Conjugation of Kuga fiber varieties (Q1195989)

A Kuga fiber variety is a family of abelian varieties parametrized by an arithmetic variety and constructed from a symplectic representation of an algebraic group. A lower bound for the field of definition of a complex algebraic variety \(X\) is given by Bot\(X\), the strong bottom field; this is a field \(k\) such that for any automorphism \(\sigma\) of \(\mathbb{C}\), \(X^ \sigma\cong X\) if and only if \(\sigma\) is the identity on \(k\). The principal result of this paper is that if \(A\to V\) is a Kuga fiber variety defined by a \(\mathbb{Q}\)-irreducible representation satisfying a certain rigidity condition, and if the generic fibers are principal abelian varieties, then Bot\(A\) is an abelian extension of Bot\(V\). In fact, the Galois group of Bot\(A\) over Bot\(V\) is embedded into the class group of a maximal order in a simple algebra.