Mathematical Sciences: The Arithmetic of Shimura Varieties

该奖项支持詹姆斯·米尔恩教授在代数几何方面的研究。几十年来，人们已经知道，一方面是阿贝尔变数及其退化的算法，另一方面是西格尔模变数及其紧化的算法，两者之间存在着密切而富有成果的关系。最近，米尔恩教授发现了阿贝尔动机与志村变种阿贝尔类型之间的类似关系。提出了对阿贝尔型退化的研究。本文提出研究阿贝尔动机的退化，并利用它们之间的关系来获得关于数域上志村变的紧化的信息。但在过去的四分之一个世纪里，它已经有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。

This award supports the research of Professor James Milne to work in algebraic geometry. It has been known for several decades that there is a close and fruitful relation between the arithmetic of abelian varieties and their degenerations, on the one hand, and the arithmetic of Siegel modular varieties and their compactifications on the other hand. Recently, Professor Milne has found a similar relation between abelian motives and Shimura varieties of abelian type. It is proposed to study the degeneration of abelian type. It is proposed to study the degeneration of abelian motives and exploit the relation in order to obtain information about the compactifications of Shimura varieties over number fields This research uses the techniques of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.