The p-adic valuations of Weil sums of binomials

Abstract

We investigate the p-adic valuation of Weil sums of the form WF,d(a)=∑x∈Fψ(xd−ax), where F is a finite field of characteristic p, ψ is the canonical additive character of F, the exponent d is relatively prime to |F×|, and a is an element of F. Such sums often arise in arithmetical calculations and also have applications in information theory. For each F and d one would like to know VF,d, the minimum p-adic valuation of WF,d(a) as a runs through the elements of F. We exclude exponents d that are congruent to a power of p modulo |F×| (degenerate d), which yield trivial Weil sums. We prove that VF,d≤(2/3)[F:Fp] for any F and any nondegenerate d, and prove that this bound is actually reached in infinitely many fields F. We also prove some stronger bounds that apply when [F:Fp] is a power of 2 or when d is not congruent to 1 modulo p−1, and show that each of these bounds is reached for infinitely many F.