On geometry of some pseudo-reductive groups

TL;DR

This paper explores the geometric properties of certain pseudo-reductive groups, focusing on their rationality and Picard groups, based on the structure of Weil restrictions of semisimple groups.

Contribution

It establishes the retract rationality of split pseudo-semisimple groups and provides new insights into their Picard groups, extending previous work by Conrad-Gabber-Prasad.

Findings

Split pseudo-semisimple groups are retract rational.

Results on the Picard groups of these groups.

Extension of geometric understanding of pseudo-reductive groups.

Abstract

Based on the work of Conrad-Gabber-Prasad, the paper deals with the geometry of particular pseudo-semisimple groups, namely those which can be written as quotient of Weil restriction of semisimple groups. We establish that these groups are retract rational when their are split, and give results on their Picard groups.