On algebraic curves with many automorphisms in characteristic p

Abstract

Let ${\mathcal {X}}$ be an irreducible, non-singular, algebraic curve defined over a field of odd characteristic p. Let g and $\gamma$ be the genus and p-rank of ${\mathcal {X}}$, respectively. The influence of g and $\gamma$ on the automorphism group $Aut({\mathcal {X}})$ of ${\mathcal {X}}$ is well-known in the literature. If $g \ge 2$ then $Aut({\mathcal {X}})$ is a finite group, and unless ${\mathcal {X}}$ is the so-called Hermitian curve, its order is upper bounded by a polynomial in g of degree four (Stichtenoth). In 1978 Henn proposed a refinement of Stichtenoth’s bound of degree 3 in g up to few exceptions, all having p-rank zero. In this paper a further refinement of Henn’s result is proposed. First, we prove that if an algebraic curve of genus $g \ge 2$ has more than $336g^2$ automorphisms then its automorphism group has exactly two short orbits, one tame and one non-tame, that is, the action of the group is completely known. Finally when $|Aut({\mathcal {X}})| \ge 900g^2$ sufficient conditions for ${\mathcal {X}}$ to have p-rank zero are provided.