On the distribution of Jacobi sums

TL;DR

This paper extends the understanding of the distribution of Jacobi sums over finite fields, showing that normalized sums are asymptotically uniformly distributed on the unit circle for multiple characters, generalizing previous results.

Contribution

It generalizes the equidistribution result of Jacobi sums from two characters to multiple characters, under broader conditions.

Findings

Normalized Jacobi sums are asymptotically equidistributed on the unit circle for multiple characters.

The result applies when two of the sets of characters are sufficiently large.

Generalizes previous results for the case m=2 to m≥2.

Abstract

Let $\mathbf{F}_q$ be a finite field of $q$ elements. For multiplicative characters $\chi_1,\dots, \chi_m$ of $\mathbf{F}_q^\times$, we let $J(\chi_1,\dots, \chi_m)$ denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for $m=2$, the normalized Jacobi sum $q^{-1/2}J(\chi_1,\chi_2)$ ($\chi_1\chi_2$ nontrivial) is asymptotically equidistributed on the unit circle as $q\to \infty$, when $\chi_1$ and $\chi_2$ run through all nontrivial multiplicative characters of $\mathbf{F}_q^\times$. In this paper, we show a similar property for $m\ge 2$. More generally, we show that the normalized Jacobi sum $q^{-(m-1)/2}J(\chi_1,\dots,\chi_m)$ ($\chi_1\dotsm \chi_m$ nontrivial) is asymptotically equidistributed on the unit circle, when $\chi_1,\dots, \chi_m$ run through arbitrary sets of nontrivial multiplicative characters of $\mathbf{F}_q^\times$ with two of the sets being sufficiently large. The case $m=2$ answers a question of Shparlinski.