$p$-ranks and automorphism groups of algebraic curves

Abstract:

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