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.