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 κ (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 overQ attached to f . One of the main results of Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016) is that κ (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 κ (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 κ (f, g, h) as a canonical point in (E(H)⊗ Vgh)Q 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).