Elliptic curves of rank two and generalised 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 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 $$\kappa (f,g,h)$$ 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 $$V_g$$ and $$V_h$$ of $$\mathrm {Gal\,}(H/\mathbb {Q})$$ 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 $$\mathbb {Q}$$ attached to f. One of the main results of Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016) is that $$\kappa (f,g,h)$$ lies in the pro-p Selmer group of E over H precisely when $$L(E,V_{gh},1)=0$$ , where $$L(E,V_{gh},s)$$ is the L-function of E twisted by $$V_{gh}:= V_g\otimes V_h$$ . In the setting of interest, parity considerations imply that $$L(E,V_{gh},s)$$ vanishes to even order at $$s=1$$ , and the Selmer class $$\kappa (f,g,h)$$ is expected to be trivial when $${\mathrm {ord}}_{s=1}L(E,V_{gh},s) >2$$ . The main new contribution of this article is a conjecture expressing $$\kappa (f,g,h)$$ as a canonical point in $$(E(H)\otimes V_{gh})^{G_\mathbb {Q}}$$ when $${\mathrm {ord}}_{s=1} L(E,V_{gh},s)=2$$ . 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).

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