Essential dimension of simple algebras with involutions

TL;DR

This paper improves bounds on the essential dimension of certain classes of central simple algebras with involutions, specifically providing exact values for some cases like degree 16 algebras with exponent 2.

Contribution

It introduces new upper bounds for the essential and 2-dimensions of classes of simple algebras with involutions, including exact values for specific cases.

Findings

Established that _{2}(\u00a0Alg_{16,2})=24 over fields of characteristic not 2.

Provided improved upper bounds for essential dimension of _{2}(_{n,m}) classes.

Analyzed the structure of central simple algebras with involutions to derive these bounds.

Abstract

Let $1\leq m \leq n$ be integers with $m|n$ and $\cat{Alg}_{n,m}$ the class of central simple algebras of degree $n$ and exponent dividing $m$. In this paper, we find new, improved upper bounds for the essential dimension and 2-dimension of $\cat{Alg}_{n,2}$. In particular, we show that $\ed_{2}(\cat{Alg}_{16,2})=24$ over a field $F$ of characteristic different from 2.