K群结构及与数论相关的一些问题的研究

本项目主要研究代数K-理论及其在数论中的应用等相关问题，主要内容包括：.(1) 代数数域的Tame核结构的研究：运用类数、同余数、reflection theorem和类域论继续深入地研究二次域和分圆域的Tame核的p-Sylow子群; .(2) 代数整数环上K2OF与理想类群的关系：运用秦在二次域的K2OF的4-rank上的精确结果、已有的用K2群计算类群的方法，Legendre定理以及Galois上同调来寻求二次域和分圆域的类群上的结果；.(3) 代数K-理论与正定三元二次型的相关研究：通过选取适当的三元二次型以及运用Minkowski-Siegel公式、模形式、椭圆曲线等来进一步研究二次域K2OF的结构；.(4)代数K-理论中密度问题的研究：将重点研究二次域和三次域的Tame核的2-rank 和3-rank的密度问题以及将函数域情形下的理想类群的密度定理推广到代数K群中去。

We will study some problems related to algebraic K-theory and its applications in number theory. Specifically, we consider the following topics: .(1) Research on tame kernels of number fields. We will use class number, congruent number, reflection theorem and class field theory to study the p-Sylow subgroups of tame kernels of quadratic fields and cyclotomic fields;.(2) Research on the relations between K2OF and ideal class group. We will investigate the class groups of quadratic fields and cyclotomic fields by the precise results on the 4-rank of K2OF of quadratic fields given by Qin, some existing methods about computing class number by K2 groups, Legendre Theory and Galois cohomology；.(3) Research on algebraic K-Theory and positive ternary quadratic forms. We will study the structure of K2OF of some quadratic fields by choosing proper ternary quadratic forms and applying Minkowski-Siegel formula, modular form, elliptic curve；.(4) Research on densities of tame kernels in K-theory. We will study densities of 2-rank and 3-rank of tame kernels for quadratic fields and cubic fields, we will also extend the density theorems for ideal class groups under function fields to algebraic K-Groups.