On Properties of Weil Sums of Binomials

Abstract

This dissertation explores questions regarding the Weil sum of binomials, a finite field character sum originated from information theory. The Weil spectrum counts distinct values of the Weil sum through invertible elements in the finite field. The value of these sums and the size of the Weil spectrum are of particular interest, as they link problems in information theory, coding theory, and cryptography to other areas of math such as number theory and arithmetic geometry. In the setting of Niho exponents, we prove the Vanishing Conjecture of Helleseth ($1971$) on the presence of zero values in the Weil spectrum and deduce bounds on the Weil sum. At certain roots of unity, we derive an exact formula for the Weil sum. Finally, we state a conjecture on when the Weil spectrum contains at least five elements, and prove it for a certain class of Niho exponents.