志村曲線之定義方程式

Abstract

在這篇論文中,我們會討論在給定不同D值下的志村曲線的方程式,這裡$W_D$代表所有在$X^D_0(1)$上的Atkin-Lehner involutions所形成的群。論文中最主要的想法是建構兩個合適且weight為0的Borcherds form以產生在$X^D_0(1)$模函數組成的體,藉由Schofer的方程式來計算不同Borcherds form在CM點上的值最後找出兩個Borcherds form的關聯性。

In the thesis, we will determine the equations of Shimura curves$X^D_0(1)/W_D$ of genus one for several D, where $W_D$ denotes the group of all Atkin-Lehner involutions on $X^D_0(1)$. The main idea is to construct two suitable Borcherds forms of weight 0 that generate the field of modular functions on $X^D_0(1)/W_D$ and use Schofer’s formula for values of Borcherds forms at CM-points to find the relation between the two Borcherds forms.