A remark on the modularity of abelian varieties of \(GL_2\)-type over \(\mathbb{Q}\)
type: Item
instance of: scholarly article
title: A remark on the modularity of abelian varieties of \(GL_2\)-type over \(\mathbb{Q}\)
zbMATH Open document ID: 0962.11025
author: Naoki Murabayashi
published in: Journal of Number Theory
publication date: 2001-06-17T00:00:00Z
review text: Let \(A/{\mathbb Q}\) be an abelian variety whose algebra of \(\mathbb Q\)-endomorphisms is a number field of degree equal to the dimension of \(A\) (\(A\) is said to be of \(GL_2\)-type over \({\mathbb Q}\)). The author proves that if \(A\) does not admit complex multiplication and \(A\) is ``modular over \(\mathbb C\)'', i.e., there exists a non-zero homomorphism \(J_1(N) \rightarrow A\) over \(\mathbb C\) (where \(J_1(N)\) is the Jacobian of the modular curve \(X_1(N)\) of level \(N\)), then \(A\) is ``modular over \(\mathbb Q\)'', i.e., isogenous over \(\mathbb Q\) to a factor of \(J_1(M)\) corresponding to a newform of weight two on \(\Gamma_1(M)\) for some integer \(M\). The proof is inspired by the case where \(A\) is one-dimensional [cf. \textit{B. Mazur}, Am. Math. Mon. 98, No. 7, 593-610 (1991; Zbl 0764.11021)].
reviewed by: Gunther Cornelissen
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 1466188
zbMATH Keywords: modularity
zbMATH Keywords: abelian variety
MaRDI profile type: Publication
full work available at URL: jnth.1999.2503
OpenAlex ID: W2042760231
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DOI: 10.1006/JNTH.1999.2503
description: scientific article; zbMATH DE number 1466188