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 rst derivatives of the associated Hasse-Weil L-series. Yet the fruitful connection of Heegner points with 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 [BDR2] and [DR2], 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 center. The generalized 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 coecients for a suitable number eld 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 [BDR2] and [DR2] 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: = V