Explicit Moduli Problems and Arithmetic Statistics

在数学中研究复杂对象的一种方法是计算对象的简单“不变量”，它表达了关于对象的一些（但不完整）信息。该项目研究与多项式方程组相关的对象所产生的不变量，以及当考虑这些对象的大家族时，某些不变量如何分布。例如，如果不变量是一个整数，人们可能会问这个整数有多大可能性，比如说，对于一个“随机”对象来说，是零。这个项目涉及使用代数几何中的模空间的显式构造的问题，以及这些参数化的应用，以获得有关算术不变量的统计信息，使用的方法从经典的代数几何构造到李理论和解析数论的筛选技术。PI建议致力于寻找代数群表示的轨道与曲线、阿贝尔簇和其他簇的模空间之间的对应关系。 PI还计划使用这些对模空间的明确描述，沿着来自数的几何和解析数论的技术，来研究与参数化相关的算术数据的渐近增长。 这些应用包括研究数域的理想类群的分布，界定椭圆曲线的平均秩，以及找到具有给定点数的曲线的密度。

One way to study complicated objects in mathematics is to compute simpler "invariants" of the object, which express some (but not complete) information about the object. This project studies invariants arising from objects associated to sets of polynomial equations, and how certain invariants are distributed when considering large families of these objects. For example, if the invariant is an integer, one may ask how likely the integer is, say, zero for a "random" object. Such results then translate into a better understanding of the initial equations and their behavior.This project involves problems using explicit constructions of moduli spaces in algebraic geometry, as well as applications of these parametrizations to obtain statistical information about arithmetic invariants, using methods ranging from classical algebraic geometry constructions to Lie theory and sieve techniques from analytic number theory. The PI proposes to work on finding correspondences between orbits of representations of algebraic groups and moduli spaces of curves, abelian varieties, and other varieties. The PI also plans to use these explicit descriptions of the moduli spaces, along with techniques from the geometry of numbers and analytic number theory, to study the asymptotic growth of arithmetic data related to the parametrizations. Such applications include studying the distributions of the ideal class groups of number fields, bounding the average rank of elliptic curves, and finding densities of curves with a given number of points.