Number of Rational points of the Generalized Hermitian Curves over ${\mathbb{F}}_{p^n}$

Abstract

In this paper we consider the curves $H_{k,t}^{(p)}:y^{p^k}+y=x^{p^{kt}+1}$ over ${\mathbb{F}}_p$ and and find an exact formula for the number of ${\mathbb{F}}_{p^n}$-rational points on $H_{k,t}^{(p)}$ for all integers $n\ge 1$. We also give the condition when the $L$-polynomial of a Hermitian curve divides the $L$-polynomial of another over ${\mathbb{F}}_p$.