Shimura Varieties and Motives

9970422本项目研究了Shimura簇、交换簇、基模及其相关对象的算术问题，包括以下具体问题：在不假设Tate猜想的情况下，定义有限域上交换基模的一个好范畴，Langlands和Rapoport关于非零特征的Shimura簇的结构的猜想; Shimura簇在不良约化情形下的典范积分模型;交换基的退化及其在Shimura簇紧化研究中的应用;有限域上具有整数系数的混合动机。该建议属于算术几何领域，它研究算术有趣环或域上多项式方程组的解，例如整数环或有理数域。 它还包括研究椭圆曲线的高维类似物，即阿贝尔簇，及其推广，动机。 这本书是算术和几何的交汇点。

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.