The $p$-Adic Valuations of Weil Sums of Binomials

TL;DR

This paper studies the $p$-adic valuations of Weil sums of binomials over finite fields, establishing upper bounds and demonstrating their sharpness in infinitely many cases, with implications for number theory and information theory.

Contribution

It provides new upper bounds for the $p$-adic valuation of Weil sums and proves these bounds are attained infinitely often, extending understanding of their behavior.

Findings

Bound $V_{F,d} \\leq (2/3)[F:\\mathbb{F}_p]$ for all nondegenerate $d$

Bounds are sharp, achieved in infinitely many fields

Stronger bounds when $[F:\\mathbb{F}_p]$ is a power of 2 or $d$ not congruent to 1 mod $p-1$

Abstract

We investigate the $p$-adic valuation of Weil sums of the form $W_{F,d}(a)=\sum_{x \in F} \psi(x^d -a x)$, where $F$ is a finite field of characteristic $p$, $\psi$ is the canonical additive character of $F$, the exponent $d$ is relatively prime to $|F^\times|$, 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 $|F^\times|$ (degenerate $d$), which yield trivial Weil sums. We prove that $V_{F,d} \leq (2/3)[F\colon{\mathbb F}_p]$ 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 $[F\colon{\mathbb F}_p]$ 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 $F$.