数域上代数簇的周群，K-群及其不变量研究

本项目研究数域上代数簇的基本算术问题，如周群（其中包括有理点与整点问题），K-群以及与此相关的不变量（如示性类）。..利用etale上同调工具，建立数域上光滑几何有理代数簇，K3曲面以及Shimura簇等以上同调为障碍的强逼近理论，并应用于研究整点和有理点问题（如Manin猜想）以及周群的算术性质（如Colliot-Thelene和Kato-Saito猜想）。..寻找在单位根作用下等变的高次算术K-群，建立等变高次算术K-群的局部化正合列（localization sequance)。建立Arakelov几何中高次算术K-群与Gillet-Soule算术周群之间的关系，构造取值在高次算术周群中的高次算术陈特征并证明相应的高次算术K-群的Riemann-Roch定理。

This research program study some fundamental arithmetic problems of algebraic varieties over a number fields, such as Chow groups (including rational points and integral points problems), K-groups and the related invariants (such characteristic classes)...By using etale cohomology theory, we will establish strong approximation with obstruction from cohomology for smooth and geometrically rational connected varieties, K3 surfaces and Shimura varieties and apply it to study the integral points and rational points (such as Manin's conjecture) and arithmetic properties of Chow groups (such as Colliot-Thelene and Kato-Saito conjecture)...We will find equivariant arithmetic higher K-groups under the action of roots of unity and provide the localization sequence for arithmetic higher K-groups. We also establish the relation between arithmetic higher K-groups and Gillet-Soule arithmetic Chow groups, construct higher Chern character with values in higher arithmetic Chow groups and prove the related Riemann-Roch theorem for higher arithmetic K-groups.