Cohomology of Exponential Sums

指数在数学中有很多用途。历史上，它们出现在数论问题中。最近，它们在密码学和编码理论中得到了应用。出现的主要问题是为指数和的绝对值找到明显的上界。基于P. Deligne关于Weil猜想的基本工作，标准的方法是计算与指数和相关的l- adicohomology群。最近，研究者和他的合作者计算了一些新的指数和类的p进上同调。计算表明，这类指数和应该有“好的”上界。研究者计划寻找更多这样的指数和类，并尝试计算它们的l进上同调，从而获得期望的上界。指数和最初出现在数论的基本问题中，比如试图找到给定方程的整数解的个数。通常很难找到这样的解的确切数量，所以下一个最好的办法就是近似这个数字。人们发现，这个问题通常可以简化为估计某些复数和的大小的问题，这些和被称为“指数和”。一个实质性的理论已经发展了多年来处理这个问题，它已经找到了现代应用领域的密码学和编码理论。研究者发现了一些新的指数和，他认为这些指数和是可以估计的。他将设法扩展现有的理论，以涵盖这些新的类别。

Exponential uses have many uses in mathematics. Historically, theyarose in problems in number theory. More recently, they have foundapplications in cryptology and coding theory. The main question thatarises is to find sharp upper bounds for the absolute value of anexponential sum. The standard approach, based on P. Deligne'sfundamental work on the Weil conjectures, is to compute the l-adiccohomology groups associated to the exponential sum. Recently, theinvestigator and his collaborator computed the p-adic cohomology ofsome new classes of exponential sums. The calculations indicate thatthese classes of exponential sums should have "good" upper bounds.The investigator plans to search for more such classes of exponentialsums and try to compute their l-adic cohomology, thus obtaining thedesired upper bounds.Exponential sums originally arose in basic problems in number theory,such as trying to find the number of integer solutions to a givenequation. Usually it is very difficult to find the exact number ofsuch solutions, so the next best thing is to approximate that number.It was discovered that this question could often be reduced to theproblem of estimating the size of certain sums of complex numbers,called "exponential sums." A substantial theory has developed overthe years to deal with this subject, and it has found modernapplications in the fields of cryptology and coding theory. Theinvestigator has discovered new classes of exponential sums which hebelieves it should be possible to estimate. He will try to extend theexisting theory to cover these new classes.