Mathematical Sciences: The Arithmetic and Analysis of Automorphic Forms

本研究分为三个部分。 第一部分 关注作为椭圆的行列式出现的自守形式， 某些流形上的算子 获得GL（n）类似物的方法 theta和其他椭圆函数的算术 兴趣将被研究。 还将研究辛 最近在数学物理中发现的自守形式 以及高阶空间中的Selberg zeta函数。 在 第二部分，余有限的尖点型的存在性问题， SL（2，R）的子群，特别强调Hecke族 将研究群体。 在这个地区， 算术群的理论刻画以及 区分一致性和非一致性的标准 将研究亚组。 本研究的第三部分 将致力于GL（n）上尖点形式之间的相互作用， Kloosterman总结。 可能的联系在某些 自同构对象，如Kloosterman zeta函数和Hecke 多项式和某些无穷和和产品附加到 将描述代数几何中的Kloosterman和。 本研究将集中在自守理论 forms. 这些是复杂的解析函数， 对数论的深刻见解 他们把基本数论 信息，使现代分析的强大工具， 有助于解决数论中的基本问题。 这方面 这是当今数学中最重要的一个 年轻的调查员已经做了很多重要的工作 在它。进一步的重要成果肯定会导致这一点 目前的赠款。

This research is divided into three parts. The first part concerns automorphic forms that arise as determinants of elliptic operators on certain manifolds. Ways of obtaining GL(n) analogs of theta and other elliptic functions which are of arithmetic interest will be studied. Also studied will be symplectic automorphic forms recently discovered in mathematical physics as well as Selberg zeta functions in higher rank spaces. In the second part, the existence problem of cusp forms for co-finite subgroups of SL(2,R) with special emphasis on the family of Hecke groups will be studied. In this area a conjectural spectral theoretic characterization of arithmetic groups as well as a criterion that should distinguish congruence and non-congruence subgroups will be investigated. The third part of this research will be devoted to the interplay between cusp forms on GL(n) and Kloosterman sums. Possible connections between certain automorphic objects such as Kloosterman zeta functions and Hecke polynomials and certain infinite sums and products attached to Kloosterman sums in algebraic geometry will be described. This research will concentrate on the theory of automorphic forms. These are complex analytic functions which lead to great insights into number theory. They encode basic number theoretic information, allowing the powerful tools of modern analysis to help solve fundamental problems in number theory. This area is one of the very most important in mathematics today and this young investigator has done much already done much of importance in it. Further important results will surely result from this current grant.