On the Zeta function and the automorphism group of the generalized Suzuki curve

Abstract

For $p$ an odd prime number, $q_0=p^t$, and $q=p^{2t-1}$, let ${\mathcal{X}}_{{\mathcal{G}}_{\mathcal{S}}}$ be the nonsingular model of $$Y^q-Y=X^{q_0}(X^q-X).$$ In the present work, the number of ${\mathbb{F}}_{q^n}$-rational points and the full automorphism group of ${\mathcal{X}}_{{\mathcal{G}}_{\mathcal{S}}}$ are determined. In addition, the L-polynomial of this curve is provided, and the number of ${\mathbb{F}}_{q^n}$-rational points on the Jacobian $J_{{\mathcal{X}}_{{\mathcal{G}}_{\mathcal{S}}}}$ is used to construct étale covers of ${\mathcal{X}}_{{\mathcal{G}}_{\mathcal{S}}}$, some with many rational points.