INVARIANT THEORY FOR THE ELLIPTIC NORMAL QUINTIC, II. THE COVERING MAP

TOM FISHER

Abstract. A genus one curve C of degree 5 is defined by the 4×4 Pfaffians of a 5 × 5 alternating matrix of linear forms on ℙ^{4}. We prove a result characterising the covariants for these models in terms of their restrictions to the family of curves parametrised by the modular curve X(5). We then construct covariants describing the covering map of degree 25 from C to its Jacobian and give a practical algorithm for evaluating them.

1. Introduction

Definition 1.1. Let n ≥ 3 be an integer.

(i) An elliptic normal curve C ⊂ ℙ^{n−1} is a smooth curve of genus one and degree n that spans ℙ^{n−1}.

(ii) A rational nodal curve C ⊂ ℙ^{n−1} is a rational curve of degree n that spans ℙ^{n−1} and has a single node.

If C ⊂ ℙ^{n−1} is an elliptic normal curve then there is a covering map π of degree n^{2} from C to its Jacobian E given by P ↦ [nP −H] ∈ Pic^{0}(C) ≅ E where H is the hyperplane section. We may also describe π : C → E as the map that quotients out by the action of E[n] on C by translation (assuming we are not in characteristic dividing n). The subgroup of SL_{n} consisting of matrices that describe this action is called the Heisenberg group of C. If n is odd then over an algebraically closed field we may change co-ordinates so that this group is generated by

(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 ( \begin{matrix}0 & 0 & ⋯ & 0 & 1 \\ 1 & 0 & ⋯ & 0 & 0 \\ 0 & 1 & ⋯ & 0 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \end{matrix} )

where ζ_{n} is a primitive nth root of unity. (If n is even then one must take scalar multiples of these matrices with determinant 1.)

In the cases n = 2, 3, 4 classical invariant theory gives formulae for the Jacobian E and for the covering map π : C → E. See [13], [14] for the cases n = 2, 3, and [2] for a survey of the cases n = 2, 3, 4. In [8] we gave a practical algorithm for evaluating the invariants in the case n = 5 and showed that they give a formula for the Jacobian. We now extend this invariant theoretic approach to give a formula for the covering map.

Date: 11th March 2013.

We work throughout over a field K of characteristic not dividing 6n, where in due course we take n = 5. Except in the following paragraph, and at the end of Section 8, we assume for simplicity that K is algebraically closed.

To explain the motivation for our work, let E be an elliptic curve over a number field K. For any integer n ≥ 2 the quotient group E(K)/nE(K) injects into the n-Selmer group S^{(n)}(E/K), which is finite and effectively computable. In an explicit n-descent calculation one represents each element of the n-Selmer group by (equations for) an elliptic normal curve C ⊂ ℙ^{n−1} with Jacobian E. It is perhaps better to call C a “genus one normal curve” as it need not have any Krational points. The Selmer group elements with C(K) ≠ ∅ make up the image of E(K)/nE(K) in S^{(n)}(E/K). Moreover if P ∈ C(K) then a coset representative for the corresponding element of E(K)/nE(K) is given by the image of P under the covering map. Having explicit formulae for the covering map can therefore help in finding generators for the Mordell-Weil group E(K).

In the case n = 5 the curves of Definition 1.1 are called elliptic normal quintics and rational nodal quintics. By the Buchsbaum-Eisenbud structure theorem [4],

[5] they are defined by the 4 × 4 Pfaffians of a 5 × 5 alternating matrix of linear forms on ℙ^{4}. We call such a matrix ϕ a genus one model and write C_{ϕ} ⊂ ℙ^{4} for the subvariety defined by the 4 × 4 Pfaffians. It is shown in [8, Proposition 5.10] that C_{ϕ} is a smooth curve of genus one if and only if it is an elliptic normal quintic. In this case we say that ϕ is non-singular.

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 acts by changing coordinates on ℙ^{4}. 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}. 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}).

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

(2)  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}.

Lemma 1.2. The action of GL(V) × GL(W) is transitive on the the genus one models ϕ for which C_{ϕ} is a rational nodal quintic, and on the genus one models ϕ for which C_{ϕ} is an elliptic normal quintic with given j-invariant.

Proof. See [8, Proposition 4.6]. □

The co-ordinate ring K[∧^{2}V ⊗ W] is a polynomial ring in 50 variables.

Theorem 1.3. 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. Moreover if we scale them as specified in [8] 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 j-invariant c_{4}(ϕ)^{3}/Δ(ϕ).

Proof. See [8, Theorem 4.4]. □

Lemma 1.4. Let ϕ ∈ ∧^{2}V ⊗ W be a genus one model with C_{ϕ} either an elliptic normal quintic or a rational nodal quintic. Then the Zariski closure of the GL(V)× GL(W)-orbit of ϕ is the zero locus of an irreducible homogeneous invariant I. Moreover we can take

I = \{ \begin{matrix}c_{4} & if j(C_{ϕ}) = 0 \\ c_{6} & if j(C_{ϕ}) = 1728 \\ Δ & if C_{ϕ} is a rational nodal quintic \\ c_{4}^{3} − j(C_{ϕ})Δ & otherwise. \end{matrix}

Proof. The existence of I is proved in [8, Lemma 4.10]. The invariants listed vanish at ϕ by Theorem 1.3 and are irreducible in K[c_{4}, c_{6}]. They are therefore irreducible in K[∧^{2}V ⊗W] since any factors would themselves have to be invariants. We use here that SL(V) × SL(W) is connected and has no 1-dimensional rational representations. Alternatively we can prove irreducibility by restricting to the Weierstrass models in [8, Section 6]. □

Lemma 1.5. Let I be a non-constant homogeneous invariant. Then there exists ϕ ∈ ∧^{2}V ⊗ W with I(ϕ) = 0 and C_{ϕ} either an elliptic normal quintic or a rational nodal quintic.

Proof. We may assume that I is irreducible in K[c_{4}, c_{6}]. So up to scalar multiples we have I = c_{4}, c_{6}, Δ or c_{4}^{3} − jΔ with j ≠ 0, 1728. We take C_{ϕ} to be an elliptic normal quintic with the appropriate j-invariant, or in the case I = Δ a rational nodal quintic. □

The covariants we need to describe the covering map are SL(V) × SL(W)- equivariant polynomial maps ∧^{2}V ⊗W → S^{5d}W for d = 1, 2, 3. More generally we defined a covariant to be an SL(V) × SL(W)-equivariant polynomial map ∧^{2}V ⊗ W → Y where Y is a rational representation of GL(V) × GL(W). In all our examples Y will be homogeneous by which we mean there exist integers r and s such that the morphism ρ_{Y} : GL(V) × GL(W) → GL(Y) satisfies ρ_{Y} (λI_{V} , μI_{W}) = λ^{r}μ^{s}I_{Y} for all λ, μ ∈ K^{×}.

Lemma 1.6. Let Y be a homogeneous rational representation of GL(V)×GL(W) with degrees (r, s). If F : ∧^{2}V ⊗ W → Y is a homogeneous covariant then there exist integers p and q called the weights of F such that

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

Proof. See [10, Lemma 2.2]. □

For example the Pfaffian map (2) is a covariant of degree 2 with weights (p, q) = (1, 0). The covariants in the case Y is the trivial representation are the invariants as described in Theorem 1.3. For general Y the covariants form a module over the ring of invariants K[c_{4}, c_{6}].

In Section 2 we recall our method [10] for studying the covariants via their restrictions to the Hesse family, i.e. the universal family over X(5). These restrictions are nearly characterised by their invariance properties under an appropriate action of SL_{2}(ℤ/5ℤ). In Sections 3 and 4 we make this relationship precise. Thus our work resolves, albeit in one particular case, what is described in [1, Chapter V,§22] as the “mysterious role of invariant theory”. We give examples for a range of different Y in Section 5. In Section 6 we show how a free basis for the K[c_{4}, c_{6}]- module of covariants for Y may be characterised in terms of its specialisations to the genus one models ϕ of the form considered in Lemma 1.4. In Section 7 we relate the covariants in the case Y = S^{5}W to work of Hulek [11] and finally in Section 8 we give our formula for the covering map.

2. Discrete covariants

In this section we recall some of the theory from [10]. We then state our main result on the relationship between covariants and discrete covariants.

We take n ≥ 5 an odd integer. 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} and its centre is a cyclic group of order n generated by ζ = [σ, τ] = στσ^{−1}τ^{−1}. In [10, Section 3] we defined a group homomorphism s_{β} : GL_{2}(ℤ/nℤ) → Aut(H_{n}) 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}.

where the exponents are read as integers mod n.

Definition 2.1. 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′).

The Schrödinger representation θ : H_{n} → SL_{n}(K) maps σ and τ to the matrices (1). These matrices have commutator θ(ζ) = ζ_{n}I_{n}.

Theorem 2.2. (i) The Schrödinger representation θ : H_{n} → SL_{n}(K) extends uniquely to a representation θ^{+} : H_{n}^{+} → SL_{n}(K). (ii) The normaliser of θ(H_{n}) in SL_{n}(K) is θ^{+}(H_{n}^{+}).

Proof. See [10, Theorem 3.6]. □

Remark 2.3. (i) The representation θ^{+} of Theorem 2.2 is given on the generators S = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and T = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) for SL_{2}(ℤ/nℤ) by suitable scalar multiples of

( \begin{matrix}1 & 1 & 1 & ⋯ & 1 \\ 1 & ζ_{n} & ζ_{n}^{2} & ⋯ & ζ_{n}^{−1} \\ 1 & ζ_{n}^{2} & ζ_{n}^{4} & ⋯ & ζ_{n}^{−2} \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ 1 & ζ_{n}^{−1} & ζ_{n}^{−2} & ⋯ & ζ_{n} \end{matrix} ) and ( \begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 0 & ζ_{n}^{1/2} & 0 & ⋯ & 0 \\ 0 & 0 & ζ_{n}^{2^{2}/2} & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & 0 & ⋯ & ζ_{n}^{1/2} \end{matrix} ) .

(ii) 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 2.2 to any one of these representations.

The Hesse family of elliptic normal quintics (studied for example in [9], [11]) is given by

(4)  \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. We define actions of the Heisenberg group H_{5} on V and W so that the Hesse models u(a, b) are H_{5}-invariant.

(5)  \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}

Since θ_{V} and θ_{W} are conjugates of the Schrödinger representation they extend by Theorem 2.2 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 homogeneous rational representation of GL(V) × GL(W). Then θ_{V} and θ_{W} define an action of H_{5}^{+} on Y and so an action of Γ = SL_{5}(ℤ/5ℤ) on Y^{H_{5}}. Taking Y = ∧^{2}V ⊗ W the action of Γ on (∧^{2}V ⊗ W)^{H_{5}} = Im(u) is described by a representation χ_{1} : Γ → GL_{2}(K).

Definition 2.4. Let π : Γ → GL(Z) be a representation. A discrete covariant for Z is a polynomial map f : 𝔸^{2} → Z satisfying f ∘ χ_{1}(γ) = π(γ) ∘ f for all γ ∈ Γ.

Theorem 2.5. 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. See [10, Theorem 4.3]. □

For any given Y the discrete covariants may be computed using invariant theory for the finite groups H_{5} and SL_{2}(ℤ/5ℤ). 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 2.5. It is important to note that not every discrete covariant is a covariant. For example, taking Y to be the trivial representation, the ring of invariants is K[c_{4}, c_{6}] as described in Theorem 1.3 whereas the ring of discrete invariants is generated by

(6)  \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}. We use the same notation for both a covariant and its restriction to the Hesse family. By the uniqueness part of Theorem 2.5 this should not cause any confusion.

There are essentially two ways in which a discrete covariant might fail to be a covariant. The first is that the weights computed using (3) might not be integers. For example D has weights (p, q) = (24/5, 12/5) and so cannot be an invariant. The second is that denominators might be introduced. More precisely we prove the following theorem in Section 3.

Theorem 2.6. Let f : 𝔸^{2} → Y^{H_{5}} be an integer weight discrete covariant. Then Δ^{k}f is a covariant for some k ≥ 0.

In Section 4 we give a practical method for computing the least such k.

Remark 2.7. 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 (3) 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. Since Δ = D^{5} and deg D = 12 is coprime to 5, an equivalent formulation of Theorem 2.6 is that if f : 𝔸^{2} → Y^{H_{5}} is a homogeneous discrete covariant then D^{m}f is a covariant for some m ≥ 0.

3. Fractional covariants

In this section we prove Theorem 2.6.

Lemma 3.1. Let ϕ ∈ ∧^{2}V ⊗ W be a non-singular Hesse model.

(i) The stabiliser of ϕ in SL(V) × SL(W) is

H = {(θ_{V} (h), θ_{W} (h)) : h ∈ H_{5}}.

(ii) The normaliser of H in SL(V) × SL(W) is

N = {(θ_{V} (h), ζθ_{W} (h)) : (ζ, h) ∈ μ_{5} × H_{5}^{+}}.

Proof. (i) It is clear by (4) and (5) that H is contained in the stabiliser of ϕ. Since any automorphism of C_{ϕ} of order 5 is translation by a 5-torsion point of its Jacobian, all such automorphisms are described by elements of H.

Now let g ∈ SL(V)×SL(W) with g(ϕ) = ϕ and let γ be the automorphism of C_{ϕ} induced by g. By [8, Proposition 5.19 and Lemma 2.4] γ preserves the invariant differential and is therefore a translation map. Since C_{ϕ} ⊂ ℙ^{4} is a curve of degree 5 this translation is by a point of order 5. Composing g with a suitable element of H reduces us to the case γ is the identity. Then g = (g_{V} , g_{W}) is a pair of scalar matrices. Since these matrices each have determinant 1 and jointly fix ϕ it follows that (g_{V} , g_{W}) = (θ_{V} (h), θ_{W} (h)) for some h in the centre of H_{5}.

(ii) We see by Theorem 2.2(ii) that N is contained in the normaliser of H, and that any element of the normaliser may be composed with an element of N to give an element of the form g = (I_{V} , g_{W}) where I_{V} is the identity. Since θ_{V} is faithful it follows that g_{W} is in the centraliser of θ_{W} (H_{5}) in SL(W), which turns out to consist only of scalar matrices. □

The following proposition will be used to explain the relationship between the covariants and the discrete covariants.

Proposition 3.2. Let G be a linear algebraic group acting on irreducible affine varieties X and Y. Let H ⊂ G be a subgroup whose normaliser N ⊂ G is of finite order coprime to char K. Suppose that A ⊂ X^{H} is an irreducible variety acted on by N/H, and U ⊂ A is a dense open subset such that

(i) the morphism G × U → X; (g, ϕ) ↦ g(ϕ) has dense image, (ii) the stabiliser in G of each element of U is H, (iii) either char K = 0 or the derivative of the map in (i) is an isomorphism

at all points of G × U. Then by restriction to A there is a bijection between

• G-equivariant rational maps F : X − → Y, and

• N/H-equivariant rational maps f : A − → Y^{H}

Proof. Let F : X − → Y be a G-equivariant rational map. Its domain of definition is a G-invariant open subset of X and hence by (i) it meets U. Therefore F restricts to a rational map f on A. By hypothesis A is acted on by N and pointwise fixed by H. Since F is N-equivariant it follows that f(A) ⊂ Y^{H} and f is N/H-equivariant.

Conversely suppose f : A − → Y^{H} is an N/H-equivariant rational map. We let δ ∈ N act on G × A via (g, a) ↦ (gδ^{−1}, δa). Since N is a finite group of order coprime to char K and G × A is an affine variety, the quotient (G × A)/N exists, and is an affine variety. We consider the maps

ψ_{id} : (G × A)/N ⟶ X;  (g, a) ↦ g(a)
ψ_{f} : (G × A)/N − → Y;  (g, a) ↦ g(f(a)).

Shrinking U if necessary, we may assume that N/H acts on U. By (i) ψ_{id} has dense image, by (ii) it is injective on the dense subset (G × U)/N, and by (iii) it is separable. It follows that ψ_{id} is birational. Then F = ψ_{f} ∘ ψ_{id}^{−1} is a G-equivariant rational map extending f. □

Proof of Theorem 2.6. We apply Proposition 3.2 with G = SL(V) × SL(W), X = ∧^{2}V ⊗ W and H ⊂ N ⊂ G as in Lemma 3.1. We also let A = X^{H} be the space of Hesse models and U ⊂ A the space of non-singular Hesse models.

We check the hypotheses (i), (ii) and (iii). By [9, Proposition 4.1] every nonsingular model is equivalent to a Hesse model and by Theorem 1.3 the nonsingular models are Zariski dense in ∧^{2}V ⊗ W. This proves (i). We checked (ii) in Lemma 3.1 and (iii) is checked in Lemma 3.3 below.

By Lemma 3.1 and the definition of H_{5}^{+} we have N/H ≅ μ_{5} × Γ where Γ = SL_{2}(ℤ/5ℤ). Now f is Γ-equivariant by definition of a discrete covariant and μ_{5}- equivariant by the assumption it has integer weights. So by Proposition 3.2 it is the restriction of a G-equivariant rational map F : ∧^{2}V ⊗ W − → Y. (We say F is a fractional covariant.)

It remains to show that Δ^{k}F is regular for some k ≥ 0. Let S ∈ K[∧^{2}V ⊗ W] be a homogeneous polynomial of least degree such that SF is regular. Then F = R/S where R is a covariant and S is an invariant. Suppose S(ϕ) = 0 for some non-singular model ϕ. By [9, Proposition 4.1] we may suppose that ϕ is a Hesse model, and so by the regularity of f we have R(ϕ) = S(ϕ) = 0. By Lemma 1.4 the Zariski closure of the GL(V) × GL(W)-orbit of ϕ is the zero locus of a homogeneous invariant I. Now both R and S are divisible by I and this contradicts the choice of S. Therefore F is regular on all non-singular models. By Theorem 1.3(i) and the Nullstellensatz it follows that Δ^{k}F is regular for some k ≥ 0. □

The following lemma completes the proof of Theorem 2.6 in the case of positive characteristic (still assuming char K ≠ 2, 3, 5).

Lemma 3.3. The derivative of the morphism

\begin{matrix}SL(V) × SL(W) × 𝔸^{2} & → ∧^{2}V ⊗ W \\ (g_{V} , g_{W} , (a, b)) & ↦ (g_{V} , g_{W})u(a, b) \end{matrix}

is an isomorphism at all (g_{V} , g_{W} , (a, b)) with D(a, b) ≠ 0.

Proof. It suffices to compute the derivative at (I_{V} , I_{W} , (a, b)). This is a linear map

(7)  𝔰𝔩(V) × 𝔰𝔩(W) × 𝔸^{2} → ∧^{2}V ⊗ W.

We write E_{ij} for the n × n matrix with (i, j) entry 1 and all other entries 0. Then 𝔰𝔩_{n} has basis {E_{ij} : i ≠ j} ∪ {E_{00} − E_{ii} : i ≠ 0}. Taking these bases for 𝔰𝔩(V) and 𝔰𝔩(W), the standard basis for 𝔸^{2} and the basis {(v_{i}∧v_{j})w_{k} : i < j} for ∧^{2}V ⊗W, we found by direct calculation that the derivative (7) has determinant 5^{4}D(a, b)^{4}. □

4. Denominators

In this section we show how to find the least value of k in Theorem 2.6. We consider the family of genus one models

(8)  \begin{matrix}u_{1} : 𝔸^{5} & → ∧^{2}V ⊗ W \\ (λ_{0}, …, λ_{4}) & ↦ ∑ λ_{0}(v_{1} ∧ v_{4})w_{0} + ∑(v_{2} ∧ v_{3})w_{0}. \end{matrix}

where the sums are taken over all cyclic permutations of the subscripts mod 5. These models are related to the Hesse family by

(9)  u_{1}(a, …, a) = u(a, 1).

Remark 4.1. If ϕ = u_{1}(λ_{0}, …, λ_{4}) then C_{ϕ} ⊂ ℙ^{4} is defined by λ_{0}x_{0}^{2} + x_{1}x_{4} − λ_{2}λ_{3}x_{2}x_{3} = 0 and its cyclic permutes. These curves were studied in [7] where it is shown that ϕ = u_{1}(λ, 1, …, 1) defines the universal family of (generalised) elliptic curves parametrised by X_{1}(5). Here λ is a co-ordinate on X_{1}(5) ≅ ℙ^{1}.

Definition 4.2. Let D ⊂ SL(V) × SL(W) be the subgroup of pairs of diagonal matrices

(10)  ( \begin{matrix}α_{1}α_{4} \\ & α_{0}α_{2} \\ & & α_{1}α_{3} \\ & & & α_{2}α_{4} \\ & & & & α_{0}α_{3} \end{matrix} ) , ( \begin{matrix}α_{0} \\ & α_{1} \\ & & α_{2} \\ & & & α_{3} \\ & & & & α_{4} \end{matrix} ) ,

with ∏ α_{i} = 1.

Lemma 4.3. The action of D on 𝔸^{5} compatible with u_{1} is

(λ_{0}, λ_{1}, λ_{2}, λ_{3}, λ_{4}) ↦ (\frac{α_{0}^{2}}{α_{1}α_{4}} λ_{0}, \frac{α_{1}^{2}}{α_{0}α_{2}} λ_{1}, \frac{α_{2}^{2}}{α_{1}α_{3}} λ_{2}, \frac{α_{3}^{2}}{α_{2}α_{4}} λ_{3}, \frac{α_{4}^{2}}{α_{0}α_{3}} λ_{4}).

In particular D acts transitively on the subsets of 𝔸^{5} defined by the condition that λ_{0}, …, λ_{4} have a fixed non-zero product.

Proof. Let g_{V} and g_{W} be the matrices (10) with (α_{0}, …, α_{4}) = (α, 1, …, 1). Then

(g_{V} , g_{W}) u_{1}(λ_{0}, …, λ_{4}) = α u_{1}(α^{2}λ_{0}, α^{−1}λ_{1}, λ_{2}, λ_{3}, α^{−1}λ_{4}).

From this calculation and the obvious cyclic symmetry it follows that the action of D on 𝔸^{5} is as stated. In the special case (α_{0}, …, α_{4}) = (β^{−2}, β^{−1}, 1, β, β^{2}) this action is given by (λ_{0}, λ_{1}, λ_{2}, λ_{3}, λ_{4}) ↦ (β^{−5}λ_{0}, λ_{1}, λ_{2}, λ_{3}, β^{5}λ_{4}). Since we are working over an algebraically closed field the final statement is clear. □

Let Y be a homogeneous rational representation of GL(V) × GL(W).

Theorem 4.4. Let f : 𝔸^{2} → Y^{H_{5}} be an integer weight discrete covariant. (i) There is a unique D-equivariant rational map f_{1} : 𝔸^{5} − → Y with

f_{1}(a, …, a) = f(a, 1).

(ii) f is a covariant if and only if f_{1} is regular.

Proof. By Theorem 2.6 there is a fractional covariant F : ∧^{2}V ⊗ W − → Y with f = F ∘u. It follows by (9) and Lemma 4.3 that f_{1} = F ∘u_{1} satisfies (i). Uniqueness is proved using the final part of Lemma 4.3.

It remains to show that if f_{1} is regular then F is regular. Theorem 2.6 already shows that R = Δ^{k}F is a covariant for some k ≥ 0. We take the least such k. Let ϕ = u_{1}(0, 1, 1, 1, 1). Then C_{ϕ} is the rational nodal quintic parametrised by

(x_{0} : …: x_{4}) = (s^{5} − t^{5} : st^{4} : s^{2}t^{3} : −s^{3}t^{2} : −s^{4}t).

If k ≥ 1 then by regularity of f_{1} we have R(ϕ) = 0. Then Lemma 1.4 shows that R is divisible by Δ contradicting our choice of k. Therefore k = 0 and F is a covariant. By the convention introduced following Theorem 2.5, we say that f is a covariant. □

What makes Theorem 4.4 useful is that we can compute f_{1} from f without going via F. Explicitly we put

(11)  f_{1}(λ_{0}, …, λ_{4}) = ρ_{Y} (g_{V} , g_{W})f(a, 1)

where g_{V} and g_{W} are given by (10) and satisfy u_{1}(λ_{0}, …, λ_{4}) = (g_{V} , g_{W})u(a, 1). We then eliminate α_{0}, …, α_{4} and a from the right hand side, using the relations

(12)  α_{i}^{2}/(α_{i+1}α_{i+4}) = λ_{i}/a
(13)  α_{i}^{5} = λ_{i}^{2}/(λ_{i+2}λ_{i+3})
(14)  a^{5} = λ_{0}λ_{1} … λ_{4}.

The first of these comes from Lemma 4.3. The other two may be deduced from the first using ∏ α_{i} = 1. One systematic way to proceed is by using (12) to eliminate α_{0}, α_{1}, α_{2}, then (13) to eliminate α_{3}, α_{4} and finally (14) to eliminate a.

Remark 4.5. It can be shown that Theorem 4.4(i) still holds if we weaken the condition that f is SL_{2}(ℤ/5ℤ)-equivariant and just require that it is equivariant for the action of T = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ).

5. Examples

We can use Theorem 4.4 in the case Y is the trivial representation to give another proof (independent of Theorem 1.3) that the discrete invariants c_{4} and c_{6} are in fact invariants. Indeed let f be an integer weight discrete invariant. The integer weight condition is that f is homogeneous of degree a multiple of 5. We construct f_{1} from f by making the substitutions a^{5} ↦ ∏ λ_{i} and b ↦ 1. Since no denominators are introduced it follows by Theorem 4.4 that f is an invariant.

In the cases Y = ∧^{2}V ⊗W and ∧^{2}V^{∗}⊗W^{∗} the following proposition was already proved in [9] using evectants. We now have a general method. In the calculations that follow all sums and products are taken over the cyclic permutations of the subscripts mod 5. Recall that we fixed bases v_{0}, …, v_{4} and w_{0}, …, w_{4} for V and W. The dual bases for V^{∗} and W^{∗} are v_{0}^{∗}, …, v_{4}^{∗} and w_{0}^{∗}, …, w_{4}^{∗}.

Proposition 5.1. Let Y be any one of

\begin{matrix}∧^{2}V ⊗ W, & V^{∗} ⊗ ∧^{2}W, & V ⊗ ∧^{2}W^{∗}, & ∧^{2}V^{∗} ⊗ W^{∗}, \\ V^{∗} ⊗ S^{2}W, & S^{2}V^{∗} ⊗ W^{∗}, & S^{2}V ⊗ W, & V ⊗ S^{2}W^{∗}. \end{matrix}

Then every integer weight discrete covariant f : ∧^{2}V ⊗ W → Y^{H_{5}} is a covariant. In particular the covariants F : ∧^{2}V ⊗ W → Y form a free K[c_{4}, c_{6}]-module of rank 2 or 3 and the generators have degrees as indicated in [10, Table 4.6].

Proof. Let f : ∧^{2}V ⊗ W → Y^{H_{5}} be an integer weight discrete covariant. In each case [10, Lemma 4.4] shows that dim Y^{H_{5}} = 2 or 3 and a basis is found by inspection. We construct f_{1} from f by making the substitutions a^{5} ↦ ∏ λ_{i}, b ↦ 1, and

a ∑(v_{1} ∧ v_{4})w_{0} ↦ ∑ λ_{0}(v_{1} ∧ v_{4})w_{0}  ∑ v_{0}^{∗}(w_{1} ∧ w_{4}) ↦ ∑ v_{0}^{∗}(w_{1} ∧ w_{4})
∑(v_{2} ∧ v_{3})w_{0} ↦ ∑(v_{2} ∧ v_{3})w_{0}  a^{2} ∑ v_{0}^{∗}(w_{2} ∧ w_{3}) ↦ ∑ λ_{2}λ_{3}v_{0}^{∗}(w_{2} ∧ w_{3})
∑ v_{0}(w_{1}^{∗} ∧ w_{4}^{∗}) ↦ ∑ v_{0}(w_{1}^{∗} ∧ w_{4}^{∗})  a^{4} ∑(v_{1}^{∗} ∧ v_{4}^{∗})w_{0}^{∗} ↦ ∑ λ_{1}λ_{2}λ_{3}λ_{4}(v_{1}^{∗} ∧ v_{4}^{∗})w_{0}^{∗}
a^{3} ∑ v_{0}(w_{2}^{∗} ∧ w_{3}^{∗}) ↦ ∑ λ_{0}λ_{1}λ_{4}v_{0}(w_{2}^{∗} ∧ w_{3}^{∗})  ∑(v_{2}^{∗} ∧ v_{3}^{∗})w_{0}^{∗} ↦ ∑(v_{2}^{∗} ∧ v_{3}^{∗})w_{0}^{∗}
a ∑ v_{0}^{∗}w_{0}^{2} ↦ ∑ λ_{0}v_{0}^{∗}w_{0}^{2}  a^{2} ∑ v_{0}^{∗2}w_{0}^{∗} ↦ ∑ λ_{2}λ_{3}v_{0}^{∗2}w_{0}^{∗}
∑ v_{0}^{∗}w_{1}w_{4} ↦ ∑ v_{0}^{∗}w_{1}w_{4}  a^{4} ∑ v_{1}^{∗}v_{4}^{∗}w_{0}^{∗} ↦ ∑ λ_{1}λ_{2}λ_{3}λ_{4}v_{1}^{∗}v_{4}^{∗}w_{0}^{∗}
a^{2} ∑ v_{0}^{∗}w_{2}w_{3} ↦ ∑ λ_{2}λ_{3}v_{0}^{∗}w_{2}w_{3}  ∑ v_{2}^{∗}v_{3}^{∗}w_{0}^{∗} ↦ ∑ v_{2}^{∗}v_{3}^{∗}w_{0}^{∗}
a^{3} ∑ v_{0}^{2}w_{0} ↦ ∑ λ_{0}λ_{1}λ_{4}v_{0}^{2}w_{0}  a^{4} ∑ v_{0}w_{0}^{∗2} ↦ ∑ λ_{1}λ_{2}λ_{3}λ_{4}v_{0}w_{0}^{∗2}
a ∑ v_{1}v_{4}w_{0} ↦ ∑ λ_{0}v_{1}v_{4}w_{0}  ∑ v_{0}w_{1}^{∗}w_{4}^{∗} ↦ ∑ v_{0}w_{1}^{∗}w_{4}^{∗}
∑ v_{2}v_{3}w_{0} ↦ ∑ v_{2}v_{3}w_{0}  a^{3} ∑ v_{0}w_{2}^{∗}w_{3}^{∗} ↦ ∑ λ_{0}λ_{1}λ_{4}v_{0}w_{2}^{∗}w_{3}^{∗}

Since these substitutions eliminate a it is clear that no denominators are introduced. It follows by Theorem 4.4 that f is a covariant. □

Proposition 5.2. Let Y be any one of S^{5}W, S^{5}V, S^{5}V^{∗}, S^{5}W^{∗}. Then the covariants F : ∧^{2}V ⊗ W → Y form a free K[c_{4}, c_{6}]-module of rank 6 with generators in degrees 10, 20, 30, 30, 40, 50 except in the case Y = S^{5}W^{∗} where the generators have degrees 30, 40, 50, 50, 60, 70.

Proof. The module of integer weight discrete covariants is computed as described in [10] and is found to be a free K[c_{4}, c_{6}]-module of rank 6 with generators in degrees 10, 10, 20, 20, 30, 30. We use Theorem 4.4 to decide which of these are covariants. We construct f_{1} from f by making the substitutions a^{5} ↦ ∏ λ_{i}, b ↦ 1 and

a^{5} ∑ w_{0}^{5} ↦ ∑ λ_{0}^{3}λ_{1}λ_{4}w_{0}^{5}  a^{5} ∑ v_{0}^{5} ↦ ∑ λ_{0}λ_{1}^{2}λ_{4}^{2}v_{0}^{5}
a^{4} ∑ w_{0}^{3}w_{1}w_{4} ↦ ∑ λ_{0}^{2}λ_{1}λ_{4}w_{0}^{3}w_{1}w_{4}  a^{3} ∑ v_{0}^{3}v_{1}v_{4} ↦ ∑ λ_{0}λ_{1}λ_{4}v_{0}^{3}v_{1}v_{4}
a^{3} ∑ w_{0}w_{1}^{2}w_{4}^{2} ↦ ∑ λ_{0}λ_{1}λ_{4}w_{0}w_{1}^{2}w_{4}^{2}  a ∑ v_{0}v_{1}^{2}v_{4}^{2} ↦ ∑ λ_{0}v_{0}v_{1}^{2}v_{4}^{2}
a^{2} ∑ w_{0}w_{2}^{2}w_{3}^{2} ↦ ∑ λ_{2}λ_{3}w_{0}w_{2}^{2}w_{3}^{2}  a^{4} ∑ v_{0}v_{2}^{2}v_{3}^{2} ↦ ∑ λ_{1}λ_{2}λ_{3}λ_{4}v_{0}v_{2}^{2}v_{3}^{2}
a ∑ w_{0}^{3}w_{2}w_{3} ↦ ∑ λ_{0}w_{0}^{3}w_{2}w_{3}  a^{2} ∑ v_{0}^{3}v_{2}v_{3} ↦ ∑ λ_{1}λ_{4}v_{0}^{3}v_{2}v_{3}
∏ w_{0} ↦ ∏ w_{0}  ∏ v_{0} ↦ ∏ v_{0}
a^{5} ∑ v_{0}^{∗5} ↦ ∑ λ_{0}λ_{2}^{2}λ_{3}^{2}v_{0}^{∗5}  a^{10} ∑ w_{0}^{∗5} ↦ ∑ λ_{1}^{2}λ_{2}^{3}λ_{3}^{3}λ_{4}^{2}w_{0}^{∗5}
a^{2} ∑ v_{0}^{∗3}v_{1}^{∗}v_{4}^{∗} ↦ ∑ λ_{2}λ_{3}v_{0}^{∗3}v_{1}^{∗}v_{4}^{∗}  a^{6} ∑ w_{0}^{∗3}w_{1}^{∗}w_{4}^{∗} ↦ ∑ λ_{1}λ_{2}^{2}λ_{3}^{2}λ_{4}w_{0}^{∗3}w_{1}^{∗}w_{4}^{∗}
a^{4} ∑ v_{0}^{∗}v_{1}^{∗2}v_{4}^{∗2} ↦ ∑ λ_{1}λ_{2}λ_{3}λ_{4}v_{0}^{∗}v_{1}^{∗2}v_{4}^{∗2} a^{2} ∑ w_{0}^{∗}w_{1}^{∗2}w_{4}^{∗2} ↦ ∑ λ_{2}λ_{3}w_{0}^{∗}w_{1}^{∗2}w_{4}^{∗2}
a ∑ v_{0}^{∗}v_{2}^{∗2}v_{3}^{∗2} ↦ ∑ λ_{0}v_{0}^{∗}v_{2}^{∗2}v_{3}^{∗2}  a^{3} ∑ w_{0}^{∗}w_{2}^{∗2}w_{3}^{∗2} ↦ ∑ λ_{0}λ_{1}λ_{4}w_{0}^{∗}w_{2}^{∗2}w_{3}^{∗2}
a^{3} ∑ v_{0}^{∗3}v_{2}^{∗}v_{3}^{∗} ↦ ∑ λ_{0}λ_{2}λ_{3}v_{0}^{∗3}v_{2}^{∗}v_{3}^{∗}  a^{4} ∑ w_{0}^{∗3}w_{2}^{∗}w_{3}^{∗} ↦ ∑ λ_{1}λ_{2}λ_{3}λ_{4}w_{0}^{∗3}w_{2}^{∗}w_{3}^{∗}
∏ v_{0}^{∗} ↦ ∏ v_{0}^{∗}  ∏ w_{0}^{∗} ↦ ∏ w_{0}^{∗}

In the cases Y = S^{5}W, S^{5}V, S^{5}V^{∗} an integer weight discrete covariant is a covariant if and only if the coefficient of ∑ w_{0}^{5}, ∑ v_{0}^{5}, ∑ v_{0}^{∗5} is divisible by a^{5}. Computing the discrete covariants we find that there is a single constraint in degree 10m for each m ≥ 1. The covariants therefore have Hilbert series

\frac{2(t^{10} + t^{20} + t^{30})}{(1 − t^{20})(1 − t^{30})} − \frac{t^{10}}{1 − t^{10}} = \frac{t^{10} + t^{20} + 2t^{30} + t^{40} + t^{50}}{(1 − t^{20})(1 − t^{30})} .

In Section 7 we give further details of the covariants in the case Y = S^{5}W.

In the case Y = S^{5}W^{∗} an integer weight discrete covariant is a covariant if and only if the coefficient of ∑ w_{0}^{∗5} is divisible by a^{10} and the coefficient of ∑ w_{0}^{∗3}w_{1}^{∗}w_{4}^{∗} is divisible by a^{5}. We find that the discrete covariants of degrees 10 and 20 are not covariants and that there are 3 constraints in degrees 10m for each m ≥ 3. The covariants therefore have Hilbert series

\frac{2(t^{10} + t^{20} + t^{30})}{(1 − t^{20})(1 − t^{30})} − 2t^{10} − 2t^{20} − \frac{3t^{30}}{1 − t^{10}} = \frac{t^{30} + t^{40} + 2t^{50} + t^{60} + t^{70}}{(1 − t^{20})(1 − t^{30})} .

□

Example 5.3. The degree 10 covariant for Y = S^{5}W is and the degree 30 covariant for Y = S^{5}W^{∗} is

S_{10} = a^{5}b^{5} ∑ w_{0}^{5} − a^{4}b(a^{5} − 3b^{5}) ∑ w_{0}^{3}w_{1}w_{4} + a^{3}b^{2}(a^{5} + 2b^{5}) ∑ w_{0}w_{1}^{2}w_{4}^{2}

+a^{2}b^{3}(2a^{5} − b^{5}) ∑ w_{0}w_{2}^{2}w_{3}^{2} − ab^{4}(3a^{5} + b^{5}) ∑ w_{0}^{3}w_{2}w_{3} + (a^{10} − 16a^{5}b^{5} − b^{10}) ∏ w_{0}

T_{30} = 125a^{10}b^{10}(3a^{10} − 8a^{5}b^{5} − 3b^{10}) ∑ w_{0}^{∗5}
−5a^{6}b^{4}(3a^{20} + 134a^{15}b^{5} + 57a^{10}b^{10} + 216a^{5}b^{15} − 22b^{20}) ∑ w_{0}^{∗3}w_{1}^{∗}w_{4}^{∗}
+a^{2}b^{3}(32a^{25} − 195a^{20}b^{5} + 4110a^{15}b^{10} + 900a^{10}b^{15} + 480a^{5}b^{20} + 9b^{25}) ∑ w_{0}^{∗}w_{1}^{∗2}w_{4}^{∗2}
−a^{3}b^{2}(9a^{25} − 480a^{20}b^{5} + 900a^{15}b^{10} − 4110a^{10}b^{15} − 195a^{5}b^{20} − 32b^{25}) ∑ w_{0}^{∗}w_{2}^{∗2}w_{3}^{∗2}
−5a^{4}b^{6}(22a^{20} + 216a^{15}b^{5} − 57a^{10}b^{10} + 134a^{5}b^{15} − 3b^{20}) ∑ w_{0}^{∗3}w_{2}^{∗}w_{3}^{∗}
+(a^{30} − 258a^{25}b^{5} + 3435a^{20}b^{10} − 23040a^{15}b^{15} − 3435a^{10}b^{20} − 258a^{5}b^{25} − b^{30}) ∏ w_{0}^{∗}.

The covariant S_{10} is (a scalar multiple of) the determinant of the Jacobian matrix of the quadrics defining C_{ϕ}. We do not know of any similar construction for T_{30}. The contraction of these two covariants is ⟨S_{10}, T_{30}⟩ = c_{4}^{2}. In [8, Section 8] we used the existence of a such a covariant T_{30} to justify our algorithm for computing the invariants in the case of a singular genus one model.

In [10, Section 7] we showed that the covariant Ω_{5} of degree 5 in the following proposition represents the invariant differential.

Proposition 5.4. The covariants for Y = ∧^{2}W^{∗} ⊗ S^{2}W form a free K[c_{4}, c_{6}]- module of rank 6 with generators in degrees 5, 15, 15, 25, 25, 35.

Proof. The module of integer weight discrete covariants is computed as described in [10] and is found to be a free K[c_{4}, c_{6}]-module of rank 6 with generators in degrees 5, 15, 15, 25, 25, 35. We use Theorem 4.4 to decide which of these are covariants. We construct f_{1} from f by making the substitutions a^{5} ↦ ∏ λ_{i}, b ↦ 1 and

a ∑(w_{1}^{∗} ∧ w_{4}^{∗})w_{0}^{2} ↦ ∑ λ_{0}(w_{1}^{∗} ∧ w_{4}^{∗})w_{0}^{2}
∑(w_{1}^{∗} ∧ w_{4}^{∗})w_{1}w_{4} ↦ ∑(w_{1}^{∗} ∧ w_{4}^{∗})w_{1}w_{4}
a^{2} ∑(w_{1}^{∗} ∧ w_{4}^{∗})w_{2}w_{3} ↦ ∑ λ_{2}λ_{3}(w_{1}^{∗} ∧ w_{4}^{∗})w_{2}w_{3}
a^{4} ∑(w_{2}^{∗} ∧ w_{3}^{∗})w_{0}^{2} ↦ ∑ λ_{0}^{2}λ_{1}λ_{4}(w_{2}^{∗} ∧ w_{3}^{∗})w_{0}^{2}
a^{3} ∑(w_{2}^{∗} ∧ w_{3}^{∗})w_{1}w_{4} ↦ ∑ λ_{0}λ_{1}λ_{4}(w_{2}^{∗} ∧ w_{3}^{∗})w_{1}w_{4}
∑(w_{2}^{∗} ∧ w_{3}^{∗})w_{2}w_{3} ↦ ∑(w_{2}^{∗} ∧ w_{3}^{∗})w_{2}w_{3}.

In this case every integer weight discrete covariant is a covariant. □

6. Independence of covariants

Let Y be a homogeneous rational representation of GL(V) × GL(W).

Theorem 6.1. Assume char K = 0.

(i) The module of covariants ∧^{2}V ⊗W → Y is a free K[c_{4}, c_{6}]-module of rank

m = dim Y^{H_{5}}.

(ii) Let F_{1}, …, F_{m} be homogeneous covariants for Y. Then F_{1}, …, F_{m} are a free basis for the module in (i) if and only if for each ϕ with C_{ϕ} either an elliptic normal quintic or rational nodal quintic, F_{1}(ϕ), …, F_{m}(ϕ) are linearly independent over K.

Proof. (i) The fact we obtain a free module is a standard result in invariant theory. We have assumed char K = 0 so that SL(V) × SL(W) is linearly reductive, i.e. it has a Reynolds operator. Applying the Reynolds operator to the free K[∧^{2}V ⊗W]- module of polynomial maps ∧^{2}V ⊗ W → Y shows that the covariants form a projective K[c_{4}, c_{6}]-module and hence a free K[c_{4}, c_{6}]-module. By Theorem 2.6 the rank is the same as for the integer weight discrete covariants. We proved in [10, Lemma 4.5] that this rank is m.

(ii) Let F_{1}, …, F_{m} be homogeneous covariants that are a basis for the module in (i) and let ϕ be a genus one model with C_{ϕ} either an elliptic normal quintic or rational nodal quintic. Suppose for a contradiction that there is a dependence relation

λ_{1}F_{1}(ϕ) + … + λ_{m}F_{m}(ϕ) = 0

for some λ_{1}, …, λ_{m} ∈ K not all zero. Let

(15)  d = \{ \begin{matrix}6 & if c_{4}(ϕ) = 0 \\ 4 & if c_{6}(ϕ) = 0 \\ 2 & otherwise. \end{matrix}

Since ϕ is equivalent to a Weierstrass model we see by [8, Proposition 4.7] that for every ζ ∈ μ_{d} there exists g = (g_{V} , g_{W}) ∈ SL(V) × GL(W) with gϕ = ϕ and det g_{W} = ζ. Let F_{i} have weights (p_{i}, q_{i}). Applying g to the above dependence relation we obtain

ζ^{q_{1}}λ_{1}F_{1}(ϕ) + … + ζ^{q_{m}}λ_{m}F_{m}(ϕ) = 0.

We may therefore reduce to the case where all the q_{i} are congruent mod d. This implies by (3) that the degrees of the F_{i} are congruent mod 5d. We recall that c_{4} and c_{6} have degrees 20 and 30. It follows by (15) that there is a homogeneous covariant

F = I_{1}F_{1} + … + I_{m}F_{m}

with F (ϕ) = 0 where each I_{i} is a monomial in c_{4} and c_{6} and I_{i}(ϕ) ≠ 0 for some i. Then F is divisible by the invariant I constructed in Lemma 1.4. Since we are assuming F_{1}, …, F_{m} are a basis for the module of covariants it follows that I divides I_{i} and so I_{i}(ϕ) = 0 for all i. This is the required contradiction.

Conversely suppose F_{1}, …, F_{m} are covariants whose specialisations at ϕ are linearly independent over K whenever C_{ϕ} is an elliptic normal quintic or rational nodal quintic. If there is a relation

I_{1}F_{1} + … + I_{m}F_{m} = 0

for some invariants I_{1}, …, I_{m} then these invariants vanish on all non-singular models and so are identically zero by Theorem 1.3. Thus F_{1}, …, F_{m} generate a free submodule of rank m. By (i) it remains to show that if

I_{1}F_{1} + … + I_{m}F_{m} = IF

for some covariant F and invariants I, I_{1}, …, I_{m} then I divides I_{i} for all i. We prove this by specialising to the genus one model in Lemma 1.5. □

Remark 6.2. (i) The result that F_{1}(ϕ), …, F_{m}(ϕ) are linearly independent for ϕ non-singular could equally be proved using discrete covariants. However this proof does not generalise to the case C_{ϕ} is a rational nodal quintic. (ii) We can remove the hypothesis char K = 0 from Theorem 6.1 (but still of course requiring char K ≠ 2, 3, 5) by applying the Reynolds operator for SL_{2}(ℤ/5ℤ) to the free K[a^{5}, b^{5}]-module of ⟨T ⟩-equivariant maps 𝔸^{2} → Y^{H_{5}} that pass the test of Theorem 4.4. See also Remark 4.5.

7. Quintic covariants

We give further details of the covariants in the case Y = S^{5}W. We already noted in the proof of Proposition 5.2 that the integer weight discrete covariants form a free K[c_{4}, c_{6}]-module of rank 6 generated in degrees 10, 10, 20, 20, 30, 30. A basis for Y^{H_{5}} is

F_{1} = ∑ w_{0}^{5} − 30 ∏ w_{0},  F_{2} = 10 ∑ w_{0}^{3}w_{1}w_{4},  F_{3} = 10 ∑ w_{0}^{3}w_{2}w_{3},
G_{1} = ∑ w_{0}^{5} + 20 ∏ w_{0},  G_{2} = 10 ∑ w_{0}w_{1}^{2}w_{4}^{2},  G_{3} = 10 ∑ w_{0}w_{2}^{2}w_{3}^{2}.

In terms of this basis we have generators

F_{10} = (a^{10} − 36a^{5}b^{5} − b^{10})F_{1} + 5a^{4}b(a^{5} − 3b^{5})F_{2} + 5ab^{4}(3a^{5} + b^{5})F_{3},
F_{20} = (a^{20} + 114a^{15}b^{5} + 114a^{5}b^{15} − b^{20})F_{1}
−a^{4}b(a^{15} + 171a^{10}b^{5} + 247a^{5}b^{10} − 57b^{15})F_{2}
−ab^{4}(57a^{15} + 247a^{10}b^{5} − 171a^{5}b^{10} + b^{15})F_{3},
F_{30} = D(10a^{4}b^{4}(9a^{10} + 26a^{5}b^{5} − 9b^{10})F_{1}
+a^{3}(a^{15} + 126a^{10}b^{5} + 117a^{5}b^{10} − 12b^{15})F_{2}
−b^{3}(12a^{15} + 117a^{10}b^{5} − 126a^{5}b^{10} + b^{15})F_{3}),
G_{10} = (a^{10} + 14a^{5}b^{5} − b^{10})G_{1} + 5a^{3}b^{2}(a^{5} + 2b^{5})G_{2} + 5a^{2}b^{3}(2a^{5} − b^{5})G_{3},
G_{20} = (a^{20} − 136a^{15}b^{5} − 136a^{5}b^{15} − b^{20})G_{1}
−a^{3}b^{2}(7a^{15} + 272a^{10}b^{5} − 221a^{5}b^{10} + 26b^{15})G_{2}
−a^{2}b^{3}(26a^{15} + 221a^{10}b^{5} + 272a^{5}b^{10} − 7b^{15})G_{3},
G_{30} = 2D^{2}(10a^{3}b^{3}G_{1} + a(a^{5} − 3b^{5})G_{2} − b(3a^{5} + b^{5})G_{3}).

We recall from Section 5 that a discrete covariant is a covariant if and only if the coefficient of ∑ w_{0}^{5} is divisible by a^{5}. Therefore the K[c_{4}, c_{6}]-module of covariants for Y = S^{5}W has basis

(16)  \begin{aligned}S_{10} & = F_{10} − G_{10}, & S_{30} & = F_{30} − G_{30}, & S_{40} & = c_{6}F_{10} + c_{4}F_{20}, \\ S_{20} & = F_{20} − G_{20}, & S′_{30} & = F_{30} + G_{30}, & S_{50} & = c_{4}^{2}F_{10} + c_{6}F_{20}. \end{aligned}

If we evaluate these covariants at a non-singular model ϕ then by Theorem 6.1 we obtain a basis for the space of Heisenberg invariant quintics. The space of Heisenberg invariant quintics relative to a fixed elliptic normal quintic was studied by Hulek [11]. We show that our basis obtained by specialising the covariants picks out some of the quintic hypersurfaces to which Hulek was able to attach a geometric meaning.

Lemma 7.1. Let ϕ ∈ ∧^{2}V ⊗ W be non-singular and write S_{10}, …, S_{50} for the quintic forms obtained by evaluating the covariants (16) at ϕ.

(i) S_{10} is (a scalar multiple of) the determinant of the Jacobian matrix of the quadrics defining C_{ϕ}.

(ii) The Heisenberg invariant quintics vanishing on the tangent variety of C_{ϕ}

are linear combinations of S_{10}, S_{20}, S′_{30}. (iii) The quintics S_{10}, S_{20}, S_{30} are singular along C_{ϕ}. (iv) The quintics S_{10}, S_{20}, S_{30}, S′_{30}, S_{40} vanish on C_{ϕ}.

Proof. For the proof we may take ϕ = u(a, b) a Hesse model. Let p_{0}, …, p_{4} be the equations for C_{ϕ}, i.e. p_{i} = abw_{i}^{2} + b^{2}w_{i+1}w_{i+4} − a^{2}w_{i+2}w_{i+3}.

(i) We compute S_{10} = 25 det(∂p_{i}/∂w_{j}). (ii) The tangent line to C_{ϕ} at P = (0 : a : b : −b : −a) also passes through

Q = (5a^{3}b^{3} : 0 : −b(2a^{5} − b^{5}) : −b(a^{5} + 2b^{5}) : a(a^{5} − 3b^{5})).

Evaluating the quintic forms at λP + Q we find that S_{10}, S_{20}, S′_{30} vanish on the tangent line whereas S_{30}, S_{40}, S_{50} give polynomials in λ of degrees 0, 2, 4. (iii) We may write these quintics as linear combinations of ∑ p_{0}^{2}w_{0}, ∑ p_{1}p_{4}w_{0} and ∑ p_{2}p_{3}w_{0}. (iv) We may write these quintics as linear combinations of ∑ p_{0}w_{0}^{3}, ∑ p_{0}w_{0}w_{1}w_{4}, ∑ p_{0}w_{0}w_{2}w_{3}, ∑ p_{0}(w_{1}^{2}w_{3} + w_{2}w_{4}^{2}) and ∑ p_{0}(w_{1}w_{2}^{2} + w_{3}^{2}w_{4}). □

Theorem 7.2. Let C = C_{ϕ} be an elliptic normal quintic. Let Tan C and Sec C be the tangent and secant varieties of C. Let F be the locus of singular lines of the rank 3 quadrics containing C. Then

(i) Sec C is the degree 5 hypersurface defined by S_{10}.

(ii) Tan C and F are irreducible surfaces of degrees 10 and 15 and their union

is the complete intersection defined by S_{10} and S_{20}. (iii) The space of Heisenberg invariant quintics containing Tan C has basis

S_{10}, S_{20}, S′_{30}. (iv) The space of Heisenberg invariant quintics containing F, equivalently that

are singular along C, has basis S_{10}, S_{20}, S_{30}.

(v) The space of Heisenberg invariant quintics containing C has basis S_{10}, S_{20}, S_{30}, S′_{30}, S_{40}.

Proof. This follows by Lemma 7.1 and work of Hulek [11]. □

8. The covering map

We call the covariants ∧^{2}V ⊗W → S^{d}W covariants of order d. The action of the Heisenberg group shows that the order must be a multiple of 5. By Theorem 6.1 and [10, Lemma 4.4], the K[c_{4}, c_{6}]-modules of covariants of orders 5, 10, 15 have ranks 6, 41, 156. Fortunately we do not need to classify all these covariants since most of them vanish on C_{ϕ} and therefore are of no use for describing the covering map.

Lemma 8.1. Let C ⊂ ℙ^{n−1} be an elliptic normal curve. Then the space of Heisenberg invariant polynomials of degree nd, quotiented out by the subspace vanishing on C, has dimension d.

Proof. Let π : C → E be the covering map of degree n^{2} from C to its Jacobian E. Then π^{∗}(d.0_{E}) ∼ ndH where H is the hyperplane section for C. So if f_{1}, …, f_{d} is a basis for the Riemann-Roch space L(d.0_{E}) then π^{∗}f_{1}, …, π^{∗}f_{d} are basis for the space of forms of degree nd in K[x_{0}, …, x_{n−1}]/I(C) that are invariant under the action of E[n]. Applying the Reynold’s operator for the Heisenberg group shows that every such form has a representative in K[x_{0}, …, x_{n−1}] that is itself Heisenberg invariant. □

Lemma 8.2. Let C ⊂ ℙ^{n−1} be either an elliptic normal curve or a rational nodal curve, and let P ∈ C be a smooth point. Suppose Z, X, Y are homogeneous polynomials in K[x_{0}, …, x_{n−1}] of degrees n, 2n, 3n with ord_{P} (Z) = 1, ord_{P} (X) = 0, ord_{P} (Y) = 0. Then for each d ≥ 1 the forms

{X^{i}Y^{j}Z^{k} : i, k ≥ 0, j ∈ {0, 1}, 2i + 3j + k = d}

are linearly independent in the co-ordinate ring K[x_{0}, …, x_{n−1}]/I(C).

Proof. This is clear since ord_{P} (X^{i}Y^{j}Z^{k}) = k and the forms listed have distinct values of k. □

Lemma 8.3. There are covariants Z, X, Y of orders 5, 10, 15 and degrees 50, 110, 165 such that whenever C_{ϕ} is an elliptic normal quintic or rational nodal quintic there is a smooth point P ∈ C_{ϕ} such that the evaluations of Z, X, Y at ϕ satisfy ord_{P} (Z) = 1, ord_{P} (X) = 0, ord_{P} (Y) = 0.

Proof. We start with the covariants U, H : ∧^{2}V ⊗ W → ∧^{2}V ⊗ W and Q_{6} : ∧^{2}V ⊗ W → S^{2}V ⊗ W where U is the identity map and (on the Hesse family)

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

There are covariants P_{2}, P_{12}, P_{22} : ∧^{2}V ⊗ W → V^{∗} ⊗ S^{2}W where P_{2} is the Pfaffian map (2) and P_{12}, P_{22} satisfy

P_{2}(λU + μH) = λ^{2}P_{2} + 2λμP_{12} + μ^{2}P_{22}.

We define covariants M_{30} : ∧^{2}V ⊗ W → S^{5}V and N_{30} : ∧^{2}V ⊗ W → S^{5}V^{∗} where M_{30} = det Q_{6} and N_{30} is the coefficient of t in det(P_{2} + tP_{22}). We also define T_{23} and T_{28} taking values in V ⊗ S^{3}W by

(⊗^{2}V ⊗ W) × (V^{∗} ⊗ S^{2}W) → V ⊗ S^{3}W
(U, P_{22}) ↦ T_{23}
(Q_{6}, P_{22}) ↦ T_{28}.

We then put

Z = (1/2)Q_{6}(P_{22}, P_{22})
X = (3^{3}/2^{6})M_{30}(P_{12}, P_{12}, P_{12}, P_{22}, P_{22})
Y = (3^{3}/2^{8})N_{30}(T_{23}, T_{28}, T_{28}, T_{28}, T_{28}).

As required these are covariants of orders 5, 10, 15 and degrees 50, 110, 165.

Suppose C_{ϕ} is a rational nodal quintic. By Lemma 1.2 we may assume that ϕ is as given in Section 4, i.e. ϕ = u_{1}(0, 1, 1, 1, 1). Then C_{ϕ} is parametrised by

(x_{0} : …: x_{4}) = (s^{5} − t^{5} : st^{4} : s^{2}t^{3} : −s^{3}t^{2} : −s^{4}t)

Evaluating Z, X, Y at ϕ we find

(17)  \begin{aligned}Z(s^{5} − t^{5}, st^{4}, s^{2}t^{3}, −s^{3}t^{2}, −s^{4}t) & = −2^{8}3^{4}s^{10}t^{10}(s^{5} − t^{5}) \\ X(s^{5} − t^{5}, st^{4}, s^{2}t^{3}, −s^{3}t^{2}, −s^{4}t) & = 2^{16}3^{9}s^{20}t^{20}(s^{10} + 10s^{5}t^{5} + t^{10}) \\ Y (s^{5} − t^{5}, st^{4}, s^{2}t^{3}, −s^{3}t^{2}, −s^{4}t) & = 2^{26}3^{15}s^{35}t^{35}(s^{5} + t^{5}). \end{aligned}

The conclusions of the lemma are satisfied for P = (0 : 1 : 1 : −1 : −1).

Now suppose C_{ϕ} is an elliptic normal quintic. Then by [9, Proposition 4.1] we may assume that ϕ = u(a, b) is a Hesse model. There is a flex (i.e. hyperosculating point) of C_{ϕ} at P = (0 : a : b : −b : −a). Evaluating Z, X, Y at ϕ we find

(18)  \begin{aligned}Z(0, a, b, −b, −a) & = 0 \\ X(0, a, b, −b, −a) & = 2^{18}3^{10}D^{10} \\ Y (0, a, b, −b, −a) & = −2^{27}3^{15}D^{15} \end{aligned}

where D = ab(a^{10} − 11a^{5}b^{5} − b^{10}). Since Δ = D^{5} it is clear that X and Y do not vanish at P. Now C_{ϕ} ⊂ ℙ^{4} is a curve of degree 5 meeting the degree 5 hypersurface defined by Z at the 25 flexes of C_{ϕ}. So by Bezout’s theorem either ord_{P} (Z) = 1 or Z vanishes identically on C_{ϕ}. To rule out the latter we write Z in terms of the basis (16). Explicitly we find

Z = (39/10)c_{4}^{2}S_{10} + 4c_{6}S_{20} − 54c_{4}S_{30} − (198/5)c_{4}S′_{30} + 12S_{50}.

By Theorem 6.1 the specialisations of S_{10}, …, S_{50} at ϕ are linearly independent. It follows by Theorem 7.2(v) that Z does not vanish identically on C_{ϕ}. □

Lemma 8.4. Let 𝕃/𝕂 be a finite Galois extension with Galois group Γ. Let 𝕍 be a finite dimensional vector space over 𝕃. Suppose there is an action of Γ on 𝕍 satisfying γ(v + w) = γ(v) + γ(w) and γ(λv) = γ(λ)γ(v) for all γ ∈ Γ, λ ∈ 𝕃 and v, w ∈ 𝕍. Then dim_{𝕂} 𝕍^{Γ} = dim_{𝕃} 𝕍.

Proof. A generalised form of Hilbert’s Theorem 90 states that H^{1}(Γ, GL_{n}(𝕃)) = {1}. See for example [12, Chapter X, Proposition 3]. We fix a basis for 𝕍 over 𝕃, and then compare this basis with its Galois conjugates. By writing the resulting cocycle as a coboundary, we find a new basis for 𝕍 over 𝕃 consisting of vectors fixed by Γ. □

Lemma 8.5. Let M_{d} be the K[c_{4}, c_{6}]-module of covariants for Y = S^{5d}W, quotiented out by the submodule of covariants that vanish on the curve. Then M_{d} is a free K[c_{4}, c_{6}]-module of rank d generated by

{X^{i}Y^{j}Z^{k} : i, k ≥ 0, j ∈ {0, 1}, 2i + 3j + k = d}

where Z, X, Y are the covariants in Lemma 8.3.

Proof. Let Z = (S^{5d}W)^{H_{5}} and m = dim Z. We apply Lemma 8.4 with 𝕂 = K(a, b)^{Γ}, 𝕃 = K(a, b) and 𝕍 either 𝕌 = 𝕃 ⊗_{K} Z or the subspace 𝕌_{0} of forms that vanish on the curve defined by the generic Hesse model u(a, b). Since the action of Γ on 𝔸^{2} (and hence on 𝕃 = K(𝔸^{2})) was defined so that u : 𝔸^{2} → (∧^{2}V ⊗ W)^{H_{5}} is Γ-equivariant, we do indeed have that Γ acts on 𝕌_{0}. By Lemmas 8.1 and 8.4 we compute

dim_{𝕂} 𝕌^{Γ} = dim_{𝕃} 𝕌 = m,
dim_{𝕂} 𝕌_{0}^{Γ} = dim_{𝕃} 𝕌_{0} = m − d.

Thus the K[a, b]^{Γ}-module of discrete covariants 𝔸^{2} → Z has rank m, and the submodule of discrete covariants vanishing on the curve has rank m−d. It follows by Theorem 2.6, and the proof of [10, Lemma 4.5], that the K[c_{4}, c_{6}]-module of covariants ∧^{2}V ⊗ W → S^{5d}W has rank m, and the submodule of covariants vanishing on the curve has rank m − d. Therefore M_{d} has rank d.

Let F_{1}, …, F_{d} be the covariants in the statement of the lemma. Lemmas 8.2 and 8.3 show that if C_{ϕ} is an elliptic normal quintic or rational nodal quintic then F_{1}(ϕ), …, F_{d}(ϕ) are linearly independent over K. An argument similar to the proof of Theorem 6.1(ii) now shows that F_{1}, …, F_{d} are a free basis for M_{d}. □

We show that the covariants Z, X, Y give a formula for the covering map. The formula for the Jacobian was already proved in [8] by a different method.

Theorem 8.6. Let ϕ ∈ ∧^{2}V ⊗ W be non-singular. Then C_{ϕ} has Jacobian elliptic curve E with Weierstrass equation

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

and the covering map C_{ϕ} → E is given by (x, y) = (X/Z^{2}, Y/Z^{3}) where Z, X, Y are the evaluations at ϕ of the covariants in Lemma 8.3.

Proof. By Lemma 8.5 the K[c_{4}, c_{6}]-module M_{6} has basis

X^{3}, XY Z, X^{2}Z^{2}, Y Z^{3}, XZ^{4}, Z^{6}.

Since Z, X, Y have degrees 50, 110, 165 and c_{4}, c_{6} have degrees 20, 30 we must therefore have

Y^{2} = λX^{3} + μc_{4}XZ^{4} + νc_{6}Z^{6}

for some λ, μ, ν ∈ K. We determine these scalars by specialising to the case C_{ϕ} is a rational nodal quintic. Using (17) we find λ = 1, μ = −27, ν = −54. Thus (x, y) = (X/Z^{2}, Y/Z^{3}) defines a morphism π : C_{ϕ} → E where E is the curve defined by (19). The fibre above the point at infinity on E is C_{ϕ}∩{Z = 0}. By (18) and Bezout’s Theorem this consists of the 25 flexes on C_{ϕ}. Thus deg π = 25. Since Z, X, Y are covariants it is clear that π quotients out by the action of the Heisenberg group on C_{ϕ}. Hence E is the Jacobian of C_{ϕ} and π is the covering map. □

We gave algorithms for computing Q_{6} and H in [8, Section 8] and [9, Section 11]. So we can evaluate the covariants Z, X, Y by following the proof of Lemma 8.3. This gives a practical algorithm for computing the covering map. Although we have been working over an algebraically closed field it is clear that Theorem 8.6 still holds without this assumption. We give an example in the case K = ℚ.

Example 8.7. Let C ⊂ ℙ^{4} be the elliptic normal quintic defined by the 4 × 4 Pfaffians of

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

The invariants of this model are c_{4} = 21288863488 and c_{6} = 3106257241074688. Our Magma function CoveringCovariants evaluates the covariants of Lemma 8.3 to give forms Z, X, Y. The first of these is

Z = 208089517036452423241728x_{1}^{5} + 481348375428118457413632x_{1}^{4}x_{2}
− 1067331097433708461809664x_{1}^{4}x_{3} − 861565401032195664871424x_{1}^{4}x_{4}

− 2713065303844178403139584x_{1}^{4}x_{5} − 1159509369215265868720128x_{1}^{3}x_{2}^{2}

+ … + 8511800259354855263252480x_{5}^{5}.

Evaluating these forms at (4013 : −2384 : −1616 : 1388 : 1021) ∈ C(ℚ) we obtain

Z = 3412377609951638022163996178720787224832,
X = 12141242195111585999097107425889311253617470393872861501219624577\
3843080932512669892608,
Y = 13341702475842976696719854379608150742217144829049714776419935109\

10164201520123953599858396067352426339710162835468918162316066816.

The Jacobian of C is the elliptic curve E with Weierstrass equation y^{2} = x^{3} − 27c_{4}x − 54c_{6} and P = (X/Z^{2}, Y/Z^{3}) ∈ E(ℚ) is a point of canonical height 164.90718 . . .. In fact E(ℚ) has rank 1 and is generated by P.

Remark 8.8. The elliptic curve E in the above example is labelled 17472bz1 in Cremona’s tables [6]. It satisfies a 5-congruence with the elliptic curve F labelled 17472bx2. In fact F has Weierstrass equation y^{2} = x(x+16)(x−26) and the genus one model in Example 8.7 may be constructed from the point (−2, 28) ∈ F (ℚ) using visibility as described in [10].

References

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 in Math. Ann., http://link.springer.com/article/10.1007/s00208-012-0850-9

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[13] A. Weil, Remarques sur un mémoire d’Hermite, Arch. Math. (Basel) 5, (1954) 197–202.

[14] A. Weil, Euler and the Jacobians of elliptic curves, Arithmetic and geometry, Vol. I, (M. Artin and J. Tate, eds.), 353–359, Progr. Math., 35, Birkhuser Boston, Boston, MA,

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

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