Teichmüller modular forms

Teichm¨ uller modular forms

Takashi Ichikawa

Department of Mathematics, Faculty of Science and Engineering, Saga University Saga 840-8502, Japan (e-mail: ichikawa@ms.saga-u.ac.jp)

1. We define Teichm¨uller modular formsas global sections of line bundles on the moduli space of algebraic curves with fixed genus, which are, over C:

automorphic functions on the Teichm¨uller space,

because the moduli space overCis the quotient of the Teichm¨uller space by the action of the mapping class group. This naming is an analogy to that of Siegel modular forms which are defined as automorphic functions on the Siegel upper half space.

Besides the analogy of the namings, Teichm¨uller modular forms and Siegel modular forms are connected by the period map and the Torelli map, i.e., the pullback by the Torelli map τ gives a map

τ^∗ :{Siegel modular forms} −→ {Teichm¨uller modular forms}.

Note that τ^∗ is injective in the degree 2 and 3 cases, however it is not injective nor surjective in general. Hence there are Teichm¨uller modular forms not obtained in this way which appear in string theory and soliton theory. Since it is shown by Royden that the Teichm¨uller space of degree ≥ 2 is not a homogeneous space, to study Teichm¨uller modular forms, we need a different method from known theories on automorphic forms associated with algebraic groups. In this note, we review some results in [3] on Teichm¨uller modular forms (the Schottky problem, the ring structure) which are obtained from studying Teichm¨uller modular forms by their evaluation on Schottky-Mumford uniformized universal curves.

For each connected and finite graph ∆ whose any vertex has degree ≥ 3, a Schottky-Mumford uniformized universal curve C∆ is constructed in [3] as a stable curve which is a universal deformation of degenerate curves with dual graph ∆. The base ringA_∆ofC_∆consists of the moduli parameters and the deformation parameters of the degenerate curves, more precisely, A∆ is the ring of formal power series of the deformation parameters whose coefficients are rational functions of the moduli parameters with integral coefficients. Furthermore, C_∆ is Mumford uniformized by a Schottky group over

B_∆ = A_∆

[1

y_i (y_i : deformation parameters)

]

,

and becomes Schottky uniformized Riemann surfaces and Mumford curves over nonarchimedean valuation fields by specializing the parameters. The construction of C∆, which had been done by Ihara and Nakamura in the case when the degenerate curve is maximally degenerate and consists of smooth projective lines, is carried out by extending Mumford’s uniformization theory on local base rings. To calculate the evaluation of Teichm¨uller modular forms on each universal curveC∆of genus g,it is important that the multiplicative periods

p_ij =p_ji (1≤i, j ≤g)

of C∆, which we call universal periods, are seen to be computable elements of B∆

using the infinite product presentation of multiplicative periods given by Schottky, Manin, and Drinfeld. For example, if C_∆ is the universal deformation with param- eters y1, ..., yg of irreducible degenerate curves obtained by identifying xi, x₋i ∈ P¹ (i= 1, ..., g) in pairs, then its multiplicative periods p_ij are computed as

p_ij = c_ij



1 + ^∑

|k|̸=i,j

(x_i−x_−i)(x_j−x_−j)(x_k−x_−k)²

(x_i−x_k)(x_−i −x_k)(x_j−x_−k)(x_−j −x_−k)y_|k|+· · ·



,

where

cij =







(x_i−x_j)(x_−i−x_−j)

(x_i−x_−j)(x_−i−x_j) (if i̸=j) y_i (if i=j).

In [1] (resp. [4]), we apply these universal periods to constructingp-adic theta func- tion solutions to the KP (resp. KdV) hierarchy. In [2], we construct multiplicative periods of Schottky uniformized analytic curves of infinite genus over (archimedean and nonarchimedean) valuation fields, and give their application to soliton theory.

2. To state our results more precisely, we prepare some notations. Let π : C → Mgbe the universal curve over the moduli stack classifying proper and smooth curves of genusg,and letλbe the Hodge line bundle, which corresponds to the automorphic factor, defined as ^∧gπ_∗(Ω_C/Mg), where Ω_C/Mg denotes the sheaf of differentials ofC overMg.Then it is shown by Harer that the Picard group ofMg forg ≥3 is a cyclic group generated by λ.For a commutative ringR, we define the space of Teichm¨uller modular forms over R of degree g and weight has

T_g,h(R) = H⁰(Mg, λ^⊗h⊗ZR).

Then using a Satake type compactification of Mg(C), one can see that if g ≥ 3, then T_g,h(C) becomes the space of holomorphic functions on the Teichm¨uller space

of degree g having the automorphy condition of weight h with respect to the action of the mapping class group. Since eachC_∆is smooth over B_∆, the evaluation onC_∆ gives an R-linear map

κ_∆:T_g,h(R) −→ B_∆⊗ZR

which is injective by the universality of C_∆ and the result of Deligne and Mumford that all the geometric fibers of Mg are irreducible. Furthermore, this map makes the following commutative diagram:

Sg,h(R) −→^{F (}Z

[

q_ij^±1 (i̸=j)

]

[[q11, ..., qgg]]

)⊗ZR

τ^∗ ↓ ↓

T_g,h(R) −→^κ∆ B_∆⊗ZR,

where S_g,h(R) denotes the space of Siegel modular forms over R of degree g and weight h, F denotes the Fourierq-expansion constructed by Chai and Faltings, and the downarrow on the right-hand side is the ring homomorphism sending the variables q_ij = q_ji (1 ≤ i, j ≤ g) to the universal periods p_ij for C_∆ (if R = C, then q_ij = exp(2π√

−1z_ij), where (z_ij)_i,j belongs to the Siegel upper half space of degree g).

Therefore, we have a solution to the Schottky problem, i.e., a characterization of Siegel modular forms vanishing on the Jacobian locus (see [4] for a solution to the hyperelliptic Schottky problem):

Theorem 1. For a Siegel modular form φ overR of degree g and weight h, let F(φ) = ^∑

T=(tij)

aT

∏

1≤i,j≤g

qijtij (aT ∈R)

be its Fourier q-expansion, where T runs through symmetric matrices of degree g defining integral positive semidefinite quadratic forms. Then we have

τ^∗(φ) = 0 ⇐⇒ F(φ)|qij=pij = 0.

This result generates relations between the Fourier coefficients of such a Siegel mod- ular form. For example, if a Siegel modular form φ with Fourier coefficients aT

vanishes on the Jacobian locus, then using the above universal periods, for integers s₁, ..., s_g ≥0 such that ^∑g_i=1s_i = min{the trace of T |a_T ̸= 0}, we have

∑

tii=si

a_T ^∏

i<j

((x_i−x_j)(x_−i−x_−j) (xi−x₋j)(x₋i−xj)

)_2tij

= 0 (x_±1, ..., x_±g : variables) which was already obtained by Brinkmann and Gerritzen.

3. Next we consider the ring structure of Teichm¨uller modular forms. Let π : C → Mg be the universal stable curve over the Deligne-Mumford compactification of Mg, and put λ =^∧gπ_∗(ω_C/Mg), where ω_C/Mg denotes the dualizing sheaf of C over Mg.By studying the behavior of Teichm¨uller modular forms at the boundary ofMg, we can show that if g ≥3, then any element ofT_g,h(Z) can be uniquely extended to a global section of λ^⊗h, and hence we have:

Theorem 2. Assume that g ≥ 3. Then T_g,h(Z) = 0 if h < 0, T_g,0(Z) = Z and each Tg,h(Z) is a Z-lattice of finite rank in the C-vector space Tg,h(C).Furthermore, the ring T_g^∗(Z) =^⊕_h≥0T_g,h(Z) of Teichm¨uller modular forms of degree g becomes a normal finitely generated Z-algebra.

We consider special Teichm¨uller modular forms and the ring structure of T₃^∗(Z).

For eachg ≥3,Tsuyumine had obtained a Teichm¨uller modular form of degree g as a square root of the product

θ_g(Z) = ^∏

a,b∈ {0,1/2}^g 4a^tb: even

∑ n_∈Z^g

exp

(

2π√

−1

[1

2(n+a)Z^t(n+a) + (n+a)^tb

])

of theta constants with even characteristic (Z belongs to the Siegel upper half space of degree g). By calculating the lowest term of κ_∆(τ^∗(θ_g)) = F(θ_g)|qij=pij and using a result of Igusa on Siegel modular forms of degree 3, we have:

Theorem 3. Put

N_g =

{ −2²⁸ (g = 3) 2^{2g−1(2g−1)} (g ≥4).

Then τ^∗(θ_g)/N_g has a square root as a primitive (i.e., not congruent to 0 modulo any prime) Teichm¨uller modular form, which we denote by f_g, over Z of degree g and weight 2^g−3(2^g+ 1). Furthermore, the ring T₃^∗(Z) is generated by Siegel modular forms over Z of degree 3 and by f₃ (f₃ has odd weight 9, and hence is not a Siegel modular form).

Note that θ_g/N_g is a primitive Siegel modular form over Z, however this fact does not imply the primitiveness of τ^∗(θ_g)/N_g because τ^∗ is not injective in general.

Remark 1. Since M2 is an affine space, each T_2,h(Z) has infinite rank. We prove that T₂^∗(Z) = ^⊕_h∈ZTg,h(Z) is generated by Siegel modular forms over Z of degree 2 and by (θ₂/2⁶)⁻² which has minus weight −10, and hence is not a Siegel modular form.

Remark 2. Using the Schottky-Mumford uniformized universal deformation of a

maximally degenerate curve, our evaluation theory and Theorems 1, 2 seem to be extended for Teichm¨uller modular forms of higher level. Tsuyumine showed that each theta constant of degree ≥3 has a square root as a Teichm¨uller modular form of higher level and weight 1/4. It would be interesting to study these Teichm¨uller modular forms.

References

[1] T. Ichikawa, P-adic theta functions and solutions of the KP hierarchy, Com- mun. Math. Phys. 176 (1996) 383-399.

[2] T. Ichikawa, Schottky uniformization theory on Riemann surfaces and Mum- ford curves of infinite genus, J. Reine Angew. Math. 486 (1997) 45-68.

[3] T. Ichikawa, Generalized Tate curve and integral Teichm¨uller modular forms, Amer. J. Math. 122 (2000) 1139-1174.

[4] T. Ichikawa, Universal periods of hyperelliptic curves and their applications, J.

Pure Appl. Algebra. 163 (2001) 277-288.