On some examples of modular QM-abelian surfaces
Let \(f = \sum^\infty_{n=1} a_n q^n\) be a newform of weight two on \(\Gamma_0 (N)\) with trivial Nebentypus character such that \(K_f = \mathbb Q( \{ a_n \}^\infty_{n=1})\) is a real quadratic field, and let \(A_f\) be the associated abelian surface. Let \(\chi\) be a quadratic Dirichlet character such that \(a_p^\sigma = \chi (p) a_p\) for all \(p \nmid N\), where \(\sigma\) is the nontrivial automorphism of \(K_f / \mathbb Q\). In this paper, the author shows that \[ \text{End}_{\overline{\mathbb Q}} (A_f) \otimes \mathbb Q = \biggl( \frac{d, \chi(-1) r} {\mathbb Q} \biggr) , \] where \(d\) is the discriminant of \(K_f\), \(r\) is the conductor of \(\chi\), and \((\frac{a,b} {\mathbb Q})\) denotes the quaternion algebra over \(\mathbb Q\) with reduced norm form \(x^2 -ay^2 -bz^2 + abw^2\). He also describes \(N\) when \(A_f\) has quaternion multiplication, that is, when \(\text{End}_{\overline{\mathbb Q}} (A_f) \otimes \mathbb Q\) is equal to a quaternion division algebra, and then displays the examples for various specific values of \(N\).
scientific article; zbMATH DE number 897708
scientific article