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On Genus Two Riemann Surfaces Formed from Sewn Tori

Geoffrey Mason·Michael P. Tuite
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Abstract

We describe the period matrix and other data on a higher genus Riemann surface in terms of data coming from lower genus surfaces via an explicit sewing procedure. We consider in detail the construction of a genus two Riemann surface by either sewing two punctured tori together or by sewing a twice-punctured torus to itself. In each case the genus two period matrix is explicitly described as a holomorphic map from a suitable domain (parameterized by genus one moduli and sewing parameters) to the Siegel upper half plane $\mathbb{H}_{2}$. Equivariance of these maps under certain subgroups of $Sp(4,\mathbb{Z)}$ is shown. The invertibility of both maps in a particular domain of $\mathbb{H}_{2}$ is also shown.

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BibTeXRIS

Geoffrey Mason, Michael P. Tuite. 2006-06-20. On Genus Two Riemann Surfaces Formed from Sewn Tori. https://doi.org/10.1007/s00220-006-0163-5

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