Dualit\'e \'etale \`a la Poitou-Tate pour les tores sur des vari\'et\'es d\'efinies sur un corps fini

TL;DR

This paper extends the classical Poitou-Tate duality to tori over schemes defined on varieties over finite fields of arbitrary dimension, focusing on the Tate-Shafarevich group and its duality properties.

Contribution

It establishes a generalized Poitou-Tate duality for tori over schemes on varieties over finite fields, broadening the scope of classical duality results.

Findings

Proves a duality theorem for the Tate-Shafarevich group of tori.

Generalizes Poitou-Tate duality to higher-dimensional varieties.

Provides a framework for studying arithmetic dualities over finite fields.

Abstract

Let $k$ be a global field of characteristic $p>0$. Denote $\Omega_k$ the set of places of $k$ and let $S$ be a non-empty subset of $\Omega_k$. 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 caract\'eristique $p>0$. Notons $\Omega_k$ l'ensemble des places de $k$ et soit $S$ un sous-ensemble non vide de $\Omega_k$. On consid\`ere un sch\'ema $\mathscr{X} \rightarrow Spec(\mathcal{O}_S)$ lisse, s\'epar\'e, de type fini et $\mathscr{T}$ un tore d\'efini sur $\mathscr{X}$. On \'etudie le groupe de Tate-Shafarevich donn\'e par les \'el\'ements 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 \'etablit une dualit\'e pour $\mathscr{T}$ qui g\'en\'eralise la dualit\'e de Poitou-Tate classique pour les tores \`a des vari\'et\'es d\'efinies sur un corps fini de dimension arbitraire.