Selmer groups, arithmetic statistics, and parity conjectures.

这个项目，基于数论，但跨越代数几何，拓扑学和概率论，是关于提高我们对两个主要猜想的理解：科恩-伦斯特拉猜想，和伯奇和斯温纳顿-戴尔猜想，后者是“千年奖问题”之一。该科恩-Lenstra prostitics关注类群体，神秘的对象已经研究了高斯超过200年前，其中衡量失败的某些广义整数承认一个独特的因式分解成素数。虽然在具体的例子中很难理解，但在20世纪80年代，科恩和伦斯特拉提出了一种激进的替代方法来研究它们，预测它们在家庭中的行为可以通过随机过程来建模。随后，这一原则被证明对理解许多相关对象是有效的，导致Bhargava在2014年获得菲尔兹奖的“算术统计”领域。这个项目旨在以这种方式研究数论中普遍存在的某些其他群：整数环的K群。这些都是自然的“更高的类似物”的阶级团体，并像阶级团体，发挥了核心作用，在一个显着的联系算术和分析;他们的秩序出现在特殊的价值公式，为某些复杂的解析函数称为戴德金zeta功能。虽然可以说比类群更神秘（完全确定整数的K-群将解决著名的库默-范德维尔猜想，例如），但有证据表明，它们也可以由随机过程建模。我的目标是从统计学的角度出发，对这些对象进行系统的研究，将科恩-伦斯特拉几何学扩展到虚二次域的整数环的K群，并利用新的突破来证明其中的一大部分，从而大大提高我们对这些重要对象的理解。上面提到的分析和算术之间的显著联系再次体现在这个项目的第二个核心假设中，即伯奇和斯温纳顿-戴尔猜想。这涉及到阿贝尔簇的算术，阿贝尔簇是某些几何对象，它们的点自然形成一个群。这种结构将它们与其他几何对象区分开来，并将它们置于现代数论，代数几何和密码学研究的最前沿。例如，无论是法尔丁斯对莫德尔猜想的解决方案，还是怀尔斯对费马大定理的证明，都对阿贝尔簇进行了关键性的使用，尽管这些问题最初似乎并不涉及它们。附在阿贝尔簇上的是两个性质非常不同的基本对象。一个是秩，它是一个度量阿贝尔簇有多少个有理点，本质上是算术的。另一个对象是L-函数，并且是复解析的。伯奇和斯温纳顿-戴尔猜想预言了两者之间的惊人关系：L-函数在临界点消失的阶数应该等于秩。这个猜想是在20世纪60年代提出的，从那以后一直是研究的焦点。一个显著的结果是奇偶猜想：一个容易计算的解析量，即根数，应该决定秩是奇数还是偶数。仅凭这一点就足以预测定义阿贝尔簇的方程何时有无穷多个解，并对许多古老的问题产生了影响。事实上，证明这一标准的有效性将解决许多重要的情况下，同余数问题，可以追溯到至少世纪。这个建议的第二个主要目的是利用我最近的工作中引入的新技术来建立一个变体，2-奇偶猜想，对于任意维数的大类阿贝尔簇，这样做提供了证据，表明伯奇和斯温纳顿-戴尔猜想以预期的方式扩展到这个设置，几乎什么都不知道。

This project, based in number theory but spanning algebraic geometry, topology, and probability theory, is about improving our understanding of two major conjectures: the Cohen--Lenstra heuristics, and the Birch and Swinnerton-Dyer conjecture, the latter one of the `millennium prize problems'. The Cohen--Lenstra heuristics concern class groups, mysterious objects studied already by Gauss over 200 years ago, which measure the failure of certain generalised integers to admit a unique factorisation into primes. Whilst notoriously hard to understand in specific examples, in the 1980s Cohen and Lenstra proposed a radical alternative approach to studying them, predicting that their behaviour in families could be modeled by a random process. Subsequently, this principle has proven effective for understanding many related objects, leading to the field of `arithmetic statistics' in which Bhargava was awarded the Fields medal in 2014. This project aims to study in this way certain other groups ubiquitous in number theory: K-groups of rings of integers. These are natural `higher analogues' of class groups and, like class groups, play a central role in a remarkable link between arithmetic and analysis; their order appears in special value formulae for certain complex analytic functions called Dedekind zeta functions. Whilst arguably more mysterious than class groups (determining completely the K-groups of the integers would solve the famous Kummer--Vandiver conjecture for example) there is evidence that they too can be modeled by random processes. I aim to initiate a systematic study of these objects from a statistical point of view, extending the Cohen--Lenstra heuristics to K-groups of rings of integers of imaginary quadratic fields, and leverage new breakthroughs to prove a big piece of this, improving significantly our understanding of these important objects. The remarkable link between analysis and arithmetic alluded to above again manifests itself in the second of the conjectures central to this project, the Birch and Swinnerton-Dyer conjecture. This concerns the arithmetic of abelian varieties, certain geometric objects whose points naturally form a group. This structure distinguishes them amongst other geometric objects and has placed them at the forefront of research in modern number theory, algebraic geometry, and cryptography. For example, both Faltings's resolution of the Mordell conjecture and Wiles's proof of Fermat's last theorem made crucial use of abelian varieties, despite the problems not initially appearing to involve them. Attached to an abelian variety are two fundamental objects of a very different nature. One, the rank, is a measure of how many rational points the abelian variety has and is arithmetic in nature. The other object is the L-function, and is complex analytic. The Birch and Swinnerton-Dyer conjecture predicts a striking relationship between the two: the order of vanishing of the L-function at its critical point should equal the rank. This conjecture was made in the 1960s and has been a focal point for research ever since. One remarkable consequence is the parity conjecture: a certain easily computable analytic quantity, the root number, should determine whether the rank is odd or even. This alone is often sufficient to predict when the equations defining an abelian variety have infinitely many solutions, and has ramifications for many ancient problems. Indeed, a proof of the validity of this criterion would settle many important cases of the Congruent Number Problem, dating back to at least the 17th century. The second major aim of this proposal is to draw on new techniques introduced in my recent work to establish a variant, the 2-parity conjecture, for large classes of abelian varieties of arbitrary dimension, in doing so providing evidence that the Birch and Swinnerton-Dyer conjecture extends in the expected way to this setting, where almost nothing is known.