Flach–Morin special value conjecture for arithmetic schemes

Let ${\mathcal X}$ be a regular scheme of pure dimension $d$ proper over $\operatorname{Spec}(\mathbb Z)$, and let $n\in\mathbb Z$. Assume that ${\mathcal X}$ satisfies Assumptions $\mathbf{L}(\overline{\mathcal{X}}_{et},n)$, $\mathbf{L}(\overline{\mathcal{X}}_{et},d-n)$, $\mathbf{AV}(\overline{\mathcal{X}}_{et},n)$ and $\mathbf{B}(\mathcal{X},n)$ of Flach–Morin. Under these assumptions, let $R\Gamma_{W,c}({\mathcal X},\mathbb Z(n))$ be the resulting perfect complex of abelian groups and define the fundamental line

$$\Delta({\mathcal X}/\mathbb S,n):={\det}_\mathbb Z R\Gamma_{W,c}(\mathcal{X},\mathbb{Z}(n))\otimes_\mathbb Z {\det}_\mathbb Z R\Gamma({\mathcal X},L\Omega_{{\mathcal X}/\mathbb S}^{<n}).$$

Let $\lambda:\mathbb R\xrightarrow{\sim}\Delta({\mathcal X}/\mathbb S,n)\otimes_\mathbb Z\mathbb R$ be the canonical trivialization, and let $\zeta({\mathcal X},s)$ be the zeta function of ${\mathcal X}$, assumed to have a meromorphic continuation to the entire complex plane. Write $\zeta^*({\mathcal X},n)\in\mathbb R^\times$ for its leading Taylor coefficient at $s=n$. Flach–Morin special value conjecture. We have

$$\lambda(\zeta^*(\mathcal{X},n)^{-1}\cdot\mathbb{Z})=\Delta({\mathcal X}/\mathbb{S},n).$$

This is the special value conjecture formulated by Flach and Morin, here expressed using the fundamental line and its canonical trivialization. Its status is not established in the supplied text.