The essential $2$-dimension of the linear groups

TL;DR

This paper calculates the essential 2-dimension of various linear algebraic groups over fields with specific properties, providing explicit values for cases involving odd primes and particular congruences.

Contribution

It determines the essential 2-dimension for several classes of linear groups under new conditions related to prime and field properties.

Findings

Computed essential 2-dimension for general linear groups and variants

Established results for special linear groups with specific prime and field conditions

Extended understanding of algebraic group invariants in these contexts

Abstract

In this paper, we compute the essential $2$-dimension when the defining prime is odd of the general linear groups, the projective general linear groups, the special linear groups when $n$ is odd or $n = 2$, as well as the special linear groups and quotients of it (such as the projective special linear groups) in the case case $q \equiv 1 \mod 4$, $s = v_2(q-1)$, and $\Gamma = \text{Gal}(k(\zeta_{2^s})/k)$ is trivial.