RESEARCH STATEMENT JIE XIA

My main interests lie primarily in algebraic geometry and arithmetic geometry. The focus of my thesis is to answer the following question. Question 0.1. What is an appropriate definition of Shimura varieties in positive characteristics? The answer is only known for Shimura varieties of PEL type which admit a natural moduli in-terpretation. Such a definition is essential in proving local Langlands conjecture(see [Har01] for a glimpse) and other subjects [Kot92][RZ96]. We consider 0.1 for Shimura varieties of Hodge type and give several answers in the generic ordinary case, in terms of crystalline Hodge cycles (3.3), l-adic monodromy (3.5) and Dieudonne crystals (3.7). I elaborate these results in Section 3. Applications in other areas of mathematics and future projects are in Section 4. 1