Mathematical Sciences: L-Independence in Arithmetic Algebraic Geometry

德容9625417 这项建议计划调查的基本问题之一，目前正在研究的算术几何，即不变性的某些上同调定义的不变量使用各种上同调理论，特别是l-进（etale）上同调的变化l和p-进（刚性）上同调。要问的问题是是否尺寸的某些空间是独立的l，是否特征值的对应作用于某些群体是独立的l，是否特征值的弗罗贝纽斯是独立的l，等等，这是假设，这些问题的答案都是积极的。 这个项目福尔斯属于算术几何的一般领域-一个融合了两个最古老的数学领域：数论和几何的主题。 事实证明，这种结合非常富有成效--最近解决了几代人都无法解决的问题。 在它的许多后果是新的纠错码。 此类代码对于现代计算机（硬盘）和光盘都是必不可少的。

de Jong 9625417 This proposal plans to investigate one of the fundamental questions currently being studied in arithmetic geometry, namely the invariance of certain cohomologically defined invariants using various cohomology theories, especially l-adic (etale) cohomologies for varying l and p-adic (rigid) cohomology. The questions to be asked are whether dimensions of certain spaces are independent of l, whether the eigenvalues of correspondences acting on certain groups are independent of l, whether the eigenvalues of Frobenius are independent of l, etc. It is conjectured that the answers to these questions are all positive. This project falls into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.