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arXiv 1809.04286math.NT
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Jacobi和的等分布

Equidistributions of Jacobi sums

Ping Xi

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8 pages. All comments are welcome
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摘要：

设$\mathbf{F}_q$是$q$元素的有限域。我们表明，规范化的雅可比和$J(χ,η)/\sqrt{q}$，为每个固定的非平凡乘法字符$η$，成为均匀分布在单位圆$q\rightarrow+\infty,$时，$χ$运行在所有非平凡乘法字符不同于$η^{-1}.$以前，类似的均匀分布得到了Katz和郑通过改变两者的$χ$和$η$。另一方面，我们也得到了当$(χ,η)$在$\mathcal{X}\times\mathcal{Y}\subseteq(\widehat{\mathrm{F}^*})^2$上运行时，只要对任何$\varepsilon>0$有$|\mathcal{X}|>q^{\frac{1}{2}+\varepsilon}$和$|\mathcal{Y}|>q^\varepsilon$，$J(χ,η)$的等分布。这更新了Lu，Zheng和Zheng最近的工作，他们需要$|\mathcal{X}||\mathcal{Y}|>q\log^2q.$。主要成分是由于Katz的超几何和的估计。

英文摘要：

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

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