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。这门课的中心原理是Cohen-Lenstra启发式[CL1，CL2]，它预测了类群在数域家族中的分布(p-部分)，以及Malle关于特定Galois类型的数域的渐近行为的猜想[Mal1，Mal2]。算术统计领域提供了一个粗略的蓝图来解决数论中的这些经典而重要的问题，但要在最有趣的情况下贯彻这一点，需要比迄今所提供的更复杂地应用代数和分析中的工具。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抵抗攻击的问题。我现在总结我正在进行的和提出的最有意义的研究方向。在即将到来的与Shankar[SV]的合作中，我们使用新的工具来计算从Dirichlet双曲线方法派生的数域，并结合传统的算术统计技术。我们证明了Galois八次场的Malle猜想，并且在D4-八次场的情况下能够精确地确定渐近常数。我们下一步的工作是标准化这一策略，以证明Malle猜想对2-群的其他突出例子。最近，Bhargava-Shnidman[BS]计算了具有固定二次Hessian协变的立方体区域。类似地，四次场在其整数环的(无迹部分)格子上的迹形式中有一个相关的协变。通过在这个二次协变上纤化四次场，我应该能够利用最近开发的方法来计算仿射齐次变种[EMS，DRS]上的点，并且我希望能够计算各种细族的四次场族，其中最值得注意的是，按判别式排序的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-Quartics[ASVW]的计数工具，结合S5-五次域[Bha10]的计数技术，在Bhargava的参数化中获得这些特殊元素上的相关轨道的渐近性，进而计算D5-五次环。值得注意的是，我们提出的策略应该允许我们计算D5-五次域的特殊族，这将相当于在二次域的类群中平均5-挠率(这是该地区的一个旗舰问题)。总而言之，算术统计学这个相对较新的领域继续受益于与更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。通过这样做，我的研究计划将解开算术对象在家庭中的行为，这样我们就可以走向算术统计的凝聚力理论。

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.