Essential dimension of simple algebras in positive characteristic (1012.4877v1)

Abstract: 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.