Zeta function of the projective curve \(aY^{2l}= bX^{2l}+ cZ^{2l}\) over a class of finite fileds, for odd primes
In a previous article [J. Number Theory 77, No. 2, 288--313 (1999; Zbl 0955.11017)], the author and \textit{S.~A.~Katre} found the number of \(\mathbb F_{q^n}\)-rational points on the nonsingular projective curves \(aY^l=bX^l+cZ^l\) and \(aY^{2l}=bX^{2l}+cZ^{2l}\) \((a,b,c\neq 0)\), defined over \(\mathbb F_q\), with \(q=p^\alpha\equiv 1\pmod e\), for \(e=l\) or \(e=2l\), and for odd primes \(l\) and primes \(p\) such that the order of \(l\) modulo \(p\) is even. They also explicitly computed the zeta function of the curve \(aY^l=bX^l+cZ^l\) over \(\mathbb F_q\) in this situation. Here, the author explicitly computes the zeta function of the curve \(aY^{2l}=bX^{2l}+cZ^{2l}\) over \(\mathbb F_q\) in this situation. There are seven distinct cases depending on the indices of \(b/c\) and \(a/c\) in \(\mathbb F_q^*\). The only such curves that are maximal or minimal curves are the Fermat curves, obtained when \(a=b=c=1\).
scientific article; zbMATH DE number 2152549
scientific article