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

Poonen, Bjorn

Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 263-274.

Nous montrons que pour tout $t\in 𝐐$, la courbe

$$5x^3+9y^3+10z^3+12{\left(\frac{t^2+82}{t^2+22}\right)}^3{(x+y+z)}^3=0$$

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 $𝐙/3\times 𝐙/3$.

We prove that for any $t\in 𝐐$, the curve

$$5x^3+9y^3+10z^3+12{\left(\frac{t^2+82}{t^2+22}\right)}^3{(x+y+z)}^3=0$$

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 $𝐙/3\times 𝐙/3$.

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