# algorithm for calculating the Chow groups of a variety over a finite field

Is there an algorithm for calculating the Chow groups of a variety over a finite field?

It is know that $H^{2i,i}_\mathrm{mot}(X,\mathbf{Z}) = CH^i(X)$. In how many cases does this help us?

• Are there any algorithms for varieties over, say, C or Q? – Kevin H. Lin Jan 10 '10 at 17:21
• @Kevin: A^1 is isomorphic to Pic. Since there is no known algorithm for computing Pic of an elliptic curve over Q, there can't be a known algorithm for computing Chow groups over Q either. – David E Speyer Jan 10 '10 at 20:08
• @David: there is an algorithm for computing Pic of an elliptic curve over Q. We just haven't proved it terminates yet! :-) – Kevin Buzzard Jan 10 '10 at 20:52
• Fair enough. As you say, the right statement is that there is an algorithm which is conjectured to always terminate and, when it terminates, it computes Pic of an elliptic curve over Q. – David E Speyer Jan 20 '10 at 22:02
• Hey, have you had any luck with this question? – Dror Speiser Jun 11 '11 at 12:51

I am not an expert, but let me point out that computing $CH^0(X)$ (which is freely generated by the irreducible components) is already quite hard. Algorithms do exist in this case, see page 206 of "Ideals, varieties and algorithms" by Cox, Little, O'Shea for references. I know of no way to compute the class groups (which can be identified with $CH^1(X)$ for smooth $X$) in general, but I will be very interested in what other people have to say about this.
Of course, in special situations, more is known. For example, the total Chow group of quadric hypersurfaces (at least up to tensoring with $\mathbb Q$).