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

Herivelto Borges; Mariana Coutinho

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arXiv · 1912.01659

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

Herivelto Borges·Mariana Coutinho

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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 \'{e}tale covers of $\mathcal{X}_{\mathcal{G}_{\mathcal{S}}}$, some with many rational points.

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Herivelto Borges, Mariana Coutinho. 2019-12-03. On the Zeta function and the automorphism group of the generalized Suzuki curve. https://arxiv.org/abs/1912.01659

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