Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve
Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve
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DOI：
10.1007/s00222-006-0510-3
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发表时间：
2004-10
影响因子：
3.1
通讯作者：
Martin Möller
中科院分区：
文献类型：
--
作者：
Martin Möller
周期点是Veech曲面上的点，其在仿射微分同构群下的轨道是有限的。如果Veech曲面被适当地映射到它的雅可比矩阵或其适当的因子，我们将这些点描述为扭转点。对于2属的原始Veech曲面，我们证明了其周期点只有Weierstraß点和奇异点。我们的主要工具是hodge理论表征的teichm<s:1> ller曲线。我们由此推导出在teichm<s:1> ller曲线上雅可比矩阵族的modell - weil群的有限性结果。通过将周期点解释为覆盖在teichm<s:1> ller曲线上的曲线族的部分，提供了与周期点分类的联系。
Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterize those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate factor thereof. For a primitive Veech surface in genus two we show that the only periodic points are the Weierstraß points and the singularities.Our main tool is the Hodge-theoretic characterization of Teichmüller curves. We deduce from it a finiteness result for the Mordell-Weil group of the family of Jacobians over a Teichmüller curve. The link to the classification of periodic points is provided by interpreting them as sections of the family of curves over a covering of the Teichmüller curve.