Generalization of Vélu’s formulae for isogenies between elliptic curves

Given an elliptic curve E and a finite subgroup G, V ́lu’s formulae concern to a separable isogeny IG : E → E ′ with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P + G as the difference between t...

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Detalles Bibliográficos
Autores: Miret, Josep M. (Josep Maria), Moreno Chiral, Ramiro, Rio, Anna
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2007
País:España
Institución:Universitat de Lleida (UdL)
Repositorio:Repositori Obert UdL
OAI Identifier:oai:repositori.udl.cat:10459.1/44519
Acceso en línea:https://doi.org/10.5565/PUBLMAT_PJTN05_07
http://hdl.handle.net/10459.1/44519
Access Level:acceso abierto
Palabra clave:Elliptic curve
Isogeny
Rational subgroup
Corbes el·líptiques
Nombres, Teoria dels
Anàlisi diofàntica
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spelling Generalization of Vélu’s formulae for isogenies between elliptic curvesMiret, Josep M. (Josep Maria)Moreno Chiral, RamiroRio, AnnaElliptic curveIsogenyRational subgroupCorbes el·líptiquesNombres, Teoria delsAnàlisi diofànticaGiven an elliptic curve E and a finite subgroup G, V ́lu’s formulae concern to a separable isogeny IG : E → E ′ with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P + G as the difference between the abscissa of IG (P ) and the first elementary symmetric polynomial on the abscissas of the nontrivial points of the kernel G. On the other hand, they express Weierstraß coefficients of E ′ as polynomials in the coefficients of E and two additional parameters: w0 = t and w1 = w. We generalize this by defining parameters wn for all n ≥ 0 and giving analogous formulae for all the elementary symmetric polynomials and the power sums on the abscissas of the points in P +G. Simultaneously, we obtain an efficient way of performing computations concerning the isogeny when G is a rational group.Universitat Autònoma de Barcelona. Departament de Matemàtiques2007info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttps://doi.org/10.5565/PUBLMAT_PJTN05_07http://hdl.handle.net/10459.1/44519reponame:Repositori Obert UdL instname:Universitat de Lleida (UdL)InglésReproducció del document publicat a https://doi.org/10.5565/PUBLMAT_PJTN05_07Reproducció del document publicat a http://ddd.uab.cat/record/52?ln=caPublicacions matemàtiques, 2007, vol. Extra, p. 147–163(c) Universitat Autònoma de Barcelona. Departament de Matemàtiques, 2007info:eu-repo/semantics/openAccessoai:repositori.udl.cat:10459.1/445192026-06-24T12:42:17Z
dc.title.none.fl_str_mv Generalization of Vélu’s formulae for isogenies between elliptic curves
title Generalization of Vélu’s formulae for isogenies between elliptic curves
spellingShingle Generalization of Vélu’s formulae for isogenies between elliptic curves
Miret, Josep M. (Josep Maria)
Elliptic curve
Isogeny
Rational subgroup
Corbes el·líptiques
Nombres, Teoria dels
Anàlisi diofàntica
title_short Generalization of Vélu’s formulae for isogenies between elliptic curves
title_full Generalization of Vélu’s formulae for isogenies between elliptic curves
title_fullStr Generalization of Vélu’s formulae for isogenies between elliptic curves
title_full_unstemmed Generalization of Vélu’s formulae for isogenies between elliptic curves
title_sort Generalization of Vélu’s formulae for isogenies between elliptic curves
dc.creator.none.fl_str_mv Miret, Josep M. (Josep Maria)
Moreno Chiral, Ramiro
Rio, Anna
author Miret, Josep M. (Josep Maria)
author_facet Miret, Josep M. (Josep Maria)
Moreno Chiral, Ramiro
Rio, Anna
author_role author
author2 Moreno Chiral, Ramiro
Rio, Anna
author2_role author
author
dc.subject.none.fl_str_mv Elliptic curve
Isogeny
Rational subgroup
Corbes el·líptiques
Nombres, Teoria dels
Anàlisi diofàntica
topic Elliptic curve
Isogeny
Rational subgroup
Corbes el·líptiques
Nombres, Teoria dels
Anàlisi diofàntica
description Given an elliptic curve E and a finite subgroup G, V ́lu’s formulae concern to a separable isogeny IG : E → E ′ with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P + G as the difference between the abscissa of IG (P ) and the first elementary symmetric polynomial on the abscissas of the nontrivial points of the kernel G. On the other hand, they express Weierstraß coefficients of E ′ as polynomials in the coefficients of E and two additional parameters: w0 = t and w1 = w. We generalize this by defining parameters wn for all n ≥ 0 and giving analogous formulae for all the elementary symmetric polynomials and the power sums on the abscissas of the points in P +G. Simultaneously, we obtain an efficient way of performing computations concerning the isogeny when G is a rational group.
publishDate 2007
dc.date.none.fl_str_mv 2007
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://doi.org/10.5565/PUBLMAT_PJTN05_07
http://hdl.handle.net/10459.1/44519
url https://doi.org/10.5565/PUBLMAT_PJTN05_07
http://hdl.handle.net/10459.1/44519
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Reproducció del document publicat a https://doi.org/10.5565/PUBLMAT_PJTN05_07
Reproducció del document publicat a http://ddd.uab.cat/record/52?ln=ca
Publicacions matemàtiques, 2007, vol. Extra, p. 147–163
dc.rights.none.fl_str_mv (c) Universitat Autònoma de Barcelona. Departament de Matemàtiques, 2007
info:eu-repo/semantics/openAccess
rights_invalid_str_mv (c) Universitat Autònoma de Barcelona. Departament de Matemàtiques, 2007
eu_rights_str_mv openAccess
dc.publisher.none.fl_str_mv Universitat Autònoma de Barcelona. Departament de Matemàtiques
publisher.none.fl_str_mv Universitat Autònoma de Barcelona. Departament de Matemàtiques
dc.source.none.fl_str_mv reponame:Repositori Obert UdL
instname:Universitat de Lleida (UdL)
instname_str Universitat de Lleida (UdL)
reponame_str Repositori Obert UdL
collection Repositori Obert UdL
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,811543

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