Principe local-global pour les zéro-cycles sur les surfaces réglées

Colliot-Thélène, Jean-Louis

doi:10.1090/S0894-0347-99-00318-5

Principe local-global pour les zéro-cycles sur les surfaces réglées
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by Jean-Louis Colliot-Thélène

J. Amer. Math. Soc. 13 (2000), 101-124

DOI: https://doi.org/10.1090/S0894-0347-99-00318-5

Published electronically: September 29, 1999

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Abstract:

Let $k$ be a number field, $C/k$ a smooth projective curve, and $X$ a smooth projective surface which is a conic bundle over $C$. Let $CH_0(X/C)$ be the relative Chow group, which is the kernel of the projection map $CH_0(X) \rightarrow CH_0(C)$ on Chow groups of zero-cycles. For each place $v$ of $k$, one may consider the relative Chow group $CH_0(X_v/C_v)=CH_0(X\times _kk_v/C\times _kk_v)$. Under minor assumptions, we identify the diagonal image of $CH_0(X/C)$ in the product of all $CH_0(X_v/C_v)$ as the kernel of the natural pairing with the Brauer group of $X$. When $C$ is an elliptic curve with finite Tate-Shafarevich group, under minor assumptions, we show that the Brauer-Manin obstruction to the existence of a zero-cycle of degree one on $X$ is the only obstruction.

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