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arXiv · 2005.14358

On the distribution of multivariate Jacobi sums

Qing Lu·Weizhe Zheng
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Abstract

Let $\mathbf{F}_q$ be a finite field of $q$ elements. We show that the normalized Jacobi sum $q^{-(m-1)/2}J(\chi_1,\dots,\chi_m)$ ($\chi_1\dotsm \chi_m$ 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.

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Qing Lu, Weizhe Zheng. 2020-05-29. On the distribution of multivariate Jacobi sums. https://doi.org/10.1016/j.ffa.2021.101840

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