TITLE: Examples of rational families of abelian varieties.
QUESTION [15 upvotes]: I'd like to know examples of non-trivial families of abelian varieties over rational bases (e.g. open subschemes of the projective line P^1).
One can generate many examples as Jacobians of rational families of curves (e.g. the hyperellitpic family, plane curves, complete intersections). Prym varieties are another example.
Are there any examples which are not obviously Jacobians of a family of curves? I would like to know both principally polarized and non principally polarized examples.

REPLY [3 votes]: There are nice examples of Gross and Popescu of Calabi-Yau threefolds fibered over $\mathbb P^1$ with generic fibers being abelian surfaces. See  arXiv:math/000108 and arXiv:0904.3354.