In the case of complex elliptic surfaces we can consider the elliptic modular surfaces, as Shioda did in his paper "On elliptic modular surfaces". The latter have Mordell-Weil rank 0.
Is there a similar theorem in the case of a family of complex abelian surfaces parametrized by a curve? In other words I am asking if there is a type of abelian schemes $\mathcal{A} \rightarrow C$, where $C$ is a curve, whose Mordell-Weil rank is 0. In particular I am interested in the case of a family of simple jacobians of genus two curves.
