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