Arithmetic Statistics: Asymptotics on number fields and their class groups

算术统计：数域及其类群的渐近

• 批准号: RGPIN-2020-06146
• 负责人: Varma, Ila
• 金额: $ 1.89万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=718246
• 关键词: Arithmetic; Statistics; Asymptotics; number; fields; Arithmetic; Statistics; Asymptotics; number; fields

Much of my work has centered on statistical questions surrounding arithmetic objects such as number fields and their class groups. The central tenets in the subject are the Cohen-Lenstra heuristics [CL1, CL2] which predict the distribution (of p-parts) of class groups in families of number fields, and Malle's conjecture [Mal1, Mal2] on the asymptotic behavior of number fields of a specific Galois type. The field of arithmetic statistics provides a rough blueprint to attacking such classical and important questions in number theory, but to follow this through in the most interesting cases requires more sophisticated applications of tools from algebra and analysis than what has been present thus far. In my research, I attempt to incorporate such methods in order to resolve questions that have long resisted attack. I now summarize the most significant of my ongoing and proposed research directions. In upcoming work with Shankar [SV], we make use of new tools for counting number fields derived from the Dirichlet hyperbola method in conjunction with traditional arithmetic statistics techniques. We prove Malle's conjecture for Galois octic fields, and we are able to determine the asymptotic constant precisely in the case of D4-octic fields. We are next working on standardizing this strategy to prove other outstanding cases of Malle's conjecture for 2-groups. Recently, Bhargava-Shnidman [BS] counted cubic fields with a fixed quadratic Hessian covariant. Analogously, quartic fields have an associated covariant arising from the trace form on the (trace-free part of the) lattice of its ring of integers. By fibering quartic fields over this quadratic covariant, I should be able to utilize recent methods developed to count points on affine homogenous varieties [EMS, DRS], and I hope to be able to count various thin families of quartic fields, including, most notably, the family of A4-quartic fields ordered by discriminant. Most ambitiously, in joint work with Altug, Shankar, and Wilson we are working to extend methods to study the family of D5-quintic fields. We plan on using counting tools from D4-quartics [ASVW] in conjunction with techniques from counting S5-quintic fields [Bha10] to obtain asymptotics for the relevant orbits on these special elements within Bhargava's parametrization, and in turn count D5-quintic rings. It is noteworthy that the strategy we propose should allow us to count special families of D5-quintic fields, which would be tantamount to averaging 5-torsion in class groups of quadratic fields (a flagship problem in the area). In conclusion, the relatively nascent field of arithmetic statistics is continuing to benefit from an influx of interactions with more classical subjects. I will develop these connections in order to tackle the deepest questions in the field. In doing so, my research program will unravel the behavior of arithmetic objects in families so that we can move towards a cohesive theory of arithmetic statistics.

我的大部分工作都集中在围绕算术对象（例如数字字段及其课程组）的统计问题上。该受试者的中心宗旨是Cohen-Lenstra启发式方法[CL1，CL2]，该启发式方法预测了数字领域家族中班级组的分布，以及Malle的猜想[MAL1，MAL2]在特定Galois类型的数字场上的渐近行为上。算术统计的领域为攻击数字理论中的经典和重要问题提供了一个粗略的蓝图，但是在最有趣的情况下遵循这一点，需要代数和分析的工具对工具的更复杂的应用，而不是迄今为止的存在。在我的研究中，我试图纳入此类方法，以解决长期以来拒绝攻击的问题。现在，我总结了我正在进行的最重要和拟议的研究方向。 在即将与Shankar [SV]的工作中，我们利用新工具来计算源自Dirichlet双曲线方法与传统算术统计技术结合使用的数字字段。我们证明了Malle对Galois Octic场的猜想，并且我们能够确定渐近常数在D4-污染场的情况下。接下来，我们将致力于标准化这种策略，以证明Malle对2组的猜想的其他杰出案例。 最近，Bhargava-Shnidman [BS]用固定的二次Hessian协变量对立方场进行了计数。类似地，四分之一的场具有相关的协变量，该协方差是由其整数环（无痕迹部分）上的（无痕迹部分）上产生的。通过在这个二次协变量上纤维化田地，我应该能够利用开发的最新方法来计算仿生均质品种[EMS，drs]的观点，并且我希望能够计算各种Quartic领域的各个稀薄家族，其中包括，包括A4 Quartic of Indicts conderss Indicts conders conders conders conders conders conders conders conders conders conders conders conders condings conders condings conders conders conders conderss conders conderss conderss conders conderss conders conderss distriminants。 最雄心勃勃的是，在与Altug，Shankar和Wilson的联合合作中，我们正在努力扩展研究D5 Quintic领域家族的方法。 我们计划将来自D4 Quartics [ASVW]的计数工具与计数S5 Quintic Fields [BHA10]的技术结合使用，以获取Bhargava参数化中这些特殊元素的相关轨道的渐近器，而反过来又计数count Count Count Count Count D5 Quintic rings。值得注意的是，我们提出的策略应该使我们能够计算D5 Quintic领域的特殊家庭，这将符合平均二次领域（该地区的旗舰问题）中的5个扭转。 总之，算术统计的相对新生的领域继续受益于与更多经典主题的相互作用的涌入。我将建立这些联系，以解决该领域中最深切的问题。这样一来，我的研究计划将揭示算术对象在家庭中的行为，以便我们可以朝着算术统计的凝聚力理论迈进。