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

Bjorn Poonen

doi:10.5802/jtnb.320

Bjorn Poonen

Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 263-274.

Abstract
Résumé

We prove that for any $t \in 𝐐$ , the curve $5 x^{3} + 9 y^{3} + 10 z^{3} + 12 {(\frac{t^{2} + 82}{t^{2} + 22})}^{3} {(x + y + z)}^{3} = 0$ 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 $𝐙 / 3 \times 𝐙 / 3$ .

Nous montrons que pour tout $t \in 𝐐$ , la courbe $5 x^{3} + 9 y^{3} + 10 z^{3} + 12 {(\frac{t^{2} + 82}{t^{2} + 22})}^{3} {(x + y + z)}^{3} = 0$ 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 $𝐙 / 3 \times 𝐙 / 3$ .

MR: 1838086 | Zbl: 1046.11038

DOI: 10.5802/jtnb.320

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     title = {An explicit algebraic family of genus-one curves violating the {Hasse} principle},
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     pages = {263--274},
     publisher = {Universit\'e Bordeaux I},
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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, Volume 13 (2001) no. 1, pp. 263-274. doi : 10.5802/jtnb.320. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.320/

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