A Quadratic Bound for Automorphism Groups of Curves of Positive $p$-Rank in Odd Characteristic

Abstract

Let $X$ be a nonsingular projective curve of genus $g\ge2$ over an algebraically closed field of odd characteristic $p$. Giulietti and Korchm\'aros established a quadratic genus bound with constant $900$ forcing zero $p$-rank for curves of even genus. We extend this bound to odd genus: for every $g\ge2$, positive $p$-rank implies $|\Aut(X)|<900g^2$. Starting from Montanucci's two-short-orbit reduction, we combine ramification estimates and a minimum-genus argument to reduce a counterexample to an almost-simple group acting on a cover of the projective line with two branch points. The main ingredients are an intrinsic description of the second ramification group via Hasse--Arf and genus bounds for intermediate quotients associated with parabolic subgroups. These, together with order and local-structure estimates, exclude every possible simple socle. The proof uses the classification of finite simple groups.