THETA-CONSTANTS: THE INFLUENCE OF IGUSA’S WORK
RICCARDO SALVATI MANNI0
1. I NTRODUCTION
We work over C.
Let X = Cg /Λτ be an abelian variety where
Λτ = hidg , τi
is a lattice generated by the columns of idg and
τ ∈ Hg := {τ = τt , Im(τ) > 0}.
A theta function of order 1 is a holomorphic function of (τ, z) ∈ Hg × Cg
m0
m0
θm (τ, z) := θ
(τ, z) m =
∈ Z2g .
m 00
m 00
The basic example is θ0 (τ, z) the Riemann theta function for m = 0. We view a theta function
θm (τ, z) for fixed τ as a section of a line bundle on the abelian variety X = Cg /Λτ .
When m 0 , m 00 ∈ {0, 1}g , we obtain 22g theta functions with the property that
θm (τ, −z) = e(m)θm (τ, z)
0
00
e(m) = (−1)hm ,m i .
We call these theta functions even or odd based on the sign of e(m).
2. E VEN CHARACTERISTIC
For m even, the theta constant θm (τ, 0) : Hg → C at z = 01 is a modular form of weight
1/2 with respect to a finite index subgroup Γg (4, 8) ⊂ Γg = Sp(2g, Z). That is,
θm (M · τ) = κ(M) det(Cτ + D)1/2 θm (τ, 0).
for M ∈ Γg (4, 8) where κ(M) is a constant. Recall that Sp(2g, R) acts on τ ∈ Hg by
A B
M·τ=
· τ = (Aτ + B)(Cτ + D)−1 .
C D
More generally for Γ ⊂ Sp(2g, Z) a subgroup of finite index, there is a space [Γ, k] of
modular forms of weight k with respect to Γ . These are holomorphic functions f : Hg → C
satisfying a similar formula for the action of M ∈ Γ . When g = 1 we also impose some
regularity at the boundary of H1 .
Date: March 27, 2017.
0
Notes by Dori Bejleri who takes all responsibility for any mistakes or inconsistencies.
1
Also known as a Thetanullwerte
1
Let
∞
M
[Γ, k]
A(Γ ) :=
k=0
denote the ring of modular forms.
Theorem 2.1. (Igusa, 1964) The ring A(Γg (4, 8)) is normal and the integral closure of the subring C[θm θn ] in A(Γg (4, 8)) generated by products of even theta functions is all of A(Γg (4, 8)).
Corollary 2.2. (Igusa)
• Gives the complete structure of A(Γ2 ):
A(Γ2 ) = C[E4 , E6 , χ10 , χ12 , χ35 ].
• Gives equations for Igusa quartic as a moduli space of principally polarized abelian
varieties with level structure.
• The theta constants θm (τ, 0) (almost) give a parametrization of ppav with level structure.
Remark 2.3. (a) The cusp forms χ5 and χ35 appear in many fields of math such as string
theory and Borcherds products.
(b) These theta constants were an important tool in Mumfords work on defining equations for
abelian varieties.
2.1. Higher dimension. Using Jacobians of theta functions (to be introducted in the next section), Igusa showed that
ϕ : C[θm θn ] ( A(Γ3 (4, 8))
The natural question then is to compare the corresponding projective varieties. There is a map
Proj A(Γg (4, 8)) → Proj C[θm θn ]
where the source is none other but the Satake compactification of Ag .
Theorem 2.4. (Igusa 1981) This map is bijective for all g but is not an isomorphism for g > 6.
This left open the case of g = 3, 4, 5.
Theorem 2.5. (Freitag - Salvati Manni, 2016) The map is an isomorphism for g = 3.
2.2. Related results.
2.2.1. Tsuyimana (1986). Gave the structure of the graded ring A(Γ3 ). The main tool was
Igusa’s paper on modular forms and projective invariants as well as Shioda’s work on the graded
ring of invariants of binary octavics.
2.2.2. Runge (1993/94). Using slightly different theta constants gave an easier description of
the structure of A(Γ3 ) using only 8 theta constants.
2.2.3. Different groups. Various people including Hammond, Freitag, Ibukiyama, Salvati Manni
and others have studied the ring of modular forms of genus 2 with respect to other groups.
2
2.2.4. I Voronoi compactification. Igusa (1968) wrote an important paper on the desingularization of the Siegel modular variety and introduced the Voronoi compactifications.
Shepherd-Barron showed that for g > 12, the I Voronoi compactification of Ag = Γg \Hg
(also known Igusa’s compactification) is a canonical model in the sense of birational geometry.
Q
2.2.5. Freitag. Used the theta constant θnull = m even θm in his proof that Γg \Hg = Ag is of
general type for g > 8.
3. O DD CHARACTERISTIC
For m odd, θm (τ, 0) is identically zero so what we did above isn’t interesting. On the other
hand, we can take the gradient gradz θm (τ, 0) which gives us a vector valued modular function.
To obtain a single modular function, take the Jacobian determinant at z = 02 :
∂(θm1 . . . θmg )
D(m1 , . . . , mg ) := π−g
(τ, 0).
∂(z1 . . . zg )
For θm1 , . . . , θmg odd theta functions, D(m1 , . . . , mg ) is a weight g/2 + 1 modular function.
1
Example 3.1. When g = 1 and m =
,
1
0
0
1
D(m)(τ) = θ
θ
θ
(τ, 0)
0
1
0
is a Jacobi form.
Igusa gave necessary conditions that m1 , . . . , mg must satisfy if D(m1 , . . . , mg ) ∈ C[θm ].
Starting at genus g = 3, there are mi such that D(m1 , m2 , m3 ) ∈
/ C[θm ]. Fay (1979) gave an
example in genus 5 with D(m1 , . . . , m5 ) ∈ C[θm ].
Igusa (1981) proved that D(m1 , . . . , mg ) ∈
/ C[θm ] for all g > 6 and any mi . This leads to
the following natural question. Let X = {D(m1 , . . . , mg )} and Y = {θn1 . . . θng+2 } where mi
are odd and ni are even. Then X and Y are spaces of modular forms of weight g/2 + 1 and we
can ask what is X ∩ Y inside the space of all such modular forms. Igusa showed there are two
possibilities for dim(X ∩ Y) and if it is nonzero conjectured a precise formula of the form
X
X
±D(m1 , . . . , mg ) =
±θn1 . . . θng+2 .
This formula is still open for g > 6.
3.1. Other applications. These functions D(m1 , . . . , mg ) have geometric applications. For
example in genus 3, they appear as a main ingredient to explain how to relate the moduli space
of plain quartics with level structure and the moduli space (P2 )7 of 7 points in P2 . These
functions are coefficients of the universal Coble’s quartic in P7 while odd theta characteristics
correspond to bitangents of the quartic.
2
This is also known as the nullwerte of Jacobian
3