Asymptotics for Rational Points

有理点的渐近

基本信息

批准号：
1700884
负责人：
Jordan Ellenberg
金额：
$ 36万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-06-01 至 2021-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700884&HistoricalAwards=false
关键词：
Asymptotics Rational Points

项目摘要

The research scope of this grant lies on the active interface between number theory and geometry. Geometry is perhaps the oldest part of mathematics, and number theory, the study of equations and their solutions in whole numbers, is hardly younger. Yet it is only in the very recent history of mathematics that researchers have understood just how interrelated these subjects are and how much they have to offer each other. One example is the "cap set problem," related to the popular card game Set. In the game, one asks: how many cards is it possible to have on the table with no legal play? It turns out that this problem has to do with the geometry of points and lines in 4-dimensional space, with equations among numbers and their last base-3 digit, and the relation between these. In 2016 the PI was part of a major breakthrough on this old problem, and his proposed research will continue investigating the new ideas that led to progress as well as other projects mixing number theory and geometry. The proposal covers several areas in number theory, algebraic geometry, topology, combinatorics, and applied math, in collaboration with a wide group of fellow researchers, including graduate students. One of the central questions of arithmetic statistics is: how many number fields are there of discriminant at most X? More particularly: how many of these have Galois group G for a specified subgroup G of a symmetric group S_n? A famous conjecture of Malle proposes a description for the asymptotic behavior of this count as X grows. Many of the major themes in contemporary number theory (e.g. Bhargava's work on counting quartic and quintic extensions, progress on Cohen-Lenstra conjectures) concern cases of this conjecture. In previous work, the PI showed that the Cohen-Lenstra conjecture over the function field F_q(t) could be approached by the methods of algebraic topology, using Grothendieck's theory of etale cohomology as the bridge between the two subjects. Now the PI proposes to prove the upper bound in the Malle conjecture in the case K = F_q(t), again using a combination of topological and arithmetic methods, but now with input from the theory of quantum shuffle algebras. In another project, the PI proposes to investigate the analogy between Malle's conjectures and the Batyrev-Manin conjectures, which study the asymptotics for rational points with bounded height on algebraic varieties. The height is a natural notion of complexity of an algebraic point just as the discriminant is for a number field. Here, the technical bridge is the theory of algebraic stacks; the PI will develop a theory of rational points of bounded height on Deligne-Mumford stacks, which first of all requires defining the height of a point on a stack. In particular, the discriminant of a number field is the height (in the novel sense) of a point on the classifying stack of a finite group. The PI will formulate a generalized Batyrev-Manin conjecture for stacks, which specializes to both Malle's conjecture and the Batyrev-Manin conjecture. The PI will also investigate properties of the new definition: for instance, the PI will aim to prove that the Faltings height is actually height on the moduli stack of abelian varieties in this sense. The PI also proposes problems in additive combinatorics, the homology of FI-modules, and the geometry of machine learning.

该资助的研究范围在于数论和几何之间的活跃界面。几何也许是数学中最古老的部分，而研究方程及其整数解的数论也并不年轻。然而，直到最近的数学史，研究人员才了解这些学科之间的相互关联性以及它们之间的相互关系有多大。一个例子是“上限设置问题”，与流行的纸牌游戏 Set 相关。在游戏中，有人会问：在不合法的情况下，桌上可能有多少张牌？事实证明，这个问题与 4 维空间中的点和线的几何形状、数字及其最后一位 3 位数之间的方程以及它们之间的关系有关。 2016 年，PI 是这个老问题重大突破的一部分，他提出的研究将继续调查导致进步的新想法以及混合数论和几何的其他项目。该提案涵盖了数论、代数几何、拓扑、组合学和应用数学等多个领域，与包括研究生在内的众多研究人员合作。算术统计的核心问题之一是：判别式X最多有多少个数域？更具体地说：其中有多少个具有对称群 S_n 的指定子群 G 的伽罗瓦群 G？ Malle 的一个著名猜想提出了随着 X 增长该计数的渐近行为的描述。当代数论中的许多主要主题（例如巴尔加瓦关于计算四次和五次扩展的工作、科恩-伦斯特拉猜想的进展）都涉及该猜想的案例。在之前的工作中，PI 证明了函数域 F_q(t) 上的 Cohen-Lenstra 猜想可以通过代数拓扑的方法来逼近，并使用 Grothendieck 的 etale 上同调理论作为两个主题之间的桥梁。现在，PI 提议再次使用拓扑和算术方法的组合来证明 K = F_q(t) 情况下马勒猜想的上限，但现在使用量子洗牌代数理论的输入。在另一个项目中，PI 提议研究 Malle 猜想和 Batyrev-Manin 猜想之间的类比，后者研究代数簇上具有有界高度的有理点的渐近性。高度是代数点复杂性的自然概念，就像数域的判别式一样。这里，技术桥梁是代数栈理论； PI 将开发 Deligne-Mumford 堆栈上有界高度的有理点理论，该理论首先需要定义堆栈上点的高度。特别是，数域的判别式是有限群的分类栈上的点的高度（在新颖的意义上）。 PI 将为堆栈制定广义的 Batyrev-Manin 猜想，专门针对 Malle 猜想和 Batyrev-Manin 猜想。 PI 还将研究新定义的属性：例如，PI 将旨在证明法尔廷斯高度实际上是这个意义上的阿贝尔簇模堆栈上的高度。 PI 还提出了加性组合学、FI 模块的同源性和机器学习几何学方面的问题。