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

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.

35 pages, updated the introduction (and the bibliography) so as to reflect recent developments in the field (mostly pertaining to the Iwasawa theory at critical slope)

References in corpus (7)

• Rankin--Eisenstein classes and explicit reciprocity laws
• Indivisibility of Heegner points in the multiplicative case
• Iwasawa theory of overconvergent modular forms, I: Critical $p$-adic $L$-functions
• On the exceptional zeros of $p$-non-ordinary $p$-adic $L$-functions and a conjecture of Perrin-Riou
• P-adic heights and p-adic Hodge theory
• Anticyclotomic p-ordinary Iwasawa Theory of Elliptic Modular Forms
• Selmer complexes and the $p$-adic Hodge theory

Cited by in corpus (1)

• On the theory of higher rank Euler, Kolyvagin and Stark systems, III: applications