Let $X$ be a smooth and proper scheme over $\text{Spec}(\mathbf{Z})$.

C. Soulé has conjectures about special values of the completed zeta function of $X$, $\zeta(X,s)$, which were first reformulated by Bloch in terms of his higher Chow groups as follows:

$$-\text{ord}_{s=\text{dim}(X)-n}\zeta(X,s) = \sum_{i}(-1)^i\text{rk}\text{CH}^n(X,i).$$

However, in the literature (see this paper by Flach and Morin, for example) one can find formulas for the order of vanishing of $\zeta(X,s)$ at integers (and for its special values) that do not involve Bloch's higher Chow groups, but rather an Arakelov version of higher Chow groups.

It seems these latter ones are typically much larger than usual higher Chow groups, so I am wondering what conjectural statement is actually expected to be true.

After some time searching the literature, I could not pinpoint which between Bloch's original statement and the Flach-Morin statement, say, was meant to be the correct conjecture, and the lack of a succinct source clearly stating such expectations is very confusing.

Specifically, can anyone please clarify what, in the expected formula:

$$\text{ord}_{s=n}\zeta(X,s) = \sum_i(-1)^i\text{rk}_{\mathbf{Z}}{C}^n_i$$

(up to a sign) $C^n_i$ are truly meant to be *in terms of higher Chow groups* and their variants? (Bloch's higher Chow groups? Arakelov Chow groups? Real Deligne cohomology groups?)

More precise questions:

- Let $X$ be a smooth projective variety defined over $\mathbf{Q}$. What is the relation between the ranks of all the variants of $\text{CH}^n(X,i)$ used in potential formulas for $\text{ord}_{s=n}\zeta(X,s)$?

We denote by $\widehat{\text{CH}}^n(X,i)$ the Arakelov Chow groups of Bloch and Soulé, and by $H^{2n-i}_{\mathcal{D}}(X,\mathbf{R}(n))$ the real Deligne cohomology of $X$.

Are the cycle maps $\text{CH}^n(X,i)\to\widehat{\text{CH}}^n(X,i)$ and $\text{CH}^n(X,i)\to H^{2n-i}_{\mathcal{D}}(X,\mathbf{R}(n))$ expected to be isomorphisms rationally? (surely not in the first case)

What features is the integral cycle map

$$\text{CH}^n(X,i)\to H^{2n-i}_{\mathcal{D}}(X,\mathbf{Z}(n))$$ expected to have? (ie. finite kernel, discrete image?) and for which $i,n$?