Modular embeddings of Teichmüller curves
Modular embeddings of Teichmüller curves
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Teichmüller 曲线的模块化嵌入
作者：
Martin Moeller;D. Zagier
具有模嵌入的Fuchsian群在非算术Fuchsian群中具有最丰富的算术性质。但它们非常罕见，所有已知的例子要么与三角形群有关，要么与泰希米勒曲线有关。在本文的第一部分中，我们研究了模嵌入的算术性质，并从头开始建立了具有模嵌入的Fuchsian群的扭曲模形式理论，证明了维数公式、系数增长估计和微分方程。在第二部分中，我们给出了Picard-Fuchs方程解的APéry类积分声明的模证明。给出了扭转模形式的显式傅立叶展开和Hilbert模曲面上Teichmüler曲线的方程。在第三部分中，我们证明了Hilbert模曲面上的亏格两条Teichmüler曲线是由一个theta导数的乘积切割出来的。我们从这个角度重新推导了这些Teichmüler曲线的大部分已知性质，而不使用平坦曲面理论。作为结果，我们给出了所有亏格两条Teichmüler曲线的模嵌入，并证明了它们的扭曲模形式的傅立叶展开是代数的，直到一个超越标度常数。此外，我们还证明了Bainbridge对Hilbert模曲面的紧化是环面的。紧凑化的策略可以用连分式来表示，在形式上类似于赫兹布鲁赫的，但每个细节都是不同的。
Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmüller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and develop from scratch a theory of twisted modular forms for Fuchsian groups with a modular embedding, proving dimension formulas, coefficient growth estimates and differential equations. In Part II we provide a modular proof for an Apéry-like integrality statement for solutions of Picard–Fuchs equations. We illustrate the theory on a worked example, giving explicit Fourier expansions of twisted modular forms and the equation of a Teichmüller curve in a Hilbert modular surface. In Part III we show that genus two Teichmüller curves are cut out in Hilbert modular surfaces by a product of theta derivatives. We rederive most of the known properties of those Teichmüller curves from this viewpoint, without using the theory of flat surfaces. As a consequence we give the modular embeddings for all genus two Teichmüller curves and prove that the Fourier developments of their twisted modular forms are algebraic up to one transcendental scaling constant. Moreover, we prove that Bainbridge’s compactification of Hilbert modular surfaces is toroidal. The strategy to compactify can be expressed using continued fractions and resembles Hirzebruch’s in form, but every detail is different.