Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

An 𝔽p2-maximal Wiman sextic and its automorphisms

An 𝔽p2-maximal Wiman sextic and its automorphisms AbstractIn 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 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Geometry de Gruyter

An 𝔽p2-maximal Wiman sextic and its automorphisms

Download PDF
11 pages

Loading next page...
 
/lp/de-gruyter/an-p2-maximal-wiman-sextic-and-its-automorphisms-20FMWuDVPr
Publisher
de Gruyter
Copyright
© 2021 Walter de Gruyter GmbH, Berlin/Boston
eISSN
1615-715X
DOI
10.1515/advgeom-2020-0012
Publisher site
See Article on Publisher Site

Abstract

AbstractIn 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 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.

Journal

Advances in Geometry – de Gruyter

Published: Oct 26, 2021

Keywords: Hermitian curve; unitary group; quotient curve; maximal curve; Wiman’s sextic; 11G30; 11G20; 14H37

There are no references for this article.