Algebraic Models and Arithmetic Geometry of Teichmüller Curves in Genus Two
Algebraic Models and Arithmetic Geometry of Teichmüller Curves in Genus Two
复制标题
二格Teichmüller曲线的代数模型和算术几何
DOI：
10.1093/imrn/rnw193
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发表时间：
2014
影响因子：
1
通讯作者：
Ronen E. Mukamel
中科院分区：
文献类型：
--
作者：
Abhinav Kumar;Ronen E. Mukamel
TeichMuller曲线是代数曲线在Riemann曲面的模空间中的代数等距浸入。给出了正亏格的TeichMuller曲线的第一个显式代数模型。我们的方法是基于对某些Hilbert模形式的研究，并利用Ahlfors变分公式来确定亏格二雅可比实乘的特征形式。我们还给出了TeichMuller曲线允许丰富的算术几何的证据，通过展示具有糟糕归约的小素数和在其尖端支持显著因子的例子。
A Teichmuller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichmuller curves of positive genus. Our methods are based on the study of certain Hilbert modular forms and the use of Ahlfors's variational formula to identify eigenforms for real multiplication on genus two Jacobians. We also present evidence that Teichmuller curves admit a rich arithmetic geometry by exhibiting examples with small primes of bad reduction and notable divisors supported at their cusps.