Holmstrom Weil-etale cohomology

Weil-etale cohomology

Introduced by Lichtenbaum

---

Weil-etale cohomology

MathSciNet

Google Scholar

Google

arXiv: Experimental full text search

arXiv: Abstract search

---

Weil-etale cohomology

AG (Algebraic geometry), AAG (Arithmetic algebraic geometry)

---

Weil-etale cohomology

Mixed, Arithmetic

---

Weil-etale cohomology

See intro to Flach-Morin for a general introduction. In intro to Morin’s thesis there are some more conjectural properties, notably the statement that integral coeff groups should vanish for i>2d+1, and a different formulation of the Taylor coefficient formula, in which the role of the torsion parts is more clear.

---

Weil-etale cohomology

Morin’s thesis: axioms for a Weil-etale topos.

---

Weil-etale cohomology

Matthias Flach - Caltech, Title: Weil-etale cohomology and the Tamagawa number conjecture Abstract: We discuss the equivariant Tamagawa number conjecture on motivic L-functions from the point of view of Weil-etale cohomology, a recent idea of Lichtenbaum. We focus on the fact that the conjecture always describes the leading Taylor coefficient of a motivic L-function and give some illustrating examples involving Dirichlet L-functions.

---

Is the following talk about Weil-etale cohomology???

Stephen Lichtenbaum - Brown University Title: Special values of L-functions of 1-motives Abstract: We propose a conjecture that for any motive over a number field, there exist a finite number of cohomology complexes such that the order of the zero of the L-function of the motive at s = 0 is given by the sum of the ranks of the complexes, and the leading term at s = 0 is given by the procuct of the Euler characteristics of the same complexes. These complexes, or rather their homology, will be made very explicit in the case of 1-motives.

---

Is there something like Weil-etale K-theory, or W-E cobordism??

---

Weil-etale cohomology

Key authors: Flach, Morin, Geisser, Lichtenbaum.

arXiv:1103.6061 Zeta functions of regular arithmetic schemes at s=0 from arXiv Front: math.NT by Baptiste Morin Lichtenbaum conjectured in [Lichtenbaum] the existence of a Weil-étale 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-Poincaré characteristics. Assuming the (conjectured) finite generation of some motivic cohomology groups we construct such a cohomology theory for regular schemes proper over $\mathrm{Spec}(\mathbb{Z})$. In particular, we compute (unconditionally) the right Weil-étale cohomology of number rings and projective spaces over number rings. We state a precise version of Lichtenbaum’s conjecture, which expresses the vanishing order (resp. the special value) of the Zeta function $\zeta(\mathcal{X},s)$ at $s=0$ as the rank (resp. the determinant) of a single perfect complex of abelian groups $R\Gamma_{W,c}(\mathcal{X},\mathbb{Z})$. Then we relate this conjecture to Soulé’s conjecture and to the Tamagawa Number Conjecture. Lichtenbaum’s conjecture for projective spaces over the ring of integers of an abelian number field follows.

Geisser

Geisser: Weil-etale cohomology over finite fields

MR2117419 (2005i:11079) Burns, David(4-LNDKC) On the values of equivariant zeta functions of curves over finite fields.

arXiv:1103.6061 Zeta functions of regular arithmetic schemes at s=0 from arXiv Front: math.AG by Baptiste Morin Lichtenbaum conjectured in [Lichtenbaum] the existence of a Weil-étale 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-Poincaré characteristics. Assuming the (conjectured) finite generation of some motivic cohomology groups we construct such a cohomology theory for regular schemes proper over $\mathrm{Spec}(\mathbb{Z})$. In particular, we compute (unconditionally) the right Weil-étale cohomology of number rings and projective spaces over number rings. We state a precise version of Lichtenbaum’s conjecture, which expresses the vanishing order (resp. the special value) of the Zeta function $\zeta(\mathcal{X},s)$ at $s=0$ as the rank (resp. the determinant) of a single perfect complex of abelian groups $R\Gamma_{W,c}(\mathcal{X},\mathbb{Z})$. Then we relate this conjecture to Soulé’s conjecture and to the Tamagawa Number Conjecture. Lichtenbaum’s conjecture for projective spaces over the ring of integers of an abelian number field follows.

arXiv:1209.4322 The full faithfulness conjectures in characteristic p from arXiv Front: math.AG by Bruno Kahn We present a triangulated version of the conjectures of Tate and Beilinson on algebraic cycles over a finite field. This sheds a new light on Lichtenbaum’s Weil-etale cohomology.

Title: On the Weil-étale cohomology of the ring of $S$-integers Authors: Yi-Chih Chiu Categories: math.NT Number Theory Abstract: In this article, we first briefly introduce the history of the Weil-étale cohomology theory of arithmetic schemes and review some important results established by Lichtenbaum, Flach and Morin. Next we generalize the Weil-etale cohomology to $S$-integers and compute the cohomology for constant sheaves $\mathbb{Z}$ or $\mathbb{R}$. We also define a Weil-étale cohomology with compact support $H_c(Y_W, -)$ for $Y=Spec \mathcal{O}_{F,S}$ where $F$ is a number field, and computed them. We verify that these cohomology groups satisfy the axioms state by Lichtenbaum. As an application, we derive a canonical representation of Tate sequence from $RGamma_c(Y_W,\mathbb{Z})$. Motivated by this result, in the final part, we define an étale complex $RGm$, such that the complexes $\mathbb{Z}$-dual of the complex $\RG(U_{et},R\Gm),\,\mathbb{Z})[2]$ is canonically quasi-isomorphic to $\tau^{\leq 3}\RG_c(U_W,\mathbb{Z})$ for arbitrary étale $U$ over $Spec \mathcal{O}_{F}$. This quasi-isomorphism provides a possible approach to define the Weil-etale cohomology for higher dimensional arithmetic schemes, as the Weil groups are not involved in the definition of $R\Gm$.

Title: Weil-étale Cohomology over $p$-adic Fields Authors: David A. Karpuk Categories: math.NT Number Theory (math.AG Algebraic Geometry) Abstract: 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 Duality. We define Weil-smooth cohomology for varieties over local fields, and prove a duality theorem for the cohomology of $\G_m$ on a smooth, proper curve with a rational point. This last theorem is analogous to, and implies, a classical duality theorem for such curves.

nLab page on Weil-etale cohomology?