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.
