Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals
type: Item
instance of: scholarly article
title: Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals
zbMATH Open document ID: 1180.14028
published in: Archiv der Mathematik
publication date: 2009-11-20T00:00:00Z
review text: Let \(S\) be a compact Riemann surface. Then \(S\) can be described as a non-singular complex projective algebraic curve \(C\) inside some projective space by homogeneous polynomials \(P_{1},\dots ,P_{r}\) with coefficients in a subfield \(K\) of \({\mathbb C}\) called a field of definition of \(S\). For a given \(\sigma \in \text{Aut} ({\mathbb C})\), we can construct new polynomials \(P_{1}^{\sigma} ,\dots ,P_{r}^{\sigma}\) by applying \(\sigma\) to the coefficients of the \(P_{i}\). Let \(C^{\sigma}\) denote the algebraic curve defined by \(P_{1}^{\sigma} ,\dots ,P_{r}^{\sigma}\) and let \(S^{\sigma}\) denote the corresponding compact Riemann surface. Let \(G\) denote the subgroup of \(\text{Aut} ({\mathbb C} )\) consisting of all \(\sigma \in \text{Aut} ({\mathbb C} )\) such that \(S\) and \(S^{\sigma}\) are conformally equivalent. Then we call the fixed field of \(G\) the moduli field of \(S\). Clearly the moduli field is a subfield of any field of definition of \(S\). The moduli field however need not be a field of definition, and there are specific examples of families of hyperelliptic surfaces where this is not the case, see for example [\textit{Y. Fuertes} and \textit{G. González-Diez}, Arch. Math. 86, No. 5, 398--408 (2006; Zbl 1095.14028)]. However, no previous examples exist for surfaces which are not hyperelliptic. In the paper under review, the author constructs an explicit example of a non-hyperelliptic family of compact Riemann surfaces where the field of moduli is not a field of definition. The results of the author rely on two main results. First, a compact Riemann surface \(S\) has moduli field contained in \({\mathbb R}\) if and only if \(S\) admits an anticonformal automorphism. Second, \(S\) has a field of definition contained in \({\mathbb R}\) if and only if \(S\) admits an anticonformal involution of order \(2\). Using these two results, it suffices to exhibit an explicit family of non-hyperelliptic compact Riemann surfaces which admit some …
reviewed by: Aaron Wootton
Mathematics Subject Classification ID: 14H37
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 30F10
zbMATH DE Number: 5636258
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: algebraic curves
zbMATH Keywords: automorphisms
zbMATH Keywords: fields of moduli
author: Rubén A. Hidalgo
MaRDI profile type: Publication
full work available at URL: s00013-009-0025-4
OpenAlex ID: W2083935089
cites work: Q3422672
cites work: Generalized Humbert curves
cites work: Q3567907
cites work: Fields of moduli and definition of hyperelliptic covers
cites work: Generalized Fermat curves
cites work: On uniqueness of automorphisms groups of Riemann surfaces
cites work: On the Field of Rationality for an Abelian Variety
cites work: Q4037342
DOI: 10.1007/S00013-009-0025-4
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description: scientific article; zbMATH DE number 5636258