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

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.