We prove that for any , the curve in is a genus curve violating the Hasse principle. An explicit Weierstrass model for its jacobian is given. The Shafarevich-Tate group of each contains a subgroup isomorphic to .
Nous montrons que pour tout , la courbe de est une courbe de genre 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 .
@article{JTNB_2001__13_1_263_0, author = {Bjorn Poonen}, 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}, doi = {10.5802/jtnb.320}, zbl = {1046.11038}, mrnumber = {1838086}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.320/} }
TY - JOUR TI - An explicit algebraic family of genus-one curves violating the Hasse principle JO - Journal de Théorie des Nombres de Bordeaux PY - 2001 DA - 2001/// SP - 263 EP - 274 VL - 13 IS - 1 PB - Université Bordeaux I UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.320/ UR - https://zbmath.org/?q=an%3A1046.11038 UR - https://www.ams.org/mathscinet-getitem?mr=1838086 UR - https://doi.org/10.5802/jtnb.320 DO - 10.5802/jtnb.320 LA - en ID - JTNB_2001__13_1_263_0 ER -
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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