Algebraic points on Shimura curves of $\Gamma_0(p)$-type

In this article, we classify the characters associated to algebraic points on Shimura curves of $\Gamma_0(p)$-type, and over a quadratic field we show that there are at most elliptic points on such a Shimura curve for every sufficiently large prime number $p$. This is an analogue of the study of rational points or points over a quadratic field on the modular curve $X_0(p)$ by Mazur and one of the author (Momose). We also apply the result to a finiteness conjecture on abelian varieties with constrained prime power torsion by Rasmussen-Tamagawa.