The Selmer groups of elliptic curves.
type: Item
instance of: scholarly article
title: The Selmer groups of elliptic curves.
zbMATH Open document ID: 1036.11024
author: Fu Zheng Wang
publication date: 2003-12-17T00:00:00Z
review text: Fix a prime \(p\) and let \(E\) be an elliptic curve with discriminant \(\Delta\), defined over some number field \(K\) containing a \(p\)-torsion point. Let \(S\) denote the set of all places of \(K\) dividing \(p\Delta\infty\), and let \(C_{K,S}\) denote the \(S\)-class group of \(K\). If \(E[p] \subset K\), then the \(p\)-Selmer group \(S^{(p)}(E/K)\) has order \(\# S^{(p)}(E/K) \leq p^{2 \# S}(\# C_{K,S}[p])^2\) (actually this bound is proved with \(S\) replaced by a slightly smaller set); if \(E(K)[p]\) has order \(p\), a similar bound is derived involving the \(S\)-class group of \(L = K(E[p])\) for a suitable set \(S\) of places in \(L\).
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 1921180
zbMATH Keywords: elliptic curves
zbMATH Keywords: Selmer group
zbMATH Keywords: class group
MaRDI profile type: Publication
full work available at URL: s10114-002-0211-9
OpenAlex ID: W2352245695
cites work: Q4288919
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cites work: Arithmetic on Curves of Genus 1 : III. The Tate-Šafarevič and Selmer Groups
cites work: Q3710637
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cites work: Q3856855
DOI: 10.1007/S10114-002-0211-9
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 1921180