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。 第一种是使用 这些GL（n）Poincare级数来获得关于GL（n）的信息 自守形式（L-函数的零点，大筛选系数 不等式）和GL（n）Kloosterman和的和。 第二 研究GL（n）Kuznietsov公式 和另外两个迹公式：Selberg迹公式和 相对迹公式 三是研究庞加莱级数 (and L-函数）上的特殊群。 四是 仅用a，得到了GL（3）扭的强匡威定理 有限数量的字符和显式水平。 第五 是研究两个乘积的谱分解 自守形式作为理解的第一步 两个自守的张量积的分解 表示为不可约分量， 赫克代数 这项研究将集中在一个最活跃， 解析数论中最重要的领域，即 自守形式理论 将这个有价值的工具推广到 更高的维度被证明是既困难又有益的。 数论中的许多问题都是以这种方式回答的。 私家侦探已经成为这方面的专家， 毫无疑问会引起很多人对这笔赠款的兴趣 期

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.