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.