Shimura Varieties and Motives

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9970422This project studies the arithmetic of Shimura varieties, abelian varieties, motives, and related objects, including the following specific problems: define a good category of abelian motives over a finite field without assuming the Tate conjecture; the conjecture of Langlands and Rapoport concerning the structure of Shimura varieties in nonzero characteristic; canonical integral models of Shimura varieties in the case of bad reduction; the degeneration of abelian motives and its application to the study of compactifications of Shimura varieties; mixed motives with integer coefficients over finite fields.The proposal is in the area of arithmetic geometry, which studies solutions of systems of polynomial equations over arithmetically interesting rings or fields, such as the ring of integers or the field of rational numbers. It also includes the study of higher dimensional analogues of elliptic curves, namely, abelian varieties, and their generalizations, motives. This material is in the meeting ground between arithmetic and geometry.