The curve \(y^n=x^\ell(x^m+1)\) over finite fields. II (Q2046815)

When \(X\) is a (projective, smooth, absolutely irreducible) curve of genus \(g\) over \(\mathbb{F}_q\), Hasse-Weil bound shows that \(\#X(\mathbb{F}_q) \leq 1+q+ 2g \sqrt{q}\). When \(q\) is a square, curves whose number of rational points reaches this bound are called maximal and have been studied extensively and partially classified. The present article deals with maximal curves of the form \(y^n= x^{\ell} (x^m+1)\) with certain conditions on \(n,\ell\) and \(m\) with respect to \(q\), divided into cases which have been handled in previous papers by the same authors and the present one. In order to show that the curve is maximal, the authors use the classical trick to reveal them as explicit sub-covers of curves which are known to be maximal (mainly the Hermitian curve). For Part I see [the authors, ibid. 19, No. 2, 263--268 (2019; Zbl 1478.11090)].