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Cohen-Lenstra heuristics, and ordinary representations of finite groups

Cohen-Lenstra 启发式和有限群的普通表示

基本信息

批准号：
EP/N006542/1
负责人：
Alex Bartel
金额：
$ 12.62万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2015
资助国家：
英国
起止时间：
2015 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN006542%2F1
关键词：
Cohen Lenstra heuristics ordinary representations

项目摘要

One of the fundamental problems in Pure Mathematics is to understand and measure symmetries. Classically, the word "symmetry" was applied to geometric shapes, e.g. referring to rotations and reflections of regular polygons or solids. However, after the ground breaking contributions of Évariste Galois in the 19th century, we have learned to understand symmetries in a much wider sense, and the notion of symmetry has been put on a powerful rigorous footing by group theory, and later by representation theory. These days, we express the idea of symmetry through the language of group actions. Two of the most fundamental group actions in pure mathematics are actions on sets ("G-sets"), and actions on vector spaces ("linear representations"). It is an old problem with many applications in and outside of algebra, and with a rich literature, to understand the natural procedure that turns a set with a group action into a vector space with the induced group action. In previous joint work with Tim Dokchitser, we have completely understood one side of this procedure, namely when distinct G-sets give rise to the same representation, thereby settling an over 60 year old problem. We have also made considerable progress on the dual question of which representations can be obtained from G-sets. In this project, I propose to settle instances of this latter problem for several further important infinite families of finite groups.The oldest branch of mathematics is the area called number theory, the biggest open problems today going back to the ancient Greeks. The second part of the proposed project, to be carried out jointly with Hendrik Lenstra, lives at the intersection of representation theory and number theory. The aim is to study symmetry groups of several classical number theoretic invariants, such as class groups. Gauss was the first to ask statistical questions about the structure of class groups, e.g. how often are they trivial, and how fast does their size grow in families. Many of these questions are open to this day. But our conceptual understanding in this area was revolutionised by a paper of Cohen and Lenstra from the early 80s, who proposed that the main factor that accounts for the frequency of algebraic objects in nature is the number of symmetries of this object (in high-browese the size of its automorphism group). Their heuristic works "out of the box" and agrees very well with numerical experiments in the easiest and most-studied family of ideal class groups (those of imaginary quadratic fields), but its generalisations to arbitrary families seem to deviate from the basic idea and to modify the postulated probability weights in ad-hoc ways. Until now, a conceptual explanation of these modifications has remained elusive. In this project, I will develop a framework that allows to compare sizes of automorphism groups, even when those groups are infinite. This will allow to recast the original Cohen-Lenstra heuristic for general families of class groups in a much more conceptual way, but it will also make it applicable in many more general situations. I plan to use this framework to investigate other statistical properties of many important number theoretic invariants, such as class groups, so-called K-groups, and also Selmer groups of elliptic curves. Those are some of the most fascinating and mysterious objects in number theory. The algebraic machine that I will develop to this end will also be of intrinsic interest, and will have applications to distribution questions in other areas, e.g. in geometry (to homology of hyperbolic manifolds) and combinatorics (to Jacobians of graphs).

纯数学的基本问题之一是理解和度量对称性。传统上，“对称”一词适用于几何形状，例如指规则多边形或实体的旋转和反射。然而，在世纪伽罗瓦的开创性贡献之后，我们已经学会了在更广泛的意义上理解对称性，对称性的概念已经被群论和后来的表示论置于一个强大而严格的基础上。如今，我们通过群作用的语言来表达对称性的概念。纯数学中两个最基本的群作用是集合上的作用（“G-集合”）和向量空间上的作用（“线性表示”）。这是一个古老的问题，在代数内外有许多应用，并且有丰富的文献，理解将具有群作用的集合变成具有诱导群作用的向量空间的自然过程。在之前与Tim Dokchitser的合作中，我们已经完全理解了这个过程的一个方面，即当不同的G集产生相同的表示时，从而解决了一个超过60年的问题。我们也取得了相当大的进展的双重问题，其中表示可以从G-集。在这个项目中，我建议解决后一个问题的几个进一步重要的无限族有限群的实例。数学最古老的分支是所谓的数论领域，今天最大的开放问题可以追溯到古希腊人。拟议项目的第二部分，将与亨德里克·伦斯特拉共同进行，生活在表示论和数论的交叉点。目的是研究几个经典数论不变量的对称群，如类群。高斯是第一个提出关于阶级结构的统计问题的人，例如，他们有多平凡，以及他们在家庭中的规模增长有多快。其中许多问题至今仍悬而未决。但是，我们在这一领域的概念性理解被科恩和伦斯特拉在80年代初的一篇论文彻底改变了，他们提出，自然界中代数对象出现频率的主要因素是该对象的对称性数量（在高浏览器中，它的自同构群的大小）。他们的启发式工程“开箱即用”，并同意非常好的数值实验中最简单和最研究家庭的理想类群（那些虚二次域），但其推广到任意家庭似乎偏离了基本思想，并修改假设的概率权重在特设的方式。到目前为止，这些修改的概念解释仍然难以捉摸。在这个项目中，我将开发一个框架，允许比较自同构群的大小，即使这些群是无限的。这将允许以一种更概念化的方式来重铸最初的科恩-伦斯特拉启发法，但它也将使其适用于许多更一般的情况。我计划使用这个框架来研究许多重要的数论不变量的其他统计性质，例如类群，所谓的K-群，以及椭圆曲线的塞尔默群。这些是数论中最迷人和神秘的对象。代数机，我将发展到这一目的也将是内在的利益，并将应用于分布问题在其他领域，例如在几何（同源的双曲流形）和组合（雅可比图）。