# On the Weil-Étale Topos of Regular Arithmetic Schemes

@article{Flach2012OnTW, title={On the Weil-{\'E}tale Topos of Regular Arithmetic Schemes}, author={Matthias Flach and Baptiste Morin}, journal={arXiv: Number Theory}, year={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 ˜R-coefficients has the expected relation to ζ(X, s) at s = 0 if the Hasse-Weil L-functions L(h^(i)(X_(Q)), s) have the expected meromorphic
continuation and functional equation. If X has characteristic p the cohomology with Z-coefficients also has the expected relation to ζ(X, s) and our cohomology…

## 19 Citations

### 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…

### Weil-étale cohomology for arbitrary arithmetic schemes and n < 0 . Part II : The special value conjecture

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Following the ideas of Flach and Morin [FM2018], 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…

### THE WEIL-ÉTALE FUNDAMENTAL GROUP OF A NUMBER FIELD I

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- 2010

Lichtenbaum has conjectured (Ann of Math. (2) 170(2) (2009), 657-683) the existence of a Grothendieck topology for an arithmetic scheme X such that the Euler characteristic of the cohomology groups…

### The Weil-étale fundamental group of a number field II

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- 2011

We define the fundamental group underlying the Weil-étale cohomology of number rings. To this aim, we define the Weil-étale topos as a refinement of the Weil-étale sites introduced by Lichtenbaum…

### On the Construction of Higher etale Regulators

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We present three approaches to define the higher etale regulator maps Φr,net : Hret(X,Z(n)) → HrD(X,Z(n)) for regular arithmetic schemes. The first two approaches construct the maps on the cohomology…

### Weil-etale cohomology and special values of L-functions at zero

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We construct the Weil-\'etale cohomology and Euler characteristics for a subclass of the class of $\mathbb{Z}$-constructible sheaves on the spectrum of the ring of integers of a totally imaginary…

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

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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…

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For a proper, flat, generically smooth scheme X over a complete discrete valuation ring with finite residue field of characteristic p, we construct a specialization morphism from the rigid cohomology…

### Special L-values of geometric motives

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This paper proposes a conjecture about special values of L-functions L(M,s) := Q p det(Id Fr 1 p s |iM`) 1 of geometric motives M over Z. This includes L-functions of mixed motives over Q and…

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