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.
The PI successfully investigated several questions in Analytic Number Theory related to automorphic forms and their L-functions. In short, Number Theory is the study of arithmetically and alegbraically rich objects and structures (e.g. automorphic forms). In Analytic Number Theory, one often first decomposes such arithmetic structure into its fundamental components (or "building blocks") and then encodes that information into objects amenable to Harmonic Analysis. Proving something non-trivial about the analytic object will usually imply something non-trivial about the arithmetic structure. 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. One of the problems studied in this proposal was the subconvexity problem, in different aspects and settings, for the Rankin-Selberg convolution L-functions associated with holomorphic and Hecke-Maass eigen cusp forms. In particular, the PI was interested in hybrid subconvexity results when multiple forms are contributing to the size of the analytic conductor and therefore to the complexity of the problem. The proposer also studied sup-norms of GL(3) Hecke-Maass cusp forms, extending the work of Iwaniec and Sarnak in the GL(2) case. The main ideas that were further developed as a result of this research were: Moment method techniques for families of L-functions Appropriate choices of averaging family The role of the arithmetic structure of the conductor in choice of family New applications of the delta-method of Duke-Friedlander-Iwaniec Analytic techniques for the study of the sup-norm problem This project led to the following outcomes and products: 6 research articles 2 Ph.D. dissertations: 1 completed and 1 nearing completion 5 research collaborations The articles are available online here: http://arxiv.org/find/math/1/au:+Holowinsky_R/0/1/0/all/0/1 As for impact, these results are shedding new light on the subconvexity problem for Rankin-Selberg convolution L-functions and providing insight on appropriate choices of family and moment methods. Furthermore, with the recent work on Dirichlet L-functions, it was demonstrated that the ``arithmetic'' structure of the conductor plays a key role in which method is appropriate for studying Rankin-Selberg convolutions. The progress on generalizing arguments of Iwaniec-Sarnak for non-trivial sup-norm estimates of classical Hecke-Maass cusp forms to the GL(3) setting is leading to a better understanding of modifying one's analysis for such higher rank groups. Finally, new research projects have been developed as a result of this funding that will continue to contribute to the collective understanding of automorphic forms, their associated L-functions and the methods and techniques of Analytic Number Theory.
