Sep 2009:
See instead the whole, long conversation named “Soule conjecture on precise values”, at gmail.
many thanks for sending me the letter. I’ve referenced it in my thesis. However, it leaves many questions open. For example (unless Soule himself has generalized it), the arithmetic Chow groups of Feliu are restricted to X over a field. The construction of her is going via Bloch’s higher Chow groups, which are defined at present only over fields. Also, choosing a Haar measure on Deligne cohomology is OK up to torsion (since you want the period map etc. to come in, you choose the measure to be induced by Betti cohomology and de Rham cohomology), but I fail to see how one can remove the ambiguity of the rational factor. Anyway, it is nice that Soule is interested in the stuff. Also, probably little surprisingly, the outcome of what I get is quite close to his ideas.
Some questions before Jakob’s seminar talk visit Nov 2009:
Why do we use the terminology higher arithmetic Chow groups, rather than arithmetic motivic cohomology?
If I could construct the groups over Z, using Voevodsky’s motivic complexes or A1-homotopy theory, hence avoiding the use of cycles, what would be bad with this approach?
Homotopy invariance: CH-hat should be HI, but zeta functions are not. Are L-functions homotopy invariant, and how should one in general interpret the non-invariance? Are for example orders of vanishing a homotopy invariant notion?
The general discussion about integral/rational/real coeffs. What can Feliu do? What would we like to have? What does Scholbach, and Soule, need for their conjectures?
How far can you go with Chow theory for schemes over Z. Do you have higher Chow groups? When do you need to tensor with Q in order to have a product, and when can you do with integral coeffs?
Is there any obvious relation between the intersection theory philosophy, or actual intersection between cycles, and the conjectures on special values of L and zeta functions?
nLab page on Scholbach