Mathematical Sciences: Shafarevich-Tate Groups

该奖项支持算术代数的研究 威廉·麦卡勒姆教授的几何学 亚利桑那麦卡勒姆博士的项目是研究 上定义的椭圆曲线群和阿贝尔簇 数字字段，以便验证 科利瓦金所考虑的椭圆曲线将是 模块化，而要研究的阿贝尔品种都将是 费马曲线的雅可比簇。 算术代数几何是一门结合了 代数几何和数论的技巧。在其 原始公式，代数几何处理的图形， 可以用最简单的方程在平面上定义，即 多项式数论始于整数， 诸如一个整数被另一个整数整除之类的问题。 这两个看似相距甚远的主题，实际上 从最早的时候就相互影响，但在过去， 世纪以来，相互间的影响大大增加。的 算术代数几何领域现在使用的技术， 现代数学的全部，并产生了相应的影响 超越自己的边界。

This award supports the research in arithmetic algebraic geometry of Professor William McCallum of the University of Arizona. Dr. McCallum's project is to study the Shafarevich-Tate group of elliptic curves and Abelian varieties defined over number fields in order to verify a number of conjectures of Kolyvagin. The elliptic curves under consideration will be modular, while the Abelian varieties to be studied will all be Jacobian varieties of Fermat curves. Arithmetic algebraic geometry is a subject that combines the techniques of algebraic geometry and number theory. In its original formulation, algebraic geometry treated figures that could be defined in the plane by the simplest equations, namely polynomials. Number theory started with the whole numbers and such questions as divisibility of one whole number by another. These two subjects, seemingly so far apart, have in fact influenced each other from the earliest times, but in the past quarter century the mutual influence has increased greatly. The field of arithmetic algebraic geometry now uses techniques from all of modern mathematics, and is having corresponding influence beyond its own borders.