Essential dimension of simple algebras with involutions

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.Comment: Sections 1 and 3 are rewritte

Essential dimension of simple algebras in positive characteristic

Let $p$ be a prime integer, $1\leq s\leq r$ integers, $F$ a field of characteristic $p$. Let \cat{Dec}_{p^r} denote the class of the tensor product of $r$ $p$-symbols and \cat{Alg}_{p^r,p^s} denote the class of central simple algebras of degree $p^r$ and exponent dividing $p^s$. For any integers $s<r$, we find a lower bound for the essential $p$-dimension of \cat{Alg}_{p^r,p^s}. Furthermore, we compute upper bounds for \cat{Dec}_{p^r} and \cat{Alg}_{8,2} over $\ch(F)=p$ and $\ch(F)=2$, respectively. As a result, we show \ed_{2}(\cat{Alg}_{4,2})=\ed(\cat{Alg}_{4,2})=\ed_{2}(\gGL_{4}/\gmu_{2})=\ed(\gGL_{4}/\gmu_{2})=3 and 3\leq \ed(\cat{Alg}_{8,2})=\ed(\gGL_{8}/\gmu_{2})\leq 10 over a field of characteristic 2.Comment: Any comments are welcom