Algebraic Families of Nonzero Elements of Shafarevich-tate Groups Jean-louis Colliot-thélène and Bjorn Poonen

The purpose of this note is to 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. More precisely, we will prove the following theorem. (Throughout this paper, P unadorned denotes PQ.) Theorem 1. There exists an open subscheme U of P containing all closed points of odd degree, and there exist smooth projective geometrically integral varieties A and X over Q equipped with dominant morphisms πA and πX to P 1 such that (1) AU := π A (U) is an abelian scheme of relative dimension 2 over U . (2) XU := π X (U) is an AU -torsor over U . (3) πX (X (kv)) = P(kv) for every local field kv ⊃ Q (archimedean or not). (4) X has no zero-cycle of odd degree over Q. (5) A is a non-constant family (i.e., there exist u, v ∈ U(Q) such that the fibers Au and Av are not isomorphic over Q) (6) The generic fiber of A → P is absolutely simple (i.e., simple over Q(t)).