ≡ s_{i}(x, y, z)^{2} mod (x^{3}y + y^{3}z + z^{3}x)

for i = 1, 2. The coefficients of r_{1} and r_{2} are homogeneous polynomials in a, b, c of degrees 10 and 6. Recall that in the proof of Theorem 3.9 we put

F(x, y, z) = \frac{1}{H^{3}} F (xx + y(∇F × ∇H) + zHe),
G(x, y, z) = \frac{1}{H^{2}} F (x∇F + y(x × e) + zH^{2}∇H).

The cubics d_{1} and d_{2} in the statement of the theorem are likewise found by putting

(25)  \begin{aligned}d_{1}(x, y, z) & = \frac{1}{2abcH^{4}} r_{1}(xx + y(∇F × ∇H) + zHe), \\ d_{2}(x, y, z) & = \frac{2H^{5}}{abc} r_{2}(x∇F + y(x × e) + zH^{2}∇H). \end{aligned}

It is clear from these constructions that {d_{1} = 0} and {d_{2} = 0} meet the corresponding twists of the Klein quartic in divisors of the form specified in Theorem 4.6. So our formulae for the families of elliptic curves parametrised by Y_{E}(7) and Y_{E}^{−}(7) are correct up to quadratic twist, say by δ ∈ K^{×}. It remains to show that δ is a square. As noted in [16, Section 7.1] it suffices to check this in the case ϕ : E[7] ≅ μ_{7} × ℤ/7ℤ is defined over K. Then (a : b : c) is a K-rational point on X(7). We write (a, b, c) = (λa_{0}, λb_{0}, λc_{0}) with a_{0}, b_{0}, c_{0} ∈ K. By our earlier choice of scaling for a, b, c we have λ^{7} ∈ K. Comparing the Weierstrass equation (18) for E with that in the statement of the theorem it follows that λ^{7}a_{0}b_{0}c_{0} ∈ (K^{×})^{2}. So a^{7}, b^{7}, c^{7} ∈ K and (abc)^{7} ∈ (K^{×})^{2}. Using (12) and (16) we compute

c_{k}(F)(x, y, z) = (2^{9}3^{6})^{k/2}c_{k}(xx + y(∇F × ∇H) + zHe)
c_{k}(G)(x, y, z) = (2^{9}3^{6}H^{7})^{k/2}c_{k}(x∇F + y(x × e) + zH^{2}∇H).

for k = 4, 6. It follows by (25) that

\frac{c_{k}(F)(x, y, z)}{d_{1}(x, y, z)^{k/2}} = ξ^{k} \frac{c_{k}(xHx + yH(∇F × ∇H) + zH^{2}e)}{((abc)^{6}r_{1}(xHx + yH(∇F × ∇H) + zH^{2}e))^{k/2}}
\frac{c_{k}(G)(x, y, z)}{d_{2}(x, y, z)^{k/2}} = η^{k} \frac{c_{k}(xH^{3}∇F + yH^{3}(x × e) + zH^{5}∇H)}{((abc)^{6}H^{10}r_{2}(xH^{3}∇F + yH^{3}(x × e) + zH^{5}∇H))^{k/2}}

for some ξ, η ∈ K^{×}. The covariant columns Hx, H(∇F × ∇H), H^{2}e have degrees 7, 14, 21 and the contravariant columns H^{3}∇F, H^{3}(x × e), H^{5}∇H have degrees 21, 28, 35. Since each column has degree a multiple of 7, its evaluation at (a, b, c) is K-rational. Therefore the families of curves in the statement of the theorem are K-isomorphic to

Y^{2} = X^{3} − 27\frac{c_{4}(x, y, z)}{((abc)^{6}r_{1}(x, y, z))^{2}} X − 54\frac{c_{6}(x, y, z)}{((abc)^{6}r_{1}(x, y, z))^{3}}

and

Y^{2} = X^{3} − 27\frac{c_{4}(x, y, z)}{((abc)^{6}H^{10}r_{2}(x, y, z))^{2}} X − 54\frac{c_{6}(x, y, z)}{((abc)^{6}H^{10}r_{2}(x, y, z))^{3}} .

To identify these with (18) we note that the cubic forms (abc)^{3}s_{1}(x, y, z) and (abc)^{3}H^{5}s_{2}(x, y, z) have coefficients in K (since the degree of each coefficient is a multiple of 7) and then use (24). □

Making the change of co-ordinates in Remark 3.10, we can replace d_{1} and d_{2} by cubic forms that satisfy the conditions of Theorem 4.6 for X_{E}(7) = {F = 0} ⊂ ℙ^{2} and X_{E}^{−}(7) = {G = 0} ⊂ ℙ^{2}, where F and G are the quartics in Theorem 1.1. Moreover having found one such form we can use the Riemann-Roch machinery in Magma to find further such forms.

In the case of X_{E}(7) we obtain a cubic form d_{11} with X_{E}(7) ∩ {d_{11} = 0} = 2D_{1} for some divisor D_{1} ∼ 2T. Then L(3H − D_{1}) has basis

d_{11} = −2(ax^{2} + 3bxz + 3y^{2} + 2ayz)z,
d_{12} = 2(ax^{2} + 3bxz + 3y^{2} + 2ayz)x,
d_{13} = 4(3bx^{2} − 2axy − 2a^{2}xz − 3byz − 2abz^{2})z,
d_{14} = 4(a^{2}x^{2} + 3bxy + 4abxz + ay^{2} + 3b^{2}z^{2})z.

More generally there are cubic forms d_{ij} for 1 ≤ i, j ≤ 4 such that the matrix (d_{ij}) is symmetric and each 2 × 2 minor vanishes mod F. The remaining d_{ij} are computed using d_{11}d_{ij} ≡ d_{1i}d_{1j} (mod F). Then X_{E}(7) ∩ {d_{ij} = 0} = D_{i} + D_{j} where D_{1}, …, D_{4} are divisors all linearly equivalent to 2T. The family of elliptic curves parametrised by Y_{E}(7) is now given by (23) with (F, d) = (F, d_{ii}) for any 1 ≤ i ≤ 4.

The A-curve is the image of X_{E}(7) → ℙ^{3}; (x : y : z) ↦ (d_{11} : …: d_{14}) with equations

rank (\begin{matrix}0 & t_{3} & −t_{4} & 2at_{1} + t_{4} \\ t_{1} & 2at_{1} + t_{4} & 2bt_{1} + at_{2} + at_{3} & 2at_{2} + at_{3} \\ t_{2} & 2bt_{1} + at_{3} & −a^{2}t_{1} + bt_{3} − at_{4} & 2bt_{2} − bt_{3} − at_{4} \end{matrix}) ≤ 2.

Our formula for the elliptic curve corresponding to P ∈ Y_{E}(7) fails when d_{ii}(P) = 0. However the zeros of d_{ii} correspond to the hyperplane section {t_{i} = 0} on the A-curve. Therefore, for any given point P, we have d_{ii}(P) ≠ 0 for some i. So unlike the treatment in [16, Theorem 5.2], where only the cubic form d_{11} was given, we have found formulae that cover all cases.

In the case of X_{E}^{−}(7) we likewise find cubic forms d′_{ij} for 1 ≤ i, j ≤ 4 such that the matrix (d′_{ij}) is symmetric and each 2 × 2 minor vanishes mod G. Explicitly

d′_{11} = −7ax^{2}y + 6x^{2}z + 3a^{2}y^{3} − 8ay^{2}z + 3yz^{2},
d′_{12} = 2ax^{3} + 12bx^{2}y − 2axyz − 3aby^{3} + 6by^{2}z,
d′_{13} = 2a^{2}xy^{2} − 10axyz + 6xz^{2} + 5aby^{3} − 12by^{2}z,
d′_{14} = 2a^{2}x^{2}y − 3ax^{2}z + 5abxy^{2} − 12bxyz − 3a^{2}y^{2}z + 8ayz^{2} − 3z^{3}.

The remaining d′_{ij} are computed using d′_{11}d′_{ij} ≡ d′_{1i}d′_{1j} (mod G). The family of elliptic curves parametrised by Y_{E}^{−}(7) is now given by (23) with (F, d) = (G, Δd′_{ii}) for any 1 ≤ i ≤ 4. Exactly as before, these formulae cover all cases.

4.5. Formulae in the case n = 11. Our approach is similar to that in the last section. As one would expect the formulae in the case n = 11 are more complicated than those in the case n = 7. There are however two further complications. One is that there is no invariant c_{6}. The other is that the form we are looking for is no longer uniquely determined by its image in the co-ordinate ring. Indeed in the case n = 7 we were looking for a cubic form, and in the case n = 11 we are looking for a quintic form. But in both cases our twist of X(n) is defined by quartics.

The action of SL_{2}(ℤ/11ℤ) on both the z-curve and the A-curve suggests we start with the forms

s_{1}(v, w, x, y, z) = (a^{3}bc^{3} + b^{4}cd^{2} − ab^{2}c^{2}de − 2bc^{2}de^{3})v^{2}wxz
+ (b^{3}cd^{3} + c^{4}de^{2} − abc^{2}d^{2}e − 2a^{3}cd^{2}e)vw^{2}xy
+ (c^{3}de^{3} + a^{2}d^{4}e − abcd^{2}e^{2} − 2ab^{3}de^{2})wx^{2}yz
+ (a^{3}d^{3}e + ab^{2}e^{4} − a^{2}bcde^{2} − 2a^{2}bc^{3}e)vxy^{2}z
+ (ab^{3}e^{3} + a^{4}bc^{2} − a^{2}b^{2}cde − 2ab^{2}cd^{3})vwyz^{2}
+ 2(a^{2}b^{2}c^{2}e + a^{2}b^{2}de^{2} + a^{2}cd^{2}e^{2} + ab^{2}c^{2}d^{2} + bc^{2}d^{2}e^{2})vwxyz,
s_{2}(v, w, x, y, z) = a^{2}bcev^{2}wxz + ab^{2}cdvw^{2}xy + bc^{2}dewx^{2}yz + acd^{2}evxy^{2}z
+ abde^{2}vwyz^{2} + 2abcdevwxyz.

We then solve for r_{1} and r_{2} satisfying

(26)  r_{i}(v, w, x, y, z)(vwxyz)^{3} ≡ s_{i}(v, w, x, y, z)^{4} (mod I, I′)

where I and I′ are the homogeneous ideals for X(11) ⊂ ℙ^{4} with respect to the two sets of variables a, b, c, d, e and v, w, x, y, z. The coefficients of r_{1} and r_{2} are homogeneous polynomials of degrees 28 and 20 in a, b, c, d, e. It is important to note that r_{1} and r_{2} are not uniquely determined by (26). However by averaging over the group we were able to choose r_{i} = (abcde)^{3}r̃_{i} in such a way that the coefficients of

r̃_{1}(vx_{1} + wx_{4} + xx_{5} + yx_{9} + zx_{14})

and

r̃_{2}(v∇F + w∇I_{7} + x∇I_{8} + y∇I_{9} + z∇c_{4})

are congruent mod I to certain polynomials in F and c_{4}. The result is a pair of quintic forms d̃_{1}(v, w, x, y, z) and d̃_{2}(v, w, x, y, z) with coefficients in ℚ[F, c_{4}]. We then put

d_{1}(v, w, x, y, z) = d̃_{1}(F v, w, F^{7}x, F^{2}y, F^{4}z)
d_{2}(v, w, x, y, z) = \frac{1}{F^{4}} d̃_{2}(F^{2}v, F^{8}w, F^{4}x, y, F^{3}z)

and replace F^{11} by Δ so that d_{1} and d_{2} have coefficients in ℚ[c_{4}, Δ].

Remark 4.9. The polynomials r̃_{i} and d_{i} would take several pages to print out, so we must refer the reader to the accompanying Magma file [13] for further details. The computation of d_{1} and d_{2} took several hours of computer time, whereas all other calculations up to this point ran in a few seconds.

Theorem 4.10. Let E/K be an elliptic curve with Weierstrass equation y^{2} = x^{3} − 27c_{4}x − 54c_{6} and let Δ = (c_{4}^{3} − c_{6}^{2})/1728. Assume j(E) ≠ 0, 1728 and let X = X_{E}(11), respectively X_{E}^{−}(11), be as given in Theorem 3.13. If (v : w : x : y : z) ∈ X(K) \ {d_{i} = 0}, not a cusp, then the corresponding elliptic curve E′/K satisfies

c_{4}(E′) ≡ d_{1}(v, w, x, y, z) c_{4}(F)(v, w, x, y, z) mod (K^{×})^{4},

respectively

c_{4}(E′) ≡ d_{2}(v, w, x, y, z) c_{4}(G)(v, w, x, y, z) mod (K^{×})^{4}.

Proof. As noted in [16, Section 7.1] we are free to extend our field K so that ϕ : E[11] ≅ μ_{11} × ℤ/11ℤ is defined over K. Let (a : b : c : d : e) be the corresponding K-point on X(11). We scale a, b, c, d, e so that c_{4}(a, b, c, d, e) = c_{4}. Then a^{11}, …, e^{11} ∈ K and by comparing the Weierstrass equation for E in the statement of the theorem with (19) we deduce that (abcde)^{11} ∈ (K^{×})^{4}. The polynomials F and G were computed in Section 3.3 as twists of F. Putting

(v′, w′, x′, y′, z′)^{T} = vF^{7}x_{1} + wF^{6}x_{4} + xF^{13}x_{5} + yF^{8}x_{9} + zF^{10}x_{14},
(v′′, w′′, x′′, y′′, z′′)^{T} = vF^{3}∇F + wF^{9}∇I_{7} + xF^{5}∇I_{8} + yF ∇I_{9} + zF^{4}∇c_{4},

it follows by (14), (15) and (17) that

c_{4}(F)(v, w, x, y, z) = \frac{(c_{4}^{3} − 1728F^{11})^{8}}{F^{22}} c_{4}(v′, w′, x′, y′, z′),
c_{4}(G)(v, w, x, y, z) = \frac{(55(c_{4}^{3} − 1728F^{11}))^{8}}{F^{11}} c_{4}(v′′, w′′, x′′, y′′, z′′).

By construction of d_{1} and d_{2} we have

d_{1}(v, w, x, y, z) = \frac{1}{(abcde)^{3}F^{30}} r_{1}(v′, w′, x′, y′, z′),
d_{2}(v, w, x, y, z) = \frac{1}{(abcde)^{3}F^{9}} r_{2}(v′′, w′′, x′′, y′′, z′′).

In view of Theorem 4.4 our aim is to show that

d_{1}(v, w, x, y, z) c_{4}(F)(v, w, x, y, z) ≡ v′w′x′y′z′c_{4}(v′, w′, x′, y′, z′) mod (K^{×})^{4},
d_{2}(v, w, x, y, z) c_{4}(G)(v, w, x, y, z) ≡ v′′w′′x′′y′′z′′c_{4}(v′′, w′′, x′′, y′′, z′′) mod (K^{×})^{4},

equivalently

(abcde)^{8}F^{36}r_{1}(v′, w′, x′, y′, z′) ≡ v′w′x′y′z′ mod (K^{×})^{4},
(abcde)^{8}F^{24}r_{2}(v′′, w′′, x′′, y′′, z′′) ≡ v′′w′′x′′y′′z′′ mod (K^{×})^{4}.

To finish the proof we note that the quintic forms

(abcde)^{2}F^{9}s_{1}(v′, w′, x′, y′, z′) and (abcde)^{2}F^{6}s_{2}(v′′, w′′, x′′, y′′, z′′)

have coefficients in K (since the degree of each coefficient is a multiple of 11) and then use (26). □

We already gave a formula for the j-invariant in Section 4.1. So (assuming j(E′) ≠ 0) Theorem 4.10 determines E′ up to quadratic twist by −1. In the case K = ℚ it is easy to decide which of the remaining two possibilities is correct by looking at traces of Frobenius.

In principle it should be possible to find alternative quintic forms to be used at points where d_{1} or d_{2} vanishes. The quintic forms in question are those meeting the z-curve in a divisor 4D where D is a hyperplane section for the A-curve. In the case n = 7 we managed to find the alternative forms using the Riemann-Roch machinery in Magma. Unfortunately the analogue of this in the case n = 11 does not appear to be practical. In the case of X_{E}^{−}(11) this is not a problem, since the 25 points with d_{2} = 0 correspond to the elliptic curves ℓ-isogenous to E for ℓ = 2, 7, 13. We can also account for 7 of the points on X_{E}(11) with d_{1} = 0 as corresponding to the elliptic curve E itself and the elliptic curves 5-isogenous to E. We are yet to encounter an example (over K = ℚ) where one of the remaining points with d_{1} = 0 is rational.

5. Examples

We use the formulae in Theorems 1.1 and 1.2 to give examples of non-trivial n-congruences for n = 7, 11 over ℚ and ℚ(T). By “non-trivial” we mean that the elliptic curves are not isogenous. The examples over ℚ illustrate the value of minimising and reducing, as mentioned in the introduction. The examples over ℚ(T) were found by setting a = b = −27j/(4(j − 1728)) to obtain a surface fibred over the j-line and then intersecting with one of the co-ordinate hyperplanes in the hope of finding a rational curve. We refer to elliptic curves over ℚ by their labels in Cremona’s tables [4]. For elliptic curves beyond the current range of Cremona’s tables we simply write the conductor followed by a ∗.

Remark 5.1. If elliptic curves E and E′ are related by an isogeny of degree d coprime to n, then they are clearly n-congruent. Since dual isogenies are adjoints with respect to the Weil pairing, the curves are directly n-congruent if d is a square in (ℤ/nℤ)^{×} and reverse n-congruent if −d is a square in (ℤ/nℤ)^{×}.

5.1. Examples in the case n = 7.

Example 5.2. Let E be the elliptic curve 162c1. Let F and G be the equations for X_{E}(7) and X_{E}^{−}(7) in Theorem 1.1 with a = 3645 and b = −13122. These have invariants Ψ(F) = −2^{11} · 3^{18} and Ψ(G) = 2^{22} · 3^{36}. Minimising and reducing suggests that we substitute

F (x, y, z) = \frac{1}{2^{10}3^{14}} F(36y − 9z, 1944x − 972y − 1215z, z)
G(x, y, z) = \frac{1}{2^{12}3^{20}} G(18x + 18y + 9z, z, −486x + 1458y + 1944z)

to give quartics

F (x, y, z) = 3x^{3}z + 3x^{2}y^{2} − 6x^{2}yz + 3x^{2}z^{2} − 3xy^{3}
+ 3xz^{3} + 2y^{4} − y^{3}z − 9y^{2}z^{2} + 4yz^{3} − 5z^{4}
G(x, y, z) = −x^{3}y − x^{3}z − 6x^{2}z^{2} + 6xy^{2}z − 6xyz^{2}
+ 6xz^{3} + 2y^{4} + 2y^{3}z − 6y^{2}z^{2} − 38yz^{3} − 8z^{4}

with invariants Ψ(F) = −2 · 3^{4} and Ψ(G) = 2^{2} · 3^{4}. We find rational points P_{1} = (1 : 0 : 0), P_{2} = (3 : −2 : −1) on {F = 0} ⊂ ℙ^{2}, and rational points P_{3} = (1 : 0 : 0), P_{4} = (1 : 1 : −1), P_{5} = (4 : −1 : 1) on {G = 0} ⊂ ℙ^{2}. The corresponding elliptic curves 7-congruent to E are

P_{1} 162c1  y^{2} + xy = x^{3} − x^{2} + 3x − 1
P_{2} 293706x2 y^{2} + xy = x^{3} − x^{2} − 62930562x − 192134303740
P_{3} 162c2  y^{2} + xy = x^{3} − x^{2} − 42x − 100
P_{4} 17334f1 y^{2} + xy = x^{3} − x^{2} − 5473977x − 4956193171
P_{5} 624186∗  y^{2} + xy = x^{3} − x^{2} − 11751402282x + 360746315347508.

Since the elliptic curves 162c1 and 162c2 are 3-isogenous, it was already clear from Remark 5.1 that they are reverse 7-congruent.

It is shown in [16, Proposition 6.3] that there are infinitely many 6-tuples of directly 7-congruent non-isogenous elliptic curves over ℚ. The following example shows that there are infinitely many pairs of reverse 7-congruent non-isogenous elliptic curves over ℚ.

Example 5.3. Let E/ℚ(T) be the elliptic curve y^{2} = x^{3} + ax + b where a = b = −27j/(4(j − 1728)) and j = 27T^{3}(5T − 56)/(T − 1). Then on X_{E}^{−}(7), with equation as given in Theorem 1.1, we find the rational point

(x : y : z) = (0 : −4(T^{2} − 12T + 8)(5T^{2} + 4T + 8) : 9T^{2}(T + 4)(5T − 56)).

Specialising T (and taking quadratic twists by d as indicated) we obtain the following pairs of reverse 7-congruent elliptic curves E_{1} and E_{2}.

\begin{matrix}T & d & E_{1} & E_{2} \\ −16 & −38 & 361a1 & 361a2 \\ 8 & −10 & 700g1 & 2100q1 \\ 2 & −2 & 2116b1 & 10580h1 \\ 16/5 & −42 & 24255r1 & 24255m2 \end{matrix}

The existence of specialisations E_{1} and E_{2} that are not isogenous is enough to show that there are infinitely many such specialisations.

5.2. Examples in the case n = 11.

Example 5.4. Let E be the elliptic curve 1782b1. Let F be the cubic form describing X_{E}(11) ⊂ ℙ^{4} in Theorem 1.2 with a = 765 and b = 15102. The invariant is Ψ(F) = −2^{28} · 3^{12} · 11^{6}. Minimising and reducing suggests that we substitute

(\begin{matrix}v \\ w \\ x \\ y \\ z \end{matrix}) ← (\begin{matrix}984 & 12900 & −9093 & −34056 & 13689 \\ −2040 & −24252 & −3315 & 0 & −16857 \\ 328 & 164 & −435 & 0 & −57 \\ −352 & 88 & −264 & 264 & −1056 \\ −8 & −4 & −13 & 0 & 25 \end{matrix}) (\begin{matrix}v \\ w \\ x \\ y \\ z \end{matrix})

so that X_{E}(11) ⊂ ℙ^{4} is the singular locus of the Hessian of

−v^{2}w + v^{2}x − v^{2}y + 2v^{2}z − vw^{2} + 4vwz − 4vx^{2} − 8vxy + 2vxz + 6vyz +3vz^{2} + 2w^{3} − 3w^{2}x − 2w^{2}y + 8w^{2}z + 6wx^{2} + 2wxy + 2wxz + 6wy^{2} − 6wyz

+9wz^{2} − x^{3} − x^{2}z − 3xy^{2} − 6xyz − 9xz^{2} − 6y^{3} + 9y^{2}z + 3yz^{2} − 7z^{3} = 0

with invariant 2^{2} · 3^{4} · 11^{2}. We find rational points P_{1} = (−1 : 5 : 1 : 2 : 1), P_{2} = (0 : 0 : 0 : 1 : 0) and P_{3} = (1 : 1 : −1 : 0 : −4). The corresponding elliptic curves directly 11-congruent to E are

P_{1} 1782b1  y^{2} + xy = x^{3} − x^{2} + 48x + 224
P_{2} 1782b2  y^{2} + xy = x^{3} − x^{2} − 447x − 7795
P_{3} 447282∗ y^{2} + xy = x^{3} − x^{2} − 17552171922x − 227953575178678

Since the elliptic curves 1782b1 and 1782b2 are 3-isogenous, it was already clear from Remark 5.1 that they are directly 11-congruent.

Example 5.5. Let E be the elliptic curve 4466c1. Let G be the cubic form describing X_{E}^{−}(11) ⊂ ℙ^{4} in Theorem 1.2 with a = 85 and b = −83162. The invariant is Ψ(G) = 2^{21} · 7 · 11^{2} · 29^{2}. Minimising and reducing suggests that we substitute

(\begin{matrix}v \\ w \\ x \\ y \\ z \end{matrix}) ← (\begin{matrix}4096 & −1408 & 128 & −1312 & 45088 \\ 0 & 128 & 128 & 32 & 110 \\ 0 & 0 & −256 & −96 & −103 \\ 0 & 0 & 0 & −32 & −11 \\ 0 & 0 & 0 & 0 & −1 \end{matrix}) (\begin{matrix}v \\ w \\ x \\ y \\ z \end{matrix})

so that X_{E}^{−}(11) ⊂ ℙ^{4} is the singular locus of the Hessian of

−2v^{2}z − 4vwy + 12vxy + 4vxz + 5vy^{2} + 6vyz − 43vz^{2} − w^{2}x + w^{2}y −4wxy − 2wxz − 3wy^{2} + 196wyz + 83wz^{2} − 11x^{3} − 12x^{2}y − 9x^{2}z

−11xy^{2} + 366xyz + 125xz^{2} + 322y^{3} + 447y^{2}z + 275yz^{2} + 632z^{3} = 0

with invariant −2^{2} · 7 · 11^{2} · 29^{2}. We find rational points P_{1} = (−7 : 11 : 3 : 1 : 1) and P_{2} = (7830 : −3553 : 510 : −281 : 71). The corresponding elliptic curves reverse 11-congruent to E are

P_{1} 4466c2 y^{2} + xy + y = x^{3} − x^{2} − 1755x − 27349
P_{2} 1174558∗ y^{2} + xy + y = x^{3} − x^{2} + 117885809240x + 16240157710556505

Since the elliptic curves 4466c1 and 4466c2 are 2-isogenous, it was already clear from Remark 5.1 that they are reverse 11-congruent.

A table of pairs of 11-congruent elliptic curves over ℚ is available from the website [13]. These were found by searching for rational points on X_{E}(11) and X_{E}^{−}(11) for all elliptic curves E/ℚ in Cremona’s tables. As happened in Examples 5.4 and 5.5, the elliptic curves 11-congruent to E that we find, often have conductor beyond the current range of Cremona’s tables.

34 TOM FISHER

The following example shows that there are infinitely many pairs of directly 11-congruent non-isogenous elliptic curves over ℚ. Example 5.6. Let E/ℚ(T) be the elliptic curve y^{2} = x^{3} + a(T)x + b(T) where

a(T) = −3(T − 3)(T^{4} − 5T^{2} − 24T − 92)/(T^{3} − T^{2} + 4T + 24)
b(T) = −2(T − 3)(T^{5} − T^{4} − 11T^{3} − 43T^{2} − 62T − 316)/(T^{3} − T^{2} + 4T + 24).

Then X_{E}(11), with equations as given in Theorem 1.2, has rational point

(\begin{matrix}v \\ w \\ x \\ y \\ z \end{matrix}) = (\begin{matrix}T^{6} + T^{5} + 31T^{4} + 259T^{3} + 520T^{2} + 676T + 1248 \\ −(T − 3)(T^{5} + 4T^{4} + 43T^{3} + 100T^{2} − 44T − 320) \\ −(T^{2} + 3T + 14)(T^{3} − T^{2} + 4T + 24) \\ 0 \\ (T + 4)(T^{3} − T^{2} + 4T + 24) \end{matrix})

Specialising T (and taking quadratic twists by d as indicated) we obtain the following pairs of directly 11-congruent elliptic curves E_{1} and E_{2}.

\begin{matrix}T & d & E_{1} & E_{2} \\ 2 & −6 & 11a3 & 11a2 \\ 1 & 42 & 49a1 & 49a4 \\ −3 & −2 & 216b1 & 1512c1 \\ 11 & −426 & 10082c1 & 70574h1 \end{matrix}

The elliptic curve 11-congruent to E is y^{2} = x^{3} + A(T)x + B(T) where

A(T) = −3(T − 3)(T^{2} − 8T − 17)(T^{3} − T^{2} + 4T + 24)(T^{12} − 250T^{11} + 3473T^{10}

− 23824T^{9} + 106654T^{8} − 354556T^{7} + 890186T^{6} − 1710568T^{5} + 2386357T^{4} − 2054170T^{3} + 1799781T^{2} + 956680T + 3570796),

B(T) = −2(T − 3)(T^{3} − T^{2} + 4T + 24)^{2}(T^{20} + 476T^{19} − 27815T^{18} + 556718T^{17}

− 6046664T^{16} + 42450848T^{15} − 213832636T^{14} + 823702888T^{13} − 2497998850T^{12} + 5954643736T^{11} − 10798748818T^{10} + 13644339892T^{9} − 7927895108T^{8} − 10398245632T^{7} + 25581636532T^{6} − 10366268760T^{5} − 60876061719T^{4} + 164062110060T^{3} − 98120800447T^{2} + 262948421518T + 141270230564).

These elliptic curves have discriminants

2^{12}3^{6}(T − 5)(T − 3)^{2}(T + 1)^{5}(T^{2} + 7)/(T^{3} − T^{2} + 4T + 24)^{3},

− 2^{12}3^{6}(T − 5)^{4}(T − 3)^{2}(T + 1)^{3}(T^{2} + 7)(T^{3} − T^{2} + 4T + 24)^{3}(T^{3} − T^{2} + 15T − 31)^{11}.

We did not find any pairs of reverse 11-congruent non-isogenous elliptic curves over ℚ(T). We note that according to [18, Theorem 4] the modular diagonal surface in this case is of general type.

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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