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

@inproceedings{Beshenov2021ZetavaluesOO, title={Zeta-values of one-dimensional arithmetic schemes at strictly negative integers}, author={Alexey Beshenov}, year={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 in terms of the étale motivic cohomology of X and a regulator. We prove it in the case when for each generic point η ∈ X with charκ(η) = 0, the extension κ(η)/Q is abelian. We conjecture that the formula holds for any onedimensional arithmetic scheme. This is a consequence of the Weil-étale formalism…

## Figures from this paper

## References

SHOWING 1-10 OF 46 REFERENCES

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

- Mathematics
- 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…

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…

Algebraic K-Theory of Rings of Integers in Local and Global Fields

- Mathematics
- 2005

The problem of computing the higher K-theory of a number field F , and of its rings of integers OF , has a rich history. Since 1972, we have known that the groups Kn(OF ) are finitely generated [48],…

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…

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…

Annihilation of motivic cohomology groups in cyclic 2-extensions.

- Mathematics
- 2008

Suppose that E/F is an abelian extension of number fields with Galois group G. The generalized Coates-Sinnott Conjecture predicts that for n 2 a natural higher Stickelberger ideal, defined using…

Stable real cohomology of arithmetic groups

- Mathematics
- 1974

Given a discrete subgroup Γ of a connected real semisimple Lie group G with finite center there is a natural homomorphism
$$j_\Gamma ^q:I_G^q \to {H^q}\left( {\Gamma ;c} \right)\quad \left( {q =…

Compatibility of Special value conjectures with the functional equation of Zeta functions

- Mathematics
- 2020

We prove that the special value conjecture for the Zeta function of a proper, regular arithmetic scheme X that we formulated in our previous article [8] is compatible with the functional equation of…

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…