Bound on the order of the decomposition groups of an algebraic curve in positive characteristic.

In this paper, X is an algebraic curve of genus g ≥ 2 defined over an algebraically closed field K of positive characteristic p , G is an automorphism group of X which fixes K element-wise, and, for a point P ∈ X , G P is the subgroup of G which fixes P. The question "how large G P can be compared to g " has been the subject of several papers. We are concerned with the case where the second ramification group G P (2) of G P is trivial. Under this condition Theorem 3.1 states that if | G P | > 12 (g − 1) then X is either an ordinary hyperelliptic curve, or it has zero p -rank and p ≠ 3. More precisely, up to birational equivalence, there exists a separable p -linearized polynomial L (T) ∈ K T of degree q such that an affine equation of X is L (y) = a x + 1 / x with a ∈ K ⁎ in the former case, and L (y) = x 3 + b x with b ∈ K in the latter case. In 1987 Nakajima proved that if X is an ordinary curve (more generally, the second ramification group of G is trivial for every P ∈ X), then the order of G does not exceed 84 g (g − 1). We show that Theorem 3.1 together with some refinements of Nakajima's computations provide a slight improvement in Nakajima's bound from 84 g (g − 1) to 48 (g − 1) 2. [ABSTRACT FROM AUTHOR]