An explicit algebraic family of genus-one curves violating the Hasse principle

Bjorn Poonen

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arXiv · math/9910124

An explicit algebraic family of genus-one curves violating the Hasse principle

Bjorn Poonen

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Abstract

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.

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Bjorn Poonen. 1999-10-25. An explicit algebraic family of genus-one curves violating the Hasse principle. https://arxiv.org/abs/math/9910124

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