The parameterization of algebraic structures, and applications

这个项目是正在进行的研究计划的一部分。它围绕着理解数学中出现的各种基本代数对象-例如环、类群、代数曲线和簇，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：1)描述这些基本代数对象相对于它们最基本的不变量是如何分布的；2)发现这些对象的新的不变量及其应用；3)开发高效而实用的算法来执行这些代数对象的计算；以及，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紧密相关的。在过去的几年里，PI已经使用了这样的参数化来获得关于基本不变量的数域分布的准确信息，例如判别式和类号可除性。例如，将精确的计数方法应用于这些参数化已经导致了高次数域的Cohen-Lenstra-Martinet类数启发式的第一个已知情况的证明，并且其他类似性质的定理即将出现。我们期望在不久的将来，椭圆曲线的秩会得到类似的定理，代数曲线和曲面也会得到类似的结果。

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. We expect similar theorems for ranks of elliptic curves, and other data of this kind for algebraic curves and surfaces, in the near future.