On QM-abelian surfaces with model of GL\(_{2}\)-type over \(\mathbb Q\)
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.
scientific article; zbMATH DE number 2164820
scientific article