Algebraic Families of Nonzero Elements of Shafarevich-tate Groups

Around 1940, Lind [Lin] and (independently, but shortly later) Reichardt [Re] discovered that some genus 1 curves over Q, such as 2y = 1− 17x, violate the Hasse principle; i.e., there exist x, y ∈ R satisfying the equation, and for each prime p there exist x, y ∈ Qp satisfying the equation, but there do not exist x, y ∈ Q satisfying the equation. In fact, even the projective nonsingular model has no rational points. We address the question of constructing algebraic families of such examples. Can one find an equation in x and y whose coefficients are rational functions of a parameter t, such that specializing t to any rational number results in a genus 1 curve violating the Hasse principle? The answer is yes, even if we impose a nontriviality condition; this is the content of our Theorem 1.2. Any genus 1 curve X is a torsor for its Jacobian E. Here E is an elliptic curve: a genus 1 curve with a rational point. If X violates the Hasse principle, then X moreover represents a nonzero element of the Shafarevich-Tate group X(E). We may relax our conditions by asking for families of torsors of abelian varieties. In this case we can find a family with stronger properties: one in which specializing the parameter to any number α ∈ Q of odd degree over Q results in a torsor of an abelian variety over Q(α) violating the Hasse principle. A precise version of this result follows. (Throughout this paper, P unadorned denotes the projective line over Q.) Theorem 1.1. 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 such that (1) AU := π−1 A (U) is an abelian scheme of relative dimension 2 over U .