Pathologies du groupe des classes de R-\'equivalence d'un groupe alg\'ebrique lin\'eaire

TL;DR

This paper provides an example of a smooth connected unipotent algebraic group over a function field where the R-equivalence classes form a non-commutative group, highlighting pathologies in R-equivalence group structures.

Contribution

It constructs a specific example of a unipotent algebraic group over a function field with non-commutative R-equivalence classes, revealing new pathological behaviors.

Findings

R-equivalence classes can form non-commutative groups

Existence of unipotent groups with non-commutative R-classes over function fields

Counterexamples to expected commutativity in R-group structures

Abstract

Let $k_0$ be a field of characteristic $p>0$ and $k=k_0(t)$, where $t$ is transcendental over $k_0$. We give an example of a smooth connected unipotent $k$-group $G$ such that $G(F)/R$ is non-commutative for some finite separable field extension $F$ containing $k$.