Bound on the order of the decomposition groups of an algebraic curve in positive characteristic
Let \(\mathcal{X}\) be a (projective, geometrically irreducible, nonsingular algebraic) curve of genus \(\mathfrak{g}\geqslant 2\) defined over an algebraically closed field \(\mathbb{K}\) of characteristic \(p>0\), and let \(G=\text{Aut}(\mathcal{X})\) be the automorphism group of \(\mathcal{X}\) fixing \(\mathbb{K}\) elementwise, which is well known to be a finite group; see [\textit{J. W. P. Hirschfeld} et al., Algebraic curves over a finite field. Princeton, NJ: Princeton University Press (2008; Zbl 1200.11042)]. For a point \(P\in \mathcal{X}\), in this paper the authors deal with the problem of how large \(G_{P}\) can be with respect to \(\mathfrak{g}\), where \(G_{P}\) is the subgroup of \(G\) which fixes \(P\). More specifically, they prove that if the second ramification group \(G^{(2)}_{P}\) is trivial and \(|G_{P}|>12(\mathfrak{g}-1)\), then \(\mathcal{X}\) is either an ordinary hyperelliptic curve, or it has zero \(p\)-rank and \(p\neq 3\). Moreover, up to birational equivalence, there exists a separable \(p\)-linearized polynomial \(L(T) \in \mathbb{K}[T]\) of degree \(q\) such that an affine equation of \(\mathcal{X}\) is \(L(y) = ax + 1/x\) with \(a \in\mathbb{K}^{\ast}\) in the former case, and \(L(y)=x^{3}+bx\) with \(b\in\mathbb{K}\) in the latter case. This result, together with some refinements of Nakajima's computations, is then used to provide an improvement in Nakajima's bound from \(84\mathfrak{g}(\mathfrak{g}-1)\) to \(48(\mathfrak{g}-1)^{2}\).
scientific article; zbMATH DE number 7313135
scientific article