On abelian surfaces with potential quaternionic multiplication (Q2475537)

An Abelian surface \(A\) over a field \(K\) has a potential quaternionic multiplication if the ring \(\text{End}_{\overline K}(A)\), where \(\overline K\) is a separable closure of \(K\), of geometric endomorphisms of \(A\) is an order in an indefinite rational division quaternion algebra. In the paper under review the authors prove that if \(A/\mathbb{Q}\) is an Abelian surface with potential quaternionic multiplication by a hereditary order \(O\) of discriminant \(D\) in a quaternion algebra \(B\), then for the minimal field of definition \(L/\mathbb{Q}\) of the quaternionic multiplication on \(A\) and for \(F:= \text{End}_{\mathbb{Q}}(A)\otimes\mathbb{Q}\) either \(L\) is an imaginary quadratic field and \(F= \mathbb{Q}(\sqrt{m})\) is a real quadratic field such that \(B\simeq({-D\delta,m\over \mathbb{Q}})\) for any possible degree \(\delta\) of a polarization on \(A/\mathbb{Q}\), or \(L\) is purely imaginary dihedral extension of \(\mathbb{Q}\) of degree \([L:\mathbb{Q}]\geq 4\) and \(F=\mathbb{Q}\). The results should be seen as a strengthen of their previous results [J. Algebra 281, No. 1, 124--143 (2004; Zbl 1064.11043 )] in the special case that the base field is \(\mathbb{Q}\). They also give examples of Abelian surfaces which show that all the cases of their result can occur and, at the end, they give non-trivial examples of Abelian surfaces \(A/K\) over a quadratic imaginary field \(K\) such that all quaternionic endomorphisms of \(A\) are defined over \(K\) itself.