An explicit algebraic family of genus-one curves violating the Hasse principle
Bjorn Poonen
Abstract
We prove that for any t in Q, the curve 5 x3 + 9 y3 + 10 z3 + 12((t12-t4-1)/(t12-t8-1))3 (x+y+z)3 = 0 in P2 is a genus 1 curve violating the Hasse principle. An explicit Weierstrass model for its Jacobian Et is given. The Shafarevich-Tate group of each Et contains a subgroup isomorphic to Z/3 x Z/3.
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