On Genus Two Riemann Surfaces Formed from Sewn Tori

Details On Genus Two Riemann Surfaces Formed from Sewn Tori On Genus Two Riemann Surfaces Formed from Sewn Tori

2006-03-03

arxiv.org/abs/math/0603088v2

Commun.Math.Phys. 270 (2007) 587-634

Mathematics

Quantum Algebra

61 pages, 1 Figure

Scientific paper

10.1007/s00220-006-0163-5

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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