On the distribution of Jacobi sums

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 $q^{-(m-1)/2}J({\chi }_1,\dots ,{\chi }_m)$ (${\chi }_1\cdots {\chi }_m$ 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.

arXiv comment18 pages. v3: fixed some typos; v2: improved some bounds

Cited by in corpus (1)

• On the distribution of multivariate Jacobi sums