Automorphic Forms on Shimura Curves

• Main Authors: Tu, Fang-Ting, 凃芳婷
• Other Authors: Yang, Yifan
• Format: Others
• Language: en_US
• Published: 2013
• Online Access: http://ndltd.ncl.edu.tw/handle/73456550841619337321

博士 === 國立交通大學 === 應用數學系所 === 102 === 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.