Homomorphisms from the group of rational points on elliptic curves to class groups of quadratic number fields
type: Item
instance of: scholarly article
title: Homomorphisms from the group of rational points on elliptic curves to class groups of quadratic number fields
zbMATH Open document ID: 0811.14035
author: Ragnar Soleng
published in: Journal of Number Theory
publication date: 1995-05-02T00:00:00Z
review text: Let \(E: Y^ 2= X^ 3+ a_ 2 X^ 2+ a_ 4 X+ a_ 6\) be an elliptic curve defined over a field \(k\) (of characteristic \(\neq 2\)). The author exhibits a homomorphism from a subgroup of the \(k\)-rational points of \(E\), called the group of primitive points on \(E\), to the ideal class group of the order \(\mathbb{Z}+ \mathbb{Z} \sqrt{a_ 6}\). A key remark is that an integral point \((x,y)\) on \(E\) gives birth to the quadratic form \((x+n)S^ 2+ 2y ST+ (n^ 2+ (a_ 2+ x)n+ x^ 2+ a_ 2 x+a_ 4) T^ 2\) of discriminant \(D= 4(-n^ 3+ a_ 2n^ 2- a_ 4 n+ a_ 6)\). This looks like a promising direction.
reviewed by: Claude Levesque
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 556366
zbMATH Keywords: elliptic curve
zbMATH Keywords: group of primitive points
zbMATH Keywords: ideal class group
zbMATH Keywords: integral point
MaRDI profile type: Publication
full work available at URL: jnth.1994.1013
OpenAlex ID: W2088028324
DOI: 10.1006/JNTH.1994.1013
description: scientific article; zbMATH DE number 556366