志村曲線上的自守型式

Abstract

在上個世紀，模型式和模曲線在數論的發展上佔了很重要地位。志村的曲線是模曲線的一個推廣，因此自守型式和志村曲線的算術性質在近代數論的發展也是舉足輕重。 我們的主要目標是研究自守型式的算術性質。這篇論文的工作是研究自守型式算術性質的一個起點。

根據楊一帆教授最近的結果，我們可以用Schwarzian微分方程的解來描述虧格為零的志村曲線上的自守型式，這提供了我們一個明確的方法來對自守型式作計算並幫助我們瞭解自守型式的算術性質。因此，如何找到的相關的Schwarzian微分方程就成為我們現在最重要的問題。

在這篇論文中，我們決定了大部分虧格為零志村曲線的Schwarzian微分方程。另外， 在學習自守型式的算術性質時，我們有個有趣的發現： 超幾何函數的代數變換。這 主要的概念是把志村曲線上的自守型式用超幾何函數來表示，並利用自守型式之間的相 等關係，我們就可以看到這些有趣的代數變換。

During the last century, modular forms and modular curves played important roles in the developments of number theory. Shimura curves are natural generalizations of classical modular curves. The arithmetic properties of automorphic forms and Shimura curves are particularly important in modern number theory. Our aim is to study the arithmetic properties of automorphic forms and automorphic functions on Shimura curves. The work in this dissertation is a starting point. Due to the recent work of Yifan Yang, if the Shimura curve is of genus zero, then one can express its automorphic forms in terms of the solutions of the associated Schwarzian differential equation. This provides a concrete space of automorphic forms. We then can do explicit computation on the spaces to study the arithmetic properties of automorphic forms and functions. Therefore, the main question is how to find the Schwarzian differential equations. In this thesis, we determine the Schwarzian differential equations for certain Shimura curves of genus zero. As a byproduct of study on automorphic forms on Shimura curves, we also obtain several algebraic transformations of -Hypergeometric functions. This discovery is achieved by interpreting Hypergeometric functions as automorphic forms on Shimura curves.