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An 𝔽p2-maximal Wiman sextic and its automorphisms

Massimo Giulietti, Motoko Kawakita, Stefano Lia and Maria Montanucci
From the journal Advances in Geometry

Abstract

In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X6+Y6+6+(X2+Y2+2)(X4+Y4+4)−12X2Y22 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S5. We show that this holds also over every algebraically closed field 𝕂 of 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 𝔽192-maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19 over the finite field 𝔽192.

MSC 2010: 11G30; 11G20; 14H37

Funding statement: This resarch was carried out within the project “Progetto Geometrie di Galois, Curve Algebriche su campi finiti e loro Applicazioni”, supported by Fondo Ricerca di Base, 2015, of Università degli Studi di Perugia.

  1. Communicated by: G. Korchmáros

Acknowledgements

The authors would like to thank the Italian Ministry MIUR, Strutture Geometriche, Combinatoria e loro Applicazioni, Prin 2012 prot. 2012XZE22K and GNSAGA of the Italian INDAM.

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Received: 2019-02-24
Revised: 2020-01-26
Published Online: 2021-10-05
Published in Print: 2021-10-26

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