Local-Global Principles in Arithmetic

数论是数学中最古老的分支之一，但它在科学中有越来越多的应用。在这个项目中，首席研究员（PI）将研究数论的两个主要焦点之间的关系，这两个焦点都被用于计算机科学和物理学：(1)素数和整数的可整除性；(2)多项式方程的代数解。其基本思想是理解全局对象可以在多大程度上由其局部部分的集合以算术方式确定。本项目将采用的策略和技术源于广泛的其他数学学科，包括分析、几何、代数，在某些情况下还包括统计学。PI感兴趣的一些具体问题是本科生和高中生都能理解的，在项目的整个过程中，PI计划利用这一点继续在教育方面的努力，支持数学多样性的增加。本项目围绕着局部-全局原理在代数和解析数论中的广泛现象，从理解整数环中唯一素因数分解的障碍，到通过具有固定p进不变量的局部扩展的数量来确定具有固定不变量的全局域的渐近数，再到证明Langlands程序中的局部-全局相容结果。首先，PI将开展研究，进一步对源于Cohen-Lenstra启发式的阶级群体进行统计研究；除此之外，这将包括证明阶族的类群的渐近性。其次，PI将从八元四元数数域的情况开始，研究数域分布和控制其渐近性的局部-全局原理。获得此类结果的策略将依赖于算术不变量理论、筛选方法和算术统计领域中经常使用的数的几何技术。在自同构方面，PI将研究由CM域上一般线性群的非共轭自对偶正则代数自同构表示产生的伽罗瓦表示的p进族的算术和几何性质。这将包括研究特征变量和加强p进插值方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Number theory is one of the oldest branches of mathematics, and yet it continues to have more and more applications within the sciences. In this project, the principal investigator (PI) will investigate the relationship between two of the main focuses of number theory, both of which are utilized in computer science as well as physics: (1) prime numbers and the divisibility of integers and (2) algebraic solutions to polynomial equations. The fundamental idea is to understand the extent to which global objects can be arithmetically determined by the collection of its local pieces. The strategies and techniques that will be utilized in this project originate in a broad range of other mathematical subjects, including analysis, geometry, algebra and in some cases, statistics. Some of the specific questions the PI is interested in are at a level accessible to undergraduate and high-school students, and throughout the course of the project, the PI plans to utilize this to continue in educational efforts supporting an increase in diversity within mathematics. This project surrounds the widespread phenomenon of local-global principles throughout algebraic and analytic number theory, ranging from understanding obstructions of unique prime factorization in rings of integers to determining the asymptotic number of global fields with fixed invariants via the number of local extensions with fixed p-adic invariants to proving local-global compatibility results within the Langlands program. First, the PI will conduct research that furthers the statistical study of class groups that originated with the Cohen-Lenstra heuristics; amongst others, this will include proving asymptotics for class groups of families of orders. Second, the PI will study number field distributions and the local-global principles that can control their asymptotics, beginning with the case of octic quaternion number fields. The strategy for obtaining such results will rely on arithmetic invariant theory, sieve methods, and geometry-of-numbers techniques utilized frequently in the field of arithmetic statistics. On the automorphic side, the PI will investigate arithmetic and geometric properties of p-adic families of Galois representations arising from non-conjugate self-dual regular algebraic automorphic representations of the general linear group over CM fields. This will involve studying eigenvarieties and strengthening p-adic interpolation methods.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.