Automorphic forms and L-functions

Ohio State University, The, Columbus OH

Investigators

Abstract

The proposer will investigate several questions in analytic number theory related to automorphic forms. One of these will be the subconvexity problem, in different aspects and settings, for the Rankin-Selberg convolution L-functions associated with holomorphic and Maass Hecke eigen cusp forms. The novelty here is that both forms in the convolution will contribute to the size of the analytic conductor and therefore to the complexity of the problem. For example, consider the case of two forms with varying co-prime levels. The goal then is to establish subconvexity bounds for these L-functions in both levels. Furthermore, this goal is meant to be achieved without the use of an amplifier when the levels vary at distinguishable rates. The main idea is to take advantage of the choice in "averaging family" as one has several spectrally complete families to choose from. This idea is quite general and carries over into higher degree L-functions as well. One might even then hope to establish subconvexity bounds for the triple product L-function appearing in Watson type formulas when all involved forms are varying at certain relative rates. A result which could certainly be linked with interesting applications. The proposer will also study sup-norms of GL(3) Maass cusp forms, extending the work of Iwaniec and Sarnak in the GL(2) case. L-functions are functions constructed out of arithmetic information encoded in objects which often have a beautiful natural structure. Many questions regarding their analytic properties, like the Riemann Hypothesis, remain open. However, the simple fact that these L-functions are built out of local data associated with natural objects means that some exciting formulas relating deep questions in science to special values of L-functions have been established. For example, such L-functions have appeared in formulas related to questions in quantum chaos (the study of the relations between quantum mechanics and classical chaos) and have been a crucial tool in the recent resolution of the Quantum Unique Ergodicity Conjecture of Rudnick and Sarnak. Motivated by the significance of these L-functions, the proposer will investigate and analyze a new class of L-functions which may also have a natural interpretation in terms of dynamics and could lead to new and interesting questions in science and nature.