Kato's Euler system and rational points on elliptic curves. I: A \(p\)-adic Beilinson formula
type: Item
instance of: scholarly article
title: Kato's Euler system and rational points on elliptic curves. I: A \(p\)-adic Beilinson formula
zbMATH Open document ID: 1317.11071
author: Massimo Bertolini
author: Henri Darmon
published in: Israel Journal of Mathematics
publication date: 2014-06-25T00:00:00Z
review text: For a modular elliptic curve \(E/\mathbb Q\), Kato's construction of an Euler system arising from \(p\)-adic families of Beilinson elements in the \(K\)-theory of modular curves in [\textit{K. Kato}, in: Cohomologies \(p\)-adiques et applications arithmétiques (III). Paris: Société Mathématique de France. 117--290 (2004; Zbl 1142.11336)] yields a global class \(\kappa\in H^1(\mathbb Q,V_p(E))\), where \(V_p(E)\) is the \(p\)-adic Galois representation attached to \(E\). Kato's reciprocity law implies that \(\kappa\) is crystalline, hence belongs to the \(p\)-adic Selmer group of \(E\), precisely when the Hasse-Weil \(L\)-series \(L(E,s)\) vanishes at \(s = 1\). In this case, \textit{B. Perrin-Riou} [Ann. Inst. Fourier 43, No. 4, 945--995 (1993; Zbl 0840.11024)] conjectures that the image \(\mathrm{res}_p(\kappa)\) in \(H_f^1(\mathbb Q_p,V_p(E))\) is non-zero if and only if \(L'(E,1)\) is non-zero, and predicts a precise relation between the logarithm of \(\mathrm{res}_p(\kappa)\) and the formal group logarithm of a global point in \(E(\mathbb Q)\). The present article is the first in a series of three, whose ultimate goal is the demonstration of Perrin-Riou's conjecture. The main step here is a proof of a \(p\)-adic Beilinson formula relating the syntomic regulators (in the sense of \textit{R. Coleman} and \textit{E. de Shalit} [Invent. Math. 93, No. 2, 239--266 (1988; Zbl 0655.14010)] and \textit{A. Besser} [Isr. J. Math. 120, Part B, 291--334 (2000; Zbl 1001.19003); ibid. 120, Part B, 335--359 (2000; Zbl 1001.19004)]) of certain distinguished elements in the \(K\)-theory of modular curves to the special values at integer points \(\geq 2\) of the Mazur and Swinnerton-Dyer \(p\)-adic \(L\)-function attached to a cusp form \(f\) of weight 2. The authors' proof is independent from Kato's reciprocity law (which can be re-derived from it), and makes heavy use of results and techniques of \textit{H. Darmon} and \textit{V. Rotger} [Ann. Sci. Éc. Norm. Supér. (4) 47, No. 4…
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 6308376
zbMATH Keywords: Kato's Euler system
zbMATH Keywords: \(p\)-adic regulators
zbMATH Keywords: special values
zbMATH Keywords: \(K\)-theory of modular curves
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
full work available at URL: s11856-013-0047-2
OpenAlex ID: W2021817402
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DOI: 10.1007/S11856-013-0047-2
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description: scientific article; zbMATH DE number 6308376