Mathematicians are trying to measure how fast certain complex functions grow as their inputs get very large—a problem that sits at the heart of number theory and mathematical physics. These functions, called L-functions, are generalised versions of the Riemann zeta function, whose zeros are the subject of the famous Riemann Hypothesis. Proving exact growth estimates (subconvexity) for these functions is notoriously difficult, so this project targets a weaker version: proving subconvexity on average for a specific family of L-functions linked to the general linear group. Success would bring mathematicians closer to resolving the full subconvexity problem, which is itself tied to the Quantum Unique Ergodicity conjecture—a problem in quantum chaos for which a Fields Medal was awarded. This is fundamental, curiosity-driven mathematics. It has no immediate practical application. However, past work on L-functions has underpinned advances in cryptography, and deeper understanding of automorphic forms has fed into theoretical physics. If successful, this project will provide new tools in analytic number theory and representation theory, potentially opening routes to future applications in areas that rely on prime numbers or high-energy waveforms.
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The proposed project focuses on the analytic and arithmetic properties of L-functions. An L-function is a far-reaching generalization of the famous Riemann zeta function. The Riemann Hypothesis, one of the Millennium Prize Problems, asks to locate the places where the Riemann zeta function becomes zero. Analogously, the Generalized Riemann Hypothesis presents a similar inquiry for general L-functions. Unfortunately, both of these problems remain unsolved given the limitations of existing technology. However, there is a relatively more tractable problem that asks for the growth estimate of the L-functions as the arguments become increasingly large. Such are commonly referred to as the subconvexity problems in literature. The subconvexity problems are profoundly interlinked with the theory of automorphic forms which have numerous connections in diverse branches of mathematics, starting from number theory, such as the distribution of prime numbers, to mathematical physics, exemplified by the equidistribution of high-energy waveforms. The project aims to prove subconvexity on average for an extremely fascinating family of L-functions. This problem is a weaker version of a subconvexity problem that is infamous for being notoriously difficult. Moreover, this subconvexity problem is closely linked with the Quantum Unique Ergodicity conjecture, a special case of which was successfully resolved by Lindenstrauss, earning him the Fields Medal in 2010. Accomplishing the goals outlined in this project will pave the way toward a resolution of the aforementioned subconvexity problem. The specific objective of the project is the asymptotic evaluation of a high moment of the standard L-functions for the general linear group with arbitrary dimensions. The project's methodology includes techniques derived from representation theory, analytic number theory, and harmonic analysis to address this problem comprehensively. The novel tools and findings of this research project will establish new avenues of exploration within the realms of both number theory and automorphic representation theory.
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