Dualité étale à la Poitou-Tate pour les tores sur des variétés définies sur un corps fini

TL;DR

The paper develops Poitou-Tate type dualities for Tate-Shafarevich groups of algebraic tori on varieties defined over global function fields of characteristic $p>0$. It constructs a cohomological framework on the total space $\mathscr{X}$ using motivic complexes and Artin-Verdier duality, defining $\Sha^{i}(\mathscr{X},\mathscr{S},\mathscr{T})$ and $\Sha^{i}_c(\mathscr{X},\mathscr{S},\mathscr{T})$ via étale cohomology with and without compact support. Under finiteness/torsion assumptions on local cohomology, it proves a duality pairing between $\Sha^{a}(\mathscr{X},\mathscr{S},\mathscr{T})\{p'\}$ and $\Sha^{2d+2-a}_c(\mathscr{X},\mathscr{S},\tilde{\mathscr{T}})\{p'\}$, generalizing known results to higher-dimensional bases; in the torus case ($a=1$) a stronger, perfect pairing with $\overline{\Sha^{2d+1}(\mathscr{X},\mathscr{S},\tilde{\mathscr{T}})}\{p'\}$ is obtained. This advances local-global principles for torsors under tori on varieties, unifying étale and motivic dualities in this setting and extending Poitou-Tate theory beyond curves to arbitrary dimension bases.

Abstract

Let $k$ be a global field of characteristic $p>0$. Denote $Ω_k$ the set of places of $k$ and let $S$ be a non-empty subset of $Ω_k$. We consider a scheme $\mathscr{X} \rightarrow Spec(\mathcal{O}_S)$ smooth, separated, of finite type and $\mathscr{T}$ a tori defined over $\mathscr{X}$. We study the Tate-Shafarevich group given by the elements of $H^1(\mathscr{X}, \mathscr{T})$ which vanish in the group $H^1(\mathscr{X} \otimes_{\mathcal{O}_S} k_v, \mathscr{T})$ for all $v \in S$. We establish a Poitou-Tate duality for $\mathscr{T}$ which generalise the classical Poitou-Tate duality for tori for varieties defined over a finite field of arbitrary dimension. Soit $k$ un corps global de caractéristique $p>0$. Notons $Ω_k$ l'ensemble des places de $k$ et soit $S$ un sous-ensemble non vide de $Ω_k$. On considère un schéma $\mathscr{X} \rightarrow Spec(\mathcal{O}_S)$ lisse, séparé, de type fini et $\mathscr{T}$ un tore défini sur $\mathscr{X}$. On étudie le groupe de Tate-Shafarevich donné par les éléments de $H^1(\mathscr{X}, \mathscr{T})$ qui s'annulent dans les groupes $H^1(\mathscr{X} \otimes_{\mathcal{O}_S} k_v, \mathscr{T})$ pour tout $v \in S$. On établit une dualité pour $\mathscr{T}$ qui généralise la dualité de Poitou-Tate classique pour les tores à des variétés définies sur un corps fini de dimension arbitraire.

Theorems & Definitions (14)

• Proposition 1.4
• Proposition 1.5
• Proposition 1.6
• Proposition 1.7
• Proposition 1.8
• Proposition 2.1
• Proposition 2.3
• Proposition 2.4
• Proposition 3.1
• Proposition 3.2