I believe that there is a conjecture that for any smooth projective variety $X$ over a number field $K$, its Chow groups $CH^i(X)$ (or at least $CH^i(X)\otimes_{\mathbf Z} \mathbf Q$) are finitely generated. Is it called Beilinson's Conjecture? What is the best reference for this conjecture? Is it know in any non-trivial case?