The Brauer–Manin obstruction for curves having split Jacobians
Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 3, p. 773-777

Let X𝒜 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.

Soit X𝒜 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.

@article{JTNB_2004__16_3_773_0,
     author = {Siksek, Samir},
     title = {The Brauer--Manin obstruction for curves having split Jacobians},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
     number = {3},
     year = {2004},
     pages = {773-777},
     doi = {10.5802/jtnb.469},
     mrnumber = {2144966},
     zbl = {1076.14033},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2004__16_3_773_0}
}
Siksek, Samir. The Brauer–Manin obstruction for curves having split Jacobians. Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 3, pp. 773-777. doi : 10.5802/jtnb.469. http://www.numdam.org/item/JTNB_2004__16_3_773_0/

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