Motivic Cohomology and Descent on Algebraic Varieties

点击翻译按钮获取中文摘要

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.