  In this paper, we seek an appropriate definition for a Shimura curve of Hodge
type in positive characteristics, i.e. a characterization of curves in positive
characteristics which are reduction of Shimura curve over the complex field.
Specifically, we study the liftablity of a curve in moduli space of principally
polarized abelian fourfolds over k, char k=p. We show that some conditions on
the crystalline Hodge cycles over such a curve imply that this curve can be
lifted to a Shimura curve.
