Motivic Cohomology and Descent on Algebraic Varieties

对于美国国家科学基金会0070850--拉斯金德、盖瑟和夏拉施金，研究人员将研究算术型数域上的代数簇的算术和K-理论，例如代数数域。他们将介绍描述数域上一大类代数簇上有理点的方法。研究这些问题的基本工具之一是Motivic上同调，Geisser和Raskind将继续发展这一工具。有理系数多项式方程组理论在密码学和密码学中具有重要的意义。对这种方程组的可解性了解很多，或者缺乏可解性，可以在开发或破解密码系统和代码方面发挥很大作用。提出者将使用数论和代数几何中的最新技术来研究这些问题。虽然这些都是非常古老的课题，但它们在最近发现了壮观的应用，这有助于刺激它们的理论发展。

Abstract for NSF 0070850--Raskind, Geisser, and ScharaschkinThe investigators will study the arithmetic and K-theory of algebraic varieties over fields of arithmetic type, such as algebraic number fields. They will introduce methods to describe the rational points on a wide class of algebraic varieties over number fields. One of the basic tools used to study these questions is motivic cohomology, which Geisser and Raskind will continue to develop.The theory of systems of polynomial equations with rational coefficients is important for questions in cryptography and coding theory. Knowing a lot about the solvability of such a system of equations, or lack thereof, can play a big role in developing or breaking cryptosystems and codes. The proposers will use the latest techniques in number theory and algebraic geometry to study these questions. Although these are very old subjects, they have found spectacular applications in recent times, which has helped spur their theoretical development.