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

Let $X$ be a nonsingular projective curve of genus $g\ge 2$ over an algebraically closed field of odd characteristic $p$. Giulietti and Korchmáros 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 $g\ge 2$, positive $p$-rank implies $\mathrm{\mid }\text{Aut}(X)\mathrm{\mid }<900g^2$. 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.