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

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.