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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 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. [ABSTRACT FROM AUTHOR]

For every q = n 3 with n a prime power greater than 2, the GK-curve is an F q 2 -maximal curve that is not F q 2 -covered by the Hermitian curve. In this paper some Galois subcovers of the GK curve are investigated. Infinitely many examples of maximal curves that cannot be Galois covered by the Hermitian curve are obtained. We also describe explicit equations for some families of quotient curves of the GK-curve. In several cases, such curves provide new values in the spectrum of genera of F q 2 -maximal curves. [ABSTRACT FROM AUTHOR]

For each prime power ℓ the plane curve X ℓ with equation Y ℓ 2 − ℓ + 1 = X ℓ 2 − X is maximal over F ℓ 6 . Garcia and Stichtenoth in 2006 proved that X 3 is not Galois covered by the Hermitian curve and raised the same question for X ℓ with ℓ > 3 ; in this paper we show that X ℓ is not Galois covered by the Hermitian curve for any ℓ > 3 . Analogously, Duursma and Mak proved that the generalized GK curve C ℓ n over F ℓ 2 n is not a quotient of the Hermitian curve for ℓ > 2 and n ≥ 5 , leaving the case ℓ = 2 open; here we show that C 2 n is not Galois covered by the Hermitian curve over F 2 2 n for n ≥ 5 . [ABSTRACT FROM AUTHOR]