Algebraic points on Shimura curves of ${\Gamma }_0(p)$-type (III)

Abstract

In previous articles, we classified the characters associated to algebraic points on Shimura curves of ${\Gamma }_0(p)$-type, and over number fields in a certain large class we showed that there are at most elliptic points on such a Shimura curve for every sufficiently large prime number $p$. In this article, we prove the non-existence of elliptic points on Shimura curves of ${\Gamma }_0(p)$-type under a mild assumption. We also give an explicit example.

11 pages. arXiv admin note: substantial text overlap with arXiv:1211.0129, arXiv:1205.3596, arXiv:1202.4841