An explicit algebraic family of genus-one curves violating the Hasse principle
Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 263-274.

We prove that for any t𝐐, the curve 5x 3 +9y 3 +10z 3 +12t 2 +82 t 2 +22 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×𝐙/3.

Nous montrons que pour tout t𝐐, la courbe 5x 3 +9y 3 +10z 3 +12t 2 +82 t 2 +22 3 (x+y+z) 3 =0 de 𝐏 2 est une courbe de genre 1 qui ne satisfait pas au principe de Hasse. On donne un modèle de Weierstrass explicite pour sa jacobienne. Le groupe de Shafarevich-Tate de chacune des ces jacobiennes contient un sous-groupe isomorphe à 𝐙/3×𝐙/3.

@article{JTNB_2001__13_1_263_0,
     author = {Poonen, Bjorn},
     title = {An explicit algebraic family of genus-one curves violating the Hasse principle},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     pages = {263-274},
     zbl = {1046.11038},
     mrnumber = {1838086},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2001__13_1_263_0/}
}
Poonen, Bjorn. An explicit algebraic family of genus-one curves violating the Hasse principle. Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 263-274. https://jtnb.centre-mersenne.org/item/JTNB_2001__13_1_263_0/

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