On algebraic curves with many automorphisms in characteristic p

Let χ be an irreducible, non-singular, algebraic curve defined over a field of odd characteristic p. Let g and γ be the genus and p-rank of χ, respectively. The influence of g and γ on the automorphism group Aut(χ) of χ is well-known in the literature. If g > 2 then Aut(χ) is a finite group, and unless χ 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 ≥ 2 has more than 336g² 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(χ)| ≥ 900g² sufficient conditions for χ to have p-rank zero are provided.