An explicit algebraic family of genus-one curves violating the Hasse principle, by Bjorn Poonen
We prove that for any t in Q, the curve
5 x^3 + 9 y^3 + 10 z^3 + 12((t^12-t^4-1)/(t^12-t^8-1)^3 (x+y+z)^3 = 0
in P^2 is a genus 1 curve violating the Hasse principle.
An explicit Weierstrass model for its Jacobian E_t is given.
The Shafarevich-Tate group of each E_t contains
a subgroup isomorphic to Z/3 x Z/3.
In addition to the DVI file "cubics.dvi"
the PostScript file "planecubics.eps"
should be downloaded for the figure.
Bjorn Poonen <email@example.com>