Precise formulation of conjectures on orders of vanishing?

Question

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 $i,n$?

Answer 1

You can find the relation between Soulé's pole order conjecture and the formulation in terms of arithmetic Chow groups in a paper of mine "Special L-values of geometric motives" (Asian Journal of Mathematics (2017), Vol. 21 (2) pp. 225–264). Briefly, I raise there the conjecture that motivic homology (of which Chow groups are instances) is related to what we call Arakelov motivic cohomology (of which the groups $\widehat{CH}(X)$ are instances) by means of a conjecturally perfect pairing of real vector spaces. I also show in this paper that the perfectness of this pairing is equivalent to the conjunction of Soulé's, Beilinson's and Tate's conjecture, in the presence of the "usual" conjectures on mixed motives. Thus, if you believe this perfectness conjecture (which is, in my opinion, a conceptually appealing statement), you can phrase the pole order conjecture equivalently in terms of usual Chow groups or in terms of arithmetic Chow groups.

In addition to David Loeffler's comment, let me also clarify that the literature offers two different types of arithmetic Chow groups. Gillet-Soulé introduce one which sits in an exact sequence roughly of the form $$\dots \to \text{first higher }CH \to A \to \widetilde{CH} \to CH \to 0,$$ where $A$ is a real vector space of the type "all differential forms (of some degree) modulo boundaries". In particular, these spaces are usually infinite-dimensional and their group $\widetilde{CH}$ is indeed much bigger than $CH$. These groups are not the ones featuring in the formulation of the conjecture in Flach-Morin and my work you mention. Instead it is a subgroup $\widehat{CH} \subset \widetilde{CH}$ (which is also considered by Gillet-Soulé), defined in a way so that there are exact sequences $$\dots \to H^*_D(X, *) \to \widehat{CH} \to CH \to H^*_D(X, *) \to \dots$$ i.e., these groups measure precisely the extent to which the Beilinson regulator is not an isomorphism.