2016/08/14 by Katz, Daniel J., Langevin, Philippe, Lee, Sangman +1
#05D99 #11L05 #11L07 #11L40 #11T23 #11T71 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1608.04047
We investigate the p-adic valuation of Weil sums of the form WF,d(a)=∑x ∈ F ψ(xd -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 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\colon\mathbb 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\colon\mathbb 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.