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},
     pages = {263--274},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     mrnumber = {1838086},
     zbl = {1046.11038},
     language = {en},
     url = {http://archive.numdam.org/item/JTNB_2001__13_1_263_0/}
}
TY  - JOUR
AU  - Poonen, Bjorn
TI  - An explicit algebraic family of genus-one curves violating the Hasse principle
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2001
SP  - 263
EP  - 274
VL  - 13
IS  - 1
PB  - Université Bordeaux I
UR  - http://archive.numdam.org/item/JTNB_2001__13_1_263_0/
LA  - en
ID  - JTNB_2001__13_1_263_0
ER  - 
%0 Journal Article
%A Poonen, Bjorn
%T An explicit algebraic family of genus-one curves violating the Hasse principle
%J Journal de théorie des nombres de Bordeaux
%D 2001
%P 263-274
%V 13
%N 1
%I Université Bordeaux I
%U http://archive.numdam.org/item/JTNB_2001__13_1_263_0/
%G en
%F 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. http://archive.numdam.org/item/JTNB_2001__13_1_263_0/

[AP] S.Y. An, S.Y. Kim, D. Marshall, S. Marshall, W. Mccallum, A. Perlis, Jacobians of genus one curves. Preprint, 1999.

[CG] J.W.S. Cassels, M.J.T. Guy, On the Hasse principle for cubic surfaces. Mathematika 13 (1966), 111-120. | MR | Zbl

[CKS] J.-L. Colliot-Thélène, D. Kanevsky, J.-J. Sansuc, Arithmétique des surfaces cubiques diagonales, pp. 1-108 in Diophantine approximation and transcendence theory (Bonn, 1985), Lecture Notes in Math. 1290, Springer, Berlin, 1987. | MR | Zbl

[CP] J.-L. Colliot-Thélène, B. Poonen, Algebraic families of nonzero elements of Shafarevich-Tate groups. J. Amer. Math. Soc. 13 (2000), 83-99. | MR | Zbl

[Lin] C.-E. Lind, Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins. Thesis, University of Uppsala, 1940. | JFM | MR | Zbl

[Ma] Yu.I. Manin, Cubic forms. Translated from the Russian by M. Hazewinkel, Second edition, North-Holland, Amsterdam, 1974. | MR | Zbl

[Mi] J. Milne, Étale cohomology. Princeton Univ. Press, Princeton, N.J., 1980. | MR | Zbl

[O'N] C. O'Neil, Jacobians of curves of genus one. Thesis, Harvard University, 1999.

[Ra] M. Raynaud, Caractéristique d'Euler-Poincaré d'un faisceau et cohomologie des variétés abéliennes. Séminaire Bourbaki, Exposé 286 (1965). | Numdam | Zbl

[Re] H. Reichardt, Einige im Kleinen überall lösbare, im Grossen unlösbare diophantische Gleichungen. J. Reine Angew. Math. 184 (1942), 12-18. | JFM | MR | Zbl

[RVT] F. Rodriguez-Villegas, J. Tate, On the Jacobian of plane cubics. in preparation, 1999.

[Se] E. Selmer, The Diophantine equation ax3 + by3 + cz3 = 0. Acta Math. 85 (1951), 203-362; 92 (1954), 191-197. | MR | Zbl

[Si] J. Silverman, The arithmetic of elliptic curves. Graduate Texts in Mathematics 106, Springer-Verlag, New York-Berlin, 1986. | MR | Zbl

[St] B. Sturmfels, Introduction to resultants. Applications of computational algebraic geometry (San Diego, CA, 1997), 25-39, Proc. Sympos. Appl. Math. 53, Amer. Math. Soc., Providence, RI, 1998. | MR | Zbl

[SD] H.P.F. Swinnerton-Dyer, Two special cubic surfaces. Mathematika 9 (1962), 54-56. | MR | Zbl