Invariants for the elliptic normal quintic

Tom Fisher

June 21, 2002

Abstract

Let k be a field with char k ≠ 2, 3, 5. Let C be a smooth curve of genus one defined over k. Suppose that C admits a k-rational divisor of degree n. For n = 2, 3, 4 classical invariant theory allows us to compute the Jacobian of C without making any field extensions. We extend to the case n = 5, where although the corresponding invariants are too large to write down as polynomials, we have found a practical algorithm for computing them.

Introduction

Let C be a smooth projective curve of genus one defined over a field k. We would like to compute the Jacobian of C, giving the answer in Weierstrass form. We recall that the Jacobian is a certain twist of C that is soluble over k. One approach is to work over an extension field, chosen to make C soluble. This reduces the problem to that of putting a curve in Weierstrass form. But there are two objections to this method. Firstly extending our field may not be practical. Secondly one has to be extremely careful not to accidently write down a quadratic twist of the Jacobian.

Another approach, described by Weil [26], makes use of classical invariant theory. This avoids the need to make any field extensions, but is limited to curves of small degree. To make this last statement precise we take D a k-rational divisor on C of degree n. It is well known that

• If n = 1 then C has a k-rational point. In particular C is its own Jacobian.

• If n ≥ 2 then the linear system associated to D defines a morphism C → P^{n−1}. We identify P^{n−1} = P(W^{∗}) where W = H^{0}(C, O_{C}(D)).

• If n ≥ 3 then this morphism is an embedding. We refer to the geometric configuration C ⊂ P^{n−1} as an elliptic normal curve.

• If n ≥ 4 then C ⊂ P^{n−1} is defined by quadrics. The space of quadrics V = H^{0}(P^{n−1}, I_{C}(2)) has dimension n(n − 3)/2 and generates the homogeneous ideal of C.

From now on we assume that char k ∤ 6n. Classical invariant theory gives formulae for the Jacobian in the cases n = 2, 3, 4. A unified account is given in [2]. The vector spaces V and W introduced above provide a convenient notation for describing the group actions involved. We take projective equations for C

n = 2 y^{2}x_{1}^{2} = U(x_{0}, x_{1})  (U a quartic)
n = 3 U(x_{0}, x_{1}, x_{2}) = 0  (U a cubic)
n = 4 U(x_{0}, x_{1}, x_{2}, x_{3}) = U′(x_{0}, x_{1}, x_{2}, x_{3}) = 0 (U, U′ quadrics)

(1)

and view these equations as elements of S^{4}W, S^{3}W respectively V^{∗} ⊗ S^{2}W. Then SL(W), respectively SL(V) × SL(W), acts on each affine space, and so on the polynomial rings k[S^{4}W], k[S^{3}W] and k[V^{∗} ⊗ S^{2}W]. In each case the invariant subring is generated by polynomials c_{4} and c_{6}. Labelling these invariants correctly, it turns out that C has Jacobian y^{2} = x^{3} − 27c_{4}x − 54c_{6}. Moreover the equations (1) describe a smooth curve of genus one precisely whenever the discriminant Δ = (c_{4}^{3} − c_{6}^{2})/1728 is non-zero.

For n = 2 the invariants may be found in [2], [7], [26] or virtually any text on classical invariant theory. For n = 3 the invariants are rather more cumbersome, but may still be found written out in [2], [20] or [23]. For n = 4 we view the quadrics U, U′ as 4 × 4 symmetric matrices and reduce to the case n = 2 by considering the quartic det(U + tU′). We refer to [2], [18] for details.

In this article we extend to the case n = 5. Let C ⊂ P^{4} be an elliptic normal quintic. In other words C is a smooth curve of genus one and degree 5 spanning P^{4}. It is known that C is defined by 5 quadrics, but that 5 quadrics do not in general define an elliptic normal quintic. However it follows from work of Buchsbaum and Eisenbud [5] that the quadrics defining an elliptic normal quintic may be written as the 4×4 Pfaffians of a 5×5 skew-symmetric matrix with entries linear forms on P^{4}. A geometric argument recalled in [11, §2] shows that generically this construction gives rise to an elliptic normal quintic. We view these matrices as elements of ∧^{2}V ⊗W and write C(ϕ) ⊂ P^{4} for the subscheme defined by the 4 × 4 Pfaffians of ϕ ∈ ∧^{2}V ⊗ W.

Theorem 1 Let V and W be 5-dimensional vector spaces over k. Then the invariant subring for the action of SL(V)×SL(W) on k[∧^{2}V ⊗W] is generated by homogeneous polynomials c_{4} and c_{6} of degrees 20 and 30 respectively. If we set Δ = (c_{4}^{3} − c_{6}^{2})/1728 then

(i) C(ϕ) ⊂ P^{4} is an elliptic normal quintic if and only if Δ(ϕ) ≠ 0. (ii) If (i) holds then C(ϕ) has Jacobian y^{2} = x^{3} − 27c_{4}(ϕ)x − 54c_{6}(ϕ).

Given what is classically known in the cases n = 2, 3, 4 the statement of Theorem 1 is not a surprise. However it should be stressed that prior to the work of Buchsbaum and Eisenbud it was not possible even to pose the problem correctly.

We define the size |p| of a polynomial p to be the number of monomials appearing in p. The size of the covariants c_{4} and c_{6} (this is independent of the bases chosen for V and W) grows rapidly with n.

n  Group action  |c_{4}| |c_{6}|
2  SL(W) acts on S^{4}W  3  5
3  SL(W) acts on S^{3}W  25 103
4  SL(V) × SL(W) acts on V^{∗} ⊗ S^{2}W  4003 81547
5  SL(V) × SL(W) acts on ∧^{2} V ⊗ W  ?  ?

As the reader might well surmise from this table, the invariants in the case n = 5 are too large to write down as explicit polynomials. See Appendix B for a more careful estimate. Instead we give a practical algorithm for computing the Jacobian. It has 2 steps

• Given 5 quadrics defining an elliptic normal quintic C ⊂ P^{4} we compute ϕ ∈ ∧^{2}V ⊗ W such that C = C(ϕ).

• Given ϕ ∈ ∧^{2}V ⊗ W we compute c_{4}(ϕ) and c_{6}(ϕ).

The first step follows from the work of Buchsbaum and Eisenbud. Indeed the matrix ϕ may be extracted from the syzygies between the original quadrics, and this reduces the problem to linear algebra. The second step is more involved and should be regarded as our main result.

We were inspired to find such an algorithm by the following observation on plane cubics. We take C ⊂ P^{2} a plane cubic with equation U(x_{0}, x_{1}, x_{2}) = 0. The Hessian is H = det(∂^{2}U/∂x_{i}∂x_{j}). Salmon [20, Chapter V. Art. 241] observes that if we write the quadrics ∂U/∂x_{i} and ∂H/∂x_{i} in terms of the basis x_{0}^{2}, x_{1}^{2}, x_{2}^{2}, x_{0}x_{1}, x_{1}x_{2}, x_{0}x_{2} then we obtain a 6 × 6 matrix whose determinant is a non-zero multiple of the discriminant. He gives equally simple recipes for c_{4} and c_{6}.

We take from this two lessons. Firstly that there are practical algorithms for computing the invariants, far more elegant than the invariants themselves written out as explicit polynomials. Secondly that we should make a definition of covariant sufficiently general to include both the original curve U and the invariants we seek. Each step of our algorithm will then construct new covariants from the ones we already know.

To check that the above algorithm for Δ is correct, it suffices to work over an algebraically closed field. Then making an appropriate choice of co-ordinates our plane cubic takes the form

U(x_{0}, x_{1}, x_{2}) = a(x_{0}^{3} + x_{1}^{3} + x_{2}^{3}) − 3bx_{0}x_{1}x_{2}.  (2)

This curve is non-singular provided D = a(a^{3} − b^{3}) is non-zero. The discriminant is Δ = D^{3}. The Hessian works out as

\frac{1}{18}H(x_{0}, x_{1}, x_{2}) = \frac{∂D}{∂b}(x_{0}^{3} + x_{1}^{3} + x_{2}^{3}) + 3\frac{∂D}{∂a}x_{0}x_{1}x_{2}.  (3)

So the 6 × 6 determinant may be put in block diagonal form, with each 2 × 2 block proportional to

det (\begin{matrix}a & b \\ −∂D/∂b & ∂D/∂a \end{matrix}) = 4D.

It follows that the 6 × 6 determinant is a non-zero multiple of Δ.

We should regard (a : b) as co-ordinates on X(3) ≃ P^{1} and (2) as the universal family X (3) → X(3). So we could at this point present our algorithm for computing the invariants in the case n = 5, observe that it works for the family X (5) → X(5) and thus reduce the length of this article considerably. However we have decided also to explain the methods we used to find our algorithm.

As the above example shows, there are huge computational savings to be made if we are willing to restrict our covariants to X(n). So it is natural to ask how one might recognise a covariant from its restriction to X(n). But to identify C ⊂ P^{n−1} as a fibre of X (n) → X(n) we must pick a basis P, Q ∈ Jac(C)[n] such that the Weil pairing e_{n}(P, Q) is a fixed primitive nth root of unity. Two such choices differ by an element of SL_{2}(Z/nZ). So the restriction of a covariant to X(n) satisfies an invariance property under SL_{2}(Z/nZ). We call the forms on X(n) satisfying this invariance property the discrete covariants. Since SL_{2}(Z/nZ) is a finite group the discrete covariants may essentially be read off from the character table. Carrying this programme through in the case n = 3 we would for instance learn that (3) is an interesting expression to consider, even if we did not know that it could be computed as the determinant of the matrix of second partial derivatives.

In §1 we recall from [11] some preliminaries on elliptic normal quintics and make our definition of covariant. From §2 until the end of §11 we work over an algebraically closed field. We begin by classifying the orbits of codimension 1 for the action of GL(V) × GL(W) on ∧^{2}V ⊗ W. This shows that the invariants c_{4}, c_{6} and Δ exist, but neither gives a way of computing them, nor explains the link with the Jacobian.

In §§3,4,5 we develop the theory of discrete covariants. Theorem 2 and its corollary give practical necessary and sufficient conditions for a discrete covariant to come from a covariant. This leads to an improved description of the ring of invariants, and in particular to a proof of Theorem 1 in the case k is algebraically closed. In §7 we give a complete classification of the discrete covariants. Then in §8 we use this classification to write down some non-trivial covariants. It was this prior knowledge of the covariants, albeit restricted to the family X (5) → X(5), that enabled us to find our algorithm for computing the invariants. We present the algorithm itself in §9. Its verification does not depend on Theorem 2. In §§10,11 we construct further covariants of geometric interest. These correspond to the invariant differential and quintic hypersurfaces respectively.

In §12 we drop our assumption that k is algebraically closed and make the necessary modifications to the proof of Theorem 1. Then in §13 we implement our algorithm for computing the Jacobian and give some examples in the case k = Q. In §14 we add to our examples by comparing our algorithm with that found by C. O’Neil [19] in the special case where the Jacobian is assumed to have a k-rational point of order 5. There are two appendices. Appendix A gives the character table for Γ = SL_{2}(Z/5Z). Appendix B gives an estimate for the size of the invariant c_{4}.

Some notation

Let V be a vector space over a field k. Let V^{∗} = Hom(V, k). We write k[V] = k ⊕ V^{∗} ⊕ S^{2}V^{∗} ⊕ … for the ring of polynomial functions on V, and k(V) for its field of fractions. The Grassmannian Gr(r, V) is the set of r-dimensional subspaces of V. We write P(V) = Gr(1, V). The pairing between a vector space and its dual, or any pairing derived from it, is denoted ⟨ , ⟩. If π : G → GL(V) is a linear representation then we sometimes write π = π_{V} or V = V_{π}. We write x ∝ y to mean x = cy for some c ∈ k^{∗}.

While working with curves of degree n we assume char k ∤ 6n. For most of this article n = 5 and k is algebraically closed. We write μ_{n} for the nth roots of unity and ζ_{n} for a generator. The vector spaces V and W introduced in §1 have bases v_{0}, …, v_{4} and x_{0}, …, x_{4}. The dual bases for V^{∗} and W^{∗} are v_{0}^{∗}, …, v_{4}^{∗} and x_{0}^{∗}, …, x_{4}^{∗}. We identify P^{4} = P(W^{∗}) so that x_{0}, …, x_{4} are co-ordinates on P^{4}.

1 The space of elliptic normal quintics

Let k be a field with char k ≠ 2, 3, 5. Let V and W be 5-dimensional vector spaces over k. We agree to view an element ϕ ∈ ∧^{2}V ⊗ W as a 5 × 5 as skew-symmetric matrix whose entries are linear forms on P^{4} := P(W^{∗}). We write C(ϕ) ⊂ P^{4} for the subscheme defined by the 4 × 4 Pfaffians of ϕ. Our starting point is

Proposition 1.1 (i) Let C ⊂ P^{4} be an elliptic normal quintic defined over k. Then C = C(ϕ) for some ϕ ∈ ∧^{2}V ⊗ W. (ii) The elliptic normal quintics are Zariski dense in ∧^{2}V ⊗ W.

(i) We gave two different proofs in [11]^{1}. The first interprets V^{∗} as the global sections of the unique indecomposable rank 2 vector bundle over C with determinant the hyperplane section. The second, which depends on a description of Gorenstein ideals in codimension 3 due to Buchsbaum and Eisenbud, interprets V as the space of quadrics containing C. (ii) This follows from properties of the Grassmannian Gr(2, V). For details see [11, §2]. □

We call the 50-dimensional affine space ∧^{2}V ⊗ W the space of elliptic normal quintics. Theorem 1 describes the invariant subring for the action of SL(V) × SL(W) on k[∧^{2}V ⊗ W]. More generally for ρ : GL(V) × GL(W) → GL(Y) a linear representation we define

1Our notation differs in that we have replaced V by V^{∗}.

Definition 1.2 A polynomial function F : ∧^{2}V ⊗ W → Y is a covariant of weight (p, q) ∈ Z^{2} if

F ∘ (g_{V} , g_{W}) = (det g_{V})^{p}(det g_{W})^{q}ρ(g_{V} , g_{W}) ∘ F  (4)

for all (g_{V} , g_{W}) ∈ GL(V) × GL(W). If Y = k is the trivial representation then F is called an invariant.

Remark 1.3 The only algebraic characters of GL(V) × GL(W) are of the form (g_{V} , g_{W}) ↦ (det g_{V})^{p}(det g_{W})^{q}. So the invariants of Definition 1.2 are precisely the homogeneous polynomials in k[∧^{2}V ⊗ W] invariant under the action of SL(V) × SL(W).

The main reason for considering covariants, and not just invariants, is that certain covariants are easy to write down. By manipulating these we construct further covariants, and if all goes well, eventually arrive at the invariants we seek. We shall take Y to be a homogeneous subspace of the tensor algebra generated by V, W, V^{∗} and W^{∗}. By homogeneous we mean that there are integers r and s such that ρ(λI_{V} , μI_{W}) = λ^{r}μ^{s}I_{Y} for all λ, μ ∈ k^{∗}. It follows from (4) that

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

The most obvious example of a covariant is the identity map U : ∧^{2}V ⊗ W → ∧^{2}V ⊗ W. It has degree 1 and weight (0,0). Identifying ∧^{4}V = V^{∗} there is a natural squaring map P : ∧^{2}V ⊗ W → V^{∗} ⊗ S^{2}W. We call this the Pfaffian map, since the quadrics ⟨P (ϕ), v⟩ for v ∈ V are the 4 × 4 Pfaffians of ϕ. By definition these are also the quadrics defining C(ϕ). It is easy to see that P is a covariant of degree 2 and weight (1, 0).

2 Orbits of codimension 1

Let n ≥ 3 be an integer. Let k be an algebraically closed field with char k ∤ 6n. We make some observations about elliptic normal curves of degree n and certain of their degenerations. We then return to the case n = 5 and, as a first step towards understanding the invariants, describe the orbits of codimension 1 for the action of GL(V) × GL(W) on ∧^{2}V ⊗ W. Our assumptions on k will remain in force until the end of §11.

Definition 2.1 (i) An elliptic normal curve C ⊂ P^{n−1} is a smooth curve of genus one and degree n spanning P^{n−1}. (ii) A rational nodal curve C ⊂ P^{n−1} is a rational curve of degree n, with a single node, spanning P^{n−1}. (iii) A Néron polygon C ⊂ P^{n−1} is the union of n lines ℓ_{0}, …, ℓ_{n−1}, spanning P^{n−1} and arranged such that ℓ_{i} meets ℓ_{j} if and only if i − j ≡ ±1 (mod n).

Lemma 2.2 The automorphisms of P^{n−1} act transitively on (i) the isomorphism classes of elliptic normal curves, (ii) the rational nodal curves and (iii) the Néron polygons.

Proof. (i) Let C, C′ ⊂ P^{n−1} be elliptic normal curves with C ≃ C′. We pick 0 ∈ C with n.0 linearly equivalent to the hyperplane section. Similarly we pick 0′ ∈ C′. Composing with a translation, if necessary, our isomorphism maps 0 ↦ 0′ and so extends to P^{n−1}. (ii) The linear system {f ∈ H^{0}(P^{1}, O_{P^{1}}(n)) | f(0) = f(∞)} defines a morphism P^{1} → P^{n−1} with image a rational nodal curve. It suffices to observe that every rational nodal curve arises in this way. (ii) A Néron polygon is uniquely determined by its n vertices. The lemma reduces to a well known result. □

Lemma 2.3 Let C ⊂ P^{n−1} be either (i) an elliptic normal curve or (ii) a rational nodal curve. Then there are only finitely many automorphisms of C that extend to automorphisms of P^{n−1}.

Proof. (i) We say that P ∈ C is a flex if n.P is linearly equivalent to the hyperplane section. It is well known that C has n^{2} flexes and that these span P^{n−1}. An automorphism of C extends to P^{n−1} if and only if it permutes the flexes, and is uniquely determined by this permutation. A more careful argument gives the number of automorphisms as dn^{2} where d = 2, 4 or 6. (ii) Let C be the image of (a : b) ↦ (a^{n} + b^{n} : ab^{n−1} : …: a^{n−1}b). The automorphisms that extend to P^{n−1} belong to the dihedral group generated by (a : b) ↦ (b : a) and (a : b) ↦ (a : ζ_{n}b). □

We return to the case n = 5. The curves of Definition 2.1 will be called elliptic normal quintics, rational nodal quintics and Néron pentagons respectively. In §1 we introduced 5-dimensional vector spaces V and W and identified ∧^{2}V ⊗ W as the space of elliptic normal quintics.

Proposition 2.4 Let G := GL(V) × GL(W) act on ∧^{2}V ⊗ W. Then there is an orbit of codimension 1 for each isomorphism class of elliptic curves, as determined by the j-invariant, and exactly one further orbit of codimension 1 corresponding to the rational nodal quintics.

We need some lemmas.

Lemma 2.5 Let C(ϕ) ⊂ P^{4} satisfy one of the conditions of Definition 2.1.

(i) If C(ϕ) = C(ϕ′) as subschemes of P^{4} then ϕ′ = (g_{V} , 1)ϕ for some g_{V} ∈ GL(V). Furthermore g_{V} is unique up to sign. (ii) If C(ϕ) ≃ C(ϕ′) as curves then ϕ′ = (g_{V} , g_{W})ϕ for some (g_{V} , g_{W}) ∈ G. (iii) If C(ϕ) is either an elliptic normal quintic or a rational nodal quintic then the stabiliser of ϕ has finite image in PGL(V) × PGL(W).

Proof. (i) For an elliptic normal quintic this is [11, Prop. 9.4]. The alternative proof given at the end of [11, §10] immediately carries over to the degenerate case. (ii) We apply Lemma 2.2 and the existence part of (i). (iii) We apply Lemma 2.3 and the uniqueness part of (i). The corresponding statement for a Néron pentagon is false. □

Lemma 2.6 Let C(ϕ) be either an elliptic normal quintic or a rational nodal quintic. Then the orbit of ϕ under the action of G has codimension 1 in ∧^{2}V ⊗ W. In particular the closure of the orbit of ϕ is the zero locus of an irreducible invariant.

Proof. The first statement follows from Proposition 2.5(iii) on counting dimensions. For the second statement we note that the only algebraic characters of G are of the form (g_{V} , g_{W}) ↦ (det g_{V})^{p}(det g_{W})^{q}. □

Lemma 2.7 Let Z′ ⊂ Gr(5, ∧^{2}V) be the set of 4-planes Π ⊂ P(∧^{2}V) for which Gr(2, V) ∩ Π is not an elliptic normal quintic. Then Z′ is closed and irreducible.

Proof. According to the proof of Proposition 1.1(ii), Z′ ⊂ Gr(5, ∧^{2}V) is the set of Π for which Gr(2, V) ∩ Π is not a smooth curve. So it suffices to prove a form of Bertini’s theorem [13, II.8.18]. For x ∈ Gr(2, V) let B_{x} ⊂ Gr(5, ∧^{2}V) be the set of 4-planes Π satisfying (i) x ∈ Π, and (ii) dim T_{x} Gr(2, V) ∩ T_{x}Π > 1. These are the bad points with respect to x. Let

B = {(x, Π) ∈ Gr(2, V) × Gr(5, ∧^{2}V) | Π ∈ B_{x}}.

We write p_{1} and p_{2} for the first and second projections. Since p_{1}(B) = Gr(2, V) and each fibre B_{x} is irreducible, it follows that B is irreducible. We deduce that Z′ = p_{2}(B) is closed and irreducible. □

Lemma 2.8 Let Z ⊂ ∧^{2}V ⊗ W be the set of ϕ for which C(ϕ) is not an elliptic normal quintic. Then Z is closed and irreducible.

Proof. There is a rational map κ : ∧^{2}V ⊗ W − → Gr(5, ∧^{2}V) that associates to ϕ the corresponding subspace Π ⊂ ∧^{2}V. Let Z_{1} ⊂ ∧^{2}V ⊗ W be the set of ϕ that fail to span W. It follows by Definition 1.2 that all invariants of positive degree vanish on Z_{1}. On the complement of Z_{1}, κ is a morphism with irreducible fibres. We write Z_{2} for the closure of κ^{−1}(Z′), so that Z_{2} ⊂ ∧^{2}V ⊗ W is closed and irreducible. Then Z = Z_{1} ∪ Z_{2} and it suffices to show that Z = Z_{2}. But applying Lemma 2.6 in the case of a rational nodal quintic shows that Z_{2} is the zero locus of an invariant. Thus Z_{1} ⊂ Z_{2} and Z = Z_{2} as required. □

Proof of Proposition 2.4. According to Lemmas 2.5 and 2.6 there are orbits of codimension 1 as specified. Let Z ⊂ ∧^{2}V ⊗ W be the subset defined in Lemma 2.8. Since the rational nodal quintics are dense in Z, there is no room left for any further orbits of codimension 1. □

We prove a weak form of Theorem 1.

Proposition 2.9 There are irreducible invariants c_{4}, c_{6} and Δ such that (i) C(ϕ) ⊂ P^{4} is an elliptic normal quintic if and only if Δ(ϕ) ≠ 0. (ii) If (i) holds then

\begin{aligned}j(C(ϕ)) = 0 & ⟺ & c_{4}(ϕ) = 0 \\ j(C(ϕ)) = 1728 & ⟺ & c_{6}(ϕ) = 0. \end{aligned}

Proof. We construct c_{4} and c_{6} by applying Lemma 2.6 with ϕ an elliptic normal quintic of the appropriate j-invariant. Likewise Δ is constructed from the rational nodal quintics. The remaining statements follow by Proposition 2.4 and Lemma 2.8. □

This construction sheds no light on the following important questions.

• How are c_{4}, c_{6} and Δ related?

• What are the degrees and weights?

• Given ϕ, how might one practically compute c_{4}(ϕ), c_{6}(ϕ) and Δ(ϕ)?

For n = 2, 3, 4 we may answer these questions by writing down c_{4}, c_{6} and Δ as explicit polynomials. However in the case n = 5 we believe that the invariants are too large to write down. To make explicit computations feasible we restrict our covariants to the modular curve X(5) ⊂ P(∧^{2}V ⊗ W). Over the next three sections we develop a practical method for deciding which forms on X(5) arise in this way.

3 Some modular curves

We review the facts we need about the modular curve X(n), with special attention to the case n = 5.

Let n ≥ 3 be an integer, and let ζ_{n} ∈ k be a primitive nth root of unity. The Heisenberg group H_{n} is the subgroup of GL_{n} generated by

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}). (6)

As our notation indicates, we prefer to view H_{n} as an abstract group with generators σ and τ. Then (6) defines a faithful representation θ : H_{n} → GL_{n} called the Schrödinger representation. We write \overline{θ} : H_{n} → PGL_{n} for the corresponding projective representation, and also put

\overline{ν}(−I) ∝ (\begin{matrix}1 & 0 & ⋯ & 0 & 0 \\ 0 & 0 & ⋯ & 0 & 1 \\ 0 & 0 & ⋯ & 1 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 1 & ⋯ & 0 & 0 \end{matrix}).  (7)

The modular curve Y (n) parametrises the isomorphism classes of triples (E, P, Q), where E is an elliptic curve and P, Q ∈ E[n] satisfy e_{n}(P, Q) = ζ_{n}. We write τ_{P} and τ_{Q} for the translation maps by P and Q respectively.

Lemma 3.1 Let n ≥ 3 be an odd integer and let (E, P, Q) be a triple as above. If we embed E ⊂ P^{n−1} via the complete linear system |n.0| then there is a unique choice of co-ordinates on P^{n−1} such that τ_{P}, τ_{Q} and [−1] are given by \overline{θ}(σ), \overline{θ}(τ) and \overline{ν}(−I).

Proof. This is well known. See for example [8], [9]. □

For n ≥ 5 we obtain an embedding X(n) ⊂ P^{n−1}, given on the open subset Y (n) by (E, P, Q) ↦ 0. In fact X(n) ⊂ P(A) where

A = {(0, a_{1}, a_{2}, …, −a_{2}, −a_{1})} ⊂ k^{n}

is an eigenspace for the action of \overline{ν}(−I). We write (x_{0} : x_{1} : …: x_{n−1}) for our co-ordinates on P^{n−1} and agree to read all subscripts mod n.

Definition 3.2 (i) Let A(n) ⊂ P(A) be the subscheme defined by the 4 × 4 Pfaffians of the matrix (a_{i−j}a_{i+j})_{i,j=0}^{n−1}. (ii) Let A(n) ⊂ A(n) × P^{n−1} be the subscheme defined by the 4 × 4 Pfaffians of the matrix (a_{i−j}x_{i+j})_{i,j=0}^{n−1}.

Proposition 3.3 Let n ≥ 5 be an odd integer and let (E, P, Q) be a triple as above. If we embed E ⊂ P^{n−1} as described in Lemma 3.1 then

(i) E ∩ P(A) = {0}, and (ii) E is the fibre of A(n) → A(n) above 0.

Proof. See [8], [11]. □

We obtain equations for X(n) for some small values of n

X(5) = P^{1} X(7) = {a^{3}b + b^{3}c + c^{3}a = 0} ⊂ P^{2}

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

In the case n = p is prime, Vélu [24] has shown that X(p) = A(p). Accordingly we write X (p) → X(p) instead of A(p) → A(p).

There is an action of PSL_{2}(Z/nZ) on X(n) given by relabelling torsion. Explicitly PSL_{2}(Z/nZ) acts on Y (n) via

(\begin{matrix}a & b \\ c & d \end{matrix}) : (E, P, Q) ↦ (E, aP + bQ, cP + dQ).  (8)

It is well known that the quotient of X(n) by this action is the j-line, and that the cusps of X(n) are the points above j = ∞. It follows by Lemma 3.1 that \overline{ν}(−I) extends uniquely to a projective representation

\overline{ν} : SL_{2}(Z/nZ) → PGL_{n}

satisfying

\overline{ν}(\begin{matrix}a & b \\ c & d \end{matrix})^{−1} \overline{θ}(σ^{u}τ^{v}) \overline{ν}(\begin{matrix}a & b \\ c & d \end{matrix}) ∝ \overline{θ}(σ^{au+cv}τ^{bu+dv}).  (9)

By construction, \overline{ν} acts on P(A) and its restriction to Y (n) ⊂ P(A) is given by (8). Fixing generators

S = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) ,  T = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix})  (10)

for SL_{2}(Z/nZ) we find

\overline{ν}(S) ∝ (ζ_{n}^{ij})_{i,j=0}^{n−1},  \overline{ν}(S^{−1}T S) ∝ (ζ_{n}^{i^{2}/2}δ_{ij})_{i,j=0}^{n−1}.  (11)

This defines an action of SL_{2}(Z/nZ) on the family A(n) → A(n). In [8] we used this action both to find the cusps of X(n) and to describe the degenerate fibres. The formulae (11) could also be used to show that the action of SL_{2}(Z/nZ) on P(A) is irreducible, and hence that X(n) spans P(A). In this article we use the action of SL_{2}(Z/5Z) to help recover the covariants from their restrictions to X(5).

Specialising to the case n = 5 we identify X(5) = P(A) where

A = {(0, a, b, −b, −a)} ⊂ k^{5}.

The family of curves X (5) → X(5) is defined by quadrics

p_{i} := abx_{i}^{2} + b^{2}x_{i+1}x_{i+4} − a^{2}x_{i+2}x_{i+3}.

These are the 4 × 4 Pfaffians of the matrix

(\begin{matrix}0 & bx_{3} & −ax_{1} & ax_{4} & −bx_{2} \\ −bx_{3} & 0 & bx_{4} & −ax_{2} & ax_{0} \\ ax_{1} & −bx_{4} & 0 & bx_{0} & −ax_{3} \\ −ax_{4} & ax_{2} & −bx_{0} & 0 & bx_{1} \\ bx_{2} & −ax_{0} & ax_{3} & −bx_{1} & 0 \end{matrix}) .

We rewrite this matrix in the notation of §1. In terms of bases v_{0}, …, v_{4} and x_{0}, …, x_{4} for V and W it is given by^{2}

u(a, b) = a ∑(v_{1} ∧ v_{4})x_{0} + b ∑(v_{2} ∧ v_{3})x_{0}.

This defines a linear map u : A → ∧^{2}V ⊗ W with the property that the fibre C(ϕ) of X (5) → X(5) above (a : b) is given by ϕ = u(a, b). The action of SL_{2}(Z/5Z) tells us that the cusps are the roots of

D(a, b) = ab(a^{10} − 11a^{5}b^{5} − b^{10})  (12)

and that the fibre at each cusp is a Néron pentagon.

To bridge the gap between the rational nodal quintics and the Néron pentagons we consider

u_{1}(λ_{0}, …, λ_{4}) = ∑ λ_{0}(v_{1} ∧ v_{4})x_{0} + ∑(v_{2} ∧ v_{3})x_{0}.  (13)

This family of curves was studied in [8], [9]. We restate [9, Lemma 2.9] in the language of invariants.

Lemma 3.4 Let F ∈ k[∧^{2}V ⊗ W] be an invariant. Then f_{1} = F ∘u_{1} is a polynomial in λ := ∏ λ_{i}.

Proof. Putting g_{V} = Diag(1, α, 1, 1, α) and g_{W} = Diag(α, 1, 1, 1, 1) we find

(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}).  (14)

2The summations are over all cyclic permutations of the subscripts.

The formulae (5) with r = s = 0 give 2p + q = deg F. By (4) we have

f_{1}(λ_{0}, …, λ_{4}) = f_{1}(α^{2}λ_{0}, α^{−1}λ_{1}, λ_{2}, λ_{3}, α^{−1}λ_{4}).

If λ ≠ 0 then repeated application of this identity, and its cyclic permutes, shows that f_{1}(λ_{0}, …, λ_{4}) only depends on λ. The lemma follows. □

Since ϕ = u_{1}(0, 0, 0, 0, 0) is a Néron pentagon and ϕ′ = u_{1}(0, 1, 1, 1, 1) is a rational nodal quintic, we deduce

Corollary 3.5 Let F ∈ k[∧^{2}V ⊗ W] be an invariant. Then F (ϕ) ≠ 0 for ϕ a Néron pentagon if and only if F (ϕ′) ≠ 0 for ϕ′ a rational nodal quintic.

The next lemma will help us describe the ring of invariants.

Lemma 3.6 Let F ∈ k(∧^{2}V ⊗ W) be a rational invariant, i.e. a rational function satisfying (4). Then F is regular if and only if f = F ∘u is regular.

Proof. Suppose that f is regular and write F = R/S where R and S are coprime invariants. We suppose for a contradiction that S has positive degree. Then S(ϕ) = 0 for some ϕ = u(a, b) and R(ϕ) = f(a, b)S(ϕ) = 0. If ϕ is an elliptic normal quintic then it follows by Lemma 2.6 that R and S have a common factor. This is a contradiction. It remains to consider ϕ a Néron pentagon. But in that case Corollary 3.5 gives R(ϕ′) = S(ϕ′) = 0 for ϕ′ a rational nodal quintic. We use Lemma 2.6 to conclude as before. □

The analogue of Lemma 3.6 for covariants is false. However the technique of Lemma 3.4 is still useful. We record the necessary formulae.

Lemma 3.7 Let F : ∧^{2}V ⊗ W → Y be a covariant. We put f = F ∘u and f_{1} = F ∘u_{1}. Let λ_{0}, …, λ_{4} ∈ k with ∏ λ_{i} = a^{5} and α_{0}, …, α_{4} ∈ k with ∏ α_{i} = 1. Suppose that the λ_{i} and α_{i} are related via

α_{i}^{2}/(α_{i+1}α_{i+4}) = λ_{i}/a  (15)
α_{i}^{5} = λ_{i}^{2}/(λ_{i+2}λ_{i+3}).  (16)

Then

f_{1}(λ_{0}, …, λ_{4}) = ρ_{Y} (g_{V} , g_{W})f(a, 1)  (17)

where g_{V} = Diag(α_{1}α_{4}, α_{0}α_{2}, …, α_{0}α_{3}) and g_{W} = Diag(α_{0}, …, α_{4}).

Proof. We apply the definition (4) to (14) and its cyclic permutes. □

Remark 3.8 In practice we are given f and wish to compute f_{1}. We use (15) to eliminate α_{2}, α_{3}, α_{4} from the right hand side of (17). Then we use (16) to eliminate α_{0} and α_{1}.

4 The extended Heisenberg group

We introduce more of the technical machinery needed to recover a covariant from its restriction to X(5).

Let n ≥ 3 be an odd integer. The Heisenberg group H_{n} ⊂ SL_{n} is the subgroup generated by the matrices θ(σ) and θ(τ) of §3. It is a non-abelian group of order n^{3} with centre generated by ζ := στσ^{−1}τ^{−1}. Let \overline{H}_{n} be the quotient of H_{n} by its centre. There is an exact sequence

0 ⟶ μ_{n} ⟶ H_{n} ⟶ \overline{H}_{n} ⟶ 0

and a bilinear pairing on \overline{H}_{n} defined by e_{n}(\overline{x}, \overline{y}) = xyx^{−1}y^{−1}. It is easy to see that e_{n} is alternating and non-degenerate. Taking \overline{σ}, \overline{τ} as our basis for \overline{H}_{n} there is a natural group homomorphism β : Aut(H_{n}) → GL_{2}(Z/nZ). The kernel of β is the group of inner automorphisms of H_{n}, i.e. a copy of \overline{H}_{n}.

Let N_{n} be the normaliser of H_{n} in SL_{n}. Then N_{n} acts on H_{n} via conjugation, and the corresponding action on \overline{H}_{n} preserves the pairing e_{n}. Conversely, Lemma 3.1 tells us that every such automorphism of \overline{H}_{n} arises in this way. We obtain a commutative diagram with exact rows

0 ⟶ H_{n} ⟶  N_{n}  ⟶^{α} SL_{2}(Z/nZ) ⟶ 0
↓  ↓  ↓
0 ⟶ \overline{H}_{n} ⟶ Aut(H_{n}) ⟶^{β} GL_{2}(Z/nZ) ⟶ 0.

(18)

The group N_{5} was used by Horrocks and Mumford [14] in their construction of an indecomposable rank 2 vector bundle on P^{4}. In fact the number |N_{5}| = 15, 000 appears in the title of their paper.

We make a choice of involution ι ∈ Aut(H_{n}) with β(ι) = −I. The second row of (18) allows us to identify Aut(H_{n}) as a group of affine transformations. Since n is odd, our choice of ι uniquely determines a section^{3} s_{β} for β with s_{β}(−I) = ι. Indeed the image of s_{β} is the centraliser of ι in Aut(H_{n}). The middle vertical map of (18) factors through \overline{N}_{n}, the image of N_{n} in PGL_{n}. So the restriction of s_{β} to SL_{2}(Z/nZ) determines a projective representation

3We require that our sections are group homomorphisms.

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

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 inclusion of H_{n}^{+} in GL_{n} will be called the extended Schrödinger representation. But in order to identify H_{n}^{+} = N_{n} we must first fix a section for the map α compatible with s_{β}. More precisely

Lemma 4.1 The following objects are in natural 1-1 correspondence.

(i) Sections s_{α} for α compatible with s_{β}. (ii) Lifts of \overline{ω} : SL_{2}(Z/nZ) → PGL_{n} to ω : SL_{2}(Z/nZ) → GL_{n}. (iii) Extensions of θ : H_{n} → GL_{n} to θ^{+} : H_{n}^{+} → GL_{n}.

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

\overline{ω}(γ) h \overline{ω}(γ)^{−1} = s_{β}(γ)h  for all h ∈ H_{n}, γ ∈ SL_{2}(Z/nZ).  (19)

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

s_{α}(γ) h s_{α}(γ)^{−1} = s_{β}(γ)h  for all h ∈ H_{n}, γ ∈ SL_{2}(Z/nZ).  (20)

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 (20) that θ^{+} is a homomorphism. Conversely, given θ^{+} we set s_{α}(γ) = θ^{+}(1, γ). This gives the correspondence between (i) and (iii). □

We fix, once and for all, our choice of involution ι ∈ Aut(H_{n}). We take

ι(σ) = σ^{−1} and ι(τ) = τ^{−1}. For γ = (\begin{matrix}a & b \\ c & d \end{matrix}) it follows that

s_{β}(γ) : σ ↦ ζ^{−ac/2}σ^{a}τ^{c}; τ ↦ ζ^{−bd/2}σ^{b}τ^{d}.

Substituting in (19) we see that \overline{ω}(γ) is uniquely determined by

\begin{aligned}\overline{ω}(γ)θ(σ)\overline{ω}(γ)^{−1} & = & ζ_{n}^{−ac/2}θ(σ^{a}τ^{c}) \\ \overline{ω}(γ)θ(τ)\overline{ω}(γ)^{−1} & = & ζ_{n}^{−bd/2}θ(σ^{b}τ^{d}). \end{aligned}

Comparison with (9) shows that the projective representations \overline{ν} and \overline{ω} differ only by the automorphism γ ↦ (γ^{T})^{−1} = S^{−1}γS of SL_{2}(Z/nZ). In particular we have explicit expressions (11) for \overline{ω}(S) and \overline{ω}(T).

We temporarily adopt the notation P^{n−1} = P(U) where U is an ndimension vector space. The action of \overline{ω}(−I) gives an eigenspace decomposition U = U_{+} ⊕ U_{−} with dim U_{±} = (n ± 1)/2. We use this decomposition to give criteria for \overline{ω} to lift to GL_{n}.

Lemma 4.2 Suppose we are given lifts ω(S) and ω(T) for \overline{ω}(S) and \overline{ω}(T), and characters χ_{±} : SL_{2}(Z/nZ) → k^{∗} such that ω(S) and ω(T) restricted to U_{±} have determinants χ_{±}(S) and χ_{±}(T). Then ω extends uniquely to a lifting of \overline{ω}.

Proof. Let γ ∈ SL_{2}(Z/nZ). We claim that there is a unique lifting ω(γ) of \overline{ω}(γ) whose restriction to U_{±} has determinant χ_{±}(γ). The existence follows from our assumptions in the cases γ = S, T. The uniqueness relies on the fact that dim U_{+} and dim U_{−} are coprime. It is evident that ω, so defined, is a group homomorphism. □

Lemma 4.3 The group SL_{2}(Z/nZ) has gcd(3, n) 1-dimensional characters. Moreover each character is uniquely determined by its value at T.

Proof. This follows from the relations (ST)^{3} = S^{4} = T^{n} = I. The characters in the case 3|n factor through PSL_{2}(Z/3Z) ≃ A_{4}. □

Lemma 4.4 The projective representation \overline{ω} lifts to GL_{n}. The number of such liftings is gcd(3, n).

Proof. Since 3(1^{2} + 2^{2} + … + (\frac{n−1}{2})^{2}) ≡ 0 (mod n) it follows from (11) and Lemma 4.3 that we may choose ω(T) and χ_{±} satisfying the hypotheses of Lemma 4.2. We scale ω(S) such that

ω(S)ω(T)^{−1}ω(S) = ω(T)ω(S)ω(T).  (21)

We check the remaining hypotheses of Lemma 4.2 by restricting to U_{±} and taking determinants in (21). This proves the existence of a lifting ω. Since any two liftings differ by a 1-dimensional character, the number of liftings is determined by Lemma 4.3. □

Remark 4.5 Explicit calculations show that ω(S) = g_{n}^{−1}(ζ_{n}^{ij})_{i,j=0}^{n−1} where the Gauss sum g_{n} = ∑_{i=0}^{n−1} ζ_{n}^{−i^{2}/2} satisfies g_{n}^{2} = (−1)^{(n−1)/2}n.

Remark 4.6 For n = p is a prime it is known [16, V, Satz 25.5] that the group of Schur multipliers H^{2}(SL_{2}(Z/pZ), k^{∗}) is trivial. This gives an alternative proof of Lemma 4.4.

Corollary 4.7 The Schrödinger representation θ : H_{n} → GL_{n} extends to a representation θ^{+} : H_{n}^{+} → GL_{n}. If n is prime to 6 then θ^{+} is unique.

Proof. We combine Lemmas 4.1 and 4.4. □

Remark 4.8 We say that two representations of H_{n} are conjugate if they differ by an automorphism of H_{n}. The Schrödinger representation has ϕ(n) conjugates, one for each primitive nth root of unity. We may apply Corollary 4.7 to any one of these representations.

5 The discrete covariants

We return to the case n = 5. In §3 we considered the family of curves

u(a, b) = a ∑(v_{1} ∧ v_{4})x_{0} + b ∑(v_{2} ∧ v_{3})x_{0}.

We choose conjugates of the Schrödinger representation that make u(a, b) Heisenberg invariant

\begin{matrix}θ_{V} : H_{5} → GL(V) ; & σ : v_{i} ↦ ζ_{5}^{2i}v_{i}; & τ : v_{i} ↦ v_{i+1} \\ θ_{W} : H_{5} → GL(W) ; & σ : x_{i} ↦ ζ_{5}^{i}x_{i}; & τ : x_{i} ↦ x_{i+1}. \end{matrix}  (22)

We continue to write θ_{V} and θ_{W} for the extensions of these representations to H_{5}^{+} as determined by Corollary 4.7.

Let Stab(ϕ) ⊂ SL(V) × SL(W) be the stabiliser of ϕ ∈ ∧^{2}V ⊗ W. If ϕ is an elliptic normal quintic, we write H(ϕ) ⊂ Stab(ϕ) for the subgroup^{4} acting on C(ϕ) by translations. We also write N(ϕ) for the normaliser of H(ϕ) in SL(V) × SL(W). In the case ϕ = u(a, b) we relate these groups to the exact sequence

0 ⟶ H_{5} ⟶ H_{5}^{+} ⟶ SL_{2}(Z/5Z) ⟶ 0.  (23)

4In fact we shall see in Lemma 12.2 that H(ϕ) = Stab(ϕ).

Lemma 5.1 Let ϕ = u(a, b) be an elliptic normal quintic. Then

(i) H(ϕ) = {(θ_{V} (h), θ_{W} (h)) | h ∈ H_{5}}. (ii) N(ϕ) = {(ζ_{V} θ_{V} (h^{+}), ζ_{W} θ_{W} (h^{+})) | ζ_{V} , ζ_{W} ∈ μ_{5}, h^{+} ∈ H_{5}^{+}}.

Proof. (i) We saw in §3 that the translation automorphisms of C(ϕ) extending to P^{4} are described by the Heisenberg group. Notice that by working with SL(V) × SL(W) we have removed the sign ambiguity of Lemma 2.5(i). (ii) The inclusion “⊃” is clear from (23). So let (g_{V} , g_{W}) ∈ N(ϕ). Then g_{V} belongs to the normaliser of H_{5} in SL(V). The construction of the extended Heisenberg group shows that g_{V} = θ_{V} (h^{+}) for some h^{+} ∈ H_{5}^{+}. This reduces us to the case g_{V} = I_{V}. But now g_{W} belongs to the centraliser of H_{5} in SL(W). It follows that g_{W} is a scalar matrix as required. □

The representations θ_{V} and θ_{W}, or more precisely their extensions to H_{5}^{+}, define actions of Γ := SL_{2}(Z/5Z) on A ⊂ W^{∗} and on (∧^{2}V ⊗ W)^{H_{5}}.

Lemma 5.2 The linear map u : A → (∧^{2}V ⊗ W)^{H_{5}} is an isomorphism of Γ-modules.

Proof. We compute the character of the H_{5}-representation ∧^{2}V ⊗ W. It is constant on the centre of H_{5} and zero elsewhere. Thus dim(∧^{2}V ⊗ W)^{H_{5}} = 2 and u is an isomorphism of vector spaces. Now suppose that γ ∈ Γ maps ϕ = u(a, b) to ϕ′ = u(a′, b′). Then γ acts on P(A) ⊂ P^{4} but maps C(ϕ) ⊂ P^{4} to C(ϕ′) ⊂ P^{4}. By Lemma 3.3(i) we have

γ : (0 : a : b : −b : −a) ↦ (0 : a′ : b′ : −b′ : −a′).

So γ(a, b) ∝ (a′, b′) and the lemma holds up to scalars. We are done since Γ has no 1-dimensional characters. □

The action of Γ on A ⊂ W^{∗} is given by χ : Γ → GL(A) where

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

and ϕ = (ζ_{5}−ζ_{5}^{4})/(ζ_{5}^{2}−ζ_{5}^{3}). Up to equivalence this is a lifting of the action (8) of PSL_{2}(Z/5Z) on X(5) = P(A). If we use the exact sequence (23) to extend χ to H_{5}^{+} then Lemma 5.2 takes the form

u∘χ(h^{+}) = (θ_{V} (h^{+}), θ_{W} (h^{+}))∘u for all h^{+} ∈ H_{5}^{+}.  (24)

Now let F : ∧^{2}V ⊗ W → Y be a covariant and let f = F ∘u be the restriction of F to X(5). We give Y the structure of H_{5}^{+}-module via

π(h^{+}) = ρ_{Y} (θ_{V} (h^{+}), θ_{W} (h^{+})) for all h^{+} ∈ H_{5}^{+}.  (25)

Then

f∘χ(h^{+}) = F ∘u∘χ(h^{+})
= F ∘(θ_{V} (h^{+}), θ_{W} (h^{+}))∘u  by (24)
= ρ_{Y} (θ_{V} (h^{+}), θ_{W} (h^{+}))∘F ∘u by Definition 1.2
= π(h^{+})∘f  by (25)

We obtain a Γ-equivariant map f : A → Y^{H_{5}}. More generally for π : Γ → GL(Z) a linear representation we define

Definition 5.3 A homogeneous polynomial function f : A → Z is a discrete covariant if f∘χ(γ) = π(γ)∘f for all γ ∈ Γ. If Z = k is the trivial representation then f is called a discrete invariant.

The discrete covariants are easy to compute. In §7 we give a complete classification, by first reducing to the case where π is irreducible.

Lemma 5.4 A covariant F : ∧^{2}V ⊗ W → Y is uniquely determined by its corresponding discrete covariant f = F ∘u : A → Y^{H_{5}}.

Proof. In §§1,2 we saw that GL(V) × GL(W) acts transitively on each isomorphism class of elliptic curves and that the elliptic normal quintics are dense in ∧^{2}V ⊗ W. We are done since X(5) covers the j-line. □

It is therefore reasonable to make the definition

Definition 5.5 Let f : A → Y^{H_{5}} be a discrete covariant. We say that f is a covariant if f = F ∘u for some covariant F : ∧^{2}V ⊗ W → Y.

There are essentially two different ways in which a discrete covariant may fail to be a covariant.

Fractional weights. Given f a discrete covariant, we may compute its weights using (5). If these fail to be integers then f is not a covariant.

Example. The discrete invariant D(a, b) = ab(a^{10} − 11a^{5}b^{5} − b^{10}) has degree 12. By (5) it has weights (24/5,12/5) and so cannot be an invariant. Another example, this time with Y = S^{5}W, is

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

By (5) it has weights (4/5, −3/5) and so cannot be a covariant.

Denominators. We generalise the notion of covariant to that of rational covariant, by which we mean a rational map F : ∧^{2}V ⊗ W − → Y satisfying (4). Let f = F ∘u be the corresponding rational discrete covariant. It may happen that f is regular even when F is not. In such cases the proof of Lemma 5.4 shows that f is not a covariant. Example. Decomposing (S^{10}W)^{H_{5}} as an Γ-module (see §11) we find that the trivial representation appears. So there is a discrete covariant of degree zero. But clearly there are no covariants of degree zero.

We prove that these are the only two obstructions.

Theorem 2 Let ρ : GL(V) × GL(W) → GL(Y) be a homogeneous representation and let f : A → Y^{H_{5}} be an integer weight discrete covariant. Then there is a rational covariant F : ∧^{2}V ⊗ W − → Y such that f = F ∘u.

Proof. We are assuming that ρ_{Y} (λI_{V} , μI_{W}) = λ^{r}μ^{s}I_{Y} for all λ, μ ∈ k^{∗} and that there are integers p, q with

\begin{aligned}2 deg f & = & 5p + r \\ deg f & = & 5q + s. \end{aligned}  (26)

Let ϕ ∈ ∧^{2}V ⊗ W be an elliptic normal quintic. Then ϕ = (g_{V} , g_{W})u(a, b) for some (g_{V} , g_{W}) ∈ SL(V) × SL(W) and (a, b) ∈ A. We define a rational map F : ∧^{2}V ⊗ W − → Y via

F (ϕ) = ρ_{Y} (g_{V} , g_{W})f(a, b).

We must check that F is well defined. So suppose that ϕ′ = (g_{V} , g_{W})ϕ with ϕ = u(a, b), ϕ′ = u(a′, b′) and (g_{V} , g_{W}) ∈ SL(V) × SL(W). We aim to show that f(a′, b′) = ρ_{Y} (g_{V} , g_{W})f(a, b). By Lemma 5.1(i) we have H(ϕ) = H(ϕ′). So (g_{V} , g_{W}) ∈ N(ϕ) and Lemma 5.1(ii) tells us that

(g_{V} , g_{W}) = (ζ_{V} θ_{V} (h^{+}), ζ_{W} θ_{W} (h^{+}))  (27)

for some ζ_{V} , ζ_{W} ∈ μ_{5}, h^{+} ∈ H_{5}^{+}. We compute

ϕ′ = ζ_{V}^{2} ζ_{W} (θ_{V} (h^{+}), θ_{W} (h^{+}))ϕ by (27)
⟹ (a′, b′) = ζ_{V}^{2} ζ_{W} χ(h^{+})(a, b)  by (24)
⟹ f(a′, b′) = ζ_{V}^{r} ζ_{W}^{s} f(χ(h^{+})(a, b))  by (26)
= ζ_{V}^{r} ζ_{W}^{s} π(h^{+})f(a, b)  by Definition 5.3
= ρ_{Y} (g_{V} , g_{W})f(a, b)  by (25) and (27).

Thus F is a well defined rational map. It is clear from our construction that F satisfies (4) and that f = F ∘u. □

Let Δ be the irreducible invariant introduced in §1. We recall that C(ϕ) is an elliptic normal quintic if and only if Δ(ϕ) ≠ 0. In the case of a specific covariant, such as Δ, we shall use the same symbol both for the covariant and for its discrete covariant. This should not cause any confusion.

Corollary 5.6 Let ρ : GL(V) × GL(W) → GL(Y) be a homogeneous representation and let f : A → Y^{H_{5}} be an integer weight discrete covariant. Let F be the rational covariant constructed in Theorem 2. Then

(i) Δ^{m}f is a covariant for m sufficiently large, (ii) f is a covariant if and only if f_{1} := F ∘u_{1} is regular. Furthermore an integer weight discrete invariant is an invariant.

Proof. (i) By construction F is regular outside {Δ = 0}. It follows that Δ^{m}F is regular for m sufficiently large. (ii) It is clear that f_{1} is regular whenever F is regular. For the converse we recall from §2 that ϕ = u_{1}(0, 1, 1, 1, 1) is a rational nodal quintic, and that the rational nodal quintics are dense in {Δ = 0}. The statement concerning invariants follows from Lemma 3.6. □

Remark 5.7 We use the formulae of Proposition 3.7 to construct f_{1} directly from f. Thus Corollary 5.6 gives a practical method for deciding which discrete covariants are covariants. We give examples in §§8,10,11.

Remark 5.8 Let f : A → Y^{H_{5}} be a discrete covariant with weights (p, q). If Y^{H_{5}} ≠ 0 then 2r + s ≡ 0 (mod 5). It follows by (26) that p is an integer if and only if q is an integer. In other words to demand that a discrete covariant has integer weights is really only one condition. This condition is a congruence mod 5 on the degree.

6 The invariants

We continue to assume that k is algebraically closed, and give a proof of Theorem 1 in this case.

Proposition 6.1 The ring of invariants is generated by invariants c_{4} and c_{6} of degrees 20 and 30 respectively. If we set Δ = (c_{4}^{3} − c_{6}^{2})/1728 then

(i) C(ϕ) ⊂ P^{4} is an elliptic normal quintic if and only if Δ(ϕ) ≠ 0. (ii) If (i) holds then C(ϕ) has Weierstrass equation

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

We begin by describing the discrete invariants. Since Γ = SL_{2}(Z/5Z) has no 1-dimensional characters, a homogeneous polynomial f ∈ k[A] is a discrete invariant if and only if its roots are invariant under the action of PSL_{2}(Z/5Z) ≃ A_{5} on X(5) ≃ P(A). We recognise this as a classical problem and recall from [17, Part I, Chapter II, §13] the discrete invariants

D(a, b) = ab(a^{10} − 11a^{5}b^{5} − b^{10})
c_{4}(a, b) = −\frac{1}{121} | \begin{matrix}\frac{∂^{2}D}{∂a^{2}} & \frac{∂^{2}D}{∂a∂b} \\ \frac{∂^{2}D}{∂a∂b} & \frac{∂^{2}D}{∂b^{2}} \end{matrix} |
= a^{20} + 228a^{15}b^{5} + 494a^{10}b^{10} − 228a^{5}b^{15} + b^{20}
c_{6}(a, b) = \frac{1}{20} | \begin{matrix}\frac{∂D}{∂a} & \frac{∂D}{∂b} \\ \frac{∂c_{4}}{∂a} & \frac{∂c_{4}}{∂b} \end{matrix} |
= −a^{30} + 522a^{25}b^{5} + 10005a^{20}b^{10} + 10005a^{10}b^{20} − 522a^{5}b^{25} − b^{30}.

The roots of these polynomials correspond under stereographic projection to the vertices, faces and edges of an icosahedron. They are also the ramification points of the quotient map j : X(5) → P^{1}. We must have

j ∝ c_{4}^{3}/D^{5} and j − 1728 ∝ c_{6}^{2}/D^{5}.  (28)

In particular D^{5}, c_{4}^{3} and c_{6}^{2} satisfy a linear dependence relation. Comparing a few coefficients we find 1728D^{5} = c_{4}^{3} − c_{6}^{2} and in fact this gives equality in (28). We arrive at the discrete analogue of Proposition 6.1.

Proposition 6.2 The ring of discrete invariants is generated by D, c_{4} and c_{6} subject to the single relation 1728D^{5} = c_{4}^{3} − c_{6}^{2}.

(i) ϕ = u(a, b) is an elliptic normal quintic if and only if D(a, b) ≠ 0. (ii) If (i) holds then C(ϕ) has Weierstrass equation

y^{2} = x^{3} − 27c_{4}(a, b)x − 54c_{6}(a, b).  (29)

Proof. The orbits for the action of PSL_{2}(Z/5Z) on X(5) are the roots of the polynomials c_{4}, c_{6}, D and c_{4}^{3} − jD^{5} for j ≠ 0, 1728. The discrete invariants are products of these polynomials.

(i) We saw in §3 that the cusps of X(5) are the roots of D. (ii) Since we are working over an algebraically closed field, it suffices to compare the j-invariants. The above discussion shows that C(ϕ) has jinvariant 1728c_{4}^{3}/(c_{4}^{3} − c_{6}^{2}). This is also the j-invariant of (29). □

Proof of Proposition 6.1. The ring of discrete invariants is k[D, c_{4}, c_{6}]. The formulae (5) with r = s = 0 tell us that a discrete invariant has integer weights if and only if it has degree d ≡ 0 (mod 5). So the ring of integer weight discrete invariants is k[D^{5}, c_{4}, c_{6}] = k[c_{4}, c_{6}]. By Corollary 5.6 this is the ring of invariants. We now set Δ = D^{5} and use Proposition 6.2 to identify c_{4}, c_{6} and Δ as non-zero multiples of the invariants of the same name introduced in §2. This proves (i). For (ii) we reduce to the case ϕ = u(a, b) and apply Proposition 6.2(ii). □

Remark 6.3 We recall from §3 that the curves C(ϕ) for ϕ = u(a, b) have rational point (0 : a : b : −b : −a). This makes it possible to prove Proposition 6.2 without the assumption that k is algebraically closed. Indeed in the notation of [8], [9], [10] we recognise C(ϕ) as the elliptic curve C_{λ} for λ = a^{5}/b^{5}. Then starting with the isogenous curve

D_{λ} : y^{2} + (1 − λ)xy − λy = x^{3} − λx^{2}

we used Vélu’s formulae [25] to compute a Weierstrass equation for C_{λ}. Substituting λ = a^{5}/b^{5} gives (29) as required.

In §9 we give a practical algorithm for computing the invariants. Then in §12 we complete the proof of Theorem 1 by showing that over a nonalgebraically closed field the invariants describe the Jacobian.

7 Computing the discrete covariants

Let Γ = SL_{2}(Z/5Z). In §5 we defined a 2-dimensional irreducible representation χ : Γ → GL(A) and so an action of Γ on k[A] = k[a, b]. Let

R = k[A]^{Γ} = k[D, c_{4}, c_{6}]

be the ring of discrete invariants. The discrete covariants attached to the representation π : Γ → GL(Z) form a graded R-module

M_{π} = ⊕_{d≥0}M_{π,d} = (k[A] ⊗ Z_{π})^{Γ}

where the grading is by degree. The aim of this section is to compute M_{π} for all representations π. We immediately reduce to the case where π is irreducible.

The character table for Γ is given in Appendix A. There are 5 even characters labelled ψ_{1}, …, ψ_{5} and 4 odd characters labelled χ_{1}, …, χ_{4}. Naturally we use the same symbol both for a representation and its character. We have arranged that ψ_{1} = 1 and that χ_{1} = χ. We use the recurrence relation S^{r+1}χ ⊕ S^{r−1}χ ≃ S^{r}χ ⊗ χ to compute the symmetric powers of χ.

χ_{1}, ψ_{4}, χ_{3}, ψ_{3}, χ_{4}, ψ_{2} ⊕ ψ_{5}, χ_{2} ⊕ χ_{4}, ψ_{2} ⊕ ψ_{3}, …

The integers dim M_{π,d} may be read off from this list. However we require more detailed information. We fix a basis z_{1}, …, z_{m} for Z_{π} and accordingly write Π : Γ → GL_{m} for the representation satisfying

π(γ)[z_{1}, …, z_{m}] = [ ∑_{i} Π(γ)_{i1}z_{i}, …, ∑_{i} Π(γ)_{im}z_{i}].

A discrete covariant takes the form [f_{1}, …, f_{m}].[z_{1}, …, z_{m}] := ∑ f_{i}z_{i} where the f_{i} ∈ k[A] are homogeneous polynomials of the same degree satisfying

[f_{1}, …, f_{m}]∘χ(γ) = [ ∑_{i} Π(γ)_{1i}f_{i}, …, ∑_{i} Π(γ)_{mi}f_{i}].

Since we assume that π is irreducible, the f_{i} are linearly independent and we may recover the matrices Π(γ) from [f_{1}, …, f_{m}]. We write computer algebra routines that

• Given [f_{1}, …, f_{m}] compute the matrices Π(S) and Π(T).

• Given matrices Π(S) and Π(T) compute a basis for M_{π,d}, writing each basis element in the form [f_{1}, …, f_{m}].

We specify a preferred choice of basis for Z_{π} by listing a single discrete covariant for each irreducible representation π. The matrices Π(S) and Π(T) may then be recovered using the first of the above routines. The notation we use to label the discrete covariants will be explained later.

\begin{matrix}ψ_{1} & [1] \\ ψ_{2} & ∂ ⊗ E = [5a^{2}b^{4}, −a(a^{5} + 2b^{5}), b(2a^{5} − b^{5}), −5a^{4}b^{2}] \\ ψ_{3} & U^{4} = [a^{4}, 4a^{3}b, 6a^{2}b^{2}, 4ab^{3}, b^{4}] \\ ψ_{4} & U^{2} = [a^{2}, 2ab, b^{2}] \\ ψ_{5} & ∂_{E}^{2}c_{4} ∝ [(3a^{5} + b^{5})b, −10a^{3}b^{3}, a(a^{5} − 3b^{5})] \\ χ_{1} & U = [a, b] \\ χ_{2} & E = [a^{2}(a^{5} + 7b^{5}), (7a^{5} − b^{5})b^{2}] \\ χ_{3} & U^{3} = [a^{3}, 3a^{2}b, 3ab^{2}, b^{3}] \\ χ_{4} & U^{5} = [a^{5}, 5a^{4}b, 10a^{3}b^{2}, 10a^{2}b^{3}, 5ab^{4}, b^{5}] \end{matrix}

The above routines allow us to compute the vector spaces M_{π,d} one at a time. Indeed this is sufficient for many applications. However we shall go further and give generators for each M_{π} as an R-module.

Proposition 7.1 Let R = k[D, c_{4}, c_{6}] and let M be a finitely generated torsion free graded R-module. The following are equivalent.

(i) M is Cohen-Macaulay as an R-module, i.e. depth(R_{+}, M) = 2. (ii) M is a free k[D, c_{4}]-module. (iii) M is a free k[D, c_{6}]-module. (iv) M is a free k[c_{4}, c_{6}]-module.

Proof. This is a special case of [4, Theorem 4.3.5]. □

Our next result is an application of [4, Theorem 4.3.6] originally due to Hochster and Eagon.

Proposition 7.2 Let π : Γ → GL(Z_{π}) with m = dim(Z_{π}). 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. Furthermore, 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. The polynomial ring k[A] is Cohen-Macaulay as an R-module, say on account of the regular sequence c_{4}, c_{6}. By Proposition 7.1, k[A] is a free k[c_{4}, c_{6}]-module and the same goes for k[A] ⊗ Z_{π}. The projection map

\frac{1}{|Γ|} ∑_{γ∈Γ} γ : k[A] ⊗ Z_{π} ⟶ M_{π}

shows that M_{π} is a projective k[c_{4}, c_{6}]-module. By [4, Theorem 4.1.1] it is therefore a free k[c_{4}, c_{6}]-module. The equivalent conditions of Proposition 7.1 are satisfied. So it only remains to compute the ranks. Let K = k(A)^{Γ} be the field of fractions of R. By the normal basis theorem the K[Γ]-module k(A) is a copy of the regular representation. Thus

K ⊗ Z_{π} ≃ (k(A) ⊗ Z_{π})^{Γ} = K ⊗ M_{π}.

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

\begin{aligned}[K : k(D, c_{4})] & = & 2 \\ [K : k(D, c_{6})] & = & 3 \\ [K : k(c_{4}, c_{6})] & = & 5. \end{aligned}

For the final statement we observe that M_{π}(r) is free of rank m_{r} with 5m = ∑_{r=0}^{4} m_{r}. But multiplication by D shows that m_{r} ≤ m_{r+2} for all r. So m_{0} = … = m_{4} = m as required. □

We use Molien’s theorem [4, Theorem 2.5.2] to compute the Poincaré series of M_{π}

p(M_{π}, t) := ∑_{d=0}^{∞}(dim_{k} M_{π,d})t^{d} = \frac{1}{|Γ|} ∑_{γ∈Γ} \frac{Tr π(γ)}{1 − Tr χ(γ)t + t^{2}} .  (30)

For example taking π = χ we find

p(M_{χ}, 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})}.

A basis for M_{χ} as a k[D, c_{4}]-module is

\begin{aligned}U & = & [a, b] & ∂c_{4} & = & [∂c_{4}/∂b, −∂c_{4}/∂a] \\ ∂D & = & [∂D/∂b, −∂D/∂a] & ∂c_{6} & = & [∂c_{6}/∂b, −∂c_{6}/∂a]. \end{aligned}  (31)

To give a basis for M_{χ} as a k[D, c_{6}]-module we need two further generators, namely c_{4}U and c_{4}∂c_{4}. Finally the k[c_{4}, c_{6}]-modules M_{χ}(r) have bases

\begin{matrix}D^{s}U, D^{s}∂D & for r = 2s + 1, s = 0, 1, 2, 3 \\ ∂c_{4}, ∂c_{6} & for r = 4. \end{matrix}

In order to construct new covariants from old, we adopt the notation

\begin{aligned}[f_{1}, …, f_{m}] ⊗ [g_{1}, …, g_{n}] & = & [f_{1}g_{1}, …, f_{m}g_{1}, f_{1}g_{2}, …, f_{m}g_{n}] \\ [f_{1}, …, f_{m}][g_{1}, …, g_{n}] & = & [f_{1}g_{1}, f_{1}g_{2} + f_{2}g_{1}, …, f_{m}g_{n}] \\ ∂_{[f_{1}(a,b),... ,f_{m}(a,b)]} & = & [f_{1}(\frac{∂}{∂b}, −\frac{∂}{∂a}), …, f_{m}(\frac{∂}{∂b}, −\frac{∂}{∂a})]. \end{aligned}  (32)

We have already used the operator ∂ := ∂_{U} in the above example. To get started in the case π = χ_{2} we put

\begin{aligned}E & = & [a^{2}(a^{5} + 7b^{5}), b^{2}(7a^{5} − b^{5})] \\ F & = & [−b^{3}(26a^{10} + 39a^{5}b^{5} − b^{10}), a^{3}(a^{10} + 39a^{5}b^{5} − 26b^{10})]. \end{aligned}

The degrees of the generators for M_{π} as a k[D, c_{4}]-module may be read off from the polynomial p(M_{π}, t)(1−t^{12})(1−t^{20}). We tabulate these polynomials, giving in each case a list of generators. We write our discrete covariants relative to the preferred basis for Z_{π} specified above.

\begin{matrix}ψ_{1} & \{ \begin{matrix}1 + t^{30} \\ 1, c_{6} \end{matrix} \\ ψ_{2} & \{ \begin{matrix}t^{6} + t^{8} + t^{12} + t^{14} + t^{16} + t^{18} + t^{22} + t^{24} \\ ∂ ⊗ E, U ⊗ E, ∂ ⊗ F, U ⊗ F, \\ ∂ ⊗ ∂_{F} c_{6}, U ⊗ ∂_{F} c_{6}, ∂ ⊗ ∂_{E}c_{6}, U ⊗ ∂_{E}c_{6} \end{matrix} \\ ψ_{3} & \{ \begin{matrix}t^{4} + t^{8} + t^{10} + t^{12} + t^{14} + t^{16} + t^{18} + t^{20} + t^{22} + t^{26} \\ U^{4}, ∂^{4}D, U∂^{3}D, …, U^{3}∂D, ∂^{4}c_{4}, U∂^{3}c_{4}, …, U^{3}∂c_{4}, ∂^{4}c_{6} \end{matrix} \\ ψ_{4} & \{ \begin{matrix}t^{2} + t^{10} + t^{12} + t^{18} + t^{20} + t^{28} \\ U^{2}, ∂^{2}D, U∂D, ∂^{2}c_{4}, U∂c_{4}, ∂^{2}c_{6} \end{matrix} \\ ψ_{5} & \{ \begin{matrix}t^{6} + t^{10} + t^{14} + t^{16} + t^{20} + t^{24} \\ ∂_{E}^{2}c_{4}, ∂_{E}∂_{F} c_{6}, E^{2}, ∂_{E}^{2}c_{6}, EF, E∂_{F} c_{6} \end{matrix} \end{matrix}

\begin{matrix}χ_{1} & \{ \begin{matrix}t + t^{11} + t^{19} + t^{29} \\ U, ∂D, ∂c_{4}, ∂c_{6} \end{matrix} \\ χ_{2} & \{ \begin{matrix}t^{7} + t^{13} + t^{17} + t^{23} \\ E, F, ∂_{F} c_{6}, ∂_{E}c_{6} \end{matrix} \\ χ_{3} & \{ \begin{matrix}t^{3} + t^{9} + t^{11} + t^{13} + t^{17} + t^{19} + t^{21} + t^{27} \\ U^{3}, ∂^{3}D, U∂^{2}D, U^{2}∂D, ∂^{3}c_{4}, U∂^{2}c_{4}, U^{2}∂c_{4}, ∂^{3}c_{6} \end{matrix} \\ χ_{4} & \{ \begin{matrix}t^{5} + t^{7} + t^{9} + t^{11} + t^{13} + 2t^{15} + t^{17} + t^{19} + t^{21} + t^{23} + t^{25} \\ U^{5}, ∂^{5}D, U∂^{4}D, …, U^{4}∂D, ∂^{5}c_{4}, U∂^{4}c_{4}, …, U^{4}∂c_{4}, ∂^{5}c_{6} \end{matrix} \end{matrix}

Since our notation (32) tends to introduce large constants, we acknowledge that this table has been constructed under the assumption char k = 0.

8 Computing the covariants

Let ρ : GL(V) × GL(W) → GL(Y) be a representation. Our method for computing the covariants F : ∧^{2}V ⊗ W → Y is as follows. First we compute Y^{H_{5}} as a Γ-module. Then we use the work of §7 to give explicit expressions for the discrete covariants f : A → Y^{H_{5}}. Finally we use Corollary 5.6 to decide which of these are covariants.

Before giving some examples, we recall from [15, Chap I, §2] the character table for the Heisenberg group H_{p} where p is an odd prime. There are p^{2} 1-dimensional characters indexed by (u, v) ∈ (Z/pZ)^{2} and p − 1 conjugates of the Schrödinger representation indexed by t ∈ (Z/pZ)^{∗}.

\begin{aligned}& ζ^{i} & σ^{j}τ^{k} \\ λ_{u,v} & 1 & ζ_{p}^{ju+kv} \\ θ_{t} & pζ_{p}^{it} & 0 \end{aligned}  (33)

Thus θ_{t} has dual θ_{−t}. We read off some further rules

Lemma 8.1 Let s, t ∈ (Z/pZ)^{∗}. Then

(i) θ_{s} ⊗ θ_{t} ≃ \{ \begin{matrix}pθ_{s+t} & if s + t ≢ 0 (mod p) \\ ⨁_{u,v} λ_{u,v} & if s + 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) ⨁_{u,v} λ_{u,v} & if d ≡ 0 (mod p). \end{matrix}

We return to the case p = 5. By (22) the characters for V and W are θ_{2} and θ_{1} respectively. Lemma 8.1 gives a quick way of computing dim Y^{H_{5}}. For example Y = ∧^{2}V ⊗ W has character

∧^{2}θ_{2} ⊗ θ_{1} ≃ 2θ_{4} ⊗ θ_{1} ≃ 2 ∑_{u,v} λ_{u,v}.

So dim Y^{H_{5}} = 2. By inspection Y^{H_{5}} has basis ∑(v_{1} ∧ v_{4})x_{0}, ∑(v_{2} ∧ v_{3})x_{0}. Since

T = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) : v_{i} ↦ ζ_{5}^{i^{2}}v_{i}; x_{i} ↦ ζ_{5}^{3i^{2}}x_{i}

acts on Y^{H_{5}} with trace ζ_{5}^{2} + ζ_{5}^{3}, reference to the character table for Γ (see Appendix A) shows that Y^{H_{5}} ≃ χ_{1} as a Γ-module. So the discrete covariants form a copy of M_{χ}. According to (5) a discrete covariant has integer weights if and only if it has degree d ≡ 1 (mod 5). We recall from §7 that M_{χ}(1) is a free k[c_{4}, c_{6}]-module with generators U and ∂D. We arrive at integer weight discrete covariants

\begin{aligned}U & = & a ∑(v_{1} ∧ v_{4})x_{0} + b ∑(v_{2} ∧ v_{3})x_{0} \\ H & = & (∂D/∂b) ∑(v_{1} ∧ v_{4})x_{0} − (∂D/∂a) ∑(v_{2} ∧ v_{3})x_{0}. \end{aligned}

We use Corollary 5.6(ii) to check that U and H are covariants. The substitutions envisaged by Proposition 3.7 are a^{5} ↦ ∏ λ_{i}, b ↦ 1 and

\begin{matrix}a ∑(v_{1} ∧ v_{4})x_{0} & ↦ & ∑ λ_{0}(v_{1} ∧ v_{4})x_{0} \\ ∑(v_{2} ∧ v_{3})x_{0} & ↦ & ∑(v_{2} ∧ v_{3})x_{0}. \end{matrix}

Since no denominators are introduced, U and H are covariants and indeed a basis for the k[c_{4}, c_{6}]-module of covariants for Y = ∧^{2}V ⊗ W.

We recognise U as the identity map, corresponding to our original choice of elliptic normal quintic. The covariant H describes a second curve, invariant under the same representation of the Heisenberg group. By analogy with the case of plane cubics we call H the Hessian, and note that the linear span of U and H in P(∧^{2}V ⊗ W) is a twist of the modular curve X(5).

We give another example, this time with Y = S^{2}V ⊗ W. It is immediate from Lemma 8.1 that dim Y^{H_{5}} = 3. The action of T on the basis

∑ v_{0}^{2}x_{0}, ∑ v_{1}v_{4}x_{0}, ∑ v_{2}v_{3}x_{0}

shows that Y^{H_{5}} ≃ ψ_{5} as a Γ-module. We recall from §7 that M_{ψ_{5}}(1) is a free k[c_{4}, c_{6}]-module with generators ∂_{E}^{2}c_{4}, ∂_{E}^{2}c_{6} and DE^{2}. We obtain integer weight discrete covariants

Q_{6} = ∑ x_{0}(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})
Q_{16} = ∑ x_{0}(25a^{3}b^{3}(a^{10} + b^{10})v_{0}^{2} − a(a^{15} − 39a^{10}b^{5} − 143a^{5}b^{10} + 3b^{15})v_{1}v_{4}
− b(3a^{15} + 143a^{10}b^{5} − 39a^{5}b^{10} − b^{15})v_{2}v_{3})
Q_{26} = D ∑ x_{0}(a^{2}b^{2}(a^{5} + 7b^{5})(7a^{5} − b^{5})v_{0}^{2}
− b^{4}(7a^{5} − b^{5})^{2}v_{1}v_{4} + a^{4}(a^{5} + 7b^{5})^{2}v_{2}v_{3})

Again we use Corollary 5.6(ii) to check that these are covariants. This time the substitutions envisaged by Proposition 3.7 are a^{5} ↦ ∏ λ_{i}, b ↦ 1 and

\begin{matrix}a^{3} ∑ v_{0}^{2}x_{0} & ↦ & ∑ λ_{0}λ_{1}λ_{4}v_{0}^{2}x_{0} \\ a ∑ v_{1}v_{4}x_{0} & ↦ & ∑ λ_{0}v_{1}v_{4}x_{0} \\ ∑ v_{2}v_{3}x_{0} & ↦ & ∑ v_{2}v_{3}x_{0}. \end{matrix}

Since no denominators are introduced, Q_{6}, Q_{16} and Q_{26} are covariants and indeed a basis for the k[c_{4}, c_{6}]-module of covariants for Y = S^{2}V ⊗ W.

We tabulate the results of these and similar calculations. In each case we give the character of Y^{H_{5}} as a Γ-module and the degrees of the generators for the k[c_{4}, c_{6}]-module of covariants for Y.

\begin{matrix}Y & Y^{H_{5}} & degrees \\ ∧^{2}V ⊗ W & χ_{1} & 1, 11 \\ V ⊗ ∧^{2}W^{∗} & χ_{2} & 13, 23 \\ ∧^{2}V^{∗} ⊗ W^{∗} & χ_{1} & 19, 29 \\ V^{∗} ⊗ ∧^{2}W & χ_{2} & 7, 17 \\ V^{∗} ⊗ S^{2}W & ψ_{4} & 2, 12, 22 \\ S^{2}V ⊗ W & ψ_{5} & 6, 16, 26 \\ V ⊗ S^{2}W^{∗} & ψ_{4} & 18, 28, 38 \\ S^{2}V^{∗} ⊗ W^{∗} & ψ_{5} & 14, 24, 34 \end{matrix}

It would be interesting to find geometric interpretations for as many of these covariants as possible.

9 Computing the invariants

Let C(ϕ) ⊂ P^{4} be an elliptic normal quintic. We give a practical algorithm for computing the invariants c_{4}(ϕ) and c_{6}(ϕ). We start with the covariants

\begin{matrix}U : ∧^{2}V ⊗ W & → & ∧^{2}V ⊗ W \\ u(a, b) & ↦ & a ∑(v_{1} ∧ v_{4})x_{0} + b ∑(v_{2} ∧ v_{3})x_{0} \\ P : ∧^{2}V ⊗ W & → & V^{∗} ⊗ S^{2}W \\ u(a, b) & ↦ & ∑ v_{0}^{∗}(abx_{0}^{2} + b^{2}x_{1}x_{4} − a^{2}x_{2}x_{3}) \end{matrix}

and write p_{i} = ⟨P (ϕ), v_{i}⟩ for the quadrics defining C(ϕ). Then

S : ∧^{2}V ⊗ W → S^{5}W; ϕ ↦ det(∂p_{i}/∂x_{j})/2

is a covariant of degree 10. According to [15, VIII.2.5], [11, §12], S(ϕ) is an equation for the secant variety of C(ϕ). In the case ϕ = u(a, b) we find

p_{i} = abx_{i}^{2} + b^{2}x_{i+1}x_{i+4} − a^{2}x_{i+2}x_{i+3}  (34)

and

S = a^{5}b^{5} ∑ x_{0}^{5} − a^{4}b(a^{5} − 3b^{5}) ∑ x_{0}^{3}x_{1}x_{4} + a^{3}b^{2}(a^{5} + 2b^{5}) ∑ x_{0}x_{1}^{2}x_{4}^{2}
+ a^{2}b^{3}(2a^{5} − b^{5}) ∑ x_{0}x_{2}^{2}x_{3}^{2} − ab^{4}(3a^{5} + b^{5}) ∑ x_{0}^{3}x_{2}x_{3}
+ (a^{10} − 16a^{5}b^{5} − b^{10}) ∏ x_{0}.

A calculation reveals that the partial derivatives of S may be written as quadrics in the p_{i}. Explicitly

∂S/∂x_{i} = 5a^{3}b^{3}p_{i}^{2} + a(a^{5} − 3b^{5})p_{i+1}p_{i+4} − b(3a^{5} + b^{5})p_{i+2}p_{i+3}.  (35)

We were led to this otherwise rather improbable looking identity by considering the covariant Q = Q_{6} found in §8. The following lemma shows that it is practical to compute Q for general ϕ.

Lemma 9.1 Let C(ϕ) be an elliptic normal quintic defined by quadrics p_{i} = ⟨P (ϕ), v_{i}⟩. Then the quartics p_{i}p_{j} for i ≤ j are linearly independent.

Proof. It suffices to treat the case ϕ = u(a, b) with ab ≠ 0. Then the p_{i} are given by (34). We suppose that there is a linear dependence relation

∑_{i≤j} λ_{ij}p_{i}p_{j} = 0.  (36)

The coefficient of x_{i}^{4} gives λ_{ii} = 0. Then the coefficient of x_{i}^{2}x_{j}^{2} gives λ_{ij} = 0 for all i < j. So (36) is the trivial relation and we are done. □

We arrive at covariants

Y  degree weights
U ∧^{2}V ⊗ W  1  (0, 0)
P V^{∗} ⊗ S^{2}W  2  (1, 0)
Q S^{2}V ⊗ W  6  (2, 1)
M  S^{5}V^{∗}  10  (5, 2)
N  S^{5}V  5  (1, 1)
N′  S^{5}V  30  (11, 6)

where M, N and N′ are the following determinants

V^{∗} ⊗ S^{2}W ⊂ V^{∗} ⊗ Hom(W^{∗}, W) M = det(P)
∧^{2}V ⊗ W ⊂ V ⊗ Hom(V^{∗}, W)  N = det(U)
S^{2}V ⊗ W ⊂ V ⊗ Hom(V^{∗}, W)  N′ = det(Q).

Now ⟨M, N⟩ and ⟨M, N′⟩ are invariants of degrees 15 and 40 respectively. By Proposition 6.1 we must have ⟨M, N⟩ = 0. It even turns out that N = 0. However computing a single example shows that ⟨M, N′⟩ is non-zero, and so proportional to c_{4}^{2}. To compute both c_{4} and c_{6} we need one further idea. Let

N_{t} := det(U + tQ)

where t is an indeterminate. Computing in the case ϕ = u(a, b) we find

⟨M, N_{t}⟩ = 20c_{4}t − 40c_{6}t^{3} + 4c_{4}^{2}t^{5}.

We extract the invariants c_{4} and c_{6} as coefficients of this polynomial. Subject to completing the proof of Theorem 1, we now have a practical algorithm for computing the Jacobian of an elliptic normal quintic.

Remark 9.2 Let C ⊂ P^{4} be an elliptic normal quintic. It is stated in [1, Example 22.6] (without proof or adequate reference) that C is the singular locus of its secant variety. The identity (35) together with [15, IV.5.6] makes it possible to give a computational proof of this result. We omit the details.

10 The invariant differential

Let C = C(ϕ) be an elliptic normal quintic. The aim of this section is to describe the invariant differential ω on C in terms of a covariant.

We begin by computing the covariants in the case Y = ∧^{2}W^{∗} ⊗ S^{2}W. Lemma 8.1 gives

∧^{2}θ_{4} ⊗ S^{2}θ_{1} ≃ 2θ_{3} ⊗ 3θ_{2} ≃ 6 ∑_{u,v} λ_{u,v}

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}

Now −I and T act with trace −6 and 1 respectively. Reference to the character table for Γ shows that Y^{H_{5}} ≃ χ_{4} as a Γ-module. We recall from §7 that M_{χ_{4}}(0) is a free k[c_{4}, c_{6}]-module with generators

U^{5}, U^{4}∂D, ∂^{5}c_{4}, ∂^{5}c_{6}, DU^{4}∂D, D^{2}U^{3}∂^{2}D.  (37)

These are the integer weight discrete covariants. We use Corollary 5.6(ii) to decide which are covariants. The substitutions envisaged by Proposition 3.7 are a^{5} ↦ ∏ λ_{i}, b ↦ 1 and

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

Invariance under T ensures that each discrete covariant has coefficient of ∑(x_{2}^{∗} ∧ x_{3}^{∗})x_{0}^{2} divisible by a^{4}. Arguing similarly for the other coefficients we see that no denominators are introduced. So for this choice of Y every integer weight discrete covariant is a covariant.

As in §9 we write p_{i} = ⟨P (ϕ), v_{i}⟩ for the quadrics defining C(ϕ). Taking ϕ = u(a, b), the Jacobian matrix (∂p_{i}/∂x_{j}) is

J(ϕ) = (\begin{matrix}2abx_{0} & b^{2}x_{4} & −a^{2}x_{3} & −a^{2}x_{2} & b^{2}x_{1} \\ b^{2}x_{2} & 2abx_{1} & b^{2}x_{0} & −a^{2}x_{4} & −a^{2}x_{3} \\ −a^{2}x_{4} & b^{2}x_{3} & 2abx_{2} & b^{2}x_{1} & −a^{2}x_{0} \\ −a^{2}x_{1} & −a^{2}x_{0} & b^{2}x_{4} & 2abx_{3} & b^{2}x_{2} \\ b^{2}x_{3} & −a^{2}x_{2} & −a^{2}x_{1} & b^{2}x_{0} & 2abx_{4} \end{matrix})

and the first of the covariants (37) works out as

Ω(ϕ) = (\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})  (38)

where

\begin{aligned}α_{i} & = & 5a^{4}bx_{i}^{2} − 10a^{3}b^{2}x_{i−1}x_{i+1} + (a^{5} − 3b^{5})x_{i−2}x_{i+2} \\ β_{i} & = & 5ab^{4}x_{i}^{2} − (3a^{5} + b^{5})x_{i−1}x_{i+1} + 10a^{2}b^{3}x_{i−2}x_{i+2}. \end{aligned}  (39)

Lemma 10.1 Let C = C(ϕ) be an elliptic normal quintic. Let J = J(ϕ) and Ω = Ω(ϕ) be the matrices defined above. Then

(i) The matrix product JΩ vanishes in the function field k(C). (ii) The skew-symmetric matrix Ω has rank 2 at all points on C.

Proof. As usual it suffices to treat the case ϕ = u(a, b).

(i) This may be checked by direct calculation, preferably using Gröbner bases. (ii) Since C ⊂ P^{4} is a smooth curve of codimension 3, the Jacobian matrix J has rank 3 on C. It follows by (i) that Ω has rank at most 2 on C. Alternatively we could observe that the 4 × 4 Pfaffians of Ω are the quartics ∂S/∂x_{i} and that by (35) these quartics vanish on C. It remains to show that Ω is non-zero on C. By (34) and (39) it suffices to show that

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(a, b)  (40)

is non-zero. But our assumption that ϕ = u(a, b) is an elliptic normal quintic gives D(a, b) ≠ 0 as required. □

Writing Ω = (Ω_{ij}), it follows that the invariant differential on C is

ω = \frac{x_{j}^{2}d(x_{i}/x_{j})}{Ω_{ij}}  for all i ≠ j.  (41)

According to (5) the covariant Ω has weights (p, q) = (2, 1). This observation has some important consequences (see Remark 12.3).

We outline a method for computing the covariant Ω. Let R = k[W^{∗}] = k[x_{0}, …, x_{4}]. Let I(ϕ) ◁ R be the ideal generated by the quadrics p_{i} = ⟨P (ϕ), v_{i}⟩ and let I′(ϕ) ◁ R be the ideal generated by the quartics ∂S/∂x_{i}. We saw in Remark 9.2 that an elliptic normal quintic is the singular locus of its secant variety. Thus I(ϕ) is the radical of I′(ϕ) and both ideals have codimension 3.

Lemma 10.2 Let C(ϕ) ⊂ P^{4} be an elliptic normal quintic. Then the ideal I′(ϕ) ◁ R is Gorenstein of codimension 3.

Proof. As usual it suffices to treat the case ϕ = u(a, b). As observed in the proof of Lemma 10.1, the quartics ∂S/∂x_{i} are the 4 × 4 Pfaffians of Ω. We obtain a complex of graded free R-modules

0 ⟶ R ⟶^{∂S/∂x_{i}} W^{∗} ⊗ R ⟶^{Ω} W ⊗ R ⟶^{∂S/∂x_{i}} R ⟶ 0.  (42)

Since I′(ϕ) has codimension 3 it follows by the Buchsbaum-Eisenbud structure theorem [5], [11, §10] that (42) is a minimal free resolution for R/I′(ϕ) and that I′(ϕ) is Gorenstein. □

It follows that we may compute the covariant Ω from the minimal resolution for R/I′(ϕ). This gives a practical method for computing the discriminant Δ, analogous to Salmon’s result for plane cubics cited in the introduction. Indeed the original 5 quadrics p_{i} together with the 10 quadrics ω_{ij} for i < j give a total of 15 elements of S^{2}W. Since dim S^{2}W = 15 we may form a 15×15 determinant. We claim that this determinant is a non-zero multiple of the discriminant. As usual it suffices to check in the case ϕ = u(a, b). But in that case the 15 × 15 matrix may be put in block diagonal form, with each 3 × 3 block a copy of (40). Our claim follows since Δ = D^{5}. It should be pointed out that the work of §9 makes this algorithm obsolete.

Remark 10.3 In view of the relation JΩ = 0 in k(C) there is a much more direct way of computing Ω = (ω_{ij}). But working in the function field means that we would only determine the ω_{ij} up to addition of linear combinations of the quadrics p_{i}. It is interesting to note that the above algorithm for computing the discriminant is not affected by such changes.

11 The quintic covariants

In the cases n = 2, 3, 4 we recall [2] that classical invariant theory provides us with equations for the covering map of degree n^{2} from a curve of genus one to its Jacobian. We would like to generalise this to the case n = 5. As a first step we compute the covariants in the case Y = S^{5}W. Lemma 8.1 gives

S^{5}θ_{1} ≃ λ_{0,0} ⊕ 5 ∑_{u,v} λ_{u,v}

and so dim Y^{H_{5}} = 6. A basis for Y^{H_{5}} is

∑ x_{0}^{5}, ∑ x_{0}^{3}x_{1}x_{4}, ∑ x_{0}^{3}x_{2}x_{3} ∑ x_{0}x_{1}^{2}x_{4}^{2}, ∑ x_{0}x_{2}^{2}x_{3}^{2}, ∏ x_{0}.

Since

R = (\begin{matrix}2 & 0 \\ 0 & 3 \end{matrix}) : x_{i} ↦ −x_{2i} and T = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) : x_{i} ↦ ζ_{5}^{3i^{2}}x_{i}

act on Y^{H_{5}} with trace −2 and 1, reference to the character table for Γ shows that Y^{H_{5}} ≃ ψ_{4} ⊕ ψ_{5} as a Γ-module. So the integer weight discrete covariants form a copy of M_{ψ_{4}}(0) ⊕ M_{ψ_{5}}(0). Each summand is a free k[c_{4}, c_{6}]-module with generators in degrees 10, 20 and 30. In the notation of §7 they are labelled ∂^{2}D, U∂c_{4}, D∂^{2}c_{4} and ∂_{E}∂_{F} c_{6}, EF, D^{2}∂_{E}^{2}c_{4}. We put

\begin{aligned}F & = & [∑ x_{0}^{5} − 30 ∏ x_{0}, 10 ∑ x_{0}^{3}x_{1}x_{4}, 10 ∑ x_{0}^{3}x_{2}x_{3}] \\ G & = & [∑ x_{0}^{5} + 20 ∏ x_{0}, 10 ∑ x_{0}x_{1}^{2}x_{4}^{2}, 10 ∑ x_{0}x_{2}^{2}x_{3}^{2}] \end{aligned}

and so obtain integer weight discrete covariants

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

We use Corollary 5.6(ii) to decide which of these are covariants. The substitutions envisaged by Proposition 3.7 are a^{5} ↦ ∏ λ_{i}, b ↦ 1 and

\begin{matrix}a^{5} ∑ x_{0}^{5} & ↦ & ∑ λ_{0}^{3}λ_{1}λ_{4}x_{0}^{5} \\ a^{4} ∑ x_{0}^{3}x_{1}x_{4} & ↦ & ∑ λ_{0}^{2}λ_{1}λ_{4}x_{0}^{3}x_{1}x_{4} \\ a^{3} ∑ x_{0}x_{1}^{2}x_{4}^{2} & ↦ & ∑ λ_{0}λ_{1}λ_{4}x_{0}x_{1}^{2}x_{4}^{2} \\ a^{2} ∑ x_{0}x_{2}^{2}x_{3}^{2} & ↦ & ∑ λ_{2}λ_{3}x_{0}x_{2}^{2}x_{3}^{2} \\ a ∑ x_{0}^{3}x_{2}x_{3} & ↦ & ∑ λ_{0}x_{0}^{3}x_{2}x_{3} \\ ∏ x_{0} & ↦ & ∏ x_{0} \end{matrix}

Thus the condition for an integer weight discrete covariant to be a covariant is that the coefficient of ∑ x_{0}^{5} be divisible by a^{5}. In particular F_{10}, F_{20}, G_{10} and G_{20} are not covariants. We put

\begin{aligned}F_{40} & := & c_{6}F_{10} + c_{4}F_{20} & G_{40} & := & c_{6}G_{10} + c_{4}G_{20} \\ F_{50} & := & c_{4}^{2}F_{10} + c_{6}F_{20} & G_{50} & := & c_{4}^{2}G_{10} + c_{6}G_{20}. \end{aligned}  (43)

Then the quintic covariants form a free k[c_{4}, c_{6}]-module with basis

F_{10} − G_{10}, F_{20} − G_{20}, F_{30}, G_{30}, F_{40}, F_{50}.  (44)

In particular the secant variety S ∝ F_{10} − G_{10} is the only quintic covariant of degree 10. A calculation in the case ϕ = u(a, b) confirms that the quintics (44) remain linearly independent when we specialise to ϕ an elliptic normal quintic. The quintic hypersurfaces through a fixed elliptic normal quintic have previously been studied by Hulek [15]. We give some examples to illustrate how our work with covariants picks out the quintic hypersurfaces to which Hulek was able to attach a geometric meaning.

Lemma 11.1 Let C(ϕ) ⊂ P^{4} be an elliptic normal quintic. Then

(i) The Heisenberg invariant quintics vanishing on C have basis F_{10} − G_{10}, F_{20} − G_{20}, F_{30}, G_{30}, F_{40}. (ii) The Heisenberg invariant quintics singular along C have basis F_{10} − G_{10}, F_{20} − G_{20}, F_{30} − G_{30}.

Proof. Hulek [15, VI] shows that these spaces have dimension 5 and 3 respectively. The latter has basis ∑ p_{0}^{2}x_{0}, ∑ p_{1}p_{4}x_{0}, ∑ p_{2}p_{3}x_{0}.

(i) We compute the covariants in the case Y = V ⊗S^{3}W. We find Y^{H_{5}} ≃ χ_{1}⊕ ψ_{3} as a Γ-module. By Molien’s theorem the k[c_{4}, c_{6}]-module of integer weight discrete covariants is generated in degrees 13, 23 and 8, 18, 28, 28, 38. All of these are covariants. Substituting v_{i} ↦ p_{i} we obtain quintic covariants vanishing on C. They have degrees 15, 25 and 10, 20, 30, 30, 40. Comparison with (44) shows the first two covariants must vanish. We conclude by a direct calculation in the case ϕ = u(a, b). (ii) In §8 we computed the covariants for Y = S^{2}V ⊗ W and so obtained a free k[c_{4}, c_{6}]-module generated in degrees 6, 16 and 26. Substituting v_{i} ↦ p_{i} we obtain the quintic covariants singular along C. We conclude by a direct calculation in the case ϕ = u(a, b). □

Remark 11.2 The condition for a quintic covariant to vanish on C may be expressed in terms of a quintic contravariant (i.e. a covariant with Y = S^{5}W^{∗}) of degree 10. The contraction of this contravariant with F_{50} is a non-zero multiple of the discriminant Δ.

Lemma 11.3 Let C = C(ϕ) be an elliptic normal quintic. Let Tan C be the locus of tangent lines of C. Let F be the locus of singular lines of rank 3 quadrics containing C. Then

(i) The quintics containing Tan C have basis F_{10} − G_{10}, F_{20} − G_{20}, F_{30} + G_{30}. (ii) The quintics containing F have basis F_{10} − G_{10}, F_{20} − G_{20}, F_{30} − G_{30}. (iii) Tan C ∪ F is the complete intersection F_{10} − G_{10} = F_{20} − G_{20} = 0.

Proof. (i) We recall [15, VII] that Tan C is an irreducible surface of degree 10, and that the space of quintics containing Tan C has dimension 3. We compute a basis by working in the case ϕ = u(a, b). Evaluating the rows of (38) at the rational point 0 = (0 : a : b : −b : −a) we obtain 5 points on the tangent line to C at 0. The quintic covariants listed are the only ones to vanish at all these points. (ii) We recall [15, IV.4.7] that F is an irreducible surface of degree 15. By [15, VIII.3.5] the quintics containing F are precisely the Heisenberg invariant quintics singular along C. We are done by Lemma 11.1(ii). (iii) The proofs of (i) and (ii) show that Tan C ∪ F is a surface of degree 25 contained in the complete intersection F_{10} − G_{10} = F_{20} − G_{20} = 0 also of degree 25. □

Remark 11.4 The character table for Γ gives S^{2}ψ_{4} ≃ S^{2}ψ_{5} ≃ ψ_{1} ⊕ ψ_{3}. But according to the table below, the trivial representation ψ_{1} appears only once in the decomposition of (S^{10}W)^{H_{5}} as a Γ-module. It follows that the quintic covariants satisfy a quadratic relation. We find

F_{10}F_{50} + 5F_{20}F_{40} + 3F_{30}^{2} = G_{10}G_{50} + 5G_{20}G_{40} + 3G_{30}^{2}.  (45)

We return to the complete intersection of Lemma 11.3(iii). Substituting F_{10} = G_{10} and F_{20} = G_{20} in (45) and recalling our definitions (43) we find (F_{30} +G_{30})(F_{30} −G_{30}) = 0. These factors correspond to Tan C and F respectively.

As explained at the start of this section, we would like to find covariants describing the covering map from an elliptic normal quintic to its Jacobian. For this we would need a practical algorithm for computing certain covariants of order r = 5, 10 and 15. Unfortunately the dimension of (S^{r}W)^{H_{5}} grows rapidly with r and this makes it difficult even to classify the discrete covariants. For example there are 41 covariants of order 10, all but 2 of which vanish on C.

Some hope is afforded by the work of Horrocks and Mumford [14, §2] who work directly^{5} with characters of the extended Heisenberg group H_{5}^{+}. In fact they give an inductive procedure for computing (virtual) characters π_{r} of Γ such that

S^{r}θ_{1} ≃ \{ \begin{matrix}π_{r} ⊗ θ_{r} & if r ≢ 0 (mod 5) \\ 1 ⊕ π_{r} ⊗ (θ_{1} ⊗ θ_{1}^{∗}) & if r ≡ 0 (mod 5). \end{matrix}

Here we have used Corollary 4.7 to view θ_{t} as a character for H_{5}^{+}, and the exact sequence (23) to view characters of Γ as characters of H_{5}^{+}. For 5 | r it follows that (S^{r}W)^{H_{5}} ≃ 1 ⊕ π_{r} as a Γ-module. Implementing their algorithm we find

\begin{matrix}r & dim(S^{r}W)^{H_{5}} & ψ_{1} & ψ_{2} & ψ_{3} & ψ_{4} & ψ_{5} & χ_{1} & χ_{2} & χ_{3} & χ_{4} \\ 5 & 6 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 10 & 41 & 1 & 1 & 4 & 1 & 1 & 1 & 1 & 0 & 1 \\ 15 & 156 & 2 & 7 & 6 & 6 & 6 & 1 & 1 & 5 & 6 \\ 20 & 426 & 6 & 16 & 22 & 11 & 11 & 6 & 6 & 12 & 18 \\ 25 & 951 & 6 & 33 & 43 & 28 & 28 & 16 & 16 & 27 & 43 \\ 30 & 1856 & 21 & 68 & 83 & 48 & 48 & 27 & 27 & 59 & 86 \end{matrix}

This gives us partial information concerning the discrete covariants. But at present we do not know of any practical algorithm in the style of §9 giving equations for the covering map.

5This is in contrast with our approach of working with H_{5} and Γ separately.

12 Computing the Jacobian

Let k be a field with char k ≠ 2, 3, 5. We drop our assumption that k is algebraically closed. To complete the proof of Theorem 1 we prove

Proposition 12.1 Let ϕ ∈ ∧^{2}V ⊗ W with C(ϕ) ⊂ P^{4} an elliptic normal quintic. Then C(ϕ) has Jacobian

y^{2} = x^{3} − 27c_{4}(ϕ)x − 54c_{6}(ϕ).  (46)

The invariants c_{4} and c_{6} were defined in §2. Their scalings were fixed §6. In the same section we gave a proof of Theorem 1 under the assumption that k is algebraically closed. So it remains to rule out the possibility that (46) might be a non-trivial quadratic twist^{6} of the Jacobian. By analogy with the cases n = 2, 3, 4 the only issue here should be the appropriate scaling of the invariants.

Let E be the elliptic curve y^{2} = x^{3} + Ax + B. If we embed E ⊂ P^{4} via (x_{0} : x_{2} : x_{3} : x_{4} : x_{5}) = (1 : x : y : x^{2} : xy) then E = C(ϕ_{A,B}) where

ϕ_{A,B} = (\begin{matrix}0 & Bx_{0} & x_{5} & Ax_{0} + x_{4} & −x_{3} \\ & 0 & −x_{4} & −x_{3} & x_{2} \\ & & 0 & −x_{2} & 0 \\ & − & & 0 & −x_{0} \\ & & & & 0 \end{matrix}) .  (47)

Running our algorithm, as described in §9, we find

c_{4}(ϕ_{A,B}) = −48A and c_{6}(ϕ_{A,B}) = −864B.  (48)

This proves Proposition 12.1 in the case ϕ = ϕ_{A,B} and shows that our covariants are scaled appropriately. (See Remark 6.3 for an alternative approach.) As comparison of (46) and (48) reminds us, we are always free to rescale (c_{4}, c_{6}) ↦ (λ^{4}c_{4}, λ^{6}c_{6}) for any λ ∈ k^{∗}.

Our proof of Proposition 12.1 relies on the following observation.

Lemma 12.2 Let C(ϕ) ⊂ P^{4} be an elliptic normal quintic. Then Stab(ϕ) ⊂ SL(V) × SL(W) acts on C(ϕ) as the 5-torsion of its Jacobian.

6The reader should modify this statement as appropriate when j = 0, 1728

Proof. For the duration of the proof we assume that k is algebraically closed. Let (g_{V} , g_{W}) ∈ SL(V) × GL(W) with (g_{V} , g_{W})ϕ = ϕ and let α be the automorphism of C(ϕ) induced by g_{W}. We must show that α is a translation map if and only if det g_{W} = 1. If α is a translation map we reduce to the case ϕ = u(a, b). In the notation of §5 we have (g_{V} , g_{W}) = (θ_{V} (h), θ_{W} (h)) for some h ∈ H_{5}. So det g_{V} = det g_{W} = 1 as required. If α is not a translation map then it has a finite number of fixed points. We reduce to the case ϕ = ϕ_{A,B} with fixed point 0 = (0 : 0 : 0 : 0 : 1). The automorphism [−1] is given by

g_{V} = Diag(1, −1, 1, −1, 1) g_{W} = Diag(−1, −1, 1, −1, 1).

If A = 0 or B = 0 there are additional automorphisms given by

g_{V} = Diag(ζ^{2}, ζ, 1, ζ^{−1}, ζ^{−2}) g_{W} = Diag(ζ^{3}, ζ, 1, ζ^{−1}, ζ^{−2})

with ζ^{6} = 1 or ζ^{4} = 1. In each case we find det g_{V} = 1 and det g_{W} ≠ 1. □

Remark 12.3 In §10 we gave a description of the invariant differential in terms of a covariant Ω. Since Ω has weights (2, 1) and the translation maps are characterised by the fact they preserve the invariant differential, this leads to an alternative proof of Lemma 12.2. In particular we avoid the need to split into cases.

It is known that the quotient of an elliptic normal quintic by the 5-torsion of its Jacobian is isomorphic to its Jacobian. So taking our universal family of elliptic normal quintics parametrised by ∧^{2}V ⊗W \{Δ = 0} and quotienting out by the action of SL(V)×SL(W) we expect to obtain the universal family of Jacobians. We are therefore confident that everything in [2] generalises to the case n = 5, even if (as acknowledged in §11) we are yet to find the covariants that describe the covering map from C(ϕ) to its Jacobian.

We take a slightly different approach, still relying on Lemma 12.2. Let \overline{k} be a separable closure for k. We put \overline{V} = V ⊗_{k} \overline{k} and \overline{W} = W ⊗_{k} \overline{k}. Since char k ≠ 2, 3, 5 it turns out that all our work in the case of an algebraically closed field carries over to a separably closed field.

Lemma 12.4 Let ϕ, ϕ′ ∈ ∧^{2}V ⊗ W with C(ϕ), C(ϕ′) ⊂ P^{4} elliptic normal quintics. Suppose that (g_{V} , g_{W})ϕ = ϕ′ for some (g_{V} , g_{W}) ∈ SL(\overline{V}) × GL(\overline{W}). If det g_{W} ∈ k then the Jacobians of C(ϕ) and C(ϕ′) are isomorphic over k.

Proof. Since det g_{W} ∈ k the cocycle σ ↦ (σ(g_{V})g_{V}^{−1}, σ(g_{W})g_{W}^{−1}) takes values in SL(\overline{V})×SL(\overline{W}). By Lemma 12.2 the cocycle σ ↦ σ(g_{W})g_{W}^{−1} acts on C(ϕ′) via translations. So g_{W} induces an isomorphism of Jacobians. □

Proof of Proposition 12.1. Let ϕ ∈ ∧^{2}V ⊗ W with C(ϕ) ⊂ P^{4} an elliptic normal quintic. Let A = −27c_{4}(ϕ) and B = −54c_{6}(ϕ). Let ϕ′ = ϕ_{A,B} be given by (47). Proposition 6.1 tells us that C(ϕ) and C(ϕ′) are isomorphic over \overline{k}. So by Lemma 2.5(i) there exists (g_{V} , g_{W}) ∈ SL(\overline{V}) × GL(\overline{W}) with (g_{V} , g_{W})ϕ = ϕ′. We apply c_{4} and c_{6} to both sides. For the left-hand side we note that c_{4} and c_{6} have weights (8, 4) and (12, 6). For the right-hand side we use (48). We obtain

\begin{aligned}(det g_{W})^{4}c_{4}(ϕ) & = & 6^{4}c_{4}(ϕ) \\ (det g_{W})^{6}c_{6}(ϕ) & = & 6^{6}c_{6}(ϕ). \end{aligned}

If c_{4}(ϕ) ≠ 0 and c_{6}(ϕ) ≠ 0 then det g_{W} = ±6 and we may apply Lemma 12.4. In the remaining cases we compose (g_{V} , g_{W}) with one of the automorphisms of C(ϕ′) described in the proof of Lemma 12.2. So in all cases we may suppose that det g_{W} = 6. Lemma 12.4 tells us that the Jacobians of C(ϕ) and C(ϕ′) are isomorphic. Since the latter is an elliptic curve with Weierstrass equation (46), this completes the proof of Proposition 12.1. □

Finally Propositions 6.1 and 12.1 combine to prove Theorem 1.

Remark 12.5 As noted above, the invariants c_{4} and c_{6} have weights (8, 4) and (12, 6). So if we change our co-ordinates on P^{4}, or change our basis for the space of quadrics defining our curve, then c_{4} and c_{6} change in a way that does not alter the Jacobian.

13 A practical algorithm

Let k be a field with char k ≠ 2, 3, 5. Let C ⊂ P^{4} be an elliptic normal quintic defined by quadrics p_{1}, …, p_{5} ∈ k[x_{1}, …, x_{5}]. We describe in detail our algorithm for computing the Jacobian of C and then give some examples in the case k = Q.

The first step is to write the quadrics p_{i} as the 4 × 4 Pfaffians of a 5 × 5 skew-symmetric matrix. This is done by computing the syzygies between the p_{i} and then appealing to the work of Buchsbaum and Eisenbud, [5], [11, §10].

Algorithm 13.1 (Writing the quadrics as Pfaffians)

INPUT: A vector p = (p_{i})_{i=1}^{5} of quadrics.
OUTPUT: A skew-symmetric matrix ϕ = (ϕ_{ij})_{i=1}^{5} of linear forms.
1.  begin

2. choose (Λ_{i})_{i=1}^{5} a basis for {Λ = (λ_{jk})_{j,k=1}^{5} : ∑_{j,k} λ_{jk}x_{j}p_{k} = 0};

3.  A = matrix(5,5,i,k,sum(j=1,5,λ_{i,jk}x_{j}));
4.    solve for B = (b_{ij})_{i,j=1}^{5} such that BA + A^{T} B^{T} = 0;
5.  ϕ = BA;
6.    check pfaffian(ϕ^{{i}}) ∝ p_{i};
7.  end

Remarks. (i) The matrix A is a matrix of linear forms, whereas the matrix B is a matrix of constants. (ii) We write ϕ^{{i}} for the 4 × 4 matrix obtained by deleting the ith row and ith column of ϕ. See [11, §1] for a definition of the Pfaffian. (iii) If the p_{i} define an elliptic normal quintic then the vector space of step 2 has dimension 5, the matrix B of step 4 is uniquely determined up to scalars, and the condition of step 6 is satisfied. If any of these conditions fail then Algorithm 13.1 should output an error message to the effect that the p_{i} do not define an elliptic normal quintic. (iv) The constants of proportionality in step 6 are uniquely determined up to squares. In general we should not expect to obtain equality.

Next we make explicit the algorithm described in §9.

Algorithm 13.2 (Computing the invariants)

INPUT: A skew-symmetric matrix ϕ = (ϕ_{ij})_{i,j=1}^{5} of linear forms.
OUTPUT: Invariants c_{4} and c_{6}.
1.  begin
2.  p = vector(5,i,(-1)^{i}*pfaffian(ϕ^{{i}}));
3.  S = det(matrix(5,5,i,j,∂p_{i}/∂x_{j}));
4.  q = vector(5,i,

5. solve for ε_{jk} such that ∂S/∂x_{i} = ∑_{j≤k} ε_{jk}p_{j}p_{k};

6.      sum(j=1,5,sum(k=j,5,ε_{jk}x_{j}x_{k})));
7.  U = matrix(5,5,i,j,sum(k=1,5,polcoeff(ϕ_{jk}, x_{i})*x_{k}));
8.  P = matrix(5,5,i,j,sum(k=1,5,polcoeff(∂p_{k}/∂x_{j}, x_{i})*x_{k}));
9.  Q = matrix(5,5,i,j,∂q_{i}/∂x_{j});
10. F = sum(I,I!*polcoeff(det(P),x_{I})*polcoeff(det(U+tQ),x_{I}));

11. c_{4} = polcoeff(F,t)/40;
12. c_{6} = -polcoeff(F,t^{3})/320;
13.   check c_{4}^{2} = polcoeff(F,t^{5})/128;
14. end

Remarks. (i) For ϕ an elliptic normal quintic, Lemma 9.1 shows that the quartics {p_{i}p_{j} | i ≤ j} are linearly independent. If there is a dependence relation Algorithm 13.2 should output an error message to the effect that the C(ϕ) is not elliptic normal quintic. (ii) We employ the multi-index notation I = (i_{1}, … i_{5}) with x_{I} = x_{1}^{i_{1}} … x_{5}^{i_{5}} and I! = i_{1}! … i_{5}!. The sum of step 10 is taken over all 126 multi-indices of degree 5.

Finally Theorem 1 tells us that the Jacobian of C is

y^{2} = x^{3} − 27c_{4}x − 54c_{6}.

Remark 13.3 We test our algorithm by applying it in cases where we already know the Jacobian. These include ϕ = u_{1}(λ_{0}, …, λ_{4}) given by (13) and ϕ = ϕ_{A,B} given by (47). We may also disguise our examples by changing co-ordinates on P^{4}, or changing our basis for the space of quadrics. These examples illustrate the need for a reduction algorithm. It is hoped that this will be the subject of future work.

We give some examples in the case k = Q.

Example 13.4 In [8, Appendix B] we constructed an elliptic normal quintic violating the Hasse principle

\begin{aligned}p_{1} & = & 11x_{1}^{2} − x_{2}^{2} + 5x_{3}^{2} + x_{4}^{2} − 5x_{5}^{2} \\ p_{2} & = & x_{2}^{2} + 10x_{2}x_{3} + 5x_{3}^{2} + 2x_{1}x_{4} − x_{2}x_{4} − 5x_{3}x_{4} − 5x_{2}x_{5} − 5x_{3}x_{5} \\ p_{3} & = & x_{2}^{2} + 2x_{2}x_{3} + 5x_{3}^{2} + x_{2}x_{4} + x_{3}x_{4} + 2x_{1}x_{5} + x_{2}x_{5} + 5x_{3}x_{5} \\ p_{4} & = & 11x_{1}x_{2} + 55x_{1}x_{3} + 2x_{2}x_{4} + 2x_{4}^{2} − 10x_{3}x_{5} + 10x_{5}^{2} \\ p_{5} & = & 11x_{1}x_{2} + 11x_{1}x_{3} − 2x_{3}x_{4} + 2x_{2}x_{5} − 4x_{4}x_{5}. \end{aligned}

Algorithm 13.1 writes these quadrics as the 4 × 4 Pfaffians of

ϕ = (\begin{matrix}0 & −2x_{5} & 2x_{4} & −x_{2} − x_{3} & x_{2} + 5x_{3} \\ 2x_{5} & 0 & −11x_{1} & x_{3} + x_{5} & x_{2} + x_{4} \\ −2x_{4} & 11x_{1} & 0 & −x_{2} + x_{4} & −5x_{3} + 5x_{5} \\ x_{2} + x_{3} & −x_{3} − x_{5} & x_{2} − x_{4} & 0 & −x_{1} \\ −x_{2} − 5x_{3} & −x_{2} − x_{4} & 5x_{3} − 5x_{5} & x_{1} & 0 \end{matrix}) .

Algorithm 13.2 computes the invariants

c_{4}(ϕ) = 2^{12} × 9384400 c_{6}(ϕ) = 2^{18} × 28748141000.

From these we compute a minimal Weierstrass equation for the Jacobian

275B3 y^{2} + y = x^{3} + x^{2} − 195508x − 33338481.

Conditional on the Birch Swinnerton-Dyer conjecture, Cremona’s tables [6] show that this is the elliptic curve over Q of smallest conductor whose Tate-Shafarevich group contains an element of order 5.

Example 13.5 C. Wuthrich [27] has constructed an element of the Tate-Shafarevich group of order 5 using methods unrelated to [8], [9]. Written as an elliptic normal quintic his example is

p_{1} = 3x_{1}^{2} + x_{1}x_{5} − x_{2}x_{4} − x_{3}^{2}
p_{2} = 17x_{1}^{2} − 10x_{1}x_{3} + 7x_{1}x_{5} − 7x_{2}x_{4} − 4x_{2}x_{5} + 4x_{3}x_{4}
p_{3} = 215x_{1}^{2} − 16x_{1}x_{2} − 80x_{1}x_{3} + 16x_{1}x_{4} + 81x_{1}x_{5} − 49x_{2}x_{4}
− 28x_{2}x_{5} − 16x_{3}x_{5} − 16x_{4}^{2}
p_{4} = 60x_{1}^{2} + 48x_{1}x_{2} − 34x_{1}x_{3} − 24x_{1}x_{4} + 20x_{1}x_{5} − 8x_{2}^{2} − 5x_{2}x_{3}
+ 16x_{2}x_{5} − 14x_{3}x_{5} − 8x_{4}x_{5} − 12x_{2}x_{4}
p_{5} = 18x_{1}^{2} + 9x_{1}x_{3} − 4x_{1}x_{4} − 4x_{1}x_{5} − 4x_{2}x_{3} − 8x_{2}x_{4} − 6x_{2}x_{5}
+ 8x_{3}x_{5} − 4x_{5}^{2}.

Algorithms 13.1 and 13.2 compute the invariants

c_{4}(ϕ) = 2^{44} × 151009, c_{6}(ϕ) = −2^{66} × 34871057.

From these we compute a minimal Weierstrass equation for the Jacobian

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

It has square-free conductor N = 1289106508910. We are able to confirm that the Jacobian does not admit a rational 5-isogeny. So the methods of [8],

[9] do not apply.

14 A special case

Let k be a field with char k ≠ 5. Let C ⊂ P^{4} be an elliptic normal quintic defined over k whose Jacobian has a k-rational point of order 5. We recall an algorithm of C. O’Neil [19] for computing the Jacobian of C, and compare it with our general algorithm.

First, to give a supply of examples, we take x_{0}, y_{0} linear forms on P^{4}, defined over k, and M ∈ GL_{5}(k) with M^{5} = bI for some b ∈ k^{∗}. We assume throughout that the image of M in PGL_{5} has order 5. Let x_{i} = M^{i}x_{0} and y_{i} = M^{i}y_{0} for i = 0, …, 4. We define C[M; x_{0}, y_{0}] ⊂ P^{4} with equations the 4 × 4 Pfaffians of

(\begin{matrix}0 & x_{3} & by_{1} & y_{4} & x_{2} \\ & 0 & x_{4} & y_{2} & y_{0} \\ & & 0 & bx_{0} & y_{3} \\ & − & & 0 & x_{1} \\ & & & & 0 \end{matrix}) .  (49)

Generically this will be an elliptic normal quintic. Explicitly writing out the Pfaffians it is clear that C[M; x_{0}, y_{0}] is preserved by the action of M on P^{4}. In other words M corresponds to a k-rational point of order 5 on the Jacobian. Conversely

Lemma 14.1 Let C ⊂ P^{4} be an elliptic normal quintic, defined over k, whose Jacobian has a k-rational point of order 5. Then C = C[M; x_{0}, y_{0}] for some x_{0}, y_{0} linear forms on P^{4}, defined over k, and M ∈ GL_{5}(k).

Proof. Proposition 1.1(i) gives C = C(ϕ) for some ϕ ∈ ∧^{2}V ⊗W. The action of a k-rational point of order 5 corresponds to (g_{V} , g_{W}) ∈ GL(V) × GL(W) with (g_{V} , g_{W})ϕ = ϕ. We choose a basis for V such that

g_{V} = (\begin{matrix}0 & a^{−1} & 0 & 0 & 0 \\ 0 & 0 & a & 0 & 0 \\ 0 & 0 & 0 & a^{−1} & 0 \\ 0 & 0 & 0 & 0 & a^{−1} \\ −a & 0 & 0 & 0 & 0 \end{matrix})

for some a ∈ k^{∗}. Then we take M = g_{W} and b = a^{2}. Writing out the identity (g_{V} , g_{W})ϕ = ϕ it follows that ϕ takes the form (49). □

The Jacobian of C[M; x_{0}, y_{0}] must belong to the universal family of elliptic curves with a specified k-rational point of order 5. Choosing a co-ordinate λ on X_{1}(5) this family has equations

D_{λ} : y^{2} + (1 − λ)xy − λy = x^{3} − λx^{2}.

The cusps are at λ = 0, ∞, ϕ^{5}, \overline{ϕ^{5}} where ϕ = 1 + ζ_{5} + ζ_{5}^{4} is the golden ratio. In [8], [9] we wrote C_{λ} for the family of curves 5-isogenous to D_{λ}, and saw that C_{λ} ≃ D_{η(λ)} over k(μ_{5}) where η is the involution λ ↦ (ϕ^{5}λ + 1)/(λ − ϕ^{5}). We give two methods for computing λ = λ(M; x_{0}, y_{0}).

Proposition 14.2 Let C ⊂ P^{4} be an elliptic normal quintic preserved by M ∈ GL_{5}(k). Let Q be a quadratic form whose translates under M define C. Let B(u, v) = Q(u + v) − Q(u) − Q(v) be the corresponding bilinear form. Let v_{0}, …, v_{4} be the eigenvectors of M, ordered such that the corresponding eigenvalues are in geometric progression. Then C has Jacobian D_{λ} where

λ = η(λ′) and λ′ = ∏_{i=0}^{4} \frac{Q(v_{i})}{B(v_{i+1}, v_{i+4})}.

Proof. This is [19, Theorem 4.2]. The only modification we have made is to put the answer in Weierstrass form. □

Remark 14.3 The proof of Proposition 14.2 exhibits an isomorphism between the Jacobian of C and D_{λ}. This isomorphism is defined over a Galois extension of k. It maps the subgroup generated by M to the subgroup generated by (x, y) = (0, 0). But an elliptic curve equipped with a rational point of order 5 has no automorphisms. So comparing the above isomorphism with its Galois conjugates shows that it must after all be defined over k.

Our second method takes greater account of the form of the equations for C[M; x_{0}, y_{0}]. We define polynomials F (x) = x^{5} + 3x^{4} + 4x^{3} + 2x^{2} + x and G(x) = x^{4} − 2x^{3} + 4x^{2} − 3x + 1 satisfying

ϕ^{5}F (x) + G(x) = (ϕx + 1)^{5} and F (x) − ϕ^{5}G(x) = (x − ϕ)^{5}. (50)

Proposition 14.4 Let C = C[M; x_{0}, y_{0}] be an elliptic normal quintic, with y_{0} = ∑_{i=0}^{4} a_{i}x_{i}. Let ω be a symbol^{7} such that

∏_{β^{5}=b}(a_{0} + βa_{1} + … + β^{4}a_{4}) = (a_{0} + ω)^{5}.  (51)

Then C[M; x_{0}, y_{0}] has Jacobian D_{λ} where λ = F (a_{0} + ω)/G(a_{0} + ω).

7For an introduction to symbolic notation see [12].

Proof. Let e_{0}, …, e_{4} be linear forms on P^{4} defined over K = k(μ_{5}, \sqrt[5]{b}) with Me_{i} = ζ_{5}^{i}\sqrt[5]{b}e_{i} for i = 0, …, 4. We further arrange that

x_{0} = e_{0} + … + e_{4} and y_{0} = l_{0}e_{0} + … + l_{4}e_{4}

where the l_{i} are the linear forms on the left hand side of (51). According to [9, Proposition 2.8], C must have equations of the form

τ_{i}e_{i}^{2} + e_{i+1}e_{i+4} − τ_{i+2}τ_{i+3}e_{i+2}e_{i+3} = 0.

Comparing a few coefficients we find

τ_{i} = \frac{(ϕl_{i} + 1)(l_{i} − ϕ)}{(l_{i+1} − ϕ)(l_{i+4} − ϕ)}.

Continuing to work over K, the Jacobian of C is isomorphic to C_{λ′} where

λ′ = ∏_{i=0}^{4} (\frac{ϕl_{i} + 1}{l_{i} − ϕ})  by [9, Proposition 2.12]

= (\frac{ϕ(a_{0} + ω) + 1}{a_{0} + ω − ϕ})^{5}  by (51)
= \frac{ϕ^{5}F (a_{0} + ω) + G(a_{0} + ω)}{F (a_{0} + ω) − ϕ^{5}G(a_{0} + ω)} by (50)

Putting λ = η(λ′) = F (a_{0} + ω)/G(a_{0} + ω) the Jacobian of C is isomorphic to D_{λ}. Remark 14.3 shows that this isomorphism is defined over k. □

We give an example in the case k = Q.

Example 14.5 Let y_{0} = x_{0} + x_{2} − x_{3} and b = 3. Then C = C[M; x_{0}, y_{0}] is defined by the quadric

Q = 12x_{0}^{2} − 9x_{0}x_{1} − 3x_{0}x_{4} + 3x_{1}x_{2} + 4x_{1}x_{4} − x_{2}^{2} − x_{3}x_{4}

and its translates under M. Using either Proposition 14.2 or Proposition 14.4 the Jacobian is D_{λ} with λ = 8. For the latter we make the substitutions

(ω, ω^{2}, ω^{3}, ω^{4}, ω^{5}) ↦ (0, 3/2, 0, 9, −18).

On the other hand Algorithm 13.2 gives

c_{4} = −48^{4} × 1055, c_{6} = 48^{6} × 15535.

Each of these methods shows that the Jacobian is

50B2 y^{2} + xy + y = x^{3} + x^{2} + 22x − 9.

Remark 14.6 Suppose we are given 5 quadrics defining an elliptic normal quintic and are told that its Jacobian has a k-rational point of order 5. To compute the Jacobian using Proposition 14.2 or Proposition 14.4 we must first find the matrix M. We do not know a good way of doing this. So even in this special case where the Jacobian has a k-rational point of order 5, our general algorithm is to be preferred.

A Character table for Γ = SL_{2}(Z/5Z).

We write ϕ = 1 + ζ_{5} + ζ_{5}^{4} and \overline{ϕ} = 1 − ϕ = 1/ϕ for its conjugate.

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

B How big are the invariants?

Let c_{4} and c_{6} be the invariants for the elliptic normal quintic, as described in Theorem 1. The smaller invariant c_{4} is a homogeneous polynomial of degree 20 in 50 variables. We aim to improve on the trivial upper bound

|c_{4}| ≤ \binom{69}{20} ≈ 1.156 × 10^{17}.

We first consider the case n = 3. The invariant subring for the action of SL(W) on k[S^{3}W] is generated by homogeneous polynomials of degrees 4 and 6 respectively. As explained in [23, §4.4-5] there are upper bounds

|c_{4}| ≤ dim(S^{4}(S^{3}W))^{T} = 25
|c_{6}| ≤ dim(S^{6}(S^{3}W))^{T} = 103 (52)

where T ⊂ SL(W) is the subgroup of diagonal matrices. But knowing explicit expressions for c_{4} and c_{6} we find |c_{4}| = 25 and |c_{6}| = 103. In other words, equality holds in (52).

Returning to the case n = 5, there are upper bounds

|c_{4}| ≤ dim(S^{20}(∧^{2}V ⊗ W))^{T}
|c_{6}| ≤ dim(S^{30}(∧^{2}V ⊗ W))^{T}

where the maximal torus T is now the subgroup of SL(V)×SL(W) consisting of pairs of diagonal matrices. Even to compute the first of these estimates is a non-trivial combinatorial problem.

Definition B.1 Let f_{r}(x_{1}, …, x_{5}) be the number of ways of placing 8r balls, equally distributed among r colours, in a 5 × 5 array of boxes such that

(i) No balls are placed on the leading diagonal. (ii) The arrangement is symmetric about the leading diagonal. (iii) The row totals are x_{1}, …, x_{5}.

It is clear that f_{r}(x_{1}, …, x_{5}) = 0 unless ∑ x_{i} = 8r and 0 ≤ x_{i} ≤ 4r. Some examples are

\begin{aligned}f_{2}(8, 8, 0, 0, 0) & = & 1 \\ f_{2}(4, 3, 3, 3, 3) & = & 2454. \end{aligned}

Let X = {x ∈ Z^{5} | f_{2}(x) ≠ 0}. We find that |X| = 3195 and list the pairs (x, f_{2}(x)) for x ∈ X. It is feasible to store the function f_{1} in a look-up table. Finally we compute

dim S^{20}(∧^{2}V ⊗ W)^{T} = f_{5}(8, 8, 8, 8, 8)
= ∑_{x,y∈X} f_{1}(8 − x − y)f_{2}(x)f_{2}(y)
= 172 786 883 860
≈ 1.728 × 10^{11}.

Although this is only an upper bound for |c_{4}|, comparison with the case n = 3 suggests that it is not far from the truth. These observations support our belief that the invariants in the case n = 5 are too large to write down as explicit polynomials.

Acknowledgements

I would like to thank Nick Shepherd-Barron for introducing me to this problem. The computer calculations in support of this work were performed using

pari [3].

References

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[3] C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, pari/gp,

a computer algebra package, http://www.parigp-home.de

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