Tensor decomposition of isocrystals characterizes Mumford curves

We seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics via characterizing curves in positive characteristics which are reduction of Shimura curve over $\mathbb{C}$. In this paper, we study the liftablity of a curve in the moduli space of principally polarized abelian varieties over $k, \text{char} k=p$. We show that in the generic ordinary case, some tensor decomposition of the isocrystal associated to the family imply that this curve can be lifted to a Shimura curve.