Automorphic Forms and L-Functions

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This research project concerns number theory. It focuses on automorphic forms---functions that are invariant under a large discrete group of symmetries. When the order of applying the symmetries does not matter, such functions may be studied by Fourier analysis, discovered in the nineteenth century; however, for more complicated symmetries new ideas are needed and our knowledge is far from complete. The fundamental Langlands Functoriality Conjectures predict that highly symmetric functions are the keys to understanding solutions to polynomial equations, making a bridge between the continuous (functions) and the discrete (solutions). A specific case of the conjectures played a key role in Wiles's proof of Fermat's Conjecture, but most cases of the Langlands Conjectures are still unproved. This project will provide new information about automorphic forms, including functoriality, and about quantities in number theory and in other areas of mathematics related to automorphic forms.In more detail, this project includes problems related to functoriality, L-functions, and covering groups. In one series of problems, the principal investigator proposes to give applications of, and to generalize, a new recent construction, the twisted doubling integral, which makes it possible to analyze L-functions for non-generic automorphic forms. A second series of problems concerns automorphic forms on general covering groups, the metaplectic groups. The principal investigator proposes to give broad examples of the principle that constructions involving automorphic forms on linear groups (or the double covers described by Weil) can be extended to general covering groups. A third set of problems concerns new constructions involving automorphic forms, representation theory, and number theory, including the study of functionals on p-adic groups related to Iwahori-Hecke algebras and quantum groups. This research will advance our knowledge of automorphic forms on reductive groups and their covers, with consequences for number theory, representation theory, and string theory.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.