Elliptic curves with trivial conductor over quadratic fields
type: Item
instance of: scholarly article
title: Elliptic curves with trivial conductor over quadratic fields
zbMATH Open document ID: 0732.11028
author: Salvador Comalada
published in: Pacific Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: This paper deals with elliptic curves defined over quadratic number fields and having good reduction at all primes. It extends work of \textit{B. Setzer} [Pac. J. Math. 74, 235-250 (1978; Zbl 0394.14018)], and Ill. J. Math. 25, 233-245 (1981; Zbl 0471.14019)] and the reviewer [Pac. J. Math. 108, 451-463 (1983; Zbl 0491.14024)]. The results include the following: Those real quadratic number fields \({\mathbb{K}}\) are characterized which admit elliptic curves having a global minimal model and a 2-division point over \({\mathbb{K}}\), the characterization being achieved through the solvability of certain diophantine equations over \({\mathbb{Z}}\). - An effective procedure is given to determine all curves with the said property for each given real quadratic field \({\mathbb{K}}\). Also the number of isomorphism classes is given. A characterization is given of the real quadratic fields \({\mathbb{K}}\) of odd class number which admit curves with good reduction everywhere and having a \({\mathbb{K}}\)-rational 2-division point. A table is added containing all those curves defined over \({\mathbb{Q}}(\sqrt{d})\) in the range \(2\leq d<100\).
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 14G25
zbMATH DE Number: 4210244
zbMATH Keywords: elliptic curves
zbMATH Keywords: good reduction
zbMATH Keywords: global minimal model
zbMATH Keywords: diophantine equations
zbMATH Keywords: effective procedure
zbMATH Keywords: real quadratic field
zbMATH Keywords: number of isomorphism classes
zbMATH Keywords: 2-division point
zbMATH Keywords: table
reviewed by: Roelof J. Stroeker
MaRDI profile type: Publication
full work available at URL: pjm.1990.144.237
OpenAlex ID: W2087785998
DOI: 10.2140/PJM.1990.144.237
description: scientific article; zbMATH DE number 4210244