Large automorphism groups of curves of zero $p$-rank in odd characteristic

Source abstract

Let $\cX$ be a curve of genus $g\ge2$ and zero $p$-rank over an algebraically closed field of odd characteristic $p$. We classify the pairs $(\cX,G)$ for which $G\le\Aut(\cX)$ has no common fixed point and $|G|>24g(g-1)$. For $g\ge4$, the curves are explicit cyclic covers of the projective line, the Hermitian curve, or the Ree curve. We determine their full automorphism groups and the possible subgroups $G$, including the central extensions. The exceptional $A_7$ action in characteristic $5$ determines the Hermitian curve of degree $6$. The cases of genus $2$ and $3$ are treated separately. These results give an odd-characteristic counterpart to the classification of large automorphism groups of zero $2$-rank curves. As an application, we obtain a new characterization of the generalized Suzuki curve in which the fixed-point hypothesis is no longer required.