Analytic Theory of Automorphic Forms and L-Functions

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This project is jointly funded by the Algebra and Number Theory program in the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR). This research project is concerned with two objects at the cornerstones of number theory: L-functions and automorphic forms. These are very special types of functions, which package a lot of information and symmetry, and for this reason are the key to solving a myriad of mathematical problems. The project will reveal new symmetries possessed by L-functions, which in turn will be useful for understanding objects such as the prime numbers. Prime numbers are essential in cryptography, a method by which computer data is securely transferred. The project will also reveal how automorphic forms are distributed. This will partially answer some open questions in Arithmetic Quantum Chaos, a multidisciplinary field at the interface of number theory and theoretical physics. The project will provide research training opportunities for graduate students. The principal investigator will also continue to be involved in an outreach program to prepare underprivileged high school students for college entrance.In more detail, the principal goals of this project are to: 1) Make progress towards the Random Wave Conjecture, which predicts that automorphic forms should behave like random waves and 2) investigate the class of reciprocity formulae for L-functions. The Random Wave Conjecture can be formulated in terms of moments of automorphic forms, with the statement for the second moment corresponding to the familiar Quantum Unique Ergodicity problem. This project will go beyond QUE, by investigating higher moments of automorphic forms, and yielding a finer understanding of the distribution of automorphic forms. The reciprocity formulae are certain exact formulae for moments of L-functions, exhibiting some surprising symmetry. An early example is Motohashi's formula, which relates the fourth moment of the Riemann Zeta function to the third moment of automorphic forms. This project will discover new examples of reciprocity formulae, which will shed light on the nature of such symmetries and yield new applications, such as subconvexity bounds for L-functions. The unifying approach for both goals will be a study of moments of automorphic forms and L-functions, using methods from the analytic theory of L-functions and the spectral theory of automorphic forms.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.