Cohomology of Exponential Sums

指数和的上同调

• 批准号: 0070510
• 负责人: Alan Adolphson
• 金额: $ 8.52万
• 依托单位: Oklahoma State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070510&HistoricalAwards=false
• 关键词: Cohomology; Exponential; Sums

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.

指数运算在数学中有很多用处。从历史上看，数论中的问题很难解决.最近，它们在密码学和编码理论中得到了应用。由此产生的主要问题是找到指数和的绝对值的精确上界。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上同调群。最近，这位研究人员和他的合作者计算了一些新的指数和的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类指数和，并试图计算它们的L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，例如试图寻找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，所以次好的办法是近似这个数字。人们发现，这个问题通常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，针对这一主题已经发展了大量的理论，并在密码学和编码理论领域找到了现代应用。这位研究人员发现了一类新的指数和，他认为应该可以估计这些和。他将尝试将现有的理论扩展到涵盖这些新的班级。