Algebraic families of nonzero elements of Shafarevich-Tate groups

Abstract

Principal homogeneous spaces under an abelian variety defined over a number field k k may have rational points in all completions of the number field without having rational points over k k . Such principal homogeneous spaces represent the nonzero elements of the Shafarevich-Tate group of the abelian variety. We produce nontrivial, one-parameter families of such principal homogeneous spaces. The total space of these families are counterexamples to the Hasse principle. We show these may be accounted for by the Brauer-Manin obstruction.