L-Functions and Automorphic Forms: Algebraic and p-adic Aspects

L 函数和自守形式：代数和 p 进方面

• 批准号: 2302011
• 负责人: Ellen Eischen
• 金额: $ 25万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302011&HistoricalAwards=false
• 关键词: Functions; Automorphic; Forms; Algebraic; adic

The PI (Principal Investigator) will conduct research in number theory, a central branch of mathematics with deep ties to many other areas of mathematics and beyond. The research focuses on building bridges between a priori disparate phenomena, to help improve understanding of families of geometric and algebraic data. Anticipated outcomes will enable substantial progress toward resolution of several open questions and unresolved conjectures about patterns in numbers, symmetries arising in associated structures, and behavior of related objects. As part of the project, the PI will develop tools to improve the community’s understanding of phenomena that are of central importance. The project’s reach includes geometry, algebra, and beyond. The PI will also carry out outreach and educational activities that will expand the impact of her work well beyond the research community. These activities, including ones incorporating approaches from the arts, will promote active engagement with core mathematical topics among both students and the broader public. The PI’s research will focus on automorphic forms and L-functions as tools to advance knowledge about behavior of families of arithmetic data. The main objective of the research is to prove new results about their algebraic and p-adic behavior, especially in the context of unitary and symplectic groups. Key components include proving algebraicity results for critical values of particular Langlands L-functions, constructing new p-adic L-functions interpolating those critical values, establishing properties of p-adic and positive characteristic automorphic forms on higher rank groups, and investigating certain differential operators related to Maass—Shimura differential operators. As a crucial step, the PI will also develop associated geometric infrastructure tied to the spaces over which the automorphic forms in her work are defined. Anticipated consequences include progress toward instances of Deligne’s conjecture about critical values of L-functions, the Iwasawa—Greenberg conjectures about p-adic behavior, and higher rank analogues of Serre’s conjectures about Galois representations. The methods bridge several different viewpoints and include analytic, geometric, and algebraic techniques.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.

PI（主要研究者）将在数字理论上进行研究，这是数学的中心分支，与许多其他数学领域及其他领域有着深厚的联系。该研究重点是在先验不同现象之间建造桥梁，以帮助提高对几何和代数数据家庭的理解。预期的结果将能够在解决数字数字，对称性的几个开放问题和未解决的猜想方面取得实质性进展，作为项目的一部分，对称性产生的对称性，PI将开发工具，以提高社区对至关重要的现象的理解。该项目的影响力包括几何，代数等。 PI还将进行外展和教育活动，以扩大其工作的影响，远远超出了研究界。这些活动，包括艺术的越来越多的方法，将促进学生和更广泛的公众的核心数学主题积极参与。 PI的研究将集中于自动形式和L功能，作为促进有关算术数据家族行为的知识的工具。该研究的主要目的是证明其代数和P-Adic行为，尤其是在单一和符号群的背景下。关键组成部分包括为特定兰兰斯L功能的关键值提供代数结果，建造新的P-ADIC L功能插入这些关键价值，建立了对较高等级组的P-ADIC和积极特征性自动形式的特性，并研究与Maass-Maass-Shimura-Shimura差异运营商相关的某些差异操作员。作为关键步骤，PI还将开发与定义工作中自动形式的空间相关的相关几何基础架构。抗后果包括朝着deLigne关于L功能临界价值的概念的进展，iWasawa的iWasawa - greenberg关于P-ADIC行为的概念以及Serre关于Galois表示的概念的较高等级类比。这些方法桥接了几个不同的观点，包括分析，几何和代数技术。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被认为是通过评估来获得的支持。