Equidistributions of Jacobi sums

Abstract

Let ${\mathbf{F}}_q$ be a finite field of $q$ elements. We show that the normalized Jacobi sum $J(\chi ,\eta )/\sqrt{q}$, for each fixed non-trivial multiplicative character $\eta$, becomes equidistributed in the unit circle as $q\to +\infty ,$ when $\chi$ runs over all non-trivial multiplicative characters different from ${\eta }^{-1}.$ Previously, the similar equidistribution was obtained by Katz and Zheng by varying both of $\chi$ and $\eta$. On the other hand, we also obtain the equidistribution of $J(\chi ,\eta )$ as $(\chi ,\eta )$ runs over $\mathcal{X}\times \mathcal{Y}\subseteq (\widehat{{\mathrm{F}}^{\ast }}{)}^2$, as long as $|\mathcal{X}|>q^{\frac{1}{2}+\epsilon }$ and $|\mathcal{Y}|>q^{\epsilon }$ for any $\epsilon >0$. This updates a recent work of Lu, Zheng and Zheng, who require $|\mathcal{X}||\mathcal{Y}|>q{\log }^{2}q.$ The main ingredient is the estimate for hypergeometric sums due to Katz.

8 pages. All comments are welcome