Shimura Varieties and Motives

9970422本课题研究了下村变元、阿贝尔变元、动机及其相关对象的算法，包括以下具体问题：在不假设泰特猜想的情况下，定义有限域上良好的阿贝尔动机范畴；朗兰兹和拉波波特关于非零特征下村变元结构的猜想；下村变元在不良约化情况下的正则积分模型；阿贝尔动机的退化及其在下村变元紧致化研究中的应用；有限域上具有整数系数的混合动机。建议在算术几何领域，研究算术感兴趣的环或域上的多项式方程组的解，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，以及它们的推广和动机。这种材料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.