Zeta function of the projective curve $\pmb{aY^{2 l} = bX^{2 l} + cZ^{2 l}}$ over a class of finite fields, for odd primes $\pmb{l}$

Abstract

Let $p$ and $l$ be rational primes such that $l$ is odd and the order of $p$ modulo $l$ is even. For such primes $p$ and $l$, and for $e=l, 2l$, we consider the non-singular projective curves $aY^e = bX^e + cZ^e$ ($abc \neq 0$) defined over finite fields $\mathbf{F}_q$ such that $q=p^\alpha \equiv 1(\bmod {e})$. We see that the Fermat curves correspond precisely to those curves among each class (for $e=l,2l$), that are maximal or minimal over $\mathbf{F}_q$. We observe that each Fermat prime gives rise to explicit maximal and minimal curves over finite fields of characteristic 2. For $e=2l$, we explicitly determine the $\zeta$-function(s) for this class of curves, over $\mathbf{F}_q$, as rational functions in the variable $t$, for distinct cases of $a,b$, and $c$, in $\mathbf{F}_q^*$. The $\zeta$-function in each case is seen to satisfy the Weil conjectures (now theorems) for this concrete class of curves.