In this paper we speculate on how to construct moduli for genus $g \ge 2$ analogous to the $g_2$ and $g_3$ invariants for $g = 1$.
Speculation
The moduli for genus one can be constructed from the periods of the holomorphic differential $dz$. Let \begin{equation} \omega(\gamma) = \int_{\gamma} dz \end{equation} then summing over all non-trival paths in the homology group we have an Eisenstein form \begin{equation} \label{eq:gn} G_n = \sum_{\gamma \ne 0} \frac 1 {\omega(\gamma)^{2n}} \end{equation} how do we generalise this to genus $g$ ? In genus $g$ choose a basis for the space of holomorphic differentials $dz_1 \ldots dz_g$ and a basis for the homology group $\gamma_1 \ldots \gamma_{2g}$. Then the Riemann period matrix $B$ is defined by \begin{equation} \label{eq:B} B_{ij} = \int_{\gamma_j} dz_i \end{equation} We want to convert this into something analogous to \eqref{eq:gn}. And we want to do it in a way which is independent of both the basis of the space of differentials and the basis of the homology group. The simplest way to construct an expression from a matrix, that is independent of its basis, is to take the determinant. But $B$ is not a square matrix - what to do?$\left|BB^T\right|$ is a possibility but it's not going to get us to equation \eqref{eq:gn} when $g = 1$.
Suppose instead we chose just $g$ linearly independent paths $\gamma=\lbrace \gamma_1 \ldots \gamma_g \rbrace$ from the homology group. Then the matrix $B(\gamma)$ would be square and we could take its determinant. We have removed the dependence on the basis for the space of differentials, but we still have dependence on $\gamma$.
Suppose we sum over all subsets of $g$ linearly independent paths. Then we no longer have dependence on the basis of the homology group. \begin{equation} \label{eq:Gn} G_n = \sum_{\left|\gamma\right| \ne 0} \frac 1 {\left|B(\gamma)\right|^{2n}} \end{equation} For genus $g=1$ equation \eqref{eq:Gn} reduces to equation \eqref{eq:gn}.
Of course this sum is not even close to absolute convergence. The problem is that for $g \ge 2$ there are an infinite number of integral unimodular matrices. Every pair of consecutive integers gives rise to several of them.
An equivalent formulation: let $B$ be the $g \times 2g$ matrix \eqref{eq:B} and $J$ be a $2g \times g$ matrix with integral coefficients, then \begin{equation} \label{eq:Gn2} G_k(B) = \sum_{\lbrace J:\rank(J)=g \rbrace} \frac 1 {\left|BJ\right|^{2k}} \end{equation} and to get the genus 1 formulae put $B = \begin{pmatrix}\omega_1 & \omega_2 \end{pmatrix}$ and $J = \begin{pmatrix}m \\ n\end{pmatrix}$. \begin{equation} \label{eq:Gn3} G_k(\omega_1, \omega_2) = \sum_{(m,n) \ne (0,0)} \frac 1 {\left(m\omega_1+n\omega_2\right)^{2k}} \end{equation}