On the field of definition for modularity of CM elliptic curves
type: Item
instance of: scholarly article
title: On the field of definition for modularity of CM elliptic curves
zbMATH Open document ID: 1079.11030
author: Naoki Murabayashi
published in: Journal of Number Theory
publication date: 2005-04-28T00:00:00Z
review text: Let \(E\) be CM elliptic curve defined over a number field \(F\) with complex multiplication by an imaginary quadratic field \(K\). It is known that there exists a new form \(f\) such that \(\Hom_{\overline {\mathbb Q}}(E,J_f)\neq\{0\}\), where \(J_f\) denotes the abelian variety over \(\mathbb Q\) associated to \(f\) by the Eichler-Shimura theory. In this paper, the author considers the problem to determine the field over which the homomorphisms are defined, and obtains a criterion for \(\Hom_F(E,J_f)\) to be nontrivial. Let \(F'= FK\) and \(\beta_{E/F'}: F^{\prime\times}_{\mathbb A}\to \mathbb C^\times\) denote the Grössen-character of \(E/F'\). He shows that \(\Hom_F(E,J_f)\neq\{0\}\) if and only if there exists a Grössen-character \(\gamma: K^\times_{\mathbb A}\to\mathbb C\) such that \(\gamma\circ N_{F'/K}= \beta_{E/F'}\) and the latter is equivalent to the condition that every torsion element of \(E\) is rational over \(FK_{ab}\). Furthermore, he determines the isogeny decomposition of \(J_f\) over \(F\) when these conditions are met.
reviewed by: Fumio Hazama
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 11G18
zbMATH DE Number: 2162040
zbMATH Keywords: complex multiplication
zbMATH Keywords: elliptic curve
zbMATH Keywords: modular variety
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.05.004
OpenAlex ID: W2065412526
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DOI: 10.1016/J.JNT.2004.05.004
description: scientific article; zbMATH DE number 2162040