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

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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.