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

For p p an odd prime number, alttext="q 0 equals p Superscript t"> q 0 = t encoding="application/x-tex">q_{0}=p^{t} , and 2 t minus 1"> 2 ? 1 encoding="application/x-tex">q=p^{2t-1} let alttext="script upper X Subscript script G Sub S"> X mathvariant="script">G mathvariant="script">S encoding="application/x-tex">\mathcal {X}_{\mathcal {G}_{\mathcal {S}}} be the nonsingular model of Y X ( stretchy="false">) . encoding="application/x-tex">\begin{equation*} Y^{q}-Y=X^{q_{0}}(X^{q}-X). \end{equation*} In present work, number alttext="double-struck F n"> mathvariant="double-struck">F n encoding="application/x-tex">\mathbb {F}_{q^{n}} -rational points full automorphism group are determined. addition, L-polynomial this curve is provided, on Jacobian J J encoding="application/x-tex">J_{\mathcal {S}}}} used to construct étale covers some with many rational points.