The Brauer–Manin obstruction for curves having split Jacobians

Siksek, Samir

doi:10.5802/jtnb.469

Siksek, Samir ¹

¹ Department of Mathematics and Statistics Faculty of Science Sultan Qaboos University P.O. Box 36 Al-Khod 123, Oman

Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 3, pp. 773-777.

Résumé
Abstract

Soit $X \to 𝒜$ un morphisme (qui n’est pas constant) d’une courbe $X$ vers une variété abélienne $𝒜$ , tous définis sur un corps de nombres $k$ . Supposons que $X$ ne satisfait pas le principe de Hasse. Nous donnons des conditions suffisantes pour que l’obstruction de Brauer-Manin soit la seule obstruction au principe de Hasse. Ces conditions suffisantes sont légèrement plus fortes que de supposer que $𝒜 (k)$ et $Ш (𝒜 / k)$ sont finis.

Let $X \to 𝒜$ be a non-constant morphism from a curve $X$ to an abelian variety $𝒜$ , all defined over a number field $k$ . Suppose that $X$ is a counterexample to the Hasse principle. We give sufficient conditions for the failure of the Hasse principle on $X$ to be accounted for by the Brauer–Manin obstruction. These sufficiency conditions are slightly stronger than assuming that $𝒜 (k)$ and $Ш (𝒜 / k)$ are finite.

MR Zbl

DOI : 10.5802/jtnb.469

Affiliations des auteurs :

Siksek, Samir ¹

¹ Department of Mathematics and Statistics Faculty of Science Sultan Qaboos University P.O. Box 36 Al-Khod 123, Oman

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     title = {The {Brauer{\textendash}Manin} obstruction for curves having split {Jacobians}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {773--777},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
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     doi = {10.5802/jtnb.469},
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Siksek, Samir. The Brauer–Manin obstruction for curves having split Jacobians. Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 3, pp. 773-777. doi : 10.5802/jtnb.469. http://www.numdam.org/articles/10.5802/jtnb.469/

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