On Genus Two Riemann Surfaces Formed from Sewn Tori

الأصل · EN

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 H₂. Equivariance of these maps under certain subgroups of Sp(4,Z) is shown. The invertibility of both maps in a particular domain of H₂ is also shown.