Shimura and Teichmüller curves
Shimura and Teichmüller curves
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DOI：
10.3934/jmd.2011.5.1
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发表时间：
2011-04
影响因子：
1.1
通讯作者：
Martin Möller
中科院分区：
文献类型：
--
作者：
Martin Möller
我们在曲线$M_g$的模空间中对同时是Shimura曲线和Teichmuller曲线的曲线进行分类：对于$g=3$和$g=4$都精确地存在一条这样的曲线，对于$g=2$和$g \geq 6$没有这样的曲线。我们从志村曲线和泰奇穆勒曲线的霍奇理论描述开始，揭示了这两类曲线的异同。分类的证明依赖于方形覆盖层的几何形状和对这些特殊纤维表面的数值不变量的估计。最后，我们将我们的主要结果转化为具有完全简并李雅普诺夫谱的Teichmuller曲线的分类。
We classify curves in the moduli space of curves $M_g$ that are both Shimura and Teichmuller curves: for both $g=3$ and $g=4$ there exists precisely one such curve, for $g=2$ and $g \geq 6$ there are no such curves. We start with a Hodge-theoretic description of Shimura curves and of Teichmuller curves that reveals similarities and differences of the two classes of curves. The proof of the classification relies on the geometry of square-tiled coverings and on estimating the numerical invariants of these particular fibered surfaces. Finally, we translate our main result into a classification of Teichmuller curves with totally degenerate Lyapunov spectrum.