A lower bound on the essential dimension of ${\mathrm{PGL}}_4$ in characteristic $2$

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 $\mathrm{ed}({\mathrm{PGL}}_4)\ge 4$ over a field of characteristic $2$.