On the distribution of multivariate Jacobi sums

Abstract

Let ${\mathbf{F}}_q$ be a finite field of $q$ elements. 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 ${\chi }_1\in {\mathcal{A}}_1,\dots ,{\chi }_m\in {\mathcal{A}}_m$ run through arbitrary sets of nontrivial multiplicative characters of ${\mathbf{F}}_q^{\times }$, if $\#{\mathcal{A}}_1\ge q^{\frac{1}{2}+\epsilon }$, $\#{\mathcal{A}}_2\ge (\log q{)}^{\frac{1}{\delta }-1}$ for $\epsilon >\delta >0$ fixed and $q\to \infty$ or if $\#{\mathcal{A}}_1\#{\mathcal{A}}_2/q\to \infty$. This extends previous results of Xi, Z. Zheng, and the authors.

6 pages. v3: added more details