$$\hbox{Vector-valued Teichmueller Modular forms}$$

Vector-valued Siegel
modular forms are the natural generalization of the classical elliptic
modular forms as seen by studying the cohomology of the universal abelian variety.
In spite of their relevance they have been studied essentially for genus $g=2$, where
correspond to suitable commutators of Siegel modular forms. 

In the case $g=2$ and $g=3$ Ichikawa introduced the concept of Teichmueller modular forms.
It turns out that the Mumford forms for $g>3$ lead to the concept of vector-valued Teichmueller modular forms.


The main steps are the following.
For each fixed positive integers
$g,n$, define  $$M_n(g)=M_n:={g+n-1\choose n}\ ,\;
N_n(g)=N_n:=(2n-1)(g-1)\ ,\quad K_n:=M_n-N_n\ . $$
Let ${\frak H}_g:=\{Z\in M_g({\Bbb C})\mid {}^tZ=Z,\mathop{\rm Im} Z>0\}$, be the Siegel
upper half-space. Denote by $\tau_{ij}$ the Riemann period matrix and by ${\cal I}_g$ the closure of the locus of Riemann
period matrices in ${\frak H}_g$.

Denote by ${\cal M}_g$ the moduli space of Riemann surfaces.
Consider the case $g\ge 2$ and a given symplectic basis for $H_1(C,{\Bbb Z})$. For each positive integer $n$, consider the basis $\tilde\omega_1^{(n)},\ldots,\tilde\omega_{M_n}^{(n)}$ of ${\rm Sym}^n H^0(K_C)$ whose elements are symmetrized tensor products of $n$-tuples of vectors of the basis $\omega_1,\ldots,\omega_g$, taken with respect to an arbitrary ordering chosen once and for all. Denote by $\omega_i^{(n)}$, $i=1,\ldots, M_n$, the image of $\tilde\omega_i^{(n)}$ under the natural map $\psi:{\rm Sym}^n H^0(K_C)\to H^0(K_C^n)$. It is well known that such a map is surjective if and only if $g=2$ or $C$ is non-hyperelliptic of genus $g>2$. For $g=2$ and $g=3$ non-hyperelliptic, this map is an isomorphism.


Consider the Thetanullwerte
$\chi_k(Z):=\prod_{\delta\hbox{ even}} \theta\[\delta\](0,Z)$,
$Z\in{\frak H}_g$, with $k=2^{g-2}(2^g+1)$.
Set
$$F_g:=2^g
\sum_{\delta\hbox{
even}}\theta^{16}\[\delta\](0,Z)-\bigl(\sum_{\delta\hbox{
even}}\theta^{8}\[\delta\](0,Z)\bigr)^2 \ .
$$
It turns out that $F_4$, the Schottky-Igusa form, vanishes only on the Jacobian. Furthermore, there is a nice relation
between $F_g$ and the theta series $\Theta_\Lambda$
corresponding to the even unimodular lattices $\Lambda=E_8$ and
$\Lambda=D_{16}^+$:
$$
F_g=2^{-2g}(\Theta_{D_{16}^+}-\Theta_{E_8}^2) \ .
$$

Let $\{\phi^n_i\}_{1\le i\le N_n}$ be a basis of
$H^0(K_C^n)$, $n\geq2$. 
The Mumford form is, up to a universal constant
$$
\mu_{g,n}={\kappa[\omega]^{(2n-1)^2}\over
\kappa[\phi^n]}{\phi^n_1\wedge\cdots\wedge\phi^n_{N_n}\over
(\omega_1\wedge\cdots\wedge\omega_g)^{c_n}} \ .
$$
where $\omega_1,\ldots,\omega_g$ is the standard (normalized) basis of $H^0(K_C)$. $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis $\phi^n$ (see Prop.1.2). In the case $n=2$ and $g<4$ one may choose the natural basis 
${\rm Sym}^2 H^0(K_C)$ for $H^0(K_C^2)$, and for $g=2$ gets
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over \pi^{12}\chi_{5}^2(\tau)}\ ,$$
whereas for $g=3$
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over 2^6\pi^{18}\chi_{18}^{1/2}(\tau)}\ . $$
For $g>3$ one has $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless one may continue to take $3g-3$ elements of ${\rm Sym}^2 H^0(K_C)$, or, more generally $N_n:=(2n-1)(g-1)$ elements of ${\rm Sym}^n H^0(K_C)$. Doing this leads to some surprise

1. To simplify notation, denote here by $\omega^{(n)}$ the basis $\{\omega^{(n)}_k\}$ with
$k=i_1,\ldots,i_{N_n}\in\{1,\ldots,M_{n}\}$.
$$[i_{N_n+1},\ldots,i_{M_n}|\tau]=\epsilon_{i_1,\ldots,i_{M_n}}
{\kappa[\omega^{(n)}]\over\kappa[\omega]^{(2n-1)^2}}\ ,$$
are vector-valued Teichmuller modular forms without poles on ${\cal M}_g$ and vanishing on the hyperelliptic locus, of weight
$$
d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ .
$$
Note that the vector-valued nature is just a consequence of the inequality $M_n-N_n>0$ for some $g$. For example, for $n=2$ one has $g(g+1)/2-(3g-3)>0$ satisfied for $g>3$. This implies that there are free indeces: the $i_{N_n+1},\ldots,i_{M_n}$.
This may be seen as a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case $g=4$  (presumably here should also appear some interesting Number Theoretical structures).

2. For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in\{1,\ldots,M_n\}$ one has
$$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x)
=0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space.

3. Such vector-valued forms seem to be a key tool to characterize the Jacobian. Such a problem has been explicitly solved only for $g=4$: there is a weight 8 Siegel modular forms vanishing only on the Jacobian, this is the Schottky-Igusa form $F_4$. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to $S_{4ij}(\tau)$, where
$$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$

4. For $g=4$ the discriminant of the quadrics is proportional to the square root of
$\chi_{68}$, the $g=4$ Thetanullwerte 
$$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ . $$
Note that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight
$34$) only when restricted to ${\cal I}_4$.


5. The $g=4$, $n=2$ Mumford form is
$$\mu_{4,2}=\pm{1\over c
S_{4ij}}{\omega_1\omega_1\wedge\cdots\wedge
\widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over
(\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ .
$$