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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*} 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*} B_{ij} = \int_{\gamma_j} dz_i \end{equation*}

We want to convert this into something analogous to 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 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*} G_n = \sum_{\left|\gamma\right| \ne 0} \frac 1 {\left|B(\gamma)\right|^{2n}} \end{equation*}

For genus $g=1$ equation Gn reduces to equation 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 B and $J$ be a $2g \times g$ matrix with integral coefficients, then

\begin{equation*} 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*} 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*}