Construction of a homomorphism concerning Euler systems for an elliptic curve
type: Item
instance of: scholarly article
title: Construction of a homomorphism concerning Euler systems for an elliptic curve
zbMATH Open document ID: 1188.11029
author: Rei Otsuki
published in: Tokyo Journal of Mathematics
publication date: 2009-09-16T00:00:00Z
review text: The aim of this paper is to generalize the main results contained in the work [\textit{M. Kurihara}, Invent. Math. 149, No. 1, 195--224 (2002; Zbl 1033.11028)] comparing Euler systems and what the author here calls ``admissible systems''. In the quoted work Kurihara constructed a pairing on cohomology groups attached to an elliptic curve \(E/\mathbb{Q}\) with values in group rings and studies the behavior of this pairing in the cyclotomic \(\mathbb{Z}_p\)-extension of \(\mathbb{Q}\) where \(p\) is a prime of good, supersingular reduction for \(E\). By fixing one explicit argument, one deduces from the pairing a group homomorphism \[ \mathcal{P}_{N}:H^1(\mathbb{Q}_{p}(\zeta_N),V_pE)\longrightarrow \mathbb{Q}_p[\mathrm{Gal}(\mathbb{Q}(\zeta_N)/\mathbb{Q})]\tag{1} \] and the main goal of the paper under review is to compare the Euler system belonging to the left-hand side of (1) constructed by \textit{K. Kato} in [Astérisque 295, 117--290 (2004; Zbl 1142.11336)] to the system of modular elements in the group rings appearing in (1) constructed by \textit{B. Mazur} and \textit{J. Tate} [ Duke Math. J. 54, 711--750 (1987; Zbl 0636.14004)], together with some integrality property of the above morphism: this is done in a greater generality than Kurihara's but with similar techniques. More precisely, let \(E/\mathbb{Q}\) be an elliptic curve and let \(p\) be an odd prime of good, supersingular reduction for \(E\). Denoting by \(T_pE\) the \(p\)-adic Tate module of \(E\) and by \(V_pE\) the representation \(T_pE\otimes\mathbb{Q}_p\), \textit{K. Kato} defined in [loc. cit.] an Euler system \((z_N)_{N\geq 1}\) of elements \(z_N\in H_{\acute{e}t}^1(\mathbb{Z}[\mu_N,1/S],V_pE)\). This means that, for every prime number \(q\), the following relation holds \[ \text{Norm}_{qN/N}(z_{qN})=\begin{cases} z_N & q\mid N \\F_q(\text{Frob}_q^{-1})z_n & q \nmid N,\end{cases} \tag{2} \] where \(F_q(T)=1-(a_q/q)T+(\varepsilon_q/q)T^2\in\mathbb{Q}[T]\) is the polynomial whose coefficient \(\var…
reviewed by: Filippo Nuccio
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11S25
zbMATH DE Number: 5604247
zbMATH Keywords: Euler systems
zbMATH Keywords: modular elements
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1249648421
OpenAlex ID: W2054291546
cites work: Q3972724
cites work: Division values in local fields
cites work: On the theory of commutative formal groups
cites work: Q4831110
cites work: Iwasawa theory for elliptic curves at supersingular primes
cites work: On the Tate-Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction. I.
cites work: Q3499979
cites work: Refined conjectures of the ``Birch and Swinnerton-Dyer type''
cites work: On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer
cites work: \(p\)-adic \(L\)-functions of an elliptic curve and rational points
DOI: 10.3836/TJM/1249648421
description: scientific article; zbMATH DE number 5604247