An Fp2-maximal Wiman sextic and its automorphisms

Massimo Giulietti; Motoko Kawakita; Stefano Lia; Maria Montanucci

doi:10.1515/advgeom-2020-0012

An F_p²-maximal Wiman sextic and its automorphisms

Massimo Giulietti^*, Motoko Kawakita, Stefano Lia, Maria Montanucci

^*Corresponding author for this work

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Abstract

In 1895 Wiman introduced the Riemann surface W of genus 6 over the complex field C defined by the equation X⁶+Y⁶+Z⁶+(X²+Y²+Z²)(X⁴+Y⁴+Z⁴)-12X²Y²Z² = 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 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 F₁₉2-maximal sextic W is not Galois covered by the Hermitian curve H₁₉ over the finite field H₁₉2.

Original language	English
Journal	Advances in Geometry
Volume	21
Issue number	4
Pages (from-to)	451-461
ISSN	1615-715X
DOIs	https://doi.org/10.1515/advgeom-2020-0012
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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abstract = "In 1895 Wiman introduced the Riemann surface W of genus 6 over the complex field C defined by the equation X6+Y6+Z6+(X2+Y2+Z2)(X4+Y4+Z4)-12X2Y2Z2 = 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 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 F192-maximal sextic W is not Galois covered by the Hermitian curve H19 over the finite field H192. ",

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AU - Giulietti, Massimo

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N2 - In 1895 Wiman introduced the Riemann surface W of genus 6 over the complex field C defined by the equation X6+Y6+Z6+(X2+Y2+Z2)(X4+Y4+Z4)-12X2Y2Z2 = 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 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 F192-maximal sextic W is not Galois covered by the Hermitian curve H19 over the finite field H192.

AB - In 1895 Wiman introduced the Riemann surface W of genus 6 over the complex field C defined by the equation X6+Y6+Z6+(X2+Y2+Z2)(X4+Y4+Z4)-12X2Y2Z2 = 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 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 F192-maximal sextic W is not Galois covered by the Hermitian curve H19 over the finite field H192.

KW - Hermitian curve

KW - Maximal curve

KW - Quotient curves

KW - Unitary groups

KW - Wiman's sextic

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DO - 10.1515/advgeom-2020-0012

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JO - Advances in Geometry

JF - Advances in Geometry

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