Related papers: Tori over number fields and special values at s=1

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 number field. Then we show that the special…

We construct the Weil-\'etale cohomology and Euler characteristics for a subclass of the class of $\mathbb{Z}$-constructible sheaves on an open subscheme of the spectrum of the ring of integers of a number field. Then we show that the…

We construct a well-behaved Weil-\'etale complex for a large class of $\mathbb{Z}$-constructible sheaves on a regular irreducible scheme $U$ of finite type over $\mathbb{Z}$ and of dimension $1$. We then give a formula for the special value…

We introduce an exact category of torsion-free constructible tori and an abelian category of constructible tori over a Dedekind scheme with perfect residue fields. The first one has an explicit description as $2$-term complexes of smooth…

Let $X$ be a variety over a finite field. Given an order $R$ in a semi-simple algebra over the rationals and a constructible \'etale sheaf $F$ of $R$-modules over $X$, one can consider a natural non-commutative $L$-function associated with…

Flach and Morin constructed in (Doc. Math. 23 (2018), 1425--1560) Weil-\'etale cohomology $H^i_\text{W,c} (X, \mathbb{Z} (n))$ for a proper, regular arithmetic scheme $X$ (i.e. separated and of finite type over $\operatorname{Spec}…

Lichtenbaum has conjectured the existence of a Grothendieck topology for an arithmetic scheme $X$ such that the Euler characteristic of the cohomology groups of the constant sheaf $\mathbb{Z}$ with compact support at infinity gives, up to…

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 Weil-\'etale cohomology complexes. This extends…

We study Torelli-type theorems in the Zariski topology for varieties of dimension at least 2, over arbitrary fields. In place of the Hodge structure, we use the linear equivalence relation on Weil divisors. Using this setup, we prove a…

We define the fundamental group underlying to Lichtenbaum's Weil-\'etale cohomology for number rings. To this aim, we define the Weil-\'etale topos as a refinement of the Weil-\'etale sites introduced in \cite{Lichtenbaum}. We show that the…

Given a double vector bundle $D\to M$, we define a bigraded `Weil algebra' $\mathcal{W}(D)$, which `realizes' the algebra of smooth functions on the supermanifold $D[1,1]$. We describe in detail the relations between the Weil algebras of…

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…

In this paper we introduce constructible analogs of the discrete complexity classes $\mathbf{VP}$ and $\mathbf{VNP}$ of sequences of functions. The functions in the new definitions are constructible functions on $\mathbb{R}^n$ or…

Let $X$ be an arithmetic scheme (i.e., separated, of finite type over $\operatorname{Spec} \mathbb{Z}$) of Krull dimension $1$. For the associated zeta function $\zeta (X,s)$, we write down a formula for the special value at $s = n < 0$ in…

Lichtenbaum conjectured 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 $\mathcal{X}$ at $s=0$ in terms of Euler-Poincar\'e…

Following the ideas of Flach and Morin (Doc. Math. 23 (2018), 1425--1560), we state a conjecture in terms of Weil-\'etale cohomology for the vanishing order and special value of the zeta function $\zeta (X, s)$ at $s = n < 0$, where $X$ is…

A reduced divisor on a nonsingular variety defines the sheaf of logarithmic 1-forms. We introduce a certain coherent sheaf whose double dual coincides with this sheaf. It has some nice properties, for example, the residue exact sequence…

We establish duality results for the cohomology of the Weil group of a $p$-adic field, analogous to, but more general than, results from Galois cohomology. We prove a duality theorem for discrete Weil modules, which implies Tate-Nakayama…

In this article, we first briefly introduce the history of the Weil-\'etale cohomology theory of arithmetic schemes and review some important results established by Lichtenbaum, Flach and Morin. Next we generalize the Weil-etale cohomology…

We combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud-Mustata-Stillman and Mustata on cohomology of toric and monomial ideals to obtain a formula for computing the Euler characteristic of a Weil…