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

TL;DR

This paper derives an exact formula for counting rational points on generalized Hermitian curves over finite fields and explores divisibility conditions of their L-polynomials, advancing understanding of their algebraic and arithmetic properties.

Contribution

It provides a closed-form expression for the number of rational points on generalized Hermitian curves over finite fields and characterizes L-polynomial divisibility conditions.

Findings

Exact formula for rational points on $H_{k,t}^{(p)}$ over $F_{p^n}$

Conditions for L-polynomial divisibility among Hermitian curves

Enhanced understanding of algebraic structure of generalized Hermitian curves

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 $\mathbb F_p$.