# Weil-\'etale cohomology and zeta-values of arithmetic schemes at negative integers

@inproceedings{Beshenov2021WeiletaleCA, title={Weil-\'etale cohomology and zeta-values of arithmetic schemes at negative integers}, author={Alexey Beshenov}, year={2021} }

Following the ideas of Flach and Morin [11], we state a conjecture in terms of Weil-étale cohomology for the vanishing order and special value of the zeta function ζ(X, s) at s = n < 0, where X is a separated scheme of finite type over SpecZ. We prove that the conjecture is compatible with closed-open decompositions of schemes and with affine bundles, and consequently, that it holds for cellular schemes over certain one-dimensional bases. This is a continuation of [2], which gives a…

## One Citation

### Zeta-values of one-dimensional arithmetic schemes at strictly negative integers

- Mathematics
- 2021

Let X be an arithmetic scheme (i.e., separated, of finite type over SpecZ) of Krull dimension 1. For the associated zeta function ζ(X, s), we write down a formula for the special value at s = n < 0…

## References

SHOWING 1-10 OF 47 REFERENCES

### Special Values of Zeta Functions of Schemes

- Mathematics
- 2017

Let X be a regular scheme, projective and flat over Spec \mathbb Z. We give two conjectural formulas, up to sign and powers of 2, for \zeta^*(X,r), the leading term in the series expansion of…

### Weil-\'{e}tale cohomology and duality for arithmetic schemes in negative weights

- Mathematics
- 2020

Flach and Morin constructed in [9] Weil-étale cohomology H i W,c(X,Z(n)) for a proper, regular arithmetic scheme X (i.e. separated and of finite type over SpecZ) and n ∈ Z. In the case when n < 0, we…

### Arithmetic cohomology over finite fields and special values of ζ-functions

- Mathematics
- 2004

We construct cohomology groups with compact support H i c(Xar, Z(n)) for separated schemes of finite type over a finite field, which generalize Lichtenbaum’s Weil-etale cohomology groups for smooth…

### ZETA FUNCTIONS OF REGULAR ARITHMETIC SCHEMES AT s = 0

- Mathematics
- 2014

Lichtenbaum conjectured in (22) the existence of a Weil-etale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme X at s = 0 in…

### The Weil-étale topology on schemes over finite fields

- MathematicsCompositio Mathematica
- 2005

We introduce an essentially new Grothendieck topology, the Weil-étale topology, on schemes over finite fields. The cohomology groups associated with this topology should behave better than the…

### On the Weil-Étale Topos of Regular Arithmetic Schemes

- Mathematics
- 2012

We define and study a Weil-etale topos for any regular,
proper scheme X over Spec(Z) which has some of the properties suggested by Lichtenbaum for such a topos. In particular, the cohomology with…

### Special Values of Zeta-Functions for Proper Regular Arithmetic Surfaces

- Mathematics
- 2019

We explicate Flach's and Morin's special value conjectures in [8] for proper regular arithmetic surfaces π : X → Spec Z and provide explicit formulas for the conjectural vanishing orders and leading…

### Motivic cohomology over Dedekind rings

- Mathematics
- 2004

Abstract.We study properties of Bloch’s higher Chow groups on smooth varieties over Dedekind rings. We prove the vanishing of for i > n, and the existence of a Gersten resolution for if the residue…

### On values of zeta functions and ℓ-adic Euler characteristics

- Mathematics
- 1978

The values of the zeta function (~(s) of a totally real number field K on the negative integers s = n are, by a theorem of Siegel, rational numbers; they are zero iff n is even. S. Lichtenbaum has…

### Weil-étale cohomology over finite fields

- Mathematics
- 2004

Abstract.We calculate the derived functors Rγ* for the base change γ from the Weil-étale site to the étale site for a variety over a finite field. For smooth and proper varieties, we apply this to…