Products of CM elliptic curves
type: Item
instance of: scholarly article
title: Products of CM elliptic curves
zbMATH Open document ID: 1237.11027
author: Ernst Kani
published in: Collectanea Mathematica
publication date: 2011-11-04T00:00:00Z
review text: The present paper concerns for an arbitrary field \(K\) the problem whether a given abelian variety \(A\) over \(K\) is isomorphic to a product of isogenous CM-elliptic curves. Previously ciriteria have been worked out over \(\mathbb C\) by \textit{T. Shioda} and \textit{N. Mitani} in dimension 2 [Lect. Notes Math. 412, 259--287 (1974; Zbl 0302.14011)] and implicitly by \textit{C. Schoen} in higher dimension [J. Reine Angew. Math. 429, 115-123 (1992; Zbl 0744.14030)]. Either approach relies on period lattices. Presently the role of the lattice is replaced by ideal classes in homomorphism rings of elliptic curves. The main advantage of this replacement is that it works over arbitrary fields. As a special case, the author treats the isomorphy problem for two abelian varieties which are already known to be of product type.
reviewed by: Matthias Schütt
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14K22
Mathematics Subject Classification ID: 14K10
zbMATH DE Number: 5968124
zbMATH Keywords: elliptic curve
zbMATH Keywords: complex multiplication
zbMATH Keywords: abelian variety
zbMATH Keywords: isomorphic
zbMATH Keywords: ideal class
zbMATH Keywords: homomorphism ring
MaRDI profile type: Publication
full work available at URL: s13348-010-0029-1
OpenAlex ID: W2050019333
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DOI: 10.1007/S13348-010-0029-1
description: scientific article; zbMATH DE number 5968124