Mathematical Sciences: Rational Points on Algebraic Varieties

本研究的主要目的是研究Mazur关于有理数域上有理点的拓扑的一个猜想。与Hasse原理、弱逼近和Brauer-Manin阻塞有关的问题在本研究中起着关键作用。这种方法依赖于对所考虑的每个品种的特殊性质的详细研究，以及在适用的情况下标准高度函数的构造。这项研究属于算术几何的一般数学领域。这个超现代的研究领域结合了两个最古老的数学分支：数论和几何。这种结合产生的新见解正在产生越来越强大的工具来解决像费马最后定理这样的长期存在的问题，该定理抵制了三个多世纪以来数学家们最强大的努力。此外，虽然算术几何有时被认为是最纯粹的数学之一，但它也一直在开发有洞察力的新技术，导致纠错码和密码学等应用领域的巨大进步。

The main goal of this research is to study a conjecture of Mazur concerning the topology of rational points on a variety over the rational number field. Questions related to the Hasse Principle, weak approximation and the Brauer-Manin obstruction play a key role in this study. The approach depends on a detailed study of special properties of each variety under consideration and the construction of canonical height functions when applicable. This research is in the general mathematical area of Arithmetic Geometry. This ultramodern research area combines two of the oldest branches of mathematics: number theory and geometry. The new insights arising out of this combination are producing increasingly powerful tools to solve long-standing problems like Fermat's Last Theorem, which have resisted the strongest efforts of over three centuries of mathematicians. In addition, though Arithmetic Geometry is sometimes regarded as among the purest of pure mathematics, it has also been developing insightful new techniques leading to dramatic progress in such applied areas as error-correcting codes and cryptography.