Class field theory of arithmetic surfaces (Q1382107)

Let \(k\) be an algebraic number field, \(X\) a smooth projective curve over \(k\) and \(Y\) a regular proper flat model of \(X\) over the ring of integers of \(k\). Take an effective reduced divisor \(D\) on \(X\) and its closure \(\overline D\) on \(Y\). For any real place \(v\) of \(k\) denote by \(\pi_{0}J_{X,D}(k_{v})\) the group of connected components of the generalized Jacobian of \(X\) relative to \(D\). Assume that \((X-D)(k)\) is not empty. Then there is an exact sequence \[ \coprod_{\text{real} v} \{ \pm \} \times \pi_{0}J_{X,D}(k_{v}) \rightarrow \pi_1^{\text{ab}}(Y-\overline D) \rightarrow CH_{}(Y,\overline D) \rightarrow 0. \] Here \(CH_{}(Y,\overline D)\) is the relative Chow group of 0-cycles. It is also proved that \(CH_{}(Y,\overline D)\) is finite.