Fields of moduli of some special curves
type: Item
instance of: scholarly article
title: Fields of moduli of some special curves
zbMATH Open document ID: 1337.14030
author: Rubén A. Hidalgo
author: Saúl Quispe
published in: Journal of Pure and Applied Algebra
publication date: 2015-10-27T00:00:00Z
review text: Let \(C\) denote a smooth projective algebraic curve given by the zeros of the polynomials \(P_1,\dots ,P_s \in {\mathbb L}[x_0,\dots ,x_n]\) where \(\mathbb{L}\) is an algebraic closure of a perfect field \(\mathbb{K}\) of characteristic \(q\neq 2\). If the polynomials \(P_1,\dots ,P_s \) are defined over a subfield \(\mathbb{F}\) of \(\mathbb{L}\), then we call \(\mathbb{F}\) a field of definition of \(C\). For \(\sigma \in \Gamma =\text{Gal} (\mathbb{L} /\mathbb{K})\), the polynomials \(P_1^{\sigma} ,\dots ,P_s^{\sigma}\) define a new smooth projective curve \(C^{\sigma}\) which is, in general, not isomorphic to \(C\). We define the moduli field \(\mathbb{M}\) of \(C\) relative to the extension \(\mathbb{L} /\mathbb{K}\) to be the fixed field of the group \[ U=\left\{ \sigma \in \Gamma =\text{Gal} (\mathbb{L} /\mathbb{K}) \bigg| C \text{ is isomorphic to } C^{\sigma} \text{ over } \mathbb{L} \right\}. \] Clearly the moduli field is a subfield of any field of definition of \(C\). The moduli field however need not be a field of definition. Necessary and sufficient conditions for when the field of moduli is a field of definition were given in [\textit{A. Weil}, Am. J. Math. 78, 509--524 (1956; Zbl 0072.16001)]. A consequence of Weil's conditions is that if the automorphism group \(\text{Aut} (C)\) of \(C\) is trivial, then the moduli field is a field of definition. Unfortunately, when \(\text{Aut} (C)\) is non-trivial, Weil's conditions are often difficult to test. This has motivated an interest in developing alternate conditions for when the moduli field is a field of definition, and this is the primary goal of the authors in the current article. In order to present these conditions, we need the following definitions. The signature of a subgroup \(H\) of \(\text{Aut} (C)\) is defined to be the tuple \((g_0;m_1,\dots ,m_r)\) where the curve \(C/H\) has genus \(g_0\) and the quotient map \(C\rightarrow C/H\) is branched over \(r\) points with branching orders \(m_1,\…
reviewed by: Aaron Wootton
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14H37
zbMATH DE Number: 6499128
zbMATH Keywords: field of moduli
zbMATH Keywords: field of definition
zbMATH Keywords: automorphism
MaRDI profile type: Publication
full work available at URL: j.jpaa.2015.05.042
OpenAlex ID: W1462066139
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DOI: 10.1016/J.JPAA.2015.05.042
description: scientific article; zbMATH DE number 6499128