Algebraic points on Shimura curves of Γ0(p)-type

Abstract

Abstract. In this article, we classify the characters associated to algebraic points on Shimura curves of Γ0(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 ) $X_0(p)$ by Mazur and the second author. We also apply the result to a finiteness conjecture on abelian varieties with constrained prime power torsion by Rasmussen–Tamagawa.