Beilinson–Kato and Beilinson–Flach elements, Coleman–Rubin–Stark classes, Heegner points and a conjecture of Perrin-Riou

Abstract

Our first goal in this article is to explain that a weak form of Perrin-Riou's conjecture on the non-triviality of Beilinson–Kato classes follows as an easy consequence of the Iwasawa main conjectures. We also explain that the refined form of this conjecture in the $p$-supersingular case also follows from the classical Gross–Zagier formula and Kobayashi's $p$-adic Gross–Zagier formula combined with this simple observation.

Our second goal is to set up a conceptual framework in the context of $\Lambda$-adic Kolyvagin systems to treat analogues of Perrin-Riou's conjectures for motives of higher rank. We apply this general discussion in order to establish a link between Heegner points on a general class of CM abelian varieties and the (conjectural) Coleman–Rubin–Stark elements we introduce here. This can ben thought of as a higher dimensional version of Rubin's results on rational points on CM elliptic curves.