# ZETA FUNCTIONS OF REGULAR ARITHMETIC SCHEMES AT s = 0

@article{Morin2014ZETAFO, title={ZETA FUNCTIONS OF REGULAR ARITHMETIC SCHEMES AT s = 0}, author={Baptiste Morin}, journal={Duke Mathematical Journal}, year={2014}, volume={163}, pages={1263-1336} }

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 terms of Euler- Poincare characteristics. Assuming the (conjectured) finite generation of some motivic cohomology groups we construct such a cohomology theory for regular schemes proper over Spec(Z). In particular, we compute (uncon- ditionally) the right Weil-etale cohomology of number rings and…

## 15 Citations

### Zeta-values of arithmetic schemes at negative integers and Weil-étale cohomology

- Mathematics
- 2018

This work is dedicated to interpreting in cohomological
terms the special values of zeta functions of arithmetic schemes. Baptiste
Morin and Matthias Flach gave a construction of Weil-etale…

### 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

- Mathematics
- 2021

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…

### Special values of L-functions on regular arithmetic schemes of dimension 1

- MathematicsJournal of Number Theory
- 2022

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

- Mathematics
- 2015

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…

### 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…

### Arakelov motivic cohomology I

- Mathematics
- 2015

We show that the constructions done in part I generalize their classical counterparts: firstly, the classical Beilinson regulator is induced by the abstract Chern class map from BGL to the Deligne…

### Weil-\'etale cohomology and Zeta-values of proper regular arithmetic schemes

- Mathematics
- 2016

We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme $\mathcal{X}$ at any integer $n$ in terms of…

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