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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 known in any non-trivial case?

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?

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?

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 known in any non-trivial case?