The extension to $g>3$ of Ichikawa work leads to vector-valued 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 ${\frak H}_g$.
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}} $$ where $\omega_1,\ldots,\omega_g$ is the standard (normalized) basis of $H^0(K_C)$. $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis $\phi^n$ (see Prop.1.2). In the case $n=2$ and $g<4$ one may choose a natural basis for $H^0(K_C^2)$: ${\rm Sym}^2 H^0(K_C)$, and for $g=2$ gets $${\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)}\ . $$ For $g>3$ one has $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless one may continue to take $3g-3$ elements of ${\rm Sym}^2 H^0(K_C)$, or, more generally $N_n:=(2n-1)(g-1)$ elements of ${\rm Sym}^n H^0(K_C)$. Doing this leads to some surprise
${\kappa[\phi^n]\over\kappa[\omega^{(n)}]^{(2n-1)^2}}$, that essentially correspond to what we denoted by $$[i_{N_n+1},\ldots,i_{M_n}|\tau] \ , $$ are vector-valued Teichmueller modular forms of weight $$d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ .$$
For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in\{1,\ldots,M_n\}$ one has $$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x) =0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space.
Such vector-valued forms seem to be a key tool to characterize the Jacobian. Such a problem has been explicitly solved only for $g=4$: there is a weight 8 Siegel modular forms vanishing only on the Jacobian, this is the Schottky-Igusa form $F_4$. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to $S_{4ij}(\tau)$, where $$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$
For $g=4$ the discriminant of the quadrics is proportional to the square root of $\chi_{68}$, the $g=4$ Thetanullwerte $$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ . $$ Note that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight $34$) only when restricted to ${\cal I}_4$.
The $g=4$, $n=2$ Mumford form is $$\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}} \ . $$