• Corpus ID: 244709400

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

  title={Weil-\'etale cohomology and zeta-values of arithmetic schemes at negative integers},
  author={Alexey Beshenov},
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… 

Figures from this paper

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

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



Special Values of Zeta Functions of Schemes

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

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

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


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

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

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

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

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

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

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