The Rankin-Selberg Method, Zeros of Special Functions and Models of Representations Over Finite Fields

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This award provides funding for the investigator's continued research into the Rankin-Selberg method. Much of this work will likely be joint work. The research will likely shift from the search for new Rankin-Selberg integrals to extracting more information from existing Rankin-Selberg constructions. An initial problem will be the nonvanishing of the archimedean local integrals for the symmetric cube L-functions on GL(2). Friedberg, Hoffstein, and Bump have been exploring reflection groups coming from double Dirichlet series with automorphic L-functions on GL(n) and quadratic characters. The corresponding question for GL(2) and cubic characters will be studied. Other problems on automorphic forms will be investigated. 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.