A characterization of Jacobi sums

TL;DR

This paper characterizes Jacobi sums by their fundamental properties, showing that any function satisfying these properties arises from the Jacobi sums of a finite field, thus providing a unique characterization.

Contribution

It demonstrates that the elementary properties of Jacobi sums nearly uniquely determine them among functions on finite abelian groups.

Findings

Jacobi sums satisfy three elementary properties.

These properties characterize Jacobi sums among functions on finite abelian groups.

Any function with these properties corresponds to Jacobi sums of a finite field.

Abstract

Let $\mathbb{M}$ be the group of multiplicative characters of a finite field $\mathbb{F}$, and let $\mathbb{J}(\alpha, \beta)$ be the Jacobi sum, for $\alpha, \beta \in \mathbb{M}$. We observe that the function $\mathbb{J} \colon \mathbb{M} \times \mathbb{M} \to \mathbf{C}$ satisfies three elementary properties. We show that these properties (very nearly) characterize Jacobi sums: if $M$ is an arbitrary non-trivial finite abelian group and $J \colon M \times M \to \mathbf{C}$ is a function satisfying these properties then $M$ is naturally the group of multiplicative characters of a finite field and $J$ is the Jacobi sum.