Cohomology of Exponential Sums

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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.