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.

研究项目在于纯数学领域，特别是代数几何和数字理论。纯数学是高度抽象的，并且通过多种应用（例如密码学和数据科学）对社会至关重要，但对社会至关重要。该项目将在品种合理点领域取得进展，这是对二磷剂方程的现代研究。基本主题将是研究这些方程式的解决方案的存在和分布，这对于数字理论家来说是感兴趣的。具体的目标是证明关于Hasse原理在Del Pezzo表面的失败的新结果，并且还证明了Manin对Del Pezzo表面和紧密相关品种的新案例。我将更多地关注数字理论。主要目的是尝试使用我的主管及其合作者开发的方法来消除David J. Wright对定理G给定G的顺序的限制。该定理是关于固定Galois组G的GALOIS扩展数量的渐近公式，该固定GALOIS G组的渐近公式不受基本场特征的排除。