TITLE: Chow Groups of varieties over number fields
QUESTION [12 upvotes]: 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. 
What is the standard name for this conjecture? In private communication people referred to it as Beilinson's conjecture. I assume that it should have been formulated before Beilinson. What is the best paper to cite for this conjecture? Is it known in any non-trivial case?

REPLY [4 votes]: Just adding to Lucifer's answer: This is Conjecture 5 in Beilinson's paper
https://mathscinet.ams.org/mathscinet-getitem?mr=902590
For a smooth projective variety over a number field and a fixed codimension, Tate's conjecture predicts that the rank of algebraic cycles modulo homological equivalence is given by the order of the pole of an appropriate L-function; Beilinson's conjecture predicts that the rank of homologically equivalent cycles (modulo rational equivalence) is the order of the zero of an appropriate L-function; special cases were stated by Swinnerton-Dyer, Tate, Bloch before. See 6.2 and 6.5 of Nekovar's article below for the precise statements.
Nice introductions to Beilinson's conjectures are
1) Nekovar's article
http://math.stanford.edu/~conrad/BSDseminar/refs/BeilinsonintroII.pdf
2) Ramakrishnan's article
https://mathscinet.ams.org/mathscinet-getitem?mr=991982
3) The Book: available at Professor Schneider's webpage
https://ivv5hpp.uni-muenster.de/u/pschnei/publ/beilinson-volume/