NSF Award
1503629
This project features two lines of research at the interface of mathematical analysis, the study of continuous change, and number theory, the study of properties of integers borne out of the notion of divisibility. The finer distribution of prime numbers, such as whether they end in certain blocks of digits more often than others, is captured by highly structured oscillating sequences known as Dirichlet characters, which give rise to the associated L-functions. This project will prove analytic results about certain special values of L-functions and sums of the values of Dirichlet characters, and it will specifically investigate the impact of the underlying modulus being composed of many smaller factors on these results and on the tools available to prove them. As a prototype of the other class of objects to be studied, signals and motions (such as light waves or vibrations of a string) are often much better understood when viewed as combinations, or superpositions, of simple periodic motions. The wave-like functions that analogously serve as building blocks of analysis on other spaces are known as eigenfunctions and are central in disciplines ranging from spectral geometry to quantum mechanics. This project will investigate the behavior of rapidly oscillating eigenfunctions on spaces with a rich set of symmetries that are arithmetic in nature, in particular how pronounced are their extreme values.<br/><br/>This research project centers around two principal themes, that of extremal behavior of high-energy eigenfunctions on arithmetic manifolds and that of the depth and smooth aspects in analytic number theory. On certain arithmetic manifolds with a specific geometric and functorial structure, the joint eigenfunctions (Hecke--Maass eigenforms) exhibit power growth, which is neither generically expected nor predicted by physical models. The PI will seek out extremal growth and investigate in detail the sup-norm and restriction norm problems on several specific classes of arithmetic manifolds to inform general conjectures and understanding of the phenomenon of concentration of mass on arithmetic manifolds, the precise structure that drives it, and its place within the framework of the correspondence principle of quantum mechanics. In number-theoretic problems involving characters and automorphic forms of large level, the depth and smooth aspects, which are concerned with highly powerful or factorable conductors, play a very distinctive role. The structural impact of the powerful or factorable structure on nonvanishing, subconvexity, and moments of L-functions, as well as exponential sums involving p-adically analytic fluctuations will be studied using non-archimedean analysis, analytic number theory, and spectral theory.
