Bounds on the number of automorphisms of curves over algebraically closed fields
type: Item
instance of: scholarly article
title: Bounds on the number of automorphisms of curves over algebraically closed fields
zbMATH Open document ID: 1271.14038
author: Barry W. Green
published in: Israel Journal of Mathematics
publication date: 2013-06-18T00:00:00Z
review text: For any prime \(p\) and \(g \geq 2\), define \(\mu(g, p)\) to be the maximum possible number of automorphisms for a smooth projective curve of genus \(g\) over a field of characteristic \(p\). The classical Hurwitz bound shows that \(\mu(g, 0) \leq 84(g-1)\). In fact \[ 8(g+1) \leq \mu(g,0) \leq 84(g-1), \] and both inequalities are equalities for infinitely many \(g\). In characteristic \(p\), the behavior of \(\mu(g,p)\) is not as well understood. Oort asked if there exists a polynomial \(M_p(T) \in \mathbb{Q}[T]\) such that \(M_p(g)\) is a lower bound on \(\mu(g, p)\) which attains equality infinitely often. As a consequence of what has been written above, \(M_0(g) = 8(g+1)\). Oort asked further, is \(M_p(g) = 8(g+1)\)? In this note, it is shown that if \(g \geq 2\), then \(\mu(g, p) \geq 8(g+1)\) for almost all \(p\). If we restrict to those \(g\) with \(\mu(g, 0) = 8(g+1)\), then \(\mu(g, p) = 8(g+1)\) for almost all \(p\). Furthermore, there exist arbitrarily long increasing sequences of \(g_i \geq 2\) such that \(\mu(g_i, p) = \mu(g_i, 0) = 8(g+1)\) for almost al \(p\), and such that \(\mu(g_i, p) = \mu(g_i,0) = 84(g_i-1)\) for almost all \(p\). The proofs of these statements are fairly straightforward consequences of the fact that smooth curves in characteristic \(p\) with prime-to-\(p\) automorphism groups lift to characteristic \(0\) (along with the automorphism groups), that a curve in characteristic zero has good reduction at a set of valuations covering almost all primes as residue characteristics, and that automorphisms of curves of genus \(\geq 2\) with good reduction do not become trivial in the reduction.
reviewed by: Andrew Obus
Mathematics Subject Classification ID: 14H37
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 12F05
zbMATH DE Number: 6176537
zbMATH Keywords: automorphisms of curves
zbMATH Keywords: Hurwitz bound
zbMATH Keywords: good reduction
MaRDI profile type: Publication
full work available at URL: s11856-012-0078-0
OpenAlex ID: W2039558577
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DOI: 10.1007/S11856-012-0078-0
description: scientific article; zbMATH DE number 6176537