Beilinson-Kato and Beilinson-Flach elements, Coleman-Rubin-Stark classes, Heegner points and a conjecture of Perrin-riou. (English) Zbl 1473.11119
Kurihara, Masato (ed.) et al., Development of Iwasawa theory – the centennial of K. Iwasawa’s birth. Proceedings of the international conference “Iwasawa 2017”, University of Tokyo, Tokyo, Japan, July 19–28, 2017. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 86, 141-193 (2020).
Summary: 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.
For the entire collection see [Zbl 1462.11006].
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.
For the entire collection see [Zbl 1462.11006].
MSC:
| 11G05 | Elliptic curves over global fields |
| 11G07 | Elliptic curves over local fields |
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
| 11R23 | Iwasawa theory |
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
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