9700757 Hoffstein In this proposal we put forward two new techniques for the study of automorphic, Dirichlet and Hecke L-series. The first technique involves the use of double Dirichlet series, associated groups of functional equations and a convexity principle for functions of two complex variables. Our purpose is to study the properties of L-series of number theoretic interest in one variable s that arise in these double Dirichlet series as coefficients in the expansion as a Dirichlet series in another variable w. Such double series often arise in the theory of metaplectic Eisenstein series on higher rank groups. A good deal of our efforts in past years have been devoted to obtaining information about collections of L-series that appear in these double series. The main tool has been generalizations of the Rankin-Selberg method applied to metaplectic Eisenstein series and cusp forms. A major obstacle to progress, however, has been the difficulty of analyzing the archimedian factors associated to these L-series that are created by the Rankin-Selberg process. It is our belief that the technique described above should be applicable in all the cases that are theoretically accessible by the Rankin-Selberg method, but the difficulties of working at the infinite place should be avoided. We hope to use this method to prove the entirety of the symmetric fourth power L-series associated to an automorphic form on GL(2) . This would have as a consequence an improvement of the best known exponent in Fourier coefficients estimates for Maass forms from 5/28 to 1/6 . Other applications should include the entirety of the symmetric cube L-series, non-vanishing of infinitely many cubic twists of a GL(2) automorphic L-series at arbitrary points, and mean value estimates for arbitrary order twists of Hecke L-series. The second technique is connected to certain generalizations of metaplectic forms. Metaplectic forms on GL(2) transform under the action of a congruence subgroup of the fu ll modular group. In this project, the investigator and his collaborators will construct a new kind of metaplectic group with the Kubota symbol replaced by a representation of the congruence subgroup. This could then be used to construct an Eisenstein series. The investigator hopes that the Artin L-series associated to extensions of the base field with Galois group a subgroup of G will be connected to this Eisenstein series. If true, an investigation of integral transforms of the Eisenstein series could lead to new information about certain classes of Artin L-series. At the very least, an investigation of such generalizations of metaplectic forms should prove interesting, as they appear to be completely new objects. 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.
