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

Authors
  • Poonen, Bjorn
Type
Preprint
Publication Date
Oct 25, 1999
Submission Date
Oct 25, 1999
Identifiers
arXiv ID: math/9910124
Source
arXiv
License
Unknown
External links

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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