Algebraic points on Shimura curves of $Gamma_{0}(p)$-type (IV) (Algebraic Number Theory and Related Topics 2013)

"Algebraic Number Theory and Related Topics 2013". December 9～13, 2013. edited by Tadashi Ochiai, Takeshi Tsuji and Iwao Kimura. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.In previous articles, we proved that there are no points rational over a fixed number field on the Shimura curve of $Gamma$_{0}(p)-type for every sufficiently large prime number p under a mild assumption. In this article, (1) we generalize the previous result to an infinite family of number fields, and (2) give examples not satisfying the mild assumption as mentioned above