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

这个研究项目主要关注数论的主题，数论是数学最古老的分支之一。数论关注整数的性质，如因数分解和相关主题。在本项目中，研究者将把数学的另一个经典分支几何学与数论结合起来，利用深刻的几何技术和素数的性质来证明数论中的新结果，特别是通过研究某些数论数据族。本研究项目中的技术建立在研究者之前的研究中发挥关键作用的技术基础上，以及在最近一些最大的数学突破中发挥关键作用的技术，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。这个项目集中在四个紧密相连的主题：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进方面，但研究者也将研究一些纯粹的阿基米德问题，这些问题是在她之前关于爱森斯坦级数和l函数值的p进插值的工作中产生的。计划工作的结果主要是在数论，但也推测在表示理论。构成本研究计划一部分的p进l函数推测地在Iwasawa理论中发挥关键作用，Iwasawa理论是研究算术数据（如类群和伽罗瓦表示）的p进理论。本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Iwasawa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.