Chow Groups of varieties over number fields 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?
 A: See Conjecture 5.0 (attributed to Swnnerton-Dyer) in the paper "Height pairing between algebraic cycles" by Beilinson. The paper by Swinnerton-Dyer that Beilinson refers to is "The conjectures of Birch-Swinnerton-Dyer and of Tate". Please take a look in these papers.
A: 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/
A: The statement you want follows fairly straightforwardly from Bass' conjecture -- sufficiently straightforwardly that it may well not have a separate name of its own.
If $\Sigma$ is a sufficiently large finite set of primes, then $X$ will admit a smooth model $\mathfrak{X}$ over $\mathcal{O}_{K, \Sigma}$. Since $\mathfrak{X}$ is a finite-type $\mathbf{Z}$-scheme, Bass' conjecture implies that all Chow groups of $\mathfrak{X}$ are finitely generated. [Edit: as user "guest" points out, this last step only works after $\otimes \mathbf{Q}$; to get finite generation with $\mathbf{Z}$ coefficients you need something slightly stronger, the "motivic Bass conjecture" which is the conjecture that finite-type $\mathbf{Z}$-schemes have finitely-generated motivic cohomology.]
So it suffices to check that the natural map $CH^i(\mathfrak{X}) \to CH^i(X)$ is surjective, which is easy, because any codimension $i$ cycle on $X$ has a scheme-theoretic closure which is a codimension $i$ cycle on $\mathfrak{X}$.
This doesn't work for motivic cohomology in other degrees, incidentally (already $H^1(\operatorname{Spec} \mathbf{Q}, \mathbf{Q}(1)) = \mathbf{Q}^\times \otimes \mathbf{Q}$ has countably infinite dimension).
