Arithmetic classification of Kuga fiber varieties of quaternion type
Let k be a totally real field, and B a quaternion algebra over k. From this (and several other data) a Kuga family \(f: A\to Y\) of abelian varieties over a base Y can be constructed (as is done by Shimura, Kuga, Mumford and many others). Some of the families thus constructed can be of Hodge type. There is a precise definition (due to Shimura) of the ``bottom field'' attached to a variety defined over \({\mathbb{C}}\). This paper discusses the bottom field for certain Kuga fiber families. It turns out that the definition of the bottom field for \(f: A\to Y\) is slightly different from the definition for Y (cf. 2.2-2.4). For a maximal order \({\mathfrak O}\) in B we find as bottom field for A/Y a finite abelian extension of the bottom field of Y, and the related Galois group is a subgroup of the class group of the Galois closure of k (cf. the main theorem in 2.5). In 2.6 we find an example (a family of abelian varieties of dimension 4), where the bottom field of Y and of A/Y are different. This paper definitely contributes to the subtle theory of fields of definition for the case of families of abelian varieties.
scientific article; zbMATH DE number 4123903
scientific article