In this paper, we explicitly evaluate some special values of ${}_{2k}F_{2k-1}$-hypergeometric series over finite fields. These values are based on expressions of certain character sums in terms of Gaussian hypergeometric series. We also find relation between the number of ${\mathbb{F}}_q$-points on certain algebraic curves and ${}_{2k}F_{2k-1}$-Gaussian hypergeometric series.

中文翻译：

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某些字符和，高斯超几何级数及其与代数曲线的关系

在本文中，我们明确评估了一些特殊值 ${}_{2ķ}F_{2ķ-\mathrm{1个}}$有限域上的超几何级数。这些值基于高斯超几何级数表示的某些字符和的表达式。我们还发现${\mathbb{F}}_q$某些代数曲线上的点 ${}_{2ķ}F_{2ķ-\mathrm{1个}}$-高斯超几何级数。