Algebraic families of nonzero elements of Shafarevich-Tate groups

Colliot-Thélène, Jean-Louis¹ ; Poonen, Bjorn¹

¹ C.N.R.S., Mathématiques, Bâtiment 425, Université de Paris-Sud, F-91405 Orsay, France

Journal of the American Mathematical Society, Tome 13 (2000) no. 1, pp. 83-99

Résumé

Principal homogeneous spaces under an abelian variety defined over a number field $k$ may have rational points in all completions of the number field without having rational points over $k$. Such principal homogeneous spaces represent the nonzero elements of the Shafarevich-Tate group of the abelian variety. We produce nontrivial, one-parameter families of such principal homogeneous spaces. The total space of these families are counterexamples to the Hasse principle. We show these may be accounted for by the Brauer-Manin obstruction.

Détail

DOI : 10.1090/S0894-0347-99-00315-X

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