Divisibility of Weil Sums of Binomials

Consider the Weil sum $W_{F,d}(u)={\sum }_{x\in F}\psi (x^d+ux)$, where $F$ is a finite field of characteristic $p$, $\psi$ is the canonical additive character of $F$, $d$ is coprime to $\mathrm{\mid }{F}^{\ast }\mathrm{\mid }$, and $u\in F^{\ast }$. We say that $W_{F,d}(u)$ is three-valued when it assumes precisely three distinct values as $u$ runs through $F^{\ast }$: this is the minimum number of distinct values in the nondegenerate case, and three-valued $W_{F,d}$ are rare and desirable. When $W_{F,d}$ is three-valued, we give a lower bound on the $p$-adic valuation of the values. This enables us to prove the characteristic $3$ case of a 1976 conjecture of Helleseth: when $p=3$ and $[F:{\mathbb{F}}_3]$ is a power of $2$, we show that $W_{F,d}$ cannot be three-valued.