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

该研究项目主要涉及数论主题，数论是数学最古老的分支之一。 数论涉及整数的性质，例如因式分解和相关主题。 在这个项目中，研究人员将把几何学（数学的另一个经典分支）与数论结合起来，使用深度几何技术和素数的性质来证明数论的新结果，特别是通过研究某些数论数据族。 该研究项目中的技术建立在研究者先前研究中发挥关键作用的技术以及在数学领域近期一些最大突破（例如费马大定理的证明）中至关重要的技术的基础上。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。 该项目重点关注四个密切相关的主题：L 函数、自同构形式、微分算子和相关的几何问题。 大多数研究涉及 p 进数方面，但研究者还将研究一些纯粹的阿基米德问题，这些问题是在她之前对爱森斯坦级数和 L 函数值的 p 进插值工作中出现的。 计划中的工作主要在数论中产生影响，但也推测在表示论中产生影响。 据推测，构成本研究项目一部分的 p 进 L 函数在岩泽理论中发挥着关键作用，岩泽理论是一种用于研究算术数据（例如类群和伽罗瓦表示）的 p 进理论。 该研究项目中的几何问题也将有助于理解 p 进 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.