Geometric Analytic Number Theory

几何解析数论

• 批准号: MR/V021362/1
• 负责人: Daniel Loughran
• 金额: $ 147.28万
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=MR%2FV021362%2F1
• 关键词: Geometric; Analytic; Number; Theory

My project concerns Diophantine equations and related arithmetic and geometric objects. Diophantine equations are polynomial equations where one seeks solutions in the whole numbers. They have been studied since antiquity, but still hold many secrets and are an important area of research: Andrew Wiles' 1995 solution of Fermat's Last Theorem was one of the crowning achievements of 20th Century Mathematics, resolving a 350 year old conjecture of Fermat. Moreover, in the modern era they have surprisingly found numerous applications to information security and underlie many cryptosystems (e.g. Elliptic Curve Cryptography).Given a Diophantine equation, the first challenge is whether a solution actually exists. My project would take this to the next level: I will develop a unifying framework for studying *families* of Diophantine equations. Namely, when one runs through a family of equations given by varying the coefficients, how many have a solution? I will propose new conjectures for such problems as well as new techniques for tackling them, which vastly generalise the case of conics originally investigated by Jean-Pierre Serre (one of the leading mathematicians of the 20th Century). This will open new research directions in mathematics.

我的项目涉及二磷剂方程以及相关的算术和几何对象。双方方程是多项式方程，其中一个人在整个数字中寻求解决方案。自上古以来，他们已经对它们进行了研究，但仍然拥有许多秘密，并且是研究的重要领域：安德鲁·威尔斯（Andrew Wiles）的1995年解决Fermat的最后一个定理解决方案是20世纪数学的最高成就之一，解决了350年的Fermat猜想。此外，在现代时代，他们出人意料地发现了许多对信息安全性的应用，并构成了许多密码系统（例如椭圆曲线密码学）。赋予了二磷剂方程，第一个挑战是解决方案是否真的存在。我的项目将把它提升到一个新的水平：我将开发一个统一的框架，用于研究 *家族 *的diophantine方程。也就是说，当一个人通过改变系数给出的方程式传播时，有多少个解决方案？我将提出有关此类问题的新猜想，以及针对它们的新技术，这大大推广了Jean-Pierre Serre（20世纪的主要数学家之一）最初调查的锥体案例。这将打开数学的新研究方向。

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