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

PI(首席调查员)将进行数论方面的研究，数论是数学的一个中心分支，与数学的许多其他领域和其他领域有着密切的联系。这项研究的重点是在先验的不同现象之间建立桥梁，以帮助提高对几何和代数数据族的理解。预期的结果将使几个悬而未决的问题和悬而未决的猜想的解决取得实质性进展，这些猜想涉及数字的模式、相关结构中出现的对称性以及相关对象的行为。作为该项目的一部分，PI将开发工具来提高社区对至关重要的现象的理解。该项目的范围包括几何、代数和更远的领域。国际和平协会还将开展外联和教育活动，将其工作的影响扩大到远远超出研究界的范围。这些活动，包括结合艺术方法的活动，将促进学生和更广泛的公众对核心数学主题的积极参与。PI的研究将专注于自同构表和L-作为工具，以促进对算术数据家族行为的了解。这项研究的主要目的是证明关于它们的代数和p-进行为的新结果，特别是在酉群和辛群的背景下。主要内容包括证明特定的朗兰兹L-函数临界值的代数性结果，构造新的p-进L-函数来插值这些临界值，建立高秩群上p-进和正特征自同构型的性质，以及研究与Maass-Shimura微分算子相关的某些微分算子。作为关键的一步，PI还将开发与其作品中定义自同构形的空间相关联的几何基础设施。预期的结果包括向Deligne关于L函数临界值的猜想，关于p-进行为的岩泽-格林伯格猜想，以及关于Galois表示的Serre猜想的更高阶类似的实例的进展。这些方法连接了几个不同的观点，包括分析、几何和代数技术。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.