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

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统计问题周围的算术对象，如数字字段和他们的类组。中心原则的主题是科恩-Lenstra prostitics [CL 1，CL 2]预测分布（p部分）的类组在家庭的数域，和马勒的猜想[Mal 1，Mal 2]的渐近行为的数域的一个特定的伽罗瓦类型。算术统计领域提供了一个粗略的蓝图来攻击数论中的这些经典和重要问题，但是在最有趣的情况下，要做到这一点，需要比迄今为止更复杂的代数和分析工具的应用。在我的研究中，我试图将这些方法结合起来，以解决长期以来一直难以解决的问题。我现在总结我正在进行的和建议的研究方向中最重要的。 在即将与Shankar [SV]合作的工作中，我们使用新工具来计算Dirichlet双曲线方法与传统算术统计技术相结合的数域。本文证明了Galois八度域的Malle猜想，并且在D4-八度域的情形下，我们能够精确地确定渐近常数。我们接下来将致力于标准化这个策略，以证明2群的Malle猜想的其他杰出案例。 最近，Bhargava-Shnidman [BS]计算了具有固定二次Hessian协变的立方域。类似地，四次域有一个相关的协变，它产生于它的整数环的格（无迹部分）上的迹形式。通过在这个二次协变量上计算四次域，我应该能够利用最近开发的方法来计算仿射齐次簇[EMS，DRS]上的点，我希望能够计算各种四次域的薄族，包括最值得注意的是，由判别式排序的A4-四次域族。 最雄心勃勃的是，在与Altug，Shankar和Wilson的联合工作中，我们正在努力扩展研究D5五次域家族的方法。 我们计划使用来自D4-quartics [ASVW]的计数工具，结合来自计数S5-quartic field [Bha 10]的技术，以获得Bhargava参数化中这些特殊元素上相关轨道的渐近性，并反过来计数D5-quartic环。值得注意的是，我们提出的策略应该允许我们计算D5-五次域的特殊家族，这相当于在二次域的类组中平均5-挠率（该领域的旗舰问题）。 总之，相对新生的算术统计领域继续受益于与更多经典学科的互动。我将发展这些联系，以解决该领域最深刻的问题。在这样做的过程中，我的研究计划将解开算术对象在家庭中的行为，使我们能够走向算术统计的凝聚力理论。