Elliptic curves of rank two and generalized Kato classes

Heegner points play an outstanding role in the study of the Birch and Swinnerton-Dyer conjecture, providing canonical Mordell–Weil generators whose heights encode first derivatives of the associated Hasse–Weil L-series. Yet the fruitful connection between Heegner points and L-series also accounts fo...

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Detalles Bibliográficos
Autores: Darmon, Henri, Rotger Cerdà, Víctor|||0000-0002-5293-4425
Tipo de recurso: artículo
Fecha de publicación:2016
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/116318
Acceso en línea:https://hdl.handle.net/2117/116318
https://dx.doi.org/10.1186/s40687-016-0074-9
Access Level:acceso abierto
Palabra clave:Arithmetical algebraic geometry
Diophantine geometry
Geometria algèbrica--Aritmètica
Aritmètica
Classificació AMS::11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry)
Classificació AMS::14 Algebraic geometry::14G Arithmetic problems. Diophantine geometry
Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria algebraica
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de nombres
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spelling Elliptic curves of rank two and generalized Kato classesDarmon, HenriRotger Cerdà, Víctor|||0000-0002-5293-4425Arithmetical algebraic geometryDiophantine geometryGeometria algèbrica--AritmèticaAritmèticaClassificació AMS::11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry)Classificació AMS::14 Algebraic geometry::14G Arithmetic problems. Diophantine geometryÀrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria algebraicaÀrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de nombresHeegner points play an outstanding role in the study of the Birch and Swinnerton-Dyer conjecture, providing canonical Mordell–Weil generators whose heights encode first derivatives of the associated Hasse–Weil L-series. Yet the fruitful connection between Heegner points and L-series also accounts for their main limitation, namely that they are torsion in (analytic) rank >1. This partly expository article discusses the generalised Kato classes introduced in Bertolini et al. (J Algebr Geom 24:569–604, 2015) and Darmon and Rotger (J AMS 2016), stressing their analogy with Heegner points but explaining why they are expected to give non-trivial, canonical elements of the idoneous Selmer group in settings where the classical L-function (of Hasse–Weil–Artin type) that governs their behaviour has a double zero at the centre. The generalised Kato class denoted ¿(f,g,h) is associated to a triple (f, g, h) consisting of an eigenform f of weight two and classical p-stabilised eigenforms g and h of weight one, corresponding to odd two-dimensional Artin representations Vg and Vh of Gal(H/Q) with p-adic coefficients for a suitable number field H. This class is germane to the Birch and Swinnerton-Dyer conjecture over H for the modular abelian variety E over Q attached to f. One of the main results of Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016) is that ¿(f,g,h) lies in the pro-p Selmer group of E over H precisely when L(E,Vgh,1)=0, where L(E,Vgh,s) is the L-function of E twisted by Vgh:=Vg¿Vh. In the setting of interest, parity considerations imply that L(E,Vgh,s) vanishes to even order at s=1, and the Selmer class ¿(f,g,h) is expected to be trivial when ords=1L(E,Vgh,s)>2. The main new contribution of this article is a conjecture expressing ¿(f,g,h) as a canonical point in (E(H)¿Vgh)GQ when ords=1L(E,Vgh,s)=2. This conjecture strengthens and refines the main conjecture of Darmon et al. (Forum Math Pi 3:e8, 2015) and supplies a framework for understanding the results of Darmon et al. (2015), Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016).Peer Reviewed20162016-08-2420182018-04-16journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/116318https://dx.doi.org/10.1186/s40687-016-0074-9reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/1163182026-05-27T15:37:01Z
dc.title.none.fl_str_mv Elliptic curves of rank two and generalized Kato classes
title Elliptic curves of rank two and generalized Kato classes
spellingShingle Elliptic curves of rank two and generalized Kato classes
Darmon, Henri
Arithmetical algebraic geometry
Diophantine geometry
Geometria algèbrica--Aritmètica
Aritmètica
Classificació AMS::11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry)
Classificació AMS::14 Algebraic geometry::14G Arithmetic problems. Diophantine geometry
Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria algebraica
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de nombres
title_short Elliptic curves of rank two and generalized Kato classes
title_full Elliptic curves of rank two and generalized Kato classes
title_fullStr Elliptic curves of rank two and generalized Kato classes
title_full_unstemmed Elliptic curves of rank two and generalized Kato classes
title_sort Elliptic curves of rank two and generalized Kato classes
dc.creator.none.fl_str_mv Darmon, Henri
Rotger Cerdà, Víctor|||0000-0002-5293-4425
author Darmon, Henri
author_facet Darmon, Henri
Rotger Cerdà, Víctor|||0000-0002-5293-4425
author_role author
author2 Rotger Cerdà, Víctor|||0000-0002-5293-4425
author2_role author
dc.subject.none.fl_str_mv Arithmetical algebraic geometry
Diophantine geometry
Geometria algèbrica--Aritmètica
Aritmètica
Classificació AMS::11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry)
Classificació AMS::14 Algebraic geometry::14G Arithmetic problems. Diophantine geometry
Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria algebraica
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de nombres
topic Arithmetical algebraic geometry
Diophantine geometry
Geometria algèbrica--Aritmètica
Aritmètica
Classificació AMS::11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry)
Classificació AMS::14 Algebraic geometry::14G Arithmetic problems. Diophantine geometry
Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria algebraica
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de nombres
description Heegner points play an outstanding role in the study of the Birch and Swinnerton-Dyer conjecture, providing canonical Mordell–Weil generators whose heights encode first derivatives of the associated Hasse–Weil L-series. Yet the fruitful connection between Heegner points and L-series also accounts for their main limitation, namely that they are torsion in (analytic) rank >1. This partly expository article discusses the generalised Kato classes introduced in Bertolini et al. (J Algebr Geom 24:569–604, 2015) and Darmon and Rotger (J AMS 2016), stressing their analogy with Heegner points but explaining why they are expected to give non-trivial, canonical elements of the idoneous Selmer group in settings where the classical L-function (of Hasse–Weil–Artin type) that governs their behaviour has a double zero at the centre. The generalised Kato class denoted ¿(f,g,h) is associated to a triple (f, g, h) consisting of an eigenform f of weight two and classical p-stabilised eigenforms g and h of weight one, corresponding to odd two-dimensional Artin representations Vg and Vh of Gal(H/Q) with p-adic coefficients for a suitable number field H. This class is germane to the Birch and Swinnerton-Dyer conjecture over H for the modular abelian variety E over Q attached to f. One of the main results of Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016) is that ¿(f,g,h) lies in the pro-p Selmer group of E over H precisely when L(E,Vgh,1)=0, where L(E,Vgh,s) is the L-function of E twisted by Vgh:=Vg¿Vh. In the setting of interest, parity considerations imply that L(E,Vgh,s) vanishes to even order at s=1, and the Selmer class ¿(f,g,h) is expected to be trivial when ords=1L(E,Vgh,s)>2. The main new contribution of this article is a conjecture expressing ¿(f,g,h) as a canonical point in (E(H)¿Vgh)GQ when ords=1L(E,Vgh,s)=2. This conjecture strengthens and refines the main conjecture of Darmon et al. (Forum Math Pi 3:e8, 2015) and supplies a framework for understanding the results of Darmon et al. (2015), Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016).
publishDate 2016
dc.date.none.fl_str_mv 2016
2016-08-24
2018
2018-04-16
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/116318
https://dx.doi.org/10.1186/s40687-016-0074-9
url https://hdl.handle.net/2117/116318
https://dx.doi.org/10.1186/s40687-016-0074-9
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
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
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
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