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围绕算术对象（例如数字字段及其类组）的统计问题上。该主题的中心原则是 Cohen-Lenstra 启发式 [CL1, CL2]，它预测数域族中类群的分布（p 部分），以及关于数的渐近行为的 Malle 猜想 [Mal1, Mal2]特定伽罗瓦类型的域。算术统计领域为解决数论中此类经典且重要的问题提供了一个粗略的蓝图，但要在最有趣的情况下遵循这一点，需要比迄今为止所存在的更复杂的代数和分析工具应用。在我的研究中，我尝试结合这些方法来解决长期抵制攻击的问题。现在，我总结了我正在进行和提议的研究方向中最重要的部分。 在即将与 Shankar [SV] 合作的工作中，我们利用狄利克雷双曲线方法与传统算术统计技术相结合的新工具来计算数字域。我们证明了伽罗瓦八次场的马勒猜想，并且能够精确地确定 D4 八次场的渐近常数。接下来我们将致力于标准化该策略，以证明 2 群的 Malle 猜想的其他杰出案例。 最近，Bhargava-Shnidman [BS] 使用固定的二次 Hessian 协变对立方域进行了计数。类似地，四次域有一个相关的协变，该协变源自其整数环晶格（无迹部分）上的迹形式。通过在这个二次协变上纤维化四次场，我应该能够利用最近开发的方法来计算仿射同质簇[EMS，DRS]上的点，并且我希望能够计算四次场的各种薄族，包括，最值得注意的是，按判别式排序的 A4 四次域族。 最雄心勃勃的是，通过与 Altug、Shankar 和 Wilson 的合作，我们正在努力扩展研究 D5 五次场系列的方法。 我们计划使用 D4-四次 [ASVW] 的计数工具与 S5-五次场 [Bha10] 计数技术相结合，以获得 Bhargava 参数化中这些特殊元素的相关轨道的渐近线，然后对 D5-五次环进行计数。值得注意的是，我们提出的策略应该允许我们计算 D5 五次场的特殊族，这相当于对二次场类组中的 5 扭转进行平均（该领域的旗舰问题）。 总之，相对新兴的算术统计领域正在继续受益于与更多经典学科的互动涌入。我将发展这些联系，以解决该领域最深层次的问题。在此过程中，我的研究计划将揭示算术对象在族中的行为，以便我们能够朝着算术统计的内聚理论迈进。