A Universal Genus-Two Curve from Siegel Modular Forms

Let $\mathfrak p$ be any point in the moduli space of genus-two curves $\mathcal M_2$ and $K$ its field of moduli. We provide a universal equation of a genus-two curve $\mathcal C_{\alpha, \beta}$ defined over $K(\alpha, \beta)$, corresponding to $\mathfrak p$, where $\alpha $ and $\beta$ satisfy a quadratic $\alpha^2+ b \beta^2= c$ such that $b$ and $c$ are given in terms of ratios of Siegel modular forms. The curve $\mathcal C_{\alpha, \beta}$ is defined over the field of moduli $K$ if and only if the quadratic has a $K$-rational point $(\alpha, \beta)$. We discover some interesting symmetries of the Weierstrass equation of $\mathcal C_{\alpha, \beta}$. This extends previous work of Mestre and others.


Introduction
Let M 2 be the moduli space of genus-two curves. It is the coarse moduli space for smooth, complete, connected curves of genus two over C. Let p ∈ M 2 (K), where K is the field of definition of p. Construction a genus 2 curve C corresponding to p is interesting from many points of view. Mestre [18] has shown how to construct equations for genus-two curves with automorphism group of order two and defined over Q. Mestre's work has recently received new attention from researchers in experimental number theory. For instance, in [3] a database of geometric and arithmetic invariants of genus-two curves defined over Q of small discriminant. In [1], the authors count the points in M 2 (Q) according to their moduli height and create a database of genus-two curves from the moduli points in M 2 (Q). In creating the database the main problem was that of constructing an equation for obstruction moduli points. This paper provides an equation over a minimal field of definition for any point p ∈ M 2 . Our work is therefore complimentary to the problem of finding an efficient construction for genustwo curves over finite fields with a prescribed number of rational points and the associated complexity analysis in [4,7]. Our equation for a genus-two curve is universal in the sense that it works for every moduli point given in terms of Igusa invariants or Siegel modular forms. It does not rely on special CM values for Siegel modular functions where the associated abelian surface has extra endomorphisms or the special invariants that can be used in theses cases (cf. [15]).
The natural question is if there exists a universal curve for the genus-two curve given in terms of a generic moduli point p ∈ M 2 . In other words, given an affine moduli point p = (x, y, z), where x, y, z are transcendentals, can we construct a curve corresponding to p? The answer is negative in the strict definition of "universal curve"; see [9, p. 39] for details. As we will show, there is a satisfactory answer in the sense that our "universal equation" applies to every moduli point p ∈ M 2 . However, the equation is often defined only over a quadratic extension of the field of moduli.
We focus mainly on constructing a genus-two curve C for any given point p = (x 1 , x 2 , x 3 ) ∈ M 2 , defined over a minimal field of definition, where x 1 , x 2 , x 3 are ratios of modular forms as defined by Igusa in [11]. Our main result is as follows: For every point p ∈ M 2 such that p ∈ M 2 (K), where K is the field of moduli, there is a genus-two curve C (α,β) given by C (α,β) : y 2 = 6 i=0 a i (α, β)x i , corresponding to p with coefficients given by equation (3.9). This curve is defined over the field of moduli K if and only if there exists a K-rational solution (α, β) to the quadratic where b and c are given in terms of the moduli point p. There are some interesting properties of the coefficients defining C (α,β) which seem to be particular to this model and not noticed before.
It must be noticed that this equation is universal in the sense that it works for every moduli point [J 2 : J 4 : J 6 : J 10 ] given in terms of the Igusa invariants J 2 , J 4 , J 6 , J 10 . The equation is defined at worst over a quadratic extension of the field of moduli K. If the equation over the field of moduli is needed, then we must search locally for a rational point in the above quadratic when evaluated at the given p. In the process we discover some interesting absolute invariants (cf. equation (2.5)) which as far as we are aware have not been used before.
The paper is organized as follows: In Section 2 we give a brief summary of Siegel modular forms, classical invariants of binary sextics and the relations among them. While this material can be found in many places in the literature, there is plenty of confusion on the labeling and normalization of such invariants and relations among them. We also introduce a set of absolute invariants that is well-suited for the construction of a universal sextic.
In Section 3 we construct the equation of the genus-two curve by determining the Clebsch conic and the cubic. We diagonalize the corresponding conic and discover a new set of invariants which make the equation of this conic short and elegant. The diagonalized conic can be quickly determined from the invariants of the curve. The intersection of this conic and the cubic gives the equation of the genus-two curve. This equation shows some interesting symmetries of the coefficients, which to the knowledge of the authors have never been discovered before. When this universal equation is restricted to loci of curves with automorphisms or the Clebsch invariant D = 0 (not covered by Mestre's approach) it shows that the field of moduli is a field of definition, results which agree with previous results of other authors.

The Siegel modular three-fold
The Siegel three-fold is a quasi-projective variety of dimension 3 obtained from the Siegel upper half-plane of degree two which by definition is the set of two-by-two symmetric matrices over C whose imaginary part is positive definite, i.e., quotiented out by the action of the modular transformations Γ 2 := Sp 4 (Z), i.e., Each τ ∈ H 2 determines a principally polarized complex abelian surface with period matrix (τ , I 2 ) ∈ Mat(2, 4; C). Two abelian surfaces A τ and A τ are isomorphic if and only if there is a symplectic matrix It follows that the Siegel three-fold A 2 is also the set of isomorphism classes of principally polarized abelian surfaces. The sets of abelian surfaces that have the same endomorphism ring form sub-varieties of A 2 . The endomorphism ring of principally polarized abelian surface tensored with Q is either a quartic CM field, an indefinite quaternion algebra, a real quadratic field or in the generic case Q. Irreducible components of the corresponding subsets in A 2 have dimensions 0, 1, 2 and are known as CM points, Shimura curves and Humbert surfaces, respectively. The Humbert surface H ∆ with invariant ∆ is the space of principally polarized abelian surfaces admitting a symmetric endomorphism with discriminant ∆. It turns out that ∆ is a positive integer ≡ 0, 1 mod 4. In fact, H ∆ is the image inside A 2 under the projection of the rational divisor associated to the equation with integers a, b, c, d, e satisfying ∆ = b 2 −4ac−4de and τ = τ 1 z z τ 2 ∈ H 2 . For example, inside of A 2 sit the Humbert surfaces H 1 and H 4 that are defined as the images under the projection of the rational divisor associated to z = 0 and τ 1 = τ 2 , respectively. In fact, the singular locus of A 2 has H 1 and H 4 as its two connected components. As analytic spaces, the surfaces H 1 and H 4 are each isomorphic to the Hilbert modular surface For a more detailed introduction to Siegel modular form, Humbert surfaces, and the Satake compactification of the Siegel modular threefold we refer to Freitag's book [6].

Igusa invariants
Suppose that C is an irreducible projective non-singular curve. If the self-intersection is C · C = 2 then C is a curve of genus two. For every curve C of genus two there exists a unique pair (Jac(C), j C ) where Jac(C) is an abelian surface, called the Jacobian variety of the curve C, and j C : C → Jac(C) is an embedding. One can always regain C from the pair (Jac(C), P) where P = [C] is the class of C in the Néron-Severi group NS(Jac(C)). Thus, if C is a genus-two curve, then Jac(C) is a principally polarized abelian surface with principal polarization P = [C], and the map sending a curve C to its Jacobian variety Jac(C) is injective. In this way, the variety of moduli of curves of genus two is also the moduli space of their Jacobian varieties with canonical polarization. We write the equation defining a genus-two curve C by a degree-six polynomial or sextic in the form The roots {α i } 6 i=1 of the sextic are the six ramification points of the map C → P 1 . Their preimages on C are the six Weierstrass points. The isomorphism class of f consists of all equivalent sextics where two sextics are considered equivalent if there is a linear transformation in GL 2 (C) which takes the set of roots to the roots of the other.
Often the Clebsch invariants (A, B, C, D) of a sextic are used instead. They are defined in terms of the transvectants of the binary sextics; see [5] for details. The invariants (A, B, C, D) are polynomial expressions in the Igusa invariants (J 2 , J 4 , J 6 , J 10 ) with rational coefficients: For formulas giving relations between all these sets of invariants see [1].

Absolute invariants
Dividing any SL 2 (C) invariant by another one of the same degree gives an invariant under GL 2 (C) action. The term absolute invariants is used first by Igusa [10] for GL 2 (C) invariants. It was the main result of [11,Theorem 3] that Since the invariants J 4 , J 6 , J 10 vanish simultaneously for sextics with triple roots all such curves are mapped to [1 : 0 : 0 : 0] ∈ WP 3 (2,4,6,10) with uniformizing affine coordinates x 1 , x 2 , x 3 around it. Blowing up this point gives a variety that parameterizes genus-two curves with J 2 = 0 and their degenerations. In the blow-up space we have to introduce additional coordinates that are obtained as ratios of x 1 , x 2 , x 3 and have weight zero. Those are precisely the coordinates y 1 , y 2 , y 3 already introduced in equation (2.1). It turns out that the coordinate ring of the blown-up space is We introduce the three absolute invariants It follows: For invariants (ρ, σ, κ) given by equation (2.5) such that ρ and κ do not vanish simultaneously, a point [J 2 : J 4 : J 6 : J 10 ] in WP 3 (2,4,6,10) is given by Proof . The proof is computational. We express ρ, σ, κ as rational functions of x 1 , x 2 , x 3 and vice versa over Q. The condition that ρ and κ do not vanish simultaneously is based on the fact that J 2 , J 4 , J 6 , J 10 must not vanish simultaneously.
In this case we have This means that under the usual morphism to P 5 the regular genus-two curves with J 2 = 0 and constant ratio J 10 /(J 4 J 6 ) are mapped to the same point.

Recovering the equation of the curve from invariants
Let p ∈ M 2 and C a genus-two curve corresponding to p defined by the sextic polynomial f in equation (2.3). Then, Aut(p) is a finite group as described in [19]. The quotient space C/ Aut(p) is a genus zero curve and therefore isomorphic to a conic. Since conics are in one to one correspondence with three-by-three symmetric matrices (up to equivalence), let M = [A ij ] be the symmetric matrix corresponding to this conic. Let X = [X 1 : X 2 : X 3 ] ∈ P 2 and A ij X i X j = 0. (2.8) Clebsch [5] determined the entries of this matrix M as follows The coefficients are obtained as follows: from the sextic f in equation (2.3) three binary quadrics y i (x) with i = 1, 2, 3 are obtained by an operation called 'Überschiebung' [18, p. 317] We define R to be 1/2 times the determinant of the three binary quadrics y i for i = 1, 2, 3 with respect to the basis x 2 , x, 1. If one extends the operation ofÜberschiebung by product rule [18, p. 317], then R can be re-written as or, equivalently, as R = − 1 8 y 1,yy y 2,xy y 3,xx − y 1,yy y 2,xx y 3,xy − y 1,xy y 2,yy y 3,xx + y 1,xy y 2,xx y 3,yy + y 1,xx y 2,yy y 3,xy − y 1,xx y 2,xy y 3,yy .
It is then obvious that under the operation f (x) →f (x) = f (−x) the determinant R changes its sign, i.e., R(f ) → R(f ) = −R(f ). A straightforward calculation shows that where A ij are the invariants in equation ( (2.10) Bolza [2] described the possible automorphism groups of genus-two curves defined by sextics and provided criteria for the cases when the automorphism group of the sextic curve in equation (2.3) is nontrivial. For a detailed discussion of the automorphism groups of genus-two curve defined over any field k and the corresponding loci in M 2 see [19]. We have the following lemma summarizing our discussion: Lemma 2.5. We have the following statements: 1. R 2 is an order 30 invariant of binary sextics expressed as a polynomial in (J 2 , J 4 , J 6 , J 10 ) as in [19, equation (17)] given by plugging Clebsch invariants and (2.9) into equation (2.10).
2. The locus of curves p ∈ M 2 such that V 4 → Aut(p) is a two-dimensional irreducible rational subvariety of M 2 given by the equation R 2 = 0 and a birational parametrization given by the u, v-invariants as in [19,Theorem 1].
We have introduced the invariant R 2 for any binary sextic f . To the corresponding symmetric matrix M with coefficients A ij = (y i y j ) 2 of order zero and invariant under the operation f (x) → f (x) = f (−x), we associated a conic Q. Similarly, there is also a cubic curve given by the equation where the coefficients a ijk are of order zero and invariant under f (x) →f (x) = f (−x). In terms of 'Überschiebung' the coefficients are obtained by The coefficients a ijk are given explicitly as follows: The relations between all aforementioned invariants and Siegel modular forms, in particular the relation between χ 35 and R 2 can be found in [1].
Since 'Überschiebung' preserves the rationality of the coefficients, we have the following corollary: Corollary 2.6. Let p ∈ M 2 and C a genus-two curve corresponding to p defined by a sextic polynomial f in equation (2.3). Then, Aut(p) is a finite group, and the quotient space C/ Aut(p) is a genus zero curve isomorphic to the conic Q in equation (2.8). Moreover, if p ∈ M 2 (K), for some number field K, the conic Q and cubic T have K-rational coefficients.
The intersection of the conic Q with the cubic T consists of six points which are the zeroes of a polynomial f (x) of degree 6 in the parameter x. The roots of this polynomial are the images of the Weierstrass points under the hyperelliptic projection. Hence, the affine equation of a genus-two curve corresponding to p is given by y 2 = f (x). The main question is if the sextic given by y 2 = f (x) provides a genus-two curve defined over a minimal field of definition. We start with the following known result. Proposition 2.7. A genus g ≥ 2 hyperelliptic curve X g with hyperelliptic involution w is defined over the K if and only if the conic Q = X g / w has a K-rational point.
The above result was briefly described in [18, Lemma 1] even though it seems as it had been known before. Mestre's method is briefly described as follows: if the conic Q has a rational point over Q, then this leads to a parametrization of Q, say (h 1 (x), h 2 (x), h 3 (x)). Substitute X 1 , X 2 , X 3 by h 1 (x), h 2 (x), h 3 (x) in the cubic T and we get the degree 6 polynomial f (x). However, if the conic has no rational point or R 2 = 1 2 det M = 0 the method obviously fails. In Section 3 we determine the intersection T ∩ Q over a quadratic extension which is always possible.

A universal genus-two curve from the moduli space
The goal of this section is to explicitly determine a universal equation of a genus-two curve corresponding to this generic point p. We have the following lemma: where (ρ, σ, κ) are the absolute invariants in equation (2.5), γ = ρ 2 + σ and Moreover, for J 2 , ρ, σ, κ ∈ Q the conic Q in equation ( Proof . For the conic Q in equation (2.8), we apply the coordinate transformation given by We then obtain the conic Q in equation ( The rational point on the conic Q is then given by Proof . If rational numbers α, β exist such that equation (3.4) is satisfied, then the point in equation (3.5) is rational and is easily checked to be on the conic. If there is a rational point on the conic then we can choose β ∈ Q in equation (3.5), thus α ∈ Q.
We have the following: Assume that a point on the conic in equation (3.1) is given by equation (3.5) with x 0 2 = 0 which is always possible if ρ = 0. Then every point on the conic is given by Proof . If a point of Q is obtained from some (rational) values (α, β) then there are three more (rational) points given by setting (α, β) → (±α, ±β). If ρ = 0, one of these points satisfies x 0 2 = α + ρ = 0. The proof then follows from the known formulas parametrizing conics for x 0 2 = 0 given by where a = 1, b = −γ, c = −Λ 6 and x 0 1 , x 0 2 , x 0 3 were given in equation (3.5).
Remark 3.7. The absolute invariants (ρ, σ, κ) in equation (2.5) such that ρ and κ do not vanish simultaneously and J 10 = 0 describe the moduli of genus-two curves with J 4 · J 6 · J 10 = 0. The discussion of Lemma 3.6 proves that only for genus-two curves with J 4 · J 6 · J 10 = 0, the conic Q in equation (2.8) is not guaranteed to have a rational point.
Substituting the parametrization of the conic Q in Lemma 3.3 into the cubic T in equation (3.7) and setting U = x and V = 1, one obtains the ramification locus of a sextic curve.
We have the following main result: Theorem 3.11. Let p ∈ M 2 such that p ∈ M 2 (K), for some number field K, and j = [J 2 : J 4 : J 6 : J 10 ] the corresponding point in WP 3 (2,4,6,10) (O K ), where O K is the ring of integers of K. A genus-two curve corresponding to p is constructed as follows: i) If J 2 · J 10 = 0 there is a genus-two curve C (α,β) given by with coefficients given in equations (3.9) and (3.10), and a pair (α, β) satisfying where Λ 6 , σ, and γ are determined by p. Moreover, C (α,β) is defined over its field of moduli K, i.e., a i (α, β) ∈ K, i = 0, . . . , 6, if and only if K-rational α and β exist.
This completes the proof.
Remark 3.12. The four pairs (±α, ±β) belong to the same conic Q . Therefore, we get four genus-two curves in Theorem 3.11, but they are all twists of each other. That is, we get one curve (over the algebraic closure), but four twists.
The main benefit of the above result is that it will give a curve defined over Q whenever possible. This is an improvement from results in [18] where a curve is provided only for curves with automorphism group of order 2 and J 2 = 0. The equation is valid even when the field of moduli is not a field of definition. Hence, for every point p ∈ M 2 we get a curve. Next we have the following result: Corollary 3.13. For every point p ∈ M 2 such that p ∈ M 2 (K), for some number field K, there is a genus-two curves C given by corresponding to p, such that a i (α, 0) ∈ K(α), i = 0, . . . , 6 as given in equation (3.9). Moreover, C (α,0) is at worst defined over the quadratic extension K(α) of the field of moduli K with α 2 = ρ 2 + σ.
We have the immediate consequence: Corollary 3.14. Let x 1 , x 2 , x 3 be transcendentals. There exists a genus-two curve C (α,0) defined over Q(x 1 , x 2 , x 3 )[α] with α 2 = ρ 2 + σ such that We have the following corollary: Corollary 3.15. Let σ = 0 and ρ = 0 for p ∈ M 2 . Then, there is a genus-two curve C given by Corollary 3.13, and it is defined over the field of moduli.
We have the following corollary: Corollary 3.18. Let D = 0 and χ 2 35 = 0 for p ∈ M 2 . Then, there is a genus-two curve C given by Corollary 3.13, and it is defined over the field of moduli.

A word about extra automorphisms
In this section we derive a sextic polynomial for the sublocus of M 2 with χ 35 = 0. We have the following proposition: Proposition 3.19. Let D = 0 and χ 35 = 0 for p ∈ M 2 . Then, there is a genus-two curve C : , (3.12) and with coefficients in Z[α, ρ, κ] given by Here, the absolute invariants α, γ, ρ, κ are subject to the constraints Λ 6 = 0 in equation (3.2) and α 2 = γ.
The curve y 2 =F (x) has extra involutions, i.e., it has automorphisms other than the hyperelliptic involution, for appropriate values of a, b (the discriminant is nonzero). In [19] for curves with automorphism the dihedral invariants were defined which give a birational parametrization of this locus L 2 which is a two-dimensional subvariety of M 2 . We have the following: , and the Igusa invariants [J 2 : J 4 : J 6 : J 10 ] given by [19, equation (16)].