Rational Points on Varieties

品种的理性点

• 批准号: 2751922
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2751922
• 关键词: Rational; Points; Varieties

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.

该研究项目属于纯数学领域，特别是代数几何和数论。纯数学是高度抽象且近乎深奥的，但通过密码学和数据科学等众多应用对社会至关重要。该项目将在簇有理点领域取得进展，即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。具体目标是证明德尔佩佐曲面哈斯原理失败的新结果，以及证明德尔佩佐曲面马宁猜想的新案例以及密切相关的簇。今年我更关注数论；主要目的是尝试使用我的导师和他的合作者开发的方法来消除对 David J. Wright 定理的给定 G 群阶的限制。该定理是关于固定伽罗瓦群G在全局函数域上伽罗瓦扩张数的渐近公式，要求G的阶数不能被基域的性质整除。