A Universal Genus-Two Curve from Siegel Modular Forms

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 089, 17 pages A Universal Genus-Two Curve from Siegel Modular Forms

Andreas MALMENDIER † and Tony SHASKA ‡

† Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA E-mail: [email protected] ‡ Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA E-mail: [email protected] Received July 18, 2017, in final form November 25, 2017; Published online November 30, 2017 https://doi.org/10.3842/SIGMA.2017.089

Abstract. Let p be any point in the moduli space of genus-two curves M2 and K its field of moduli. We provide a universal equation of a genus-two curve Cα,β defined over K(α, β), corresponding to p, where α and β satisfy a quadratic α2 + bβ2 = c such that b and c are given in terms of ratios of Siegel modular forms. The curve Cα,β is defined over the field of moduli K if and only if the quadratic has a K-rational point (α, β). We discover some interesting symmetries of the Weierstrass equation of Cα,β. This extends previous work of Mestre and others. Key words: genus-two curves; Siegel modular forms

2010 Mathematics Subject Classiﬁcation: 14H10; 14H45

1 Introduction

Let M2 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 ∈ M2(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 M2(Q) according to their moduli height and create a database of genus-two curves from the moduli points in M2(Q). In creating the database the main problem was that of constructing an equation for obstruction moduli points. This

arXiv:1607.08294v3 [math.AG] 30 Nov 2017 paper provides an equation over a minimal field of definition for any point p ∈ M2. Our work is therefore complimentary to the problem of finding an efficient construction for genus- two 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 ∈ M2. 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,

This paper is a contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui. The full collection is available at http://www.emis.de/journals/SIGMA/modular-forms.html 2 A. Malmendier and T. Shaska there is a satisfactory answer in the sense that our “universal equation” applies to every moduli point p ∈ M2. 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 = (x1, x2, x3) ∈ M2, defined over a minimal field of definition, where x1, x2, x3 are ratios of modular forms as defined by Igusa in [11]. Our main result is as follows: For every point p ∈ M2 such that p ∈ M2(K), where K is the field of moduli, there is a genus-two curve C(α,β) given by

6 2 X i C(α,β) : y = ai(α, β)x , i=0 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 α2 + b · β2 = c 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 [J2 : J4 : J6 : J10] given in terms of the Igusa invariants J2, J4, J6, J10. 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 Section2 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 Section3 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.

2 Preliminaries

2.1 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., τ1 z 2 H2 = τ = τ1, τ2, z ∈ C, Im (τ1) Im (τ2) > Im (z) , Im (τ2) > 0 , z τ2 quotiented out by the action of the modular transformations Γ2 := Sp4(Z), i.e.,

A2 = H2/Γ2. A Universal Genus-Two Curve from Siegel Modular Forms 3

Each τ ∈ H2 determines a principally polarized complex abelian surface 2 2 2 Aτ = C / Z ⊕ τZ with period matrix (τ, I2) ∈ Mat(2, 4; C). Two abelian surfaces Aτ and Aτ 0 are isomorphic if and only if there is a symplectic matrix AB M = ∈ Γ , CD 2

0 −1 such that τ = M(τ) := (Aτ + B)(Cτ + D) . It follows that the Siegel three-fold A2 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 A2. 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 A2 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 A2 under the projection of the rational divisor associated to the equation

2 aτ1 + bz + cτ3 + d z − τ1τ2 + e = 0,

2 τ1 z with integers a, b, c, d, e satisfying ∆ = b −4ac−4de and τ = z τ2 ∈ H2. For example, inside of A2 sit the Humbert surfaces H1 and H4 that are defined as the images under the projection of the rational divisor associated to z = 0 and τ1 = τ2, respectively. In fact, the singular locus of A2 has H1 and H4 as its two connected components. As analytic spaces, the surfaces H1 and H4 are each isomorphic to the Hilbert modular surface (SL2(Z) × SL2(Z)) o Z2 \ H × H . 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].

2.2 Siegel modular forms

In general, we can define the Eisenstein series ψ2k of degree g and weight 2k (where we assume 2k > g + 1 for convergence) by setting

X −2k ψ2k(τ) = det(C · τ + D) , (C,D) where the sum runs over non-associated bottom rows (C,D) of elements in Sp2g(Z) where non- associated means with respect to the multiplication by GLg(Z). In the following, we will always assume g = 2 in the definition of ψ2k. Using Igusa’s definition [11, Section 8, p. 195] we define a cusp form of weight 10 by 43867 χ (τ) = − (ψ (τ)ψ (τ) − ψ (τ)) . 10 21235527 · 53 4 6 10 Based on Igusa’s definition [11, Section 8, p. 195] and the work in [16] we define a second cusp form χ12 of weight 12 by 131 · 593 χ (τ) = 3272ψ3(τ) + 2 · 53ψ2(τ) − 691ψ (τ). 12 213375372337 4 6 12 4 A. Malmendier and T. Shaska

Moreover, Igusa proved [12, 13] that the ring of Siegel modular forms is generated by ψ4, ψ6, χ10, χ12 and by one more cusp form χ35 of odd weight 35 whose square is the following polynomial [12, p. 849] in the even generators 1 χ2 = χ 224315χ5 − 21339ψ3χ4 − 21339ψ2χ4 + 33ψ6χ3 − 2 · 33ψ3ψ2χ3 35 21239 10 12 4 12 6 12 4 12 4 6 12 14 8 2 3 23 12 2 2 3 3 4 3 11 6 4 2 2 − 2 3 ψ4ψ6χ10χ12 − 2 3 5 ψ4χ10χ12 + 3 ψ6χ12 + 2 3 37ψ4χ10χ12 11 6 2 2 2 23 9 3 3 2 2 7 2 2 4 2 2 + 2 3 5 · 7ψ4ψ6χ10χ12 − 2 3 5 ψ6χ10χ12 − 3 ψ4χ10χ12 + 2 · 3 ψ4ψ6χ10χ12 11 5 3 3 20 8 3 2 4 2 4 2 11 5 2 3 3 + 2 3 5 · 19ψ4ψ6χ10χ12 + 2 3 5 11ψ4χ10χ12 − 3 ψ4ψ6χ10χ12 + 2 3 5 ψ6χ10χ12 6 3 12 4 5 4 2 3 3 3 12 4 2 2 2 4 21 7 4 5 − 2ψ4ψ6χ10 − 2 3 ψ4χ10 + 2 ψ4ψ6χ10 + 2 3 5 ψ4ψ6χ10 + 2 3 5 ψ4ψ6χ10 5 3 32 9 5 6 − 2ψ6χ10 + 2 3 5 χ10 . 12 9 2 Hence, the expression Q := 2 3 χ35/χ10 is a polynomial of degree 60 in the even generators. Igusa also proved that each Siegel modular form (with trivial character) of odd weight is divisible by the form χ35. The following fact is known[8]:

Proposition 2.1. The vanishing divisor of Q is the Humbert surface H4, i.e., a period point τ is equivalent to a point with τ1 = τ2 if and only if Q = 0. Accordingly, the vanishing divisor of χ35 is the formal sum H1 + H4 of Humbert surfaces, that constitutes the singular locus of A2. In accordance with Igusa [11, Theorem 3] we also introduce the following ratios of Siegel modular forms 2 3 6 ψ4χ10 ψ6χ10 χ10 x1 = 2 , x2 = 3 , x3 = 5 , χ12 χ12 χ12 as well as 3 3 2 2 2 2 x1 ψ4 x2 ψ6 x1x2 ψ4ψ6χ10 y1 = = , y2 = = , y3 = = , (2.1) x3 χ12 x3 χ12 x3 χ12 where we have suppressed the dependence of each Siegel modular form on τ. These ratios have the following asymptotic expansion for z → 0 [11, pp. 180–182] in terms of ordinary Eisenstein series E4 and E6 and the Dedekind η-function 4 5 x1 = E4(τ1)E4(τ2)(πz) + O z , 6 7 x2 = E6(τ1)E6(τ2)(πz) + O z , 24 24 12 13 x3 = η (τ1)η (τ2)(πz) + O z , and 2 y1 = j(τ1)j(τ2) + O z , 2 y2 = (1728 − j(τ1))(1728 − j(τ2)) + O z , 2 2 E4 (τ1)E4 (τ2)E6(τ1)E6(τ2) 2 3 y3 = 24 24 (πz) + O z , (2.2) η (τ1)η (τ2) where we have set 3 3 1728E4 (τj) E4 (τj) j(τj) = 3 2 = 24 , E4 (τj) − E6 (τj) η (τj) 2 2 1728E6 (τj) E6 (τj) 1728 − j(τj) = 3 2 = 24 . E4 (τj) − E6 (τj) η (τj) The following fact follows from the above asymptotic analysis [8]:

Proposition 2.2. The modulus point τ is equivalent to a point with z = 0 or [τ] ∈ H1 ⊂ A2 such that the principally polarized abelian surface is a product of two elliptic curves Aτ = Eτ1 × Eτ2 if and only if χ10(τ) = 0. The elliptic modular parameters are determined by equation (2.2). A Universal Genus-Two Curve from Siegel Modular Forms 5

2.3 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), jC) where Jac(C) is an abelian surface, called the Jacobian variety of the curve C, and jC : 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 N´eron–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 6 6 2 Y X i C : y = f(x) = a0 (x − αi) = aix . (2.3) i=1 i=0 6 1 The roots {αi}i=1 of the sextic are the six ramification points of the map C → P . Their pre- images 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 GL2(C) which takes the set of roots to the roots of the other. The ring of invariants of binary sextics is generated by the Igusa invariants (J2,J4,J6,J10) as defined in [14, equation (9)], which are the same invariants as the ones denoted by (A0,B0,C0,D0) in [18, p. 319] and also the same invariants as (A, B, C, D) in [11, p. 176]. For expressions of such invariants in terms of the coefficients a0, . . . , a6 of the binary sextic, or Jk ∈ Z[a0, . . . , a6] for k ∈ {2, 4, 6, 10}; see [14, equation (11)] and in terms of thetanulls see [17]. One can then ask what the Igusa invariants of a genus-two curve C defined by a sextic curve f are in terms of τ such that (τ, I2) ∈ Mat(2, 4; C) is the period matrix of the principally polarized abelian surface Aτ = Jac(C). Based on the asymptotic behavior in Equations (2.2) and (2.2), Igusa [12, p. 848] proved that the relations are as follows

3 χ12(τ) J2 = −2 · 3 , χ10(τ) 2 J4 = 2 ψ4(τ), 3 2 5 ψ4(τ)χ12(τ) J6 = − ψ6(τ) − 2 , 3 χ10(τ) 14 J10 = −2 χ10(τ).

Thus, the invariants of a sextic define a point in a weighted projective space [J2 : J4 : J6 : 3 J10] ∈ WP(2,4,6,10) that equals 3 2 2 2 3 2 2 2 6 2 3(3χ12) : 2 3 ψ4χ10 : 2 3 4ψ4(3χ12) + ψ6χ10 χ10 : 2 χ10 . Torelli’s theorem states that the map sending a curve C to its Jacobian variety Jac(C) induces a birational map from the moduli space M2 of genus-two curves to the complement of the Humbert surface H1 in A2, i.e., A2 − supp (χ10)0. In other words, points in the projective variety Proj C[J2,J4,J6,J10] which are not on J10 = 0 are in one-to-one correspondence with isomorphism classes of regular sextics [11]. 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 (J2,J4,J6,J10) with rational coefficients: 1 A = − J , 233 · 5 2 6 A. Malmendier and T. Shaska 1 B = J 2 + 20J , 233354 2 4 1 C = − J 3 + 80J J − 600J , (2.4) 253556 2 2 4 6 1 D = − 9J 5 + 700J 3J − 3600J 2J − 12400J J 2 + 48000J J + 10800000J . 2839510 2 2 4 2 6 2 4 4 6 10 For formulas giving relations between all these sets of invariants see [1].

2.3.1 Absolute invariants

Dividing any SL2(C) invariant by another one of the same degree gives an invariant under GL2(C) action. The term absolute invariants is used first by Igusa [10] for GL2(C) invariants. It was the main result of [11, Theorem 3] that

J4 J2J4 − 3J6 J10 x1 = 144 2 , x2 = −1728 3 , x3 = 486 5 , J2 J2 J2 3 for J2 6= 0. We use x1, x2, x3 to write the point [J2 : J4 : J6 : J10] ∈ WP(2,4,6,10) as 1 1 1 1 1 : x : x + x : x . 2432 1 2634 2 2433 1 2 · 35 3

Since the invariants J4, J6, J10 vanish simultaneously for sextics with triple roots all such curves 3 are mapped to [1 : 0 : 0 : 0] ∈ WP(2,4,6,10) with uniformizing affine coordinates x1, x2, x3 around it. Blowing up this point gives a variety that parameterizes genus-two curves with J2 6= 0 and their degenerations. In the blow-up space we have to introduce additional coordinates that are obtained as ratios of x1, x2, x3 and have weight zero. Those are precisely the coordinates y1, y2, y3 already introduced in equation (2.1). It turns out that the coordinate ring of the blown-up space is C[x1, x2, x3, y1, y2, y3]. We introduce the three absolute invariants 2 2 2 4 9J − 320J4 J + 20J4 ρ = − 2 2 , 3 2 3J2 + 140J2J4 − 800J6 2 2 48 J + 20J4 σ = − 2 3 3 3J2 + 140J2J4 − 800J6 5 3 2 2 × 9J2 − 700J2 J4 + 2400J2 J6 − 262400J2J4 + 768000J4J6 + 172800000J10 , 4 2 2 2 2 27J + 2380J J4 − 12000J2J6 + 12800J J + 20J4 κ = 2 2 4 2 . (2.5) 3 2 3J2 + 140J2J4 − 800J6 It follows: Lemma 2.3. For invariants (ρ, σ, κ) given by equation (2.5) such that ρ and κ do not vanish 3 simultaneously, a point [J2 : J4 : J6 : J10] in WP(2,4,6,10) is given by 9 J = 8(κ − ρ),J = (κ − ρ)2 + 45ρ, 2 4 5 111 J = (κ − ρ)3 − 30(κ − ρ)2 + 63ρ(κ − ρ) − 270ρ, 6 25 6 4 46 J = (κ − ρ)5 + (κ − ρ)4 + ρ(κ − ρ)3 10 3125 15 75 1 42 3 + − σ + ρ (κ − ρ)2 + 12ρ2(κ − ρ) + ρ(36ρ − σ). (2.6) 6 5 2

In particular, for J2 6= 0 we have Q(x1, x2, x3) = Q(ρ, σ, κ). A Universal Genus-Two Curve from Siegel Modular Forms 7

Proof. The proof is computational. We express ρ, σ, κ as rational functions of x1, x2, x3 and vice versa over Q. The condition that ρ and κ do not vanish simultaneously is based on the fact that J2, J4, J6, J10 must not vanish simultaneously. 3 Remark 2.4. Consider the image of [J2 : J4 : J6 : J10] in WP(2,4,6,10) under the morphism 3 5 WP(2,4,6,10) → P given by

2 3 2 5 486J4J6 : 486J10 : −1728(J2J4 − 3J6)J2 : 144J2 J4 : 20736J2J4 : J2 , (2.7)

5 which is a linear transformation of the usual morphism to P given by

2 3 2 5 [J2 : J4 : J6 : J10] 7→ J10 : J4J6 : J2 J6 : J2 J4 : J2J4 : J2 .

For J2 6= 0, points in equation (2.7) equal

1 x (x + 12x ): x : x : x : x2 : 1 . 1536 1 2 1 3 2 1 1

The invariants x1, x2, x3 are not defined for J2 = 0, but ρ, σ, κ remain well-defined if ρ = κ 6= 0. In this case we have 3 J (0) = 0,J (0) = 45ρ, J (0) = −270ρ, J (0) = ρ(36ρ − σ), 2 4 6 10 2 and the invariants ρ and σ with

3 3 3 4 J4 144 J4 J4 J10 ρ = κ = 2 , σ = 2 + 6480 2 , 5 J6 5 J6 J6 J4J6 determine genus-two curves with J2 = 0, J4 · J6 6= 0 up to isomorphism. In addition to J2 = 0, we have J10 = 0 if and only if σ = 36ρ. Using = (κ − ρ) in equation (2.6), one checks that points in equation (2.7) up to terms of order O2 equal " # 7 J (0) 2 512 1 − : 10 − : 0 : 0 : − : 0 . 30 (0) (0) 2025 9 J4 J6

5 This means that under the usual morphism to P the regular genus-two curves with J2 = 0 and constant ratio J10/(J4J6) are mapped to the same point.

2.4 Recovering the equation of the curve from invariants

Let p ∈ M2 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 = [Aij] 2 be the symmetric matrix corresponding to this conic. Let X = [X1 : X2 : X3] ∈ P and

3 t X Q: X · M · X = AijXiXj = 0. (2.8) i,j=1

Clebsch [5] determined the entries of this matrix M as follows 1 A = 2C + AB, 11 3 8 A. Malmendier and T. Shaska

A22 = A13 = D, 1 2 A = BD + CB2 + AC, 33 2 9 1 1 1 A = BB2 + AC + C 2C + AB , 23 3 3 3 2 A = B2 + AC. (2.9) 12 3 The coefficients are obtained as follows: from the sextic f in equation (2.3) three binary quadrics yi(x) with i = 1, 2, 3 are obtained by an operation called ‘Uberschiebung’¨ [18, p. 317] or transvection. The quadrics yi for i = 1, 2, 3 have the property that their coefficients are polynomial expressions in the coefficients of f with rational coefficients. Moreover, under the operation f(x) 7→ f˜(x) = f(−x) the quadrics change according to yi(x) 7→ ˜yi(x) = yi(−x) for i = 1, 2, 3. Hence, they are not invariants of the sextic f. The coefficients Aij in equation (2.9) 1 satisfy Aij = (yiyj)2. Therefore, the coefficients Aij are invariant under the operation f(x) 7→ f˜(x) = f(−x), and the locus D = 0 is equivalent to

D = 0 ⇔ (y1y3)2 = (y2y2)2 = 0.

We define R to be 1/2 times the determinant of the three binary quadrics yi for i = 1, 2, 3 with respect to the basis x2, x, 1. If one extends the operation of Uberschiebung¨ by product rule [18, p. 317], then R can be re-written as

R = −(y1y2)1(y2y3)1(y3y1)1, or, equivalently, as 1 R = − y y y − y y y − y y y 8 1,yy 2,xy 3,xx 1,yy 2,xx 3,xy 1,xy 2,yy 3,xx + y1,xyy2,xxy3,yy + y1,xxy2,yyy3,xy − y1,xxy2,xyy3,yy .

It is then obvious that under the operation f(x) 7→ f˜(x) = f(−x) the determinant R changes its sign, i.e., R(f) 7→ R(f˜) = −R(f). A straightforward calculation shows that

A11 A12 A13 2 1 R = A12 A22 A23 , 2 A13 A23 A33

2 where Aij are the invariants in equation (2.9). Like the coefficients Aij, R is invariant under the operation f(x) 7→ f˜(x) = f(−x) and must be a polynomial in (J2,J4,J6,J10). Substituting (2.3) into the Clebsch invariants and then equation (2.9) it follows that

2 2 9 −9 −10 χ35(τ) R = 2 3 5 i 2 . (2.10) χ10(τ)

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 M2 see [19]. We have the following lemma summarizing our discussion:

1For two binary forms f, g of degree m and n, respectively, we denote the Uberschiebung¨ of order k by k m (fg)k = (−1) (gf)k. For f˜(x) = f(−x) andg ˜(x) = g(−x) and m = n = k, we have (fg)m = (−1) (f˜g˜)m. A Universal Genus-Two Curve from Siegel Modular Forms 9

Lemma 2.5. We have the following statements:

2 1.R is an order 30 invariant of binary sextics expressed as a polynomial in (J2,J4,J6,J10) as in [19, equation (17)] given by plugging Clebsch invariants and (2.9) into equation (2.10).

2. The locus of curves p ∈ M2 such that V4 ,→ Aut(p) is a two-dimensional irreducible 2 rational subvariety of M2 given by the equation R = 0 and a birational parametrization given by the u, v-invariants as in [19, Theorem 1]. We have introduced the invariant R2 for any binary sextic f. To the corresponding symmetric matrix M with coefficients Aij = (yiyj)2 of order zero and invariant under the operation f(x) 7→ f˜(x) = f(−x), we associated a conic Q. Similarly, there is also a cubic curve given by the equation X T : aijkXiXjXk = 0, (2.11) 1≤i,j,k≤3 ˜ where the coefficients aijk are of order zero and invariant under f(x) 7→ f(x) = f(−x). In terms of ‘Uberschiebung’¨ the coefficients are obtained by

aijk = (fyi)2(fyj)2(fyk)2.

The coefficients aijk are given explicitly as follows: 2 36a111 = 8 A C − 6BC + 9D , 3 2 36a112 = 4 2B + 4ABC + 12C + 3AD , 3 2 2 2 36a113 = 36a122 = 4 AB + 4/3A BC + 4B C + 6AC + 3BD , 4 2 2 2 2 36a123 = 2 2B + 4AB C + 4/3A C + 4BC + 3ABD + 12CD , 4 2 2 3 2 3 2 36a133 = 2 AB + 4/3A B C + 16/3B C + 26/3ABC + 8C + 3B D + 2ACD , 4 2 2 2 2 36a222 = 4 3B + 6AB C + 8/3A C + 2BC − 3CD , 3 2 3 2 36a223 = 2 − 2/3B C − 4/3ABC − 4C + 9B D + 8ACD , 5 3 2 2 2 2 2 36a233 = 2 B + 2AB C + 8/9A BC + 2/3B C − BCD + 9D , 4 2 2 2 3 3 3 2 36a333 = −2B C − 4AB C − 16/9A C − 4/3BC + 9B D + 12ABCD + 20C D. The relations between all aforementioned invariants and Siegel modular forms, in particular the 2 relation between χ35 and R can be found in [1]. Since ‘Uberschiebung’¨ preserves the rationality of the coefficients, we have the following corollary:

Corollary 2.6. Let p ∈ M2 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 ∈ M2(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 y2 = f(x). The main question is if the sextic given by y2 = 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 Xg with hyperelliptic involution w is deﬁned over the K if and only if the conic Q = Xg/hwi has a K-rational point. 10 A. Malmendier and T. Shaska

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 (h1(x), h2(x), h3(x)). Substitute X1, X2, X3 by h1(x), h2(x), h3(x) in the cubic T and we get the degree 6 polynomial f(x). 2 1 However, if the conic has no rational point or R = 2 det M = 0 the method obviously fails. In Section3 we determine the intersection T ∩ Q over a quadratic extension which is always possible.

3 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:

Lemma 3.1. The conic Q in equation (2.8) for J4 · J6 · J10 6= 0 is equivalent over Q[J2, ρ, σ, κ] to the conic 0 2 2 2 Q : x1 − γx2 − Λ6x3 = 0, (3.1) where (ρ, σ, κ) are the absolute invariants in equation (2.5), γ = ρ2 + σ and 3 2 2 2 2 Λ6 = −γ − 27ργ − 81ρ (ρ + 12)γ + 729ρ (ρ + 12) + −6ργ2 + 54ρ(5ρ + 36)γ − 1944ρ2(ρ + 12)κ + 9γ2 − 9ρ(ρ + 36)γ + 162ρ(ρ2 + 32ρ + 144)κ2 + ((30ρ + 216)γ − 432ρ(ρ + 12))κ3 + 9ρ2 − 24γ + 504ρ + 1296κ4 + (−24ρ − 288)κ5 + 16κ6. (3.2)

Moreover, for J2, ρ, σ, κ ∈ Q the conic Q in equation (2.8) has a rational point if and only if the conic Q0 in equation (3.1) does. Proof. For the conic Q in equation (2.8), we apply the coordinate transformation given by 4 2 X1 = 2(AB + 6C) AC + B x1 2 2 2 2 4 + 108B(AB + 6C) 4A C + 8AB C + 4B − 3ABD − 18CD x2 3 2 2 3 5 3 2 2 2 + 41990B 8A BC + 14AB C + 6B + 12AC + 12B C − 27D x3, 5 3 X2 = −(AB + 6C) x1 − 419904B x3 × 4A2B2C + 3AB4 + 30ABC2 + 18B3C − 18ACD − 18B2D + 36C3, 6 9 3 2 2 2 4 X3 = −2 3 B 4A C + 8AB C + 4B − 3ABD − 18CD x3. (3.3) We then obtain the conic Q0 in equation (3.1). Equation (3.3) can be rewritten as transformation over Q[J2, ρ, σ, κ] using equations (2.4) and (2.5). We have the following lemma: 0 Lemma 3.2. Assume ρ, σ, κ ∈ Q. The conic Q in equation (3.1) has a rational point if and only if there are rational numbers α, β ∈ Q such that 2 2 α + Λ6β σ = γ. (3.4) The rational point on the conic Q0 is then given by 0 0 0 x1 : x2 : x3 = [αρ + γ : α + ρ : βσ]. (3.5) Conversely, every rational point on the conic Q0 can be written in the form of equation (3.5) for some rational numbers α, β ∈ Q satisfying equation (3.4). A Universal Genus-Two Curve from Siegel Modular Forms 11

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:

Lemma 3.3. Assume that a point on the conic in equation (3.1) is given by equation (3.5) with 0 x2 6= 0 which is always possible if ρ 6= 0. Then every point on the conic is given by

2 2 x1 = (αρ + γ)U + 2Λ6βσUV + Λ6(αρ + γ)V , 2 2 x2 = (α + ρ)U − Λ6(α + ρ)V , 2 2 x3 = βσU + 2(αρ + γ)UV + Λ6βσV , (3.6)

1 for some [U : V ] ∈ P . The parametrization in equation (3.6) is a rational parametrization of 0 the conic Q if and only if α, β, ρ, κ, σ ∈ Q. Proof. If a point of Q0 is obtained from some (rational) values (α, β) then there are three more (rational) points given by setting (α, β) 7→ (±α, ±β). If ρ 6= 0, one of these points satisfies 0 x2 = α + ρ 6= 0. The proof then follows from the known formulas parametrizing conics for 0 x2 6= 0 given by

0 2 0 0 2 x1 = ax1U − 2cx3UV − cx1V , 0 2 0 2 x2 = ax2U + cx2V , 0 2 0 0 2 x3 = ax3U + 2ax1UV − cx3V ,

0 0 0 where a = 1, b = −γ, c = −Λ6 and x1, x2, x3 were given in equation (3.5). Remark 3.4. If α = ρ = 0 and γ 6= 0, a formula similar to equation (3.6) can be found using 0 the fact that x1 6= 0 in equation (3.5) in this case. Remark 3.5. If a point of Q0 is obtained for some (rational) values (α, β) then three more (rational) points on Q0 are given by setting (α, β) 7→ (±α, ±β) in equation (3.5).

Changing from coordinates [X1 : X2 : X3] to coordinates [x1 : x2 : x3] transforms the conic Q in equation (2.8) into the conic Q0 in equation (3.1). Similarly, under the same change of coordinates the cubic T in equation (2.11) becomes

3 3 3 2 3 2 2 Λ1(18γ + Λ3)x1 + γ Λ2x2 − (γ − Λ1)Λ6x3 + 3Λ1Λ6x1x3 + 3γ(9γρ + κΛ3)x1x2 2 2 2 2 2 2 + 3γ Λ3x1x2 + 3Λ5Λ6x1x3 + 3γΛ4Λ6x2x3 + 3γ Λ6x2x3 + 6γκΛ6x1x2x3, (3.7) with coefficients given by

2 Λ1 = 9ρ + κ , 2 Λ2 = γ + 18ρ + 3ρκ − 4κ , 2 3 Λ3 = 27ρ(ρ + 12) + (γ − 36ρ)κ + 3(ρ + 12)κ − 4κ , 2 2 3 4 Λ4 = −γ − 9γρ − 3(γρ − 9ρ(ρ + 12))κ + (5γ − 36ρ)κ + 3(ρ + 12)κ − 4κ , 2 2 2 Λ5 = −27γρ(ρ + 6) + 243ρ (ρ + 12) − γ − 45γρ + 324ρ κ − (3γ(ρ − 6) − 54ρ(ρ + 12))κ2 + (5γ − 72ρ)κ3 + 3(ρ + 12)κ4 − 4κ5. (3.8)

We also discuss the conic, cubic, rational point and parametrization in the cases where J2 = 0 and J4 · J6 = 0: 12 A. Malmendier and T. Shaska

Lemma 3.6. If J2 = J4 = 0 and J6 · J10 6= 0, the conic Q in equation (2.8) is equivalent over Q[J6,J10] to the conic

0 2 2 2 Q : x1 − µx2 − (1 − µ)x3 = 0,

5 4 4 5 3 0 0 0 with µ = J6 / 2 3 5 J10 and a rational point given by x1 : x2 : x3 = [1 : 1 : 1]. A rational parametrization of Q0 is then given by

2 2 2 2 x1 = U + 2(1 − µ)UV + (1 − µ)V , x2 = U − (1 − µ)V , 2 2 x3 = U + 2UV + (1 − µ)V

1 with [U : V ] ∈ P . Under the same change of coordinates the cubic T in equation (2.11) becomes

0 3 2 3 2 3 2 T : 0 = 2x1 − µ x2 − 2(1 − µ) x3 − 6µx1x2 2 2 2 − 6(1 − µ)x1x3 + 6(1 − µ)x1x3 − 3µ(1 − µ)x2x3.

If J2 = J6 = 0 and J4 · J10 6= 0, the conic Q in equation (2.8) is equivalent over Q[J4,J10] to the conic

0 2 2 2 Q : x1 − x2 − (1 − ν)x3 = 0,

5 2 5 5 2 0 0 0 with ν = J4 / 2 3 5 J10 and a rational point given by x1 : x2 : x3 = [1 : 1 : 0]. A rational parametrization of Q0 is then given by

2 2 2 2 x1 = U + (1 − ν)V , x2 = U − (1 − ν)V , x3 = 2UV

1 with [U : V ] ∈ P . Under the same change of coordinates the cubic T in equation (2.11) becomes 0 2 3 2 3 2 3 2 T : 0 = 1 − ν x1 − ν x2 − (1 − ν) x3 + ν(1 − 3ν)x1x2 2 2 2 − (1 − ν)(3 + ν)x1x3 + ν(1 − 3ν)x1x2 − ν(1 − ν)x2x3 2 2 + (1 − ν)(3 − ν)x1x3 + ν(1 − ν)x2x3 − 2ν(1 − ν)x1x2x3.

Proof. The proof is analogous to the proofs of Lemmas 3.2 and 3.3. Remark 3.7. The absolute invariants (ρ, σ, κ) in equation (2.5) such that ρ and κ do not vanish simultaneously and J10 6= 0 describe the moduli of genus-two curves with J4 · J6 · J10 6= 0. The discussion of Lemma 3.6 proves that only for genus-two curves with J4 · J6 · J10 6= 0, the conic Q in equation (2.8) is not guaranteed to have a rational point.

Substituting the parametrization of the conic Q0 in Lemma 3.3 into the cubic T 0 in equation (3.7) and setting U = x and V = 1, one obtains the ramification locus of a sextic curve. 6 P i The ramification locus is equivalent to f(x) = ai(α, β)x = 0, where we write the sextic i=0 polynomial in the form

(1) (2) 6 (1) (2) 5 (1) (2) 4 f(x) = d0 + d0 x + d1 + d1 Λ6x + d2 + d2 Λ6x (1) 2 3 (1) (2) 2 2 (1) (2) 3 (1) (2) 3 + d3 Λ6x + d2 − d2 Λ6x + d1 − d1 Λ6x + d0 − d0 Λ6. (3.9)

0 0 0 In terms of the coordinates of the point x1 : x2 : x3 in equation (3.5) we have set

(1) (1) 2 0 0 (1) 0 0 δj 0 dj = cj,0 ρ + γ x1 + 2γρx2 γ + 3cj,1 x1 + ρx2 γΛ6 x3 (1) 2 0 0 2 −2 02 (1) 0 0 2 −2 1+δj 03 − cj,0 ρ + γ x1 − 2γρx2 ρ σ Λ6 x3 − 3cj,1 x1 − ρx2 ρ σ Λ6 x3 A Universal Genus-Two Curve from Siegel Modular Forms 13

(1) 2 (1) −1 0 02 (1) 1+δj 03 + 3σcj,2 − 2ρ cj,0 σ Λ6x1 x3 + cj,3 Λ6 x3 , (2) (2) 0 2 0 2 (2) 0 0 δj 0 dj = cj,0 2ρx1 + ρ + γ x2 γ + 6cj,1 ρx1 + γx2 γΛ6 x3 (2) 0 2 0 −2 02 (2) 0 0 −2 1+δj 03 − cj,0 2γx1 − ρ ρ + γ x2 γρσ Λ6 x3 − 6cj,1 x1 − ρx2 γρσ Λ6 x3 (2) δj 0 02 + cj,2 γΛ6 x1 x3 . (3.10) 3 All coefficients remain regular and in general non-vanishing for σ = 0 since x0/σ = β. Here, (n) (α, β) is a pair solving equation (3.4), and the coefficients cj,k are given by

(1) (1) c0,0 = 18γΛ1 + 3γΛ3 + Λ1Λ3, c0,1 = γ + Λ1, (1) (1) c0,2 = −γΛ3 + Λ5, c0,3 = −4γ + Λ1, (2) (2) c0,0 = 27γρ + γΛ2 + 3κΛ3, c0,1 = κ, (2) c0,2 = −γΛ2 + 3Λ4, (1) (1) c1,0 = 6(γ + Λ1), c1,1 = 2(18γΛ1 + γΛ3 + Λ1Λ3 + 2Λ5), (1) (1) c1,2 = 2(−2γ + 3Λ1), c1,3 = 6(−γΛ3 + Λ5), (2) (2) c1,0 = 12κ, c1,1 = 18γρ + 2κΛ3 + 2Λ4, (2) c1,2 = 12κΛ6, (1) (1) c2,0 = 3(18γΛ1 − γΛ3 + Λ1Λ3 + 4Λ5), c2,1 = 5(−γ + 3Λ1), (1) (1) c2,2 = 72γΛ1 + γΛ3 + 4Λ1Λ3 + 11Λ5, c2,3 = 15Λ1, (2) (2) c2,0 = 3(9γρ − γΛ2 + κΛ3 + 4Λ4), c2,1 = 5κ, (2) c2,2 = 3(36γρ + γΛ2 + 4κΛ3 + Λ4), (1) (1) c3,0 = 20(−γ + Λ1), c3,1 = 4(18γΛ1 − γΛ3 + Λ1Λ3 + 4Λ5), (1) (1) c3,2 = 20Λ1, c3,3 = 4(36γΛ1 + 3γΛ3 + 2Λ1Λ3 + 3Λ5).

The coefficients Λ1,..., Λ5 and Λ6 were given in equations (3.8) and (3.2), respectively. Remark 3.8. Equation (3.10) allows to easily describe the change in the sextic polynomial 0 under the action of the automorphism of the conic Q given by [x1 : x2 : x3] 7→ [±x1 : ±x2 : x3]. We make the following remarks:

Λ6 (1) (2) (1) (2) Remark 3.9. The transformation x → x maps the coefficients dk ± dk 7→ dk ∓ dk for k = 0, 1, 2 and a3 7→ a3. This is to be expected since the coefficients are in terms of invariants 1 of the binary sextic f(x, z) and x → x just permutes x and z.

Remark 3.10. The fact that the coefficients defining C(α,β) are polynomials of the new absolute invariants γ, κ, σ and appear in the particular pattern given by equation (3.10) opens up the question about their meaning. It turns out that in the context of the F-theory/heterotic string theory duality the new invariants parameterize the physical defects of a certain class of 6d N = (1, 0) non-geometric vacua of the heterotic string when dualizing to F-theory. We will address this question in more detail in future work. We have the following main result:

Theorem 3.11. Let p ∈ M2 such that p ∈ M2(K), for some number ﬁeld K, and j = [J2 : 3 J4 : J6 : J10] the corresponding point in WP(2,4,6,10)(OK ), where OK is the ring of integers of K. A genus-two curve corresponding to p is constructed as follows: 14 A. Malmendier and T. Shaska

i) If J2 · J10 6= 0 there is a genus-two curve C(α,β) given by

6 2 X i C(α,β) : y = ai(α, β)x , (3.11) i=0

with coeﬃcients given in equations (3.9) and (3.10), and a pair (α, β) satisfying

2 2 α + Λ6β σ = γ,

where Λ6, σ, and γ are determined by p. Moreover, C(α,β) is deﬁned over its ﬁeld of moduli K, i.e., ai(α, β) ∈ K, i = 0,..., 6, if and only if K-rational α and β exist.

ii) If J2 = 0 and J4 · J6 · J10 6= 0, there is a genus-two curve given by setting ρ = κ 6= 0 in equation (3.11).

iii) If J2 = J6 = 0 and J4 · J10 6= 0, there is only one genus-two curve given by

y2 = (4ν + 1)(2ν − 1)x6 + 2(1 − ν)(4ν + 3)x5 − 15(1 − ν)x4 + 20(1 − ν)2x3 + 5(2ν − 3)(1 − ν)2x2 + 6(1 − ν)3x − (1 − ν)3

5 2 5 5 2 with ν = J4 / 2 3 5 J10 .

iv) If J2 = J4 = 0 and J6 · J10 6= 0, there is only one genus-two curve given by

y2 = 5x6 + 12(1 − µ)x5 − 15(1 − µ)x4 − 80(1 − µ)2x3 + 15(4µ − 7)(1 − µ)2x2 − 60(1 − µ)3x + (4µ − 13)(1 − µ)3.

5 4 4 5 3 with µ = J6 / 2 3 5 J10 .

v) If J2 = J4 = J6 = 0 and J10 6= 0, there is only one genus-two curve given by

y2 = x6 − x.

Proof. We already proved that there are genus-two curves y2 = f(x) corresponding to p, where f(x) is given in equation (3.9). We obtain coefficients ai(α, β) ∈ Q[α, β, J2,J4,J6,J10] for 0 ≤ i ≤ 6. The field of moduli K of the point p is K = Q(x1, x2, x3). For J2 6= 0 the invariants (ρ, σ, κ) are birationally equivalent to (x1, x2, x3) over Q by Lemma 2.3. By Lemma 3.1 the conic Q in equation (2.8) had a K-rational point if and only if the conic Q0 in equation (3.1) does. By Lemma 3.2 the conic Q0 has a K-rational point, i.e., there is a K- rational solution (α, β) of equation (3.4). Therefore, ai(α, β) ∈ K, for i = 0,..., 6. The cases with J4 · J6 = 0 are similarly obtained by applying Lemmas 2.3 and 3.6. This completes the proof.

Remark 3.12. The four pairs (±α, ±β) belong to the same conic Q0. 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 J2 6= 0. The equation is valid even when the field of moduli is not a field of definition. Hence, for every point p ∈ M2 we get a curve. Next we have the following result: A Universal Genus-Two Curve from Siegel Modular Forms 15

Corollary 3.13. For every point p ∈ M2 such that p ∈ M2(K), for some number ﬁeld K, there is a genus-two curves C given by

6 2 X i C(α,0) : y = ai(α, 0)x , i=0 corresponding to p, such that ai(α, 0) ∈ K(α), i = 0,..., 6 as given in equation (3.9). Moreover, 2 C(α,0) is at worst deﬁned over the quadratic extension K(α) of the ﬁeld of moduli K with α = ρ2 + σ.

We have the immediate consequence:

Corollary 3.14. Let x1, x2, x3 be transcendentals. There exists a genus-two curve C(α,0) deﬁned 2 2 over Q(x1, x2, x3)[α] with α = ρ + σ such that

x1(C(α,0)) = x1, x2(C(α,0)) = x2, x3(C(α,0)) = x3.

We have the following corollary:

Corollary 3.15. Let σ = 0 and ρ 6= 0 for p ∈ M2. Then, there is a genus-two curve C given by Corollary 3.13, and it is deﬁned over the ﬁeld of moduli.

Proof. For σ = 0 and ρ 6= 0, we have γ = ρ2, and we choose the K-rational solution (α, β) = (ρ, 0) in equation (3.4).

Remark 3.16. It is easy to check using equation (2.5) that the locus σ = 0 and ρ 6= 0 for p ∈ M2 corresponds to the locus

−11 −3 −5 5 3 2 2 J10 = −2 3 5 9J2 − 700J2 J4 + 2400J2 J6 − 262400J2J4 + 768000J4J6 .

We have the following lemma:

Lemma 3.17. In terms of the invariants ρ, σ, κ and γ = ρ2 + σ, we have

J 5(κ − ρ)2 + 9ρ(2κ − ρ)2 − γ D = − 2 , 2173755(κ − ρ)5 15 2 3 J (κ − ρ) + 9ρ Λ6 R2 = 2 . 254321515(κ − ρ)15

2 In particular, the locus D = 0 and χ35 6= 0 is given by γ = (2κ − ρ) or, equivalently, σ = 4κ(κ − ρ).

We have the following corollary:

2 Corollary 3.18. Let D = 0 and χ35 6= 0 for p ∈ M2. Then, there is a genus-two curve C given by Corollary 3.13, and it is deﬁned over the ﬁeld of moduli.

Proof. For γ = (2κ − ρ)2 we can choose (α, β) = (ρ − 2κ, 0) in equation (3.4). As κ − ρ 6= 0 we have y0 6= 0 in equation (3.5). 16 A. Malmendier and T. Shaska

3.1 A word about extra automorphisms

In this section we derive a sextic polynomial for the sublocus of M2 with χ35 = 0. We have the following proposition:

Proposition 3.19. Let D 6= 0 and χ35 = 0 for p ∈ M2. Then, there is a genus-two curve C : y2 = F (x) with

(1) (2) 6 (1) (2) 4 (1) (2) 2 (1) (2) F (x) = d0 + d0 x + d2 + d2 x + d2 − d2 x + d0 − d0 , (3.12) and with coeﬃcients in Z[α, ρ, κ] given by

(1) 2 2 2 2 d0 = 3κγ − κ + 9ρ (11κ − 9ρ − 126)γ − κ + 9ρ (4κ − 3ρ − 36), (2) 2 2 2 d0 = γ + −κ + 3κρ + 45ρ γ − 3κ κ + 9ρ (4κ − 3ρ − 36) α, (1) 2 2 2 2 d2 = −15κγ + 15 κ + 9ρ (5κ − 3ρ − 18)γ − 15 κ + 9ρ) (4κ − 3ρ − 36), (2) 2 2 2 d2 = −15γ + 75κ − 45κρ − 135ρ γ − 15κ κ + 9ρ (4κ − 3ρ − 36) α. (3.13)

Here, the absolute invariants α, γ, ρ, κ are subject to the constraints Λ6 = 0 in equation (3.2) and α2 = γ.

2 Proof. For Λ6 = with → 0, we rescale the polynomial in equation (3.12) according to 6 2 f(x)/ before setting = 0. If we substitute Λ6 = 0 into equation (3.4) we obtain α = γ, β = 0. Therefore, we will use the absolute invariants α, γ, ρ, κ subject to the constraints 2 Λ6 = 0 in equation (3.2) and α = γ. The sextic polynomial in equation (3.12) has coefficients in Z[α, ρ, κ]. The remainder of the proof then follows from specializing the formulas in equation (3.13) to β = Λ6 = 0.

The polynomial in equation (3.12) is a twist of the polynomial given by

Fˆ(x) = x6 + ax4 + bx2 + 1.

The curve y2 = Fˆ(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

u = ab, v = a3 + b3, were defined which give a birational parametrization of this locus L2 which is a two-dimensional subvariety of M2. We have the following:

2 Corollary 3.20. For the genus-two curve C : y = F (x) given by equation (3.12) with χ35 = 0 we obtain the dihedral invariants

d(1) + d(2)d(1) − d(2) u = 2 2 2 2 , (1) (2) (1) (2) d0 + d0 d0 − d0 d(1) − d(2)3 d(1) + d(2)3 v = 2 2 + 2 2 , (1) (2) (1) (2)2 (1) (2)2 (1) (2) d0 + d0 d0 − d0 d0 + d0 d0 − d0 and the Igusa invariants [J2 : J4 : J6 : J10] given by [19, equation (16)]. A Universal Genus-Two Curve from Siegel Modular Forms 17 References

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