dc.contributor |
Universitat Politècnica de Catalunya. Departament de Matemàtiques |
dc.contributor |
Universitat Politècnica de Catalunya. TN - Grup de Recerca en Teoria de Nombres |
dc.contributor.author |
Darmon, Henri |
dc.contributor.author |
Rotger Cerdà, Víctor |
dc.date |
2016-08-24 |
dc.identifier.citation |
Darmon, H., Rotger, V. Elliptic curves of rank two and generalized Kato classes. "Research in the Mathematical Sciences", 24 Agost 2016, vol. 3, núm. 27, p. 1-32. |
dc.identifier.citation |
2197-9847 |
dc.identifier.citation |
10.1186/s40687-016-0074-9 |
dc.identifier.uri |
http://hdl.handle.net/2117/116318 |
dc.language.iso |
eng |
dc.relation |
https://resmathsci.springeropen.com/articles/10.1186/s40687-016-0074-9 |
dc.rights |
info:eu-repo/semantics/openAccess |
dc.subject |
Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria algebraica |
dc.subject |
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de nombres |
dc.subject |
Arithmetical algebraic geometry |
dc.subject |
Diophantine geometry |
dc.subject |
Geometria algèbrica--Aritmètica |
dc.subject |
Aritmètica |
dc.subject |
Classificació AMS::11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry) |
dc.subject |
Classificació AMS::14 Algebraic geometry::14G Arithmetic problems. Diophantine geometry |
dc.title |
Elliptic curves of rank two and generalized Kato classes |
dc.type |
info:eu-repo/semantics/publishedVersion |
dc.type |
info:eu-repo/semantics/article |
dc.description.abstract |
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). |
dc.description.abstract |
Peer Reviewed |