The p-adic valuations of Weil sums of binomials

Abstract We investigate the p -adic valuation of Weil sums of the form W F , d ( a ) = ∑ x ∈ F ψ ( x d − a x ) , 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 V F , d , the minimum p -adic valuation of W F , 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 V F , d ≤ ( 2 / 3 ) [ F : F p ] 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 : F p ] 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 .