由缝合环面构成的两个亏格Riemann Surfaces
On Genus Two Riemann Surfaces Formed from Sewn Tori
摘要：
我们描述的周期矩阵和其他数据的高亏格黎曼曲面的数据来自低亏格曲面通过显式缝合程序。我们详细考虑了通过缝合两个穿孔环面或缝合两次穿孔环面来构造亏格为2的黎曼曲面。在每种情况下，属两个周期矩阵被明确地描述为一个全纯映射从一个合适的域（参数化属一模和缝纫参数）的西格尔上半平面$\mathbb{H}_{2}$。这些地图下的某些子群的$Sp(4,\mathbb{Z)}$的等方差。还示出了在$\mathbb{H}_{2}$的特定域中的两个映射的可逆性。
英文摘要：
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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