In this talk, I shall explain some results on the geometry and arithmetic of Shimura curves X_B modulo p. Particularly we shall answer the follow questions:

(1) When the Fq-rational points of X_B exist.

(2) The field of definition of irreducible components and how two irreducible components.

(3) Classification of superspecial points.

(4) The 2T-H theorem for the superspecial locus of Shimura curves and explicit formula for each term. This is joint work in progress with Yasuhiro Terakado and Jiangwei Xue.