On Genus Two Riemann Surfaces Formed from Sewn Tori
Mason, Geoffrey Tuite, Michael P.
Mason, Geoffrey
Tuite, Michael P.
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2006
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journal article
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Geoffrey Mason and Michael P. Tuite(2006)On Genus Two Riemann Surfaces Formed from Sewn Tori, Commun.Math.Phys. 270 (2007) 587-634
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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CC BY-NC-ND