Rational Points on Varieties

研究项目是在纯数学领域，特别是代数几何和数论。纯数学是高度抽象的，接近深奥，但通过许多应用，如密码学和数据科学，对社会至关重要。本项目将在变化的有理点方面取得进展，这是丢番图方程的现代研究。基本主题将是研究这些方程解的存在性和分布，这是数论学家感兴趣的。具体目标将是证明del Pezzo曲面的Hasse原理失效的新结果，并证明del Pezzo曲面及其密切相关变种的Manin猜想的新情况。今年我更关注数论；主要目的是利用我的导师和他的合作者开发的方法，试图消除David J. Wright定理中给定群G的阶限制。该定理是关于固定伽罗瓦群G在全局函数域上伽罗瓦扩展数的渐近公式，该定理要求G的阶数不能被基域的性质整除。

The research project lies in the area of pure mathematics, specifically algebraic geometry and number theory. Pure mathematics is highly abstract and bordering upon the esoteric, but fundamentally crucial for society through numerous applications, such as cryptography and data science. The project will make progress in the area of rational points on varieties, which is the modern study of Diophantine equations. The fundamental theme will be to study the existence and distribution of solutions to these equations, which are of interest to number theorists. Specific goals will be to prove new results on the failure of the Hasse principle for del Pezzo surfaces and also prove new cases of Manin's conjecture for del Pezzo surfaces and closely related varieties.This year I am focusing more on number theory; the main aim is to try to remove the restriction on the order of given group G of a theorem by David J. Wright, using the method developed by my supervisor and his collaborators. The theorem is about the asymptotic formula of the number of Galois extensions over a global function field for fixed Galois group G, which requires the order of G is not divisible by the character of the base field.