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On Genus Two Riemann Surfaces Formed from Sewn Tori

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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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Authors and Affiliations

  1. Department of Mathematics, University of California Santa Cruz, Santa Cruz, CA, 95064, U.S.A.

    Geoffrey Mason

  2. Department of Mathematical Physics, National University of Ireland, Galway, Ireland

    Michael P. Tuite

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  1. Geoffrey Mason
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  2. Michael P. Tuite
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Correspondence to Geoffrey Mason.

Additional information

Communicated by Y. Kawahigashi

Support provided by the National Science Foundation DMS-0245225, and the Committee on Research at the University of California, Santa Cruz.

Supported by The Millenium Fund, National University of Ireland, Galway.

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Mason, G., Tuite, M.P. On Genus Two Riemann Surfaces Formed from Sewn Tori. Commun. Math. Phys. 270, 587–634 (2007). https://doi.org/10.1007/s00220-006-0163-5

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  • DOI: https://doi.org/10.1007/s00220-006-0163-5

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