A lower bound on the essential dimension of $\operatorname{\mathbf{PGL}}_{4}$ in characteristic $2$

TL;DR

This paper establishes a lower bound of 4 for the essential dimension of PGL_4 over fields of characteristic 2, using Kato's cohomology and Tignol's results on trace forms.

Contribution

It introduces a new lower bound for the essential dimension of PGL_4 in characteristic 2 by leveraging Kato's cohomology and existing trace form results.

Findings

Essential dimension of PGL_4 in characteristic 2 is at least 4.

Utilizes Kato's cohomology group to derive bounds.

Combines cohomological methods with algebraic trace form results.

Abstract

In the present paper, we provide a lower bound of the essential dimension over a field of positive characteristic via Kato's cohomology group, defined by cokernel of a general Artin-Schreier operator. Combining this with Tignol's result on the second trace form of simple algebras of degree $4$, we show that $\operatorname{ed}(\operatorname{\mathbf{PGL}}_{4})\geq 4$ over a field of characteristic $2$.