CAREER: Synergistic activities in automorphic forms and education

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). A large swath of modern number theory is concerned with the study of symmetry. One way in which symmetry arises in number theory is through Galois theory: this is the study of the symmetry of solution sets of polynomial equations over the rational numbers. Another way in which symmetry arises in number theory is through analysis: there are a class of special functions, called automorphic forms that satisfy a certain set of differential equations and possess an infinite group of discrete symmetries. The conjectural Langlands Program relates Galois theory to automorphic forms, even though on first appearance the two areas of mathematics have nothing to do with one another. This project concerns the study of automorphic forms, especially those that exhibit "exceptional" groups of symmetries. The Principal Investigator will investigate the subtle and surprising connections that automorphic forms have to arithmetic. He will also study L-functions of automorphic forms, which are generalizations of the Riemann zeta function. The grant includes funding for graduate students, who will receive training in automorphic forms. While supported by this grant, the Principal Investigator will write a graduate textbook on "exceptional algebraic structures", which will fill a need in the literature. He will also help train the US workforce through mentoring of undergraduates and early-career colleagues.This award has three related research projects. In one project, already underway, the Principal Investigator will develop the theory of half-integral weight modular forms on exceptional groups. It is expected that the Fourier coefficients of these half-integral weight modular forms will be highly interesting arithmetic quantities. The second project involves the production of modular forms on the exceptional group G_2 with rational Fourier coefficients and development of arithmetic consequences of the existence of such modular forms. This will lead to a partial database of modular forms on G_2. In a third project, the Principal Investigator will construct new automorphic L-functions. Techniques involved in these projects include the theta correspondence and the Rankin-Selberg method.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.