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.

数学无疑是科学和自然的通用语言，其过程通常受方程式的支配。该提案中心对涉及多项式的方程式系统，具有整数系数。自2000年前古希腊人时代以来，对二元方程的理性或整体解决方案的研究既古老又困难。它与许多主题领域具有深刻的相互作用，从代数几何形状到通过数学逻辑以及两者之间的所有内容进行复杂分析。本提案以核心使用分析方法来解决有关解决方案解决方案的存在和分布的一系列长期猜想，这些猜测是在理性和积分点的背景下工作的，以及“ campana-points”，这两者之间的插值。在相反的方向上，通过研究适当的二苯胺方程系统上的理性和积分点的分布，PI希望阐明其他基本研究领域，包括兰兰兹信函以及Malle对有规定的Galois组的数量字段密度的猜想。

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