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

TL;DR

This paper investigates the properties of a specific generalized Suzuki curve, including its rational points, automorphism group, and L-polynomial, and uses these to construct covers with many rational points.

Contribution

It determines the rational points, automorphism group, and L-polynomial of the generalized Suzuki curve, and constructs étale covers with many rational points.

Findings

Exact count of rational points over finite fields.

Full automorphism group characterization.

Construction of étale covers with many rational points.

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.