We show that there exist non-trivial families of algebraic varieties for which all the fibers above rational points (or even above points of odd degree) are torsors of abelian varieties representing nonzero elements of their Shafarevich-Tate groups.

Jean-Louis Colliot-Thelene and Bjorn Poonen <colliot@math.u-psud.fr, poonen@math.berkeley.edu>