Jie Xia: Towards a definition of Shimura curves in positive characteristics

Shimura varieties are defined over complex numbers and have number fields as the field of definition. Motivated by an example constructed by Mumford, we find conditions which guarantee that a curve in char. $p$ lifts to a Shimura curve of Hodge type. These conditions, in terms of Dieudonne crystals, crystalline Tate cycles and $\ell$-adic monodromy, are intrinsic in char. $p$. One of the key ingredients in the proofs is a deformation result on Barsotti-Tate groups over a proper curve.