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 54 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…
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 for number rings
- Mathematics
- 2005
There should be a Grothendieck topology for an arithmetic scheme X such that the Euler characteristic of the cohomology groups of the constant sheaf Z with compact support at infinity gives, up to…
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…
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…