Mathematical Sciences: The Arithmetic and Analysis of Automorphic Forms

本研究分为三个部分。第一部分讨论在某些流形上作为椭圆算子行列式出现的自同构形式。我们将研究如何获得GL(n)和其他具有算术意义的椭圆函数的类似函数。还将研究最近在数学物理中发现的辛自同构形式以及高秩空间中的Selberg zeta函数。第二部分研究了SL（2，R）的共有限子群的尖形的存在性问题，重点研究了Hecke群族。在这一领域，我们将研究算术群的推测谱理论表征以及区分同余子群和非同余子群的判据。本研究的第三部分将致力于研究GL(n)和Kloosterman和上的尖形之间的相互作用。将描述代数几何中某些自同构对象（如Kloosterman zeta函数和Hecke多项式）与Kloosterman和的无穷和和积之间的可能联系。本研究将集中于自同构形式的理论。这些都是复杂的分析函数，对数论有着深刻的见解。它们编码基本的数论信息，允许强大的现代分析工具帮助解决数论中的基本问题。这个领域是当今数学界最重要的领域之一，这位年轻的研究者已经在这个领域做了很多重要的工作。目前的赠款肯定会产生进一步的重要成果。

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.