arXiv subjects

Anis Zidani

Publications and source records attributed to Anis Zidani.

4 recordsLinked to original sources

arXiv2026

Answer to a decomposition question on tori raised by Colliot-Th\'el\`ene and Sansuc

The aim of this note is to present a simple strategy to answer negatively a decomposition question on tori posed by Colliot-Th\'el\`ene and Sansuc in the article \textit{Principal Homogeneous Spaces under Flasque Tori: Applications} of 1987. We then deduce a torus $T$ over $\mathbb{Q}$ and a prime number $p$ such that $T(\mathbb{Z}_p)\,T(\mathbb{Q})\not=T(\mathbb{Q}_p)$, where $T(\mathbb{Z}_p)$ denotes the maximal compact subgroup of $T(\mathbb{Q}_p)$.

math.AG↗

arXiv2026

Conjecture I for unirational algebraic groups over imperfect fields

Serre's Conjecture I states that the first Galois cohomology set of any smooth connected linear algebraic group is trivial over a perfect field of cohomological dimension at most 1. We prove that this result remains valid for any unirational algebraic group, dropping the perfection assumption on the field. To do so, we rely on the theory of pseudo-reductive groups, combined with the structure of unirational wound unipotent groups and the recent theory of permawound unipotent groups. Finally, we extend several related results on Galois cohomology associated with the Conjecture.

math.AG↗

arXiv2025

Arithm\'etique des sous-groupes de Bruhat-Tits sur un anneau de valuation discr\`ete hens\'elien

The goal of this paper is to refine some aspects of the cohomological study of Bruhat-Tits subgroups. It aims to complement the work done in the 1987 article. As an application, we obtain a result "\`a la Grothendieck-Serre" for Bruhat-Tits subgroups of a semisimple simply connected group and the exact calculation of the obstruction in the case of quasi-split adjoint groups.

math.GR↗

arXiv2025

Arithm\'etique des sch\'emas en groupes de Bruhat-Tits sur un anneau de Dedekind semi-local

The aim of this paper is to lay the foundations for the cohomological study of Bruhat-Tits group schemes over a semi-local Dedekind ring. In particular, we obtain a simplified proof of the Grothendieck-Serre conjecture in this case and also an analogous result for Bruhat-Tits group schemes of a semisimple simply connected group.

math.AG↗