Thời gian: 16h30, thứ năm, 01/08/2024

Tóm tắt: The Brauer-Manin obstruction to the local-global principle and to weak approximation for varieties over number fields possesses an analogue when the set of rational points is replaced by the Chow group of zero-cycles (finite formal sum of closed points modulo rational equivalence). Experience suggests that these objects are somewhat easier to manipulate than rational points themselves. In particular, their arithmetics are governed by a conjecture due to Colliot-Thélène, Sansuc, Kato, and Saito. We present some known results in this directions. Then, we explain Liang's trick, allowing one to pass from results for rational points to those for zero-cycles. If the time permits, we shall discuss the descent method (with a formalism in the context of zero-cycles, developped recently by Balestrieri and Berg) and the proof of an analogue of Wittenberg's descent conjecture.

Reference:

1. F. Balestrieri and J. Berg. Descent and étale-Brauer obstructions for 0-cyc[les, 2022. Preprint, 34 pages, https://arxiv.org/abs/2202.08120, to appear in International Mathematics Research Notices.
2. J.-L. Colliot-Thélène. L’arithmétique du groupe de Chow des zéro-cycles. Journal de théorie des nombres de Bordeaux, 7(1):51–73, 1995.
3. Y. Harpaz and O. Wittenberg. On the fibration method for zero-cycles and rational points. Annals of Mathematics, 183(1):229–295, 2016.
4. Y. Liang. Arithmetic of 0-cycles on varieties defined over number fields. Annales scientifiques de l’École Normale Supérieure, 4e série, 46(1):35–56, 2013.
5. N. M. Linh. On the descent conjecture for rational points and zero-cycles, 2023. Preprint, 41 pages, https://arxiv.org/abs/2305.13228.
6. O. Wittenberg. Zéro-cycles sur les fibrations au-dessus d’une courbe de genre quelconque. Duke Mathematical Journal, 161(11):2113–2166, 2012.

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