On elliptic Galois representations and genus-zero modular units

Given an odd prime \,$p$\, and a representation $\varrho$\, of the absolute Galois group of a number field $k$ onto $\mathrm{PGL}_2(\mathbb{F}_p)$ with cyclotomic determinant, the moduli space of elliptic curves defined over $k$ with $p$-torsion giving rise to $\varrho$ consists of two twists of the...

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Detalhes bibliográficos
Autores: Fernández González, Julio|||0000-0002-9915-6704, Lario Loyo, Joan Carles|||0000-0002-6459-1837
Formato: artículo
Fecha de publicación:2006
País:España
Recursos:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/473
Acesso em linha:https://hdl.handle.net/2117/473
Access Level:acceso abierto
Palavra-chave:Number theory
Galois representations
Modular curves
Elliptic curves
Modular units
Galois, Teoria de
Nombres, Teoria dels
Classificació AMS::11 Number theory
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repository_id_str
spelling On elliptic Galois representations and genus-zero modular unitsFernández González, Julio|||0000-0002-9915-6704Lario Loyo, Joan Carles|||0000-0002-6459-1837Number theoryGalois representationsModular curvesElliptic curvesModular unitsGalois, Teoria deNombres, Teoria delsClassificació AMS::11 Number theoryGiven an odd prime \,$p$\, and a representation $\varrho$\, of the absolute Galois group of a number field $k$ onto $\mathrm{PGL}_2(\mathbb{F}_p)$ with cyclotomic determinant, the moduli space of elliptic curves defined over $k$ with $p$-torsion giving rise to $\varrho$ consists of two twists of the modular curve $X(p)$. We make here explicit the only genus-zero cases $p=3$ and $p=5$, which are also the only \emph{symmetric} cases: $\mathrm{PGL}_2(\mathbb{F}_p)\simeq\mathcal{S}_n$ for $n=4$ or $n=5$, respectively. This is done by studying the corresponding twisted Galois actions on the function field of the curve, for which a description in terms of modular units is given. As a consequence of this twisting process, we recover an equivalence between the \emph{ellipticity} of $\varrho$ and its \emph{principality}, that is, the existence in its fixed field of an element $\alpha$ of degree $n$ over~$k$\, such that $\alpha$ and $\alpha^2$ have both trace zero over $k$.20062006-07-0320062006-07-25journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/473reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 2.5 Spainhttp://creativecommons.org/licenses/by-nc-nd/2.5/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/4732026-05-27T15:37:01Z
dc.title.none.fl_str_mv On elliptic Galois representations and genus-zero modular units
title On elliptic Galois representations and genus-zero modular units
spellingShingle On elliptic Galois representations and genus-zero modular units
Fernández González, Julio|||0000-0002-9915-6704
Number theory
Galois representations
Modular curves
Elliptic curves
Modular units
Galois, Teoria de
Nombres, Teoria dels
Classificació AMS::11 Number theory
title_short On elliptic Galois representations and genus-zero modular units
title_full On elliptic Galois representations and genus-zero modular units
title_fullStr On elliptic Galois representations and genus-zero modular units
title_full_unstemmed On elliptic Galois representations and genus-zero modular units
title_sort On elliptic Galois representations and genus-zero modular units
dc.creator.none.fl_str_mv Fernández González, Julio|||0000-0002-9915-6704
Lario Loyo, Joan Carles|||0000-0002-6459-1837
author Fernández González, Julio|||0000-0002-9915-6704
author_facet Fernández González, Julio|||0000-0002-9915-6704
Lario Loyo, Joan Carles|||0000-0002-6459-1837
author_role author
author2 Lario Loyo, Joan Carles|||0000-0002-6459-1837
author2_role author
dc.subject.none.fl_str_mv Number theory
Galois representations
Modular curves
Elliptic curves
Modular units
Galois, Teoria de
Nombres, Teoria dels
Classificació AMS::11 Number theory
topic Number theory
Galois representations
Modular curves
Elliptic curves
Modular units
Galois, Teoria de
Nombres, Teoria dels
Classificació AMS::11 Number theory
description Given an odd prime \,$p$\, and a representation $\varrho$\, of the absolute Galois group of a number field $k$ onto $\mathrm{PGL}_2(\mathbb{F}_p)$ with cyclotomic determinant, the moduli space of elliptic curves defined over $k$ with $p$-torsion giving rise to $\varrho$ consists of two twists of the modular curve $X(p)$. We make here explicit the only genus-zero cases $p=3$ and $p=5$, which are also the only \emph{symmetric} cases: $\mathrm{PGL}_2(\mathbb{F}_p)\simeq\mathcal{S}_n$ for $n=4$ or $n=5$, respectively. This is done by studying the corresponding twisted Galois actions on the function field of the curve, for which a description in terms of modular units is given. As a consequence of this twisting process, we recover an equivalence between the \emph{ellipticity} of $\varrho$ and its \emph{principality}, that is, the existence in its fixed field of an element $\alpha$ of degree $n$ over~$k$\, such that $\alpha$ and $\alpha^2$ have both trace zero over $k$.
publishDate 2006
dc.date.none.fl_str_mv 2006
2006-07-03
2006
2006-07-25
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
NA
http://purl.org/coar/version/c_be7fb7dd8ff6fe43
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/473
url https://hdl.handle.net/2117/473
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300719

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