A classification of zero $p$-rank curves with large automorphism groups in odd characteristic

Abstract

Let $\cX$ be a curve of genus $g\ge 2$ and $p$-rank zero over an algebraically closed field of odd characteristic $p$, and let $G\le\Aut(\cX)$ satisfy $|G|>24g(g-1)$. We prove that $G$ fixes a point unless $(\cX,G)$ belongs to an explicit list. For $g\ge 4$, the curve is a cyclic quotient of a Hermitian curve, of a Giulietti--Korchmáros curve, or of Skabelund's cover of a Ree curve. We determine the equations and all the groups $G$ that occur, including their central kernels. The exceptional $A_7$ action in characteristic $5$ is on the Hermitian curve of degree $6$; genera $2$ and $3$ are treated separately. The equations are recovered from the group action by two rigidity lemmas for equivariant one-point $p$-group covers of the affine line. If $G$ has no common fixed point and its Sylow $p$-subgroups are noncyclic, then, unless $G$ induces ${}^2{G}_2(3)$ on the orbit of Sylow fixed points, the bound $|G|>\max \{84(g-1),2g^2\}$ gives a rational quotient with exactly one wild and one tame short orbit, and a rational Sylow quotient.