Examples of rational families of abelian varieties. 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.
 A: It is known that $A_g$ is unirational for $g\leq 5$, so at least for these g, we have some nontrivial families over open subsets of projective spaces. For $g> 3$, $\dim M_g < \dim A_g$, so these families are not Jacobians.
A: 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.
A: Section 5 of Faltings: Arakelov's theorem for abelian varieties is dedicated to the construction of an elaborate example of a family of abelian varieties that do not satisfy a condition Faltings calls $(*)$. I believe the condtion implies that the abelian varieties in the family, at least the general fiber, cannot be a Jacobian. The base of the family is $\mathbb H$ the upper-half plane, so it won't give you a family over a rational curve, but the construction itself is pretty involved so undertsanding it might help with understanding families of abelian varieties in general and might also give you some ideas for this problem.
A: David, I don't know if you are still interested in this, it's been over a year. I just stumbled upon your question in the depths of MO. I often found that Weil restrictions of elliptic curves give nice families of examples on which you can test things. 
E.g. take a family of elliptic curves $y^2=x^3+t$ and Weil restrict it from ${\mathbb Q}(i)$ to ${\mathbb Q}$. Writing $x=x_1+x_2i$ and similarly for $y$ and $t$, expanding the equation and breaking it into real and imaginary parts, you get a family of 2-dimensional abelian varieties over ${\mathbb Q}(t_1,t_2)$ given by two equations in a 4-dimensional space,
$$
  y_1^2-y_2^2 = x_1^3 - 3x_1 x_2^2 + t_1, \qquad
  2y_1y_2 = x_1^2x_2 - 3x_2^3 + t_2.
$$
Alternatively, you fix the elliptic curve, but you let the extension vary with $t$ (e.g. ${\mathbb Q}(t^{1/3})$), or both, and you also get interesting families.
The really nice thing is that as opposed to Jacobians, Weil restrictions are trivial to write down in terms of equations. Over the algebraic closure they are isogenous isomorphic to products of elliptic curves (making them boring), but for arithmetic applications they are interesting. There is a small extension of this construction, when you do not base change the elliptic curve but you "tensor it with a ${\mathbb Z}-$module with a Galois action", which is not necessarily a permutation module. This is explained in Milne's paper "On the arithmetic of abelian varieties" (Invent. Math. 1972) section 2, and it is useful if you want to write down non-principally polarised examples.
A: One can construct some families over rational bases which are not Jacobians by taking quotients:
For example, let $A$ be a fixed abelian variety of dimension $> 1$ and let $S$ be the space of all smooth complete interesection curves for some very ample line bundle on $A$. For any $s \in S$, let $C_s$ denote the corresponding curve in $A$. The inclusion of $C_s$ in $A$ induces a surjective morphism from $J(C_s)$ (the Jacobian of $C_s$) to $A$ and so by duality a morphism $A^{t}$ to $J(C_s)$ where $A^{t}$ is the dual abelian variety of $A$. The quotient of $J(C_s)$ by the image of $A^t$ gives a family of abelian varieties over $S$. It can be shown using monodromy that this family is non-trivial.
A: How about intermediate Jacobians of cubic threefolds?  It's easy to write down rational families of cubic hypersurfaces in P^4; it might be much harder to say anything about the corresponding family of abelian varieties, depending on what features you're looking for.
A: Just for the curiosity, into which category falls the fibration which has an elliptic curve with given j-invariant over a point $j \in \mathbb{P}^1$?
