On algebraic curves with many automorphisms in characteristic p.

Let X 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 X , respectively. The influence of g and γ on the automorphism group A u t (X) of X is well-known in the literature. If g ≥ 2 then A u t (X) is a finite group, and unless 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 ≥ 2 has more than 336 g 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 | A u t (X) | ≥ 900 g 2 sufficient conditions for X to have p-rank zero are provided. [ABSTRACT FROM AUTHOR]