Mathematical Sciences: Poincare series and Automorphic Forms

本文主要研究GL(N)上的Poincare级数。将开展五个具体项目。第一个是利用这些GL(N)Poincare级数得到关于GL(N)自同构型(L函数的零点，大筛子系数不等式)和GL(N)Klosterman和的信息。二是研究了该GL(N)Kuznietsov公式与另两个迹公式：Selberg迹公式和相对迹公式之间的关系。三是研究例外群上的Poincare级数(和L函数)。四是得到了GL(3)有限特征标显水平扭曲的强逆定理。第五是研究两个自同构形式的乘积的谱分解，作为理解两个自同构表示的张量积在Hecke代数作用下分解成不可约分量的第一步。这项研究将集中在解析数论中最活跃和最重要的领域之一，即自同构形理论。事实证明，将这一有价值的工具推广到更高的层面既困难又有益。数论中的许多问题都以这种方式得到了解答。私家侦探已经确立了自己在这一领域的专家地位，毫无疑问，他将在这一赠款期间产生极大的兴趣。

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.