Kloosterman Sums and Related Exponential Sums

叶9701225本奖资助叶教授从事Kloosterman和的研究。经典的Kloosterman和在数论中有重要的应用。一个例子是库兹涅佐夫对Kloosterman和的加权和的估计，这是向林尼克-塞尔伯格猜想迈出的一步；证明是基于库兹涅佐夫迹公式的。广义Kloosterman和出现在各种相关的迹公式中；这种相对迹公式实际上是广义库兹涅佐夫迹公式与它的一个相对版本之间的等式。上述Kloosterman和的出现表明了它们在数论和群表示理论中的核心作用。本研究项目以Kloosterman和及相关指数和、它们的推广及应用为中心。目标列在下面。(1)朗兰兹泛函猜想预测了不同群的表征之间的泛函关系。如果一个泛函提升可以用一个相对的迹公式来研究，那么它的基本引理可以简化为Kloosterman型指数和的恒等式。第一个目标是研究具体的提升问题，并推导相应的指数和恒等式。（ii）反过来说，这种同一性不仅包含相对迹公式的基本引理，而且根据最近关于相对迹公式局部轨道积分的Shalika胚根展开和指数求和展开的结果，可以用来推导整个相对迹公式。第二个目标是从指数和的恒等式推导出一个相对的迹公式。（iii）经典Kloosterman和的重要应用通常基于对其值和加权和的估计。最后一个目标是估计广义Kloosterman和及相关的高阶指数和的某些加权和。这个建议属于数学中被称为朗兰兹纲领的部分。这个程序代表了数论和表示理论的一个分支，它已经刺激了这两个领域最近的大量研究。数论是数学最古老的分支之一，它关注的是最基本的数学对象——普通整数。然而，事实证明，为了表达数学家发现的许多模式和关系，有必要使用20世纪数学中一些最先进和最具技术性的理论。另一方面，数论的问题为该学科其他不同部分的研究提供了强大的刺激。Langland的程序提供了一个框架，用于使用无限维表示理论的工具来研究和广泛推广所谓的数论的互易律。虽然非常技术性和深度，这个程序在理论计算机科学（构建扩展图）和编码理论（寻找最佳的Goppa代码）等领域发现了惊人的应用。它也在最近数论本身的一些惊人发展中发挥了作用，比如费马大定理的证明。

Ye 9701225 This award funds research of Professor Ye, who works on Kloosterman sums. The classical Kloosterman sum has important applications in number theory. An example is Kuznetsov's estimate of a weighted sum of Kloosterman sums which is a step toward the Linnik--Selberg conjecture; the proof is based on the Kuznetsov trace formula. Generalized Kloosterman sums appear in various relative trace formulas; such a relative trace formula is indeed an equality between a generalized Kuznetsov trace formula and one of its relative versions. The above appearances of the Kloosterman sums suggest their central role in number theory and group representation theory. This research project is centered at Kloosterman sums and related exponential sums, their generalizations, and their applications. The objectives are listed below. (i) The Langlands functoriality conjecture predicts functorial relationships between representations of different groups. If a functorial lifting can be studied by a relative trace formula, its fundamental lemma may be reduced to an identity of exponential sums of the Kloosterman type. The first objective is to study specific lifting problems and deduce the corresponding identities of exponential sums. (ii) Conversely, such an identity not only will imply the fundamental lemma of the relative trace formula, but also might be used to deduce the whole relative trace formula, based on recent results on Shalika germ expansions and exponential sum expansions of local orbital integrals of relative trace formulas. The second objective is to deduce a relative trace formula from its identity of exponential sums. (iii) Important applications of the classical Kloosterman sum are usually based on estimation of its values and weighted sums. The last objective is to estimate certain weighted sums of generalized Kloosterman sums and related high ranking exponential sums. This proposal is in the part of mathematics known as the Langlands program. This program represents a fus ion of Number Theory and Representation Theory , and it has been a stimulus to a great deal of recent research in both fields. Number Theory is one of the oldest branches of mathematics and is concerned with the most basic of mathematical objects, the ordinary whole numbers. However, it turns out that in order to express many of the patterns and relations discovered by mathematicians, it is necessary to use some of the most advanced and technical theories of twentieth century mathematics. On the other hand, the problems of number theory have provided a powerful stimulus to research in other diverse parts of the discipline. The Langland's program provides a framework for investigating and vastly generalizing the so-called reciprocity laws of number theory using the tools of infinite-dimensional representation theory. Although very technical and deep, this program has found astonishing applications in areas like theoretical computer science (construction of expanding graphs) and coding theory (finding optimal Goppa codes). It has also played a role in some of the recent spectacular developments in number theory itself, such as the proof of Fermat's Last Theorem.