Cohomology of Exponential Sums

指数函数在数学中有许多用途。 从历史上看，他们在数论问题上有很大的优势。 最近，它们在密码学和编码理论中得到了应用。 出现的主要问题是找到尖锐的上限的绝对值的指数和。 标准的方法，基于P. Deligne的基础工作的Weil结构，是计算l-自上同调群相关的指数和。 最近，研究者和他的合作者计算了几类新的指数和的p-adic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的这类指数和，并试图计算它们的l-adic上同调，从而获得所需的上界.指数和最初出现在数论的基本问题中，例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目，所以下一个最好的办法是近似这个数目。人们发现，这个问题往往可以归结为估计某些复数和的大小的问题，称为“指数和”。“多年来，一个实质性的理论已经发展到处理这个问题，它已经在密码学和编码理论领域找到了现代应用。 研究人员发现了新的指数和类，他认为这应该是可能的估计。 他将试图扩展现有的理论以涵盖这些新的类。

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.