𝑝-ranks and automorphism groups of algebraic curves

Abstract

Let X X be an irreducible complete nonsingular curve of genus g g over an algebraically closed field k k of positive characteristic p p . If g ⩾ 2 g \geqslant 2 , the automorphism group Aut ⁡ ( X ) \operatorname {Aut} (X) of X X is known to be a finite group, and moreover its order is bounded from above by a polynomial in g g of degree four (Stichtenoth). In this paper we consider the p p -rank γ \gamma of X X and investigate relations between γ \gamma and Aut ⁡ ( X ) \operatorname {Aut} (X) . We show that γ \gamma affects the order of a Sylow p p -subgroup of Aut ⁡ ( X ) ( § 3 ) \operatorname {Aut} (X)\;(\S 3) and that an inequality | Aut ⁡ ( X ) | ⩽ 84 ( g − 1 ) g |\operatorname {Aut} (X)| \leqslant 84(g - 1)g holds for an ordinary (i.e. γ = g \gamma = g ) curve X ( § 4 ) X\,(\S 4) .