Bound on the order of the decomposition groups of an algebraic curve in positive characteristic
type: Item
instance of: scholarly article
title: Bound on the order of the decomposition groups of an algebraic curve in positive characteristic
zbMATH Open document ID: 1461.14040
author: Stefano Lia
author: Marco Timpanella
published in: Finite Fields and their Applications
publication date: 2021-02-19T00:00:00Z
review text: 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 \(48(\mathfrak{g}-1)^{2}\).
reviewed by: Mariana Coutinho
Mathematics Subject Classification ID: 14H37
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 7313135
zbMATH Keywords: algebraic curves
zbMATH Keywords: positive characteristic
zbMATH Keywords: automorphism groups
MaRDI profile type: Publication
full work available at URL: j.ffa.2020.101771
OpenAlex ID: W3093601023
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DOI: 10.1016/J.FFA.2020.101771
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description: scientific article; zbMATH DE number 7313135