Number theory, representation theory, and arithmetic geometry

这个项目是一个正在进行的研究项目的一部分。它围绕着理解数学中出现的各种基本代数对象——如环、类群、代数曲线和变量，以及与这些结构相关的映射——是如何参数化的。我们理解这些参数化的目标至少有四个方面：1)描述这些基本代数对象是如何根据它们最基本的不变量分布的；2)发现这些对象的新不变量及其应用；3)开发高效实用的算法，对这些代数对象进行计算；也许最重要的是，4)发现和理解各种看似不同的代数结构实际上是如何彼此密切相关的。在过去的几年中，PI已经使用了这种参数化来获得关于诸如判别式和类数可除性等基本不变量的数域分布的精确信息。将精细计数方法应用于这些参数化已经导致，例如，证明了Cohen-Lenstra-Martinet类数启发式的第一个已知案例，并且这种性质的其他定理即将出现。在最近与Arul Shankar， Wei Ho和Dick Gross的合作作品中，获得了椭圆曲线和超椭圆曲线族中Selmer群的平均大小定理，并应用于理解这些曲线上有理点的分布。在不久的将来，我们期望在代数曲线和曲面的有理点、除数和其他此类数据上得到更多类似的结果。申请这项资助的主要目的是为研究生提供支持。上面描述的许多项目将以一种必不可少的方式涉及研究生（和本科生）。所有这些项目都涉及数论中非常基本的问题，是介绍和引导学生进入这门学科的绝佳方式。来自不同机构的许多其他数学家也将参与这些项目，这将增加来自许多不同但相关学科的研究人员之间的互动。

This project is part of an ongoing research program. It revolves around understanding how various fundamental algebraic objects occurring in mathematics---such as rings, class groups, algebraic curves and varieties, and maps relating such structures---are parametrized. Our goals in understanding these parametrizations are at least fourfold: 1) to describe how these fundamental algebraic objects are distributed with respect to their most basic invariants; 2) to discover new invariants of these objects, and their applications; 3) to develop efficient and practical algorithms for performing computations with these algebraic objects; and, perhaps most importantly, 4) to discover and understand how various seemingly different algebraic structures are in fact closely related to each other. Such parametrizations have been already used by the PI over the past few years to obtain precise information on the distribution of number fields with respect to basic invariants such as discriminant and class number divisibility. Applying refined counting methods to these parametrizations has led, for example, to a proof of the first known case of the Cohen-Lenstra-Martinet class number heuristics for higher degree number fields, and other theorems of this nature are forthcoming. In more recent joint works with Arul Shankar, Wei Ho, and Dick Gross, theorems on the average sizes of Selmer groups in families of elliptic curves and hyperelliptic curves have been obtained, with applications to understanding the distributions of rational points on such curves. We expect several further analogous results on rational points, divisors, and other data of this kind for algebraic curves and surfaces in the near future.The primary purpose of applying for this grant is the support of graduate students. Many of the projects described above will involve graduate (and undergraduate) students in an essential way. All these projects involve very fundamental questions in number theory, and are an excellent way to introduce and bring students into the subject. Many other mathematicians from various institutions will also be involved in these projects, which will thus increase interactions among researchers from many different but related subjects.