Towards a definition of Shimura curves in positive characteristics

In the thesis, we present some answers to the question What is an appropriate definition of Shimura curves in positive characteristics ? The answer is obvious for Shimura curves of PEL type due to the moduli interpretation. Thus what is more interesting is the answer on Shimura curves of Hodge type. Inspired by an example constructed by David Mumford, we find conditions on a proper smooth curve over a field of positive characteristic which guarantee that it lifts to a Shimura curve of Hodge type over the complex numbers. These conditions are in terms of geometry mod p, such as Barsotti-Tate groups, Dieudonne isocrystals, crystalline Hodge cycles and l-adic monodromy. Thus one can take them as definitions of Shimura curves in positive characteristics. More generally, We define ``weak" Shimura curves in characteristic p. Along the way, we prove if a Barsotti-Tate group is versally deformed over a proper curve over an algebraically closed field of positive characteristic, then it admits a unique deformation to the corresponding Witt ring. This deformation result serves as one of the key ingredients in the proofs.