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

Borges, Herivelto; Coutinho, Mariana

doi:10.1090/tran/8286

On the Zeta function and the automorphism group of the generalized Suzuki curve
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by Herivelto Borges and Mariana Coutinho PDF

Trans. Amer. Math. Soc. 374 (2021), 1899-1917 Request permission

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 \begin{equation*} Y^{q}-Y=X^{q_{0}}(X^{q}-X). \end{equation*} 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.

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