# Chow groups of arithmetic surfaces

Given an arithmetic surface $S$, I would like to know the following properties of its first and second Chow groups $CH^1(S), CH^2(S)$:

1. Are they finitely generated? If so, what is the rank?
2. What is the size of the torsion subgroup?
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The example you gave at the end doesn't seem to be homogeneous. Is that intentional? – S. Carnahan Apr 24 '10 at 0:24