On QM-abelian surfaces with model of GL\(_{2}\)-type over \(\mathbb Q\)
type: Item
instance of: scholarly article
title: On QM-abelian surfaces with model of GL\(_{2}\)-type over \(\mathbb Q\)
zbMATH Open document ID: 1081.14064
author: Naoki Murabayashi
published in: Journal of Number Theory
publication date: 2005-05-04T00:00:00Z
review text: An abelian variety of dimension \(2d\) defined over an algebraic number field is called a \(QM\)-abelian variety, if there exists a totally real field \(F\) of degree \(d\) over \(\mathbb{Q}\) and a totally indefinite quaternion algebra \(B\) over \(F\) such that the endomorphism algebra \(\text{End}^0(A)\) contains \(B\) as a \(\mathbb{Q}\)-subalgebra. An abelian variety \(A\) defined over \(\mathbb{Q}\) is called of \(\text{GL}_2\)-type over \(\mathbb{Q}\) if \(\text{End}^0_{\mathbb{Q}}(A)\) is a number field of degree equal to the dimension of \(A\) over \(\mathbb{Q}\). This paper is concerned with characterizing those \(QM\)-abelian varieties which have a model of \(\text{GL}_2\)-type over \(\mathbb{Q}\). In fact, it gives such a characterization in the case of abelian surfaces. This implies that the Hasse-Weil \(L\)-function of the surfaces are related to \(L\)-functions of modular forms of one variable.
reviewed by: Herbert Lange
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 11G15
zbMATH DE Number: 2164820
zbMATH Keywords: abelian variety
zbMATH Keywords: quaternion multiplication
zbMATH Keywords: GL\(_2\)-type over \(\mathbb Q\)
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.07.018
OpenAlex ID: W2012409133
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DOI: 10.1016/J.JNT.2004.07.018
description: scientific article; zbMATH DE number 2164820