Beilinson-Kato and Beilinson-Flach elements, Coleman-Rubin-Stark classes, Heegner points and the Perrin-Riou Conjecture

TL;DR

This paper links Beilinson-Kato classes, Heegner points, and Coleman-Rubin-Stark elements within Iwasawa theory, providing new insights into Perrin-Riou's conjecture and higher-dimensional analogues for CM abelian varieties.

Contribution

It demonstrates that a weak form of Perrin-Riou's conjecture follows from Iwasawa main conjectures and develops a framework for higher-rank motives using $ ext{Lambda}$-adic Kolyvagin systems.

Findings

Weak Perrin-Riou conjecture follows from Iwasawa main conjectures.

Established a connection between Heegner points and Coleman-Rubin-Stark elements.

Extended Rubin's results to higher-dimensional CM abelian varieties.

Abstract

Our first goal in this note 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, and deduce its refined versions in the supersingular case from this fact and a variety of Gross-Zagier formulae. 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 higher 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 is a higher dimensional version of Rubin's results on rational points on CM elliptic curves.