Zeta functions of Kuga fiber varieties (Q1116994)

Let \(A_ 1\to V\) and \(A_ 2\to V\) be two families of abelian varieties of Kuga's type [cf. \textit{M. Kuga}, Proc. Symp. Pure Math., 9, 338-346 (1966; Zbl 0173.489)] with same base V, both defined over a number field k. The author obtains relations between the zeta functions of \(A_ 1\) and \(A_ 2\) by showing that any rational cycle on a fiber of a Kuga family which is invariant under monodromy is a Hodge cycle, and then applying Deligne's theorem on absolute Hodge cycles. An interesting illustration of his result, based on constructions of Kuga, Ohta and Addington, is provided by a non-split family over a product of two Shimura curves.