We construct non-trivial families of torsors of abelian varieties parameterized by an open subset of ${\mathbf P}^1$ over $\mathbf Q$, such that every rational fiber (and in fact each fiber above a closed point of odd degree) represents a nonzero element of the Shafarevich-Tate group of the corresponding abelian variety. This is joint work with J.-L. Colliot-Thélène.