Rational Points on Varieties

该研究项目涉及纯数学领域，特别是代数几何和数论。纯数学是高度抽象的，近乎深奥，但通过密码学和数据科学等众多应用，对社会至关重要。该项目将在变种上的有理点领域取得进展，这是丢番图方程的现代研究。基本的主题将是研究这些方程解的存在和分布，这是数学家感兴趣的。具体的目标将是证明关于Del Pezzo曲面的Hasse原理失败的新结果，以及证明关于Del Pezzo曲面和密切相关变种的Manin猜想的新情况。今年我更专注于数论；主要目标是尝试使用我的主管和他的合作者开发的方法来消除对David J.Wright定理的给定群G的阶数的限制。该定理是关于固定Galois群G的整体函数域上的Galois扩张个数的渐近公式，它要求G的阶不能被基域的特征整除。

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.