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Some equations for the universal Kummer variety

We give a method to find quartic Heisenberg invariant equations for Kummer varieties and we give some explicit examples. From these equations for g-dimensional Kummer varieties one obtains equations for the moduli space of g+1-dimensional Kummer varieties. These again define modular forms which vanish on the period matrices of Riemann surfaces. The modular forms that we find for g=5 appear to be new and of lower weight than known before.


Introduction
The classical Kummer surface is a quartic surface in P 3 with 16 nodes. It is the quotient of an abelian surface by the involution x → −x. The translations by points of order two on the abelian surface induce projective transformations that map the Kummer surface into itself. This group action lifts to an action of a finite (non-abelian) Heisenberg group on C 4 .
Using classical theta function theory, the equation of this surface can be written as a Heisenberg invariant quartic polynomial whose coefficients are polynomials of degree 12 in the second order theta constants. Thus this polynomial is best seen as a polynomial of (bi)degree (12,4) in C[u, x] which, upon substituting the second order theta constants for the variables u, gives the equation of the Kummer surface. Such a polynomial is called an equation for the universal Kummer variety.
More generally, the Kummer variety of a (principally polarized) abelian variety of dimension g admits a map to a projective space of dimension 2 g − 1. It is defined by second order theta functions. For a general Kummer variety, the ideal of the image is generated by homogeneous polynomials of degree at most four. Equations of degree four were given in [K,Theorem 3.6].
A classical result asserts that Kummer surfaces are tangent hyperplane sections of the Igusa quartic threefold in P 4 . Somewhat remarkably, generalizations of the Igusa quartic are quite useful for finding equations for Kummer varieties and their moduli space. In Proposition 3.4 we show that a generalized Igusa equation (see Definition 3.2) provides a Heisenberg invariant quartic equation for the universal Kummer variety. In case of dimension g = 3, 4 we showed (with computer computations) that there exist generalized Igusa equations of degree four. This implies that there are equations of degree (12, 4) for the universal Kummer variety in these cases (see Sections 3.7, 3.8).
After having read a first version of this paper, R. Salvati Manni suggested that our method would also apply to the non-Heisenberg invariant quartic equations. This is indeed the case. A generalized Igusa equation of degree d, for g-dimensional abelian varieties, will provide a universal non-Heisenberg invariant quartic equation of bidegree (4(d − 1), 4) for the Kummer varieties of dimension g + 1 (see Proposition 4.2). Thus we find explicit quartic, non-Heisenberg invariant, equations for the universal Kummer variety in genus 3, 4, 5 of degree (12, 4), see Section 4.3.
The equations for the moduli space of Kummer varieties are the universal Kummer equations of degree (d, 0) for some d. In section 5 we show that a generalized Igusa equation of degree d, for g-dimensional abelian varieties, provides an equation of degree 4d for the moduli space of g + 1-dimensional abelian varieties. Finally we recall that such an equation leads to an equation for the locus of Jacobians inside the moduli space of abelian varieties of dimension g + 2.
This paper was motivated by the papers [RSSS] and [GSM] where an explicit equation of the Coble quartic was given. The singular locus of this quartic hypersurface in P 7 is the Kummer variety of the Jacobian of a non-hyperelliptic curve. The Coble quartic is also the moduli space of rank two bundles with trivial determinant on the curve. The equation for the Coble quartic is in fact an equation for the universal Kummer variety of degree (28, 4). It is not (yet) clear how it is related to the 27 equations of bidegree (12, 4) given in Section 3.7 of this paper.
It is a pleasure to acknowledge the discussions I had with Q. Ren, R. Salvati Manni and B. Sturmfels.
1. Basics 1.1. Theta functions. For τ ∈ H g , the Siegel upper halfspace of g × g symmetric complex matrices with positive definite imaginary part, one defines a (principally polarized) abelian variety A τ by A τ := C g /(Z g + Z g τ ), were the vectors are row vectors. We denote by L τ a symmetric line bundle on A τ defining the principal polarization.
The second order theta functions are defined as: They provide a basis of H 0 (A τ , L ⊗2 τ ). The (totally symmetric) line bundle L ⊗2 τ is intrinsically defined by the principal polarization on A τ .
1.2. The Kummer variety. The second order theta functions are all even functions, which they define thus factors over the quotient variety A τ /{±1} which is known as the Kummer variety of the abelian variety A τ . The singular locus of the Kummer variety consists of 2 2g points which are the images of the fixed points of the map x → −x on A τ , these are the 2-torsion points of A τ . In case the ppav A τ is not a product of ppav's of lower dimension, the image of Θ τ is isomorphic to its Kummer variety.
1.3. The universal Kummer variety. The coefficients of the equations for the Kummer variety which we will give in this paper depend only on the coordinates of the point Θ τ (0). This leads us (similar to [RSSS,Section 3]) to consider the map The image of this map is a quasi-projective variety of dimension g + g(g + 1)/2. The closure of the image is called the universal Kummer variety K g (2, 4) and we will exhibit polynomials in C[u, x] (actually in Q [u, x]) which are in the ideal I g of K g (2, 4). In particular, for genus 2, 3, 4 we show that there are such polynomials of bidegree (12, 4) (these are new for g = 3, 4). As Θ τ (0) lies in the image of the Kummer variety, any universal Kummer equation F (u, x) gives an equation F (u, u) for the image of the Siegel upper half space H g in P 2 g −1 under the map . The new equations of bidegree (12, 4) for K g (2, 4) have the extra property that F (u, u) is identically zero as polynomial in u if the equation is Heisenberg invariant (as polynomial in x). The non-Heisenberg invariant equations do give non-trivial equations for the image and we find equations for K g (2, 4) of bidegree (16, 0) for g = 3, 4, 5 (which are new for g = 4, 5).
1.4. The Heisenberg group. Any a ∈ A τ [2], the subgroup of two torsion points of A τ , defines a biholomorphic map t a : A τ → A τ , x → x + a. The line bundles L τ and t * a L τ are isomorphic and such an isomorphism is unique up to scalar multiple. However, one cannot choose the isomorphisms for the various a in such a way as to obtain a action of A τ [2] on H 0 (A τ , L ⊗2 τ ). Instead one obtains a representation of a non-commutative Heisenberg group H g , an extension of A τ [2] ∼ = (Z/2Z) 2g by C × , on H 0 (A τ , L ⊗2 τ ). The map Θ τ is equivariant for the action of H g , where the action on P 2 g −1 is generated by the following projective transformations (which we refer to as sign changes and translations): where the x σ are the homogeneous coordinates on P 2 g −1 and α, β ∈ (Z/2Z) g correspond to the elements α/2, βτ /2 ∈ 1 2 Λ τ /Λ τ = A τ [2] respectively. 2. Equations for the Kummer variety 2.1. The general theory. We will write a (homogeneous) polynomial of degree d in the 2 g variables x σ as The polynomial F is an equation for the Kummer variety of A τ if the theta function is zero as function of z ∈ C g . So Θ * τ substitutes x σ := Θ[σ](τ, z). Since F is homogeneous of degree d, the function (Θ * τ F )(z)) is a theta function of order 2d, equivalently, it corresponds to a global section of L ⊗2d τ . Let ϑ 1 , . . . , ϑ N , where N = N d , be theta functions which provide a basis of H 0 (A τ , L ⊗2d τ ). Then (Θ * τ F )(z)) = N i=1 a i ϑ i (z) for certain complex numbers a i . Moreover, F is an equation for the Kummer variety if and only if a i = 0 for all i. Thus the C-vector space I d = I d (Θ τ (A τ )) of equations of degree d of the Kummer variety is .) Finding the equations of degree d is thus a problem in linear algebra once the matrix of the C-linear map Θ * τ w.r.t. bases of its domain and image are known. For degree d = 2 this is matrix is the diagonal matrix with entries the θ[ ǫ ǫ ′ ](τ, 0). Unfortunately, for d = 3 this matrix is not known explicitly. For d = 4 the matrix was given in [vG,Proposition 4] and we recall part of the result in Section 2.4.
2.2. The map Θ * τ in degree two. we recall that for degree d = 2 the matrix of Θ * τ , w.r.t. suitable bases is a diagonal matrix. For ǫ, ǫ ′ ∈ (Z/2Z) g we define a polynomial in C[x] 2 by: with summation over σ ∈ (Z/2Z) g . In case ǫ t ǫ ′ = 0 this polynomial is not identically zero. The 2 g−1 (2 g + 1) polynomials one finds in this way are a basis of C[x] 2 . (They are also eigenvectors for the action of the Heisenberg group.) A classical theta function formula shows that ( [RSSS,(2.8 , with ǫ t ǫ ′ = 0, are a basis of the subspace of even theta functions in H 0 (A τ , L ⊗4 τ ). Therefore the matrix of Θ * τ is now a diagonal matrix with entries the even theta constants θ[ ǫ ǫ ′ ](τ, 0). In particular, there are quadratic equations for the Kummer variety of A τ if and only if at least one of these theta constants is zero.
2.3. The Heisenberg invariant quartics. The vector space C[x] 4 has the obvious basis given by the monomials σ x ασ σ with |α| = 4. As the map Θ * τ is equivariant w.r.t. the action of the Heisenberg group, the matrix will be in block form if we choose bases adapted to this action.
A basis for C[x] 4,0 , the quartic Heisenberg invariants, can be found as follows. Let T ⊂ (Z/2Z) g be a subgroup of order at most 4. So we have the following possibilities for T : In all cases we can in fact write T = {0, α, β, α + β}, possibly with α = β = 0 or β = 0. There are 1 + (2 g − 1) + (2 g − 1)(2 g − 2)/3 such subgroups. For each such subgroup T the monomial x 0 x α x β x α+β is invariant under the sign changes in the Heisenberg group. Now we simply take the sum of the monomials in the orbit of this monomial under the translations in the Heisenberg group. Then we obtain the polynomial These P T 's are a basis of C[x] 4,0 .
We will need to know the partial derivative of In case ♯T = 4, there are four monomials which contribute to the partial derivative, and each gives x σ+α x σ+β x σ+α+β . So we find the following result, which holds for all subgroups T : 2.4. The map Θ * τ in degree four. The decomposition of the image of Θ * τ into eigenspaces for the Heisenberg group is as follows: τ ) χ is 2 g (but if χ = 0, the subspace of even theta functions has dimension 2 g−1 ). With these decompositions, we have, for all characters χ of H g :

The 'multiplication by two' map [2] on the abelian variety
gives the following isomorphism: are a basis of H 0 (A τ , L ⊗8 τ ) 0 . We now consider the multiplication map for the case χ = 0: So, given a subgroup T = {0, α, β, α + β}, we want to find the complex numbers a σ,T such that Riemann's theta formula implies that (see [vG,Proposition 4]): Using the derivatives of the P T , there is an attractive way of to write this result: that is, the entries of the matrix, up to a factor 4 which does not affect the kernel, are just the partial derivatives of the P T 's evaluated in the point Θ τ (0) which has coordinates x σ = Θ[σ](τ, 0). In other words, the matrix (a σ,T ) is the transposed of the Jacobi matrix of the polynomials P T , evaluated in Θ τ (0). Using Cramer's rule, applied to a 2 g × (2 g + 1) submatrix of rank 2 g , one obtains quartic Heisenberg equations for the Kummer variety whose coefficients are of degree 3 · 2 g in the Θ[σ](τ, 0). The quartic equations given by Khaled ([K, Theorem 3.6(b)]) appear to be of degree (2 8g , 4) (see the definition of theq's on p.208 of [K]), but he also considers more general embeddings of Kummer varieties.
In Section 2.6 we recall that the classical equation for the Kummer surface in P 3 is obtained in this way. In Section 2.7 we recall a result from [RSSS] which shows that there equations of lower degree, (16, 4) rather than (24, 4), in case g = 3. However, for g = 3, 4 there also exist equations of degree (12, 4) as we will show in Sections 3.7, 3.8.
Before discussing the cases g = 2, 3, we give a characterization of the quartic, Heisenberg invariant, equations of the Kummer variety which will be exploited in the next section. Proof.
As P is an equation for the Kummer variety if and only if Θ * τ (P ) = 0 on C g , we find, using the determination of the a σ,T above, that this is the case exactly when This again is equivalent to Θ τ (0) being a singular point on the variety (better: subscheme) defined by P = 0.
In this case the multiplication matrix has size 8 × 15, but it has submatrices of rank 6. This was suggested by [RSSS,Lemma 8.2]. We checked that any 7 × 7 minor of the 8 × 7 submatrix formed by the Jacobian matrix (∂P T /∂x σ ) of the 7 quartics P 2 , P 3 , P 24 , P 34 , P 25 , P 35 , P 12 is identically zero. This leads to an equation for the universal Kummer threefold which is a linear combination of these seven P T 's. Now we consider the following 7 × 7 matrix, the last six rows are the partial derivatives (divided by 4) of the polynomials in the first row, w.r.t. the six variables x 001 , . . . , x 110 , and the derivatives are evaluated in the point with coordinates x σ = u σ : The determinant of this matrix is a linear combination T a T P T where each a T is a degree 18 = 3 · 6 polynomial in the u σ . Each coefficient a T turns out to be a multiple of u 000 u 111 . After dividing the determinant by this monomial, we obtain a polynomial of bidegree (16, 4) which is an equation for the universal Kummer threefold. Writing the polynomial as s 010 P 2 + s 011 P 3 + t 001 P 24 + t 011 P 34 + t 101 P 25 + t 111 P 35 + t 100 P 12 one obtains the coefficients s σ , t σ ∈ C[u] of the polynomial in [RSSS,Lemma 8.2].

Generalized Igusa quartics
3.1. Singular Heisenberg invariant quartics. In view of Proposition 2.5, we want to find quartic Heisenberg invariant polynomials F such that all partial derivatives of F vanish in the point Θ τ (0) with τ ∈ H g . We consider the following variation: find quartic Heisenberg invariant polynomials, whose coefficients are polynomials in the coordinates of a point u = (. . . : u σ : . . .) ∈ P 2 g −1 , such that all their partial derivatives vanish in u. So we do not require that u = Θ τ (0) for some τ . Such a polynomial F ∈ C[u, x] will be an equation for the universal Kummer variety.
The following proposition provides a method for finding such polynomials. It uses polynomials in variables p T , 3.2. Definition. A homogeneous polynomial R ∈ C[p] of degree d such that x * (R) = 0 is called a generalized Igusa equation of degree d (in dimension g). In case d = 4, such an R will be called a generalized Igusa quartic (cf. Section 3.6).

Definition.
Let R ∈ C[p] d , so R is a homogeneous polynomial of degree d in the variables p T . Then we define For each u ∈ C 2 g , the polynomial F R (u, x) is a linear combination of the P T (x), hence it is a Heisenberg invariant quartic in C[x]. The coefficient of P T is the partial derivative of R w.r.t. p T , in which one substitutes p S := P S (u) for all indices S. As ∂R/∂p T is homogeneous of degree d − 1, this coefficient is a homogeneous polynomial in the u σ of degree 4(d − 1). Upon substituting x := u in F R (x, u) one finds R F (u, u) = du * R, in fact: where we used Euler's relation.

Proposition.
Let R ∈ C[p] d be a generalized Igusa equation, so x * R = 0. Then F R ∈ C[u, x] is an equation for the universal Kummer variety which moreover satisfies F R (u, u) = 0.
Proof. By assumption x * R = 0, hence also u * R = 0. The partial derivatives ∂F R /∂x σ of F R , evaluated in x = u, are then where we used the chain rule for differentiation. Thus, for any u = (. . . , u σ , . . .) ∈ C 2 g , the Heisenberg invariant quartic F (u, x) defines a variety in P 2 g −1 which is singular in the point x = (. . . : u σ : . . .). In particular, considering the case u = Θ τ (0), we find that F R (u, x) is an equation for the Kummer variety of A τ by Proposition 2.5. Thus F R is an equation for the universal Kummer variety. Finally, F R (u, u) = du * R = 0.
3.5. Remark. Proposition 3.4 shows that the kernel of the ring homomorphism x * provides elements of the ideal I g of the universal Kummer variety: The kernel of the map x * has the following geometrical interpretation. Consider the map given by the Heisenberg invariant quartics: where N + 1 = (2 g + 1)(2 g−1 + 1)/3. For any g, this map factors over the Heisenberg quotient P 2 g −1 /H g . In case g = 2, one can show that it embeds P 3 /H 2 into P 4 as the Igusa quartic (see also Section 3.6). (For g ≥ 3 however, the Heisenberg invariant quartics do not generate the ring of all Heisenberg invariant polynomials in C[x]). In any case, the homogeneous elements R in the kernel of x * , that is, the generalized Igusa equations, are the equations for the image of this map. As 2 g − 1 < N for g ≥ 2, the kernel is non-trivial. A final observation is that for all g this map has no base points: in fact, the polynomials Q[ ǫ ǫ ′ ] from Section 2.2 are a basis of the quadratic polynomials, hence they have no base points, and the Q[ ǫ ǫ ′ ] 2 are Heisenberg invariant quartics. 3.6. The case g = 2. In this case there is a unique quartic R 2 in the p T 's which is identically zero as polynomial in the x σ , i.e. x * (R 2 ) = 0. It is the well-known Igusa quartic: . From R 2 one finds the equation of the universal Kummer surface: In fact, ∂R 2 /∂p 0 = 2(p 0 p 2 12 − p 1 p 2 p 3 ) etc. As u * (p T ) = P T (u), the polynomial F R 2 has bidegree (12, 4) in C[u, x]. An explicit computation verifies that F R 2 coincides with the determinant from Section 2.6, up to a scalar multiple.
The corresponding basis of Heisenberg invariant quartics is given Section 2.7. The polynomial F R 3 (u, x) has 728 terms.
Using the action of Sp(6, Z) (or the normalizer of the Heisenberg group) one can find a basis of the 27-dimensional vector space. As a consequence, for each R in this 27-dimensional vector space we find an equation of bidegree (12, 4) of the Kummer threefold. Therefore [RSSS,Conjecture 8.6] cannot be correct as the proposed generators for the ideal of the universal Kummer threefold have bidegree (a, b) with a > 12.
3.8. The case g = 4. In this case we found that the kernel of x * restricted to C[p] 4 has dimension 510. This implies by Proposition 3.4 that there are equations of degree (12, 4) for the Kummer fourfold in P 15 .
The space of Heisenberg invariant quartics in 2 4 = 16 variables has dimension 51, and they define a map P 15 → P 50 . The quartics in the kernel of x * are equations for the image of this map. We verified the variety Z (better: subscheme) in P 50 defined by these equations has the image as an irreducible component. For this we only had to find a point x ∈ P 15 such that tangent space T y Z has dimension 15, where y is the image of x. We do not know whether Z is irreducible.
3.9. The general case. In Section 3.7, 3.8 we showed that there exist generalized Igusa quartics for g = 3, 4. We were not able to show that such Igusa quartics exist for any g > 2. Using representation theory, one can make some guesses as to what to expect in general.
The action of Γ g := Sp(2g, Z) on the Siegel space H g induces a projective projective representation of Γ g on the 2 g -dimensional vector space of second order theta functions. Identifying this vector space with C[x] 1 , the linear forms in the 2 g variables x σ , we obtain a projective representation of Γ g on C[x]. On the subspace of Heisenberg invariant polynomials, one obtains a linear representation of Sp(2g, F 2 ) = Γ g /Γ g (2), where Γ g (2) is the subgroup of matrices congruent to I mod 2.
Similarly, identifying C[p] 1 = C[x] 4,0 by p T → P T , we obtain a representation of Sp(2g, F 2 ) on C [p]. The representation on C[p] 1 is irreducible and it is the representation denoted by V g in [F]. The kernel of the map x * : 4d,0 is a subrepresentation. This should be helpful in understanding the kernel.
This suggest that dim ker(x * : C[p] 4 −→ C[x] 16 ) might have this dimension also for g > 4.
In any case, if this kernel is non-trivial then we would have equations of bidegree (12, 4) for the universal Kummer variety for all g ≥ 2. One still has to check that not all coefficients, that are degree 12 polynomials in the u σ , are zero on the image of H g in P 2 g −1 .
4. Non-Heisenberg invariant quartic equations 4.1. The map Θ * τ in degree four (bis). In section 2.4 we considered the map Θ * τ : τ ) 0 , whose kernel are the quartic Heisenberg invariant equations for the Kummer variety of A τ . We now consider the case of a non-trivial character χ of the Heisenberg group H g . The action of Sp(2g, Z) on the non-trivial characters of H g is transitive. Thus we concentrate on a fixed character.
It will be convenient to consider g + 1-dimensional abelian varieties and to consider the character χ : H g+1 → {±1} which is trivial on the sign changes and which is (−1) β g+1 on the translation v σ → v σ+β with β ∈ (Z/2Z) g+1 . We introduce a polynomial ring C[v] in 2 g+1 variables v σ , σ ∈ (Z/2Z) g+1 , which is the analogue of the ring C[u] in the 2 g variables u σ for σ ∈ (Z/2Z) g . Now we define a ring homomorphism which is defined by the assignment where the P T are a basis of the Heisenberg invariant quartics (in the 2 g variables x σ ). These are thus evaluated first in x σ := v σ0 and next in x σ := v σ1 . As the P T are H g -invariants, it is easy to see thatp T,χ ∈ C[v] 4,χ . Moreover, this homomorphism induces an isomorphism C[p] 1 ∼ = C[v] 4,χ . Thus the polynomialsp T,χ , where T runs over the (2 g + 1)(2 g−1 + 1)/3 subgroups of order at most four of (Z/2Z) g , form a basis of A basis of H 0 (A τ , L ⊗8 τ ) χ,+ , the even theta functions in H 0 (A τ , L ⊗8 τ ) χ , is given by the θ[ σ0 0 1 ](2τ, 4z), where σ runs over (Z/2Z) g (here τ ∈ H g+1 , z ∈ C g+1 ). W.r.t. these bases, the map Using Riemann's theta formula one finds (see [vG,Proposition 4]): is a quartic equation for for the universal Kummer variety K g+1 which, as polynomial in y, lies in (C[v])[y] 4,χ .
Proof. We use the Heisenberg group H g+1 acting on P 2 g+1 −1 . The coordinates on this projective space, in terms of which the action of H g+1 is by translations and signs changes, will be denoted by y σ , σ ∈ (Z/2Z) g+1 . If a ∈ H g+1 acts non-trivially on P 2 g+1 −1 , then it is has two eigenspaces, each isomorphic to P 2 g −1 . We choose one of them and call it P 2 g −1 a . The subgroup of elements in H g+1 which map this eigenspace into itself induce the action of H g on P 2 g −1 a . This allows one to choose coordinates x σ , σ ∈ (Z/2Z) g , adapted to H g on this eigenspace.
From the proof of Proposition 3.4 and Section 2.3 we have for all σ ∈ (Z/2Z) g . The monomial in front of c T (u) is similar to b σ,T and we exploit this fact. Let τ ∈ H g+1 . Then we substitute is an equation for the Kummer variety of the g + 1-dimensional abelian variety A τ . To see that this is a universal Kummer equation, we need to show that the coefficients are actually polynomials in the Θ[σ](τ, 0).
Next we use the formulae (2.4a,b) for Θ * τ (PT ): The Θ[σ](τ, 2b) are the coordinates of the point Θ τ (2b). As a = 2b is a 2-torsion point, the point Θ τ (2b) is obtained from the point Θ τ (0) by the action of the sign change of a ∈ H g+1 above. In fact, one computes that As far as I know, this equation for the universal Kummer threefold was not known before. For each non-trivial character of H 3 one finds such a quartic relation in the corresponding eigenspace, thus we get 63 such equations.
4.4. Remark. The quartic polynomials Q[ ǫ ǫ ′ ] 2 are Heisenberg invariant. It is not hard to find their image in C[v]:

5.
Equations for the moduli space 5.1. Equations for the moduli space. In this section we show that the generalized Igusa equations defined in Section 3.2 also produce equations for the moduli space Θ(H g+1 ) ⊂ P 2 g+1 −1 (note that g increases by 1 in this process, like for the non-Heisenberg invariant equations for the universal Kummer variety in Section 4).

Proposition.
Let R ∈ C[p] d be a generalized Igusa equation of degree d, so R ∈ ker(x * : . By Proposition 4.2 it defines the the equatioñ F R,χ ∈ C[v, y] for the universal Kummer variety. ThusF R,χ (Θ τ (0), Θ τ (z)) = 0 for all τ ∈ H g and all z ∈ C g+1 . Putting z = 0, we see thatF R,χ (v, v) is an equation for the moduli space. Euler's relation shows that hence also R χ (v) is an equation for the moduli space.
5.3. Example g = 2. In case g = 2, the kernel of x * is generated by the Igusa quartic R 2 , see Section 3.6. It is also well-known that the image of the map Θ : H 3 → P 7 , known as the Satake hypersurface, is defined by an irreducible homogeneous polynomial f 16 of degree 16 (cf. [RSSS,Proposition 3.1]). The space C[x] 4,0 is five dimensional, its basis P T was given explicitly in Section 3.6. Thep T,χ are given in Section 4.3. The polynomialR 2,χ in the eight variables v σ can then be computed by substituting p T :=p T,χ in the polynomial R 2 from Section 3.6. One finds thatR 2,χ is, up to multiplication by a non-zero constant, the polynomial f 16 . Thus Proposition 5.2 produces non-trivial equations. 5.4. Example g = 3. We checked that the polynomialR 3,χ in the 16 variables obtained from the generalized Igusa equation R 3 from Section 3.7 is not identically zero. It is thus an equation of degree 16 for the image of H 4 in P 15 .
In [FO], Freitag and Oura show that there exists an equation of degree 24 for Θ(H 4 ) in P 15 .
Schottky verified that F is not identically zero on H 4 and Igusa [I] (and independently Freitag [Fr]) showed that F (τ ) = 0 implies that τ is in the Jacobi locus J 4 ⊂ H 4 , which is the closure of the locus of period matrices of Riemann surfaces in H 4 .