## Abstract

In 1895 Wiman introduced the Riemann surface W of genus 6 over the complex field C defined by the equation X^{6}+Y^{6}+Z^{6}+(X^{2}+Y^{2}+Z^{2})(X^{4}+Y^{4}+Z^{4})-12X^{2}Y^{2}Z^{2} = 0, and showed that its full automorphism group is isomorphic to the symmetric group S_{5}. We show that this holds also over every algebraically closed field Kof characteristic p ≥ 7. For p = 2, 3 the above polynomial is reducible over K, and for *p* = 5 the curve W is rational and Aut(W) ≅ PGL(2,K). We also show that Wiman's F_{19}2-maximal sextic W is not Galois covered by the Hermitian curve H_{19} over the finite field H_{19}2.

Original language | English |
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Journal | Advances in Geometry |

Volume | 21 |

Issue number | 4 |

Pages (from-to) | 451-461 |

ISSN | 1615-715X |

DOIs | |

Publication status | Published - 1 Oct 2021 |

### Bibliographical note

Publisher Copyright:© 2021 Walter de Gruyter GmbH, Berlin/Boston.

## Keywords

- Hermitian curve
- Maximal curve
- Quotient curves
- Unitary groups
- Wiman's sextic

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_{p}^{2}