The Weil-étale cohomological formula for $\operatorname{ord}_{s=0}\zeta(\mathcal{X},s)$ and $\zeta^*(\mathcal{X},0)$ for regular arithmetic schemes (Morin's Conjecture 4.2 and its conjectural inputs)

For a regular scheme $\mathcal{X}$, proper and flat over $\mathrm{Spec}(\mathbb{Z})$ of pure dimension $d$, Morin constructed — conditional on Lichtenbaum's finite-generation conjecture for étale motivic cohomology — a perfect complex of abelian groups $\mathrm{R}\Gamma_{W,c}(\mathcal{X},\mathbb{Z})$ (integral compact-support Weil-étale cohomology) and conjectured that $\operatorname{ord}_{s=0}\zeta(\mathcal{X},s)$ is an alternating rank sum of its cohomology and that the leading coefficient $\zeta^*(\mathcal{X},0)$ is, up to sign, the inverse of the determinant of this complex. Under Beilinson's regulator conjecture these statements are equivalent to Soulé's conjecture on vanishing orders and to the Tamagawa number conjecture. Unconditionally the conjecture is known only for rings of integers, for geometrically cellular schemes over abelian number rings in an explicitly constructed class, and in cases following from the Tamagawa number conjecture. The open problem is to make the Weil-étale cohomological description of the special value at $s=0$ unconditional for general regular arithmetic schemes, beyond these classes.

Background

An arithmetic scheme is a separated scheme $\mathcal{X}$ of finite type over $\mathrm{Spec}(\mathbb{Z})$; its zeta function is the Euler product $\zeta(\mathcal{X},s)=\prod_{x\in|\mathcal{X}|}(1-N(x)^{-s})^{-1}$ over closed points, with $N(x)$ the size of the residue field. It is expected to admit a meromorphic continuation to all of $\mathbb{C}$, and one writes $\zeta^*(\mathcal{X},0)$ for the leading coefficient of its Taylor expansion at $s=0$. For $\mathcal{X}=\mathrm{Spec}(\mathcal{O}_F)$, the ring of integers of a number field, the analytic class number formula together with Dirichlet's unit theorem gives an unconditional cohomological description of $\operatorname{ord}_{s=0}\zeta_F(s)$ and $\zeta_F^*(0)$ in terms of unit and class groups — the prototype of the general question.

Lichtenbaum proposed that for any arithmetic scheme these values should be computed by a suitably defined Weil-étale cohomology: a cohomology theory for a topology mixing the étale site with the Weil group of class field theory, whose integral cohomology groups are finitely generated and whose Euler characteristics produce the order of vanishing and special value at $s=0$. Soulé later conjectured that for regular proper $\mathcal{X}$ over $\mathbb{Z}$ the order of vanishing of $\zeta(\mathcal{X},s)$ at $s=0$ equals an alternating sum of the ranks of the motivic cohomology groups $H^{2d-i}(\mathcal{X},\mathbb{Q}(d))$, where $d=\dim\mathcal{X}$ and motivic cohomology is the cycle-complex cohomology of Bloch (with $\mathbb{Z}(d)$ the complex of codimension-$d$ cycles, a concrete model for the motivic complex in weight $d$).

Morin (Duke Math. J. 163 (2014), no. 7, 1263-1336; arXiv preprint posted in 2011) made this precise for regular $\mathcal{X}$ proper over $\mathrm{Spec}(\mathbb{Z})$. Conditional on Conjecture 1.2 of Lichtenbaum — finite generation of the étale motivic cohomology groups $H^i(\mathcal{X}_{et},\mathbb{Z}(d))$ for $0\le i\le 2d$, taken on the étale topos of $\mathcal{X}$ — he constructed integral Weil-étale cohomology complexes $\mathrm{R}\Gamma_W(\mathcal{X},\mathbb{Z})$ and $\mathrm{R}\Gamma_{W,c}(\mathcal{X},\mathbb{Z})$ as perfect complexes of abelian groups (bounded complexes of finitely generated free abelian groups up to quasi-isomorphism), whose determinant is the alternating tensor product of the determinant lattices of their cohomology groups. He proved the construction unconditionally for schemes in an explicitly constructed class $\mathcal{L}(\mathbb{Z})$, which contains all geometrically cellular schemes over number rings (schemes built from affine spaces by stratifications) and Soulé's class of varieties built from products of curves over finite fields. Assuming in addition Conjecture 1.4 of Beilinson — the regulator map from motivic cohomology to Deligne cohomology (the cohomology theory interpolating between singular and de Rham cohomology over $\mathbb{R}$, whose periods detect regulators) is an isomorphism in the relevant range — he proved that the resulting special-value conjecture, Conjecture 4.2, is equivalent to Soulé's conjecture on the vanishing order and, for $\mathcal{X}$ smooth projective over $\mathcal{O}_F$ with vanishing $H_f^1$-terms, equivalent to the Tamagawa number conjecture of Bloch–Kato in the formulation of Fontaine–Perrin-Riou (the conjecture that the determinant of the motivic complex of a motive, with its integral structure, computes the leading term of its $L$-function at integer points; it contains the analytic class number formula and Birch–Swinnerton-Dyer-type statements as special cases). Flach and Morin subsequently extended the whole framework to all integer points $s=n$, still under analogous finiteness and regulator assumptions.

Problem Statement

Let $\mathcal{X}$ be a regular connected scheme of pure dimension $d$, proper and flat over $\mathrm{Spec}(\mathbb{Z})$, and assume $\zeta(\mathcal{X},s)$ admits a meromorphic continuation to a neighborhood of $s=0$. The open problem, in the full generality of Morin's 2014 Duke Mathematical Journal paper, is to make the Weil-étale cohomological description of the special value at $s=0$ unconditional for general $\mathcal{X}$. In Morin's construction, this description is Conjecture 4.2 of that paper for the resulting perfect complex $\mathrm{R}\Gamma_{W,c}(\mathcal{X},\mathbb{Z})$, namely

$$\operatorname{ord}_{s=0}\zeta(\mathcal{X},s)=\sum_{i\in\mathbb{Z}}(-1)^i\,i\,\operatorname{rank}_\mathbb{Z}H^i_{W,c}(\mathcal{X},\mathbb{Z})$$

and

$$\mathbb{Z}\cdot\lambda(\zeta^*(\mathcal{X},0)^{-1})=\det{}_{\mathbb{Z}}\,\mathrm{R}\Gamma_{W,c}(\mathcal{X},\mathbb{Z}),$$

where $\lambda:\mathbb{R}\xrightarrow{\sim}(\det{}_{\mathbb{Z}}\,\mathrm{R}\Gamma_{W,c}(\mathcal{X},\mathbb{Z}))_{\mathbb{R}}$ is the canonical isomorphism of one-dimensional $\mathbb{R}$-vector spaces furnished by Morin's construction via the $\tilde{\mathbb{R}}$-cohomology of the complex. Morin's construction states these identities as a conjecture, formulated under two conjectural inputs: Lichtenbaum's finite-generation conjecture $L(\mathcal{X}_{et},d)$: the étale motivic cohomology groups $H^i(\mathcal{X}_{et},\mathbb{Z}(d))$ (cohomology of Bloch's cycle complex $\mathbb{Z}(d)$ on the étale topos of $\mathcal{X}$) are finitely generated for $0\le i\le 2d$, open beyond the explicitly constructed class $\mathcal{L}(\mathbb{Z})$ where it is known; and the Beilinson regulator conjecture $B(\mathcal{X},d)$: the regulator from motivic cohomology to Deligne cohomology is an isomorphism in the relevant degrees. Currently open for all $\mathcal{X}$ outside rings of integers, geometrically cellular schemes in $\mathcal{L}(\mathbb{Z})$ over abelian number rings, and cases following from known instances of the Tamagawa number conjecture; it remains open, for instance, for any regular proper flat arithmetic surface whose generic fiber is a smooth projective curve of genus at least 1. The target is all regular proper flat arithmetic schemes; by the cited equivalences (Morin's Theorem 1.5, which runs through his construction under the hypotheses $L(\mathcal{X}_{et},d)$ and $B(\mathcal{X},d)$), a resolution of the identities of Conjecture 4.2 for general $\mathcal{X}$ simultaneously proves Soulé's vanishing-order conjecture and the Tamagawa number conjecture for the motives $h^i(\mathcal{X}_F)$.

References

1. Baptiste Morin, Zeta functions of regular arithmetic schemes at s=0, Duke Mathematical Journal 163 (2014), no. 7, 1263-1336. DOI: 10.1215/00127094-2681387; arXiv:1103.6061. https://arxiv.org/abs/1103.6061
2. Baptiste Morin, Matthias Flach, Weil-étale cohomology and Zeta-values of proper regular arithmetic schemes, Documenta Mathematica 23 (2018), 1425-1560. DOI: 10.4171/dm/651; arXiv:1605.01277. https://arxiv.org/abs/1605.01277
3. Matthias Flach, Baptiste Morin, On the Weil-étale topos of regular arithmetic schemes, Documenta Mathematica 17 (2012), 313-399. DOI: 10.4171/dm/369; arXiv:1010.3833. https://arxiv.org/abs/1010.3833
4. Christophe Soulé, K-théorie et zéros aux points entiers de fonctions zéta, Proceedings of the International Congress of Mathematicians, Vol. 1 (Warsaw, 1983), 1984, pp. 437-445. zbMATH: 0574.14010. https://zbmath.org/0574.14010
5. Thomas Geisser, Weil-étale cohomology over finite fields, Mathematische Annalen 330 (2004), no. 4, 665-692. DOI: 10.1007/s00208-004-0564-8. https://doi.org/10.1007/s00208-004-0564-8
6. Stephen Lichtenbaum, Special Values of Zeta Functions of Schemes, arXiv:1704.00062 (2017). https://arxiv.org/abs/1704.00062
7. Baptiste Morin, Topological Hochschild homology and Zeta-values, arXiv:2011.11549 (2020). https://arxiv.org/abs/2011.11549
8. Alexey Beshenov, Weil-étale cohomology and zeta-values of arithmetic schemes at negative integers, Documenta Mathematica 31 (2026), 583-624. DOI: 10.4171/dm/1059; arXiv:2102.12114. https://arxiv.org/abs/2102.12114
9. Adrien Morin, Weil-étale cohomology and the equivariant Tamagawa number conjecture for constructible sheaves in characteristic p, arXiv:2411.07896 (2024). https://arxiv.org/abs/2411.07896