Selmer groups, arithmetic statistics, and parity conjectures.

这个项目以数论为基础，但跨越了代数几何、拓扑学和概率论，旨在提高我们对两个主要猜想的理解：科恩-列斯特拉启发式猜想和伯奇和斯温纳顿-戴尔猜想，后者是“千禧年奖问题”之一。科恩-伦斯特拉启发式涉及类群，这是高斯在200多年前就已经研究过的神秘对象，它衡量了某些广义整数未能接受唯一的因式分解为素数。虽然在具体例子中很难理解是出了名的，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他们，他们预测他们在家庭中的行为可以用随机过程来建模。随后，这一原则被证明对许多相关对象的理解是有效的，从而导致了Bhargava在2014年被授予菲尔兹奖的“算术统计”领域。这个项目的目的是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。它们是类群的天然“高级类似物”，与类群一样，在算术和分析之间的显著联系中发挥着核心作用；它们的顺序出现在称为DedekindZeta函数的某些复杂分析函数的特值公式中。虽然可以说比类群更神秘(例如，完全确定整数的K-群将解决著名的Kummer-Vandiver猜想)，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开始对这些对象进行系统的研究，将Cohen-Lenstra启发式扩展到虚二次场的整数环的K-群，并利用新的突破来证明这一点的一大部分，显著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核心猜想的第二个--Birch和Swinnerton-Dyer猜想中。这涉及到阿贝尔变种的算术，即某些几何对象，其点自然地形成一组。这种结构将它们与其他几何对象区分开来，并使它们处于现代数论、代数几何和密码学研究的前沿。例如，Faltings对Mordell猜想的解决方案和Wiles对费马最后定理的证明都使用了阿贝尔变种，尽管最初看起来并不涉及这些问题。依附于阿贝尔变种的是两个性质截然不同的基本物体。一个是排名，衡量阿贝尔变种有多少有理点，本质上是算术。另一个对象是L函数，是复杂解析的。Birch和Swinnerton-Dyer猜想预测了两者之间的显著关系：L函数在其临界点的消失顺序应等于其秩次。这一猜想是在20世纪60年代提出的，此后一直是研究的焦点。一个值得注意的结果是奇偶猜想：某个容易计算的分析量，即根数，应该决定排名是奇数还是偶数。仅仅这一点通常就足以预测定义阿贝尔变化的方程何时有无限多个解，并对许多古老的问题产生影响。事实上，证明这一标准的有效性将解决许多重要的同余数问题，至少可以追溯到17世纪。这项建议的第二个主要目的是利用我最近工作中引入的新技术，为任意维度的大类阿贝尔变种建立一个变体，即2-奇偶猜想，从而提供证据，证明Birch和Swinnerton-Dyer猜想以预期的方式扩展到这种几乎一无所知的环境。

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.