On Weil Sums, Conjectures of Helleseth, and Niho Exponents

Let $F$ be a finite field, $\mu$ be a fixed additive character and $s$ be an integer coprime to $|F^\times|$. For any $a\in F$, the corresponding Weil sum is defined to be $W_{F,s}(a)=\displaystyle\sum_{x \in F} \mu(x^s-ax)$. The Weil spectrum counts distinct values of the Weil sum as $a$ runs through the 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 coding and information theory to other areas of math such as number theory and arithmetic geometry. In the setting of Niho exponents, we examine the Weil sum, its bounds and its spectrum. As a consequence, we give a new proof to the Vanishing Conjecture of Helleseth ($1971$) on the presence of zero in the Weil spectrum in the case of Niho exponents. We also state a conjecture for when the Weil spectrum contains at least five elements and prove it for a certain class of Weil sums.