INVARIANT THEORY FOR THE ELLIPTIC NORMAL QUINTIC, I. TWISTS OF X(5)

TOM FISHER

Abstract. A genus one curve of degree 5 is defined by the 4 × 4 Pfaffians of a 5 × 5 alternating matrix of linear forms on ℙ^{4}. We describe a general method for investigating the invariant theory of such models. We use it to explain how we found our algorithm for computing the invariants [12] and to extend our method in [14] for computing equations for visible elements of order 5 in the Tate-Shafarevich group of an elliptic curve. As a special case of the latter we find a formula for the family of elliptic curves 5-congruent to a given elliptic curve in the case the 5-congruence does not respect the Weil pairing. We also give an algorithm for doubling elements in the 5-Selmer group of an elliptic curve, and make a conjecture about the matrices representing the invariant differential on a genus one normal curve of arbitrary degree.

1. Introduction

A genus one normal curve C ⊂ ℙ^{n−1} of degree n ≥ 3 is a genus one curve embedded by a complete linear system of degree n. If n ≥ 4 then the homogeneous ideal of C is generated by a vector space of quadrics of dimension n(n − 3)/2.

Definition 1.1. A genus one model (of degree 5) is a 5 × 5 alternating matrix of linear forms on ℙ^{4}. We write C_{ϕ} ⊂ ℙ^{4} for the subvariety defined by the 4 × 4 Pfaffians of ϕ, and say that ϕ is non-singular if C_{ϕ} is a smooth curve of genus one.

It is a classical fact that (over ℂ) every genus one normal curve of degree 5 is of the form C_{ϕ} for some ϕ. Importantly for us, the proof using the Buchsbaum-Eisenbud structure theorem [5],[6] shows that this is still true over an arbitrary ground field.

There is a natural action of GL_{5} × GL_{5} on the space of genus one models. The first factor acts as M : ϕ ↦ MϕM^{T} and the second factor by changing co-ordinates on ℙ^{4}. To describe this situation we adopt the following notation. Let V and W be 5-dimensional vector spaces with bases v_{0}, …, v_{4} and w_{0}, …, w_{4}. The dual bases for V^{∗} and W^{∗} will be denoted v_{0}^{∗}, …, v_{4}^{∗} and w_{0}^{∗}, …, w_{4}^{∗}. We identify the space of genus one models with ∧^{2}V ⊗ W via

ϕ = (ϕ_{ij}) ⟷ ∑_{i<j}(v_{i} ∧ v_{j}) ⊗ ϕ_{ij}(w_{0}, …, w_{4}).

Date: 14th October 2011.

With this identification the action of GL_{5} × GL_{5} becomes the natural action of GL(V) × GL(W) on ∧^{2}V ⊗ W. By squaring and then identifying ∧^{4}V ≅ V^{∗} there is a natural map

(1)  P_{2} : ∧^{2}V ⊗ W → V^{∗} ⊗ S^{2}W = Hom(V, S^{2}W).

Explicitly P_{2}(ϕ) = (v_{i} ↦ p_{i}(w_{0}, …, w_{4})) where p_{0}, …, p_{4} are the 4 × 4 Pfaffians of ϕ. Thus V may be thought of as the space of quadrics defining C_{ϕ} and W as the space of linear forms on ℙ^{4}. Despite this clear geometric distinction, we show in this paper that certain covariants that mix up the roles of V and W have interesting arithmetic applications.

We work over a perfect field K with characteristic not equal to 2, 3 or 5. The co-ordinate ring K[∧^{2}V ⊗ W] is a polynomial ring in 50 variables.

Theorem 1.2. The ring of invariants for SL(V) × SL(W) acting on K[∧^{2}V ⊗ W] is generated by invariants c_{4} and c_{6} of degrees 20 and 30. If we scale them as specified in [12] and put Δ = (c_{4}^{3} − c_{6}^{2})/1728 then

(i) A genus one model ϕ is non-singular if and only if Δ(ϕ) ≠ 0.

(ii) If ϕ is non-singular then C_{ϕ} has Jacobian elliptic curve

y^{2} = x^{3} − 27c_{4}(ϕ)x − 54c_{6}(ϕ).

Proof: See [12, Theorem 4.4]. □

The invariants c_{4} and c_{6} are too large to write down as explicit polynomials. Nonetheless we gave an algorithm for evaluating them in [12, Section 8]. By Theorem 1.2 this gives an algorithm for computing the Jacobian. In [14] we studied a covariant we call the Hessian. Explicitly it is a 50-tuple of homogeneous polynomials of degree 11 defining a map H : ∧^{2}V ⊗ W → ∧^{2}V ⊗ W. Again rather than write down these polynomials we gave an algorithm for evaluating them. The Hessian allows us to compute certain twists of the universal family of elliptic curves parametrised by X(5). We used it to find equations for visible elements of order 5 in the Tate-Shafarevich group of an elliptic curve, and to recover the formulae of Rubin and Silverberg [20] for families of 5-congruent elliptic curves. However in both these applications we were restricted to 5-congruences that respect the Weil pairing. In this paper we remove this restriction.

In Sections 2, 3, 4 we explain a general method for investigating the covariants associated to a genus one model. In particular we explain how we found the algorithm for computing the invariants in [12, Section 8]. One key result on the existence of covariants is left to a sequel to this paper [16]. Our account is still however self-contained, since in Section 8 we give explicit constructions of each of the covariants used in the second half of this paper. In Section 5 we use the covariants to write down families of 5-congruent elliptic curves. In Section 6 we give a formula for doubling in the 5-Selmer group of an elliptic curve and extend our method in [14] for computing visible elements of the Tate-Shafarevich group.

In particular we check local solubility for each of the visible elements of order 5 in the Weil-Châtelet group that were considered in [8]. In Section 7 we study a covariant that describes the invariant differential. This is needed not only for some of the constructions in Section 8, but also leads us to make a conjecture about the matrices representing the invariant differential on a genus one normal curve of arbitrary degree.

2. Covariants

We recall that a rational representation of a linear algebraic group G is a morphism of group varieties ρ_{Y} : G → GL(Y).

Definition 2.1. Let Y be a rational representation of GL(V) × GL(W). A covariant (for Y) is a polynomial map F : ∧^{2}V ⊗ W → Y such that F ∘ g = ρ_{Y} (g)F for all g ∈ SL(V) × SL(W).

The covariants in the case Y = K is the trivial representation are the invariants as described in Theorem 1.2. For general Y the covariants form a module over the ring of invariants. In all our examples Y will be homogeneous by which we mean there exist integers r and s such that

ρ_{Y} (λI_{V} , μI_{W}) = λ^{r}μ^{s}I_{Y}

for all λ, μ ∈ K^{×}. A polynomial map F : ∧^{2}V ⊗ W → Y is homogeneous of degree d if F (λϕ) = λ^{d}F (ϕ) for all λ ∈ K, equivalently F is represented by a tuple of homogeneous polynomials of degree d.

Lemma 2.2. Let F : ∧^{2}V ⊗ W → Y be a covariant and suppose that both Y and F are homogeneous. Then there exist integers p and q called the weights of F such that

F ∘ g = (det g_{V})^{p}(det g_{W})^{q}ρ_{Y} (g) ∘ F

for all g = (g_{V} , g_{W}) ∈ GL(V) × GL(W). Moreover if Y has degree (r, s) then

(2)  \begin{aligned}2 deg F & = & 5p + r \\ deg F & = & 5q + s. \end{aligned}

Proof: The only 1-dimensional rational representations of GL_{n} are integer powers of the determinant. This proves the first statement. The second statement follows from the special case where g_{V} and g_{W} are scalar matrices. □

The first example of a covariant is the identity map

U : ∧^{2}V ⊗ W → ∧^{2}V ⊗ W.

It has degree 1 and weights (p, q) = (0, 0). The Pfaffian map P_{2} defined in (1) is a covariant of degree 2 with weights (p, q) = (1, 0). Subject to picking a basis for V, ϕ ∈ ∧^{2}V ⊗ W is a 5 × 5 alternating matrix of linear forms and P_{2}(ϕ) is its vector of 4 × 4 Pfaffians. The determinant of the Jacobian matrix of these 5 quadrics defines a covariant

(3)  S_{10} : ∧^{2}V ⊗ W → S^{5}W.

It has degree 10 and weights (p, q) = (4, 1). It is shown in [19, VIII.2.5] that if ϕ ∈ ∧^{2}V ⊗ W is non-singular then S_{10}(ϕ) is an equation for the secant variety of C_{ϕ} ⊂ ℙ^{4}.

Our initial motivation for studying the covariants was that by constructing a large enough supply of covariants we might eventually arrive at an algorithm for computing the invariants, and so by Theorem 1.2 an algorithm for computing the Jacobian. This programme was successful, leading to the algorithm in [12]. We have subsequently found that some of the covariants have interesting arithmetic applications in their own right.

In the next two sections we explain our methods for studying the covariants. The key idea is that although the covariants are routinely too large to write down, their restrictions to the Hesse family, i.e. the universal family of elliptic curves over X(5), are much easier to write down and are (nearly) characterised by their invariance properties under an appropriate action of SL_{2}(ℤ/5ℤ). Thus our work resolves, albeit in one particular case, what is described in [1, Chapter V,§22] as the “mysterious role of invariant theory”.

3. The extended Heisenberg group

We take n ≥ 5 an odd integer. In this section we work over an algebraically closed field K of characteristic not dividing n, and let ζ_{n} ∈ K be a primitive nth root of unity. We write E[n] for the n-torsion subgroup of an elliptic curve E and e_{n} : E[n] × E[n] → μ_{n} for the Weil pairing.

Definition 3.1. (i) The modular curve Y (n) = X(n) \ {cusps} parametrises triples (E, P_{1}, P_{2}) where E is an elliptic curve and P_{1}, P_{2} are a basis for E[n] with e_{n}(P_{1}, P_{2}) = ζ_{n}. (ii) Let Z(n) ⊂ ℙ^{n−1} be the subvariety defined by a_{0} = 0, a_{n−i} = −a_{i} and rank(a_{i−j}a_{i+j}) ≤ 2 where (a_{0} : …: a_{n−1}) are co-ordinates on ℙ^{n−1} and the subscripts are read mod n.

There is an action of SL_{2}(ℤ/nℤ) on Y (n) given by

(\begin{matrix}a & b \\ c & d \end{matrix}) : (E, P_{1}, P_{2}) ↦ (E, dP_{1} − cP_{2}, −bP_{1} + aP_{2}).

Let S = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and T = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) be the usual generators for SL_{2}(ℤ). By abuse of notation we also write S and T for their images in SL_{2}(ℤ/nℤ).

Theorem 3.2. There is an embedding X(n) ⊂ ℙ^{n−1} such that (i) X(n) ⊂ Z(n) with equality if n is prime.

(ii) The action of SL_{2}(ℤ/nℤ) on X(n) is given by \overline{ρ} : SL_{2}(ℤ/nℤ) → PGL_{n}(K) where

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

Moreover \overline{ρ} lifts uniquely to a representation ρ : SL_{2}(ℤ/nℤ) → SL_{n}(K).

Proof: The condition for equality in (i) is due to Vélu [26]. The remaining statements are proved in [15, Section 2]. Proofs that \overline{ρ} lifts in the case n = 5 may also be found in [18], [22]. □

The Heisenberg group of level n is

H_{n} = ⟨σ, τ|σ^{n} = τ^{n} = [σ, [σ, τ]] = [τ, [σ, τ]] = 1⟩.

It is a non-abelian group of order n^{3}. The centre is a cyclic group of order n generated by ζ = [σ, τ] = στσ^{−1}τ^{−1}. We write \overline{H}_{n} for the quotient of H_{n} by its centre and identify \overline{H}_{n} ≅ (ℤ/nℤ)^{2} via \overline{σ} ↦ (1, 0) and \overline{τ} ↦ (0, 1). Since each automorphism of H_{n} induces an automorphism of \overline{H}_{n} there is a natural group homomorphism β : Aut(H_{n}) → GL_{2}(ℤ/nℤ). The kernel of β is \overline{H}_{n} acting as the group of inner automorphisms. We may thus identify Aut(H_{n}) as a group of affine transformations. Let ι ∈ Aut(H_{n}) be the involution given by ι(σ) = σ^{−1} and ι(τ) = τ^{−1}. (Any involution ι with β(ι) = −I would do, but we have picked one for definiteness.) Since n is odd there is a unique section s_{β} for β with s_{β}(−I) = ι. (This means that s_{β} : GL_{2}(ℤ/nℤ) → Aut(H_{n}) is a group homomorphism with β ∘ s_{β} = id.) Indeed the image of s_{β} is the centraliser of ι in Aut(H_{n}).

Definition 3.3. The extended Heisenberg group is the semi-direct product

H_{n}^{+} = H_{n} ⋉ SL_{2}(ℤ/nℤ),

with group law (h, γ)(h′, γ′) = (h s_{β}(γ)h′, γγ′).

Remark 3.4. (i) The map s_{β} is explicitly given by

s_{β}(( \begin{matrix}a & b \\ c & d \end{matrix})) : σ ↦ ζ^{−ac/2}σ^{a}τ^{c}; τ ↦ ζ^{−bd/2}σ^{b}τ^{d}.

(ii) The group H_{5}^{+} was used by Horrocks and Mumford [18] in their construction of an indecomposable rank 2 vector bundle on ℙ^{4}. In fact the order of H_{5}^{+} appears in the title of their paper.

We now identify H_{n} as a subgroup of SL_{n}(K) via the Schrödinger representation θ : H_{n} → SL_{n}(K) where

(4)  θ(σ) = (\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}), θ(τ) = (\begin{matrix}0 & 0 & ⋯ & 0 & 1 \\ 1 & 0 & ⋯ & 0 & 0 \\ 0 & 1 & ⋯ & 0 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \end{matrix}).

Since θ(ζ) = ζ_{n}I_{n} this identifies the centre of H_{n} with μ_{n}. Let N_{n} be the normaliser of H_{n} in SL_{n}(K). It may be checked that the automorphisms of \overline{H}_{n} induced by conjugation by elements of N_{n} are precisely those that preserve the commutator pairing \overline{H}_{n} × \overline{H}_{n} → μ_{n}. This proves the surjectivity of the map α in the following commutative diagram with exact rows and columns.

[figure: 0; 0; μ_{n}; μ_{n}; 0; 0; H_{n}; N_{n}; α; SL_{2}(ℤ/nℤ); 0; 0; _{\overline{H}_{n}}; Aut(H_{n}); β; GL_{2}(ℤ/nℤ); det; 0; 0; (ℤ/nℤ)^{×}; (ℤ/nℤ)^{×}; 0; 0]

The restriction of s_{β} to SL_{2}(ℤ/nℤ) defines a projective representation

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

Comparison with the proof of Theorem 3.2 shows that this is the same as the projective representation considered there.

Lemma 3.5. The following objects are in natural 1-1 correspondence. (i) Sections s_{α} for α compatible with s_{β}. (ii) Lifts of \overline{ρ} : SL_{2}(ℤ/nℤ) → PGL_{n}(K) to ρ : SL_{2}(ℤ/nℤ) → SL_{n}(K). (iii) Extensions of θ : H_{n} → SL_{n}(K) to θ^{+} : H_{n}^{+} → SL_{n}(K).

Proof. The projective representation \overline{ρ} is defined by the requirement

\overline{ρ}(γ) h \overline{ρ}(γ)^{−1} = s_{β}(γ)h for all h ∈ H_{n}, γ ∈ SL_{2}(ℤ/nℤ).

But to say that s_{α} is a section for α compatible with s_{β} means

(5)  s_{α}(γ) h s_{α}(γ)^{−1} = s_{β}(γ)h for all h ∈ H_{n}, γ ∈ SL_{2}(ℤ/nℤ).

The correspondence between (i) and (ii) is clear. Now given s_{α} compatible with s_{β} we define θ^{+}(h, γ) = hs_{α}(γ) and check using (5) that θ^{+} is a homomorphism. Conversely, given θ^{+} we set s_{α}(γ) = θ^{+}(1, γ). This gives the correspondence between (i) and (iii). □

From the final statement of Theorem 3.2 we immediately deduce

Theorem 3.6. The Schrödinger representation θ : H_{n} → SL_{n}(K) extends uniquely to a representation θ^{+} : H_{n}^{+} → SL_{n}(K). Moreover the normaliser of θ(H_{n}) in SL_{n}(K) is θ^{+}(H_{n}^{+}).

Remark 3.7. The Schrödinger representation has ϕ(n) conjugates obtained by either changing our choice of ζ_{n} or precomposing with an automorphism of H_{n}. We may apply Theorem 3.6 to any one of these representations. This is important for our applications and explains why we were careful to define H_{n}^{+} before introducing the Schrödinger representation.

4. Discrete covariants

In this section we work over an algebraically closed field of characteristic not equal to 2, 3 or 5. The Hesse family of elliptic normal quintics (studied for example in [14], [19]) is given by

\begin{aligned}u : & 𝔸^{2} & → & ∧^{2}V ⊗ W \\ & (a, b) & ↦ & a ∑(v_{1} ∧ v_{4})w_{0} + b ∑(v_{2} ∧ v_{3})w_{0} \end{aligned}

where the sums are taken over all cyclic permutations of the subscripts mod 5. The models u(a, b), called the Hesse models, are representative of all genus one models in the following sense.

Lemma 4.1. Every non-singular genus one model is GL(V) × GL(W)-equivalent to a Hesse model.

Proof: See [14, Proposition 4.1] □

The Hesse models are invariant under the following actions of the Heisenberg group H_{5} on V and W.

(6)  \begin{matrix}θ_{V} : H_{5} → SL(V) ; & σ : v_{i} ↦ ζ_{5}^{2i}v_{i}; & τ : v_{i} ↦ v_{i+1} \\ θ_{W} : H_{5} → SL(W) ; & σ : w_{i} ↦ ζ_{5}^{i}w_{i}; & τ : w_{i} ↦ w_{i+1}. \end{matrix}

Our definition of the Hesse family differs from that in [14, Section 4] by a change of co-ordinates. This is to make the formulae (6) more transparent than those immediately preceding [14, Lemma 7.7].

Since θ_{V} and θ_{W} are conjugates of the Schrödinger representation they extend by Theorem 3.6 to representations of H_{5}^{+}. By abuse of notation we continue to write these representations as θ_{V} and θ_{W}. Let Y be a rational representation of GL(V) × GL(W). Then θ_{V} and θ_{W} define an action θ_{Y} of H_{5}^{+} on Y. We write Y^{H_{5}} for the subspace of Y fixed by H_{5}. Since H_{5}^{+} sits in an exact sequence

0 ⟶ H_{5} ⟶ H_{5}^{+} ⟶ Γ ⟶ 0

there is an action of Γ = SL_{2}(ℤ/5ℤ) on Y^{H_{5}}. In the case Y = ∧^{2}V ⊗ W we find (by using Lemma 4.4 below to compute dim Y^{H_{5}}) that Im(u) = (∧^{2}V ⊗ W)^{H_{5}}.

The action of Γ is then described by a representation χ_{1} : Γ → GL_{2}(K) with the defining property that

(7)  u ∘ χ_{1}(γ) = θ_{∧^{2}V ⊗W} (γ) ∘ u

for all γ ∈ Γ.

Definition 4.2. Let π : Γ → GL(Z) be a representation. A discrete covariant (for Z) is a polynomial map f : 𝔸^{2} → Z satisfying

u ∘ χ_{1}(γ) = π(γ) ∘ u

for all γ ∈ Γ.

Theorem 4.3. Let F : ∧^{2}V ⊗W → Y be a covariant. Then f = F ∘u : 𝔸^{2} → Y^{H_{5}} is a discrete covariant. Moreover F is uniquely determined by f.

Proof: Since F is a covariant it is SL(V) × SL(W)-equivariant (by definition) and therefore H_{5}^{+}-equivariant. So its restriction to (∧^{2}V ⊗ W)^{H_{5}} takes values in Y^{H_{5}} and this restriction is Γ-equivariant.

If F_{1} and F_{2} restrict to the same discrete covariant f then by Lemma 4.1 they agree on all non-singular models. By Theorem 1.2 the non-singular models are Zariski dense in ∧^{2}V ⊗ W and from this we deduce that F_{1} = F_{2}. □

For any given Y it is easy to compute the discrete covariants using invariant theory for the finite groups H_{5} and Γ. We say that a discrete covariant f : 𝔸^{2} → Y^{H_{5}} is a covariant if it arises from a covariant F : ∧^{2}V ⊗ W → Y as described in Theorem 4.3. It is important to note that not every discrete covariant is a covariant. We give examples below.

We recall the character table of H_{p} for p an odd prime. There are p^{2}+p−1 conjugacy classes with representatives ζ^{i} and σ^{j}τ^{k} for i, j, k ∈ ℤ/pℤ with (j, k) ≠ (0, 0). There are p^{2} one-dimensional characters indexed by (r, s) ∈ (ℤ/pℤ)^{2}. The remaining p − 1 irreducible characters are conjugates of the Schrödinger representation. These are indexed by t ∈ (ℤ/pℤ)^{×}.

ζ^{i} σ^{j}τ^{k}
λ_{r,s} 1 ζ_{p}^{jr+ks}
θ_{t} pζ_{p}^{it} 0

The dual of θ_{t} is θ_{−t}. From the character table we also deduce

Lemma 4.4. Let t, t′ ∈ (ℤ/pℤ)^{×}.

(i) θ_{t} ⊗ θ_{t′} ≅ \{ \begin{matrix}pθ_{t+t′} & if t + t′ ≢ 0 (mod p) \\ ⨁_{r,s} λ_{r,s} & if t + t′ ≡ 0 (mod p) \end{matrix}

(ii) ∧^{d}θ_{t} ≅ \{ \begin{matrix}λ_{0,0} & if d = 0 or d = p \\ \frac{1}{p} \binom{p}{d}θ_{dt} & if 1 ≤ d ≤ p − 1 \end{matrix}

(iii) S^{d}θ_{t} ≅ \{ \begin{matrix}\frac{1}{p} \binom{p+d−1}{d}θ_{dt} & if d ≢ 0 (mod p) \\ λ_{0,0} ⊕ \frac{1}{p^{2}} (\binom{p+d−1}{d} − 1) ⨁_{r,s} λ_{r,s} & if d ≡ 0 (mod p). \end{matrix}

By (6) the representations W, V, V^{∗}, W^{∗} are equivalent to θ_{t} for t = 1, 2, 3, 4. In the examples at the end of this section we use Lemma 4.4 to compute the dimension of Y^{H_{5}} and then find a basis by inspection.

The representation χ_{1} : Γ → GL_{2}(K) defined by (7) works out as

\begin{matrix}χ_{1}(S) : & (a, b) & ↦ & (φa + b, a − φb)/(ζ_{5}^{4} − ζ_{5}) \\ χ_{1}(T) : & (a, b) & ↦ & (ζ_{5}^{2}a, ζ_{5}^{3}b). \end{matrix}

where φ = 1 + ζ_{5} + ζ_{5}^{4}. To fix our notation for the other irreducible characters we recall the character table for Γ = SL_{2}(ℤ/5ℤ). In the first row we list the sizes of the conjugacy classes. The same symbols are used to denote both a representation and its character. We have written \overline{φ} = 1 − φ.

1 1 20 20 30 12 12 12 12
I −I ST −ST S T −T T^{2} −T^{2}
ψ_{1} 1 1 1  1  1 1 1 1 1
ψ_{2} 4 4 1  1  0 −1 −1 −1 −1
ψ_{3} 5 5 −1 −1 1 0 0 0 0
ψ_{4} 3 3 0  0 −1 φ φ \overline{φ} \overline{φ}
ψ_{5} 3 3 0  0 −1 \overline{φ} \overline{φ} φ φ
χ_{1} 2 −2 −1  1 0 −φ φ −\overline{φ} \overline{φ}
χ_{2} 2 −2 −1 1  0 −\overline{φ} \overline{φ} −φ φ
χ_{3} 4 −4 1 −1 0 −1 1 −1 1
χ_{4} 6 −6 0  0  0 1 −1 1 −1

The discrete covariants in the case Y = K is the trivial representation form the ring of discrete invariants R = K[a, b]^{Γ}. This ring was already studied by Klein. We noted in [14, Section 3] that R is generated by

(8)  \begin{aligned}D & = ab(a^{10} − 11a^{5}b^{5} − b^{10}) \\ c_{4} & = a^{20} + 228a^{15}b^{5} + 494a^{10}b^{10} − 228a^{5}b^{15} + b^{20} \\ c_{6} & = −a^{30} + 522a^{25}b^{5} + 10005a^{20}b^{10} + 10005a^{10}b^{20} − 522a^{5}b^{25} − b^{30} \end{aligned}

subject only to the relation c_{4}^{3} − c_{6}^{2} = 1728D^{5}. The discrete invariants c_{4} and c_{6} are the restrictions of the invariants c_{4} and c_{6} in Theorem 1.2. Our use of the same notation for both a covariant and its restriction to the Hesse family should not cause any confusion in view of the uniqueness statement in Theorem 4.3.

For an arbitrary representation π : Γ → GL_{m}(K) the discrete covariants form an R-module M_{π}. We write M_{π} = ⊕_{d≥0}M_{π,d} for the grading by degree. For any given π and d it is easy to compute a basis for M_{π,d} by linear algebra.

Lemma 4.5. Let π : Γ → GL_{m}(K) be a representation. Then (i) M_{π} is a free K[D, c_{4}]-module of rank 2m. (ii) M_{π} is a free K[D, c_{6}]-module of rank 3m. (iii) M_{π} is a free K[c_{4}, c_{6}]-module of rank 5m.

Moreover if M_{π}(r) ⊂ M_{π} is the direct sum of the graded pieces M_{π,d} with d ≡ r (mod 5), then M_{π}(r) is a free K[c_{4}, c_{6}]-module of rank m.

Proof. In [14, Lemma 5.3] we showed that M_{χ_{1}} is a free K[c_{4}, c_{6}]-module. Since the same method (recalled from [3]) works in general it only remains to compute the ranks. Let 𝕂 = K(a, b)^{Γ} be the field of fractions of R. By the normal basis theorem the 𝕂[Γ]-module K(a, b) is a copy of the regular representation. So if Γ acts on Z = K^{m} via π then

𝕂 ⊗ Z ≅ (K(a, b) ⊗ Z)^{Γ} = 𝕂 ⊗ M_{π}.

In particular dim_{𝕂}(𝕂 ⊗ M_{π}) = m. Statements (i)-(iii) follow since

[𝕂 : K(D, c_{4})] = 2,  [𝕂 : K(D, c_{6})] = 3,  [𝕂 : K(c_{4}, c_{6})] = 5.

Finally we observe that the M_{π}(r) for r ∈ ℤ/5ℤ are free K[c_{4}, c_{6}]-modules with ranks m_{r} (say) adding up to 5m. Multiplication by D shows that m_{r} ≤ m_{r+2} for all r. Therefore m_{0} = … = m_{4} = m as required. □

The Hilbert series of M_{π} can be computed using Molien’s theorem:

(9)  h(M_{π}, t) = ∑_{d=0}^{∞}(dim M_{π,d})t^{d} = \frac{1}{|Γ|} ∑_{γ∈Γ} \frac{Tr π(γ)}{1 − Tr χ_{1}(γ)t + t^{2}} .

For example taking π = χ_{1} we find

h(M_{χ_{1}}, t) = \frac{t + t^{11} + t^{19} + t^{29}}{(1 − t^{12})(1 − t^{20})}

= \frac{t + t^{11} + t^{19} + t^{21} + t^{29} + t^{39}}{(1 − t^{12})(1 − t^{30})}

= \frac{(t + t^{11}) + (t^{13} + t^{23}) + (t^{25} + t^{35}) + (t^{37} + t^{47}) + (t^{19} + t^{29})}{(1 − t^{20})(1 − t^{30})}.

The numerators of these three expression give the degrees of the generators in each part of Lemma 4.5.

There are essentially two ways in which a discrete covariant can fail to be a covariant. The first is that the weights computed using (2) might fail to be integers. For example the discrete invariant D has weights (p, q) = (24/5, 12/5) and so cannot be an invariant. (However Theorem 1.2 tells us that c_{4}, c_{6} and Δ = D^{5} are invariants.) Likewise taking Y = S^{5}W the discrete covariant

(a, b) ↦ ab ∑ w_{0}^{5} − 5b^{2} ∑ w_{0}^{3}w_{1}w_{4} + 5a^{2} ∑ w_{0}^{3}w_{2}w_{3} − 30ab ∏ w_{0}.

has weights (p, q) = (4/5, −3/5) and so cannot be a covariant. The second is that the discrete covariant f might arise from a fractional covariant by which we mean an SL(V) × SL(W)-equivariant rational map F : ∧^{2}V ⊗ W − → Y. It can happen that f is regular even when F is not. For example decomposing (S^{10}W)^{H_{5}} as a Γ-module we find it contains a copy of the trivial representation. So there is a discrete covariant of degree 0. But there are clearly no covariants ∧^{2}V ⊗ W → S^{10}W of degree 0.

In [16] we prove that these are the only two obstructions. More precisely we show that if f : 𝔸^{2} → Y^{H_{5}} is an integer weight discrete covariant then Δ^{k}f is a covariant for some integer k ≥ 0. Moreover we give a practical method for determining the least such k.

If Y is homogeneous of degree (r, s) and Y^{H_{5}} ≠ 0 then the action of the centre of H_{5} shows that 2r +s ≡ 0 (mod 5). We see by (2) that p is an integer if and only if q is an integer. So the integer weight condition is just a congruence mod 5 on the degree of a covariant. In particular Lemma 4.5 shows that the K[c_{4}, c_{6}]-module of integer weight discrete covariants is a free module of rank m = dim Y^{H_{5}}.

In this article we are primarily concerned with the rational representations Y in the following table. In each case Lemma 4.4 shows that dim Y^{H_{5}} = 2 or 3. We list a basis for Y^{H_{5}} (the sums are taken over all cyclic permutations of the subscripts mod 5) followed by its character as a Γ-module. In the final column we list the degrees of the generators for the K[c_{4}, c_{6}]-module of integer weight discrete covariants, as computed using Molien’s theorem.

Table 4.6

Y  basis for Y^{H_{5}}  character degrees
∧^{2}V ⊗ W ∑(v_{1} ∧ v_{4})w_{0}, ∑(v_{2} ∧ v_{3})w_{0}  χ_{1}  1, 11
V^{∗} ⊗ ∧^{2}W ∑ v_{0}^{∗}(w_{1} ∧ w_{4}), ∑ v_{0}^{∗}(w_{2} ∧ w_{3})  χ_{2}  7, 17
V ⊗ ∧^{2}W^{∗} ∑ v_{0}(w_{1}^{∗} ∧ w_{4}^{∗}), ∑ v_{0}(w_{2}^{∗} ∧ w_{3}^{∗})  χ_{2}  13, 23
∧^{2}V^{∗} ⊗ W^{∗} ∑(v_{1}^{∗} ∧ v_{4}^{∗})w_{0}^{∗}, ∑(v_{2}^{∗} ∧ v_{3}^{∗})w_{0}^{∗}  χ_{1}  19, 29
V^{∗} ⊗ S^{2}W ∑ v_{0}^{∗}w_{0}^{2}, ∑ v_{0}^{∗}w_{1}w_{4}, ∑ v_{0}^{∗}w_{2}w_{3}  ψ_{4}  2, 12, 22
S^{2}V^{∗} ⊗ W^{∗} ∑ v_{0}^{∗2}w_{0}^{∗}, ∑ v_{1}^{∗}v_{4}^{∗}w_{0}^{∗}, ∑ v_{2}^{∗}v_{3}^{∗}w_{0}^{∗}  ψ_{5}  14, 24, 34
S^{2}V ⊗ W ∑ v_{0}^{2}w_{0}, ∑ v_{1}v_{4}w_{0}, ∑ v_{2}v_{3}w_{0}  ψ_{5}  6, 16, 26
V ⊗ S^{2}W^{∗} ∑ v_{0}w_{0}^{∗2}, ∑ v_{0}w_{1}^{∗}w_{4}^{∗}, ∑ v_{0}w_{2}^{∗}w_{3}^{∗}  ψ_{4}  18, 28, 38

Checking the conditions in [16] it turns out that each of these discrete covariants is a covariant. In [14] we gave an alternative proof in the cases Y = ∧^{2}V ⊗ W and Y = ∧^{2}V^{∗} ⊗ W^{∗} using evectants. The explicit constructions in Section 8 also show that each of these covariants exists at least as a fractional covariant.

The discrete covariant of degree 2 for Y = V^{∗} ⊗ S^{2}W is P_{2} = ∑ v_{i}^{∗}p_{i} where

(10)  p_{i} = abw_{i}^{2} + b^{2}w_{i−1}w_{i+1} − a^{2}w_{i−2}w_{i+2}

and the discrete covariant of degree 6 for Y = S^{2}V ⊗ W is

Q_{6} = ∑(5a^{3}b^{3}v_{0}^{2} + a(a^{5} − 3b^{5})v_{1}v_{4} − b(3a^{5} + b^{5})v_{2}v_{3})w_{0}.

Substituting v_{i} = p_{i} in Q_{6} gives a covariant of degree 10 for Y = S^{5}W which turns out to be (a scalar multiple of) the secant variety covariant (3). This suggested to us the algorithm for computing Q_{6} in [12, Section 8] that is the key step in our algorithm for computing the invariants.

In the remainder of this article we are concerned with arithmetic applications of the covariants in Table 4.6 and in algorithms for evaluating them on (non-singular) genus one models.

5. Families of 5-congruent elliptic curves

From now on K will be a field of characteristic 0 with algebraic closure \overline{K}.

Definition 5.1. (i) Elliptic curves E and E′ over K are n-congruent if E[n] and E′[n] are isomorphic as Galois modules. (ii) The modular curve Y_{E}^{(r)}(n) = X_{E}^{(r)}(n) \ {cusps} parametrises the family of elliptic curves n-congruent to E via an isomorphism ψ with e_{n}(ψS, ψT) = e_{n}(S, T)^{r} for all S, T ∈ E[n].

The curves X_{E}^{(r)}(n) depend only on the class of r ∈ (ℤ/nℤ)^{×} modulo squares. In the cases r = ±1 we denote them X_{E}(n) and X_{E}^{−}(n). Rubin and Silverberg [20], [21], [23] computed formulae for the families of elliptic curves parametrised by Y_{E}(n) for n = 2, 3, 4, 5. In [14] we gave a new proof of their result and extended to Y_{E}^{−}(n) for n = 3, 4, 5. In the case n = 5 this is not so interesting since −1 is a square mod 5. In Theorem 5.8 below we remedy this by giving a formula for the family of elliptic curves parametrised by Y_{E}^{(2)}(5).

First we need some preliminaries on Heisenberg groups. Since we have dropped our earlier assumption that K is algebraically closed our point of view is slightly different from that in Section 3.

Definition 5.2. A Heisenberg group is a Galois invariant subgroup H ⊂ SL_{n}(\overline{K}) such that

(i) H is the inverse image of a subgroup Δ ⊂ PGL_{n}(\overline{K}) with Δ ≅ (ℤ/nℤ)^{2}. (ii) Taking commutators in H induces a non-degenerate pairing Δ × Δ → μ_{n}.

Let C ⊂ ℙ^{n−1} be a genus one normal curve of degree n. The Heisenberg group defined by C is the group of all matrices in SL_{n}(\overline{K}) that act on C as translation by an n-torsion point of its Jacobian E. In this case the commutator pairing is the Weil pairing e_{n} : E[n] × E[n] → μ_{n}. If C is a curve of degree n = 5 then there is another Heisenberg group determined by C coming instead from the action of E[5] on the space of quadrics defining C.

Lemma 5.3. Let ϕ ∈ ∧^{2}V ⊗ W be non-singular and let E = Jac(C_{ϕ}). Then there are projective representations χ_{V} : E[5] → PGL(V) and χ_{W} : E[5] → PGL(W) such that

(i) The action of E[5] on C_{ϕ} ⊂ ℙ(W^{∗}) is given by χ_{W} and
(ii) (χ_{V} (T), χ_{W} (T))ϕ ∝ ϕ for all T ∈ E[5].

Proof: If g_{W} ∈ GL(W) describes an automorphism of C_{ϕ} then by [14, Lemma 7.6] there exists g_{V} ∈ GL(V), unique up to sign, such that (g_{V} , g_{W})ϕ = ϕ. So once we have used (i) to define χ_{W}, condition (ii) uniquely determines χ_{V}. □

The projective representations χ_{V} and χ_{W} determine Heisenberg groups

(11)  H_{1} ⊂ SL(W^{∗}) H_{2} ⊂ SL(V^{∗}) H_{3} ⊂ SL(V) H_{4} ⊂ SL(W)

where the first of these is the Heisenberg group defined by C_{ϕ}. It follows by (6) that the commutator pairing on E[5] induced by H_{r} is the rth power of the Weil pairing.

Theorem 5.4. Let ϕ ∈ ∧^{2}V ⊗ W be a non-singular genus one model determining Heisenberg groups H_{1}, …, H_{4} as above. Then the genus one normal curves with Heisenberg group H_{r} are the C_{ϕ′} for ϕ′ a non-singular member of the pencil spanned by F_{1}(ϕ) and F_{2}(ϕ) where F_{1} and F_{2} are a basis for the K[c_{4}, c_{6}]-module of covariants ∧^{2}V ⊗ W → Y and

Y = \{ \begin{matrix}∧^{2}V ⊗ W & if r = 1 \\ V ⊗ ∧^{2}W^{∗} & if r = 2 \\ V^{∗} ⊗ ∧^{2}W & if r = 3 \\ ∧^{2}V^{∗} ⊗ W^{∗} & if r = 4. \end{matrix}

Proof: This generalises [14, Theorem 8.2] where we treated the case r = 1.

For the proof we may assume that K is algebraically closed. By Lemma 4.1 and the covariance of F_{1} and F_{2} we may assume that ϕ = u(a, b) is a Hesse model. Then each H_{r} is the standard Heisenberg group generated by the matrices (4). By [14, Lemma 7.5] the genus one normal curves with this Heisenberg group are the C_{ϕ′} for ϕ′ a non-singular Hesse model. Splitting into the cases r = 1, 2, 3, 4 we checked by computing the discrete covariants (see Remark 5.5 below for the case r = 3) that F_{1}(ϕ) and F_{2}(ϕ) are linearly independent. They therefore span the space of Hesse models. □

In [14] we studied the cases r = 1, 4. We now work out explicit formulae in the case r = 3. According to the table at the end of Section 4 the K[c_{4}, c_{6}]-module of covariants for Y = V^{∗} ⊗ ∧^{2}W is generated by covariants Ψ_{7} and Ψ_{17} of degrees 7 and 17. The corresponding discrete covariants are

(12)  (a, b) ↦ f_{d}(a, b) ∑ v_{0}^{∗}(w_{1} ∧ w_{4}) + g_{d}(a, b) ∑ v_{0}^{∗}(w_{2} ∧ w_{3})

where

f_{7}(a, b) = b^{2}(7a^{5} − b^{5}), f_{17}(a, b) = b^{2}(17a^{15} + 187a^{10}b^{5} + 119a^{5}b^{10} + b^{15}),
g_{7}(a, b) = a^{2}(a^{5} + 7b^{5}), g_{17}(a, b) = −a^{2}(a^{15} − 119a^{10}b^{5} + 187a^{5}b^{10} − 17b^{15}).

Remark 5.5. Direct calculation shows that f_{7}g_{17} − g_{7}f_{17} = −24D^{2}. We deduce that if ϕ is non-singular then Ψ_{7}(ϕ) and Ψ_{17}(ϕ) are linearly independent. In [16] we generalise this to arbitrary Y.

We recall that the ring of discrete invariants K[a, b]^{Γ} is generated by the polynomials D, c_{4} and c_{6} in (8).

Lemma 5.6. There are polynomials D(λ, μ), c_{4}(λ, μ) and c_{6}(λ, μ) with coefficients in K[c_{4}, c_{6}] such that

D(λ, μ) = 27 · D(λf_{7} + μf_{17}, λg_{7} + μg_{17})/D(a, b)^{2}
c_{4}(λ, μ) = 54^{2} · c_{4}(λf_{7} + μf_{17}, λg_{7} + μg_{17})
c_{6}(λ, μ) = 54^{3} · c_{6}(λf_{7} + μf_{17}, λg_{7} + μg_{17})

Proof: The coefficients are discrete invariants of degree a multiple of 5. We can then therefore write them as polynomials in c_{4} and c_{6}. (The factors 27, 54^{2}, 54^{3} are included to make D, c_{4}, c_{6} primitive polynomials in ℤ[c_{4}, c_{6}, λ, μ].) □

The polynomials D(λ, μ), c_{4}(λ, μ) and c_{6}(λ, μ) are easily computed from the description in Lemma 5.6. We find

D(λ, μ) = −(125c_{4}^{3} + 64c_{6}^{2})λ^{12} − 1620c_{4}^{2}c_{6}λ^{11}μ − 66(25c_{4}^{4} + 56c_{4}c_{6}^{2})λ^{10}μ^{2}

− 220(11c_{4}^{3}c_{6} + 16c_{6}^{3})λ^{9}μ^{3} + 1485(5c_{4}^{5} + 4c_{4}^{2}c_{6}^{2})λ^{8}μ^{4} + 792(53c_{4}^{4}c_{6} + 28c_{4}c_{6}^{3})λ^{7}μ^{5}

+ 660(9c_{4}^{6} + 164c_{4}^{3}c_{6}^{2} + 16c_{6}^{4})λ^{6}μ^{6} + 2376(19c_{4}^{5}c_{6} + 44c_{4}^{2}c_{6}^{3})λ^{5}μ^{7}

+ 495(27c_{4}^{7} + 104c_{4}^{4}c_{6}^{2} + 112c_{4}c_{6}^{4})λ^{4}μ^{8} + 220(81c_{4}^{6}c_{6} + 136c_{4}^{3}c_{6}^{3} + 80c_{6}^{5})λ^{3}μ^{9}

− 594(9c_{4}^{8} − 32c_{4}^{5}c_{6}^{2} − 16c_{4}^{2}c_{6}^{4})λ^{2}μ^{10} − 60(135c_{4}^{7}c_{6} − 328c_{4}^{4}c_{6}^{3} + 112c_{4}c_{6}^{5})λμ^{11}

− (729c_{4}^{9} + 108c_{4}^{6}c_{6}^{2} − 2896c_{4}^{3}c_{6}^{4} + 1600c_{6}^{6})μ^{12},

c_{4}(λ, μ) = \frac{−1}{11^{2} · 12^{2}} |\begin{matrix}\frac{∂^{2}D}{∂λ^{2}} & \frac{∂^{2}D}{∂λ∂μ} \\ \frac{∂^{2}D}{∂λ∂μ} & \frac{∂^{2}D}{∂μ^{2}} \end{matrix} | and c_{6}(λ, μ) = \frac{−1}{12 · 20} |\begin{matrix}\frac{∂D}{∂λ} & \frac{∂D}{∂μ} \\ \frac{∂c_{4}}{∂λ} & \frac{∂c_{4}}{∂μ} \end{matrix} | .

These polynomials satisfy the relation

c_{4}(λ, μ)^{3} − c_{6}(λ, μ)^{2} = (c_{4}^{3} − c_{6}^{2})^{2} D(λ, μ)^{5}.

We have contributed them to Magma [4] as HessePolynomials(5,2,[c4,c6]).

Lemma 5.7. The K[c_{4}, c_{6}]-module of covariants ∧^{2}V ⊗ W → V^{∗} ⊗ ∧^{2}W is generated by covariants Ψ_{7} and Ψ_{17} satisfying

c_{4}(λΨ_{7} + μΨ_{17}) = c_{4}(λ, μ)/54^{2}
c_{6}(λΨ_{7} + μΨ_{17}) = c_{6}(λ, μ)/54^{3}.

Proof: By Lemma 4.1 and the covariance of Ψ_{7} and Ψ_{17} it suffices to check these identities on the Hesse family. But in that case we are done by the definitions of c_{4} and c_{6} in Lemma 5.6. □

We say that elliptic curves E and E′ are indirectly 5-congruent if there is an isomorphism of Galois modules ψ : E[5] ≅ E′[5] with e_{5}(ψS, ψT) = e_{5}(S, T)^{r} for some r ∈ {2, 3}. In the notation introduced at the start of this section the elliptic curves indirectly 5-congruent to E are parametrised by Y_{E}^{(2)}(5).

Theorem 5.8. Let E be an elliptic curve over K with Weierstrass equation

y^{2} = x^{3} − 27c_{4}x − 54c_{6}.

Then the family of elliptic curves parametrised by Y_{E}^{(2)}(5) is

E_{λ,μ} : y^{2} = x^{3} − 12c_{4}(λ, μ)x − 16c_{6}(λ, μ)

where the coefficients of c_{4}(λ, μ) and c_{6}(λ, μ) are evaluated at c_{4}, c_{6} ∈ K.

Proof: We embed E ⊂ ℙ^{4} via the complete linear system |5.0_{E}|. The image is defined by some ϕ ∈ ∧^{2}V ⊗ W with invariants c_{4} and c_{6}. Let H_{1}, …, H_{4} be the Heisenberg groups (11) determined by ϕ.

(i) Suppose that ϕ′ = λΨ_{7}(ϕ)+μΨ_{17}(ϕ) is non-singular. By Theorem 5.4 the genus one normal curves C_{ϕ} ⊂ ℙ^{4} and C_{ϕ′} ⊂ ℙ^{4} have Heisenberg groups H_{1} and H_{3}. By Theorem 1.2 and Lemma 5.7 the Jacobians of these curves are E and E_{λ,μ}. Since the Heisenberg group carries the information of both the action of Galois on the 5-torsion of the Jacobian, and the Weil pairing (via the commutator), it follows that E and E_{λ,μ} are indirectly 5-congruent. (ii) Let E′ be an elliptic curve indirectly 5-congruent to E. By [9, Theorem 5.2] there is a genus one normal curve C′ ⊂ ℙ^{4} with Jacobian E′ and Heisenberg group H_{3}. Then Theorem 5.4 shows that C′ = C_{ϕ′} for some ϕ′ = λΨ_{7}(ϕ) + μΨ_{17}(ϕ). Taking Jacobians gives E′ ≅ E_{λ,μ}. □

We also worked out formulae corresponding to the case r = 2 of Theorem 5.4. We omit the details since the family of elliptic curves obtained is the same as that in Theorem 5.8. (We encountered a similar situation in [14] with the cases r = ±1.)

Example 5.9. Let F/ℚ be the elliptic curve y^{2} + xy = x^{3} − 607x + 5721 labelled 2834c1 in Cremona’s tables [7]. The invariants of this Weierstrass equation are c_{4} = 29137 and c_{6} = −4986649. Substituting into the above expression for D and then making a change of variables^{1} to simplify we obtain

𝔇(ξ, η) = \frac{1}{2^{89} · 3^{15} · 13^{9}} D(1663ξ + 2850η, 7ξ + 18η)
= −60647ξ^{12} + 74183ξ^{11}η − 366344ξ^{10}η^{2} − 965800ξ^{9}η^{3}
+ 1640430ξ^{8}η^{4} − 166188ξ^{7}η^{5} + 1473362ξ^{6}η^{6} − 1041216ξ^{5}η^{7}

+ 1224300ξ^{4}η^{8} + 816860ξ^{3}η^{9} − 474188ξ^{2}η^{10} + 22692ξη^{11} − 51256η^{12}.

1This change of variables was found by minimising to make the numerical factor on the right hand side of (13) a small integer, and reducing as described in [24].

By Theorem 5.8 the family of elliptic curves indirectly 5-congruent to F is y^{2} = x^{3} − 27𝔠_{4}(ξ, η)x − 54𝔠_{6}(ξ, η) where

𝔠_{4}(ξ, η) =\frac{−1}{11^{2}} |\begin{matrix}\frac{∂^{2}𝔇}{∂ξ^{2}} & \frac{∂^{2}𝔇}{∂ξ∂η} \\ \frac{∂^{2}𝔇}{∂ξ∂η} & \frac{∂^{2}𝔇}{∂η^{2}} \end{matrix} | and 𝔠_{6}(ξ, η) =\frac{−1}{20} |\begin{matrix}\frac{∂𝔇}{∂ξ} & \frac{∂𝔇}{∂η} \\ \frac{∂𝔠_{4}}{∂ξ} & \frac{∂𝔠_{4}}{∂η} \end{matrix} | .

These polynomials satisfy the relation

(13)  𝔠_{4}(ξ, η)^{3} − 𝔠_{6}(ξ, η)^{2} = 2 · 13 · 109^{2} · 1728 𝔇(ξ, η)^{5}.

We specialise ξ, η to integers with max(|ξ|, |η|) ≤ 100 and sort by conductor to obtain a list of elliptic curves that begins

ξ η conductor  [a_{1}, …, a_{6}]
3 2  2834 [1, −1, 1, −8109, −279017]
0 1  18157438 [1, −1, 1, 68377761, 119969009527]
1 0 171873598 [1, 0, 0, 895245563, 21917334070263]
2 1 205326134 [1, 0, 0, −637387852699482, −6550975667615204649116]
1 1 1506404198 [1, 0, 0, −793652608607, −207340288851298727]
1 −1 6582143542 [1, 0, 0, −2705846635122, −1178369764561303100]

The first curve in this list is the elliptic curve E labelled 2834d1 in Cremona’s tables. We discuss the elliptic curves E and F further in Example 6.7.

6. Doubling in the 5-Selmer group and visibility

Let E/K be an elliptic curve. In [9] we interpreted the group H^{1}(K, E[n]) as parametrising Brauer-Severi diagrams [C → S] as twists of [E → ℙ^{n−1}]. We also studied the obstruction map Ob_{n} : H^{1}(K, E[n]) → Br(K)[n] that sends the class of [C → S] to the class of the Brauer-Severi variety S. The diagrams with trivial obstruction, i.e. S ≅ ℙ^{n−1}, are genus one normal curves of degree n. Strictly speaking a diagram includes the choice of an action of E on C. So in general a genus one normal curve of degree n with Jacobian E represents a pair of inverse elements in H^{1}(K, E[n]).

In the case n = 5 we obtain the following partial interpretation of H^{1}(K, E[5]) in terms of genus one models. We say that genus one models are properly equivalent if they are related by (g_{V} , g_{W}) ∈ GL(V) × GL(W) with (det g_{V})^{2} det g_{W} = 1.

Theorem 6.1. Let c_{4} and c_{6} be the invariants of a Weierstrass equation for E. Then the genus one models over K with invariants c_{4} and c_{6}, up to proper Kequivalence, are parametrised by ker(Ob_{5}) ⊂ H^{1}(K, E[5]).

Proof: This is analogous to the case n = 3 treated in [11, Theorem 2.5]. The proof given there relies on a statement about invariant differentials which is generalised to the case n = 5 in [12, Proposition 5.19]. □

The obstruction map is not a group homomorphism and its kernel is not a group. So given two genus one models with the same invariants, their sum in H^{1}(K, E[5]) need not be represented by a genus one model. However the obstruction map is quadratic and in particular satisfies Ob_{n}(aξ) = a^{2}Ob_{n}(ξ) for a ∈ ℤ. If ϕ is a genus one model representing ξ ∈ ker(Ob_{5}) then −ϕ (which has the same invariants) represents −ξ. In this section we show how to find a model ϕ′ representing ±2ξ. It turns out that C_{ϕ′} sits inside an ambient space ℙ^{4} that is naturally the dual of the ambient space for C_{ϕ}.

First we need to recall another of the interpretations of H^{1}(K, E[n]) given in [9]. A theta group for E[n] is a central extension of E[n] by 𝔾_{m} with commutator given by the Weil pairing. The base theta group Θ_{E} ⊂ GL_{n}(\overline{K}) is the set of all matrices that act on the base diagram [E → ℙ^{n−1}] as translation by an n-torsion point of E. Then H^{1}(K, E[n]) parametrises the theta groups for E[n] as twists of Θ_{E}.

In Section 5 we saw that a non-singular genus one model ϕ determines Heisenberg groups H_{1}, …, H_{4}. We write Θ_{r} for the theta group generated by H_{r} and the scalar matrices. Writing E = Jac(C_{ϕ}) we see that Θ_{r} is a theta group for E[5] where the latter is equipped with the rth power of the Weil pairing.

Lemma 6.2. Let ϕ, ϕ′ ∈ ∧^{2}V ⊗ W be non-singular genus one models determining theta groups Θ_{1}, …, Θ_{4} and Θ′_{1}, …, Θ′_{4}. If Jac(C_{ϕ}) = Jac(C_{ϕ′}) = E then there exists ξ ∈ H^{1}(K, E[5]) and isomorphisms γ_{r} : Θ_{r} ≅ Θ′_{r} such that

(14)  σ(γ_{r})γ_{r}^{−1} : x ↦ e_{5}(ξ_{σ}, x)^{r}x

for all σ ∈ Gal(\overline{K}/K) and r ∈ (ℤ/5ℤ)^{×}.

Proof: The curves C_{ϕ} and C_{ϕ′} are isomorphic over \overline{K}. So by [12, Proposition 4.6] there exists g = (g_{V} , g_{W}) ∈ GL(V)×GL(W) with gϕ = ϕ′. In fact we can choose g so that it induces the identity map on the Jacobian E. The isomorphisms γ_{1}, …, γ_{4} are conjugation by g_{W}^{−T} , g_{V}^{−T} , g_{V} , g_{W} where the superscript −T indicates inverse transpose. Then σ(γ_{r})γ_{r}^{−1} is conjugation by an element of Θ′_{r} above ξ_{σ} ∈ E[5], with ξ_{σ} independent of r. The conclusion (14) follows since the commutator pairing for Θ_{r} is the rth power of the Weil pairing. □

Theorem 6.3. Let E and F be elliptic curves over K and ψ : E[5] ≅ F [5] an isomorphism of Galois modules with e_{5}(ψS, ψT) = e_{5}(S, T)^{r} for some r ∈ (ℤ/5ℤ)^{×}. Let Θ_{1}, …, Θ_{4} be the theta groups determined by a non-singular genus one model ϕ ∈ ∧^{2}V ⊗ W with Jac(C_{ϕ}) = E. If Θ_{1} is the twist of Θ_{E} by ξ ∈ H^{1}(K, E[5]) then Θ_{r} is the twist of Θ_{F} by ψ_{∗}(ξ) ∈ H^{1}(K, F [5])

Proof: We first prove the case ξ = 0. We claim that if ϕ describes the image of E ⊂ ℙ^{4} embedded by |5.0_{E}| then Θ_{r} → E[5] has a Galois equivariant section T ↦ M_{T} with M_{S}M_{T} = e_{5}(S, T)^{r/2}M_{S+T}. We recall the proof of this in the case r = 1 from [9, Lemma 3.11]. The [−1]-map on E lifts to ι ∈ PGL_{5}(K). Then there is a unique scaling of M_{T} such that M_{T}^{5} = I and ιM_{T} ι^{−1} = M_{T}^{−1}. The uniqueness ensures that T ↦ M_{T} is Galois equivariant. Now if M_{S}M_{T} = λM_{S+T} then conjugating by ι gives M_{S}^{−1}M_{T}^{−1} = λM_{S+T}^{−1} and so λ^{2} = M_{S}M_{T} M_{S}^{−1}M_{T}^{−1} = e_{5}(S, T). This proves the claim when r = 1. The cases r = 2, 3, 4 are similar using that the [−1]-map induces involutions in PGL(V) and PGL(W). Applying the claim to both E and F we see that if Θ_{1} ≅ Θ_{E} then Θ_{r} ≅ Θ_{F}. This proves the theorem in the case ξ = 0. The general case follows by Lemma 6.2. □

According to Table 4.6 the K[c_{4}, c_{6}]-module of covariants for Y = ∧^{2}V^{∗} ⊗ W^{∗} is generated by covariants Π_{19} and Π_{29} of degrees 19 and 29. The corresponding discrete covariants are

(15)  (a, b) ↦ \frac{1}{deg c_{k}} (\frac{∂c_{k}}{∂a} ∑(v_{1}^{∗} ∧ v_{4}^{∗})w_{0}^{∗} + \frac{∂c_{k}}{∂b} ∑(v_{2}^{∗} ∧ v_{3}^{∗})w_{0}^{∗})

for k = 4, 6. As noted in [14] these are the evectants of c_{4} and c_{6}.

Theorem 6.4. Let Π_{49} = \frac{1}{144}(c_{6}Π_{19} − c_{4}Π_{29}) be the covariant of degree 49 whose restriction to the Hesse family is

(a, b) ↦ D^{4}(b ∑(v_{1}^{∗} ∧ v_{4}^{∗})w_{0}^{∗} − a ∑(v_{2}^{∗} ∧ v_{3}^{∗})w_{0}^{∗})

If ϕ ∈ ∧^{2}V ⊗ W is non-singular and ϕ′ = Π_{49}(ϕ) then (i) C_{ϕ} and C_{ϕ′} have the same Jacobian elliptic curve E, and (ii) the class of [C_{ϕ′} → ℙ^{4}] is twice the class of [C_{ϕ} → ℙ^{4}] in H^{1}(K, E[5]).

Proof: (i) By considering ϕ a Hesse model we deduce

c_{4}(ϕ′) = Δ(ϕ)^{16}c_{4}(ϕ)
c_{6}(ϕ′) = Δ(ϕ)^{24}c_{6}(ϕ)

It follows by Theorem 1.2 that the Jacobians are isomorphic. (ii) We apply Theorem 6.3 in the case ψ : E[5] → E[5] is multiplication by 2. This shows that the double of [C_{ϕ} → ℙ^{4}] has theta group Θ_{4}. By (i) and Theorem 5.4 this double is [C_{ϕ′} → ℙ^{4}]. □

Remarks 6.5. (i) Whether Theorem 6.4 is a formula for doubling or tripling in H^{1}(K, E[5]) depends on the choice of isomorphism Jac(C_{ϕ}) ≅ Jac(C_{ϕ′}). We have not attempted to resolve these sign issues. (ii) If K is a number field then the n-Selmer group S^{(n)}(E/K) is by definition a subgroup H^{1}(K, E[n]). It is well known that S^{(n)}(E/K) ⊂ ker(Ob_{n}). Thus Theorem 6.4 gives a formula for doubling/tripling in the 5-Selmer group. (iii) Let g = (g_{V} , g_{W}) ∈ GL(V) × GL(W) be any element defined over K with (deg g_{V})^{2}(det g_{W}) = Δ(ϕ)^{4}, for example a pair of diagonal matrices. Then in terms of Theorem 6.1 the double/triple of ϕ is ±g^{−1}Π_{49}(ϕ).

Example 6.6. Wuthrich [27] constructed an element of order 5 in the Tate-Shafarevich group of the elliptic curve E/ℚ with Weierstrass equation

y^{2} + xy + y = x^{3} + x^{2} − 3146x + 39049.

His example (also discussed in [12, Section 9]) is defined by the 4 × 4 Pfaffians of

(\begin{matrix}0 & 310x_{1} + 3x_{2} + 162x_{5} & −34x_{1} − 5x_{2} − 14x_{5} & 10x_{1} + 28x_{4} + 16x_{5} & 80x_{1} − 32x_{4} \\ & 0 & 6x_{1} + 3x_{2} + 2x_{5} & −6x_{1} + 7x_{3} − 4x_{4} & −14x_{2} − 8x_{3} \\ & & 0 & −x_{3} & 2x_{2} \\ & − & & 0 & −4x_{1} \\ & & & & 0 \end{matrix})

The algorithms in [17] suggest making a change of co-ordinates

(\begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{matrix}) ← (\begin{matrix}0 & 4 & −8 & 4 & 8 \\ 0 & 0 & 0 & 0 & 16 \\ 0 & −4 & 4 & 0 & 12 \\ 4 & 5 & −15 & 2 & 7 \\ 4 & −12 & 20 & −12 & −8 \end{matrix}) (\begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{matrix})

so that Wuthrich’s example becomes

(\begin{matrix}0 & x_{2} + x_{5} & −x_{5} & −x_{1} + x_{2} & x_{4} \\ & 0 & x_{2} − x_{3} + x_{4} & x_{1} + x_{2} + x_{3} − x_{4} − x_{5} & x_{1} − x_{2} − x_{3} − x_{4} − x_{5} \\ & & 0 & x_{1} − x_{2} + 2x_{3} − x_{4} − x_{5} & −x_{2} − x_{4} + x_{5} \\ & − & & 0 & −x_{3} − x_{4} − 2x_{5} \\ & & & & 0 \end{matrix})

Our Magma function DoubleGenusOneModel uses the algorithms in Section 8 to evaluate Π_{19} and Π_{29} and then returns Π_{49} = \frac{1}{144}(c_{6}Π_{19} − c_{4}Π_{29}). Running it on the above model ϕ gives a model ϕ′ with entries

ϕ′_{12} = 3534132778x_{1} + 3583651940x_{2} − 881947110x_{3} − 323014538x_{4} + 3395115339x_{5}
ϕ′_{13} = 5079379222x_{1} − 2965539950x_{2} + 11022202860x_{3} + 12821590868x_{4} + 640276471x_{5}
ϕ′_{14} = −10098238458x_{1} − 1274966110x_{2} − 7873816170x_{3} − 3456923272x_{4} − 62353929x_{5}
ϕ′_{15} = −12929747724x_{1} − 6790511810x_{2} − 11113305270x_{3} − 15161763156x_{4} + 3241937033x_{5}
ϕ′_{23} = −3381247332x_{1} + 3810679160x_{2} + 5919634530x_{3} + 75326852x_{4} − 1245085426x_{5}
ϕ′_{24} = −3572860258x_{1} − 5569480730x_{2} − 953739600x_{3} − 2138046812x_{4} − 858145244x_{5}
ϕ′_{25} = −4674149266x_{1} − 943631490x_{2} − 6754488160x_{3} + 751535046x_{4} + 117685567x_{5}
ϕ′_{34} = −1851228934x_{1} + 5238146110x_{2} − 165588410x_{3} − 2070411506x_{4} + 678105748x_{5}
ϕ′_{35} = −6992835070x_{1} − 3744630360x_{2} + 3130208220x_{3} − 4523781310x_{4} + 433739425x_{5}
ϕ′_{45} = 780078472x_{1} + 2039763820x_{2} − 450062790x_{3} − 7105731722x_{4} + 1625466111x_{5}

The algorithms in [17] suggest making a change of co-ordinates

(\begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{matrix}) ← (\begin{matrix}92 & −36 & −153 & 129 & −131 \\ −54 & 84 & 5 & −206 & 139 \\ −63 & −174 & −60 & −79 & 53 \\ −111 & 106 & 206 & −115 & −162 \\ 314 & −466 & 158 & −328 & −12 \end{matrix}) (\begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{matrix})

whereupon the model ϕ′ simplifies to

(\begin{matrix}0 & −x_{4} + x_{5} & x_{3} − x_{4} + x_{5} & x_{2} − x_{5} & x_{1} − x_{2} + x_{3} − x_{4} − 2x_{5} \\ & 0 & x_{1} + x_{5} & −x_{2} − x_{3} & −x_{2} + x_{5} \\ & & 0 & x_{4} & −x_{1} \\ & − & & 0 & x_{1} + x_{4} − x_{5} \\ & & & & 0 \end{matrix})

This is the double in � (E/ℚ)[5] of Wuthrich’s example. If we double again then we get back to the original example. Moreover the matrices needed to minimise and reduce are the transposes of those used above.

Let E and F be a pair of n-congruent elliptic curves. In terminology introduced by Mazur [8] the visible subgroup of H^{1}(K, E) explained by F (K) is the image of the composite

\frac{F (K)}{nF (K)} ⟶^{δ} H^{1}(K, F [n]) ≅ H^{1}(K, E[n]) →^{ι} H^{1}(K, E)[n]

where the maps δ and ι come from the Kummer exact sequences for E and F, and the middle isomorphism is induced by the congruence. Our interest is in using visibility to compute explicit elements of H^{1}(K, E). In [14] we gave examples in the cases n = 2, 3, 4, 5 assuming in the case n = 5 that the congruence E[5] ≅ F [5] respects the Weil pairing. Using Theorems 5.4 and 6.3 and the explicit constructions in Section 8 we may now remove this restriction.

Example 6.7. We start with the pair of elliptic curves E = 2834d1 and F = 2834c1 taken from [8, Table 1]. We have already seen in Example 5.9 that E and F are indirectly 5-congruent. Alternatively this may be checked as follows. Let c_{4}(λ, μ) and c_{6}(λ, μ) be the polynomials defined in Section 5 with coefficients specialised to the invariants c_{4} = 29137 and c_{6} = −4986649 of F. Then writing j_{E} = −389217^{3}/(2 · 13 · 109^{3}) for the j-invariant of E we find that the binary form of degree 60

(1728 − j_{E})c_{4}(λ, μ)^{3} + j_{E}c_{6}(λ, μ)^{2} = 0

has a unique ℚ-rational root. Substituting this root (λ : μ) = (3563 : 19) into Theorem 5.8 confirms that E and F are indirectly 5-congruent.

Our method is now the same as that in [14, Section 15] except that in place of the Hessian we use the covariants

Ψ_{7}, Ψ_{17} : ∧^{2}V ⊗ W → V^{∗} ⊗ ∧^{2}W.

We have F (ℚ) ≅ ℤ^{2} generated by P_{1} = (−10, 109) and P_{2} = (−28, 45). If we embedding F ⊂ ℙ^{4} via the complete linear system |4.0_{F} +P | with P = P_{1} then the image is defined by a genus one model ϕ. Our Magma function GenusOneModel(5,P) computes such a model

(\begin{matrix}0 & x_{2} + 2x_{4} − 3x_{5} & 3x_{1} − x_{2} + 8x_{3} + 2x_{4} − 3x_{5} & x_{1} − 2x_{4} + 3x_{5} & x_{3} − x_{4} + x_{5} \\ & 0 & 3x_{2} + 2x_{3} + 2x_{4} + 3x_{5} & x_{2} + x_{3} & x_{5} \\ & & 0 & x_{3} + 2x_{4} + 2x_{5} & x_{4} + x_{5} \\ & − & & 0 & 0 \\ & & & & 0 \end{matrix})

with the same invariants as F. The algorithms in Section 8 for evaluating Ψ_{7} and Ψ_{17} are implemented in our Magma function HesseCovariants(phi,2). We use them to compute ϕ′ = 3563Ψ_{7}(ϕ) + 19Ψ_{17}(ϕ). The algorithms in [17] suggest making a change of co-ordinates

(\begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{matrix}) ← (\begin{matrix}−16 & 16 & −9 & −4 & −8 \\ 0 & 0 & −2 & −4 & 0 \\ 24 & 0 & 7 & −4 & 8 \\ 8 & 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{matrix}) (\begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{matrix})

so that ϕ′ becomes

(\begin{matrix}0 & −x_{2} + x_{3} & x_{1} − x_{5} & x_{2} − 2x_{5} & x_{2} − x_{4} − x_{5} \\ & 0 & −x_{1} + x_{2} + x_{3} + 2x_{5} & −x_{2} + x_{4} + x_{5} & −x_{1} − x_{2} \\ & & 0 & −x_{1} + x_{2} + x_{4} & x_{3} + x_{4} \\ & − & & 0 & x_{1} \\ & & & & 0 \end{matrix}) .

This genus one model has the same invariants as E. In particular its 4 × 4 Pfaffians define a curve C ⊂ ℙ^{4} with good reduction at all primes p ≠ 2, 13, 109. For p = 2, 13, 109 we checked directly that C(ℚ_{p}) ≠ ∅. Since E(ℚ) = 0 it follows that C represents a non-trivial element of � (E/ℚ)[5]. Repeating for P = r_{1}P_{1} + r_{2}P_{2} for 0 ≤ r_{1}, r_{2} ≤ 4 we similarly find equations for all elements in a subgroup of � (E/ℚ) isomorphic to (ℤ/5ℤ)^{2}.

The following corollary was already proved in many (but not all) cases in the appendix to [2].

Corollary 6.8. Let (E, F, n) be any of the triples listed in [8, Table 1]. Then E and F are n-congruent, and the visible subgroup of H^{1}(ℚ, E) explained by F (ℚ) is contained in � (E/ℚ).

Proof: The examples in this table all have n = 3 or 5. Using the methods in [14] and in this paper, we verified the congruences and computed equations for all relevant elements of H^{1}(ℚ, E). We then checked directly that these curves are everywhere locally soluble. □

7. The invariant differential

The following definition is suggested by the discussion in [12, Section 2].

Definition 7.1. Let C ⊂ ℙ^{n−1} be a genus one normal curve with hyperplane section H. An Ω-matrix for C is an n × n alternating matrix of quadratic forms representing the linear map

∧^{2}L(H) → L(2H) ; f ∧ g ↦ \frac{fdg − gdf}{ω}

where ω is an invariant differential on C.

Since the natural map S^{2}L(H) → L(2H) is surjective it is clear that Ω-matrices exist. However for n > 3 their entries are only determined up to the addition of quadrics vanishing on C. Nonetheless we claim in Conjecture 7.4 below that there is a canonical choice.

From an Ω-matrix we may recover the invariant differential ω using the rule

ω = \frac{x_{j}^{2}d(x_{i}/x_{j})}{Ω_{ij}} for any i ≠ j.

We may also characterise Ω-matrices as alternating matrices of quadratic forms such that

(∂f/∂x_{0} ⋯ ∂f/∂x_{n−1}) Ω = 0

in K(C) for all f ∈ I(C), and Ω has rank 2 at all points on C.

Returning to the case n = 5 this suggests looking for covariants in the case Y = ∧^{2}W^{∗} ⊗ S^{2}W. By Lemma 4.4 we have ∧^{2}θ_{4} ⊗ S^{2}θ_{1} ≅ 2θ_{3} ⊗ 3θ_{2} ≅ 6 ∑_{r,s} λ_{r,s} and so dim Y^{H_{5}} = 6. A basis for Y^{H_{5}} is

\begin{matrix}∑(x_{1}^{∗} ∧ x_{4}^{∗})x_{0}^{2}, & ∑(x_{1}^{∗} ∧ x_{4}^{∗})x_{1}x_{4}, & ∑(x_{1}^{∗} ∧ x_{4}^{∗})x_{2}x_{3}, \\ ∑(x_{2}^{∗} ∧ x_{3}^{∗})x_{0}^{2}, & ∑(x_{2}^{∗} ∧ x_{3}^{∗})x_{1}x_{4}, & ∑(x_{2}^{∗} ∧ x_{3}^{∗})x_{2}x_{3}. \end{matrix}

Since −I, T ∈ Γ act on Y^{H_{5}} with traces −6 and 1 it is easy to see from the character table for Γ that Y^{H_{5}} has character χ_{4} = S^{5}χ_{1}. The K[c_{4}, c_{6}]-module of integer weight discrete covariants is generated in degrees 5, 15, 15, 25, 25, 35. Checking the conditions in [16] shows that all of these are covariants.

The discrete covariant of degree 5 is

Ω_{5} = (\begin{matrix}0 & α_{3} & β_{1} & −β_{4} & −α_{2} \\ −α_{3} & 0 & α_{4} & β_{2} & −β_{0} \\ −β_{1} & −α_{4} & 0 & α_{0} & β_{3} \\ β_{4} & −β_{2} & −α_{0} & 0 & α_{1} \\ α_{2} & β_{0} & −β_{3} & −α_{1} & 0 \end{matrix})

where

(16)  \begin{aligned}α_{i} & = & 5a^{4}bw_{i}^{2} − 10a^{3}b^{2}w_{i−1}w_{i+1} + (a^{5} − 3b^{5})w_{i−2}w_{i+2} \\ β_{i} & = & 5ab^{4}w_{i}^{2} − (3a^{5} + b^{5})w_{i−1}w_{i+1} + 10a^{2}b^{3}w_{i−2}w_{i+2}. \end{aligned}

Proposition 7.2. If ϕ ∈ ∧^{2}V ⊗ W is non-singular then Ω_{5}(ϕ) is an Ω-matrix for C_{ϕ} ⊂ ℙ^{4}.

Proof: By Lemma 4.1 and the covariance of Ω_{5} it suffices to prove this for ϕ a Hesse model. Let p_{0}, …, p_{4} be the quadrics (10) defining C_{ϕ} and J = (∂p_{i}/∂w_{j}) the Jacobian matrix. We checked by direct calculation that all the entries of JΩ_{5} belong to the homogeneous ideal I(C_{ϕ}) = (p_{0}, …, p_{4}). Since C_{ϕ} ⊂ ℙ^{4} is a smooth curve the Jacobian matrix J has rank 3 at all points of C_{ϕ}. So Ω_{5} has rank at most 2 on C_{ϕ}. Since an alternating matrix always has even rank it only remains to show that Ω_{5} is non-zero on C_{ϕ}. By (10) and (16) it suffices to show that

(17)  det (\begin{matrix}ab & b^{2} & −a^{2} \\ 5a^{4}b & −10a^{3}b^{2} & a^{5} − 3b^{5} \\ 5ab^{4} & −3a^{5} − b^{5} & 10a^{2}b^{3} \end{matrix}) = 18D

is non-zero. Since Δ = D^{5} this is clear by Theorem 1.2 and our assumption that ϕ is non-singular. □

Taking the 4 × 4 Pfaffians of Ω_{5} and identifying ∧^{4}W^{∗} = W gives a covariant for Y = W ⊗ S^{4}W. This is a scalar multiple of the vector of partial derivatives of the secant variety covariant (3). This observation not only gives an algorithm for computing Ω_{5} (used in Section 8) but also suggested to us the following conjecture about Ω-matrices for genus one normal curves of arbitrary degree. The rth higher secant variety Sec^{r}C of a curve C is the Zariski closure of the locus of all (r − 1)- planes spanned by r points on C. Thus the usual secant variety is Sec^{2}C.

Lemma 7.3. Let C ⊂ ℙ^{n−1} be a genus one normal curve of degree n ≥ 3.

(a) If n = 2r + 1 is odd then Sec^{r}C ⊂ ℙ^{n−1} is a hypersurface {F = 0} of

degree n.

(b) If n = 2r + 2 is even then Sec^{r}C ⊂ ℙ^{n−1} is a complete intersection {F_{1} =

F_{2} = 0} where F_{1} and F_{2} each have degree n/2.

Proof: See [10] or [25]. □

Let R = K[x_{0}, …, x_{n−1}] be the co-ordinate ring of ℙ^{n−1} and write R = ⊕_{d≥0}R_{d} for its usual grading by degree. We require that morphisms of graded R-modules have degree 0 and write R(d) for the R-module with eth graded piece R_{d+e}.

Conjecture 7.4. Let C ⊂ ℙ^{n−1} be a genus one normal curve of degree n ≥ 3, and let F, respectively F_{1} and F_{2}, be as in Lemma 7.3.

(a) If n is odd then there is a minimal free resolution

0 ⟶ R(−2n) ⟶^{P^{T}} R(−n − 1)^{n} ⟶^{Ω} R(−n + 1)^{n} ⟶^{P} R

where Ω is an alternating matrix of quadratic forms and

P = (∂F/∂x_{0} ⋯ ∂F/∂x_{n−1}) .

Moreover P is (a scalar multiple of) the vector of (n−1)×(n−1) Pfaffians of Ω, and Ω is an Ω-matrix for C.

(b) If n is even then there is a minimal free resolution

0 ⟶ R(−n)^{2} ⟶^{P^{T}} R(\frac{−n+2}{2})^{n} ⟶^{Ω} R(\frac{−n+2}{2})^{n} ⟶^{P} R^{2}

where Ω is an alternating matrix of quadratic forms and

P = (\begin{matrix}∂F_{1}/∂x_{0} & ⋯ & ∂F_{1}/∂x_{n−1} \\ ∂F_{2}/∂x_{0} & ⋯ & ∂F_{2}/∂x_{n−1} \end{matrix}) .

Moreover the 2 × 2 minors of P are (a fixed scalar multiple of) the (n − 2) × (n − 2) Pfaffians of Ω, and Ω is an Ω-matrix for C.

Remarks 7.5. (i) If n = 3, 4 then the equations in Lemma 7.3 are the equations for Sec^{1}C = C. The conjecture reduces to some well known formulae for the invariant differential. (ii) If n = 2r + 1 is odd then it is known (see [25, Section 8]) that Sec^{r}C has singular locus Sec^{r−1}C and the latter is Gorenstein of codimension 3. It follows by the Buchsbaum-Eisenbud structure theorem [5], [6] that a minimal free resolution of the stated form exists. The content of the conjecture is that the alternating matrix constructed in this way is an Ω-matrix. (iii) If n = 5 then it suffices to take C = C_{ϕ} with ϕ a Hesse model. We have already observed that the 4 × 4 Pfaffians of Ω_{5}(ϕ) are the partial derivatives of F = S_{10}(ϕ). Combined with (ii) this proves the conjecture in the case n = 5. (iv) We have tested the conjecture in some numerical examples over finite fields for n = 6, 7, 8, 10, 12.

8. Explicit constructions

We give evaluation algorithms for each of the covariants in Table 4.6 (as reproduced below). This is mainly of interest for the covariants Π_{19} and Π_{29} used in Example 6.6 and the covariants Ψ_{7} and Ψ_{17} used in Example 6.7.

Y  degrees covariants
∧^{2}V ⊗ W 1, 11  U, H
V^{∗} ⊗ ∧^{2}W 7, 17  Ψ_{7}, Ψ_{17}
V ⊗ ∧^{2}W^{∗} 13, 23  Ξ_{13}, Ξ_{23}
∧^{2}V^{∗} ⊗ W^{∗} 19, 29  Π_{19}, Π_{29}
V^{∗} ⊗ S^{2}W 2, 12, 22 P_{2}, P_{12}, P_{22}
S^{2}V ⊗ W 6, 16, 26 Q_{6}, Q_{16}, Q_{26}
V ⊗ S^{2}W^{∗} 18, 28, 38 R_{18}, R_{28}, R_{38}
S^{2}V^{∗} ⊗ W^{∗} 14, 24, 34 S_{14}, S_{28}, S_{38}

We fix our choice of generators by specifying them on the Hesse family. The covariant U is the identity map, whereas the Hessian is given by

H = −\frac{∂D}{∂b} ∑(v_{1} ∧ v_{4})w_{0} + \frac{∂D}{∂a} ∑(v_{2} ∧ v_{3})w_{0}.

The corresponding formulae for Ψ_{7}, Ψ_{17}, Π_{19} and Π_{29} are given in (12) and (15). We put

Ξ_{d} = f_{d}(a, b) ∑ v_{0}(w_{1}^{∗} ∧ w_{4}^{∗}) + g_{d}(a, b) ∑ v_{0}(w_{2}^{∗} ∧ w_{3}^{∗})

where f_{13}(a, b) = b^{3}(26a^{10} + 39a^{5}b^{5} − b^{10}), g_{13}(a, b) = a^{3}(a^{10} + 39a^{5}b^{5} − 26b^{10}) and

f_{23}(a, b) = −b^{3}(46a^{20} + 1173a^{15}b^{5} − 391a^{10}b^{10} + 207a^{5}b^{15} + b^{20}),
g_{23}(a, b) = a^{3}(a^{20} − 207a^{15}b^{5} − 391a^{10}b^{10} − 1173a^{5}b^{15} + 46b^{20}).

We recall that P_{2} is the map taking a genus one model to its vector of 4 × 4 Pfaffians. The remaining generators in the above table are uniquely determined by the following “Pfaffian identities”.

\begin{aligned}P_{2}(λU + μH) & = λ^{2}P_{2} + 2λμP_{12} + μ^{2}P_{22} \\ P_{2}(λΨ_{7} + μΨ_{17}) & = λ^{2}S_{14} + 2λμS_{24} + μ^{2}S_{34} \\ P_{2}(λΞ_{13} + μΞ_{23}) & = λ^{2}Q_{26} − λμ(c_{6}Q_{6} + c_{4}Q_{16}) + μ^{2}(c_{4}^{2}Q_{6} + c_{6}Q_{16} − c_{4}Q_{26}) \\ P_{2}(λΠ_{19} + μΠ_{29}) & = λ^{2}(c_{4}R_{18} + R_{38}) + λμ(c_{6}R_{18} + c_{4}R_{28}) + μ^{2}(c_{6}R_{28} − c_{4}R_{38}) \end{aligned}

The evaluation algorithms below are justified by checking them on the Hesse family, and then appealing to Lemma 4.1 and the appropriate covariance properties to show that they work for all non-singular models. (We do not consider the case where the input is singular.)

Lemma 8.1. Let (F, G) = (P_{2}, P_{12}) or (Q_{6}, Q_{16}). If ϕ ∈ ∧^{2}V ⊗W is non-singular and F (ϕ) and G(ϕ) are represented by quadratic forms f_{0}, …, f_{4} and g_{0}, …, g_{4} in variables x_{0}, …, x_{4} then the 75 quintic forms {(f_{i}g_{j} + f_{j}g_{i})x_{k} : i ≤ j} are linearly independent.

Proof: It suffices to check this for ϕ = u(a, b) a Hesse model.

We arrange the coefficients of the quintic forms in a 75×126 matrix. The entries are homogeneous polynomials in ℚ[a, b]. In principle we could finish the proof by computing the GCD of the 75 × 75 minors. In practice we first decompose the space of quintic forms into its eigenspaces for the action of x_{i} ↦ ζ_{5}^{i}x_{i}. This leaves us with one 15 × 26 matrix and four 15 × 25 matrices. Rather than compute all 15 × 15 minors we compute just those that correspond to sets of monomials that are invariant under cyclic permutations of the x_{i}. In each case we find that the GCD of these minors divides a power of the discriminant. □

We gave algorithms for evaluating the invariants c_{4} and c_{6} in [12, Section 8] and the Hessian H in [14, Section 11]. We compute P_{2}, P_{12} and P_{22} by taking 4 × 4 Pfaffians of linear combinations of U and H as indicated in the first of the Pfaffian identities above. We compute Q_{6} as described in [12, Section 8]. Lemma 8.1 shows that we can solve for Q_{16} and Q′_{26} = 5(4Q_{26} + 3c_{4}Q_{6}) using the identities

Q_{16}(P_{2}, P_{12}) = Q_{6}(P_{12}, P_{12}),
Q′_{26}(P_{2}, P_{12}) = Q_{6}(P_{12}, P_{22}) + 4Q_{16}(P_{12}, P_{12}).

Next we use the determinant map V ⊗(V ⊗W) → S^{5}V to compute some covariants taking values in S^{5}V:

det(λU + μQ_{6}) = λ^{4}μM_{10} − 2λ^{2}μ^{3}M_{20} + μ^{5}M_{30},
det(λH + μQ_{6}) = λ^{4}μM_{50} + 2λ^{2}μ^{3}M_{40} + μ^{5}M_{30},
det(λU + μQ_{16}) = λ^{4}μM_{20} + 2λ^{2}μ^{3}M′_{50} + μ^{5}M_{80}.

Lemma 8.1 shows that we can solve for R_{18} and R_{28} using the identities

R_{18}(Q_{6}, Q_{16}) = \frac{−1}{18} (5c_{6}M_{10} + 14c_{4}M_{20} + M_{40}),
R_{28}(Q_{6}, Q_{16}) = \frac{−1}{792}(9c_{4}^{2}M_{10} + 620c_{6}M_{20} − 270c_{4}M_{30} + M_{50} − 216M′_{50}).

Then we compute Π_{19} and Π_{29} using the natural map

(∧^{2}V ⊗ W) × (V ⊗ S^{2}W^{∗}) → ∧^{2}V^{∗} ⊗ W^{∗}
(U, R_{18}) ↦ 2Π_{19}
(U, R_{28}) ↦ 2Π_{29}.

In Section 7 we constructed a covariant Ω_{5} : ∧^{2}V ⊗W → ∧^{2}W^{∗} ⊗S^{2}W. Conjecture 7.4 (which is a theorem in the case n = 5) gives an algorithm for computing Ω_{5} (up to sign) using minimal free resolutions. We may represent P_{12} and P_{22} as 5-tuples of quadrics and Ω_{5} as a 5 × 5 alternating matrix of quadrics. Then Ψ_{7} and Ψ_{17} tell us how to write the quadrics in P_{12} and P_{22} as linear combinations of the quadrics in Ω_{5}. In basis-free language there is a natural map

(V^{∗} ⊗ ∧^{2}W) × (∧^{2}W^{∗} ⊗ S^{2}W) → V^{∗} ⊗ S^{2}W
(Ψ_{7}, Ω_{5}) ↦ P_{12}
(Ψ_{17}, Ω_{5}) ↦ \frac{1}{2}(P_{22} + c_{4}P_{2}).

The proof of Proposition 7.2 shows that if ϕ is non-singular then the entries of Ω_{5}(ϕ) above the diagonal are linearly independent. We can therefore solve for Ψ_{7} and Ψ_{17} by linear algebra. We then compute Ξ_{13} and Ξ_{23} using the natural map

(V^{∗} ⊗ ∧^{2}W) × (S^{2}V ⊗ W) → V ⊗ ∧^{2}W^{∗}
(Ψ_{7}, Q_{6}) ↦ 2 Ξ_{13}
(Ψ_{7}, Q_{16}) ↦ −2 Ξ_{23}.

Since we only computed Ω_{5} up to sign we have only computed Ψ_{7}, Ψ_{17}, Ξ_{13} and Ξ_{23} up to sign. Fortunately this does not matter for our applications. (See Example 6.7.)

The remaining covariants S_{14}, S_{24}, S_{34} and R_{38} may be computed using the Pfaffian identities.

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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