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

We prove that for any t∈𝐐, 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 t∈𝐐, 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.

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

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