Determination of elliptic curves with everywhere good reduction over real quadratic fields
type: Item
instance of: scholarly article
title: Determination of elliptic curves with everywhere good reduction over real quadratic fields
zbMATH Open document ID: 0955.11015
author: Takaaki Kagawa
published in: Archiv der Mathematik
publication date: 2001-02-28T00:00:00Z
review text: Let \(K={\mathbb Q}(\sqrt{29})\), let \(\varepsilon=(5+\sqrt{29})/2\) be a fundamental unit of \(K\), let \(E\) be the elliptic curve defined by the equation \(y^2+xy+\varepsilon^2y=x^3\). \textit{T. Nakamura} [J. Math. Soc. Japan 36, 701-707 (1984; Zbl 0546.14022)] proved that any elliptic curve \(\tilde{E}\) defined over \(K\) and having everywhere good reduction over \(K\), is in fact \(K\)-isogenous to \(E\). In this text the author determines explicitly one representative for each class of \(K\)-isomorphism of elliptic curves which are \(K\)-isogenous to \(E\): there are in this way exactly \(4\) of these representatives and they have everywhere good reduction over \(K\). The phenomenon above is in accordance with a conjecture of Pinch predicting the existence of a correspondence between elliptic curves having everywhere good reduction over real quadratic number fields, and the elliptic curves associated to some \({\mathbb Q}\)-simple factors of spaces of cusp forms of nebentype and weight \(2\), constructed by \textit{G. Shimura} [Introduction to the arithmetic theory of automorphic functions. Princeton Univ. Press (1994; Zbl 0872.11023)].
reviewed by: Federico Pellarin
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 11D25
zbMATH DE Number: 1346230
zbMATH Keywords: elliptic curve over quadratic number field
zbMATH Keywords: cubic diophantine equation
zbMATH Keywords: isogeny
zbMATH Keywords: Shimura elliptic curve
MaRDI profile type: Publication
full work available at URL: s000130050016
OpenAlex ID: W1980301429
DOI: 10.1007/S000130050016
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description: scientific article; zbMATH DE number 1346230