A Universal Genus-Two Curve from Siegel Modular Forms
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 form...
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irk-123456789-1492682019-02-20T01:23:42Z A Universal Genus-Two Curve from Siegel Modular Forms Malmendier, A. Shaska, T. 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. 2017 Article A Universal Genus-Two Curve from Siegel Modular Forms / A. Malmendier, T. Shaska // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 19 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14H10; 14H45 DOI:10.3842/SIGMA.2017.089 http://dspace.nbuv.gov.ua/handle/123456789/149268 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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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. |
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Malmendier, A. Shaska, T. |
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Malmendier, A. Shaska, T. A Universal Genus-Two Curve from Siegel Modular Forms Symmetry, Integrability and Geometry: Methods and Applications |
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Malmendier, A. Shaska, T. |
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A Universal Genus-Two Curve from Siegel Modular Forms |
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A Universal Genus-Two Curve from Siegel Modular Forms |
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A Universal Genus-Two Curve from Siegel Modular Forms |
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A Universal Genus-Two Curve from Siegel Modular Forms |
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A Universal Genus-Two Curve from Siegel Modular Forms |
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universal genus-two curve from siegel modular forms |
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Інститут математики НАН України |
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A Universal Genus-Two Curve from Siegel Modular Forms / A. Malmendier, T. Shaska // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 19 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT malmendiera auniversalgenustwocurvefromsiegelmodularforms AT shaskat auniversalgenustwocurvefromsiegelmodularforms AT malmendiera universalgenustwocurvefromsiegelmodularforms AT shaskat universalgenustwocurvefromsiegelmodularforms |
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2025-07-12T21:12:07Z |
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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: andreas.malmendier@usu.edu
‡ Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA
E-mail: shaska@oakland.edu
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
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
mailto:andreas.malmendier@usu.edu
mailto:shaska@oakland.edu
https://doi.org/10.3842/SIGMA.2017.089
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
C(α,β) : y2 =
6∑
i=0
ai(α, β)xi,
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 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.
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.,
H2 =
{
τ =
(
τ1 z
z τ2
)∣∣∣∣ τ1, τ2, z ∈ C, Im (τ1) Im (τ2) > Im (z)2, Im (τ2) > 0
}
,
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
Aτ = C2/
〈
Z2 ⊕ τZ2
〉
with period matrix (τ , I2) ∈ Mat(2, 4;C). Two abelian surfaces Aτ and Aτ ′ are isomorphic if
and only if there is a symplectic matrix
M =
(
A B
C D
)
∈ Γ2,
such that τ ′ = M(τ) := (Aτ +B)(Cτ +D)−1. 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
aτ1 + bz + cτ3 + d
(
z2 − τ1τ2
)
+ e = 0,
with integers a, b, c, d, e satisfying ∆ = b2−4ac−4de and τ =
(
τ1 z
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
ψ2k(τ) =
∑
(C,D)
det(C · τ +D)−2k,
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
χ10(τ) = − 43867
21235527 · 53
(ψ4(τ)ψ6(τ)− ψ10(τ)) .
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
χ12(τ) =
131 · 593
213375372337
(
3272ψ3
4(τ) + 2 · 53ψ2
6(τ)− 691ψ12(τ)
)
.
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
χ2
35 =
1
21239
χ10
(
224315χ5
12 − 21339ψ3
4χ
4
12 − 21339ψ2
6χ
4
12 + 33ψ6
4χ
3
12 − 2 · 33ψ3
4ψ
2
6χ
3
12
− 21438ψ2
4ψ6χ10χ
3
12 − 22331252ψ4χ
2
10χ
3
12 + 33ψ4
6χ
3
12 + 2113637ψ4
4χ
2
10χ
2
12
+ 211365 · 7ψ4ψ
2
6χ
2
10χ
2
12 − 2233953ψ6χ
3
10χ
2
12 − 32ψ7
4χ
2
10χ12 + 2 · 32ψ4
4ψ
2
6χ
2
10χ12
+ 211355 · 19ψ3
4ψ6χ
3
10χ12 + 220385311ψ2
4χ
4
10χ12 − 32ψ4ψ
4
6χ
2
10χ12 + 2113552ψ3
6χ
3
10χ12
− 2ψ6
4ψ6χ
3
10 − 21234ψ5
4χ
4
10 + 22ψ3
4ψ
3
6χ
3
10 + 2123452ψ2
4ψ
2
6χ
4
10 + 2213754ψ4ψ6χ
5
10
− 2ψ5
6χ
3
10 + 2323955χ6
10
)
.
Hence, the expression Q := 21239χ2
35/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
x1 =
ψ4χ
2
10
χ2
12
, x2 =
ψ6χ
3
10
χ3
12
, x3 =
χ6
10
χ5
12
,
as well as
y1 =
x3
1
x3
=
ψ3
4
χ12
, y2 =
x2
2
x3
=
ψ2
6
χ12
, y3 =
x2
1x2
x3
=
ψ2
4ψ6χ10
χ12
, (2.1)
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
x1 = E4(τ1)E4(τ2)(πz)4 +O
(
z5
)
,
x2 = E6(τ1)E6(τ2)(πz)6 +O
(
z7
)
,
x3 = η24(τ1)η24(τ2)(πz)12 +O
(
z13
)
,
and
y1 = j(τ1)j(τ2) +O
(
z2
)
,
y2 = (1728− j(τ1))(1728− j(τ2)) +O
(
z2
)
,
y3 =
E2
4(τ1)E2
4(τ2)E6(τ1)E6(τ2)
η24(τ1)η24(τ2)
(πz)2 +O
(
z3
)
, (2.2)
where we have set
j(τj) =
1728E3
4(τj)
E3
4(τj)− E2
6(τj)
=
E3
4(τj)
η24(τj)
,
1728− j(τj) =
1728E2
6(τj)
E3
4(τj)− E2
6(τj)
=
E2
6(τj)
η24(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 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
C : y2 = f(x) = a0
6∏
i=1
(x− αi) =
6∑
i=0
aix
i. (2.3)
The roots {αi}6i=1 of the sextic are the six ramification points of the map C → P1. 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 (A′, B′, C ′, D′)
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
J2 = −23 · 3χ12(τ)
χ10(τ)
,
J4 = 22ψ4(τ),
J6 = −23
3
ψ6(τ)− 25ψ4(τ)χ12(τ)
χ10(τ)
,
J10 = −214χ10(τ).
Thus, the invariants of a sextic define a point in a weighted projective space [J2 : J4 : J6 :
J10] ∈WP3
(2,4,6,10) that equals[
233(3χ12) : 2232ψ4χ
2
10 : 2332
(
4ψ4(3χ12) + ψ6χ10
)
χ2
10 : 22χ6
10
]
.
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 ProjC[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:
A = − 1
233 · 5
J2,
6 A. Malmendier and T. Shaska
B =
1
233354
(
J2
2 + 20J4
)
,
C = − 1
253556
(
J3
2 + 80J2J4 − 600J6
)
, (2.4)
D = − 1
2839510
(
9J5
2 + 700J3
2J4 − 3600J2
2J6 − 12400J2J
2
4 + 48000J4J6 + 10800000J10
)
.
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
x1 = 144
J4
J2
2
, x2 = −1728
J2J4 − 3J6
J3
2
, x3 = 486
J10
J5
2
,
for J2 6= 0. We use x1, x2, x3 to write the point [J2 : J4 : J6 : J10] ∈WP3
(2,4,6,10) as[
1 :
1
2432
x1 :
1
2634
x2 +
1
2433
x1 :
1
2 · 35
x3
]
.
Since the invariants J4, J6, J10 vanish simultaneously for sextics with triple roots all such curves
are mapped to [1 : 0 : 0 : 0] ∈WP3
(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 coordi-
nates 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
ρ = −
4
(
9J2
2 − 320J4
)(
J2
2 + 20J4
)2(
3J3
2 + 140J2J4 − 800J6
)2 ,
σ = −
48
(
J2
2 + 20J4
)2(
3J3
2 + 140J2J4 − 800J6
)3
×
(
9J5
2 − 700J3
2J4 + 2400J2
2J6 − 262400J2J
2
4 + 768000J4J6 + 172800000J10
)
,
κ =
2
(
27J4
2 + 2380J2
2J4 − 12000J2J6 + 12800J2
4
)(
J2
2 + 20J4
)(
3J3
2 + 140J2J4 − 800J6
)2 . (2.5)
It follows:
Lemma 2.3. For invariants (ρ, σ, κ) given by equation (2.5) such that ρ and κ do not vanish
simultaneously, a point [J2 : J4 : J6 : J10] in WP3
(2,4,6,10) is given by
J2 = 8(κ− ρ), J4 =
9
5
(κ− ρ)2 + 45ρ,
J6 =
111
25
(κ− ρ)3 − 30(κ− ρ)2 + 63ρ(κ− ρ)− 270ρ,
J10 =
6
3125
(κ− ρ)5 +
4
15
(κ− ρ)4 +
46
75
ρ(κ− ρ)3
+
(
−1
6
σ +
42
5
ρ
)
(κ− ρ)2 + 12ρ2(κ− ρ) +
3
2
ρ(36ρ− σ). (2.6)
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. �
Remark 2.4. Consider the image of [J2 : J4 : J6 : J10] in WP3
(2,4,6,10) under the morphism
WP3
(2,4,6,10) → P5 given by[
486J4J6 : 486J10 : −1728(J2J4 − 3J6)J2
2 : 144J3
2J4 : 20736J2J
2
4 : J5
2
]
, (2.7)
which is a linear transformation of the usual morphism to P5 given by
[J2 : J4 : J6 : J10] 7→
[
J10 : J4J6 : J2
2J6 : J3
2J4 : J2J
2
4 : J5
2
]
.
For J2 6= 0, points in equation (2.7) equal[
1
1536
x1(x2 + 12x1) : x3 : x2 : x1 : x2
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
J
(0)
2 = 0, J
(0)
4 = 45ρ, J
(0)
6 = −270ρ, J
(0)
10 =
3
2
ρ(36ρ− σ),
and the invariants ρ and σ with
ρ = κ =
4
5
J3
4
J2
6
, σ =
144
5
J3
4
J2
6
+ 6480
J3
4
J2
6
J10
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 ε = (κ − ρ) in equation (2.6), one checks that
points in equation (2.7) up to terms of order O
(
ε2
)
equal[
1− 7
30
ε :
J
(0)
10
J
(0)
4 J
(0)
6
− 2
2025
ε : 0 : 0 : −512
9
ε : 0
]
.
This means that under the usual morphism to P5 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 ]
be the symmetric matrix corresponding to this conic. Let X = [X1 : X2 : X3] ∈ P2 and
Q : Xt ·M ·X =
3∑
i,j=1
AijXiXj = 0. (2.8)
Clebsch [5] determined the entries of this matrix M as follows
A11 = 2C +
1
3
AB,
8 A. Malmendier and T. Shaska
A22 = A13 = D,
A33 =
1
2
BD +
2
9
C
(
B2 +AC
)
,
A23 =
1
3
B
(
B2 +AC
)
+
1
3
C
(
2C +
1
3
AB
)
,
A12 =
2
3
(
B2 +AC
)
. (2.9)
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 ‘Überschiebung’ [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→ f̃(x) = f(−x) the quadrics change according to yi(x) 7→ ỹi(x) = yi(−x) for
i = 1, 2, 3. Hence, they are not invariants of the sextic f . The coefficients Aij in equation (2.9)
satisfy Aij = (yiyj)2.1 Therefore, the coefficients Aij are invariant under the operation f(x) 7→
f̃(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 Überschiebung by product
rule [18, p. 317], then R can be re-written as
R = −(y1y2)1(y2y3)1(y3y1)1,
or, equivalently, as
R = −1
8
(
y1,yyy2,xyy3,xx − y1,yyy2,xxy3,xy − y1,xyy2,yyy3,xx
+ y1,xyy2,xxy3,yy + y1,xxy2,yyy3,xy − y1,xxy2,xyy3,yy
)
.
It is then obvious that under the operation f(x) 7→ f̃(x) = f(−x) the determinant R changes
its sign, i.e., R(f) 7→ R(f̃) = −R(f). A straightforward calculation shows that
R2 =
1
2
∣∣∣∣∣∣
A11 A12 A13
A12 A22 A23
A13 A23 A33
∣∣∣∣∣∣ ,
where Aij are the invariants in equation (2.9). Like the coefficients Aij , R
2 is invariant under the
operation f(x) 7→ f̃(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
R2 =
(
293−95−10i
χ35(τ)
χ10(τ)2
)2
. (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 equa-
tion (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 inM2 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 Überschiebung of order k by
(fg)k = (−1)k(gf)k. For f̃(x) = f(−x) and g̃(x) = g(−x) and m = n = k, we have (fg)m = (−1)m(f̃ g̃)m.
A Universal Genus-Two Curve from Siegel Modular Forms 9
Lemma 2.5. We have the following statements:
1. R2 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
rational subvariety of M2 given by the equation R2 = 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→
f̃(x) = f(−x), we associated a conic Q. Similarly, there is also a cubic curve given by the
equation
T :
∑
1≤i,j,k≤3
aijkXiXjXk = 0, (2.11)
where the coefficients aijk are of order zero and invariant under f(x) 7→ f̃(x) = f(−x). In terms
of ‘Überschiebung’ the coefficients are obtained by
aijk = (fyi)2(fyj)2(fyk)2.
The coefficients aijk are given explicitly as follows:
36a111 = 8
(
A2C − 6BC + 9D
)
,
36a112 = 4
(
2B3 + 4ABC + 12C2 + 3AD
)
,
36a113 = 36a122 = 4
(
AB3 + 4/3A2BC + 4B2C + 6AC2 + 3BD
)
,
36a123 = 2
(
2B4 + 4AB2C + 4/3A2C2 + 4BC2 + 3ABD + 12CD
)
,
36a133 = 2
(
AB4 + 4/3A2B2C + 16/3B3C + 26/3ABC2 + 8C3 + 3B2D + 2ACD
)
,
36a222 = 4
(
3B4 + 6AB2C + 8/3A2C2 + 2BC2 − 3CD
)
,
36a223 = 2
(
− 2/3B3C − 4/3ABC2 − 4C3 + 9B2D + 8ACD
)
,
36a233 = 2
(
B5 + 2AB3C + 8/9A2BC2 + 2/3B2C2 −BCD + 9D2
)
,
36a333 = −2B4C − 4AB2C2 − 16/9A2C3 − 4/3BC3 + 9B3D + 12ABCD + 20C2D.
The relations between all aforementioned invariants and Siegel modular forms, in particular the
relation between χ35 and R2 can be found in [1].
Since ‘Überschiebung’ 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/〈w〉 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).
However, if the conic has no rational point or R2 = 1
2 detM = 0 the method obviously fails.
In Section 3 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
Q′ : x2
1 − γx2
2 − Λ6x
2
3 = 0, (3.1)
where (ρ, σ, κ) are the absolute invariants in equation (2.5), γ = ρ2 + σ and
Λ6 = −γ3 − 27ργ2 − 81ρ2(ρ+ 12)γ + 729ρ2(ρ+ 12)2
+
(
−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+ 1296
)
κ4
+ (−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 Q′ in equation (3.1) does.
Proof. For the conic Q in equation (2.8), we apply the coordinate transformation given by
X1 = 2(AB + 6C)4
(
AC +B2
)
x1
+ 108B(AB + 6C)2
(
4A2C2 + 8AB2C + 4B4 − 3ABD − 18CD
)
x2
+ 41990B3
(
8A2BC2 + 14AB3C + 6B5 + 12AC3 + 12B2C2 − 27D2
)
x3,
X2 = −(AB + 6C)5x1 − 419904B3x3
×
(
4A2B2C + 3AB4 + 30ABC2 + 18B3C − 18ACD − 18B2D + 36C3
)
,
X3 = −2639B3
(
4A2C2 + 8AB2C + 4B4 − 3ABD − 18CD
)
x3. (3.3)
We then obtain the conic Q′ 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:
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
α2 + Λ6β
2σ = γ. (3.4)
The rational point on the conic Q′ is then given by[
x0
1 : x0
2 : x0
3
]
= [αρ+ γ : α+ ρ : βσ]. (3.5)
Conversely, every rational point on the conic Q′ 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
x0
2 6= 0 which is always possible if ρ 6= 0. Then every point on the conic is given by
x1 = (αρ+ γ)U2 + 2Λ6βσUV + Λ6(αρ+ γ)V 2,
x2 = (α+ ρ)U2 − Λ6(α+ ρ)V 2,
x3 = βσU2 + 2(αρ+ γ)UV + Λ6βσV
2, (3.6)
for some [U : V ] ∈ P1. The parametrization in equation (3.6) is a rational parametrization of
the conic Q′ if and only if α, β, ρ, κ, σ ∈ Q.
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) 7→ (±α,±β). If ρ 6= 0, one of these points satisfies
x0
2 = α + ρ 6= 0. The proof then follows from the known formulas parametrizing conics for
x0
2 6= 0 given by
x1 = ax0
1U
2 − 2cx0
3UV − cx0
1V
2,
x2 = ax0
2U
2 + cx0
2V
2,
x3 = ax0
3U
2 + 2ax0
1UV − cx0
3V
2,
where a = 1, b = −γ, c = −Λ6 and x0
1, x0
2, x0
3 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
the fact that x0
1 6= 0 in equation (3.5) in this case.
Remark 3.5. If a point of Q′ is obtained for some (rational) values (α, β) then three more
(rational) points on Q′ 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 Q′ in equation (3.1). Similarly, under the same change of
coordinates the cubic T in equation (2.11) becomes
Λ1(18γ + Λ3)x3
1 + γ3Λ2x
3
2 − (γ − Λ1)Λ2
6x
3
3 + 3Λ1Λ6x
2
1x3 + 3γ(9γρ+ κΛ3)x2
1x2
+ 3γ2Λ3x1x
2
2 + 3Λ5Λ6x1x
2
3 + 3γΛ4Λ6x2x
2
3 + 3γ2Λ6x
2
2x3 + 6γκΛ6x1x2x3, (3.7)
with coefficients given by
Λ1 = 9ρ+ κ2,
Λ2 = γ + 18ρ+ 3ρκ− 4κ2,
Λ3 = 27ρ(ρ+ 12) + (γ − 36ρ)κ+ 3(ρ+ 12)κ2 − 4κ3,
Λ4 = −γ2 − 9γρ− 3(γρ− 9ρ(ρ+ 12))κ+ (5γ − 36ρ)κ2 + 3(ρ+ 12)κ3 − 4κ4,
Λ5 = −27γρ(ρ+ 6) + 243ρ2(ρ+ 12)−
(
γ2 − 45γρ+ 324ρ2
)
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
Q′ : x2
1 − µx2
2 − (1− µ)x2
3 = 0,
with µ = J5
6/
(
243455J3
10
)
and a rational point given by
[
x0
1 : x0
2 : x0
3
]
= [1 : 1 : 1]. A rational
parametrization of Q′ is then given by
x1 = U2 + 2(1− µ)UV + (1− µ)V 2, x2 = U2 − (1− µ)V 2,
x3 = U2 + 2UV + (1− µ)V 2
with [U : V ] ∈ P1. Under the same change of coordinates the cubic T in equation (2.11) becomes
T ′ : 0 = 2x3
1 − µ2x3
2 − 2(1− µ)2x3
3 − 6µx2
1x2
− 6(1− µ)x2
1x3 + 6(1− µ)x1x
2
3 − 3µ(1− µ)x2x
2
3.
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
Q′ : x2
1 − x2
2 − (1− ν)x2
3 = 0,
with ν = J5
4/
(
223555J2
10
)
and a rational point given by
[
x0
1 : x0
2 : x0
3
]
= [1 : 1 : 0]. A rational
parametrization of Q′ is then given by
x1 = U2 + (1− ν)V 2, x2 = U2 − (1− ν)V 2, x3 = 2UV
with [U : V ] ∈ P1. Under the same change of coordinates the cubic T in equation (2.11) becomes
T ′ : 0 =
(
1− ν2
)
x3
1 − ν2x3
2 − (1− ν)2x3
3 + ν(1− 3ν)x2
1x2
− (1− ν)(3 + ν)x2
1x3 + ν(1− 3ν)x1x
2
2 − ν(1− ν)x2
2x3
+ (1− ν)(3− ν)x1x
2
3 + ν(1− ν)x2x
2
3 − 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 Q′ in Lemma 3.3 into the cubic T ′ in equa-
tion (3.7) and setting U = x and V = 1, one obtains the ramification locus of a sextic curve.
The ramification locus is equivalent to f(x) =
6∑
i=0
ai(α, β)xi = 0, where we write the sextic
polynomial in the form
f(x) =
(
d
(1)
0 + d
(2)
0
)
x6 +
(
d
(1)
1 + d
(2)
1
)
Λ6x
5 +
(
d
(1)
2 + d
(2)
2
)
Λ6x
4
+ d
(1)
3 Λ2
6x
3 +
(
d
(1)
2 − d
(2)
2
)
Λ2
6x
2 +
(
d
(1)
1 − d
(2)
1
)
Λ3
6x+
(
d
(1)
0 − d
(2)
0
)
Λ3
6. (3.9)
In terms of the coordinates of the point
[
x0
1 : x0
2 : x0
3
]
in equation (3.5) we have set
d
(1)
j = c
(1)
j,0
[(
ρ2 + γ
)
x0
1 + 2γρx0
2
]
γ + 3c
(1)
j,1
[
x0
1 + ρx0
2
]
γΛ
δj
6 x
0
3
− c(1)
j,0
[(
ρ2 + γ
)
x0
1 − 2γρx0
2
]
ρ2σ−2Λ6
(
x0
3
)2 − 3c
(1)
j,1
[
x0
1 − ρx0
2
]
ρ2σ−2Λ
1+δj
6
(
x0
3
)3
A Universal Genus-Two Curve from Siegel Modular Forms 13
+
[
3σc
(1)
j,2 − 2ρ2c
(1)
j,0
]
σ−1Λ6x
0
1
(
x0
3
)2
+ c
(1)
j,3Λ
1+δj
6
(
x0
3
)3
,
d
(2)
j = c
(2)
j,0
[
2ρx0
1 +
(
ρ2 + γ
)
x0
2
]
γ2 + 6c
(2)
j,1
[
ρx0
1 + γx0
2
]
γΛ
δj
6 x
0
3
− c(2)
j,0
[
2γx0
1 − ρ
(
ρ2 + γ
)
x0
2
]
γρσ−2Λ6
(
x0
3
)2 − 6c
(2)
j,1
[
x0
1 − ρx0
2
]
γρσ−2Λ
1+δj
6
(
x0
3
)3
+ c
(2)
j,2γΛ
δj
6 x
0
1
(
x0
3
)2
. (3.10)
All coefficients remain regular and in general non-vanishing for σ = 0 since x3
0/σ = β. Here,
(α, β) is a pair solving equation (3.4), and the coefficients c
(n)
j,k are given by
c
(1)
0,0 = 18γΛ1 + 3γΛ3 + Λ1Λ3, c
(1)
0,1 = γ + Λ1,
c
(1)
0,2 = −γΛ3 + Λ5, c
(1)
0,3 = −4γ + Λ1,
c
(2)
0,0 = 27γρ+ γΛ2 + 3κΛ3, c
(2)
0,1 = κ,
c
(2)
0,2 = −γΛ2 + 3Λ4,
c
(1)
1,0 = 6(γ + Λ1), c
(1)
1,1 = 2(18γΛ1 + γΛ3 + Λ1Λ3 + 2Λ5),
c
(1)
1,2 = 2(−2γ + 3Λ1), c
(1)
1,3 = 6(−γΛ3 + Λ5),
c
(2)
1,0 = 12κ, c
(2)
1,1 = 18γρ+ 2κΛ3 + 2Λ4,
c
(2)
1,2 = 12κΛ6,
c
(1)
2,0 = 3(18γΛ1 − γΛ3 + Λ1Λ3 + 4Λ5), c
(1)
2,1 = 5(−γ + 3Λ1),
c
(1)
2,2 = 72γΛ1 + γΛ3 + 4Λ1Λ3 + 11Λ5, c
(1)
2,3 = 15Λ1,
c
(2)
2,0 = 3(9γρ− γΛ2 + κΛ3 + 4Λ4), c
(2)
2,1 = 5κ,
c
(2)
2,2 = 3(36γρ+ γΛ2 + 4κΛ3 + Λ4),
c
(1)
3,0 = 20(−γ + Λ1), c
(1)
3,1 = 4(18γΛ1 − γΛ3 + Λ1Λ3 + 4Λ5),
c
(1)
3,2 = 20Λ1, c
(1)
3,3 = 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
under the action of the automorphism of the conic Q′ given by [x1 : x2 : x3] 7→ [±x1 : ±x2 : x3].
We make the following remarks:
Remark 3.9. The transformation x → Λ6
x maps the coefficients d
(1)
k ± d
(2)
k 7→ d
(1)
k ∓ d
(2)
k for
k = 0, 1, 2 and a3 7→ a3. This is to be expected since the coefficients are in terms of invariants
of the binary sextic f(x, z) and x→ 1
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 :
J4 : J6 : J10] the corresponding point in WP3
(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
C(α,β) : y2 =
6∑
i=0
ai(α, β)xi, (3.11)
with coefficients given in equations (3.9) and (3.10), and a pair (α, β) satisfying
α2 + Λ6β
2σ = 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
with ν = J5
4/
(
223555J2
10
)
.
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.
with µ = J5
6/
(
243455J3
10
)
.
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 Q′
in equation (3.1) does. By Lemma 3.2 the conic Q′ 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 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 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
C(α,0) : y2 =
6∑
i=0
ai(α, 0)xi,
corresponding to p, such that ai(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 α2 =
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
over Q(x1,x2,x3)[α] with α2 = ρ2 + 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
J10 = −2−113−35−5
(
9J5
2 − 700J3
2J4 + 2400J2
2J6 − 262400J2J
2
4 + 768000J4J6
)
.
We have the following lemma:
Lemma 3.17. In terms of the invariants ρ, σ, κ and γ = ρ2 + σ, we have
D = −
J5
2
(
(κ− ρ)2 + 9ρ
)(
(2κ− ρ)2 − γ
)
2173755(κ− ρ)5
,
R2 =
J15
2
(
(κ− ρ)2 + 9ρ
)3
Λ6
254321515(κ− ρ)15
.
In particular, the locus D = 0 and χ35 6= 0 is given by γ = (2κ − ρ)2 or, equivalently, σ =
4κ(κ− ρ).
We have the following corollary:
Corollary 3.18. Let D = 0 and χ2
35 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 ofM2 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
F (x) =
(
d
(1)
0 + d
(2)
0
)
x6 +
(
d
(1)
2 + d
(2)
2
)
x4 +
(
d
(1)
2 − d
(2)
2
)
x2 +
(
d
(1)
0 − d
(2)
0
)
, (3.12)
and with coefficients in Z[α, ρ, κ] given by
d
(1)
0 = 3κγ2 −
(
κ2 + 9ρ
)
(11κ− 9ρ− 126)γ −
(
κ2 + 9ρ
)2
(4κ− 3ρ− 36),
d
(2)
0 =
(
γ2 +
(
−κ2 + 3κρ+ 45ρ
)
γ − 3κ
(
κ2 + 9ρ
)
(4κ− 3ρ− 36)
)
α,
d
(1)
2 = −15κγ2 + 15
(
κ2 + 9ρ
)
(5κ− 3ρ− 18)γ − 15
(
κ2 + 9ρ)2(4κ− 3ρ− 36),
d
(2)
2 =
(
−15γ2 +
(
75κ2 − 45κρ− 135ρ
)
γ − 15κ
(
κ2 + 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.
Proof. For Λ6 = ε2 with ε → 0, we rescale the polynomial in equation (3.12) according to
f(εx)/ε6 before setting ε = 0. If we substitute Λ6 = 0 into equation (3.4) we obtain α2 = γ,
β = 0. Therefore, we will use the absolute invariants α, γ, ρ, κ subject to the constraints
Λ6 = 0 in equation (3.2) and α2 = 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 equa-
tion (3.13) to β = Λ6 = 0. �
The polynomial in equation (3.12) is a twist of the polynomial given by
F̂ (x) = x6 + ax4 + bx2 + 1.
The curve y2 = F̂ (x) has extra involutions, i.e., it has automorphisms other than the hyperel-
liptic 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:
Corollary 3.20. For the genus-two curve C : y2 = F (x) given by equation (3.12) with χ35 = 0
we obtain the dihedral invariants
u =
(
d
(1)
2 + d
(2)
2
)(
d
(1)
2 − d
(2)
2
)(
d
(1)
0 + d
(2)
0
)(
d
(1)
0 − d
(2)
0
) ,
v =
(
d
(1)
2 − d
(2)
2
)3(
d
(1)
0 + d
(2)
0
)(
d
(1)
0 − d
(2)
0
)2 +
(
d
(1)
2 + d
(2)
2
)3(
d
(1)
0 + d
(2)
0
)2(
d
(1)
0 − d
(2)
0
) ,
and the Igusa invariants [J2 : J4 : J6 : J10] given by [19, equation (16)].
A Universal Genus-Two Curve from Siegel Modular Forms 17
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https://doi.org/10.2307/2369402
https://doi.org/10.1112/S146115701600019X
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https://doi.org/10.1112/S1461157014000461
https://arxiv.org/abs/1403.6911
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https://arxiv.org/abs/1003.4759
https://doi.org/10.1070/SM1996v187n11ABEH000171
https://doi.org/10.1007/b98867
https://doi.org/10.2307/1970233
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https://doi.org/10.2307/2373243
https://doi.org/10.2307/2373943
https://doi.org/10.1007/0-387-23534-5_6
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https://arxiv.org/abs/1209.0446
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https://doi.org/10.1007/978-1-4612-0441-1_21
https://doi.org/10.1007/978-3-642-18487-1_42
https://arxiv.org/abs/math.AG/0107142
1 Introduction
2 Preliminaries
2.1 The Siegel modular three-fold
2.2 Siegel modular forms
2.3 Igusa invariants
2.3.1 Absolute invariants
2.4 Recovering the equation of the curve from invariants
3 A universal genus-two curve from the moduli space
3.1 A word about extra automorphisms
References
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