The local-global principle for zero cycles on certain fibrations over a curve. I.
The main object of the paper under review is a smooth projective variety \(X\), defined over a number field, which admits a structure of fibration \(\pi: X\to C\) over a smooth projective curve (\(X\), \(C\) and the generic fibre of \(\pi\) are assumed geometrically integral). The author considers the conjecture raised by \textit{J.-L.~Colliot-Thélène} [Proc. Symp. Pure Math. 67, 1--12 (1999; Zbl 0981.14003)] which asserts that the Brauer-Manin obstruction to the Hasse principle for zero-cycles of degree 1 is the only one. The main result of the paper establishes this conjecture under the assumptions that all fibres of \(\pi\) are geometrically integral, the Tate-Shafarevich group of the Jacobian of \(C\) is finite and all fibres over some ``generalized Hilbertian'' subset of \(C\) satisfy the Hasse principle. Some parallel results are obtained for the Brauer--Manin obstruction to weak approximation. This is applied to the example of \textit{B.~Poonen} [Ann. Math. (2) 171, No. 3, 2157--2169 (2010; Zbl 1284.11096)] where there is no Brauer-Manin obstruction for points but, according to \textit{J.-L.~Colliot-Thélène} [Bull. Soc. Math. Fr. 138, No. 2, 249--257 (2010; Zbl 1205.11075)], there is a zero-cycle of degree 1. In particular, the author shows that there are many such cycles (in the sense of weak approximation). The author mentions some parallel results obtained by O.~Wittenberg under different assumptions.
scientific article; zbMATH DE number 6065235
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