## Abstract

Let **f** be the finite field of order *q*^{2}. It is sometimes attributed to Serre that any curve **F**-covered by the Hermitian curve *H*_{q+1} : *Y*^{q+1} = x^{q} + x is also **F**-maximal. For prime numbers *q* we show that every **F**-maximal curve *X* of genus *g* ≤ 2 with | Aut(*X*) | > 84(*g* - 1) is Galois-covered by *H*_{q+1}. The hypothesis on | Aut(*X*) | is sharp, since there exists an **F**-maximal curve *X* for *q* = 71 of genus *g* = 7 with | Aut(*X*) | = 84(7 - 1) which is not Galois-covered by the Hermitian curve *H*_{72}.

Original language | English |
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Journal | Advances in Geometry |

Volume | 21 |

Issue number | 3 |

Pages (from-to) | 325-336 |

ISSN | 1615-715X |

DOIs | |

Publication status | Published - 1 Jul 2021 |

### Bibliographical note

Publisher Copyright:© 2021 De Gruyter. All rights reserved.

## Keywords

- Finite field
- Galois-covering
- Hermitian curve
- Maximal curve

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