  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.
