On the distribution of Jacobi sums

Abstract

Let $\mathbf{F}_q$ be a finite field of $q$ elements. For multiplicative characters $χ_1,\dots, χ_m$ of $\mathbf{F}_q^\times$, we let $J(χ_1,\dots, χ_m)$ denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for $m=2$, the normalized Jacobi sum $q^{-1/2}J(χ_1,χ_2)$ ($χ_1χ_2$ nontrivial) is asymptotically equidistributed on the unit circle as $q\to \infty$, when $χ_1$ and $χ_2$ 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 $q^{-(m-1)/2}J(χ_1,\dots,χ_m)$ ($χ_1\dotsm χ_m$ nontrivial) is asymptotically equidistributed on the unit circle, when $χ_1,\dots, χ_m$ 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.