Special groups, versality and the Grothendieck-Serre conjecture

TL;DR

This paper proves the equivalence of two definitions of special algebraic groups and proposes a strengthened Grothendieck-Serre conjecture using the concept of essential dimension.

Contribution

It establishes the equivalence of Serre's original and weaker definitions of special groups and introduces a generalized conjecture based on essential dimension.

Findings

Proved the equivalence of two definitions of special groups.

Generalized the Grothendieck-Serre conjecture.

Connected essential dimension with the concept of special groups.

Abstract

Let $k$ be a base field and $G$ be an algebraic group over $k$. J.-P. Serre defined $G$ to be special if every $G$-torsor $T \to X$ is locally trivial in the Zariski topology for every reduced algebraic variety $X$ defined over $k$. In recent papers an a priori weaker condition is used: $G$ is called special if every $G$-torsor $T \to Spec(K)$ is split for every field $K$ containing $k$. We show that these two definitions are equivalent. We also generalize this fact and propose a strengthened version of the Grothendieck-Serre conjecture based on the notion of essential dimension.