Towards a Definition of Shimura Curves in Positive Characteristics
نویسنده
چکیده
In this paper, we seek an appropriate definition of Shimura curves of Hodge type in positive characteristics. The basic idea is to characterize the reduction of Shimura curves. Specifically, we study the liftablity of a curve in the moduli space of principally polarized abelian varieties An,1,k over k, char k = p. We show that in the generic ordinary case, a tensor decomposition of the isocrystal associated to the family implies that this curve can be lifted to a Shimura curve. In the special case A4,1,k, we have two variants. We show that some conditions on the crystalline Tate cycles or l-adic monodromy over a curve imply that this curve can be lifted to a Shimura curve.
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