Automorphic Forms and L-functions: P-adic Aspects and Applications

这个研究项目主要涉及数论的主题，数论是数学最古老的分支之一。 数论涉及整数的性质，如因子分解和相关主题。 在这个项目中，研究人员将联合收割机几何，数学的另一个经典分支，与数论，使用深刻的几何技术与素数的性质一起证明数论的新结果，特别是通过研究某些家庭的数论数据。 本研究项目中的技术建立在研究人员先前研究中发挥关键作用的技术基础上，以及在数学领域最近一些最大突破中至关重要的技术，例如费马大定理的证明。该项目的其他组成部分深入研究了该领域令人兴奋的新领域。 该项目侧重于四个密切相关的主题：L-函数，自守形式，微分算子和相关的几何问题。 大多数的研究关注的p进方面，但调查员也将工作的一些纯粹的阿基米德问题，已经出现在她以前的工作背景下对p进插值的价值爱森斯坦系列和L-函数。 计划中的工作主要在数论中产生影响，但也在表示论中产生影响。 p-adic L-函数是本研究计划的一部分，它在岩泽理论中起着关键作用，岩泽理论是一种研究算术数据的p-adic理论，例如类群和伽罗瓦表示。 本研究计划中的几何问题也将有助于理解p-adic L-函数，伽罗瓦表示及其在岩泽理论中的作用。

This research project primarily concerns topics in number theory, one of the oldest branches of mathematics. Number theory is concerned with properties of whole numbers, such as factorization and related topics. In this project, the investigator will combine geometry, another classical branch of mathematics, with number theory, using deep geometric techniques together with properties of prime numbers to prove new results in number theory, in particular by studying certain families of number theoretic data. The techniques in this research project build on ones that have played a key role in the investigator's prior research, as well as ones that have been essential in some of the largest recent breakthroughs in mathematics, such as the proof of Fermat's Last Theorem. Other components of the project delve into exciting new areas in the field. This project focuses on four closely connected topics: L-functions, automorphic forms, differential operators, and related geometric problems. Most of the research concerns p-adic aspects, but the investigator will also work on some purely Archimedean problems that have arisen in the context of her previous work on p-adic interpolation of values of Eisenstein series and L-functions. The planned work has consequences primarily in number theory but also conjecturally in representation theory. The p-adic L-functions that form part of this research program conjecturally play a key role in Iwasawa theory, a p-adic theory for studying arithmetic data, such as class groups and Galois representations. The geometric problems in this research program also will contribute to the understanding of p-adic L-functions, Galois representations, and their role in Iwasawa theory.