Pseudorandom majorants over number fields with applications in arithmetic geometry

数域上的伪随机主数及其在算术几何中的应用

• 批准号: EP/T01170X/2
• 负责人: Christopher Frei
• 金额: $ 2.68万
• 依托单位: Graz University of Technology
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FT01170X%2F2
• 关键词: Pseudorandom; majorants; over; number; fields

A Diophantine equation, named after the ancient Hellenistic mathematician Diophantus of Alexandria, is a polynomial equation in which all the coefficients are integers (whole numbers) or rational numbers (fractions). The most fundamental question, given a Diophantine equation, is whether it has a solution, that is a collection of integers or rational numbers which satisfy this equation. To decide whether a given Diophantine equation has a solution can be extremely hard, in spite of extensive mathematical machinery that was developed over centuries to attack these questions. A famous example is Fermat's Last Theorem. Despite the relative simplicity of its statement that for any integer n greater than two, the sum of two positive nth powers can not be an nth power, a proof has eluded the efforts of mathematicians for more than 350 years. It has spawned numerous new developments and was finally completed by Andrew Wiles at the end of the 20th century.Equations define not just number theoretic, but also geometric objects. A particularly successful approach, developed in the 20th century, tries to investigate solutions to Diophantine equations via the corresponding geometric objects. The modern study of Diophantine equations using these geometric techniques is called arithmetic (or Diophantine) geometry.Another branch of number theory, in which UK mathematicians play a world leading role, is called additive combinatorics. One of the aims of this discipline is to understand subsets of the integers by decomposing them into structured and random looking parts, with the main challenge arising from the fact that this is usually not a clean dichotomy, but rather a full spectrum.Extremely fruitful connections between these two fields were initiated very recently by applying certain results and techniques from additive combinatorics to questions in arithmetic geometry, thus expanding our knowledge of Diophantine equations significantly. The central aim of this project is to enhance the impact of these techniques by making them available in a much wider context that is natural in arithmetic geometry.

丢番图方程以亚历山大的古希腊数学家丢番图命名，是一个多项式方程，其中所有的系数都是整数（整数）或有序数（分数）。最基本的问题是，给定一个丢芬图方程，它是否有解，即满足这个方程的整数或有理数的集合。尽管几个世纪以来发展了广泛的数学机制来解决这些问题，但要决定给定的丢芬图方程是否有解是极其困难的。一个著名的例子是费马大定理。对于任何大于2的整数n，两个正n次方的和不可能是n次方，尽管这个命题相对简单，但数学家们350多年来一直没有找到证明。它催生了许多新的发展，最终由安德鲁·怀尔斯在20世纪末完成。方程不仅定义了数论，也定义了几何对象。20世纪发展起来的一种特别成功的方法，试图通过相应的几何对象来研究丢芬图方程的解。使用这些几何技巧的丢番图方程的现代研究被称为算术（或丢番图）几何。数论的另一个分支被称为加性组合学，英国数学家在其中发挥着世界领先的作用。这门学科的目标之一是通过将整数分解成结构化的和随机的部分来理解整数的子集，主要的挑战来自这样一个事实，即这通常不是一个干净的二分类，而是一个完整的范围。这两个领域之间非常富有成效的联系是最近通过将加性组合学的某些结果和技术应用于算术几何问题而开始的，从而大大扩展了我们对丢番图方程的认识。该项目的中心目标是通过使这些技术在算术几何中自然的更广泛的背景下可用来增强这些技术的影响。

项目成果

Distribution of genus numbers of abelian number fields

• DOI: 10.1112/jlms.12737
• 发表时间: 2022-09
• 期刊: Journal of the London Mathematical Society
• 作者: C. Frei;D. Loughran;Rachel Newton