代数簇上整点和有理点的上同调方法

研究代数簇的整点和有理点是算术代数几何核心问题之一，而上同调是现代数学研究的重要手段。本研究计划试图利用各种Grothendick拓扑定义的上同调方法,建立数域上代数簇在这些上同调障碍下的强逼近定理。由此判定这些代数簇的有理解和整数解的存在性。进一步研究这些代数簇有理解和整数解个数的渐进公式，确定上同调障碍如何通过扰动局部解个数贡献给解个数的渐进公式。研究渐进公式中体现L-函数的特殊值与代数簇的算术不变量的联系。

It is one of the core problems in arithmetic algebraic geometry to study integral points and rational points in algebraic varieties and cohomology theory is also one of important methods in modern mathematics. This research program tries to establish strong approximation of algebraic varieties over algebraic number fields with various cohomology obstructions given by various Grothendieck topology. Then one can determine the existence of integral points and rational points by using these obstructions. We further plan to study the asymptotic formula for integral points or rational points for such varieties and decide precisely how these obstruction twist the local solutions to contribute the asymptotic formula. Investigate the relation between the special values of L-functions involved in the asymptotic formula and the arithmetic invariant of algebraic varieties.