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（首席研究员）将开展数论研究，数论是数学的一个核心分支，与许多其他数学领域及其他领域有着深厚的联系。该研究的重点是在先验的不同现象之间建立桥梁，以帮助提高对几何族的理解。预期结果将在解决有关数字模式、相关结构中出现的对称性以及相关对象的行为的几个悬而未决的问题和未解决的猜想方面取得实质性进展。开发工具来提高社区对至关重要的现象的理解，该项目的影响范围包括几何、代数等。PI 还将开展外展和教育活动，将其工作的影响力扩大到研究界之外。这些活动，包括融入艺术方法的活动，将促进学生和更广泛的数学公众积极参与核心主题，PI 的研究将集中于自守形式和 L 函数作为工具，以增进对算术族行为的了解。数据。该研究的主要目标是证明有关其代数和 p-adic 行为的新结果，特别是在酉群和辛群的背景下，关键组成部分包括证明特定 Langlands L 函数临界值的代数性结果，构造新的。 p-adic L-函数对这些临界值进行插值，在较高阶群上建立 p-adic 和正特征自守形式的性质，并研究与 Maass-Shimura 微分算子相关的某些微分算子。 PI 还将开发与她的工作中定义自守形式的空间相关的相关几何基础设施。预期的结果包括关于 L 函数临界值的德利涅猜想、关于 p 的岩泽-格林伯格猜想的实例的进展。 adic 行为，以及塞尔关于伽罗瓦表示的猜想的更高级别的类似物。这些方法弥合了几种不同的观点，包括分析、几何和代数技术。该奖项反映了这一点。通过使用基金会的智力价值和更广泛的影响审查标准进行评估，NSF 的法定使命被认为值得支持。