Mathematical Sciences: Sums of L-functions, the Metaplectic Group, and Non-Generic Representations

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Friedberg 9531957 This investigation will deal with four main directions of research. First, the principal investigator proposes to continue his work, joint with D. Bump and J. Hoffstein, to obtain information about automorphic L-functions through the systematic study of certain naturally occurring Dirichlet series in two complex variables. These L-functions are not themselves Euler products, but their individual coefficients are Eulerian. These series arise as integrals of Rankin-Selberg type, possessing meromorphic continuation and functional equation. The integrals may be analyzed by local representation- theoretic methods, and the Dirichlet series coefficients related to L-functions. The properties of the integral are then used to obtain properties of the L-functions. The principal investigator will investigate integrals on certain orthogonal and symplectic groups and on their metaplectic covers, which should give analytic information concerning such objects as twists of GL(2) automorphic forms by cubic characters, mean squares of twists by ideal class characters of quadratic extensions, and the nonvanishing and mean size of quadratic twists of the standard L-function associated to an automorphic representation on GL(3).The study of more general sums of L-functions, involving more than two complex variables, is also anticipated. Second, the principal investigator proposes to investigate the Euler products associated with higher degree metaplectic automorphic forms. The existence of such Euler products is predicted by the hypothetical correspondence between metaplectic forms and non-metaplectic ones; however, they have only been exhibited in a few cases. There are two such Euler products known on covers of GSp(4), one on the double cover due to the principal investigator and Wong, and the second on the triple cover due to the principal investigator's student T. Goetze. The principal investigator proposes to first give these works less computational foundations by connecting them to the theory of non-unique models presented by Rallis and Piatetski-Shapiro, and then to use this new approach to generalize them to covers of degree higher than 3, and to groups other than GSp(4). This work will probably be joint with D. Bump. Third, the principal investigator will continue to work on the relative trace formula. This formula, which combines period considerations arising from integral expressions of $L$-functions with Langlands functoriality, should ultimately allow one to establish in many cases that L-packets contain generic members.In a recently completed massive project, the principal investigator and Jacquet have proved the fundamental lemma for one such formula. Fourth, in recent work with D. Goldberg, the principal investigator has begun to use certain models which are not Whittaker models to deal directly with non-generic representations on orthogonal and unitary groups. It is proposed to use these models to establish both local results (e.g. Langlands conjecture on Plancherel measure) and the global continuation of many L-functions arising from Eisenstein series a la Langlands-Shahidi, even for non-generic representations of these groups. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.