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 is an element of 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 vertical bar G(P)vertical bar > 12(g - 1) then X is either an ordinary hyperelliptic curve, or it has zero p-rank and p not equal 3. More precisely, up to birational equivalence, there exists a separable p-linearized polynomial L(T) is an element of K[T] of degree q such that an affine equation of X is L(y) = ax + 1/x with a is an element of K* in the former case, and L(y) = x(3) + bx with b is an element of 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 is an element of X), then the order of G does not exceed 84g(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 84g(g - 1) to 48(g - 1)(2). (c) 2020 Elsevier Inc. All rights reserved.