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 <poonen@math.berkeley.edu>