Mathematical Sciences: Poincare series and Automorphic Forms

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This research will concentrate on Poincare series on GL(n). Five specific projects will be undertaken. The first is to use these GL(n) Poincare series to obtain information on GL(n) automorphic forms (zeroes of L-functions, large sieve coefficient inequalities) and on sums of GL(n) Kloosterman sums. The second is to study the relation between this GL(n) Kuznietsov formula and two other trace formulas: the Selberg trace formula and the relative trace formula. The third is to study Poincare series (and L-functions) on the exceptional groups. The fourth is to obtain strong converse theorems on GL(3) twisting by only a finite number of characters and with explicit level. The fifth is to study the spectral decomposition of the product of two automorphic forms as a first step towards understanding the decomposition of the tensor product of two automorphic representations into irreducible components under the action of the Hecke algebra. This research will concentrate on one of the most active and most important areas of analytic number theory, namely, the theory of automorphic forms. Generalizing this valuable tool to higher dimensions has proven to be both difficult and rewarding. Many problems in number theory have been answered in this manner. The P.I. has established himself as an expert in this area and will no doubt produce much of great interest in this grant period.