Let ${\mathbb{F}}_q$ denote the finite field with q elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over ${\mathbb{F}}_q$ in terms of the number of rational points on elliptic curves. In the case where q is a prime number, we give a way to calculate these numbers. As a consequence of these results, we characterize maximal and minimal curves given by equations of the forms $a{x}^3+b{y}^3+c{z}^3=0$ and $a{x}^4+b{y}^4+c{z}^4=0$.

中文翻译：

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有限域上三次，四次和六边形曲线上的有理点

让 ${\mathbb{F}}_q$表示带有q个元素的有限域。在这项工作中，我们使用字符来给出适当的低度曲线上有理点的数量。${\mathbb{F}}_q$椭圆曲线上有理点的数量 在q是素数的情况下，我们提供了一种计算这些数字的方法。这些结果的结果是，我们表征了形式方程式给出的最大和最小曲线$\mathrm{一种}{X}^3+b{ÿ}^3+C{ž}^3=0$ 和 $\mathrm{一种}{X}^4+b{ÿ}^4+C{ž}^4=0$。