The 2-class groups of cubic fields and 2-descents on elliptic curves
type: Item
instance of: scholarly article
title: The 2-class groups of cubic fields and 2-descents on elliptic curves
zbMATH Open document ID: 0770.11049
author: Mayumi Kawachi
author: Shin-Ichi Nakano
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1993-06-17T00:00:00Z
review text: Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) with no rational point of order two and \(K\) the cubic field generated by a point of order two. Under some local conditions, there is an exact sequence which relates the 2-class group \(C(K)\) of \(K\) to the 2-Selmer group of \(E/\mathbb{Q}\). Combining the 2-descent exact sequence of \(E/\mathbb{Q}\), one can connect \(C(K)\) with the Mordell-Weil group \(E(\mathbb{Q})\) of \(E/\mathbb{Q}\), and give an inequality on their ranks. The topic of the simplest cubic fields is emphasized in this paper. For such fields, it is possible to describe more precisely the relation of two groups \(C(K)\) and \(E(\mathbb{Q})\) by means of the twists of elliptic curves, and a better estimation of the 2-rank of \(C(K)\) is obtained.
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11R16
zbMATH DE Number: 203562
zbMATH Keywords: ideal class group
zbMATH Keywords: elliptic curve
zbMATH Keywords: cubic field
zbMATH Keywords: exact sequence
zbMATH Keywords: Mordell- Weil group
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227250
OpenAlex ID: W2022500759
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cites work: Q3710637
cites work: Class numbers of cubic cyclic fields
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DOI: 10.2748/TMJ/1178227250
description: scientific article; zbMATH DE number 203562