Cohomology of Exponential Sums

Oklahoma State University, Stillwater OK

Investigators

Abstract

Exponential uses have many uses in mathematics. Historically, they arose in problems in number theory. More recently, they have found applications in cryptology and coding theory. The main question that arises is to find sharp upper bounds for the absolute value of an exponential sum. The standard approach, based on P. Deligne's fundamental work on the Weil conjectures, is to compute the l-adic cohomology groups associated to the exponential sum. Recently, the investigator and his collaborator computed the p-adic cohomology of some new classes of exponential sums. The calculations indicate that these classes of exponential sums should have "good" upper bounds. The investigator plans to search for more such classes of exponential sums and try to compute their l-adic cohomology, thus obtaining the desired upper bounds. Exponential sums originally arose in basic problems in number theory, such as trying to find the number of integer solutions to a given equation. Usually it is very difficult to find the exact number of such solutions, so the next best thing is to approximate that number. It was discovered that this question could often be reduced to the problem of estimating the size of certain sums of complex numbers, called "exponential sums." A substantial theory has developed over the years to deal with this subject, and it has found modern applications in the fields of cryptology and coding theory. The investigator has discovered new classes of exponential sums which he believes it should be possible to estimate. He will try to extend the existing theory to cover these new classes.