A note on the Schottky problem

TL;DR

This paper surveys recent advances in the Schottky problem, exploring connections with various conjectures and mathematical structures such as André-Oort, stable forms, and Siegel-Jacobi spaces, highlighting ongoing research directions.

Contribution

It provides a comprehensive overview of recent progress and discusses the interplay between the Schottky problem and several key conjectures and theories in algebraic geometry and number theory.

Findings

Summarizes recent progress in the Schottky problem

Explores relations with André-Oort and Coleman's conjecture

Discusses connections to stable modular and Jacobi forms

Abstract

In this article, we discuss and survey the recent progress towards the Schottky problem, and make some comments on the relations between the Andr{\'e}-Oort conjecture, Okounkov convex bodies, Coleman's conjecture, stable modular forms, Siegel-Jacobi spaces, stable Jacobi forms and the Schottky problem.

Equations

$${\mathbb{H}}_g=\left\{\tau \in {\mathbb{C}}^{(g,g)}|\tau ={}^t\tau ,\mathrm{Im}\tau >0\right\}$$

$$Sp(2g,\mathbb{R})=\{M\in {\mathbb{R}}^{(2g,2g)}|{}^{t}M{J}_{g}M=J_g\}$$

$$J_g=\left(\begin{array}{cc}0& I_g\\ -I_g& 0\end{array}\right).$$

$$M\cdot \tau =(A\tau +B)(C\tau +D{)}^{-1},$$

$$\Gamma_{g}=Sp(2g,\mathbb{Z})=\left\{\begin{pmatrix}A&B\\ C&D\end{pmatrix}\in Sp(2g,\mathbb{R})\,\big{|}\ A,B,C,D\ \textrm{integral}\right\}$$

$$T_g:{\mathcal{M}}_g⟶{\mathcal{A}}_g$$

$$C⟼J(C):=\mathrm{the}\mathrm{Jacobian}\mathrm{of}C$$

$${\Gamma }_g(\ell ):=\left\{\gamma \in Sp(2g,\mathbb{Z})|\gamma \equiv I_{2g}(\mathrm{mod}\ell )\right\}.$$

$$\Gamma_{g}(\ell,2\ell):=\left\{\begin{pmatrix}A&B\\ C&D\end{pmatrix}\in\Gamma_{g}(\ell)\,\Big{|}\ {\rm diag}(\,{}^{t}\!AB)\equiv{\rm diag}(\,{}^{t}CD)\equiv 0\ ({\rm mod}\,\ell)\,\right\}.$$

$${\mathcal{A}}_g(\ell ):={\Gamma }_g(\ell )\backslash {\mathbb{H}}_g\mathrm{and}{\mathcal{A}}_g(\ell ,2\ell ):={\Gamma }_g(\ell ,2\ell )\backslash {\mathbb{H}}_g.$$

$$\theta \left[\begin{array}{c}ϵ\\ \delta \end{array}\right](\tau ,z):=\sum _{N\in {\mathbb{Z}}^g}{e}^{\pi i\{(N+ϵ)\tau {}^t(N+ϵ)+2(N+ϵ){}^t(z+\delta )\}},(\tau ,z)\in {\mathbb{H}}_g\times {\mathbb{C}}^g.$$

$$\theta (\tau ,z):=\theta \left[\begin{array}{c}0\\ 0\end{array}\right](\tau ,z),(\tau ,z)\in {\mathbb{H}}_g\times {\mathbb{C}}^g.$$

$$\theta (\tau ,z+a\tau +b):=e^{-\pi i(a\tau {}^{t}a+2a{}^{t}z)}\theta (\tau ,z)\mathrm{for}\mathrm{all}a,b\in {\mathbb{Z}}^g.$$

$$\theta \left[\begin{array}{c}ϵ\\ \delta \end{array}\right](\tau ):=\theta \left[\begin{array}{c}ϵ\\ \delta \end{array}\right](\tau ,0),\tau \in {\mathbb{H}}_g$$

$$\Theta [ϵ](\tau ,z):=\theta \left[\begin{array}{c}2ϵ\\ 0\end{array}\right](2\tau ,2z),(\tau ,z)\in {\mathbb{H}}_g\times {\mathbb{C}}^g.$$

$$\Theta [ϵ](\tau ):=\theta \left[\begin{array}{c}2ϵ\\ 0\end{array}\right](2\tau ,0)=\Theta [ϵ](\tau ,0),\tau \in {\mathbb{H}}_g.$$

$${\left(\theta \left[\begin{array}{c}ϵ\\ \delta \end{array}\right](\tau ,z)\right)}^2=\sum _{\sigma \in \frac{1}{2}{\mathbb{Z}}^g/{\mathbb{Z}}^g}(-1{)}^{4{}^t\sigma \delta }\Theta [\sigma +ϵ](\tau ,0)\cdot \Theta [\sigma ](\tau ,z).$$

$${\Phi }_{\ell }:{\mathcal{A}}_g(2\ell ,4\ell )⟶{\mathbb{P}}^N(\mathbb{C}),N:={\ell }^{2g}-1$$

$$\Phi_{\ell}(\tau):=\left\{\theta\left[\begin{matrix}\epsilon\\ \delta\end{matrix}\right](\tau)\,\Big{|}\ \epsilon,\delta\in{\frac{1}{\ell}}\mathbb{Z}^{g}/\mathbb{Z}^{g}\,\right\}$$

$$Th:{\mathcal{A}}_g(2,4)⟶{\mathbb{P}}^{2^g-1}(\mathbb{C})$$

$$Th(\tau):=\left\{\Theta[\epsilon](\tau)\,\big{|}\ \epsilon\in{\frac{1}{2}}\mathbb{Z}^{g}/\mathbb{Z}^{g}\,\right\}.$$

$$A_{\tau }[\ell ]:=\left\{\tau ϵ+\delta \in A_{\tau }|ϵ,\delta \in \frac{1}{\ell }{\mathbb{Z}}^g/{\mathbb{Z}}^g\right\}$$

$${\theta }_m(\tau ,z):=\theta \left[\begin{array}{c}ϵ\\ \delta \end{array}\right](\tau ,z).$$

$$F_g(\tau ):=2^g\sum _{m\in A_{\tau }[2]}{\theta }_m^{16}(\tau )-{\left(\sum _{m\in A_{\tau }[2]}{\theta }_m^8(\tau )\right)}^2,\tau \in {\mathbb{H}}_g.$$

$$Prym(\tilde{C}⟶C):=Prym(C,\eta ):={\mathrm{Ker}}_0(J(\tilde{C})⟶J(C))\in {\mathcal{A}}_{g-1},$$

$${\left(\theta \left[\begin{array}{c}ϵ\\ \delta \end{array}\right]({\tau }_{\ast })\right)}^2=C\theta \left[\begin{array}{cc}0& ϵ\\ 0& \delta \end{array}\right](\tau )\cdot \theta \left[\begin{array}{cc}0& ϵ\\ 1& \delta \end{array}\right](\tau ).$$

$${\mathcal{S}}_g(2,4):=\bigcap _{\eta }{\mathcal{S}}_g^{\eta }(2,4),$$

$$N_{k,g}:=\{(A,\Theta )\in {\mathcal{A}}_g|\dim \mathrm{Sing}\Theta \ge k\}.$$

$${\mathcal{A}}_g^{\mathrm{dec}}:=\left(\bigcup _{k=1}^{g-1}{\mathcal{A}}_k\times {\mathcal{A}}_{g-k}\right)\subset {\mathcal{A}}_g$$

$$u(x,y,z;t):=\frac{{\partial }^2}{\partial x^2}\log \theta (x{W}_1+y{W}_2+t{W}_3+z)$$

Full text

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The Riemann theta function $\theta(\tau,z)$ is defined to be

[TABLE]

For each $\tau\in\mathbb{H}_{g},$ we have the transformation behavior

[TABLE]

The function

[TABLE]

is called the theta constant of order $\ell.$ It is known that the theta constants $\theta\left[\begin{matrix}\epsilon\\ \delta\end{matrix}\right](\tau)$ of order $\ell$ are Siegel modular forms of weight ${\frac{1}{2}}$ for $\Gamma_{g}(\ell,2\ell)$ [100, p. 200].

For a fixed $\tau\in\mathbb{H}_{g}$, we let $\Lambda_{\tau}:=\mathbb{Z}^{g}\tau+\mathbb{Z}^{g}$ be the lattice in $\mathbb{C}^{g}.$ According to the formula (2.3), the zero locus $\left\{z\in\mathbb{C}^{g}\,|\ \theta(\tau,z)=0\,\right\}$ is invariant under the action of the lattice $\Lambda_{\tau}$ on $\mathbb{C}^{g}$, and thus descends to a well-defined subvariety $\Theta_{\tau}\subset A_{\tau}:=\mathbb{C}^{g}/\Lambda_{\tau}.$ In fact $A_{\tau}$ is a principally polarized abelian variety with ample divisor $\Theta_{\tau}.$

[TABLE]

We define the theta constant of the second order to be

[TABLE]

Then we see that $\Theta[\epsilon](\tau)$ is a Siegel modular form of weight ${\frac{1}{2}}$ for $\Gamma_{g}(2,4).$

We have the following results.

[TABLE]

[TABLE]

defined by

[TABLE]

is an embedding.

Proof. See Igusa [72] for $\ell=4n^{2}$, and Salvati Manni [120] for $\ell\geq 2.$ $\Box$

[TABLE]

defined by

[TABLE]

We observe that according to Theorem 2.1, $\Phi_{2}(\tau)$ can be recovered uniquely up to signs from $\Theta(\tau).$ Since $\Phi_{2}$ is injective on ${\mathcal{A}}_{g}(4,8),$ the theta map $Th$ is finite-to-one on ${\mathcal{A}}_{g}(2,4).$ In fact, it is known that the theta map $Th$ is generically injective, and it is conjectured that $Th$ is an embedding. **

Now we briefly survey some approaches to the Schottky problem. As mentioned before, most of the following materials in this section comes from a good survey paper [60].

(A) Classical Approach

For $\tau\in\mathbb{H}_{g}$ and a positive integer $\ell\in\mathbb{Z}^{g}$,

[TABLE]

denotes the subgroup of $A_{\tau}$ consisting of torsion points of order $\ell$. For $m=\tau\epsilon+\delta\in A_{\tau}[\ell]$, we briefly write

[TABLE]

We define the Igusa modular form to be

[TABLE]

It was proved that $F_{g}(\tau)$ is a Siegel modular form of weight $8$ for the Siegel modular group $\Gamma_{g}$ such that when rewritten in terms of theta constants of the second order using Theorem 2.1,

($F_{g}$–1)  $F_{g}\equiv 0$  for $g=1,2$ ;

($F_{g}$–2)  $F_{3}$ is the defining equation for $\overline{Th(J_{3}(2,4))}=\overline{Th({\mathcal{A}}_{3}(2,4))}\subset{\mathbb{P}}^{7}(\mathbb{C})$ ;

($F_{g}$–3)  $F_{4}$ is the defining equation for $\overline{Th(J_{4}(2,4))}\subset\overline{Th({\mathcal{A}}_{4}(2,4))}\subset{\mathbb{P}}^{15}(\mathbb{C})$.

For more detail, we refer to [47, 73, 125] for the case $g=4$ and refer to [117] for the case $g=5.$ For $g\geq 5$, no similar solution is known or has been proposed.

The above theorem was proved by Grushevsky and Salvati Manni [64].

(B) The Schottky-Jung Approach

[TABLE]

where ${\rm Ker}_{0}$ denotes the connected component of 0 in the kernel and the map $J({\tilde{C}})\longrightarrow J(C)$ is the norm map corresponding to the cover ${\tilde{C}}\longrightarrow C$. We denote by ${\mathcal{P}}_{g-1}\subset{\mathcal{A}}_{g-1}$ the locus of Pryms of all étale double covers of curves in ${\mathcal{M}}_{g}.$ The problem of describing ${\mathcal{P}}_{g-1}$ is called the Prym-Schottky problem. **

[TABLE]

Here the constant $C$ is independent of $\epsilon,\delta.$

Proof. See Schottky-Jung [126] and also Farkas [41] for a rigorous proof. $\Box$

[TABLE]

where $\eta$ runs over the set ${\frac{1}{2}}\mathbb{Z}^{2g}/\mathbb{Z}^{2g}\backslash\{0\}.$ We note that the action of $\Gamma_{g}$ permutes the different $\eta$ and the ideals $S_{g}^{\eta}$. Therefore the ideal defining $\mathcal{S}_{g}(2,4)$ is $\Gamma_{g}$-invariant, and the locus $\mathcal{S}_{g}(2,4)$ is a preimage of some $\mathcal{S}_{g}\subset\mathcal{A}_{g}$ under the level cover. **

(b)* $J_{g}(2,4)$ is an irreducible component of the big Schottky-Jung locus ${\mathcal{S}}_{g}^{\eta}(2,4)$ for any $\eta.$*

Proof. The statement (a) was proved by van Geeman [53] and the statement (b) was proved by Donagi [31]. $\Box$

Donagi [32] conjectured the following.

(C) The Andreotti-Mayer Approach

We let ${\rm Sing}\,\Theta$ be the singularity set of the theta divisor $\Theta$ for a principally polarized abelian variety $(A,\Theta)$.

Proof. The proof was given by Andreotti and Mayer [5]. $\Box$

[TABLE]

[TABLE]

denotes the locus of decomposable ppavs $($product of lower-dimensional ppavs$)$ of dimension $g$.

Proof. The proof was given by Ein and Lazasfeld [38]. $\Box$

Proof. The proof was given by Andreotti and Mayer [5]. $\Box$

Proof. The proof was given by Debarre [26]. $\Box$

Proof. See Grushevsky-Salvati Manni [63] and Smith-Varley [132]. $\Box$ Problem. Can it happen that $N_{k,g}=N_{k+1,g}$ for some $k,g$?

(D) The Approach via the K-P Equation

In his study of solutins of nonlinear equations of Korteveg de Vrie type, I. Krichever [79] proved the following fact :

[TABLE]

satisfies the so-called Kadomstev-Petriashvili equation $($briefly the K-P equation$)$

[TABLE]

S. P. Novikov conjectured that $\tau\in\mathbb{H}_{g}$ is the period matrix of a curve if and only if the Riemann theta function corresponding to $\tau\in\mathbb{H}_{g}$ satisfies the K-P equation in the sense we just explained in Theorem 2.11. Shiota [129] proved that the Novikov conjecture is true, following the work of Mulase [96] and Mumford [98]. Arbarello and De Concini [7] gave another proof of the Novikov conjecture.

(E) The Approach via Geometry of the Kummer Variety

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We call the image of ${\rm Kum}$ the Kummer variety. Note that the involution $\pm 1$ has $2^{2g}$ fixed points on $A_{\tau}$ which are precisely $A_{\tau}[2]$, and thus the Kummer variety singular at their images in ${\mathbb{P}}^{2^{g}-1}(\mathbb{C}).$ **

[TABLE]

Proof. See Fay [42] and Gunning [65]. $\Box$

[TABLE]

are linearly dependent.

Proof. See Gunning [66]. $\Box$

I. Krichever [80] gave a complete proof of a conjecture of Welters concerning a condition for an indecomposable principally polarized abelian variety to be the Jacobian of a curve :

(W2)* a line tangent to it at one point, and intersecting it another point*

$($this is a semi-degenerate trisecant, when two points of secancy coincides$)$**

(W3)* a flex line *$($this is a most degenerate trisecant when all three points of

secancy coincide$)$* such that none of the points of intersection of this line with the Kummer variety are $A[2]$ $($where ${\rm Kum}(A)$ is singular$)$, then $A\in J_{g}.$*

For the Prym-Schottky problem, it will be natural whether the Kummer varieties of Pryms have any special geometric properties. Indeed, Beauville-Debarre [14] and Fay [43] obtained the following.

[TABLE]

lie on a $2$-plane in ${\mathbb{P}}^{2^{g}-1}(\mathbb{C}).$

A suitable analog of the trisecant conjecture was found for Pryms using ideas of integrable systems by Grushevsky and Krichever [62]. They proved the following.

(F) The Approach via the $\Gamma_{00}$ Conjecture

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We define the base locus

[TABLE]

[TABLE]

Proof. The above theorem was proved by Welters [143] set theoretically and also by Izadi scheme-theoretically. Originally Theorem 2.17 was conjectured by van Geeman and van der Geer [54]. $\Box$

van Geeman and van der Geer [54] conjectured the following. $\Gamma_{00}$**  Conjecture.** Let $(A,\Theta)\in{\mathcal{A}}_{g}^{\rm ind}.$ If $F_{A}\neq 0$, then $A\in J_{g}.$

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We define the Seshadri constant of $(A,\Theta)$ by

[TABLE]

Proof. See O. Debarre [28]. $\Box$

[TABLE]

for some $c,c_{ab}\in\mathbb{C}\ (1\leq a\leq b\leq g)$ and for all $\epsilon\in{\frac{1}{2}}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ holds with ${\rm rank}(c_{ab})=1$, then $A\in J_{g}.$

Proof. See S. Grushevsky [59]. $\Box$

(G) Subvarieties of a ppav: Minimal Cohomology Classes

The existence of some special subvarieties of a ppav $(A,\Theta)\in{\mathcal{A}}_{g}$ gives a criterion that $A$ is the Jacobian of a curve. We start by observing that for the Jacobian $J(C)$ of a curve $C\in{\mathcal{M}}_{g}$ we can map the symmetric product ${\rm Sym}^{d}(C)\,(1\leq d<g)$ to $J(C)={\rm Pic}^{g-1}(C)$ by fixing a divisor $D\in{\rm Pic}^{g-1-d}(C)$ and mapping

[TABLE]

The image $W^{d}(C)$ of the map $\Phi_{(d)}$ is independent of $D$ up to translation, and we can compute its cohomology class

[TABLE]

where $[\Theta]$ is the cohomology class of the polarization of $J(C)$. One can show that the cohomology class is indivisible in cohomology with $\mathbb{Z}$-coefficients, and we thus call this class minimal. We note that $W^{1}(C)\simeq C.$ These subvarieties $W^{d}(C)\ (1\leq d<g)$ are very special. We have the following criterion.

Proof. See Matsusaka [92] and Ran [116]. $\Box$

Debarre [27] proved that $J_{g}$ is an irreducible component of the locus of ppavs for which there is a subvariety of the minimal cohomology class. He conjectured the following.

This approach to the Schottky problem gives a complete geometric solution to the weaker version of the problem : determining whether a given ppav is the Jacobian of a given curve.

3. Logarithmical Canonical Line Bundles on Toroidal Compactifications of the Siegel Modular Varieties

In this section, we review the interesting results obtained by S.-T. Yau and Y. Zhang [177] concerning the asymptotic behaviors of the logarithmical canonical line bundle on a toroidal compactification of the Siegel modular variety.

Let $\Gamma$ be a neat arithmetic subgroup of $\Gamma_{g}.$ Let $\mathcal{A}_{g,\Gamma}:=\Gamma\backslash\mathbb{H}_{g}$ and $\overline{\mathcal{A}}_{g,\Gamma}$ be the toroidal compactification of $\mathcal{A}_{g,\Gamma}$ constructed by a $GL(g,\mathbb{Z})$-admissible family of polyhedral decompositions $\Sigma_{{\mathcal{F}}_{0}}$ of the cones. Here ${\mathcal{F}}_{0}$ denotes the standard minimal cusps of $\mathbb{H}_{g}$. $\overline{\mathcal{A}}_{g,\Gamma}$ is an algebraic space, but a projective variety in general. Y.-S. Tai proved that if $\Sigma_{tor}^{\Gamma}$ is projective (see [9, Chapter IV, Corollary 2.3, p. 200]), then $\overline{\mathcal{A}}_{g,\Gamma}$ is a projective variety. It is known that $\overline{\mathcal{A}}_{g,\Gamma}$ is the unique Hausdorff analytic variety containing $\mathcal{A}_{g,\Gamma}$ as an open dense subset (cf. [9]).

Assume the boundary divisor $D_{\infty,\Gamma}$:=$\overline{\mathcal{A}}_{g,\Gamma}\backslash\mathcal{A}_{g,\Gamma}$ is simple normal crossing. We put $N=g(g+1)/2.$ For each irreducible component $D_{i}$ of $D_{\infty,\Gamma}=\bigcup_{j}D_{j}$, let $s_{i}$ a global section of the line bundle $[D_{i}]$ defining $D_{i}$. Let $\sigma_{\rm max}$ be an arbitrary top-dimensional cone in $\Sigma_{{\mathcal{F}}_{0}}$ and renumber all components $D_{i}^{\prime}s$ of $D_{\infty,\Gamma}$ such that $D_{1},\cdots,D_{N}$ correspond to the edges of $\sigma_{\rm max}$ with marking order. Yau and Zhang [177, Theorem 3.2] showed that the volume form $\Phi_{g,\Gamma}$ on $\mathcal{A}_{g,\Gamma}$ may be written by

[TABLE]

where $d{\mathcal{V}}_{g}$ is a continuous volume form on a partial compactification $\mathcal{U}_{\sigma_{\rm max}}$ of $\mathcal{A}_{g,\Gamma}$ with $\mathcal{A}_{g,\Gamma}\subset\mathcal{U}_{\sigma_{\rm max}}\subset\overline{\mathcal{A}}_{g,\Gamma}$, and each $\|\cdot\|_{j}$ is a suitable Hermitian metric of the line bundle $[D_{j}]$ on $\overline{\mathcal{A}}_{g,\Gamma}\ (1\leq j\leq N)$ and $F_{\sigma_{\rm max}}\in\mathbb{Z}[x_{1},\cdots,x_{N}]$ is a homogeneous polynomial of degree $g$. Moreover the coefficients of $F_{\sigma_{\rm max}}$ depends only on both $\Gamma$ and $\sigma_{\rm max}$ with marking order of edges. Using the above volume form formula they showed that the unique invariant Kähler-Einstein metric on $\mathcal{A}_{g,\Gamma}$ endows some restraint combinatorial conditions for all smooth toroidal compactifications of $\mathcal{A}_{g,\Gamma}$.

Let $E_{1},\cdots,E_{d}$ be any different irreducible components of the boundary divisor $D_{\infty,\Gamma}$ such that $\bigcap_{k=1}^{d}E_{k}\neq\emptyset$. Let $K_{g,\Gamma}$ be the canonical line bundle on $\overline{\mathcal{A}}_{g,\Gamma}$. Yau and Zhang [177] also proved the following facts (a) and (b):

(a) Let $i_{1},\cdots,i_{d}\in\mathbb{Z}^{+}.$ If $d\geq g-1$ and $N-\sum_{k=1}^{d}i_{k}>2$ (or if $d\geq g$ and $N-\sum_{k=1}^{d}i_{k}=1)$,

  then we have

[TABLE]

(b) $K_{g,\Gamma}+D_{\infty,\Gamma}$ is not ample on $\overline{\mathcal{A}}_{g,\Gamma}$.

They also showed that if $d<g-1,$ then the intersection number

[TABLE]

can be expressed explicitly using the above volume form formula. The proofs of (a) and (b) can be found in [177, Theorem 4.15].

4. Brief Review on the André-Oort Conjecture

In this section we review recent progress on the André-Oort conjecture quite briefly.

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be the Shimura variety associated to $(G,X).$ An algebraic subvariety $Z$ of the Shimura variety $Sh_{K}(G,X)$ is said to be weakly special if there exist a Shimura sub-datum $(H,X_{H})$ of $(G,X)$, and a decomposition

[TABLE]

and $y_{2}\in X_{2}$ such that $Z$ is the image of $X_{1}\times\{y_{2}\}$ in $Sh_{K}(G,X)$. Here $(H^{\rm ad},X_{H}^{\rm ad})$ denotes the adjoint Shimura datum associated to $(G,X)$ and $(H_{i},X_{i})\ (i=1,2)$ are Shimura data. In this definition, a weakly special subvariety is said to be special if it contains a special point and $y_{2}$ is special. **

André [4] and Oort [108] made conjectures analogous to the Manin-Mumford conjecture where the ambient variety is a Shimura variety (the latter partially motivated by a conjecture of Coleman [25]). A combination of these has become known as the André-Oort conjecture (briefly the A-O conjecture).

A-O Conjecture. Let $S$ be a Shimura variety and let $\Sigma$ be a set of special points in $S$. Then every irreducible component of the Zariski closure of $\Sigma$ is a special subvariety.

(1) $\Sigma_{n}$ is a Boolean algebra (under the usual set-theoretic operations);

(2) $\Sigma_{n}$ contains every semi-algebraic subset of $\mathbb{R}^{n}$;

(3) if $A\in\Sigma_{m}$ and $B\in\Sigma_{n}$, then $A\times B\in\Sigma_{m+n}$;

(4) if $n\geq m$ and $A\in\Sigma_{n}$, then $\pi_{n,m}(A)\in\Sigma_{m}$, where $\pi_{n,m}:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ is a coordinate

     projection on the first $m$ coordinates. If $\Sigma$ is a structure, and, in addition,

(5) the boundary of every set in $\Sigma_{1}$ is finite,

then $\Sigma$ is called an $o$-minimal structure over the real field. If $\Sigma$ is a structure and $Z\subset\mathbb{R}^{n}$, then we say that $Z$ is definable in $\Sigma$ if $Z\in\Sigma_{n}$. A function $f:A\longrightarrow B$ is definable in a structure $\Sigma$ if its graph is definable, in which case the domain $A$ of $f$ and image $f(A)$ are also definable by the definition. If $A,\cdots,f,\cdots$ are sets or functions, then we denote by $\mathbb{R}_{A,\cdots,f,\cdots}$ the smallest structure containing $A,\cdots,f,\cdots$. By a definable family of sets we mean a definable subset $Z\subset\mathbb{R}^{n}\times\mathbb{R}^{m}$ which we view as a family of fibres $Z_{y}\subset\mathbb{R}^{n}$ as $y$ varies over the projection of $Z$ onto $\mathbb{R}^{m}$ which is definable, along with all the fibres $Z_{y}$. A family of functions is said to be definable if the family of their graphs is. A definable set usually means a definable set in some o-minimal structure over the real field. **

In 2011 Pila gave a unconditional proof of the A-O conjecture for arbitrary products of modular curves using the theory of o-minimality.

[TABLE]

where $n,m,\ell\geq 0,\ Y_{i}=\Gamma_{(i)}\backslash\mathbb{H}_{1}(1\leq i\leq n)$ are modular curves corresponding to congruence subgroups $\Gamma_{(i)}$ of $SL(2,\mathbb{Z})$ and $E_{j}\,(1\leq j\leq m)$ are elliptic curves defined over ${\overline{\mathbb{Q}}}$ and ${\mathbb{G}}_{m}$ is the multiplicative group. Suppose $V$ is a subset of $X$. Then $V$ contains only a finite number of maximal special subvarieties.

Proof. See Pila [111, Theorem 1.1]. $\Box$

In 2013 Peterzil and Starchenko proved the following theorem using the theory of o-minimality.

Proof. See Peterzil and Starchenko [109, 110]. $\Box$

In 2014 Pila and Tsimerman gave a conditional proof of the A-O conjecture for the Siegel modular variety $\mathcal{A}_{g}.$

Proof. See Pila-Tsimerman [113, 114]. $\Box$

Quite recently using Galois-theoretic techniques and geometric properties of Hecke correpondences, Klingler and Yafaev proved the A-O conjecture for a general Shimura variety, and independently using Galois-theoretic and ergodic techniques Ullmo and Yafaev proved the A-O conjecture for a general Shimura variety, under the assumption of the GRH for CM fields or another suitable assumption. The explicit statement is given as follows.

Proof. See Klingler-Yafaev [76] and Ullmo-Yafaev [136]. $\Box$

5. Okounkov Bodies Associated to Divisors

In this section, we briefly review the theory of Okounkov convex bodies associated to pseudoeffective divisors on a smooth projective variety. For more details of this theory, we refer to [20, 81].

Let $X$ be a smooth projective variety of dimension $d$. We fix an admissible flag $Y_{\bullet}$ on $X$

[TABLE]

where each $Y_{k}$ is a subvariety of $X$ of codimension $k$ which is nonsingular at $x$. We let $\mathbb{Z}_{+}$ denote the set of all non-negative integers. We first assume that $D$ is a big Cartier divisor on $X$. For a section $s\in H^{0}(X,\mathcal{O}_{X}(D))\backslash\{0\},$ we define the function

[TABLE]

as follows:

First we set $\nu_{1}=\nu_{1}(s):={\rm ord}_{Y_{1}}(s).$ Using a local equation $f_{1}$ for $Y_{1}$ in $X$, we define naturally a section

[TABLE]

that does not vanish along $Y_{1}$, its restriction $s_{1}^{\prime}|_{Y_{1}}$ defines a nonzero section

[TABLE]

We now take

[TABLE]

and continue in this manner to define the remaining $\nu_{i}(s).$

Next we define

[TABLE]

be the set of valuation vectors of non-zero sections of $\mathcal{O}_{X}(D).$ Then we finally set

[TABLE]

Therefore $\Delta(D)$ is a convex body in $\mathbb{R}^{d}$ that is called the Okounkov body of $D$ with respect to the fixed flag $Y_{\bullet}$. We refer to [81, §1.2] for some properties and examples of $\Delta(D)$.

We recall that a graded linear series $W_{\bullet}(D)=\{W_{m}(D)\}_{m\geq 0}$ associated to $D$ consists of subspaces

[TABLE]

satisfying the inclusion

[TABLE]

Here the product on the left denotes the image of $W_{k}\otimes W_{l}$ under the multiplication map $H^{0}(X,\mathcal{O}_{X}(kD))\otimes H^{0}(X,\mathcal{O}_{X}(lD))\longrightarrow H^{0}(X,\mathcal{O}_{X}((k+l)D)).$

[TABLE]

Under the above notations, we associate the convex body $\Delta_{Y_{\bullet}}(W_{\bullet})$ of a graded linear series $W_{\bullet}$ with respect to $Y_{\bullet}$ on $X$ as follows:

[TABLE]

where $\mathbb{R}_{+}$ denotes the set of all non-negative real numbers and $\sum\left(\Gamma(W_{\bullet})\right)$ denotes the closure of the convex cone in $\mathbb{R}_{+}^{d}\times\mathbb{R}_{+}$ spanned by $\Gamma(W_{\bullet})$. $\Delta_{Y_{\bullet}}(W_{\bullet})$ is called the Okounkov body of $W_{\bullet}$ with respect to $Y_{\bullet}$. If $W_{\bullet}$ is a complete graded linear series, that is, $W_{m}=H^{0}(X,{\mathcal{O}}(mD))$ for each $m$, then we define

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(I) We say that a graded linear series $W_{\bullet}$ satisfies Condition (B) if $W_{m}\neq 0$ for all

$m\gg 0$, and for all sufficiently large $m$, the rational map $\phi_{m}:X-->\mathbb{P}(W_{m})$

defined by $|W_{m}|$ is birational on its image.

(II) We say that a graded linear series $W_{\bullet}$ satisfies Condition (C) if

(1) for any $m\gg 0$, there exists an effective divisor $F_{m}$ such that $A_{m}:=mD-F_{m}$

     is ample, and

(2) for all sufficiently large $t$, we have

[TABLE]

If $W_{\bullet}$ is complete, that is, $W_{m}=H^{0}(X,\mathcal{O}_{X}(mA_{m}))$ for all $m\geq 0$ and $D$ is big,

then it satisfies Condition (C). **

Lazarsfeld and Mustată [81] proved the following.

[TABLE]

and

[TABLE]

where

[TABLE]

Proof. See [81, Theorem 2.13]. $\Box$

[TABLE]

For $m\in\mathbb{N}(D),$ we let

[TABLE]

be the rational map defined by the linear system $|\lfloor mD\rfloor|$. We define the Iitaka dimension of $D$ as the following value

[TABLE]

[TABLE]

is injective for every non-negative integer $m$. **

[TABLE]

For a divisor $D$ with $\kappa(D)=-\infty,$ we define $\Delta_{Y_{\bullet}}^{\rm val}(D):=\emptyset.$ **

Recently Choi, Hyun, Park and Won [20] showed the following.

[TABLE]

and

[TABLE]

Proof. See [20, Theorem 3.12]. $\Box$

[TABLE]

where $A$ is any ample divisor on $X$. If $D$ is not a pseudo-effective divisor, then we define $\Delta_{Y_{\bullet}}^{\rm lim}(D):=\emptyset.$ **

[TABLE]

for a fixed ample Cartier divisor $A$ if $h^{0}(X,\mathcal{O}_{X}(|mD|+A))\neq\emptyset$ for infinitely many $m>0$, and we define $\kappa_{\nu}(D)=-\infty$ otherwise. **

Let $D$ be a pseudo-effective Cartier divisor on a projective variety $X$ of dimension $n$. Let $V\subseteq X$ be a positive volume subvariety of $D$. Fix an admissible flag $V_{\bullet}$ on $V$

[TABLE]

let $A$ be an ample Catier divisor on $X$. For each positive integer $k$, we consider the restricted graded linear series $W_{\bullet}^{k}:=W_{\bullet}(kD+A|V)$ of $kD+A$ along $V$ given by

[TABLE]

We define the restricted limiting Okounkov body of a Cartier divisor $D$ with respect to a positive volume subvariety $V$ of $D$ as

[TABLE]

By the continuity, we can extend this definition for any pseudo-effective $\mathbb{R}$-divisor.

[TABLE]

where $A$ is any ample divisor on $X$. If $D$ is not a pseudo-effective divisor, then we define $\Delta_{V_{\bullet}}^{\rm lim}(D):=\emptyset.$ **

Recently Choi, Hyun, Park and Won [20] proved the following.

[TABLE]

and

[TABLE]

Here ${\rm Vol}_{X|V}^{+}(D)$ denotes the augmented restricted volume of $D$ along $V$ (see [20, Definition 2.2]) for the precise definition of ${\rm Vol}_{X|V}^{+}(D))$.

Proof. See [20, Theorem 3.20]. $\Box$

6. The Relations of the Schottky Problem to the André-Oort Conjecture, Okounkov Bodies and Coleman’s Conjecture

In this section, we discuss the relations among logarithmical line bundles on toroidal compactifications, the André-Oort conjecture, Okounkov convex bodies, Coleman’s conjecture and the Schottky problem.

For $\tau=(\tau_{ij})\in\mathbb{H}_{g},$ we write $\tau=X+i\,Y$ with $X=(x_{ij}),\ Y=(y_{ij})$ real. We put $d\tau=(d\tau_{ij})$ and $d{\overline{\tau}}=(d{\overline{\tau}}_{ij})$. We also put

[TABLE]

C. L. Siegel [130] introduced the symplectic metric $ds_{g;A}^{2}$ on $\mathbb{H}_{g}$ invariant under the action (1.1) of $Sp(2g,\mathbb{R})$ that is given by

[TABLE]

and H. Maass [89] proved that its Laplacian is given by

[TABLE]

Here ${\rm tr}(M)$ denotes the trace of a square matrix $M$. And

[TABLE]

is a $Sp(2g,\mathbb{R})$-invariant volume element on $\mathbb{H}_{g}$ (cf. [131, p. 130]).

Siegel proved the following theorem for the Siegel space $(\mathbb{H}_{g},ds^{2}_{g;1}).$

[TABLE]

For brevity, we put $R_{*}=R(\tau_{0},\tau_{1}).$ Then the symplectic length $\rho(\tau_{0},\tau_{1})$ of the geodesic joining $\tau_{0}$ and $\tau_{1}$ is given by

[TABLE]

where

[TABLE]

(2)* For $M\in Sp(2g,\mathbb{R})$, we set*

[TABLE]

Then $R(\tau_{1},\tau_{0})$ and $R({\tilde{\tau}}_{1},{\tilde{\tau}}_{0})$ have the same eigenvalues.

(3)* All geodesics are symplectic images of the special geodesics*

[TABLE]

where $a_{1},a_{2},\cdots,a_{g}$ are arbitrary positive real numbers satisfying the condition

[TABLE]

The proof of the above theorem can be found in [90] or [130, pp. 289-293].

B. Moonen [94] proved the following fact.

Proof. See [94, Theorem 4.3, pp. 553–554]. $\Box$

In the 1980s Coleman [25] proposed the following conjecture. Coleman’s Conjecture. For a sufficiently large integer $g$, the Jacobian locus $J_{g}$ contains only a finite number of special points in ${\mathcal{A}}_{g}.$

We also have the following conjecture.

The stronger version of Conjecture 6.1 is given as follows:

Proof. Let $g$ be a sufficiently large integer $g$. Suppose $J_{g}$ contains an infinite set $\Sigma$ of special points. Then

[TABLE]

The truth of the André-Oort conjecture implies that ${\overline{\Sigma}}$ contains an irreducible special subvariety $Y$. According to Theorem 6.2, $Y$ is a totally geodesic subvariety of ${\overline{J}}_{g}$. From the truth of Conjecture 6.1, we get a contradiction. Therefore $J_{g}$ contains only finitely many special points. $\Box$

Now we propose the following problems.

Problem 6.1. Develop the spectral theory of the Laplace operator $\Box_{g;A,B}$ on $\mathbb{H}_{g}$ and $\Gamma\backslash\mathbb{H}_{g}$ for a congruence subgroup $\Gamma$ of $\Gamma_{g}$ explicitly.

Problem 6.2. Construct all the geodesics contained in ${\overline{J}}_{g}$ with respect to the Siegel’s metric $ds^{2}_{g;A}.$

Problem 6.3. Study variations of $g$-dimensional principally polarized abelian varieties along a geodesic inside $J_{g}$.

Problem 6.4. Prove the A-O conjecture for ${\mathcal{A}}_{g}$ unconditionally.

From now on, we will adopt the notations in Section 3.

Problem 6.5. Let $p_{4,\Gamma}:\mathcal{A}_{4,\Gamma}\longrightarrow\mathcal{A}_{4}$ be a covering map and let $J_{4,\Gamma}:=p_{4,\Gamma}^{-1}(J_{4}).$ Let $\overline{\mathcal{A}}_{4,\Gamma}$ be a toroidal compactification of $\mathcal{A}_{4,\Gamma}$ which is projective. Then $J_{4,\Gamma}$ is a divisor on $\overline{\mathcal{A}}_{4,\Gamma}$. Compute the Okounkov bodies $\Delta_{Y_{\bullet}}(J_{4,\Gamma})$, $\Delta_{Y_{\bullet}}^{\rm val}(J_{4,\Gamma})$ and $\Delta_{Y_{\bullet}}^{\rm lim}(J_{4,\Gamma})$ explicitly. Describe the relations among $J_{4},\ J_{4,\Gamma}$ and these Okounkov bodies explicitly. Describe the relations between these Okounkov bodies and the $GL(4,\mathbb{Z})$-admissible family of polyhedral decompositions defining the toroidal compactification $\overline{\mathcal{A}}_{4,\Gamma}$.

Problem 6.6. Assume that a toroidal compactification $\overline{\mathcal{A}}_{g,\Gamma}$ is a projective variety. Compute the Okoukov convex bodies $\Delta_{Y_{\bullet}}(K_{g,\Gamma})$, $\Delta_{Y_{\bullet}}^{\rm val}(K_{g,\Gamma})$, $\Delta_{Y_{\bullet}}^{\rm lim}(K_{g,\Gamma})$, $\Delta_{Y_{\bullet}}(D_{\infty,\Gamma})$, $\Delta_{Y_{\bullet}}^{\rm val}(D_{\infty,\Gamma})$, $\Delta_{Y_{\bullet}}^{\rm lim}(D_{\infty,\Gamma})$, $\Delta_{Y_{\bullet}}(K_{g,\Gamma}+D_{\infty,\Gamma})$, $\Delta_{Y_{\bullet}}^{\rm val}(K_{g,\Gamma}+D_{\infty,\Gamma})$, $\Delta_{Y_{\bullet}}^{\rm lim}(K_{g,\Gamma}+D_{\infty,\Gamma})$ explicitly. Describe the relations between these Okounkov bodies and the $GL(g,\mathbb{Z})$-admissible family of polyhedral decompositions defining the toroidal compactification $\overline{\mathcal{A}}_{g,\Gamma}$.

Problem 6.7. Assume that $g\geq 5.$ Let $p_{g,\Gamma}:\mathcal{A}_{g,\Gamma}\longrightarrow\mathcal{A}_{g}$ be a covering map and let $J_{g,\Gamma}:=p_{g,\Gamma}^{-1}(J_{g}).$ Assume that $\overline{\mathcal{A}}_{g,\Gamma}$ is a toroidal compactification of $\mathcal{A}_{g,\Gamma}$ which is a projective variety. Let $D_{J,\Gamma}$ be a divisor on $\overline{\mathcal{A}}_{g,\Gamma}$ containing $J_{g,\Gamma}$. Describe the Okounkov bodies $\Delta_{Y_{\bullet}}(D_{J,\Gamma}),\ \Delta_{Y_{\bullet}}^{\rm val}(D_{J,\Gamma}),\ \Delta_{Y_{\bullet}}^{\rm lim}(D_{J,\Gamma})$. Study the relations between $J_{g,\Gamma},\ D_{J,\Gamma}$ and these Okounkov bodies.

We have the following diagram:

[TABLE]

Here $p_{g,\Gamma}\;:\;{\mathcal{A}}_{g,\Gamma}\longrightarrow{\mathcal{A}}_{g}$ is a covering map.

Finally we propose the following questions.

Question 6.1. Let $\Gamma$ be a neat arithmetic subgroup of $\Gamma_{g}$. Does the closure $\overline{J}_{g,\Gamma}$ of $J_{g,\Gamma}$ intersect the infinity boundary divisor $D_{\infty,\Gamma}$? If $g$ is sufficient large, it is probable that $\overline{J}_{g,\Gamma}$ will not intersect the boundary divisor $D_{\infty,\Gamma}$.

Question 6.2. Let $\Gamma$ be a neat arithmetic subgroup of $\Gamma_{g}$. Does the closure $\overline{J}_{g,\Gamma}$ of $J_{g,\Gamma}$ intersect the canonical divisor $K_{g,\Gamma}$?

Question 6.3. Let $\Gamma$ be a neat arithmetic subgroup of $\Gamma_{g}$. How curved is the closure $\overline{J}_{g,\Gamma}$ of $J_{g,\Gamma}$ along the boundary of $J_{g,\Gamma}$?

Quite recently using the good curvature properties of the moduli space $(\mathcal{M}_{g},\omega_{{}_{\rm WP}})$ endowed with the Weil-Petersson metric $\omega_{{}_{\rm WP}}$, Liu, Sun and Yau [87] obtained interesting results related to Conjecture 6.2. Let us explain their results briefly. We consider the coarse moduli space $(\mathcal{M}_{g},\omega_{{}_{\rm WP}})$ endowed with the Weil-Petersson metric $\omega_{{}_{\rm WP}}$ and the Siegel modular variety $(\mathcal{A}_{g},\omega_{{}_{\rm H}})$ endowed with the Hodge metric $\omega_{{}_{\rm H}}$. Let $T_{g}:\mathcal{M}_{g}\longrightarrow\mathcal{A}_{g}$ be the Torelli map (see (1.2)). Assume that $V$ is a submanifold in $\mathcal{M}_{g}$ such that the image $T_{g}(V)$ is totally geodesic in $(\mathcal{A}_{g},\omega_{{}_{\rm H}})$, and also that $T_{g}(V)$ has finite volume. Under these two assumptions they proved that $V$ must be a ball quotient. As a corollary of this fact, it can be shown that there is no higher rank locally symmetric subspace in $\mathcal{M}_{g}$. A precise statement is as follows.

[TABLE]

then $\Omega$ must be of rank $1$, i.e., $X$ must be a ball quotient.

Proof. The proof of the above theorem can be found in [87]. $\Box$

7. Final Remarks and Open Problems

In this final section we give some remarks and propose some open problems about the relations among the Schottky problem, the André-Oort conjecture, Okounkov convex bodies, stable Schottky-Siegel forms, stable Schottky-Jacobi forms and the geometry of the Siegel-Jacobi space. We define the notion of stable Schottky-Jacobi forms and the concept of stable Jacobi equations for the universal hyperelliptic locus.

For two positive integers $g$ and $h$, we consider the Heisenberg group

[TABLE]

endowed with the following multiplication law

[TABLE]

with \big{(}\lambda,\mu;\kappa\big{)},\big{(}\lambda^{\prime},\mu^{\prime};\kappa^{\prime}\big{)}\in H_{\mathbb{R}}^{(g,h)}. We refer to [146, 151, 154, 157, 160, 167, 170, 173] for more details on the Heisenberg group $H_{\mathbb{R}}^{(g,h)}.$ We define the Jacobi group $G^{J}$ of degree $g$ and index $h$ that is the semidirect product of $Sp(2g,\mathbb{R})$ and $H_{\mathbb{R}}^{(g,h)}$

[TABLE]

endowed with the following multiplication law

[TABLE]

with $M,M^{\prime}\in Sp(2g,\mathbb{R}),(\lambda,\mu;\kappa),\,(\lambda^{\prime},\mu^{\prime};\kappa^{\prime})\in H_{\mathbb{R}}^{(g,h)}$ and $(\tilde{\lambda},\tilde{\mu})=(\lambda,\mu)M^{\prime}$. Then $G^{J}$ acts on $\mathbb{H}_{g}\times\mathbb{C}^{(h,g)}$ transitively by

[TABLE]

where $M=\begin{pmatrix}A&B\\ C&D\end{pmatrix}\in Sp(2g,\mathbb{R}),\ (\lambda,\mu;\kappa)\in H_{\mathbb{R}}^{(g,h)}$ and $(\tau,z)\in\mathbb{H}_{g}\times\mathbb{C}^{(h,g)}$.

We note that the Jacobi group $G^{J}$ is not a reductive Lie group and the homogeneous space ${\mathbb{H}}_{g}\times\mathbb{C}^{(h,g)}$ is not a symmetric space. From now on, for brevity we write $\mathbb{H}_{g,h}=\mathbb{H}_{g}\times\mathbb{C}^{(h,g)}.$ The homogeneous space $\mathbb{H}_{g,h}$ is called the Siegel-Jacobi space of degree $g$ and index $h$.

For $\tau=(\tau_{ij})\in\mathbb{H}_{g},$ we write $\tau=X+iY$ with $X=(x_{ij}),\ Y=(y_{ij})$ real. We put $d\tau=(d\tau_{ij})$ and $d{\overline{\tau}}=(d{\overline{\tau}}_{ij})$. We also put

[TABLE]

For a coordinate $z\in\mathbb{C}^{(h,g)},$ we set

[TABLE]

[TABLE]

The author proved the following theorems in [163].

[TABLE]

is a Riemannian metric on $\mathbb{H}_{g,h}$ which is invariant under the action (7.1) of $G^{J}.$ In fact, $ds_{g,h}^{2}$ is a Kähler metric of $\mathbb{H}_{g,h}.$

Proof. See [163, Theorem 1.1]. $\Box$

[TABLE]

where

[TABLE]

and

[TABLE]

Furthermore ${\mathbb{M}}_{1}$ and ${\mathbb{M}}_{2}$ are differential operators on $\mathbb{H}_{g,h}$ invariant under the action (7.1) of $G^{J}.$

Proof. See [163, Theorem 1.2]. $\Box$

For a fixed $\tau\in\mathbb{H}_{g}$ and a fixed polarization type $D={\rm diag}(d_{1},\cdots,d_{g})$, we let $L_{\tau}^{D}:=\mathbb{Z}^{g}\tau+\mathbb{Z}^{g}D$ be a lattice in $\mathbb{C}^{g}$ and $A_{\tau}^{D}:=\mathbb{C}^{g}/L_{\tau}^{D}$ be a complex torus of a polarization type $D$. Let $\{c_{0},\cdots,c_{N}\}$ be the set of representatives in $\mathbb{Z}^{g}D^{-1}$ whose components of each $c_{i}\,(0\leq i\leq N)$ lie in the interval $[0,1).$ Here $N=d_{1}\cdots d_{g}-1.$

We recall Lefschetz theorem (see [100, p. 128, Theorem 1.3]).

[TABLE]

is a well-defined holomorphic mapping. For each $\tau\in\mathbb{H}_{g}$, the map $\varphi_{\tau}^{D}:\mathbb{C}^{g}\longrightarrow{\mathbb{P}}^{N}(\mathbb{C})$ defined by

[TABLE]

induces a holomorphic mapping from the complex torus $A_{\tau}^{D}$ into ${\mathbb{P}}^{N}(\mathbb{C})$. (2) If $d_{1}\geq 3,$ for each $\tau\in\mathbb{H}_{g},$ the map $\varphi_{\tau}^{D}:\mathbb{C}^{g}\longrightarrow{\mathbb{P}}^{N}(\mathbb{C})$ is an analytic embedding, whose image is an algebraic subvariety of ${\mathbb{P}}^{N}(\mathbb{C})$.

[TABLE]

and define the map $\Phi^{D}:\mathbb{H}_{g}\times\mathbb{C}^{g}\longrightarrow{\mathbb{P}}^{N}(\mathbb{C})\times{\mathbb{P}}^{N}(\mathbb{C})$ by

[TABLE]

We have the following theorem proved by Baily [10].

Proof. See [10, Section 5.1] or [110, Theorem 8.11]. $\hfill\square$

Let

[TABLE]

be the arithmetic subgroup of $G^{J},$ where

[TABLE]

We let

[TABLE]

be the universal family of principal polarized abelian varieties of dimension $gh$. Let $\pi_{g,h}:{\mathcal{A}}_{g,h}\longrightarrow{\mathcal{A}}_{g}$ be the natural projection. We define the universal Jacobian locus

[TABLE]

Problem 7.1. Characterize $J_{g,h}=\pi_{g,h}^{-1}(J_{g}).$ Describe $J_{g,h}$ in terms of Jacobi forms. We refer to [15, 37, 147, 149, 150, 152, 153, 155, 158, 159, 161, 162, 168, 171, 179] for more details about Jacobi forms.

Problem 7.2. Compute the geodesics, the distance between two points and curvatures explicitly in the Siegel-Jacobi space $(\mathbb{H}_{g,h},ds^{2}_{g,h;A,B}).$ See Theorem 6.1 for the Siegel space $\mathbb{H}_{g}$.

Problem 7.3. Find the analogue of the Hirzebruch-Mumford Proportionality Theorem for ${\mathcal{A}}_{g,\Gamma}^{u}$ (see (7.8) below). Let us give some remarks for this problem. Before we describe the proportionality theorem for the Siegel modular variety, first of all we review the compact dual of the Siegel upper half plane $\mathbb{H}_{g}$. We note that $\mathbb{H}_{g}$ is biholomorphic to the generalized unit disk $\mathbb{D}_{g}$ of degree $g$ through the Cayley transform. We suppose that $\Lambda=(\mathbb{Z}^{2g},\langle\ ,\ \rangle)$ is a symplectic lattice with a symplectic form $\langle\ ,\ \rangle.$ We extend scalars of the lattice $\Lambda$ to $\mathbb{C}$. Let

[TABLE]

be the complex Lagrangian Grassmannian variety parameterizing totally isotropic subspaces of complex dimension $g$. For the present time being, for brevity, we put $G=Sp(2g,\mathbb{R})$ and $K=U(g).$ The complexification $G_{\mathbb{C}}=Sp(2g,\mathbb{C})$ of $G$ acts on ${\mathfrak{Y}}_{g}$ transitively. If $H$ is the isotropy subgroup of $G_{\mathbb{C}}$ fixing the first summand $\mathbb{C}^{g}$, we can identify ${\mathfrak{Y}}_{g}$ with the compact homogeneous space $G_{\mathbb{C}}/H.$ We let

[TABLE]

be an open subset of ${\mathfrak{Y}}_{g}$. We see that $G$ acts on ${\mathfrak{Y}}_{g}^{+}$ transitively. It can be shown that ${\mathfrak{Y}}_{g}^{+}$ is biholomorphic to $G/K\cong\mathbb{H}_{g}.$ A basis of a lattice $L\in{\mathfrak{Y}}_{g}^{+}$ is given by a unique $2g\times g$ matrix ${}^{t}(-I_{g}\,\,\tau)$ with $\tau\in\mathbb{H}_{g}$. Therefore we can identify $L$ with $\tau$ in $\mathbb{H}_{g}$. In this way, we embed $\mathbb{H}_{g}$ into ${\mathfrak{Y}}_{g}$ as an open subset of ${\mathfrak{Y}}_{g}$. The complex projective variety ${\mathfrak{Y}}_{g}$ is called the compact dual of $\mathbb{H}_{g}.$

Let $\Gamma$ be an arithmetic subgroup of $\Gamma_{g}$. Let $E_{0}$ be a $G$-equivariant holomorphic vector bundle over $\mathbb{H}_{g}=G/K$ of rank $r$. Then $E_{0}$ is defined by the representation $\tau:K\longrightarrow GL(r,\mathbb{C}).$ That is, $E_{0}\cong G\times_{K}\mathbb{C}^{r}$ is a homogeneous vector bundle over $G/K$. We naturally obtain a holomorphic vector bundle $E$ over $\mathcal{A}_{g,\Gamma}:=\Gamma\backslash G/K.$ $E$ is often called an automorphic or arithmetic vector bundle over $\mathcal{A}_{g,\Gamma}$. Since $K$ is compact, $E_{0}$ carries a $G$-equivariant Hermitian metric $h_{0}$ which induces a Hermitian metric $h$ on $E$. According to Main Theorem in [97], $E$ admits a unique extension ${\tilde{E}}$ to a smooth toroidal compactification ${\tilde{\mathcal{A}}}_{g,\Gamma}$ of $\mathcal{A}_{g,\Gamma}$ such that $h$ is a singular Hermitian metric good on ${\tilde{\mathcal{A}}}_{g,\Gamma}$. For the precise definition of a good metric on $\mathcal{A}_{g,\Gamma}$ we refer to [97, p. 242]. According to Hirzebruch-Mumford’s Proportionality Theorem (cf. [97, p. 262]), there is a natural metric on $G/K=\mathbb{H}_{g}$ such that the Chern numbers satisfy the following relation

[TABLE]

for all $\alpha=(\alpha_{1},\cdots,\alpha_{r})$ with nonegative integers $\alpha_{i}\,(1\leq i\leq r)$ and $\sum_{i=1}^{r}\alpha_{i}={\frac{1}{2}}g(g+1),$ where ${\check{E}}_{0}$ is the $G_{\mathbb{C}}$-equivariant holomorphic vector bundle on the compact dual ${\mathfrak{Y}}_{g}$ of $\mathbb{H}_{g}$ defined by a certain representation of the stabilizer $\textrm{Stab}_{G_{\mathbb{C}}}(e)$ of a point $e$ in ${\mathfrak{Y}}_{g}$. Here $\textmd{vol}\left(\Gamma\backslash\mathbb{H}_{g}\right)$ is the volume of $\Gamma\backslash\mathbb{H}_{g}$ that can be computed (cf. [130]).

Problem 7.4. Compute the cohomology $H^{\bullet}({\mathcal{A}}_{g,h},*)$ of ${\mathcal{A}}_{g,h}.$ Investigate the intersection cohomology of ${\mathcal{A}}_{g,h}.$

Problem 7.5. Generalize the trace formula on the Siegel modular variety obtained by Sophie Morel to the universal abelian variety. For her result on the trace formula on the Siegel modular variety, we refer to her paper [95, Cohomologie d’intersection des variétés modulaires de Siegel, suite].

Problem 7.6. Construct all the geodesics contained in $J_{g,h}.$

Problem 7.7. Develop the theory of variations of abelian varieties along the geodesic joining two points in $J_{g,h}$.

Problem 7.8. Discuss the André-Oort conjecture for ${\mathcal{A}}_{g,h}$. Gao proved the Ax-Lindemann-Weierstras theorem for ${\mathcal{A}}_{g,h}$, and using this theorem proved the André-Oort conjecture for ${\mathcal{A}}_{g,h}$ under the assumption of the Generalized Riemann Hypothesis for CM fields in his paper [52].

Let $\Gamma$ be a neat arithmetic subgroup of $\Gamma_{g}$. We put $\Gamma^{J}:=\Gamma\ltimes H_{\mathbb{Z}}^{(g,h)}$. We let

[TABLE]

Let ${\mathcal{A}}_{g,h,\Gamma}^{tor}$ be a toroidal compactification of ${\mathcal{A}}_{g,h,\Gamma}$. Let $K_{g,h,\Gamma}$ be the canonical line bundle over ${\mathcal{A}}_{g,h,\Gamma}^{tor}$ and let

[TABLE]

be the infinity boundary divisor on ${\mathcal{A}}_{g,h,\Gamma}^{tor}$. Let $\pi_{g,h,\Gamma}:{\mathcal{A}}_{g,h,\Gamma}\longrightarrow{\mathcal{A}}_{g,\Gamma}$ be a projection and let $p_{g,\Gamma}:{\mathcal{A}}_{g,\Gamma}\longrightarrow{\mathcal{A}}_{g}$ be a covering map. We define

[TABLE]

Problem 7.9. Assume that ${\mathcal{A}}_{4,h,\Gamma}^{tor}$ is a toroidal compactification of ${\mathcal{A}}_{4,h,\Gamma}$ which is projective. Compute the Okounkov bodies $\Delta_{Y_{\bullet}}(J_{4,h,\Gamma})$, $\Delta_{Y_{\bullet}}^{\rm val}(J_{4,h,\Gamma})$ and $\Delta_{Y_{\bullet}}^{\rm lim}(J_{4,h,\Gamma})$ explicitly. Describe the relations among $J_{4},\ J_{4,h,\Gamma}$ and these Okounkov bodies explicitly. Describe the relations between these Okounkov bodies and the $GL(4,\mathbb{Z})$-admissible family of polyhedral decompositions defining the toroidal compactification $\overline{\mathcal{A}}_{4,\Gamma}$.

Problem 7.10. Assume that a toroidal compactification ${\mathcal{A}}_{g,h,\Gamma}^{tor}$ is a projective variety. Let $K_{g,h,\Gamma}$ be the canonical line bundle over ${\mathcal{A}}_{g,h,\Gamma}^{tor}$ and $D_{\infty,g,h,\Gamma}$ be the infinity boundary divisor on ${\mathcal{A}}_{g,h,\Gamma}^{tor}$. Compute the Okoukov convex bodies $\Delta_{Y_{\bullet}}(K_{g,h,\Gamma}),\ \Delta_{Y_{\bullet}}^{\rm val}(K_{g,h,\Gamma}),\Delta_{Y_{\bullet}}^{\rm lim}(K_{g,h,\Gamma}),\Delta_{Y_{\bullet}}(D_{\infty,g,h,\Gamma}),\Delta_{Y_{\bullet}}^{\rm val}(D_{\infty,\Gamma}^{u}),\ \Delta_{Y_{\bullet}}^{\rm lim}(D_{\infty,g,h,\Gamma}),\\ \Delta_{Y_{\bullet}}(K_{g,h,\Gamma}+D_{\infty,g,h,\Gamma}),\ \Delta_{Y_{\bullet}}^{\rm val}(K_{g,h,\Gamma}+D_{\infty,g,h,\Gamma})$ and $\Delta_{Y_{\bullet}}^{\rm lim}(K_{g,h,\Gamma}+D_{\infty,g,h,\Gamma})$ explicitly. Describe the relations between these Okounkov bodies and the $GL(g,\mathbb{Z})$-admissible family of polyhedral decompositions defining the toroidal compactification ${\mathcal{A}}_{g,h,\Gamma}^{tor}$.

Problem 7.11. Assume that a toroidal compactification ${\mathcal{A}}_{g,h,\Gamma}^{tor}$ of ${\mathcal{A}}_{g,h,\Gamma}$ is a projective variety. Let $D_{J,\Gamma}$ be a divisor on ${\mathcal{A}}_{g,h,\Gamma}^{tor}$ containing $J_{g,h,\Gamma}$. Describe the Okounkov bodies $\Delta_{Y_{\bullet}}(D_{J,\Gamma}),\ \Delta_{Y_{\bullet}}^{\rm val}(D_{J,\Gamma})$ and $\Delta_{Y_{\bullet}}^{\rm lim}(D_{J,\Gamma})$. Study the relations among $J_{g,\Gamma},\ J_{g,h,\Gamma},\ D_{J,\Gamma}$ and these Okounkov bodies.

We have the following diagram:

[TABLE]

Here $p_{g,h,\Gamma}\;:\;{\mathcal{A}}_{g,h,\Gamma}\longrightarrow{\mathcal{A}}_{g,h}$ is a covering map.

We propose the following questions. Question 7.1. Let $\Gamma$ be a neat arithmetic subgroup of $\Gamma_{g}$. Does the closure $\overline{J}_{g,h,\Gamma}$ of $J_{g,h,\Gamma}$ intersect the infinity boundary divisor $D_{\infty,g,h,\Gamma}$? If $g$ is sufficient large, it is probable that $\overline{J}_{g,h,\Gamma}$ will not intersect the boundary divisor $D_{\infty,g,h,\Gamma}^{u}$.

Question 7.2. Let $\Gamma$ be a neat arithmetic subgroup of $\Gamma_{g}$. Does the closure $\overline{J}_{g,h,\Gamma}$ of $J_{g,h,\Gamma}$ intersect the canonical divisor $K_{g,h,\Gamma}$?

Question 7.3. Let $\Gamma$ be a neat arithmetic subgroup of $\Gamma_{g}$. How curved is the closure $\overline{J}_{g,h,\Gamma}$ of $J_{g,h,\Gamma}$ along the boundary of $J_{g,h,\Gamma}$?

Now we make some conjectures.

Finally we discuss the connection between the universal Jacobian locus $J_{g,h}$ and stable Jacobi forms. We refer to Appendix E in this article for more details on stable Jacobi forms. First we review the concept of stable modular forms introduced in [45]. The Siegel $\Phi$-operator

[TABLE]

defined by

[TABLE]

where $[\Gamma_{g},k]$ denotes the vector space of all Siegel modular forms on $\mathbb{H}_{g}$ of weight $k$. Using the theory of Poincaré series, H. Maass [88] proved that if $k$ is even and $k>2g$, then $\Phi_{g,k}$ is a surjective linear map. In 1977, using the theory of singular modular forms, E. Freitag [45] proved the following facts (a) and (b) : (a) for a fixed even integer $k$, $\Phi_{g,k}$ is an isomorphism if $g>2k$ ; (b) $[\Gamma_{g},k]=0$ if $g>2k,\ k\not\equiv 0~{}({\rm mod}~{}4).$**

The fact (a) means that the vector spaces $[\Gamma_{g},k]$ stabilize to the infinity vector space $[\Gamma_{\infty},k]$ as $g$ increases. In this sense, he introduced the notion of the stability of Siegel modular forms.

(SM1) $g+m>2k$; (SM2) $f=\Phi_{g+1,k}\circ\Phi_{g+2,k}\circ\cdots\circ\Phi_{g+m,k}(F)\quad$ for some $F\in[\Gamma_{g+m},k]$. **

Scalar-valued Siegel modular forms on $\mathcal{A}_{g}$ vanishing on the Jacobian locus, equivalently, forms on the Satake compactification $\mathcal{A}_{g}^{\rm Sat}$ that vanish on the closure $J_{g}^{\rm Sat}$ of $J_{g}$ in $\mathcal{A}_{g}^{\rm Sat}$ are called Schottky-Siegel forms. The normalization $\nu:\mathcal{A}_{g}^{\rm Sat}\longrightarrow\partial\mathcal{A}_{g+1}^{\rm Sat}$ gives a restriction map which coincides with the Siegel operator $\Phi_{g,k}~{}(k\in\mathbb{Z}_{+}).$

We let

[TABLE]

be the graded ring of Siegel modular forms on $\mathbb{H}_{g}.$ It is known that $A(\Gamma_{g})$ is a finitely generated $\mathbb{C}$-algebra and the field of modular functions $K(\Gamma_{g})$ is an algebraic function field of transcendence degree ${\frac{1}{2}}g(g+1)$.

The ring

[TABLE]

is an inverse limit in the category

[TABLE]

Freitag [45] proved that ${\mathbb{A}}$ is the polynomial ring over $\mathbb{C}$ on the set of theta series $\theta_{S}$, where $S$ runs over the set of equivalence classes of indecomposable positive definite unimodular even integral matrices. In general, $A(\Gamma_{g})$ is not a polynomial ring (cf. [45, p. 204]).

We define the stable Satake compactification $\mathcal{A}_{\infty}^{\rm Sat}$ by

[TABLE]

and the stable Jacobian locus $J_{\infty}^{\rm Sat}$ by

[TABLE]

G. Codogni and N. I. Shepherd-Barron [24] proved the following theorem.

[TABLE]

induced by the inclusion $J_{\infty}^{\rm Sat}\hookrightarrow\mathcal{A}_{\infty}^{\rm Sat}$ is injective, where $\omega_{{}_{J}}$ is the restriction of the canonical line bundle $\omega$ on $\mathcal{A}_{\infty}^{\rm Sat}$ to $J_{\infty}^{\rm Sat}$.

Proof. See Theorem 1.3 and Corollary 1.4 in [24]. $\Box$

We refer to Appendix D in this paper for the definition of Jacobi forms. Now we consider the special case $\rho=\det^{k}$ with $k\in\mathbb{Z}_{+}$. We define the Siegel-Jacobi operator

[TABLE]

by

[TABLE]

where $F\in J_{k,{\mathcal{M}}}(\Gamma_{g}),\ \tau\in\mathbb{H}_{g-1}$ and $z\in\mathbb{C}^{(h,g-1)}.$ We observe that the above limit exists and $\Psi_{g,{\mathcal{M}}}$ is a well-defined linear map (cf. [179]).

The author [149] proved the following theorems.

Proof. See [149, Theorem 3.5]. $\Box$

Proof. See [149, Theorem 3.6]. $\Box$

Proof. See [149, Theorem 3.7]. $\Box$

Example. Let $S$ be a positive even unimodular symmetric integral matrix of degree $2k$ and let $c\in\mathbb{Z}^{(2k,h)}$ be an integral matrix. We define the theta series $\vartheta_{S,c}^{(g)}$ by

[TABLE]

It is easily seen that $\vartheta_{S,c}^{(g)}\in J_{k,{\mathcal{M}}}(\Gamma_{g})$ with ${\mathcal{M}}:={\frac{1}{2}}~{}\!{}^{t}\!cSc$ for all $g\geq 0$ and $\Psi_{g,{\mathcal{M}}}\vartheta_{S,c}^{(g)}=\vartheta_{S,c}^{(g-1)}$ for all $g\geq 1.$ Thus the collection

[TABLE]

is a stable Jacobi form of weight $k$ and index ${\mathcal{M}}$.

      $g\geq 0.$ (2) $\Psi_{g,{\mathcal{M}}}F_{g}=F_{g-1}$ for all $g\geq 1.$ **

We expect to prove the following claim : Claim: There are no stable Schottky-Jacobi forms for the universal Jacoban locus.

The author [174] proved the following.

Proof. See [174, Theorem 4.1]. $\Box$

Let $(\Lambda,Q)$ be an even unimodular positive definite quadratic form of rank $m$. That is, $\Lambda$ is a finitely generated free group of rank $m$ and $Q$ is an integer-valued bilinear form on $\Lambda$ such that $Q$ is even and unimodular. For a positive integer $g$, the theta series $\theta_{Q,g}$ associated to $(\Lambda,Q)$ is defined to be

[TABLE]

It is well known that $\theta_{Q,g}(\tau)$ is a Siegel modular form on $\mathbb{H}_{g}$ of weight ${\frac{m}{2}}.$ We easily see that

[TABLE]

Therefore the collection of all theta series associated to $(\Lambda,Q)$

[TABLE]

is a stable modular form.

Recently G. Codogni [23] proved the following.

[TABLE]

where $P$ and $Q$ are even unimodular positive definite quadratic forms of the same rank.

Proof. See Theorem 1.2 or Theorem 4.2 in [23]. $\Box$

In a similar way we may define the concept of stable Jacobi equation.

The author [174] proved the following.

Proof. See [174, Theorem 4.2]. $\Box$

Problem 7.12. Find the ideal of stable Jacobi equations of the universal hyperelliptic locus.

Appendix A. Subvarieties of the Siegel Modular Variety

In this appendix A, we give a brief remark on subvarieties of the Siegel modular variety and present several problems. This appendix was written on the base of the review [121] of G. K. Sankaran for the paper [165]. In fact, Sankaran made a critical review on Section 10. Subvarieties of the Siegel modular variety of the author’s paper [165] and corrected some wrong statements and information given by the author. In this sense the author would like to give his deep thanks to the reviewer, Sankanran.

Here we assume that the ground field is the complex number field $\mathbb{C}.$

Definition A.1. A nonsingular variety $X$ is said to be rational if $X$ is birational to a projective space $\Bbb P^{n}(\mathbb{C})$ for some integer $n$. A nonsingular variety $X$ is said to be stably rational if $X\times\Bbb P^{k}(\mathbb{C})$ is birational to $\Bbb P^{N}(\mathbb{C})$ for certain nonnegative integers $k$ and $N$. A nonsingular variety $X$ is called unirational if there exists a dominant rational map $\varphi:\Bbb P^{n}(\mathbb{C})\longrightarrow X$ for a certain positive integer $n$, equivalently if the function field $\mathbb{C}(X)$ of $X$ can be embedded in a purely transcendental extension $\mathbb{C}(z_{1},\cdots,z_{n})$ of $\mathbb{C}.$

Remarks A.2. (1) It is easy to see that the rationality implies the stably rationality and that the stably rationality implies the unirationality.

(2) If $X$ is a Riemann surface or a complex surface, then the notions of rationality, stably rationality and unirationality are equivalent one another.

(3) H. Clemens and P. Griffiths [22] showed that most of cubic threefolds in $\Bbb P^{4}(\mathbb{C})$ are unirational but not rational.

The following natural questions arise :

Question 1. Is a stably rational variety rational ?

Question 2. Is a general hypersurface $X\subset\Bbb P^{n+1}(\mathbb{C})$ of degree $d\leq n+1$ unirational ?

Question 1 is a famous one raised by O. Zariski (cf. B. Serge, Algebra and Number Theory (French), CNRS, Paris (1950), 135–138; MR0041480). In [12], A. Beauville, J.-L. Colliot-Thélène, J.-J. Sansuc and P. Swinnerton-Dyer gave counterexamples, e.g., the Ch${\hat{\rm}a}$telot surfaces $V_{d,P}\subset{\bf A}_{\mathbb{C}}^{3}$ defined by $y^{2}-dz^{2}=P(x),$ where $P\in\mathbb{C}[x]$ is an irreducible polynomial of degree 3, and $d$ is the discriminant of $P$ such that $d$ is not a square and hence answered negatively to Question 1.

Definition A.3. Let $X$ be a nonsingular variety of dimension $n$ and let $K_{X}$ be the canonical divisor of $X$. For each positive integer $m\in\mathbb{Z}^{+}$, we define the $m$-genus $P_{m}(X)$ of $X$ by

[TABLE]

The number $p_{g}(X):=P_{1}(X)$ is called the geometric genus of $X$. We let

[TABLE]

For the present, we assume that $N(X)$ is nonempty. For each $m\in N(X),$ we let $\left\{\phi_{0},\cdots,\phi_{N_{m}}\right\}$ be a basis of the vector space $H^{0}(X,{\mathcal{O}}(mK_{X})).$ Then we have the mapping $\Phi_{mK_{X}}\,:X\longrightarrow\Bbb P^{N_{m}}(\mathbb{C})$ by

[TABLE]

We define the Kodaira dimension $\kappa(X)$ of $X$ by

[TABLE]

If $N(X)$ is empty, we put $\kappa(X):=-\infty.$ Obviously $\kappa(X)\leq\text{dim}_{\mathbb{C}}\,X.$ A nonsingular variety $X$ is said to be of general type if $\kappa(X)=\text{dim}_{\mathbb{C}}X.$ A singular variety $Y$ in general is said to be rational, stably rational, unirational or of general type if any nonsingular model $X$ of $Y$ is rational, stably rational, unirational or of general type respectively. We define

[TABLE]

A variety $Y$ of dimension $n$ is said to be of logarithmic general type if there exists a smooth compactification ${\tilde{Y}}$ of $Y$ such that $D:={\tilde{Y}}-Y$ is a divisor with normal crossings only and the transcendence degree of the logarithmic canonical ring

[TABLE]

is $n+1$, i.e., the logarithmic Kodaira dimension of $Y$ is $n$. We observe that the notion of being of logarithmic general type is weaker than that of being of general type.

Let $\mathcal{A}_{g}:=\Gamma_{g}\backslash\mathbb{H}_{g}$ be the Siegel modular variety of degree $g$, that is, the moduli space of principally polarized abelian varieties of dimension $g$. So far it has been proved that $\mathcal{A}_{g}$ is of general type for $g\geq 7.$ At first Freitag [44] proved this fact when $g$ is a multiple of $24$. Tai [133] proved this for $g\geq 9$ and Mumford [99] proved this fact for $g\geq 7.$ On the other hand, $\mathcal{A}_{g}$ is known to be unirational for $g\leq 5\,:$ Donagi [30] for $g=5,$ Clemens [21] for $g=4$ and classical for $g\leq 3.$ For $g=3,$ using the moduli theory of curves, Riemann [118], Weber [139] and Frobenius [51] showed that $\mathcal{A}_{3}(2):=\Gamma_{3}(2)\backslash\mathbb{H}_{3}$ is a rational variety and moreover gave $6$ generators of the modular function field $K(\Gamma_{3}(2))$ written explicitly in terms of derivatives of odd theta functions at the origin. So $\mathcal{A}_{3}$ is a unirational variety with a Galois covering of a rational variety of degree $[\Gamma_{3}:\Gamma_{3}(2)]=1,451,520.$ Here $\Gamma_{3}(2)$ denotes the principal congruence subgroup of $\Gamma_{3}$ of level $2.$ Furthermore it was shown that $\mathcal{A}_{3}$ is stably rational (cf. [16, 77]). For a positive integer $k$, we let $\Gamma_{g}(k)$ be the principal congruence subgroup of $\Gamma_{g}$ of level $k$. Let $\mathcal{A}_{g}(k)$ be the moduli space of abelian varieties of dimension $g$ with $k$-level structure. It is classically known that $\mathcal{A}_{g}(k)$ is of logarithmic general type for $k\geq 3$ (cf. [99]). Wang [137, 138] gave a different proof for the fact that $\mathcal{A}_{2}(k)$ is of general type for $k\geq 4.$ On the other hand, the relation between the Burkhardt quartic and abelian surfaces with 3-level structure was established by H. Burkhardt [17] in 1890. We refer to [70, §  IV.2, pp. 132–135] for more detail on the Burkhardt quartic. In 1936, J. A. Todd [134] proved that the Burkhardt quartic is rational. van der Geer [56] gave a modern proof for the rationality of $\mathcal{A}_{2}(3)$. The remaining unsolved problems are summarized as follows:

Problem 1. Are $\mathcal{A}_{4},\ \mathcal{A}_{5}$ stably rational or rational?

Problem 2. Discuss the (uni)rationality of $\mathcal{A}_{6}.$

We already mentioned that $\mathcal{A}_{g}$ is of general type if $g\geq 7.$ It is natural to ask if the subvarieties of $\mathcal{A}_{g}\,(g\geq 7)$ are of general type, in particular the subvarieties of $\mathcal{A}_{g}$ of codimension one. Freitag [49] showed that there exists a certain bound $g_{0}$ such that for $g\geq g_{0},$ each irreducible subvariety of $\mathcal{A}_{g}$ of codimension one is of general type. Weissauer [141] proved that every irreducible divisor of $\mathcal{A}_{g}$ is of general type for $g\geq 10.$ Moreover he proved that every subvariety of codimension $\leq g-13$ in $\mathcal{A}_{g}$ is of general type for $g\geq 13.$ We observe that the smallest known codimension for which there exist subvarieties of $\mathcal{A}_{g}$ for large $g$ which are not of general type is $g-1.\ \mathcal{A}_{1}\times\mathcal{A}_{g-1}$ is a subvariety of $\mathcal{A}_{g}$ of codimension $g-1$ which is not of general type.

Remark A.4. Let $\mathcal{M}_{g}$ be the coarse moduli space of curves of genus $g$ over $\mathbb{C}.$ Then $\mathcal{M}_{g}$ is an analytic subvariety of $\mathcal{A}_{g}$ of dimension $3g-3.$ It is known that $\mathcal{M}_{g}$ is rational for $g=2,4,5,6.$ In 1915 Severi proved that $\mathcal{M}_{g}$ is unirational for $g\leq 10$ (see E. Arbarello and E. Sernesi’s paper [8] for a modern rigorous proof). The unirationality of $\mathcal{M}_{12}$ was proved by E. Sernesi [127] in 1981. Three years later the unirationality of $\mathcal{M}_{11}$ and $\mathcal{M}_{13}$ was proved by M. C. Chang and Z. Ran [19]. So the Kodaira dimension $\kappa(\mathcal{M}_{g})$ of $\mathcal{M}_{g}$ is $-\infty$ for $g\leq 13.$ In 1982 Harris and Mumford [69] proved that $\mathcal{M}_{g}$ is of general type for odd $g$ with $g\geq 25$ and $\kappa(\mathcal{M}_{23})\geq 0.$ J. Harris [67] proved that if $g\geq 40$ and $g$ is even, $\mathcal{M}_{g}$ is of general type. In 1987 D. Eisenbud and J. Harris [39] proved that $\mathcal{M}_{g}$ is of general type for all $g\geq 24$ and $\mathcal{M}_{23}$ has the Kodaira dimension at least one. In 1996 P. Katsylo [74] showed that $\mathcal{M}_{3}$ is rational and hence $\mathcal{A}_{3}.$

Remark A.5. For more details on the geometry and topology of $\mathcal{A}_{g}$ and compactifications of $\mathcal{A}_{g}$, we refer to [1, 40, 48, 55, 57, 58, 61, 71, 82, 91, 122, 123, 124, 137].

Appendix B. Extending of the Torelli Map to Toroidal Compactifications of the Siegel Modular Variety

Let ${\mathcal{M}}_{g}^{\rm DM}$ be the Deligne-Mumford compactification of $\mathcal{M}_{g}$ consisting of isomorphism classes of stable curves of genus $g$. We recall ([29, 102, 105]) that a complete curve $C$ is said to be a stable curve of genus $g\geq 1$ if

(S1) $C$ is reduced;

(S2) $C$ has only ordinary double points as possible singularities;

(S3) $\dim_{\mathbb{C}}H^{1}(C,\mathcal{O}_{C})=1$;

(S4) each nonsingular rational component of $C$ meets the other components at more than two points.

P. Deligne and D. Mumford [29] proved that the coarse moduli space ${\mathcal{M}}_{g}^{\rm DM}$ is an irreducible projective variety,and contains $\mathcal{M}_{g}$ as a Zariski open subset.

We have three standard explicit toroidal compactifications $\mathcal{A}_{g}^{\rm VI},\ \mathcal{A}_{g}^{\rm VII}$ and $\mathcal{A}_{g}^{\rm cent}$ constructed from

(VI) the 1st Voronoi (or perfect) cone decomposition;

(VII) the 2nd Voronoi cone decomposition;

(cent) the central cone decomposition

respectively. We refer to [93, 128] for more details on the perfect cone decomposition and the 2nd Voronoi cone decomposition. In 1973, Y. Namikawa [102] proposed a natural question if the Torelli map

[TABLE]

extends to a regular map

[TABLE]

In fact, $\mathcal{A}_{g}^{\rm cent}$ is the normalization of the Igusa blow-up of the Satake compactification $\mathcal{A}_{g}^{\rm Sat}$ along the boundary $\partial\mathcal{A}_{g}^{\rm cent}$. In the 1970s, Mumford and Namikawa [103, 104] showed that the Torelli map $T_{g}$ extends to a regular map

[TABLE]

In 2012, V. Alexeev and A. Brunyate [2] proved that the Torelli map $T_{g}$ can be extended to a regular map

[TABLE]

and that the extended Torelli map

[TABLE]

is regular for $g\leq 6$ but not regular for $g\geq 9$. Furthermore they also showed that the two compactifications $\mathcal{A}_{g}^{\rm VI}$ and $\mathcal{A}_{g}^{\rm VII}$ are equal near the closure of the Jacobian locus $J_{g}$. Almost at the same time the extended Torelli map $T_{g}^{\rm cent}$ is regular for $g\leq 8$ by Alexeev and et al. [3].

I would like to mention that K. Liu, X. Sun and S.-T. Yau [83, 84, 85, 86] showed the goodness of the Hermitian metrics on the logarithmic tangent bundle on $\mathcal{M}_{g}$ which are induced by the Ricci and the perturbed Ricci metrics on $\mathcal{M}_{g}$. They also showed that the Ricci metric on $\mathcal{M}_{g}$ extends naturally to the divisor $D_{g}:={\mathcal{M}}_{g}^{\rm DM}\,\backslash\,\mathcal{M}_{g}$ and coincides with the Ricci metric on each component of $D_{g}$.

Liu, Sun and Yau [84] showed that the existence of Kähler-Einstein metric on $\mathcal{M}_{g}$ is related to the stability of the logarithmic cotangent bundle over $\mathcal{M}_{g}^{\rm DM}$.

Let $E$ be a holomorphic vector bundle over a complex manifold $X$ of dimension $n$. Let $\Phi:=\Phi_{X}$ be a Kähler class (or form) of $X$. Then $\Phi$-degree of $E$ is defined by

[TABLE]

and the slope of $E$ is defined to be

[TABLE]

A bundle $E$ is said to be $\Phi$-stable if for any proper coherent subsheaf $\mathcal{F}\subset E$, we have

[TABLE]

Let $U$ be a local chart of $\mathcal{M}_{g}$ near the boundary with pinching coordinates $(t_{1},\cdots,t_{m},s_{m+1},\cdots,s_{n})$ such that $(t_{1},\cdots,t_{m})$ represent the degeneration direction. Let

[TABLE]

Then the logarithmic cotangent bundle $(T^{*}\mathcal{M}_{g})^{\rm DM}$ is the unique extension of the cotangent bundle $T^{*}\mathcal{M}_{g}$ over $\mathcal{M}_{g}$ to $\mathcal{M}_{g}^{\rm DM}$ such that on $U$ $F_{1},F_{2},\cdots,F_{n}$ is a local holomorphic frame of $(T^{*}\mathcal{M}_{g})^{\rm DM}$.

Liu, Sun and Yau [84] proved the following.

Theorem B.1. The first Chern class $c_{1}\left((T^{*}\mathcal{M}_{g})^{\rm DM}\right)$ is positive and $(T^{*}\mathcal{M}_{g})^{\rm DM}$ is stable with respect to $c_{1}\left((T^{*}\mathcal{M}_{g})^{\rm DM}\right)$.

Remark B.2. We refer to [18, 135, 144, 145] for some topics related to $\mathcal{M}_{g}$ and $\mathcal{M}_{g}^{\rm DM}$.

Appendix C. Singular Modular Forms

Let $\rho$ be a rational representation of $GL(g,\mathbb{C})$ on a finite dimensional complex vector space $V_{\rho}.$ A holomorphic function $f:\mathbb{H}_{g}\longrightarrow V_{\rho}$ with values in $V_{\rho}$ is called a modular form of type $\rho$ if it satisfies

[TABLE]

for all $\begin{pmatrix}A&B\\ C&D\end{pmatrix}\in\Gamma_{g}$ and $\tau\in\mathbb{H}_{g}.$ We denote by $[\Gamma_{g},\rho]$ the vector space of all modular forms of type $\rho.$ A modular form $f\in[\Gamma_{g},\rho]$ of type $\rho$ has a Fourier series

[TABLE]

where $T$ runs over the set of all semipositive half-integral symmetric matrices of degree $g.$ A modular form $f$ of type $\rho$ is said to be singular if a Fourier coefficient $a(T)$ vanishes unless $\text{det}\,(T)=0.$

For $\tau=(\tau_{ij})\in\mathbb{H}_{g},$ we write $\tau=X+i\,Y$ with $X=(x_{ij}),\ Y=(y_{ij})$ real. We put

[TABLE]

H. Maass [90] introduced the following differential operator

[TABLE]

characterizing singular modular forms. Using the differential operator $M_{g}$, Maass [90, pp. 202–204] proved that if a nonzero singular modular form of degree $g$ and type $\rho:=\det^{k}$ (or weight $k$) exists, then $gk\equiv 0~{}({\rm mod}~{}2)$ and $0<2k\leq g-1.$ The converse was proved by R. Weissauer [140].

Freitag [46] proved that every singular modular form can be written as a finite linear combination of theta series with harmonic coefficients and proposed the problem to characterize singular modular forms. Weissauer [140] gave the following criterion.

Theorem C.1. *Let $\rho$ be an irreducible rational representation of $GL(g,\mathbb{C})$ with its highest weight $(\lambda_{1},\cdots,\lambda_{g}).$ Let $f$ be a modular form of type $\rho.$ Then the following are equivalent :

(a) $f$ is singular.

(b) $2\lambda_{g}<g.$*

Now we describe how the concept of singular modular forms is closely related to the geometry of the Siegel modular variety. Let $X:=\mathcal{A}_{g}^{\rm Sat}$ be the Satake compactification of the Siegel modular variety $\mathcal{A}_{g}=\Gamma_{g}\backslash\mathbb{H}_{g}.$ Then $\mathcal{A}_{g}$ is embedded in $X$ as a quasiprojective algebraic subvariety of codimension $g$. Let $X_{s}$ be the smooth part of $\mathcal{A}_{g}$ and ${\tilde{X}}$ the desingularization of $X.$ Without loss of generality, we assume $X_{s}\subset{\tilde{X}}.$ Let $\Omega^{p}({\tilde{X}})$ (resp. $\Omega^{p}(X_{s}))$ be the space of holomorphic $p$-form on ${\tilde{X}}$ (resp. $X_{s}$). Freitag and Pommerening [50] showed that if $g>1$, then the restriction map

[TABLE]

is an isomorphism for $p<\text{dim}_{\mathbb{C}}\,{\tilde{X}}={{g(g+1)}\over 2}.$ Since the singular part of $\mathcal{A}_{g}$ is at least codimension $2$ for $g>1,$ we have an isomorphism

[TABLE]

Here $\Omega^{p}(\mathbb{H}_{g})^{\Gamma_{g}}$ denotes the space of $\Gamma_{g}$-invariant holomorphic $p$-forms on $\mathbb{H}_{g}.$ Let $\text{Sym}^{2}(\mathbb{C}^{g})$ be the symmetric power of the canonical representation of $GL(g,\mathbb{C})$ on $\mathbb{C}^{n}.$ Then we have an isomorphism

[TABLE]

Theorem C.2. [140] Let $\rho_{\alpha}$ be the irreducible representation of $GL(g,\mathbb{C})$ with highest weight

[TABLE]

such that $\text{corank}(\rho_{\alpha})=\alpha$ for $1\leq\alpha\leq g.$ If $\alpha=-1,$ we let $\rho_{\alpha}=(g+1,\cdots,g+1).$ Then

[TABLE]

Remark C.3. If $2\alpha>g,$ then any $f\in[\Gamma_{g},\rho_{\alpha}]$ is singular. Thus if $p<{{g(3g+2)}\over 8},$ then any $\Gamma_{g}$-invariant holomorphic $p$-form on $\mathbb{H}_{g}$ can be expressed in terms of vector valued theta series with harmonic coefficients. It can be shown with a suitable modification that the just mentioned statement holds for a sufficiently small congruence subgroup of $\Gamma_{g}.$

Thus the natural question is to ask how to determine the $\Gamma_{g}$-invariant holomorphic $p$-forms on $\mathbb{H}_{g}$ for the nonsingular range $\dfrac{g(3g+2)}{8}\leq p\leq\dfrac{g(g+1)}{2}.$ Weissauer [142] answered the above question for $g=2.$ For $g>2,$ the above question is still open. It is well know that the vector space of vector valued modular forms of type $\rho$ is finite dimensional. The computation or the estimate of the dimension of $\Omega^{p}(\mathbb{H}_{g})^{\Gamma_{g}}$ is interesting because its dimension is finite even though the quotient space $\mathcal{A}_{g}$ is noncompact.

Finally we will mention the results due to Weisauer [141]. We let $\Gamma$ be a congruence subgroup of $\Gamma_{2}.$ According to Theorem C.2, $\Gamma$-invariant holomorphic forms in $\Omega^{2}(\mathbb{H}_{2})^{\Gamma}$ are corresponded to modular forms of type (3,1). We note that these invariant holomorphic $2$-forms are contained in the nonsingular range. And if these modular forms are not cusp forms, they are mapped under the Siegel $\Phi$-operator to cusp forms of weight $3$ with respect to some congruence subgroup ( dependent on $\Gamma$ ) of the elliptic modular group. Since there are finitely many cusps, it is easy to deal with these modular forms in the adelic version. Observing these facts, he showed that any $2$-holomorphic form on $\Gamma\backslash\mathbb{H}_{2}$ can be expressed in terms of theta series with harmonic coefficients associated to binary positive definite quadratic forms. Moreover he showed that $H^{2}(\Gamma\backslash\mathbb{H}_{2},\mathbb{C})$ has a pure Hodge structure and that the Tate conjecture holds for a suitable compactification of $\Gamma\backslash\mathbb{H}_{2}.$ If $g\geq 3,$ for a congruence subgroup $\Gamma$ of $\Gamma_{g}$ it is difficult to compute the cohomology groups $H^{\ast}(\Gamma\backslash\mathbb{H}_{g},\mathbb{C})$ because $\Gamma\backslash\mathbb{H}_{g}$ is noncompact and highly singular. Therefore in order to study their structure, it is natural to ask if they have pure Hodge structures or mixed Hodge structures.

Appendix D. Singular Jacobi Forms

In this section, we discuss the notion of singular Jacobi forms. First of all we define the concept of Jacobi forms.

Let $\rho$ be a rational representation of $GL(g,\mathbb{C})$ on a finite dimensional complex vector space $V_{\rho}.$ Let ${\mathcal{M}}\in\mathbb{R}^{(h,h)}$ be a symmetric half-integral semi-positive definite matrix of degree $m$. The canonical automorphic factor

[TABLE]

for $G^{J}$ on $\mathbb{H}_{g,h}$ is given as follows :

[TABLE]

where $g=\left(\begin{matrix}A&B\\ C&D\end{matrix}\right)\in Sp(2g,\mathbb{R}),\ (\lambda,\mu;\kappa)\in H_{\mathbb{R}}^{(g,h)}$ and $(\tau,z)\in\mathbb{H}_{g,h}.$ We refer to [152] for a geometrical construction of $J_{\rho,\mathcal{M}}.$

Let $C^{\infty}(\mathbb{H}_{g,h},V_{\rho})$ be the algebra of all $C^{\infty}$ functions on $\mathbb{H}_{g,h}$ with values in $V_{\rho}.$ For $f\in C^{\infty}(\mathbb{H}_{g,h},V_{\rho}),$ we define

[TABLE]

where $g=\left(\begin{matrix}A&B\\ C&D\end{matrix}\right)\in Sp(2g,\mathbb{R}),\ (\lambda,\mu;\kappa)\in H_{\mathbb{R}}^{(g,h)}$ and $(\tau,z)\in\mathbb{H}_{g,h}.$

Definition D.1. Let $\rho$ and $\mathcal{M}$ be as above. Let

[TABLE]

be the discrete subgroup of $H_{\mathbb{R}}^{(g,h)}$. A Jacobi form of index $\mathcal{M}$ with respect to $\rho$ on a subgroup $\Gamma$ of $\Gamma_{g}$ of finite index is a holomorphic function $f\in C^{\infty}(\mathbb{H}_{g,h},V_{\rho})$ satisfying the following conditions (A) and (B):

(A)   $f|_{\rho,{\mathcal{M}}}[\tilde{\gamma}]=f$ for all $\tilde{\gamma}\in{\widetilde{\Gamma}}:=\Gamma\ltimes H_{\mathbb{Z}}^{(g,h)}$.

(B)   For each $M\in\Gamma_{g}$, $f|_{\rho,{\mathcal{M}}}[M]$ has a Fourier expansion of

       the following form :

[TABLE]

with $\lambda_{\Gamma}(\neq 0)\in\mathbb{Z}$ and $c(T,R)\neq 0$ only if $\left(\begin{matrix}{1\over{\lambda_{\Gamma}}}T&\frac{1}{2}R\\ \frac{1}{2}\,^{t}\!R&{\mathcal{M}}\end{matrix}\right)\geq 0$.

If $g\geq 2,$ the condition (B) is superfluous by Köcher principle (cf. [179, Lemma 1.6]). We denote by $J_{\rho,\mathcal{M}}(\Gamma)$ the vector space of all Jacobi forms of index $\mathcal{M}$ with respect to $\rho$ on $\Gamma$. Ziegler (cf. [37, Theorem 1.1] or [179, Theorem 1.8]) proves that the vector space $J_{\rho,\mathcal{M}}(\Gamma)$ is finite dimensional. In the special case $\rho(A)=(\det(A))^{k}$ with $A\in GL(g,\mathbb{C})$ and a fixed $k\in\mathbb{Z}$, we write $J_{k,{\mathcal{M}}}(\Gamma)$ instead of $J_{\rho,{\mathcal{M}}}(\Gamma)$ and call $k$ the weight of the corresponding Jacobi forms. For more results about Jacobi forms with $g>1$ and $h>1$, we refer to [147, 149, 150, 152, 159, 179]. Jacobi forms play an important role in lifting elliptic cusp forms to Siegel cusp forms of degree $2g$.

Without loss of generality we may assume that $\mathcal{M}$ is positive definite. For simplicity, we consider the case that $\Gamma$ is the Siegel modular group $\Gamma_{g}$ of degree $g.$

Let $g$ and $h$ be two positive integers. We recall that $\mathcal{M}$ is a symmetric positive definite, half-integral matrix of degree $h$. We let

[TABLE]

be the open convex cone of positive definite matrices of degree $g$ in the Euclidean space $\mathbb{R}^{{g(g+1)}\over 2}.$ We define the differential operator $M_{g,h,\mathcal{M}}$ on ${\mathcal{P}}_{g}\times\mathbb{R}^{(h,g)}$ defined by

[TABLE]

where

[TABLE]

and

[TABLE]

We note that this differential operator $M_{g,h,{\mathcal{M}}}$ generalizes the Maass operator $M_{g}$ (see Formula (C.1)). The author [153] characterized singular Jacobi forms as follows :

Theorem D.2. *Let $f\in J_{\rho,\mathcal{M}}(\Gamma_{g})$ be a Jacobi form of index $\mathcal{M}$ with respect to a finite dimensional rational representation $\rho$ of $GL(g,\mathbb{C}).$ Then the following conditions are equivalent :

(1) $f$ is a singular Jacobi form.

(2) $f$ satisfies the differential equation $M_{g,h,\mathcal{M}}f=0.$*

Theorem D.3. Let $\rho$ be an irreducible finite dimensional representation of $GL(g,\mathbb{C}).$ Then there exists a nonvanishing singular Jacobi form in $J_{\rho,\mathcal{M}}(\Gamma_{g})$ if and only if $2k(\rho)<g+h.$ Here $k(\rho)$ denotes the weight of $\rho.$

For the proofs of the above theorems we refer to Theorems 4.1 and 4.5 in [153].

Exercise D.4. Compute the eigenfunctions and the eigenvalues of $M_{g,h,\mathcal{M}}$ (cf. [153, pp. 2048–2049]).

Now we consider the following group $GL(g,\mathbb{R})\ltimes H_{\mathbb{R}}^{(g,h)}$ equipped with the multiplication law

[TABLE]

where $A,B\in GL(g,\mathbb{R})$ and $(\lambda,\mu,\kappa),(\lambda^{\prime},\mu^{\prime},\kappa^{\prime})\in H_{\mathbb{R}}^{(g,h)}.$ We observe that $GL(g,\mathbb{R})$ acts on $H_{\mathbb{R}}^{(g,h)}$ on the right as automorphisms. And we have the canonical action of $GL(g,\mathbb{R})\ltimes H_{\mathbb{R}}^{(g,h)}$ on ${\mathcal{P}}_{g}\times\mathbb{R}^{(h,g)}$ defined by

[TABLE]

where $A\in GL(g,\mathbb{R}),(\lambda,\mu,\kappa)\in H_{\mathbb{R}}^{(g,h)}$ and $(Y,V)\in{\mathcal{P}}_{g}\times\mathbb{R}^{(h,g)}.$

Lemma D.5. The differential operator $M_{g,h,\mathcal{M}}$ defined by the formula (D.1) is invariant under the action (D.2) of $GL(g,\mathbb{R})\ltimes\left\{(0,\mu,0)\,|\ \mu\in{\mathbb{R}}^{(h,g)}\right\}.$

Proof. It follows immediately from the direct calculation.

We have the following natural questions.

Problem D.6. Develop the invariant theory for the action of $GL(g,\mathbb{R})\ltimes H_{\mathbb{R}}^{(g,h)}$ on ${\mathcal{P}}_{g}\times\mathbb{R}^{(h,g)}.$ We refer to [169, 172] for related topics.

Problem D.7. Discuss the application of the theory of singular Jacobi forms to the geometry of the universal abelian variety as that of singular modular forms to the geometry of the Siegel modular variety (see Appendix C).

Appendix E. Stable Jacobi Forms

Throughout this appendix we put

[TABLE]

For a commutative ring $R$ and an integer $m$, we denote by $S_{m}(R)$ the set of all $m\times m$ symmetric matrices with entries in $R$.

We know that the Siegel-Jacobi space

[TABLE]

is a non-symmetric homogeneous space. Here

[TABLE]

is a subgroup of $G^{J}.$ Let $\mathfrak{g}^{J}$ be the Lie algebra of the Jacobi group $G^{J}$. Then $\mathfrak{g}^{J}$ has a decomposition

[TABLE]

where

[TABLE]

and

[TABLE]

We observe that $\mathfrak{k}^{J}$ is the Lie algebra of $K^{J}$. The complexification ${\mathfrak{p}}_{\mathbb{C}}^{J}:=\mathfrak{p}\otimes_{\mathbb{R}}\mathbb{C}$ of ${\mathfrak{p}}^{J}$ has a decomposition

[TABLE]

where

[TABLE]

and

[TABLE]

We define a complex structure $I^{J}$ on the tangent space ${\mathfrak{p}}^{J}$ of $\mathbb{H}_{g,h}$ at $(iI_{g},0)$ by

[TABLE]

Identifying $\mathbb{R}^{(h,g)}\times\mathbb{R}^{(h,g)}$ with $\mathbb{C}^{(h,g)}$ via

[TABLE]

we may regard the complex structure $I^{J}$ as a real linear map

[TABLE]

where $X+i~{}Y\in S_{g}(\mathbb{R}),\ Q+i\,P\in\mathbb{C}^{(h,g)}.\ I^{J}$ extends complex linearly on the complexification ${\mathfrak{p}}_{\mathbb{C}}^{J}.$ With respect to this complex structure $I^{J},$ we may say that a function $f$ on $\mathbb{H}_{g,h}$ is holomorphic if and only if $\xi f=0$ for all $\xi\in{\mathfrak{p}}_{-}^{J}.$

Since the space $\mathbb{H}_{g,h}$ is diffeomorphic to the homogeneous space $G^{J}/K^{J}$, we may lift a function $f$ on $\mathbb{H}_{g,h}$ with values in $V_{\rho}$ to a function $\Phi_{f}$ on $G^{J}$ with values in $V_{\rho}$ in the following way. We define the lifting

[TABLE]

by

[TABLE]

where $x\in G^{J}$ and ${\mathcal{F}}(\mathbb{H}_{g,h},V_{\rho})\,(\text{resp.}\ {\mathcal{F}}(G^{J},V_{\rho}))$ denotes the vector space consisting of functions on $\mathbb{H}_{g,h}\,(\text{resp.}\ G^{J})$ with values in $V_{\rho}$.

We see easily that the vector space $J_{\rho,\mathcal{M}}(\Gamma_{g})$ is isomorphic to the space $A_{\rho,\mathcal{M}}(\Gamma_{g,h})$ of smooth functions $\Phi$ on $G^{J}$ with values in $V_{\rho}$ satisfying the following conditions: (1a)   $\Phi(\gamma x)=\Phi(x)$  for all $\gamma\in\Gamma^{J}$ and $x\in G^{J}.$

(1b)   $\Phi(x\,r(k,\kappa))=e^{2\pi i\,\sigma(\mathcal{M}\kappa)}\rho(k)^{-1}\Phi(x)$    for all $x\in G^{J},\ r(k,\kappa)\in K^{J}.$

(2)   $Y^{-}\Phi=0$   for all $Y^{-}\in{\mathfrak{p}}^{J}_{-}$.

(3)  For all $M\in Sp(2g,\mathbb{R}),$ the function $\psi:G^{J}\longrightarrow V_{\rho}$ defined by

[TABLE]

is bounded in the domain $Y\geq Y_{0}.$ Here $x\cdot(iI_{g},0)=(\tau,z)$ with $\tau=$

$X+i\,Y,\ Y>0.$

Clearly $J_{\rho,\mathcal{M}}^{\text{cusp}}(\Gamma_{g})$ is isomorphic to the subspace $A^{0}_{\rho,\mathcal{M}}(\Gamma_{g,h})$ of $A_{\rho,\mathcal{M}}(\Gamma_{g,h})$ with the condition (3+) that the function $g\longmapsto\Phi(g)$ is bounded.

Let $\mathcal{M}$ be a fixed positive definite symmetric half-integral matrix of degree $h$. Let $\rho_{\infty}:=\,(\rho_{n})$ be a stable representation of $GL(\infty,\mathbb{C}).$ That is, for each $n\in\mathbb{Z}^{+},\ \rho_{n}$ is a finite dimensional rational representation of $GL(n,\mathbb{C})$ and $\rho_{\infty}$ is compatible with the embeddings $\alpha_{kl}:GL(k,\mathbb{C})\longrightarrow GL(l,\mathbb{C})\,(\,k<l\,)$ defined by

[TABLE]

For two positive integers $m$ and $n$, we put

[TABLE]

For $k,l\in\mathbb{Z}^{+}$ with $k<l,$ we define the mapping $\Phi_{l,k,\mathcal{M}}$ of $A_{\rho_{l},\mathcal{M}}(\Gamma_{l,m})$ into the functions on $G_{k,m}^{J}$ by

[TABLE]

where $F\in A_{\rho_{l},\mathcal{M}}(\Gamma_{l,m}^{J}),\ x=(M,(\lambda,\mu;\kappa))\in G_{k,m}^{J}$ with $M=\begin{pmatrix}A&B\\ C&D\end{pmatrix}\in Sp(2k,\mathbb{R})$ and

[TABLE]

Proposition E.1. The limit (E.2) always exists and the image of $A_{\rho_{l},\mathcal{M}}(\Gamma_{l,h})$ under $\Phi_{l,k,\mathcal{M}}$ is contained in $A_{\rho_{k},\mathcal{M}}(\Gamma_{k,h})$. Obviously the mapping

[TABLE]

is a linear mapping. The mapping $\Phi_{l,k,\mathcal{M}}$ is called the Siegel-Jacobi operator. For any $g\in\mathbb{Z}^{+},$ we put

[TABLE]

where $\rho$ runs over all isomorphism classes of irreducible rational representations of $GL(g,\mathbb{C})$. For $g=0,$ we set $A_{0,\mathcal{M}}:=\mathbb{C}.$ For each $g\in\mathbb{Z}^{+},$ we put

[TABLE]

where $\rho_{\ast}$ runs over all isomorphism classes of irreducible rational representations of $GL(g,\mathbb{C})$ with highest weight $\lambda(\rho_{\ast})\in(2\mathbb{Z})^{g}.$ It is obvious that if $k<l,$ then the Siegel-Jacobi operator $\Phi_{l,k,\mathcal{M}}$ maps $A_{l,\mathcal{M}}\,$( resp.  $A_{l,\mathcal{M}}^{\ast}\,)$ into $A_{k,\mathcal{M}}$ ( resp. $A_{k,\mathcal{M}}^{\ast}\,).$

We let

[TABLE]

be the inverse limits of $(\,A_{k,\mathcal{M}},\,\Phi_{l,k,\mathcal{M}}\,)$ and $(\,A_{k,\mathcal{M}}^{\ast},\,\Phi_{l,k,\mathcal{M}}\,)$ respectively.

Proposition E.2. $A_{\infty,\mathcal{M}}$* has a commutative ring structure compatible with the Siegel-Jacobi operators. Obviously $A_{\infty,\mathcal{M}}^{\ast}$ is a subring of $A_{\infty,\mathcal{M}}.$*

For a stable irreducible representation $\rho_{\infty}=(\rho_{g})$ of $GL(\infty,\mathbb{C})$, we define

[TABLE]

Proposition E.3. We have

[TABLE]

where $\rho_{\infty}$ runs over all isomorphism classes of stable irreducible representations of $GL(\infty,\mathbb{C}).$

Definition E.4. Elements in $A_{\infty,\mathcal{M}}$ are called stable automorphic forms on $G_{\infty,h}^{J}$ of index $\mathcal{M}$ and elements of $A_{\infty,\mathcal{M}}^{\ast}$ are called even stable automorphic forms on $G_{\infty,h}^{J}$ of index $\mathcal{M}$.

For $g\geq 1,$ we define

[TABLE]

where $\rho$ runs over all isomorphism classes of irreducible rational representations of $GL(g,\mathbb{C})$ and $\mathcal{M}$ runs over all equivalence classes of positive definite symmetric, half-integral matrices of any degree $\geq 1.$ We set ${\mathbb{A}_{0}}:=\mathbb{C}.$

For $g\geq 1,$ we also define

[TABLE]

where $\rho_{\ast}$ runs over all isomorphism classes of irreducible rational representations of $GL(g,\mathbb{C})$ with highest weight $\lambda(\rho_{\ast})\in(2\mathbb{Z})^{g}$ and $\mathcal{M}$ runs over all equivalence classes of positive definite symmetric half-integral matrices of any degree $\geq 1.$

Let $\rho_{\infty}=(\rho_{g})$ be a stable irreducible rational representation of $GL(\infty,\mathbb{C}).$ For each irreducible rational representation $\rho_{g}$ of $GL(g,\mathbb{C})$ appearing in $\rho_{\infty},$ we put

[TABLE]

where $\mathcal{M}$ runs over all equivalence classes of positive definite symmetric half-integral matrices of any degree $\geq 1.$ Clearly the Siegel-Jacobi operator $\Phi_{l,k}:=\,\bigoplus_{\mathcal{M}}\Phi_{l,k,\mathcal{M}}\,(\,k<l\,)$ maps $A(\rho_{l};\rho_{\infty})$ into $A(\rho_{k};\rho_{\infty}).$

Using the Siegel-Jacobi operators, we can define the inverse limits

[TABLE]

Theorem E.5.

[TABLE]

where $\rho_{\infty}$ runs over all equivalence classes of stable irreducible representations of $GL(\infty,\mathbb{C}).$

Let $\rho$ and $\mathcal{M}$ be the same as in the previous sections. For positive integers $r$ and $g$ with $r<g,$ we let $\rho^{(r)}:GL(r,\mathbb{C})\longrightarrow GL(V_{\rho})$ be a rational representation of $GL(r,\mathbb{C})$ defined by

[TABLE]

The Siegel-Jacobi operator $\Psi_{g,r}:J_{\rho,\mathcal{M}}(\Gamma_{g})\longrightarrow J_{\rho^{(r)},\mathcal{M}}(\Gamma_{r})$ is defined by

[TABLE]

where $f\in J_{\rho,\mathcal{M}}(\Gamma_{g}),\ \tau\in\mathbb{H}_{r}$ and $z\in\mathbb{C}^{(h,r)}.$ It is easy to check that the above limit always exists and the Siegel-Jacobi operator is a linear mapping. Let $V_{\rho}^{(r)}$ be the subspace of $V_{\rho}$ spanned by the values $\{\,(\Psi_{g,r}f)(\tau,z)\,|\ f\in J_{\rho,\mathcal{M}}(\Gamma_{g}),\ (\tau,z)\in\mathbb{H}_{r}\times\mathbb{C}^{(h,r)}\,\}.$ Then $V_{\rho}^{(r)}$ is invariant under the action of the group

[TABLE]

We can show that if $V_{\rho}^{(r)}\neq 0$ and $(\rho,V_{\rho})$ is irreducible, then $(\rho^{(r)},V_{\rho}^{(r)})$ is also irreducible.

Theorem E.6. The action of the Siegel-Jacobi operator is compatible with that of that of the Hecke operator.

We refer to [149] for a precise detail on the Hecke operators and the proof of the above theorem.

Problem E.7. Discuss the injectivity, surjectivity and bijectivity of the Siegel-Jacobi operator.

This problem was partially discussed by the author [149] and Kramer [78] in the special cases. For instance, Kramer [78] showed that if $g$ is arbitrary, $h=1$ and $\rho:GL(g,\mathbb{C})\longrightarrow\mathbb{C}^{\times}$ is a one-dimensional representation of $GL(g,\mathbb{C})$ defined by $\rho(a):=(\text{det}\,(a))^{k}$ for some $k\in\mathbb{Z}^{+},$ then the Siegel-Jacobi operator

[TABLE]

is surjective for $k\gg m\gg 0.$

Theorem E.8. Let $1\leq r\leq g-1$ and let $\rho$ be an irreducible finite dimensional representation of $GL(g,\mathbb{C}).$ Assume that $k(\rho)>g+r+\text{rank}\,(\mathcal{M})+1$ and that $k$ is even. Then

[TABLE]

Here $J_{\rho^{(r)},\mathcal{M}}^{\text{cusp}}(\Gamma_{r})$ denotes the subspace consisting of all cuspidal Jacobi forms in $J_{\rho^{(r)},\mathcal{M}}(\Gamma_{r}).$

Idea of Proof. For each $f\in J_{\rho^{(r)},\mathcal{M}}^{\text{cusp}}(\Gamma_{r}),$ we can show by a direct computation that

[TABLE]

where $E_{\rho,\mathcal{M}}^{(g)}(\tau,z;f)$ is the Eisenstein series of Klingen’s type associated with a cusp form $f.$ For a precise detail, we refer to [179].

Remark E.9. Dulinski [35] decomposed the vector space $J_{k,\mathcal{M}}(\Gamma_{g})\,(k\in\mathbb{Z}^{+})$ into a direct sum of certain subspaces by calculating the action of the Siegel-Jacobi operator on Eisenstein series of Klingen’s type explicitly.

For two positive integers $r$ and $g$ with $r\leq g-1,$ we consider the bigraded ring

[TABLE]

and

[TABLE]

where $\Gamma_{r}(\ell)$ denotes the principal congruence subgroup of $\Gamma_{r}$ of level $\ell$ and $\mathcal{M}$ runs over the set of all symmetric semi-positive half-integral matrices of degree $h$. Let

[TABLE]

be the Siegel-Jacobi operator defined by (E.11).

Problem E.10. Investigate $\text{Proj}\,J_{\ast,\ast}^{(r)}(\ell)$ over $M_{\ast}^{(r)}(\ell)$ and the quotient space

[TABLE]

for $1\leq r\leq g-1.$

The difficulty to this problem comes from the following facts (A) and (B) :

(A) $J_{\ast,\ast}^{(r)}(\ell)$ is not finitely generated over $M_{\ast}^{(r)}(\ell).$

(B) $J_{k,\mathcal{M}}^{\text{cusp}}(\Gamma_{r}(\ell))\neq\text{ker}\,\Psi_{r,r-1,\ell}$ in general.

These are the facts different from the theory of Siegel modular forms. We remark that Runge (cf. [119, pp. 190–194]) discussed some parts about the above problem.