On torsion subgroups of elliptic curves with integral \(j\)-invariant over imaginary cyclic quartic fields
type: Item
instance of: scholarly article
title: On torsion subgroups of elliptic curves with integral \(j\)-invariant over imaginary cyclic quartic fields
zbMATH Open document ID: 0918.11029
author: Toshikaru Kishi
published in: Tokyo Journal of Mathematics
publication date: 1998-05-13T00:00:00Z
review text: The author considers the problem of determining the torsion group of an elliptic curve \(E\) with integral modular invariant \(j\) over a number field \(K\). He solves the problem for imaginary cyclic quartic fields \(K\). However, he has to impose the additional condition that the residue degrees \(f_2\) and \(f_3\) of a prime of \(K\) lying over 2 resp. 3 both are less than 4. Under this condition, he obtains ten groups as possible torsion groups of \(E\) over \(K\), and all these groups do really occur. The problem was solved earlier by the reviewer and his coauthors if \(K\) is a quadratic or cubic or real biquadratic or totally imaginary biquadratic field. The real biquadratic case is treated in the later paper with the title ``Computing the torsion group of elliptic curves by the method of Gröbner bases'' [\textit{A. Pethő}, \textit{J. Stein}, \textit{Th. Weis}, and \textit{H. G. Zimmer}, Prog. Comput. Sci. Appl. 15, 245-265 (1998)]. On the other hand, the cubic case is divided into the subcases, where \(K\) is pure cubic or cyclic cubic [see the reviewer and \textit{A. Pethő}, On a system of norm-equations over cyclic cubic number fields, Publ. Math. 53, 317-332 (1998; Zbl 0911.11020)] or general cubic [see the reviewer, \textit{A. Pethő}, \textit{J. Stein} and \textit{Th. Weis}, Int. J. Algebra Comput. 7, 353-413 (1997; Zbl 0868.11030)]. The integrality condition on \(j\) is stipulated here in order to exclude multiplicative reduction of \(E\) over \(K\) when the order of the torsion group depends on \(E\) over \(K\). Of course, the integrality condition on \(j\) is removed by Mazur for \(K=\mathbb{Q}\) and by Kamienny, Kenku and Momose if \(K\) is quadratic over \(\mathbb{Q}\). We mention in this connection also a paper of \textit{L. Merel} [Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124, 437-449 (1996)] in which the famous boundedness conjecture for the torsion group of \(E\) over an arbitrary number field \(K\) …
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 1130798
zbMATH Keywords: complex cyclic quartic number field
zbMATH Keywords: torsion group
zbMATH Keywords: elliptic curve
zbMATH Keywords: integral modular invariant
zbMATH Keywords: imaginary cyclic quartic fields
reviewed by: Horst G. Zimmer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270042106
OpenAlex ID: W1994272915
DOI: 10.3836/TJM/1270042106
description: scientific article; zbMATH DE number 1130798