Rational points on algebraic varieties

本课题将重点研究代数变种上的有理点。给定一个数字域上的代数变化，自然会有这样的问题：是否存在一个有理点？如果有，是否有无限多个？如果也是，那么是否可以对有理点的分布进行更精细的定量描述呢？一般来说，这些问题都是非常困难的，但在本项目中，我们的目标是在一些特殊类型的变种上取得一些进展（特别是解决一些新的Manin猜想）。同时，本文也希望对当前一个流行的研究主题——在科中考虑这些问题，如研究具有理性点的科中品种的分布，或控制哈塞原理在科中的失效——有所贡献。为了解决这类问题，人们通常会结合使用代数几何、代数数论和解析数论的技术。

This project will focus on the study of rational points on algebraic varieties. Given an algebraic variety over a number field, natural questions are: Is there a rational point? If yes, are there infinitely many? If also yes, can one obtain a finer quantitative description of the distribution of the rational points? These problems are very difficult in general, but in this project the aim is to make some progress for some special classes of varieties (in particular solve some new cases of Manin's conjecture). It also expects to contribute to a popular current research theme, that of considering these problems in families, such as studying the distribution of varieties in a family with a rational point, or controlling failures of the Hasse principle in families. To solve such problems one normally uses a combination of techniques from algebraic geometry, algebraic number theory and analytic number theory.