Rationality of Four-Valued Families of Weil Sums of Binomials

TL;DR

This paper studies the rationality of Weil sums of binomials over finite fields, showing that four-valued spectra are generally rational integers except in a specific small field case.

Contribution

It proves that four-valued Weil spectra are rational integers, except for one specific case involving the field of size 5 and a particular exponent.

Findings

Four-valued Weil spectra are rational integers.

The exception occurs only when |K|=5 and s ≡ 3 mod 4.

Weil spectrum always has at least three distinct values.

Abstract

We investigate the rationality of Weil sums of binomials of the form $W^{K,s}_u=\sum_{x \in K} \psi(x^s - u x)$, where $K$ is a finite field whose canonical additive character is $\psi$, and where $u$ is an element of $K^{\times}$ and $s$ is a positive integer relatively prime to $|K^\times|$, so that $x \mapsto x^s$ is a permutation of $K$. The Weil spectrum for $K$ and $s$, which is the family of values $W^{K,s}_u$ as $u$ runs through $K^\times$, is of interest in arithmetic geometry and in several information-theoretic applications. The Weil spectrum always contains at least three distinct values if $s$ is nondegenerate (i.e., if $s$ is not a power of $p$ modulo $|K^\times|$, where $p$ is the characteristic of $K$). It is already known that if the Weil spectrum contains precisely three distinct values, then they must all be rational integers. We show that if the Weil spectrum contains precisely four distinct values, then they must all be rational integers, with the sole exception of the case where $|K|=5$ and $s \equiv 3 \pmod{4}$.