Local-Global Principles in Arithmetic

算术中的局部全局原理

• 批准号: 1844206
• 负责人: Kiran Kedlaya
• 金额: $ 15万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-09-01 至 2022-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1844206&HistoricalAwards=false
• 关键词: Local; Global; Principles; Arithmetic; Local; Global; Principles; Arithmetic

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.

数字理论是数学最古老的分支之一，但是它在科学中仍然具有越来越多的应用。在该项目中，首席研究员（PI）将研究数字理论的两个主要重点之间的关系，它们都用于计算机科学以及物理学：（1）整数和（2）对多种方程式的代数解决方案。基本的想法是了解可以通过其本地作品收集来确定全球对象的程度。该项目中将使用的策略和技术源于广泛的其他数学主题，包括分析，几何，几何，代数，在某些情况下，统计数据。 PI感兴趣的一些特定问题是本科生和高中生可以解决的水平，在整个项目过程中，PI计划利用这一点继续进行教育工作，以支持数学内多样性的增加。该项目围绕着整个代数和分析数理论的局部全球原则的广泛现象，从了解整数环的独特质量分解的障碍物到确定具有固定不变性的固定不变性的渐近全球次数通过固定的padic adadic gunvariants的数量来证明固定的不变性的局部性局部性局部范围的局部性综合性。首先，PI将进行研究，以进一步发展起源于Cohen-Lenstra启发式方法的课堂群体的统计研究。除其他外，这将包括证明对命令家庭阶级群体的渐近学。其次，PI将研究可以控制其渐近药的局部全球原则，从八粒四元数字段开始。获得此类结果的策略将依赖于算术不变理论，筛分方法和数量的数量技术在算术统计领域经常使用。在自态方面，PI将研究由CM场上一般线性组的非偶联的自偶常规代数自动形式表示，在CM领域的非偶联的自偶常规代数自动形式表示引起的GALOIS表示的P-ADIC家族的算术和几何特性。这将涉及研究特征值并加强P-ADIC插值方法。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响评估标准通过评估来支持的。