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.References
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Additional Information
- Herivelto Borges
- Affiliation: Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador São-carlense, 400, CEP 13566–590, São Carlos, SP, Brazil
- MR Author ID: 857653
- ORCID: 0000-0002-8100-3486
- Email: hborges@icmc.usp.br
- Mariana Coutinho
- Affiliation: Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda, 651, CEP 13083–859, Campinas, SP, Brazil
- ORCID: 0000-0002-8710-0519
- Email: mariananery@alumni.usp.br
- Received by editor(s): November 24, 2019
- Received by editor(s) in revised form: July 8, 2020
- Published electronically: January 12, 2021
- Additional Notes: The first author was supported by FAPESP (Brazil), grant 2017/04681–3, and partially funded by the 2019 IMPA Post-doctoral Summer Program. The second author was financed in part by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, CNPq (Brazil), grant 154359/2016–5, FAPESP (Brazil), grant 2018/23839–0, and also supported by UNICAMP Postdoctoral Research Program.
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 1899-1917
- MSC (2020): Primary 11G20, 14G05, 14G10, 14H37
- DOI: https://doi.org/10.1090/tran/8286
- MathSciNet review: 4216727