An annihilator for the \(p\)-Selmer group by means of Heegner points
type: Item
instance of: scholarly article
title: An annihilator for the \(p\)-Selmer group by means of Heegner points
zbMATH Open document ID: 0853.11049
author: Massimo Bertolini
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 1994-10-10T00:00:00Z
full work available at URL: https://eudml.org/doc/244212
review text: Let \(E/ \mathbb{Q}\) be a modular elliptic curve, with conductor \(N\), and let \(K\) be an imaginary quadratic field such that the primes dividing \(N\) split in \(K\), and \({\mathcal O}^\times_K= \{\pm 1\}\). Denote by \(K_\infty/ K\) the anticyclotomic \(\mathbb{Z}_p\)-extension of \(K\) and by \(K_n\) the subextension of degree \(p^n\), where \(p\) is a prime not dividing \(6N \text{disc} (K)\). The Galois group of \(K_n/ K\) will be denoted by \(G_n\). Under assumptions clearly, albeit succinctly stated, there is a family of Heegner points defined over \(K_\infty\) and, if \(\alpha_n\) is the standard Heegner point over \(K_n\), \({\mathcal E}_n= R_n \alpha_n\) is the submodule of \(E(K_n)/ pE(K_n)\) generated by \(\alpha_n\) and \(R_n= \mathbb{Z}/ p\mathbb{Z} [G_n]\). Fix \(n\) (and delete the subscript \(n\)) and let \(\gamma\) be a generator for \(G\), \(\omega= \gamma-1\) and \({\mathcal E} \cong (\omega)^{t_0}\). The main theorem is that the ideal \({\mathcal L} R= \omega^{t_0} R\) (the characteristic ideal of \(U/{\mathcal E}\), where \(U\) is some \(R\)-submodule of the Selmer group and is, in a sense a \(\text{mod } p\) universal norm sub-module of \(\text{Sel}_p (E/H)\)), annihilates \(\text{Sel}_p (E/H)/U\) provided that \({\mathcal E}\neq 0\). The paper concludes with a reformulation of the main result in terms of Iwasawa theory. The proofs rely heavily on Kolyvagin theory for elliptic curves, which the reviewer found clearly presented in \textit{B. H. Gross} [Proc. Symp. Durham, Lond. Math. Soc. Lect. Note Ser. 153, 235-256 (1991; Zbl 0743.14021)].
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14G25
zbMATH DE Number: 650741
zbMATH Keywords: annihilator
zbMATH Keywords: anticyclotomic \(\mathbb{Z}_ p\)-extension of an imaginary quadratic field
zbMATH Keywords: modular elliptic curve
zbMATH Keywords: Heegner points
zbMATH Keywords: Selmer group
zbMATH Keywords: Iwasawa theory
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 650741