Divisibility of Weil Sums of Binomials

Consider the Weil sumWF,d(u) = ∑ x∈F ψ(x +ux), where F is a finite field of characteristic p, ψ is the canonical additive character of F , d is coprime to |F |, and u ∈ F . We say that WF,d(u) is threevalued when it assumes precisely three distinct values as u runs through F : this is the minimum number of distinct values in the nondegenerate case, and three-valuedWF,d are rare and desirable. WhenWF,d is threevalued, 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 : F3] is a power of 2, we show that WF,d cannot be three-valued.