本文简要介绍了丢番图方程研究的背景，介绍了关于哈赛原则和Manin obstruction的基础知识和一些研究成果，整理了Birch，Davenport，Lewis和Heath-Brown，Skorobogatov的方法：利用Hardy-Littlewood圆法来解决有理数域Q上含范数形式的方程aN(x)+bN(y)=z^n的整数解问题，从而得到方程aN(x)+bN(y)=1的有理解，进一步地，利用代数簇的双有理等价，可以将结论用于探讨原本无法直接利用圆法的变量个数较少的方程P(t)=N(x_1,...,x_n).当P(t)在Q上恰好有2个根，并且在Q以外没有其它根时，在这一方程给出的任意光滑射影簇上，所有的哈赛原则和弱近似的反例都可以归结为Manin obstruction；并且，如果哈塞原则对这样的一个簇成立，那么在Zariski拓扑下，有理点在簇中稠密。

The study of Diophantine equations, or the study of rational solutions of certain equations over number field, is important in number theory together with algebraic geometry. In this paper, we are concerned about two important related open problems of this: the Hasse principle, a local to global principle for existence of rational solutions; and the Brauer-Manin obstruction, ertain cases when Hasse principle fails. We shall introduce some results on this in lower dimension cases and show an approach of circle method to this problem; the main result is that for certain equations involving norms over $\Q$, the Brauer-Manin obstruction is the only obstruction to the Hasse principle and when the Hasse principle holds, the rational solutions will be Zariski dense.