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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]

In this paper, X is an algebraic curve of genus g ≥ 2 defined over an algebraically closed field K of positive characteristic p , G is an automorphism group of X which fixes K element-wise, and, for a point P ∈ X , G P is the subgroup of G which fixes P. The question "how large G P can be compared to g " has been the subject of several papers. We are concerned with the case where the second ramification group G P (2) of G P is trivial. Under this condition Theorem 3.1 states that if | G P | > 12 (g − 1) then X is either an ordinary hyperelliptic curve, or it has zero p -rank and p ≠ 3. More precisely, up to birational equivalence, there exists a separable p -linearized polynomial L (T) ∈ K T of degree q such that an affine equation of X is L (y) = a x + 1 / x with a ∈ K ⁎ in the former case, and L (y) = x 3 + b x with b ∈ K in the latter case. In 1987 Nakajima proved that if X is an ordinary curve (more generally, the second ramification group of G is trivial for every P ∈ X), then the order of G does not exceed 84 g (g − 1). We show that Theorem 3.1 together with some refinements of Nakajima's computations provide a slight improvement in Nakajima's bound from 84 g (g − 1) to 48 (g − 1) 2. [ABSTRACT FROM AUTHOR]