Good reduction of elliptic curves defined over \({\mathbb Q}(^ 3\sqrt{2})\)
type: Item
instance of: scholarly article
title: Good reduction of elliptic curves defined over \({\mathbb Q}(^ 3\sqrt{2})\)
zbMATH Open document ID: 0615.14016
DOI: 10.1007/BF01313493
author: Massimo Bertolini
author: Giuseppe Canuto
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: A classical theorem of J. Tate states that there are no elliptic curves over \({\mathbb{Q}}\) with good reduction everywhere. Generalizations to complex quadratic fields are due to Stroeker and Setzer. Stroeker proved that if an elliptic curve on a imaginary quadratic field has a global minimal equation then it has bad reduction somewhere. Setzer proved that there are no elliptic curves with a good reduction everywhere on imaginary quadratic fields with class number prime to 6. In these papers the finiteness of the group of units is crucial. We prove that on \(K={\mathbb{Q}}(^ 3\sqrt{2})\) there are no elliptic curves with good reduction everywhere. In this field the group of units is infinite. We show first that any such curve must have a point of order two K-rational. We show then that the solutions of the diophantine equations we have to consider do not correspond to elliptic curves with good reduction everywhere using Tate's algorithm.
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 3995864
zbMATH Keywords: no elliptic curves with good reduction everywhere group of units
zbMATH Keywords: cubic ground field
MaRDI profile type: Publication
cites work: Ellitpic curves and modular functions
cites work: Q3669562
cites work: Q5547218
cites work: Elliptic curves over complex quadratic fields
cites work: Reduction of elliptic curves over imaginary quadratic number fields
cites work: Q4109764
cites work: The arithmetic of elliptic curves
cites work: Algorithm for determining the type of a singular fiber in an elliptic pencil
full work available at URL: https://doi.org/10.1007/bf01313493
OpenAlex ID: W2044452554
description: scientific article; zbMATH DE number 3995864