Arithmetic Statistics: Asymptotics on number fields and their class groups

我的大部分工作都集中在围绕算术对象（如数字字段及其类组）的统计问题上。该学科的核心原则是Cohen-Lenstra启发式[CL1， CL2]，它预测了数域族中类群的分布（p部分），以及关于特定伽罗瓦类型数域的渐近行为的Malle猜想[Mal1， Mal2]。算术统计领域为解决数论中这些经典而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原则，需要比目前所提供的更复杂的代数和分析工具的应用。在我的研究中，我试图结合这些方法来解决长期抵制攻击的问题。我现在总结一下我正在进行和提议的研究方向中最重要的一点。在即将与Shankar [SV]合作的工作中，我们将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数数域。我们证明了伽罗瓦频域的Malle猜想，并能精确地确定d4频域的渐近常数。我们下一步的工作是标准化这一策略，以证明马尔猜想的其他杰出案例。最近，Bhargava-Shnidman [BS]用一个固定的二次Hessian协变计算了三次场。类似地，四次域有一个相关的协变，由整数环的格（无迹部分）上的迹形式产生。通过对这个二次协变的四次场进行纤维化，我应该能够利用最近开发的方法来计数仿射齐次变种[EMS， DRS]上的点，并且我希望能够计数四次场的各种薄族，包括，最值得注意的是，由判判式排序的a4四次场族。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 -五次场族的方法。我们计划使用d4 -四分位数的计数工具[ASVW]结合s5 -五次场的计数技术[Bha10]来获得这些特殊元素在Bhargava参数化中的相关轨道的渐近性，进而对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 -五次域的特殊族，这相当于在二次域的类群中平均5-扭转（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经典学科的大量互动。我将发展这些联系，以解决该领域最深刻的问题。在此过程中，我的研究计划将揭示家庭中算术对象的行为，以便我们能够走向算术统计的内聚理论。

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.