On the topology of $\mathcal{H}(2)$

Abstract

The space $\mathcal{H}(2)$ consists of pairs $(M,\omega )$, where $M$ is a Riemann surface of genus two, and $\omega$ is a holomorphic 1-form which has only one zero of order two. There exists a natural action of ${\mathbb{C}}^{\ast }$ on $\mathcal{H}(2)$ by multiplication to the holomorphic 1-form. In this paper, we single out a proper subgroup $\Gamma$ of $\mathrm{Sp}(4,\mathbb{Z})$ generated by three elements, and show that the space $\mathcal{H}(2)/{\mathbb{C}}^{\ast }$ can be identified with the quotient $\Gamma \backslash {\mathcal{J}}_2$, where ${\mathcal{J}}_2$ is the Jacobian locus in the Siegel upper half space ${\mathfrak{H}}_2$. A direct consequence of this result is that $[\mathrm{Sp}(4,\mathbb{Z}):\Gamma ]=6$. The group $\Gamma$ can also be interpreted as the image of the fundamental group of $\mathcal{H}(2)/{\mathbb{C}}^{\ast }$ in the symplectic group $\mathrm{Sp}(4,\mathbb{Z})$.