Between rational and integral points

有理点和积分点之间

• 批准号: EP/P026710/2
• 负责人: Tim Browning
• 金额: $ 12.44万
• 依托单位: Institute of Science and Technology Austria
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FP026710%2F2
• 关键词: Between; rational; integral; points

Mathematics is undeniably the universal language of science and nature, whose processes are often governed by equations. This proposalcentres on systems of equations involving polynomials with integer coefficients. The study of rational or integral solutions to Diophantine equations is a subject that is both ancient and difficult, having commanded our attention since the time of the ancient Greeks nearly 2000 years ago. It has had profound interactions with a host of subject areas, ranging from algebraic geometry to complex analysis via mathematical logic and everything in between. At its core this proposal uses analytic methods to tackle a range of long-standing conjectures of about the existence and distribution of solutions to Diophantine equations, working in the context of both rational and integral points, as well as 'Campana-points', which interpolate between the two. In the reverse direction, by studying the distribution of rational and integral points on appropriate systems of Diophantine equations, the PI hopes to shed light on other fundamental areas of research, including the Langlands correspondence and Malle's conjecture about the density of number fields with prescribed Galois group.

无可否认，数学是科学和自然的通用语言，其过程通常由方程控制。该提案的重点是涉及具有整数系数的多项式的方程组。对丢番图方程的有理解或积分解的研究是一门既古老又困难的学科，自近 2000 年前的古希腊时代起就引起了我们的关注。它与许多学科领域有着深刻的相互作用，从代数几何到通过数理逻辑的复杂分析以及介于两者之间的一切。该提案的核心是使用分析方法来解决一系列关于丢番图方程解的存在性和分布的长期猜想，在有理点和积分点以及“坎帕纳点”的背景下工作，在两者之间进行插值。相反，通过研究适当的丢番图方程组上有理点和积分点的分布，PI 希望阐明其他基础研究领域，包括朗兰兹对应和马勒关于规定伽罗瓦数域密度的猜想团体。

项目成果

Sieving rational points on varieties

筛选品种理性点

• DOI: 10.1090/tran/7514
• 发表时间: 2018
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Browning T

The geometric sieve for quadrics

• DOI: 10.1515/forum-2020-0074
• 发表时间: 2021-01-01
• 期刊: FORUM MATHEMATICUM
• 影响因子: 0.8
• 作者: Browning, Tim D.;Heath-Brown, Roger
• 通讯作者: Heath-Brown, Roger

A geometric version of the circle method

圆法的几何版本

• DOI: 10.4007/annals.2020.191.3.4
• 发表时间: 2020
• 期刊: Annals of Mathematics
• 影响因子: 4.9
• 作者: Browning T

Free rational points on smooth hypersurfaces

光滑超曲面上的自由有理点

• DOI: 10.4171/cmh/499
• 发表时间: 2020
• 期刊: Commentarii Mathematici Helvetici
• 影响因子: 0.9
• 作者: Browning T

Density of rational points on a quadric bundle in P 3 × P 3

• DOI: 10.1215/00127094-2020-0031
• 发表时间: 2018-05
• 期刊: arXiv: Number Theory
• 作者: T. Browning;D. R. Heath-Brown